# Properties

 Label 784.4.a.e.1.1 Level $784$ Weight $4$ Character 784.1 Self dual yes Analytic conductor $46.257$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 784.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.2574974445$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 784.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.00000 q^{3} +2.00000 q^{5} -11.0000 q^{9} +O(q^{10})$$ $$q-4.00000 q^{3} +2.00000 q^{5} -11.0000 q^{9} +44.0000 q^{11} -22.0000 q^{13} -8.00000 q^{15} -50.0000 q^{17} +44.0000 q^{19} +56.0000 q^{23} -121.000 q^{25} +152.000 q^{27} +198.000 q^{29} -160.000 q^{31} -176.000 q^{33} -162.000 q^{37} +88.0000 q^{39} +198.000 q^{41} -52.0000 q^{43} -22.0000 q^{45} +528.000 q^{47} +200.000 q^{51} -242.000 q^{53} +88.0000 q^{55} -176.000 q^{57} -668.000 q^{59} -550.000 q^{61} -44.0000 q^{65} -188.000 q^{67} -224.000 q^{69} -728.000 q^{71} -154.000 q^{73} +484.000 q^{75} +656.000 q^{79} -311.000 q^{81} +236.000 q^{83} -100.000 q^{85} -792.000 q^{87} -714.000 q^{89} +640.000 q^{93} +88.0000 q^{95} +478.000 q^{97} -484.000 q^{99} +O(q^{100})$$

## 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
$$3$$ −4.00000 −0.769800 −0.384900 0.922958i $$-0.625764\pi$$
−0.384900 + 0.922958i $$0.625764\pi$$
$$4$$ 0 0
$$5$$ 2.00000 0.178885 0.0894427 0.995992i $$-0.471491\pi$$
0.0894427 + 0.995992i $$0.471491\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −11.0000 −0.407407
$$10$$ 0 0
$$11$$ 44.0000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ −22.0000 −0.469362 −0.234681 0.972072i $$-0.575405\pi$$
−0.234681 + 0.972072i $$0.575405\pi$$
$$14$$ 0 0
$$15$$ −8.00000 −0.137706
$$16$$ 0 0
$$17$$ −50.0000 −0.713340 −0.356670 0.934230i $$-0.616088\pi$$
−0.356670 + 0.934230i $$0.616088\pi$$
$$18$$ 0 0
$$19$$ 44.0000 0.531279 0.265639 0.964072i $$-0.414417\pi$$
0.265639 + 0.964072i $$0.414417\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 56.0000 0.507687 0.253844 0.967245i $$-0.418305\pi$$
0.253844 + 0.967245i $$0.418305\pi$$
$$24$$ 0 0
$$25$$ −121.000 −0.968000
$$26$$ 0 0
$$27$$ 152.000 1.08342
$$28$$ 0 0
$$29$$ 198.000 1.26785 0.633925 0.773394i $$-0.281443\pi$$
0.633925 + 0.773394i $$0.281443\pi$$
$$30$$ 0 0
$$31$$ −160.000 −0.926995 −0.463498 0.886098i $$-0.653406\pi$$
−0.463498 + 0.886098i $$0.653406\pi$$
$$32$$ 0 0
$$33$$ −176.000 −0.928414
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −162.000 −0.719801 −0.359900 0.932991i $$-0.617189\pi$$
−0.359900 + 0.932991i $$0.617189\pi$$
$$38$$ 0 0
$$39$$ 88.0000 0.361315
$$40$$ 0 0
$$41$$ 198.000 0.754205 0.377102 0.926172i $$-0.376920\pi$$
0.377102 + 0.926172i $$0.376920\pi$$
$$42$$ 0 0
$$43$$ −52.0000 −0.184417 −0.0922084 0.995740i $$-0.529393\pi$$
−0.0922084 + 0.995740i $$0.529393\pi$$
$$44$$ 0 0
$$45$$ −22.0000 −0.0728793
$$46$$ 0 0
$$47$$ 528.000 1.63865 0.819327 0.573327i $$-0.194347\pi$$
0.819327 + 0.573327i $$0.194347\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 200.000 0.549129
$$52$$ 0 0
$$53$$ −242.000 −0.627194 −0.313597 0.949556i $$-0.601534\pi$$
−0.313597 + 0.949556i $$0.601534\pi$$
$$54$$ 0 0
$$55$$ 88.0000 0.215744
$$56$$ 0 0
$$57$$ −176.000 −0.408978
$$58$$ 0 0
$$59$$ −668.000 −1.47400 −0.737002 0.675891i $$-0.763759\pi$$
−0.737002 + 0.675891i $$0.763759\pi$$
$$60$$ 0 0
$$61$$ −550.000 −1.15443 −0.577215 0.816592i $$-0.695861\pi$$
−0.577215 + 0.816592i $$0.695861\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −44.0000 −0.0839620
$$66$$ 0 0
$$67$$ −188.000 −0.342804 −0.171402 0.985201i $$-0.554830\pi$$
−0.171402 + 0.985201i $$0.554830\pi$$
$$68$$ 0 0
$$69$$ −224.000 −0.390818
$$70$$ 0 0
$$71$$ −728.000 −1.21687 −0.608435 0.793604i $$-0.708202\pi$$
−0.608435 + 0.793604i $$0.708202\pi$$
$$72$$ 0 0
$$73$$ −154.000 −0.246909 −0.123454 0.992350i $$-0.539397\pi$$
−0.123454 + 0.992350i $$0.539397\pi$$
$$74$$ 0 0
$$75$$ 484.000 0.745167
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 656.000 0.934250 0.467125 0.884191i $$-0.345290\pi$$
0.467125 + 0.884191i $$0.345290\pi$$
$$80$$ 0 0
$$81$$ −311.000 −0.426612
$$82$$ 0 0
$$83$$ 236.000 0.312101 0.156050 0.987749i $$-0.450124\pi$$
0.156050 + 0.987749i $$0.450124\pi$$
$$84$$ 0 0
$$85$$ −100.000 −0.127606
$$86$$ 0 0
$$87$$ −792.000 −0.975992
$$88$$ 0 0
$$89$$ −714.000 −0.850380 −0.425190 0.905104i $$-0.639793\pi$$
−0.425190 + 0.905104i $$0.639793\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 640.000 0.713601
$$94$$ 0 0
$$95$$ 88.0000 0.0950380
$$96$$ 0 0
$$97$$ 478.000 0.500346 0.250173 0.968201i $$-0.419513\pi$$
0.250173 + 0.968201i $$0.419513\pi$$
$$98$$ 0 0
$$99$$ −484.000 −0.491352
$$100$$ 0 0
$$101$$ −1566.00 −1.54280 −0.771400 0.636350i $$-0.780443\pi$$
−0.771400 + 0.636350i $$0.780443\pi$$
$$102$$ 0 0
$$103$$ −968.000 −0.926018 −0.463009 0.886354i $$-0.653230\pi$$
−0.463009 + 0.886354i $$0.653230\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 780.000 0.704724 0.352362 0.935864i $$-0.385379\pi$$
0.352362 + 0.935864i $$0.385379\pi$$
$$108$$ 0 0
$$109$$ −1994.00 −1.75221 −0.876103 0.482123i $$-0.839866\pi$$
−0.876103 + 0.482123i $$0.839866\pi$$
$$110$$ 0 0
$$111$$ 648.000 0.554103
$$112$$ 0 0
$$113$$ −942.000 −0.784212 −0.392106 0.919920i $$-0.628253\pi$$
−0.392106 + 0.919920i $$0.628253\pi$$
$$114$$ 0 0
$$115$$ 112.000 0.0908179
$$116$$ 0 0
$$117$$ 242.000 0.191221
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 605.000 0.454545
$$122$$ 0 0
$$123$$ −792.000 −0.580587
$$124$$ 0 0
$$125$$ −492.000 −0.352047
$$126$$ 0 0
$$127$$ −1408.00 −0.983778 −0.491889 0.870658i $$-0.663693\pi$$
−0.491889 + 0.870658i $$0.663693\pi$$
$$128$$ 0 0
$$129$$ 208.000 0.141964
$$130$$ 0 0
$$131$$ −2692.00 −1.79543 −0.897714 0.440578i $$-0.854773\pi$$
−0.897714 + 0.440578i $$0.854773\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 304.000 0.193809
$$136$$ 0 0
$$137$$ 1626.00 1.01400 0.507002 0.861945i $$-0.330754\pi$$
0.507002 + 0.861945i $$0.330754\pi$$
$$138$$ 0 0
$$139$$ −684.000 −0.417382 −0.208691 0.977982i $$-0.566920\pi$$
−0.208691 + 0.977982i $$0.566920\pi$$
$$140$$ 0 0
$$141$$ −2112.00 −1.26144
$$142$$ 0 0
$$143$$ −968.000 −0.566072
$$144$$ 0 0
$$145$$ 396.000 0.226800
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 302.000 0.166046 0.0830228 0.996548i $$-0.473543\pi$$
0.0830228 + 0.996548i $$0.473543\pi$$
$$150$$ 0 0
$$151$$ −1352.00 −0.728637 −0.364319 0.931274i $$-0.618698\pi$$
−0.364319 + 0.931274i $$0.618698\pi$$
$$152$$ 0 0
$$153$$ 550.000 0.290620
$$154$$ 0 0
$$155$$ −320.000 −0.165826
$$156$$ 0 0
$$157$$ −3142.00 −1.59719 −0.798595 0.601868i $$-0.794423\pi$$
−0.798595 + 0.601868i $$0.794423\pi$$
$$158$$ 0 0
$$159$$ 968.000 0.482814
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −3036.00 −1.45888 −0.729441 0.684043i $$-0.760220\pi$$
−0.729441 + 0.684043i $$0.760220\pi$$
$$164$$ 0 0
$$165$$ −352.000 −0.166080
$$166$$ 0 0
$$167$$ −264.000 −0.122329 −0.0611645 0.998128i $$-0.519481\pi$$
−0.0611645 + 0.998128i $$0.519481\pi$$
$$168$$ 0 0
$$169$$ −1713.00 −0.779700
$$170$$ 0 0
$$171$$ −484.000 −0.216447
$$172$$ 0 0
$$173$$ 2826.00 1.24195 0.620973 0.783832i $$-0.286737\pi$$
0.620973 + 0.783832i $$0.286737\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 2672.00 1.13469
$$178$$ 0 0
$$179$$ −3084.00 −1.28776 −0.643880 0.765127i $$-0.722676\pi$$
−0.643880 + 0.765127i $$0.722676\pi$$
$$180$$ 0 0
$$181$$ 2418.00 0.992975 0.496488 0.868044i $$-0.334623\pi$$
0.496488 + 0.868044i $$0.334623\pi$$
$$182$$ 0 0
$$183$$ 2200.00 0.888681
$$184$$ 0 0
$$185$$ −324.000 −0.128762
$$186$$ 0 0
$$187$$ −2200.00 −0.860320
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 960.000 0.363681 0.181841 0.983328i $$-0.441794\pi$$
0.181841 + 0.983328i $$0.441794\pi$$
$$192$$ 0 0
$$193$$ 2882.00 1.07488 0.537438 0.843304i $$-0.319392\pi$$
0.537438 + 0.843304i $$0.319392\pi$$
$$194$$ 0 0
$$195$$ 176.000 0.0646340
$$196$$ 0 0
$$197$$ 1086.00 0.392763 0.196381 0.980528i $$-0.437081\pi$$
0.196381 + 0.980528i $$0.437081\pi$$
$$198$$ 0 0
$$199$$ 88.0000 0.0313475 0.0156738 0.999877i $$-0.495011\pi$$
0.0156738 + 0.999877i $$0.495011\pi$$
$$200$$ 0 0
$$201$$ 752.000 0.263890
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 396.000 0.134916
$$206$$ 0 0
$$207$$ −616.000 −0.206836
$$208$$ 0 0
$$209$$ 1936.00 0.640746
$$210$$ 0 0
$$211$$ 3476.00 1.13411 0.567056 0.823679i $$-0.308082\pi$$
0.567056 + 0.823679i $$0.308082\pi$$
$$212$$ 0 0
$$213$$ 2912.00 0.936746
$$214$$ 0 0
$$215$$ −104.000 −0.0329895
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 616.000 0.190070
$$220$$ 0 0
$$221$$ 1100.00 0.334815
$$222$$ 0 0
$$223$$ 928.000 0.278670 0.139335 0.990245i $$-0.455503\pi$$
0.139335 + 0.990245i $$0.455503\pi$$
$$224$$ 0 0
$$225$$ 1331.00 0.394370
$$226$$ 0 0
$$227$$ 156.000 0.0456127 0.0228064 0.999740i $$-0.492740\pi$$
0.0228064 + 0.999740i $$0.492740\pi$$
$$228$$ 0 0
$$229$$ 1634.00 0.471519 0.235759 0.971811i $$-0.424242\pi$$
0.235759 + 0.971811i $$0.424242\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −902.000 −0.253614 −0.126807 0.991927i $$-0.540473\pi$$
−0.126807 + 0.991927i $$0.540473\pi$$
$$234$$ 0 0
$$235$$ 1056.00 0.293131
$$236$$ 0 0
$$237$$ −2624.00 −0.719186
$$238$$ 0 0
$$239$$ −1616.00 −0.437365 −0.218683 0.975796i $$-0.570176\pi$$
−0.218683 + 0.975796i $$0.570176\pi$$
$$240$$ 0 0
$$241$$ −4818.00 −1.28778 −0.643889 0.765119i $$-0.722680\pi$$
−0.643889 + 0.765119i $$0.722680\pi$$
$$242$$ 0 0
$$243$$ −2860.00 −0.755017
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −968.000 −0.249362
$$248$$ 0 0
$$249$$ −944.000 −0.240255
$$250$$ 0 0
$$251$$ −2140.00 −0.538150 −0.269075 0.963119i $$-0.586718\pi$$
−0.269075 + 0.963119i $$0.586718\pi$$
$$252$$ 0 0
$$253$$ 2464.00 0.612294
$$254$$ 0 0
$$255$$ 400.000 0.0982313
$$256$$ 0 0
$$257$$ −770.000 −0.186892 −0.0934461 0.995624i $$-0.529788\pi$$
−0.0934461 + 0.995624i $$0.529788\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2178.00 −0.516532
$$262$$ 0 0
$$263$$ 7400.00 1.73499 0.867497 0.497442i $$-0.165727\pi$$
0.867497 + 0.497442i $$0.165727\pi$$
$$264$$ 0 0
$$265$$ −484.000 −0.112196
$$266$$ 0 0
$$267$$ 2856.00 0.654623
$$268$$ 0 0
$$269$$ 2794.00 0.633283 0.316642 0.948545i $$-0.397445\pi$$
0.316642 + 0.948545i $$0.397445\pi$$
$$270$$ 0 0
$$271$$ 8624.00 1.93310 0.966551 0.256474i $$-0.0825608\pi$$
0.966551 + 0.256474i $$0.0825608\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −5324.00 −1.16745
$$276$$ 0 0
$$277$$ −1874.00 −0.406490 −0.203245 0.979128i $$-0.565149\pi$$
−0.203245 + 0.979128i $$0.565149\pi$$
$$278$$ 0 0
$$279$$ 1760.00 0.377665
$$280$$ 0 0
$$281$$ 3338.00 0.708642 0.354321 0.935124i $$-0.384712\pi$$
0.354321 + 0.935124i $$0.384712\pi$$
$$282$$ 0 0
$$283$$ 7172.00 1.50647 0.753235 0.657751i $$-0.228492\pi$$
0.753235 + 0.657751i $$0.228492\pi$$
$$284$$ 0 0
$$285$$ −352.000 −0.0731603
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −2413.00 −0.491146
$$290$$ 0 0
$$291$$ −1912.00 −0.385166
$$292$$ 0 0
$$293$$ −5214.00 −1.03961 −0.519804 0.854286i $$-0.673995\pi$$
−0.519804 + 0.854286i $$0.673995\pi$$
$$294$$ 0 0
$$295$$ −1336.00 −0.263678
$$296$$ 0 0
$$297$$ 6688.00 1.30666
$$298$$ 0 0
$$299$$ −1232.00 −0.238289
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6264.00 1.18765
$$304$$ 0 0
$$305$$ −1100.00 −0.206511
$$306$$ 0 0
$$307$$ 396.000 0.0736186 0.0368093 0.999322i $$-0.488281\pi$$
0.0368093 + 0.999322i $$0.488281\pi$$
$$308$$ 0 0
$$309$$ 3872.00 0.712849
$$310$$ 0 0
$$311$$ −4056.00 −0.739533 −0.369766 0.929125i $$-0.620562\pi$$
−0.369766 + 0.929125i $$0.620562\pi$$
$$312$$ 0 0
$$313$$ −2154.00 −0.388982 −0.194491 0.980904i $$-0.562305\pi$$
−0.194491 + 0.980904i $$0.562305\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −7386.00 −1.30864 −0.654320 0.756217i $$-0.727045\pi$$
−0.654320 + 0.756217i $$0.727045\pi$$
$$318$$ 0 0
$$319$$ 8712.00 1.52909
$$320$$ 0 0
$$321$$ −3120.00 −0.542497
$$322$$ 0 0
$$323$$ −2200.00 −0.378982
$$324$$ 0 0
$$325$$ 2662.00 0.454342
$$326$$ 0 0
$$327$$ 7976.00 1.34885
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1132.00 0.187977 0.0939884 0.995573i $$-0.470038\pi$$
0.0939884 + 0.995573i $$0.470038\pi$$
$$332$$ 0 0
$$333$$ 1782.00 0.293252
$$334$$ 0 0
$$335$$ −376.000 −0.0613226
$$336$$ 0 0
$$337$$ −3342.00 −0.540209 −0.270104 0.962831i $$-0.587058\pi$$
−0.270104 + 0.962831i $$0.587058\pi$$
$$338$$ 0 0
$$339$$ 3768.00 0.603686
$$340$$ 0 0
$$341$$ −7040.00 −1.11800
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −448.000 −0.0699116
$$346$$ 0 0
$$347$$ −2244.00 −0.347159 −0.173580 0.984820i $$-0.555533\pi$$
−0.173580 + 0.984820i $$0.555533\pi$$
$$348$$ 0 0
$$349$$ 6522.00 1.00033 0.500164 0.865931i $$-0.333273\pi$$
0.500164 + 0.865931i $$0.333273\pi$$
$$350$$ 0 0
$$351$$ −3344.00 −0.508517
$$352$$ 0 0
$$353$$ 11230.0 1.69324 0.846618 0.532200i $$-0.178635\pi$$
0.846618 + 0.532200i $$0.178635\pi$$
$$354$$ 0 0
$$355$$ −1456.00 −0.217680
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1848.00 −0.271682 −0.135841 0.990731i $$-0.543374\pi$$
−0.135841 + 0.990731i $$0.543374\pi$$
$$360$$ 0 0
$$361$$ −4923.00 −0.717743
$$362$$ 0 0
$$363$$ −2420.00 −0.349909
$$364$$ 0 0
$$365$$ −308.000 −0.0441684
$$366$$ 0 0
$$367$$ 7120.00 1.01270 0.506350 0.862328i $$-0.330994\pi$$
0.506350 + 0.862328i $$0.330994\pi$$
$$368$$ 0 0
$$369$$ −2178.00 −0.307269
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 6350.00 0.881476 0.440738 0.897636i $$-0.354717\pi$$
0.440738 + 0.897636i $$0.354717\pi$$
$$374$$ 0 0
$$375$$ 1968.00 0.271006
$$376$$ 0 0
$$377$$ −4356.00 −0.595081
$$378$$ 0 0
$$379$$ 7900.00 1.07070 0.535351 0.844630i $$-0.320179\pi$$
0.535351 + 0.844630i $$0.320179\pi$$
$$380$$ 0 0
$$381$$ 5632.00 0.757313
$$382$$ 0 0
$$383$$ 10368.0 1.38324 0.691619 0.722263i $$-0.256898\pi$$
0.691619 + 0.722263i $$0.256898\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 572.000 0.0751328
$$388$$ 0 0
$$389$$ 8830.00 1.15090 0.575448 0.817838i $$-0.304828\pi$$
0.575448 + 0.817838i $$0.304828\pi$$
$$390$$ 0 0
$$391$$ −2800.00 −0.362154
$$392$$ 0 0
$$393$$ 10768.0 1.38212
$$394$$ 0 0
$$395$$ 1312.00 0.167124
$$396$$ 0 0
$$397$$ −9878.00 −1.24877 −0.624386 0.781116i $$-0.714651\pi$$
−0.624386 + 0.781116i $$0.714651\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −13134.0 −1.63561 −0.817806 0.575494i $$-0.804810\pi$$
−0.817806 + 0.575494i $$0.804810\pi$$
$$402$$ 0 0
$$403$$ 3520.00 0.435096
$$404$$ 0 0
$$405$$ −622.000 −0.0763146
$$406$$ 0 0
$$407$$ −7128.00 −0.868113
$$408$$ 0 0
$$409$$ −906.000 −0.109533 −0.0547663 0.998499i $$-0.517441\pi$$
−0.0547663 + 0.998499i $$0.517441\pi$$
$$410$$ 0 0
$$411$$ −6504.00 −0.780581
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 472.000 0.0558303
$$416$$ 0 0
$$417$$ 2736.00 0.321301
$$418$$ 0 0
$$419$$ −5412.00 −0.631011 −0.315505 0.948924i $$-0.602174\pi$$
−0.315505 + 0.948924i $$0.602174\pi$$
$$420$$ 0 0
$$421$$ −4642.00 −0.537381 −0.268690 0.963227i $$-0.586591\pi$$
−0.268690 + 0.963227i $$0.586591\pi$$
$$422$$ 0 0
$$423$$ −5808.00 −0.667600
$$424$$ 0 0
$$425$$ 6050.00 0.690513
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 3872.00 0.435762
$$430$$ 0 0
$$431$$ −656.000 −0.0733142 −0.0366571 0.999328i $$-0.511671\pi$$
−0.0366571 + 0.999328i $$0.511671\pi$$
$$432$$ 0 0
$$433$$ −9490.00 −1.05326 −0.526629 0.850096i $$-0.676544\pi$$
−0.526629 + 0.850096i $$0.676544\pi$$
$$434$$ 0 0
$$435$$ −1584.00 −0.174591
$$436$$ 0 0
$$437$$ 2464.00 0.269723
$$438$$ 0 0
$$439$$ 5544.00 0.602735 0.301368 0.953508i $$-0.402557\pi$$
0.301368 + 0.953508i $$0.402557\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −7652.00 −0.820672 −0.410336 0.911935i $$-0.634588\pi$$
−0.410336 + 0.911935i $$0.634588\pi$$
$$444$$ 0 0
$$445$$ −1428.00 −0.152121
$$446$$ 0 0
$$447$$ −1208.00 −0.127822
$$448$$ 0 0
$$449$$ −446.000 −0.0468776 −0.0234388 0.999725i $$-0.507461\pi$$
−0.0234388 + 0.999725i $$0.507461\pi$$
$$450$$ 0 0
$$451$$ 8712.00 0.909605
$$452$$ 0 0
$$453$$ 5408.00 0.560905
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1562.00 0.159885 0.0799423 0.996799i $$-0.474526\pi$$
0.0799423 + 0.996799i $$0.474526\pi$$
$$458$$ 0 0
$$459$$ −7600.00 −0.772849
$$460$$ 0 0
$$461$$ −10582.0 −1.06910 −0.534548 0.845138i $$-0.679518\pi$$
−0.534548 + 0.845138i $$0.679518\pi$$
$$462$$ 0 0
$$463$$ 10768.0 1.08085 0.540423 0.841394i $$-0.318264\pi$$
0.540423 + 0.841394i $$0.318264\pi$$
$$464$$ 0 0
$$465$$ 1280.00 0.127653
$$466$$ 0 0
$$467$$ −9876.00 −0.978601 −0.489301 0.872115i $$-0.662748\pi$$
−0.489301 + 0.872115i $$0.662748\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 12568.0 1.22952
$$472$$ 0 0
$$473$$ −2288.00 −0.222415
$$474$$ 0 0
$$475$$ −5324.00 −0.514278
$$476$$ 0 0
$$477$$ 2662.00 0.255523
$$478$$ 0 0
$$479$$ −352.000 −0.0335768 −0.0167884 0.999859i $$-0.505344\pi$$
−0.0167884 + 0.999859i $$0.505344\pi$$
$$480$$ 0 0
$$481$$ 3564.00 0.337847
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 956.000 0.0895046
$$486$$ 0 0
$$487$$ 15176.0 1.41209 0.706047 0.708165i $$-0.250477\pi$$
0.706047 + 0.708165i $$0.250477\pi$$
$$488$$ 0 0
$$489$$ 12144.0 1.12305
$$490$$ 0 0
$$491$$ 8844.00 0.812880 0.406440 0.913677i $$-0.366770\pi$$
0.406440 + 0.913677i $$0.366770\pi$$
$$492$$ 0 0
$$493$$ −9900.00 −0.904409
$$494$$ 0 0
$$495$$ −968.000 −0.0878957
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −19404.0 −1.74077 −0.870383 0.492375i $$-0.836129\pi$$
−0.870383 + 0.492375i $$0.836129\pi$$
$$500$$ 0 0
$$501$$ 1056.00 0.0941689
$$502$$ 0 0
$$503$$ 16488.0 1.46156 0.730779 0.682614i $$-0.239157\pi$$
0.730779 + 0.682614i $$0.239157\pi$$
$$504$$ 0 0
$$505$$ −3132.00 −0.275984
$$506$$ 0 0
$$507$$ 6852.00 0.600213
$$508$$ 0 0
$$509$$ 12954.0 1.12805 0.564024 0.825759i $$-0.309253\pi$$
0.564024 + 0.825759i $$0.309253\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 6688.00 0.575599
$$514$$ 0 0
$$515$$ −1936.00 −0.165651
$$516$$ 0 0
$$517$$ 23232.0 1.97629
$$518$$ 0 0
$$519$$ −11304.0 −0.956051
$$520$$ 0 0
$$521$$ −10970.0 −0.922465 −0.461233 0.887279i $$-0.652593\pi$$
−0.461233 + 0.887279i $$0.652593\pi$$
$$522$$ 0 0
$$523$$ −16940.0 −1.41632 −0.708159 0.706053i $$-0.750474\pi$$
−0.708159 + 0.706053i $$0.750474\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8000.00 0.661263
$$528$$ 0 0
$$529$$ −9031.00 −0.742254
$$530$$ 0 0
$$531$$ 7348.00 0.600520
$$532$$ 0 0
$$533$$ −4356.00 −0.353995
$$534$$ 0 0
$$535$$ 1560.00 0.126065
$$536$$ 0 0
$$537$$ 12336.0 0.991318
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 198.000 0.0157351 0.00786755 0.999969i $$-0.497496\pi$$
0.00786755 + 0.999969i $$0.497496\pi$$
$$542$$ 0 0
$$543$$ −9672.00 −0.764393
$$544$$ 0 0
$$545$$ −3988.00 −0.313444
$$546$$ 0 0
$$547$$ 15268.0 1.19344 0.596721 0.802449i $$-0.296470\pi$$
0.596721 + 0.802449i $$0.296470\pi$$
$$548$$ 0 0
$$549$$ 6050.00 0.470324
$$550$$ 0 0
$$551$$ 8712.00 0.673582
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 1296.00 0.0991210
$$556$$ 0 0
$$557$$ 20854.0 1.58638 0.793189 0.608976i $$-0.208419\pi$$
0.793189 + 0.608976i $$0.208419\pi$$
$$558$$ 0 0
$$559$$ 1144.00 0.0865582
$$560$$ 0 0
$$561$$ 8800.00 0.662275
$$562$$ 0 0
$$563$$ −19316.0 −1.44595 −0.722977 0.690872i $$-0.757227\pi$$
−0.722977 + 0.690872i $$0.757227\pi$$
$$564$$ 0 0
$$565$$ −1884.00 −0.140284
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 7018.00 0.517065 0.258532 0.966003i $$-0.416761\pi$$
0.258532 + 0.966003i $$0.416761\pi$$
$$570$$ 0 0
$$571$$ −24420.0 −1.78975 −0.894873 0.446320i $$-0.852734\pi$$
−0.894873 + 0.446320i $$0.852734\pi$$
$$572$$ 0 0
$$573$$ −3840.00 −0.279962
$$574$$ 0 0
$$575$$ −6776.00 −0.491441
$$576$$ 0 0
$$577$$ −23234.0 −1.67633 −0.838166 0.545415i $$-0.816372\pi$$
−0.838166 + 0.545415i $$0.816372\pi$$
$$578$$ 0 0
$$579$$ −11528.0 −0.827439
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −10648.0 −0.756424
$$584$$ 0 0
$$585$$ 484.000 0.0342067
$$586$$ 0 0
$$587$$ −10604.0 −0.745611 −0.372806 0.927909i $$-0.621604\pi$$
−0.372806 + 0.927909i $$0.621604\pi$$
$$588$$ 0 0
$$589$$ −7040.00 −0.492493
$$590$$ 0 0
$$591$$ −4344.00 −0.302349
$$592$$ 0 0
$$593$$ 13838.0 0.958277 0.479139 0.877739i $$-0.340949\pi$$
0.479139 + 0.877739i $$0.340949\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −352.000 −0.0241313
$$598$$ 0 0
$$599$$ 3960.00 0.270119 0.135059 0.990837i $$-0.456877\pi$$
0.135059 + 0.990837i $$0.456877\pi$$
$$600$$ 0 0
$$601$$ 5942.00 0.403293 0.201647 0.979458i $$-0.435371\pi$$
0.201647 + 0.979458i $$0.435371\pi$$
$$602$$ 0 0
$$603$$ 2068.00 0.139661
$$604$$ 0 0
$$605$$ 1210.00 0.0813116
$$606$$ 0 0
$$607$$ −3040.00 −0.203278 −0.101639 0.994821i $$-0.532409\pi$$
−0.101639 + 0.994821i $$0.532409\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −11616.0 −0.769121
$$612$$ 0 0
$$613$$ −2530.00 −0.166698 −0.0833489 0.996520i $$-0.526562\pi$$
−0.0833489 + 0.996520i $$0.526562\pi$$
$$614$$ 0 0
$$615$$ −1584.00 −0.103859
$$616$$ 0 0
$$617$$ −19206.0 −1.25317 −0.626584 0.779354i $$-0.715547\pi$$
−0.626584 + 0.779354i $$0.715547\pi$$
$$618$$ 0 0
$$619$$ 10996.0 0.714001 0.357000 0.934104i $$-0.383799\pi$$
0.357000 + 0.934104i $$0.383799\pi$$
$$620$$ 0 0
$$621$$ 8512.00 0.550040
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 14141.0 0.905024
$$626$$ 0 0
$$627$$ −7744.00 −0.493247
$$628$$ 0 0
$$629$$ 8100.00 0.513463
$$630$$ 0 0
$$631$$ 6680.00 0.421437 0.210718 0.977547i $$-0.432420\pi$$
0.210718 + 0.977547i $$0.432420\pi$$
$$632$$ 0 0
$$633$$ −13904.0 −0.873040
$$634$$ 0 0
$$635$$ −2816.00 −0.175984
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 8008.00 0.495761
$$640$$ 0 0
$$641$$ 6274.00 0.386596 0.193298 0.981140i $$-0.438082\pi$$
0.193298 + 0.981140i $$0.438082\pi$$
$$642$$ 0 0
$$643$$ 9084.00 0.557135 0.278568 0.960417i $$-0.410140\pi$$
0.278568 + 0.960417i $$0.410140\pi$$
$$644$$ 0 0
$$645$$ 416.000 0.0253953
$$646$$ 0 0
$$647$$ −23656.0 −1.43742 −0.718712 0.695308i $$-0.755268\pi$$
−0.718712 + 0.695308i $$0.755268\pi$$
$$648$$ 0 0
$$649$$ −29392.0 −1.77771
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −6762.00 −0.405234 −0.202617 0.979258i $$-0.564945\pi$$
−0.202617 + 0.979258i $$0.564945\pi$$
$$654$$ 0 0
$$655$$ −5384.00 −0.321176
$$656$$ 0 0
$$657$$ 1694.00 0.100592
$$658$$ 0 0
$$659$$ −15276.0 −0.902987 −0.451494 0.892274i $$-0.649109\pi$$
−0.451494 + 0.892274i $$0.649109\pi$$
$$660$$ 0 0
$$661$$ −11054.0 −0.650455 −0.325228 0.945636i $$-0.605441\pi$$
−0.325228 + 0.945636i $$0.605441\pi$$
$$662$$ 0 0
$$663$$ −4400.00 −0.257740
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 11088.0 0.643672
$$668$$ 0 0
$$669$$ −3712.00 −0.214520
$$670$$ 0 0
$$671$$ −24200.0 −1.39230
$$672$$ 0 0
$$673$$ −21278.0 −1.21873 −0.609366 0.792889i $$-0.708576\pi$$
−0.609366 + 0.792889i $$0.708576\pi$$
$$674$$ 0 0
$$675$$ −18392.0 −1.04875
$$676$$ 0 0
$$677$$ −8926.00 −0.506727 −0.253363 0.967371i $$-0.581537\pi$$
−0.253363 + 0.967371i $$0.581537\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −624.000 −0.0351127
$$682$$ 0 0
$$683$$ −8116.00 −0.454685 −0.227343 0.973815i $$-0.573004\pi$$
−0.227343 + 0.973815i $$0.573004\pi$$
$$684$$ 0 0
$$685$$ 3252.00 0.181391
$$686$$ 0 0
$$687$$ −6536.00 −0.362975
$$688$$ 0 0
$$689$$ 5324.00 0.294381
$$690$$ 0 0
$$691$$ −11764.0 −0.647646 −0.323823 0.946118i $$-0.604968\pi$$
−0.323823 + 0.946118i $$0.604968\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −1368.00 −0.0746636
$$696$$ 0 0
$$697$$ −9900.00 −0.538005
$$698$$ 0 0
$$699$$ 3608.00 0.195232
$$700$$ 0 0
$$701$$ −4698.00 −0.253126 −0.126563 0.991959i $$-0.540395\pi$$
−0.126563 + 0.991959i $$0.540395\pi$$
$$702$$ 0 0
$$703$$ −7128.00 −0.382415
$$704$$ 0 0
$$705$$ −4224.00 −0.225653
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 24638.0 1.30508 0.652538 0.757756i $$-0.273704\pi$$
0.652538 + 0.757756i $$0.273704\pi$$
$$710$$ 0 0
$$711$$ −7216.00 −0.380620
$$712$$ 0 0
$$713$$ −8960.00 −0.470624
$$714$$ 0 0
$$715$$ −1936.00 −0.101262
$$716$$ 0 0
$$717$$ 6464.00 0.336684
$$718$$ 0 0
$$719$$ 16624.0 0.862268 0.431134 0.902288i $$-0.358114\pi$$
0.431134 + 0.902288i $$0.358114\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 19272.0 0.991332
$$724$$ 0 0
$$725$$ −23958.0 −1.22728
$$726$$ 0 0
$$727$$ 30216.0 1.54147 0.770735 0.637155i $$-0.219889\pi$$
0.770735 + 0.637155i $$0.219889\pi$$
$$728$$ 0 0
$$729$$ 19837.0 1.00782
$$730$$ 0 0
$$731$$ 2600.00 0.131552
$$732$$ 0 0
$$733$$ 3322.00 0.167395 0.0836977 0.996491i $$-0.473327\pi$$
0.0836977 + 0.996491i $$0.473327\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −8272.00 −0.413437
$$738$$ 0 0
$$739$$ 14692.0 0.731331 0.365666 0.930746i $$-0.380841\pi$$
0.365666 + 0.930746i $$0.380841\pi$$
$$740$$ 0 0
$$741$$ 3872.00 0.191959
$$742$$ 0 0
$$743$$ −28600.0 −1.41216 −0.706078 0.708134i $$-0.749537\pi$$
−0.706078 + 0.708134i $$0.749537\pi$$
$$744$$ 0 0
$$745$$ 604.000 0.0297032
$$746$$ 0 0
$$747$$ −2596.00 −0.127152
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 29616.0 1.43902 0.719509 0.694483i $$-0.244367\pi$$
0.719509 + 0.694483i $$0.244367\pi$$
$$752$$ 0 0
$$753$$ 8560.00 0.414268
$$754$$ 0 0
$$755$$ −2704.00 −0.130343
$$756$$ 0 0
$$757$$ 2894.00 0.138949 0.0694744 0.997584i $$-0.477868\pi$$
0.0694744 + 0.997584i $$0.477868\pi$$
$$758$$ 0 0
$$759$$ −9856.00 −0.471344
$$760$$ 0 0
$$761$$ −14762.0 −0.703183 −0.351591 0.936154i $$-0.614359\pi$$
−0.351591 + 0.936154i $$0.614359\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 1100.00 0.0519877
$$766$$ 0 0
$$767$$ 14696.0 0.691841
$$768$$ 0 0
$$769$$ 7678.00 0.360047 0.180023 0.983662i $$-0.442383\pi$$
0.180023 + 0.983662i $$0.442383\pi$$
$$770$$ 0 0
$$771$$ 3080.00 0.143870
$$772$$ 0 0
$$773$$ −27390.0 −1.27445 −0.637225 0.770678i $$-0.719918\pi$$
−0.637225 + 0.770678i $$0.719918\pi$$
$$774$$ 0 0
$$775$$ 19360.0 0.897331
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 8712.00 0.400693
$$780$$ 0 0
$$781$$ −32032.0 −1.46760
$$782$$ 0 0
$$783$$ 30096.0 1.37362
$$784$$ 0 0
$$785$$ −6284.00 −0.285714
$$786$$ 0 0
$$787$$ 19756.0 0.894823 0.447411 0.894328i $$-0.352346\pi$$
0.447411 + 0.894328i $$0.352346\pi$$
$$788$$ 0 0
$$789$$ −29600.0 −1.33560
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 12100.0 0.541846
$$794$$ 0 0
$$795$$ 1936.00 0.0863684
$$796$$ 0 0
$$797$$ −38854.0 −1.72682 −0.863412 0.504499i $$-0.831677\pi$$
−0.863412 + 0.504499i $$0.831677\pi$$
$$798$$ 0 0
$$799$$ −26400.0 −1.16892
$$800$$ 0 0
$$801$$ 7854.00 0.346451
$$802$$ 0 0
$$803$$ −6776.00 −0.297783
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −11176.0 −0.487502
$$808$$ 0 0
$$809$$ −14278.0 −0.620504 −0.310252 0.950654i $$-0.600413\pi$$
−0.310252 + 0.950654i $$0.600413\pi$$
$$810$$ 0 0
$$811$$ −716.000 −0.0310014 −0.0155007 0.999880i $$-0.504934\pi$$
−0.0155007 + 0.999880i $$0.504934\pi$$
$$812$$ 0 0
$$813$$ −34496.0 −1.48810
$$814$$ 0 0
$$815$$ −6072.00 −0.260973
$$816$$ 0 0
$$817$$ −2288.00 −0.0979767
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −23538.0 −1.00059 −0.500293 0.865856i $$-0.666775\pi$$
−0.500293 + 0.865856i $$0.666775\pi$$
$$822$$ 0 0
$$823$$ 6616.00 0.280218 0.140109 0.990136i $$-0.455255\pi$$
0.140109 + 0.990136i $$0.455255\pi$$
$$824$$ 0 0
$$825$$ 21296.0 0.898705
$$826$$ 0 0
$$827$$ −27236.0 −1.14521 −0.572605 0.819831i $$-0.694067\pi$$
−0.572605 + 0.819831i $$0.694067\pi$$
$$828$$ 0 0
$$829$$ −12070.0 −0.505680 −0.252840 0.967508i $$-0.581365\pi$$
−0.252840 + 0.967508i $$0.581365\pi$$
$$830$$ 0 0
$$831$$ 7496.00 0.312916
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −528.000 −0.0218829
$$836$$ 0 0
$$837$$ −24320.0 −1.00433
$$838$$ 0 0
$$839$$ −42024.0 −1.72924 −0.864618 0.502429i $$-0.832440\pi$$
−0.864618 + 0.502429i $$0.832440\pi$$
$$840$$ 0 0
$$841$$ 14815.0 0.607446
$$842$$ 0 0
$$843$$ −13352.0 −0.545513
$$844$$ 0 0
$$845$$ −3426.00 −0.139477
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −28688.0 −1.15968
$$850$$ 0 0
$$851$$ −9072.00 −0.365434
$$852$$ 0 0
$$853$$ −2414.00 −0.0968978 −0.0484489 0.998826i $$-0.515428\pi$$
−0.0484489 + 0.998826i $$0.515428\pi$$
$$854$$ 0 0
$$855$$ −968.000 −0.0387192
$$856$$ 0 0
$$857$$ 37686.0 1.50213 0.751067 0.660226i $$-0.229539\pi$$
0.751067 + 0.660226i $$0.229539\pi$$
$$858$$ 0 0
$$859$$ 40644.0 1.61438 0.807192 0.590289i $$-0.200986\pi$$
0.807192 + 0.590289i $$0.200986\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 18656.0 0.735872 0.367936 0.929851i $$-0.380065\pi$$
0.367936 + 0.929851i $$0.380065\pi$$
$$864$$ 0 0
$$865$$ 5652.00 0.222166
$$866$$ 0 0
$$867$$ 9652.00 0.378084
$$868$$ 0 0
$$869$$ 28864.0 1.12675
$$870$$ 0 0
$$871$$ 4136.00 0.160899
$$872$$ 0 0
$$873$$ −5258.00 −0.203845
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −13002.0 −0.500623 −0.250311 0.968165i $$-0.580533\pi$$
−0.250311 + 0.968165i $$0.580533\pi$$
$$878$$ 0 0
$$879$$ 20856.0 0.800291
$$880$$ 0 0
$$881$$ −49490.0 −1.89258 −0.946289 0.323323i $$-0.895200\pi$$
−0.946289 + 0.323323i $$0.895200\pi$$
$$882$$ 0 0
$$883$$ −1100.00 −0.0419229 −0.0209615 0.999780i $$-0.506673\pi$$
−0.0209615 + 0.999780i $$0.506673\pi$$
$$884$$ 0 0
$$885$$ 5344.00 0.202979
$$886$$ 0 0
$$887$$ −14104.0 −0.533896 −0.266948 0.963711i $$-0.586015\pi$$
−0.266948 + 0.963711i $$0.586015\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −13684.0 −0.514513
$$892$$ 0 0
$$893$$ 23232.0 0.870581
$$894$$ 0 0
$$895$$ −6168.00 −0.230361
$$896$$ 0 0
$$897$$ 4928.00 0.183435
$$898$$ 0 0
$$899$$ −31680.0 −1.17529
$$900$$ 0 0
$$901$$ 12100.0 0.447402
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 4836.00 0.177629
$$906$$ 0 0
$$907$$ 12716.0 0.465521 0.232761 0.972534i $$-0.425224\pi$$
0.232761 + 0.972534i $$0.425224\pi$$
$$908$$ 0 0
$$909$$ 17226.0 0.628548
$$910$$ 0 0
$$911$$ 39632.0 1.44135 0.720673 0.693275i $$-0.243833\pi$$
0.720673 + 0.693275i $$0.243833\pi$$
$$912$$ 0 0
$$913$$ 10384.0 0.376408
$$914$$ 0 0
$$915$$ 4400.00 0.158972
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −5704.00 −0.204742 −0.102371 0.994746i $$-0.532643\pi$$
−0.102371 + 0.994746i $$0.532643\pi$$
$$920$$ 0 0
$$921$$ −1584.00 −0.0566716
$$922$$ 0 0
$$923$$ 16016.0 0.571152
$$924$$ 0 0
$$925$$ 19602.0 0.696767
$$926$$ 0 0
$$927$$ 10648.0 0.377267
$$928$$ 0 0
$$929$$ −8162.00 −0.288252 −0.144126 0.989559i $$-0.546037\pi$$
−0.144126 + 0.989559i $$0.546037\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 16224.0 0.569293
$$934$$ 0 0
$$935$$ −4400.00 −0.153899
$$936$$ 0 0
$$937$$ 55110.0 1.92141 0.960707 0.277564i $$-0.0895270\pi$$
0.960707 + 0.277564i $$0.0895270\pi$$
$$938$$ 0 0
$$939$$ 8616.00 0.299438
$$940$$ 0 0
$$941$$ −16374.0 −0.567245 −0.283622 0.958936i $$-0.591536\pi$$
−0.283622 + 0.958936i $$0.591536\pi$$
$$942$$ 0 0
$$943$$ 11088.0 0.382900
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −8460.00 −0.290299 −0.145149 0.989410i $$-0.546366\pi$$
−0.145149 + 0.989410i $$0.546366\pi$$
$$948$$ 0 0
$$949$$ 3388.00 0.115889
$$950$$ 0 0
$$951$$ 29544.0 1.00739
$$952$$ 0 0
$$953$$ −20502.0 −0.696878 −0.348439 0.937331i $$-0.613288\pi$$
−0.348439 + 0.937331i $$0.613288\pi$$
$$954$$ 0 0
$$955$$ 1920.00 0.0650573
$$956$$ 0 0
$$957$$ −34848.0 −1.17709
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −4191.00 −0.140680
$$962$$ 0 0
$$963$$ −8580.00 −0.287110
$$964$$ 0 0
$$965$$ 5764.00 0.192280
$$966$$ 0 0
$$967$$ 36520.0 1.21448 0.607241 0.794518i $$-0.292276\pi$$
0.607241 + 0.794518i $$0.292276\pi$$
$$968$$ 0 0
$$969$$ 8800.00 0.291741
$$970$$ 0 0
$$971$$ 20244.0 0.669064 0.334532 0.942384i $$-0.391422\pi$$
0.334532 + 0.942384i $$0.391422\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −10648.0 −0.349753
$$976$$ 0 0
$$977$$ 50034.0 1.63841 0.819206 0.573499i $$-0.194414\pi$$
0.819206 + 0.573499i $$0.194414\pi$$
$$978$$ 0 0
$$979$$ −31416.0 −1.02560
$$980$$ 0 0
$$981$$ 21934.0 0.713862
$$982$$ 0 0
$$983$$ 37128.0 1.20468 0.602339 0.798240i $$-0.294235\pi$$
0.602339 + 0.798240i $$0.294235\pi$$
$$984$$ 0 0
$$985$$ 2172.00 0.0702596
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −2912.00 −0.0936261
$$990$$ 0 0
$$991$$ −27808.0 −0.891373 −0.445686 0.895189i $$-0.647040\pi$$
−0.445686 + 0.895189i $$0.647040\pi$$
$$992$$ 0 0
$$993$$ −4528.00 −0.144705
$$994$$ 0 0
$$995$$ 176.000 0.00560761
$$996$$ 0 0
$$997$$ 28514.0 0.905765 0.452882 0.891570i $$-0.350396\pi$$
0.452882 + 0.891570i $$0.350396\pi$$
$$998$$ 0 0
$$999$$ −24624.0 −0.779849
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 784.4.a.e.1.1 1
4.3 odd 2 392.4.a.e.1.1 1
7.6 odd 2 16.4.a.a.1.1 1
21.20 even 2 144.4.a.e.1.1 1
28.3 even 6 392.4.i.g.177.1 2
28.11 odd 6 392.4.i.b.177.1 2
28.19 even 6 392.4.i.g.361.1 2
28.23 odd 6 392.4.i.b.361.1 2
28.27 even 2 8.4.a.a.1.1 1
35.13 even 4 400.4.c.i.49.2 2
35.27 even 4 400.4.c.i.49.1 2
35.34 odd 2 400.4.a.g.1.1 1
56.13 odd 2 64.4.a.b.1.1 1
56.27 even 2 64.4.a.d.1.1 1
77.76 even 2 1936.4.a.l.1.1 1
84.83 odd 2 72.4.a.c.1.1 1
112.13 odd 4 256.4.b.g.129.2 2
112.27 even 4 256.4.b.a.129.2 2
112.69 odd 4 256.4.b.g.129.1 2
112.83 even 4 256.4.b.a.129.1 2
140.27 odd 4 200.4.c.e.49.2 2
140.83 odd 4 200.4.c.e.49.1 2
140.139 even 2 200.4.a.g.1.1 1
168.83 odd 2 576.4.a.k.1.1 1
168.125 even 2 576.4.a.j.1.1 1
252.83 odd 6 648.4.i.e.433.1 2
252.139 even 6 648.4.i.h.217.1 2
252.167 odd 6 648.4.i.e.217.1 2
252.223 even 6 648.4.i.h.433.1 2
280.69 odd 2 1600.4.a.bm.1.1 1
280.139 even 2 1600.4.a.o.1.1 1
308.307 odd 2 968.4.a.a.1.1 1
364.363 even 2 1352.4.a.a.1.1 1
420.83 even 4 1800.4.f.u.649.1 2
420.167 even 4 1800.4.f.u.649.2 2
420.419 odd 2 1800.4.a.d.1.1 1
476.475 even 2 2312.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
8.4.a.a.1.1 1 28.27 even 2
16.4.a.a.1.1 1 7.6 odd 2
64.4.a.b.1.1 1 56.13 odd 2
64.4.a.d.1.1 1 56.27 even 2
72.4.a.c.1.1 1 84.83 odd 2
144.4.a.e.1.1 1 21.20 even 2
200.4.a.g.1.1 1 140.139 even 2
200.4.c.e.49.1 2 140.83 odd 4
200.4.c.e.49.2 2 140.27 odd 4
256.4.b.a.129.1 2 112.83 even 4
256.4.b.a.129.2 2 112.27 even 4
256.4.b.g.129.1 2 112.69 odd 4
256.4.b.g.129.2 2 112.13 odd 4
392.4.a.e.1.1 1 4.3 odd 2
392.4.i.b.177.1 2 28.11 odd 6
392.4.i.b.361.1 2 28.23 odd 6
392.4.i.g.177.1 2 28.3 even 6
392.4.i.g.361.1 2 28.19 even 6
400.4.a.g.1.1 1 35.34 odd 2
400.4.c.i.49.1 2 35.27 even 4
400.4.c.i.49.2 2 35.13 even 4
576.4.a.j.1.1 1 168.125 even 2
576.4.a.k.1.1 1 168.83 odd 2
648.4.i.e.217.1 2 252.167 odd 6
648.4.i.e.433.1 2 252.83 odd 6
648.4.i.h.217.1 2 252.139 even 6
648.4.i.h.433.1 2 252.223 even 6
784.4.a.e.1.1 1 1.1 even 1 trivial
968.4.a.a.1.1 1 308.307 odd 2
1352.4.a.a.1.1 1 364.363 even 2
1600.4.a.o.1.1 1 280.139 even 2
1600.4.a.bm.1.1 1 280.69 odd 2
1800.4.a.d.1.1 1 420.419 odd 2
1800.4.f.u.649.1 2 420.83 even 4
1800.4.f.u.649.2 2 420.167 even 4
1936.4.a.l.1.1 1 77.76 even 2
2312.4.a.a.1.1 1 476.475 even 2