# Properties

 Label 3920.2.a.p.1.1 Level $3920$ Weight $2$ Character 3920.1 Self dual yes Analytic conductor $31.301$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} +6.00000 q^{11} -4.00000 q^{13} +1.00000 q^{15} -2.00000 q^{19} +3.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} -3.00000 q^{29} -8.00000 q^{31} -6.00000 q^{33} -4.00000 q^{37} +4.00000 q^{39} +9.00000 q^{41} +7.00000 q^{43} +2.00000 q^{45} -6.00000 q^{53} -6.00000 q^{55} +2.00000 q^{57} +6.00000 q^{59} +5.00000 q^{61} +4.00000 q^{65} -5.00000 q^{67} -3.00000 q^{69} +6.00000 q^{71} -16.0000 q^{73} -1.00000 q^{75} -2.00000 q^{79} +1.00000 q^{81} -3.00000 q^{83} +3.00000 q^{87} -15.0000 q^{89} +8.00000 q^{93} +2.00000 q^{95} +14.0000 q^{97} -12.0000 q^{99} +O(q^{100})$$

## 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 $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 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
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ −3.00000 −0.557086 −0.278543 0.960424i $$-0.589851\pi$$
−0.278543 + 0.960424i $$0.589851\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ 9.00000 1.40556 0.702782 0.711405i $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ 0 0
$$43$$ 7.00000 1.06749 0.533745 0.845645i $$-0.320784\pi$$
0.533745 + 0.845645i $$0.320784\pi$$
$$44$$ 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
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ −6.00000 −0.809040
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ −5.00000 −0.610847 −0.305424 0.952217i $$-0.598798\pi$$
−0.305424 + 0.952217i $$0.598798\pi$$
$$68$$ 0 0
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ −16.0000 −1.87266 −0.936329 0.351123i $$-0.885800\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −2.00000 −0.225018 −0.112509 0.993651i $$-0.535889\pi$$
−0.112509 + 0.993651i $$0.535889\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −3.00000 −0.329293 −0.164646 0.986353i $$-0.552648\pi$$
−0.164646 + 0.986353i $$0.552648\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 3.00000 0.321634
$$88$$ 0 0
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 0 0
$$99$$ −12.0000 −1.20605
$$100$$ 0 0
$$101$$ 15.0000 1.49256 0.746278 0.665635i $$-0.231839\pi$$
0.746278 + 0.665635i $$0.231839\pi$$
$$102$$ 0 0
$$103$$ 1.00000 0.0985329 0.0492665 0.998786i $$-0.484312\pi$$
0.0492665 + 0.998786i $$0.484312\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 15.0000 1.45010 0.725052 0.688694i $$-0.241816\pi$$
0.725052 + 0.688694i $$0.241816\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ −3.00000 −0.279751
$$116$$ 0 0
$$117$$ 8.00000 0.739600
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 0 0
$$123$$ −9.00000 −0.811503
$$124$$ 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
$$129$$ −7.00000 −0.616316
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 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
$$139$$ 10.0000 0.848189 0.424094 0.905618i $$-0.360592\pi$$
0.424094 + 0.905618i $$0.360592\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −24.0000 −2.00698
$$144$$ 0 0
$$145$$ 3.00000 0.249136
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ −22.0000 −1.75579 −0.877896 0.478852i $$-0.841053\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 0 0
$$165$$ 6.00000 0.467099
$$166$$ 0 0
$$167$$ 3.00000 0.232147 0.116073 0.993241i $$-0.462969\pi$$
0.116073 + 0.993241i $$0.462969\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −6.00000 −0.450988
$$178$$ 0 0
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 0 0
$$181$$ 11.0000 0.817624 0.408812 0.912619i $$-0.365943\pi$$
0.408812 + 0.912619i $$0.365943\pi$$
$$182$$ 0 0
$$183$$ −5.00000 −0.369611
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ −4.00000 −0.286446
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 5.00000 0.352673
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −9.00000 −0.628587
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ −12.0000 −0.830057
$$210$$ 0 0
$$211$$ 10.0000 0.688428 0.344214 0.938891i $$-0.388145\pi$$
0.344214 + 0.938891i $$0.388145\pi$$
$$212$$ 0 0
$$213$$ −6.00000 −0.411113
$$214$$ 0 0
$$215$$ −7.00000 −0.477396
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 16.0000 1.08118
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 28.0000 1.87502 0.937509 0.347960i $$-0.113126\pi$$
0.937509 + 0.347960i $$0.113126\pi$$
$$224$$ 0 0
$$225$$ −2.00000 −0.133333
$$226$$ 0 0
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −12.0000 −0.786146 −0.393073 0.919507i $$-0.628588\pi$$
−0.393073 + 0.919507i $$0.628588\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 2.00000 0.129914
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ −16.0000 −1.02640
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ 0 0
$$249$$ 3.00000 0.190117
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 18.0000 1.13165
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ 21.0000 1.29492 0.647458 0.762101i $$-0.275832\pi$$
0.647458 + 0.762101i $$0.275832\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ 0 0
$$267$$ 15.0000 0.917985
$$268$$ 0 0
$$269$$ 15.0000 0.914566 0.457283 0.889321i $$-0.348823\pi$$
0.457283 + 0.889321i $$0.348823\pi$$
$$270$$ 0 0
$$271$$ −2.00000 −0.121491 −0.0607457 0.998153i $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 6.00000 0.361814
$$276$$ 0 0
$$277$$ 8.00000 0.480673 0.240337 0.970690i $$-0.422742\pi$$
0.240337 + 0.970690i $$0.422742\pi$$
$$278$$ 0 0
$$279$$ 16.0000 0.957895
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ −2.00000 −0.118470
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ −14.0000 −0.820695
$$292$$ 0 0
$$293$$ 12.0000 0.701047 0.350524 0.936554i $$-0.386004\pi$$
0.350524 + 0.936554i $$0.386004\pi$$
$$294$$ 0 0
$$295$$ −6.00000 −0.349334
$$296$$ 0 0
$$297$$ 30.0000 1.74078
$$298$$ 0 0
$$299$$ −12.0000 −0.693978
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −15.0000 −0.861727
$$304$$ 0 0
$$305$$ −5.00000 −0.286299
$$306$$ 0 0
$$307$$ −5.00000 −0.285365 −0.142683 0.989769i $$-0.545573\pi$$
−0.142683 + 0.989769i $$0.545573\pi$$
$$308$$ 0 0
$$309$$ −1.00000 −0.0568880
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ 0 0
$$319$$ −18.0000 −1.00781
$$320$$ 0 0
$$321$$ −15.0000 −0.837218
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −4.00000 −0.221880
$$326$$ 0 0
$$327$$ −11.0000 −0.608301
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 0 0
$$333$$ 8.00000 0.438397
$$334$$ 0 0
$$335$$ 5.00000 0.273179
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −48.0000 −2.59935
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 3.00000 0.161515
$$346$$ 0 0
$$347$$ −9.00000 −0.483145 −0.241573 0.970383i $$-0.577663\pi$$
−0.241573 + 0.970383i $$0.577663\pi$$
$$348$$ 0 0
$$349$$ 17.0000 0.909989 0.454995 0.890494i $$-0.349641\pi$$
0.454995 + 0.890494i $$0.349641\pi$$
$$350$$ 0 0
$$351$$ −20.0000 −1.06752
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ −6.00000 −0.318447
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ −25.0000 −1.31216
$$364$$ 0 0
$$365$$ 16.0000 0.837478
$$366$$ 0 0
$$367$$ −35.0000 −1.82699 −0.913493 0.406855i $$-0.866625\pi$$
−0.913493 + 0.406855i $$0.866625\pi$$
$$368$$ 0 0
$$369$$ −18.0000 −0.937043
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 34.0000 1.74646 0.873231 0.487306i $$-0.162020\pi$$
0.873231 + 0.487306i $$0.162020\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ 15.0000 0.766464 0.383232 0.923652i $$-0.374811\pi$$
0.383232 + 0.923652i $$0.374811\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −14.0000 −0.711660
$$388$$ 0 0
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 2.00000 0.100631
$$396$$ 0 0
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 15.0000 0.749064 0.374532 0.927214i $$-0.377803\pi$$
0.374532 + 0.927214i $$0.377803\pi$$
$$402$$ 0 0
$$403$$ 32.0000 1.59403
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ −13.0000 −0.642809 −0.321404 0.946942i $$-0.604155\pi$$
−0.321404 + 0.946942i $$0.604155\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 3.00000 0.147264
$$416$$ 0 0
$$417$$ −10.0000 −0.489702
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 17.0000 0.828529 0.414265 0.910156i $$-0.364039\pi$$
0.414265 + 0.910156i $$0.364039\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 24.0000 1.15873
$$430$$ 0 0
$$431$$ −30.0000 −1.44505 −0.722525 0.691345i $$-0.757018\pi$$
−0.722525 + 0.691345i $$0.757018\pi$$
$$432$$ 0 0
$$433$$ −22.0000 −1.05725 −0.528626 0.848855i $$-0.677293\pi$$
−0.528626 + 0.848855i $$0.677293\pi$$
$$434$$ 0 0
$$435$$ −3.00000 −0.143839
$$436$$ 0 0
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −21.0000 −0.997740 −0.498870 0.866677i $$-0.666252\pi$$
−0.498870 + 0.866677i $$0.666252\pi$$
$$444$$ 0 0
$$445$$ 15.0000 0.711068
$$446$$ 0 0
$$447$$ −15.0000 −0.709476
$$448$$ 0 0
$$449$$ −9.00000 −0.424736 −0.212368 0.977190i $$-0.568118\pi$$
−0.212368 + 0.977190i $$0.568118\pi$$
$$450$$ 0 0
$$451$$ 54.0000 2.54276
$$452$$ 0 0
$$453$$ −4.00000 −0.187936
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 32.0000 1.49690 0.748448 0.663193i $$-0.230799\pi$$
0.748448 + 0.663193i $$0.230799\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 13.0000 0.604161 0.302081 0.953282i $$-0.402319\pi$$
0.302081 + 0.953282i $$0.402319\pi$$
$$464$$ 0 0
$$465$$ −8.00000 −0.370991
$$466$$ 0 0
$$467$$ 15.0000 0.694117 0.347059 0.937843i $$-0.387180\pi$$
0.347059 + 0.937843i $$0.387180\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 22.0000 1.01371
$$472$$ 0 0
$$473$$ 42.0000 1.93116
$$474$$ 0 0
$$475$$ −2.00000 −0.0917663
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 0 0
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −14.0000 −0.635707
$$486$$ 0 0
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 12.0000 0.539360
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 22.0000 0.984855 0.492428 0.870353i $$-0.336110\pi$$
0.492428 + 0.870353i $$0.336110\pi$$
$$500$$ 0 0
$$501$$ −3.00000 −0.134030
$$502$$ 0 0
$$503$$ −21.0000 −0.936344 −0.468172 0.883637i $$-0.655087\pi$$
−0.468172 + 0.883637i $$0.655087\pi$$
$$504$$ 0 0
$$505$$ −15.0000 −0.667491
$$506$$ 0 0
$$507$$ −3.00000 −0.133235
$$508$$ 0 0
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −10.0000 −0.441511
$$514$$ 0 0
$$515$$ −1.00000 −0.0440653
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 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
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ −36.0000 −1.55933
$$534$$ 0 0
$$535$$ −15.0000 −0.648507
$$536$$ 0 0
$$537$$ −24.0000 −1.03568
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −25.0000 −1.07483 −0.537417 0.843317i $$-0.680600\pi$$
−0.537417 + 0.843317i $$0.680600\pi$$
$$542$$ 0 0
$$543$$ −11.0000 −0.472055
$$544$$ 0 0
$$545$$ −11.0000 −0.471188
$$546$$ 0 0
$$547$$ 19.0000 0.812381 0.406191 0.913788i $$-0.366857\pi$$
0.406191 + 0.913788i $$0.366857\pi$$
$$548$$ 0 0
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −4.00000 −0.169791
$$556$$ 0 0
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ −28.0000 −1.18427
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −27.0000 −1.13791 −0.568957 0.822367i $$-0.692653\pi$$
−0.568957 + 0.822367i $$0.692653\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 22.0000 0.920671 0.460336 0.887745i $$-0.347729\pi$$
0.460336 + 0.887745i $$0.347729\pi$$
$$572$$ 0 0
$$573$$ −6.00000 −0.250654
$$574$$ 0 0
$$575$$ 3.00000 0.125109
$$576$$ 0 0
$$577$$ 26.0000 1.08239 0.541197 0.840896i $$-0.317971\pi$$
0.541197 + 0.840896i $$0.317971\pi$$
$$578$$ 0 0
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −36.0000 −1.49097
$$584$$ 0 0
$$585$$ −8.00000 −0.330759
$$586$$ 0 0
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 0 0
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −4.00000 −0.163709
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ −46.0000 −1.87638 −0.938190 0.346122i $$-0.887498\pi$$
−0.938190 + 0.346122i $$0.887498\pi$$
$$602$$ 0 0
$$603$$ 10.0000 0.407231
$$604$$ 0 0
$$605$$ −25.0000 −1.01639
$$606$$ 0 0
$$607$$ −23.0000 −0.933541 −0.466771 0.884378i $$-0.654583\pi$$
−0.466771 + 0.884378i $$0.654583\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ 0 0
$$615$$ 9.00000 0.362915
$$616$$ 0 0
$$617$$ −12.0000 −0.483102 −0.241551 0.970388i $$-0.577656\pi$$
−0.241551 + 0.970388i $$0.577656\pi$$
$$618$$ 0 0
$$619$$ −14.0000 −0.562708 −0.281354 0.959604i $$-0.590783\pi$$
−0.281354 + 0.959604i $$0.590783\pi$$
$$620$$ 0 0
$$621$$ 15.0000 0.601929
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 12.0000 0.479234
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −14.0000 −0.557331 −0.278666 0.960388i $$-0.589892\pi$$
−0.278666 + 0.960388i $$0.589892\pi$$
$$632$$ 0 0
$$633$$ −10.0000 −0.397464
$$634$$ 0 0
$$635$$ 8.00000 0.317470
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ −3.00000 −0.118493 −0.0592464 0.998243i $$-0.518870\pi$$
−0.0592464 + 0.998243i $$0.518870\pi$$
$$642$$ 0 0
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ 0 0
$$645$$ 7.00000 0.275625
$$646$$ 0 0
$$647$$ −3.00000 −0.117942 −0.0589711 0.998260i $$-0.518782\pi$$
−0.0589711 + 0.998260i $$0.518782\pi$$
$$648$$ 0 0
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −48.0000 −1.87839 −0.939193 0.343391i $$-0.888424\pi$$
−0.939193 + 0.343391i $$0.888424\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 32.0000 1.24844
$$658$$ 0 0
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ 41.0000 1.59472 0.797358 0.603507i $$-0.206231\pi$$
0.797358 + 0.603507i $$0.206231\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −9.00000 −0.348481
$$668$$ 0 0
$$669$$ −28.0000 −1.08254
$$670$$ 0 0
$$671$$ 30.0000 1.15814
$$672$$ 0 0
$$673$$ 8.00000 0.308377 0.154189 0.988041i $$-0.450724\pi$$
0.154189 + 0.988041i $$0.450724\pi$$
$$674$$ 0 0
$$675$$ 5.00000 0.192450
$$676$$ 0 0
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 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
$$687$$ −14.0000 −0.534133
$$688$$ 0 0
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ 22.0000 0.836919 0.418460 0.908235i $$-0.362570\pi$$
0.418460 + 0.908235i $$0.362570\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −10.0000 −0.379322
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 12.0000 0.453882
$$700$$ 0 0
$$701$$ −3.00000 −0.113308 −0.0566542 0.998394i $$-0.518043\pi$$
−0.0566542 + 0.998394i $$0.518043\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −31.0000 −1.16423 −0.582115 0.813107i $$-0.697775\pi$$
−0.582115 + 0.813107i $$0.697775\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 0 0
$$713$$ −24.0000 −0.898807
$$714$$ 0 0
$$715$$ 24.0000 0.897549
$$716$$ 0 0
$$717$$ 12.0000 0.448148
$$718$$ 0 0
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −2.00000 −0.0743808
$$724$$ 0 0
$$725$$ −3.00000 −0.111417
$$726$$ 0 0
$$727$$ 19.0000 0.704671 0.352335 0.935874i $$-0.385388\pi$$
0.352335 + 0.935874i $$0.385388\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −34.0000 −1.25582 −0.627909 0.778287i $$-0.716089\pi$$
−0.627909 + 0.778287i $$0.716089\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −30.0000 −1.10506
$$738$$ 0 0
$$739$$ −26.0000 −0.956425 −0.478213 0.878244i $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ 0 0
$$743$$ −39.0000 −1.43077 −0.715386 0.698730i $$-0.753749\pi$$
−0.715386 + 0.698730i $$0.753749\pi$$
$$744$$ 0 0
$$745$$ −15.0000 −0.549557
$$746$$ 0 0
$$747$$ 6.00000 0.219529
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 0 0
$$753$$ −12.0000 −0.437304
$$754$$ 0 0
$$755$$ −4.00000 −0.145575
$$756$$ 0 0
$$757$$ −28.0000 −1.01768 −0.508839 0.860862i $$-0.669925\pi$$
−0.508839 + 0.860862i $$0.669925\pi$$
$$758$$ 0 0
$$759$$ −18.0000 −0.653359
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ 50.0000 1.80305 0.901523 0.432731i $$-0.142450\pi$$
0.901523 + 0.432731i $$0.142450\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −12.0000 −0.431610 −0.215805 0.976436i $$-0.569238\pi$$
−0.215805 + 0.976436i $$0.569238\pi$$
$$774$$ 0 0
$$775$$ −8.00000 −0.287368
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −18.0000 −0.644917
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ 0 0
$$783$$ −15.0000 −0.536056
$$784$$ 0 0
$$785$$ 22.0000 0.785214
$$786$$ 0 0
$$787$$ 43.0000 1.53278 0.766392 0.642373i $$-0.222050\pi$$
0.766392 + 0.642373i $$0.222050\pi$$
$$788$$ 0 0
$$789$$ −21.0000 −0.747620
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ 0 0
$$795$$ −6.00000 −0.212798
$$796$$ 0 0
$$797$$ −48.0000 −1.70025 −0.850124 0.526583i $$-0.823473\pi$$
−0.850124 + 0.526583i $$0.823473\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ 0 0
$$803$$ −96.0000 −3.38777
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −15.0000 −0.528025
$$808$$ 0 0
$$809$$ 21.0000 0.738321 0.369160 0.929366i $$-0.379645\pi$$
0.369160 + 0.929366i $$0.379645\pi$$
$$810$$ 0 0
$$811$$ 16.0000 0.561836 0.280918 0.959732i $$-0.409361\pi$$
0.280918 + 0.959732i $$0.409361\pi$$
$$812$$ 0 0
$$813$$ 2.00000 0.0701431
$$814$$ 0 0
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ −14.0000 −0.489798
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ 19.0000 0.662298 0.331149 0.943578i $$-0.392564\pi$$
0.331149 + 0.943578i $$0.392564\pi$$
$$824$$ 0 0
$$825$$ −6.00000 −0.208893
$$826$$ 0 0
$$827$$ −15.0000 −0.521601 −0.260801 0.965393i $$-0.583986\pi$$
−0.260801 + 0.965393i $$0.583986\pi$$
$$828$$ 0 0
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ −8.00000 −0.277517
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −3.00000 −0.103819
$$836$$ 0 0
$$837$$ −40.0000 −1.38260
$$838$$ 0 0
$$839$$ 30.0000 1.03572 0.517858 0.855467i $$-0.326730\pi$$
0.517858 + 0.855467i $$0.326730\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 6.00000 0.206651
$$844$$ 0 0
$$845$$ −3.00000 −0.103203
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ −46.0000 −1.57501 −0.787505 0.616308i $$-0.788628\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ 0 0
$$855$$ −4.00000 −0.136797
$$856$$ 0 0
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 0 0
$$859$$ −32.0000 −1.09183 −0.545913 0.837842i $$-0.683817\pi$$
−0.545913 + 0.837842i $$0.683817\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −27.0000 −0.919091 −0.459545 0.888154i $$-0.651988\pi$$
−0.459545 + 0.888154i $$0.651988\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 17.0000 0.577350
$$868$$ 0 0
$$869$$ −12.0000 −0.407072
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 0 0
$$873$$ −28.0000 −0.947656
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ 0 0
$$879$$ −12.0000 −0.404750
$$880$$ 0 0
$$881$$ 57.0000 1.92038 0.960189 0.279350i $$-0.0901189\pi$$
0.960189 + 0.279350i $$0.0901189\pi$$
$$882$$ 0 0
$$883$$ 52.0000 1.74994 0.874970 0.484178i $$-0.160881\pi$$
0.874970 + 0.484178i $$0.160881\pi$$
$$884$$ 0 0
$$885$$ 6.00000 0.201688
$$886$$ 0 0
$$887$$ 21.0000 0.705111 0.352555 0.935791i $$-0.385313\pi$$
0.352555 + 0.935791i $$0.385313\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 6.00000 0.201008
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −24.0000 −0.802232
$$896$$ 0 0
$$897$$ 12.0000 0.400668
$$898$$ 0 0
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −11.0000 −0.365652
$$906$$ 0 0
$$907$$ 25.0000 0.830111 0.415056 0.909796i $$-0.363762\pi$$
0.415056 + 0.909796i $$0.363762\pi$$
$$908$$ 0 0
$$909$$ −30.0000 −0.995037
$$910$$ 0 0
$$911$$ −18.0000 −0.596367 −0.298183 0.954509i $$-0.596381\pi$$
−0.298183 + 0.954509i $$0.596381\pi$$
$$912$$ 0 0
$$913$$ −18.0000 −0.595713
$$914$$ 0 0
$$915$$ 5.00000 0.165295
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −14.0000 −0.461817 −0.230909 0.972975i $$-0.574170\pi$$
−0.230909 + 0.972975i $$0.574170\pi$$
$$920$$ 0 0
$$921$$ 5.00000 0.164756
$$922$$ 0 0
$$923$$ −24.0000 −0.789970
$$924$$ 0 0
$$925$$ −4.00000 −0.131519
$$926$$ 0 0
$$927$$ −2.00000 −0.0656886
$$928$$ 0 0
$$929$$ −21.0000 −0.688988 −0.344494 0.938789i $$-0.611949\pi$$
−0.344494 + 0.938789i $$0.611949\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −18.0000 −0.589294
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −28.0000 −0.914720 −0.457360 0.889282i $$-0.651205\pi$$
−0.457360 + 0.889282i $$0.651205\pi$$
$$938$$ 0 0
$$939$$ −8.00000 −0.261070
$$940$$ 0 0
$$941$$ −6.00000 −0.195594 −0.0977972 0.995206i $$-0.531180\pi$$
−0.0977972 + 0.995206i $$0.531180\pi$$
$$942$$ 0 0
$$943$$ 27.0000 0.879241
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 3.00000 0.0974869 0.0487435 0.998811i $$-0.484478\pi$$
0.0487435 + 0.998811i $$0.484478\pi$$
$$948$$ 0 0
$$949$$ 64.0000 2.07753
$$950$$ 0 0
$$951$$ −12.0000 −0.389127
$$952$$ 0 0
$$953$$ 60.0000 1.94359 0.971795 0.235826i $$-0.0757795\pi$$
0.971795 + 0.235826i $$0.0757795\pi$$
$$954$$ 0 0
$$955$$ −6.00000 −0.194155
$$956$$ 0 0
$$957$$ 18.0000 0.581857
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ −30.0000 −0.966736
$$964$$ 0 0
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ −35.0000 −1.12552 −0.562762 0.826619i $$-0.690261\pi$$
−0.562762 + 0.826619i $$0.690261\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 4.00000 0.128103
$$976$$ 0 0
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ 0 0
$$979$$ −90.0000 −2.87641
$$980$$ 0 0
$$981$$ −22.0000 −0.702406
$$982$$ 0 0
$$983$$ −39.0000 −1.24391 −0.621953 0.783054i $$-0.713661\pi$$
−0.621953 + 0.783054i $$0.713661\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 21.0000 0.667761
$$990$$ 0 0
$$991$$ 28.0000 0.889449 0.444725 0.895667i $$-0.353302\pi$$
0.444725 + 0.895667i $$0.353302\pi$$
$$992$$ 0 0
$$993$$ −28.0000 −0.888553
$$994$$ 0 0
$$995$$ −4.00000 −0.126809
$$996$$ 0 0
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ 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.p.1.1 1
4.3 odd 2 490.2.a.c.1.1 1
7.2 even 3 560.2.q.g.81.1 2
7.4 even 3 560.2.q.g.401.1 2
7.6 odd 2 3920.2.a.bc.1.1 1
12.11 even 2 4410.2.a.bm.1.1 1
20.3 even 4 2450.2.c.g.99.2 2
20.7 even 4 2450.2.c.g.99.1 2
20.19 odd 2 2450.2.a.w.1.1 1
28.3 even 6 490.2.e.h.471.1 2
28.11 odd 6 70.2.e.c.51.1 yes 2
28.19 even 6 490.2.e.h.361.1 2
28.23 odd 6 70.2.e.c.11.1 2
28.27 even 2 490.2.a.b.1.1 1
84.11 even 6 630.2.k.b.541.1 2
84.23 even 6 630.2.k.b.361.1 2
84.83 odd 2 4410.2.a.bd.1.1 1
140.23 even 12 350.2.j.b.249.2 4
140.27 odd 4 2450.2.c.l.99.1 2
140.39 odd 6 350.2.e.e.51.1 2
140.67 even 12 350.2.j.b.149.2 4
140.79 odd 6 350.2.e.e.151.1 2
140.83 odd 4 2450.2.c.l.99.2 2
140.107 even 12 350.2.j.b.249.1 4
140.123 even 12 350.2.j.b.149.1 4
140.139 even 2 2450.2.a.bc.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.c.11.1 2 28.23 odd 6
70.2.e.c.51.1 yes 2 28.11 odd 6
350.2.e.e.51.1 2 140.39 odd 6
350.2.e.e.151.1 2 140.79 odd 6
350.2.j.b.149.1 4 140.123 even 12
350.2.j.b.149.2 4 140.67 even 12
350.2.j.b.249.1 4 140.107 even 12
350.2.j.b.249.2 4 140.23 even 12
490.2.a.b.1.1 1 28.27 even 2
490.2.a.c.1.1 1 4.3 odd 2
490.2.e.h.361.1 2 28.19 even 6
490.2.e.h.471.1 2 28.3 even 6
560.2.q.g.81.1 2 7.2 even 3
560.2.q.g.401.1 2 7.4 even 3
630.2.k.b.361.1 2 84.23 even 6
630.2.k.b.541.1 2 84.11 even 6
2450.2.a.w.1.1 1 20.19 odd 2
2450.2.a.bc.1.1 1 140.139 even 2
2450.2.c.g.99.1 2 20.7 even 4
2450.2.c.g.99.2 2 20.3 even 4
2450.2.c.l.99.1 2 140.27 odd 4
2450.2.c.l.99.2 2 140.83 odd 4
3920.2.a.p.1.1 1 1.1 even 1 trivial
3920.2.a.bc.1.1 1 7.6 odd 2
4410.2.a.bd.1.1 1 84.83 odd 2
4410.2.a.bm.1.1 1 12.11 even 2