# Properties

 Label 784.6.a.bf.1.3 Level $784$ Weight $6$ Character 784.1 Self dual yes Analytic conductor $125.741$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 784.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$125.740914733$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{113})$$ Defining polynomial: $$x^{4} - 2x^{3} - 59x^{2} + 60x + 674$$ x^4 - 2*x^3 - 59*x^2 + 60*x + 674 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}\cdot 7$$ Twist minimal: no (minimal twist has level 49) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$7.22929$$ of defining polynomial Character $$\chi$$ $$=$$ 784.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+6.54802 q^{3} +45.9910 q^{5} -200.123 q^{9} +O(q^{10})$$ $$q+6.54802 q^{3} +45.9910 q^{5} -200.123 q^{9} +551.781 q^{11} -1094.10 q^{13} +301.150 q^{15} +1180.71 q^{17} -1166.13 q^{19} -44.3851 q^{23} -1009.82 q^{25} -2901.58 q^{27} +3329.02 q^{29} +8784.01 q^{31} +3613.07 q^{33} -2557.12 q^{37} -7164.17 q^{39} +12761.3 q^{41} +96.7714 q^{43} -9203.89 q^{45} +7679.15 q^{47} +7731.33 q^{51} -11953.3 q^{53} +25377.0 q^{55} -7635.81 q^{57} +9857.24 q^{59} +38517.9 q^{61} -50318.7 q^{65} +67548.9 q^{67} -290.634 q^{69} +61374.6 q^{71} -1850.40 q^{73} -6612.34 q^{75} +8.52913 q^{79} +29630.4 q^{81} -95039.3 q^{83} +54302.3 q^{85} +21798.5 q^{87} -53605.6 q^{89} +57517.9 q^{93} -53631.4 q^{95} -3110.79 q^{97} -110424. q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 220 q^{9}+O(q^{10})$$ 4 * q + 220 * q^9 $$4 q + 220 q^{9} + 1952 q^{11} + 4096 q^{15} + 7136 q^{23} + 2764 q^{25} - 3352 q^{29} - 9208 q^{37} - 2464 q^{39} - 20448 q^{43} + 67408 q^{51} - 102920 q^{53} - 15576 q^{57} - 63168 q^{65} + 22896 q^{67} + 153824 q^{71} + 90688 q^{79} - 17204 q^{81} + 272656 q^{85} + 247760 q^{93} - 108224 q^{95} + 42272 q^{99}+O(q^{100})$$ 4 * q + 220 * q^9 + 1952 * q^11 + 4096 * q^15 + 7136 * q^23 + 2764 * q^25 - 3352 * q^29 - 9208 * q^37 - 2464 * q^39 - 20448 * q^43 + 67408 * q^51 - 102920 * q^53 - 15576 * q^57 - 63168 * q^65 + 22896 * q^67 + 153824 * q^71 + 90688 * q^79 - 17204 * q^81 + 272656 * q^85 + 247760 * q^93 - 108224 * q^95 + 42272 * q^99

## 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
$$3$$ 6.54802 0.420055 0.210028 0.977695i $$-0.432645\pi$$
0.210028 + 0.977695i $$0.432645\pi$$
$$4$$ 0 0
$$5$$ 45.9910 0.822713 0.411356 0.911475i $$-0.365055\pi$$
0.411356 + 0.911475i $$0.365055\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −200.123 −0.823553
$$10$$ 0 0
$$11$$ 551.781 1.37494 0.687472 0.726211i $$-0.258720\pi$$
0.687472 + 0.726211i $$0.258720\pi$$
$$12$$ 0 0
$$13$$ −1094.10 −1.79555 −0.897776 0.440453i $$-0.854818\pi$$
−0.897776 + 0.440453i $$0.854818\pi$$
$$14$$ 0 0
$$15$$ 301.150 0.345585
$$16$$ 0 0
$$17$$ 1180.71 0.990883 0.495442 0.868641i $$-0.335006\pi$$
0.495442 + 0.868641i $$0.335006\pi$$
$$18$$ 0 0
$$19$$ −1166.13 −0.741074 −0.370537 0.928818i $$-0.620826\pi$$
−0.370537 + 0.928818i $$0.620826\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −44.3851 −0.0174951 −0.00874757 0.999962i $$-0.502784\pi$$
−0.00874757 + 0.999962i $$0.502784\pi$$
$$24$$ 0 0
$$25$$ −1009.82 −0.323143
$$26$$ 0 0
$$27$$ −2901.58 −0.765993
$$28$$ 0 0
$$29$$ 3329.02 0.735057 0.367529 0.930012i $$-0.380204\pi$$
0.367529 + 0.930012i $$0.380204\pi$$
$$30$$ 0 0
$$31$$ 8784.01 1.64168 0.820841 0.571157i $$-0.193505\pi$$
0.820841 + 0.571157i $$0.193505\pi$$
$$32$$ 0 0
$$33$$ 3613.07 0.577552
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2557.12 −0.307077 −0.153539 0.988143i $$-0.549067\pi$$
−0.153539 + 0.988143i $$0.549067\pi$$
$$38$$ 0 0
$$39$$ −7164.17 −0.754231
$$40$$ 0 0
$$41$$ 12761.3 1.18559 0.592794 0.805354i $$-0.298025\pi$$
0.592794 + 0.805354i $$0.298025\pi$$
$$42$$ 0 0
$$43$$ 96.7714 0.00798135 0.00399067 0.999992i $$-0.498730\pi$$
0.00399067 + 0.999992i $$0.498730\pi$$
$$44$$ 0 0
$$45$$ −9203.89 −0.677548
$$46$$ 0 0
$$47$$ 7679.15 0.507071 0.253535 0.967326i $$-0.418407\pi$$
0.253535 + 0.967326i $$0.418407\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 7731.33 0.416226
$$52$$ 0 0
$$53$$ −11953.3 −0.584520 −0.292260 0.956339i $$-0.594407\pi$$
−0.292260 + 0.956339i $$0.594407\pi$$
$$54$$ 0 0
$$55$$ 25377.0 1.13118
$$56$$ 0 0
$$57$$ −7635.81 −0.311292
$$58$$ 0 0
$$59$$ 9857.24 0.368659 0.184330 0.982864i $$-0.440989\pi$$
0.184330 + 0.982864i $$0.440989\pi$$
$$60$$ 0 0
$$61$$ 38517.9 1.32537 0.662686 0.748897i $$-0.269416\pi$$
0.662686 + 0.748897i $$0.269416\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −50318.7 −1.47722
$$66$$ 0 0
$$67$$ 67548.9 1.83836 0.919182 0.393833i $$-0.128851\pi$$
0.919182 + 0.393833i $$0.128851\pi$$
$$68$$ 0 0
$$69$$ −290.634 −0.00734893
$$70$$ 0 0
$$71$$ 61374.6 1.44492 0.722458 0.691415i $$-0.243012\pi$$
0.722458 + 0.691415i $$0.243012\pi$$
$$72$$ 0 0
$$73$$ −1850.40 −0.0406404 −0.0203202 0.999794i $$-0.506469\pi$$
−0.0203202 + 0.999794i $$0.506469\pi$$
$$74$$ 0 0
$$75$$ −6612.34 −0.135738
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.52913 0.000153758 0 7.68788e−5 1.00000i $$-0.499976\pi$$
7.68788e−5 1.00000i $$0.499976\pi$$
$$80$$ 0 0
$$81$$ 29630.4 0.501794
$$82$$ 0 0
$$83$$ −95039.3 −1.51429 −0.757143 0.653249i $$-0.773405\pi$$
−0.757143 + 0.653249i $$0.773405\pi$$
$$84$$ 0 0
$$85$$ 54302.3 0.815212
$$86$$ 0 0
$$87$$ 21798.5 0.308765
$$88$$ 0 0
$$89$$ −53605.6 −0.717357 −0.358678 0.933461i $$-0.616773\pi$$
−0.358678 + 0.933461i $$0.616773\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 57517.9 0.689597
$$94$$ 0 0
$$95$$ −53631.4 −0.609691
$$96$$ 0 0
$$97$$ −3110.79 −0.0335693 −0.0167846 0.999859i $$-0.505343\pi$$
−0.0167846 + 0.999859i $$0.505343\pi$$
$$98$$ 0 0
$$99$$ −110424. −1.13234
$$100$$ 0 0
$$101$$ −21835.9 −0.212994 −0.106497 0.994313i $$-0.533963\pi$$
−0.106497 + 0.994313i $$0.533963\pi$$
$$102$$ 0 0
$$103$$ 65341.4 0.606870 0.303435 0.952852i $$-0.401866\pi$$
0.303435 + 0.952852i $$0.401866\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 108957. 0.920020 0.460010 0.887914i $$-0.347846\pi$$
0.460010 + 0.887914i $$0.347846\pi$$
$$108$$ 0 0
$$109$$ 86728.7 0.699192 0.349596 0.936901i $$-0.386319\pi$$
0.349596 + 0.936901i $$0.386319\pi$$
$$110$$ 0 0
$$111$$ −16744.1 −0.128989
$$112$$ 0 0
$$113$$ −101496. −0.747746 −0.373873 0.927480i $$-0.621970\pi$$
−0.373873 + 0.927480i $$0.621970\pi$$
$$114$$ 0 0
$$115$$ −2041.32 −0.0143935
$$116$$ 0 0
$$117$$ 218955. 1.47873
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 143411. 0.890470
$$122$$ 0 0
$$123$$ 83560.9 0.498013
$$124$$ 0 0
$$125$$ −190165. −1.08857
$$126$$ 0 0
$$127$$ 3094.61 0.0170253 0.00851267 0.999964i $$-0.497290\pi$$
0.00851267 + 0.999964i $$0.497290\pi$$
$$128$$ 0 0
$$129$$ 633.661 0.00335261
$$130$$ 0 0
$$131$$ 253431. 1.29027 0.645136 0.764067i $$-0.276801\pi$$
0.645136 + 0.764067i $$0.276801\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −133447. −0.630193
$$136$$ 0 0
$$137$$ −97152.9 −0.442236 −0.221118 0.975247i $$-0.570971\pi$$
−0.221118 + 0.975247i $$0.570971\pi$$
$$138$$ 0 0
$$139$$ 210308. 0.923249 0.461624 0.887076i $$-0.347267\pi$$
0.461624 + 0.887076i $$0.347267\pi$$
$$140$$ 0 0
$$141$$ 50283.2 0.212998
$$142$$ 0 0
$$143$$ −603702. −2.46878
$$144$$ 0 0
$$145$$ 153105. 0.604741
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 140406. 0.518109 0.259055 0.965863i $$-0.416589\pi$$
0.259055 + 0.965863i $$0.416589\pi$$
$$150$$ 0 0
$$151$$ −119696. −0.427205 −0.213603 0.976921i $$-0.568520\pi$$
−0.213603 + 0.976921i $$0.568520\pi$$
$$152$$ 0 0
$$153$$ −236289. −0.816045
$$154$$ 0 0
$$155$$ 403986. 1.35063
$$156$$ 0 0
$$157$$ −97616.9 −0.316065 −0.158032 0.987434i $$-0.550515\pi$$
−0.158032 + 0.987434i $$0.550515\pi$$
$$158$$ 0 0
$$159$$ −78270.6 −0.245531
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −182678. −0.538539 −0.269270 0.963065i $$-0.586782\pi$$
−0.269270 + 0.963065i $$0.586782\pi$$
$$164$$ 0 0
$$165$$ 166169. 0.475160
$$166$$ 0 0
$$167$$ −451674. −1.25324 −0.626619 0.779326i $$-0.715562\pi$$
−0.626619 + 0.779326i $$0.715562\pi$$
$$168$$ 0 0
$$169$$ 825757. 2.22400
$$170$$ 0 0
$$171$$ 233369. 0.610314
$$172$$ 0 0
$$173$$ −371647. −0.944095 −0.472047 0.881573i $$-0.656485\pi$$
−0.472047 + 0.881573i $$0.656485\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 64545.4 0.154857
$$178$$ 0 0
$$179$$ −85003.4 −0.198291 −0.0991457 0.995073i $$-0.531611\pi$$
−0.0991457 + 0.995073i $$0.531611\pi$$
$$180$$ 0 0
$$181$$ 379442. 0.860892 0.430446 0.902616i $$-0.358356\pi$$
0.430446 + 0.902616i $$0.358356\pi$$
$$182$$ 0 0
$$183$$ 252216. 0.556730
$$184$$ 0 0
$$185$$ −117605. −0.252636
$$186$$ 0 0
$$187$$ 651496. 1.36241
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 922196. 1.82911 0.914555 0.404462i $$-0.132541\pi$$
0.914555 + 0.404462i $$0.132541\pi$$
$$192$$ 0 0
$$193$$ 505107. 0.976090 0.488045 0.872818i $$-0.337710\pi$$
0.488045 + 0.872818i $$0.337710\pi$$
$$194$$ 0 0
$$195$$ −329488. −0.620516
$$196$$ 0 0
$$197$$ 251505. 0.461723 0.230861 0.972987i $$-0.425846\pi$$
0.230861 + 0.972987i $$0.425846\pi$$
$$198$$ 0 0
$$199$$ −208033. −0.372392 −0.186196 0.982513i $$-0.559616\pi$$
−0.186196 + 0.982513i $$0.559616\pi$$
$$200$$ 0 0
$$201$$ 442311. 0.772215
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 586904. 0.975399
$$206$$ 0 0
$$207$$ 8882.50 0.0144082
$$208$$ 0 0
$$209$$ −643446. −1.01893
$$210$$ 0 0
$$211$$ 640577. 0.990525 0.495262 0.868744i $$-0.335072\pi$$
0.495262 + 0.868744i $$0.335072\pi$$
$$212$$ 0 0
$$213$$ 401882. 0.606945
$$214$$ 0 0
$$215$$ 4450.62 0.00656636
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −12116.4 −0.0170712
$$220$$ 0 0
$$221$$ −1.29182e6 −1.77918
$$222$$ 0 0
$$223$$ 390135. 0.525354 0.262677 0.964884i $$-0.415395\pi$$
0.262677 + 0.964884i $$0.415395\pi$$
$$224$$ 0 0
$$225$$ 202089. 0.266126
$$226$$ 0 0
$$227$$ 291353. 0.375279 0.187639 0.982238i $$-0.439916\pi$$
0.187639 + 0.982238i $$0.439916\pi$$
$$228$$ 0 0
$$229$$ 1.23040e6 1.55045 0.775227 0.631682i $$-0.217635\pi$$
0.775227 + 0.631682i $$0.217635\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 114279. 0.137903 0.0689517 0.997620i $$-0.478035\pi$$
0.0689517 + 0.997620i $$0.478035\pi$$
$$234$$ 0 0
$$235$$ 353172. 0.417174
$$236$$ 0 0
$$237$$ 55.8489 6.45867e−5 0
$$238$$ 0 0
$$239$$ 1.14782e6 1.29981 0.649906 0.760014i $$-0.274808\pi$$
0.649906 + 0.760014i $$0.274808\pi$$
$$240$$ 0 0
$$241$$ −812708. −0.901346 −0.450673 0.892689i $$-0.648816\pi$$
−0.450673 + 0.892689i $$0.648816\pi$$
$$242$$ 0 0
$$243$$ 899104. 0.976775
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.27586e6 1.33064
$$248$$ 0 0
$$249$$ −622318. −0.636084
$$250$$ 0 0
$$251$$ −406772. −0.407537 −0.203768 0.979019i $$-0.565319\pi$$
−0.203768 + 0.979019i $$0.565319\pi$$
$$252$$ 0 0
$$253$$ −24490.8 −0.0240548
$$254$$ 0 0
$$255$$ 355572. 0.342434
$$256$$ 0 0
$$257$$ 1.69712e6 1.60281 0.801403 0.598125i $$-0.204087\pi$$
0.801403 + 0.598125i $$0.204087\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −666215. −0.605359
$$262$$ 0 0
$$263$$ −205694. −0.183372 −0.0916859 0.995788i $$-0.529226\pi$$
−0.0916859 + 0.995788i $$0.529226\pi$$
$$264$$ 0 0
$$265$$ −549746. −0.480892
$$266$$ 0 0
$$267$$ −351010. −0.301330
$$268$$ 0 0
$$269$$ −1.73425e6 −1.46127 −0.730635 0.682769i $$-0.760776\pi$$
−0.730635 + 0.682769i $$0.760776\pi$$
$$270$$ 0 0
$$271$$ 369042. 0.305248 0.152624 0.988284i $$-0.451228\pi$$
0.152624 + 0.988284i $$0.451228\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −557201. −0.444304
$$276$$ 0 0
$$277$$ 1.22767e6 0.961351 0.480676 0.876899i $$-0.340391\pi$$
0.480676 + 0.876899i $$0.340391\pi$$
$$278$$ 0 0
$$279$$ −1.75789e6 −1.35201
$$280$$ 0 0
$$281$$ 2.00671e6 1.51607 0.758035 0.652214i $$-0.226159\pi$$
0.758035 + 0.652214i $$0.226159\pi$$
$$282$$ 0 0
$$283$$ −1.78581e6 −1.32547 −0.662734 0.748855i $$-0.730604\pi$$
−0.662734 + 0.748855i $$0.730604\pi$$
$$284$$ 0 0
$$285$$ −351179. −0.256104
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −25770.8 −0.0181503
$$290$$ 0 0
$$291$$ −20369.5 −0.0141009
$$292$$ 0 0
$$293$$ −853248. −0.580639 −0.290319 0.956930i $$-0.593762\pi$$
−0.290319 + 0.956930i $$0.593762\pi$$
$$294$$ 0 0
$$295$$ 453345. 0.303301
$$296$$ 0 0
$$297$$ −1.60104e6 −1.05320
$$298$$ 0 0
$$299$$ 48561.6 0.0314134
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −142982. −0.0894693
$$304$$ 0 0
$$305$$ 1.77148e6 1.09040
$$306$$ 0 0
$$307$$ −1.96068e6 −1.18730 −0.593652 0.804722i $$-0.702314\pi$$
−0.593652 + 0.804722i $$0.702314\pi$$
$$308$$ 0 0
$$309$$ 427857. 0.254919
$$310$$ 0 0
$$311$$ 863604. 0.506307 0.253153 0.967426i $$-0.418532\pi$$
0.253153 + 0.967426i $$0.418532\pi$$
$$312$$ 0 0
$$313$$ −1.10047e6 −0.634918 −0.317459 0.948272i $$-0.602830\pi$$
−0.317459 + 0.948272i $$0.602830\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.49591e6 0.836097 0.418048 0.908425i $$-0.362714\pi$$
0.418048 + 0.908425i $$0.362714\pi$$
$$318$$ 0 0
$$319$$ 1.83689e6 1.01066
$$320$$ 0 0
$$321$$ 713454. 0.386459
$$322$$ 0 0
$$323$$ −1.37686e6 −0.734318
$$324$$ 0 0
$$325$$ 1.10485e6 0.580221
$$326$$ 0 0
$$327$$ 567901. 0.293699
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 2.74015e6 1.37469 0.687345 0.726331i $$-0.258776\pi$$
0.687345 + 0.726331i $$0.258776\pi$$
$$332$$ 0 0
$$333$$ 511741. 0.252894
$$334$$ 0 0
$$335$$ 3.10665e6 1.51245
$$336$$ 0 0
$$337$$ −2.31353e6 −1.10968 −0.554842 0.831956i $$-0.687221\pi$$
−0.554842 + 0.831956i $$0.687221\pi$$
$$338$$ 0 0
$$339$$ −664600. −0.314095
$$340$$ 0 0
$$341$$ 4.84685e6 2.25722
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −13366.6 −0.00604606
$$346$$ 0 0
$$347$$ 3.05926e6 1.36393 0.681966 0.731384i $$-0.261125\pi$$
0.681966 + 0.731384i $$0.261125\pi$$
$$348$$ 0 0
$$349$$ 210232. 0.0923921 0.0461961 0.998932i $$-0.485290\pi$$
0.0461961 + 0.998932i $$0.485290\pi$$
$$350$$ 0 0
$$351$$ 3.17461e6 1.37538
$$352$$ 0 0
$$353$$ 3.76790e6 1.60939 0.804697 0.593686i $$-0.202328\pi$$
0.804697 + 0.593686i $$0.202328\pi$$
$$354$$ 0 0
$$355$$ 2.82268e6 1.18875
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1.00722e6 −0.412465 −0.206232 0.978503i $$-0.566120\pi$$
−0.206232 + 0.978503i $$0.566120\pi$$
$$360$$ 0 0
$$361$$ −1.11625e6 −0.450809
$$362$$ 0 0
$$363$$ 939058. 0.374047
$$364$$ 0 0
$$365$$ −85101.8 −0.0334354
$$366$$ 0 0
$$367$$ −1.52650e6 −0.591603 −0.295802 0.955249i $$-0.595587\pi$$
−0.295802 + 0.955249i $$0.595587\pi$$
$$368$$ 0 0
$$369$$ −2.55383e6 −0.976396
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 4.86297e6 1.80980 0.904898 0.425629i $$-0.139947\pi$$
0.904898 + 0.425629i $$0.139947\pi$$
$$374$$ 0 0
$$375$$ −1.24520e6 −0.457258
$$376$$ 0 0
$$377$$ −3.64227e6 −1.31983
$$378$$ 0 0
$$379$$ −630878. −0.225604 −0.112802 0.993617i $$-0.535983\pi$$
−0.112802 + 0.993617i $$0.535983\pi$$
$$380$$ 0 0
$$381$$ 20263.5 0.00715159
$$382$$ 0 0
$$383$$ 565644. 0.197036 0.0985182 0.995135i $$-0.468590\pi$$
0.0985182 + 0.995135i $$0.468590\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −19366.2 −0.00657306
$$388$$ 0 0
$$389$$ 592212. 0.198428 0.0992140 0.995066i $$-0.468367\pi$$
0.0992140 + 0.995066i $$0.468367\pi$$
$$390$$ 0 0
$$391$$ −52406.1 −0.0173356
$$392$$ 0 0
$$393$$ 1.65947e6 0.541986
$$394$$ 0 0
$$395$$ 392.264 0.000126498 0
$$396$$ 0 0
$$397$$ −1.34312e6 −0.427698 −0.213849 0.976867i $$-0.568600\pi$$
−0.213849 + 0.976867i $$0.568600\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −3.68716e6 −1.14507 −0.572534 0.819881i $$-0.694040\pi$$
−0.572534 + 0.819881i $$0.694040\pi$$
$$402$$ 0 0
$$403$$ −9.61057e6 −2.94772
$$404$$ 0 0
$$405$$ 1.36273e6 0.412832
$$406$$ 0 0
$$407$$ −1.41097e6 −0.422214
$$408$$ 0 0
$$409$$ −1.45630e6 −0.430470 −0.215235 0.976562i $$-0.569052\pi$$
−0.215235 + 0.976562i $$0.569052\pi$$
$$410$$ 0 0
$$411$$ −636159. −0.185764
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −4.37096e6 −1.24582
$$416$$ 0 0
$$417$$ 1.37710e6 0.387816
$$418$$ 0 0
$$419$$ −2.92192e6 −0.813080 −0.406540 0.913633i $$-0.633265\pi$$
−0.406540 + 0.913633i $$0.633265\pi$$
$$420$$ 0 0
$$421$$ 2.01999e6 0.555450 0.277725 0.960661i $$-0.410420\pi$$
0.277725 + 0.960661i $$0.410420\pi$$
$$422$$ 0 0
$$423$$ −1.53678e6 −0.417600
$$424$$ 0 0
$$425$$ −1.19231e6 −0.320197
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −3.95305e6 −1.03703
$$430$$ 0 0
$$431$$ 5.43800e6 1.41009 0.705043 0.709164i $$-0.250928\pi$$
0.705043 + 0.709164i $$0.250928\pi$$
$$432$$ 0 0
$$433$$ 3.77335e6 0.967179 0.483590 0.875295i $$-0.339333\pi$$
0.483590 + 0.875295i $$0.339333\pi$$
$$434$$ 0 0
$$435$$ 1.00253e6 0.254025
$$436$$ 0 0
$$437$$ 51758.6 0.0129652
$$438$$ 0 0
$$439$$ 2.35150e6 0.582350 0.291175 0.956670i $$-0.405954\pi$$
0.291175 + 0.956670i $$0.405954\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −4.80377e6 −1.16298 −0.581491 0.813553i $$-0.697530\pi$$
−0.581491 + 0.813553i $$0.697530\pi$$
$$444$$ 0 0
$$445$$ −2.46538e6 −0.590179
$$446$$ 0 0
$$447$$ 919383. 0.217634
$$448$$ 0 0
$$449$$ −2.76805e6 −0.647975 −0.323987 0.946061i $$-0.605024\pi$$
−0.323987 + 0.946061i $$0.605024\pi$$
$$450$$ 0 0
$$451$$ 7.04142e6 1.63012
$$452$$ 0 0
$$453$$ −783770. −0.179450
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 241566. 0.0541061 0.0270530 0.999634i $$-0.491388\pi$$
0.0270530 + 0.999634i $$0.491388\pi$$
$$458$$ 0 0
$$459$$ −3.42594e6 −0.759010
$$460$$ 0 0
$$461$$ −990579. −0.217088 −0.108544 0.994092i $$-0.534619\pi$$
−0.108544 + 0.994092i $$0.534619\pi$$
$$462$$ 0 0
$$463$$ −6.20488e6 −1.34518 −0.672591 0.740014i $$-0.734819\pi$$
−0.672591 + 0.740014i $$0.734819\pi$$
$$464$$ 0 0
$$465$$ 2.64531e6 0.567340
$$466$$ 0 0
$$467$$ −6.88497e6 −1.46086 −0.730432 0.682985i $$-0.760681\pi$$
−0.730432 + 0.682985i $$0.760681\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −639197. −0.132765
$$472$$ 0 0
$$473$$ 53396.6 0.0109739
$$474$$ 0 0
$$475$$ 1.17758e6 0.239473
$$476$$ 0 0
$$477$$ 2.39214e6 0.481383
$$478$$ 0 0
$$479$$ −5.41288e6 −1.07793 −0.538963 0.842329i $$-0.681184\pi$$
−0.538963 + 0.842329i $$0.681184\pi$$
$$480$$ 0 0
$$481$$ 2.79774e6 0.551373
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −143069. −0.0276179
$$486$$ 0 0
$$487$$ 3.01043e6 0.575182 0.287591 0.957753i $$-0.407146\pi$$
0.287591 + 0.957753i $$0.407146\pi$$
$$488$$ 0 0
$$489$$ −1.19618e6 −0.226216
$$490$$ 0 0
$$491$$ 7.24498e6 1.35623 0.678115 0.734956i $$-0.262797\pi$$
0.678115 + 0.734956i $$0.262797\pi$$
$$492$$ 0 0
$$493$$ 3.93062e6 0.728356
$$494$$ 0 0
$$495$$ −5.07853e6 −0.931591
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 5.12788e6 0.921906 0.460953 0.887425i $$-0.347508\pi$$
0.460953 + 0.887425i $$0.347508\pi$$
$$500$$ 0 0
$$501$$ −2.95757e6 −0.526429
$$502$$ 0 0
$$503$$ 1.05978e7 1.86766 0.933830 0.357718i $$-0.116445\pi$$
0.933830 + 0.357718i $$0.116445\pi$$
$$504$$ 0 0
$$505$$ −1.00426e6 −0.175233
$$506$$ 0 0
$$507$$ 5.40707e6 0.934205
$$508$$ 0 0
$$509$$ −8.78840e6 −1.50354 −0.751770 0.659425i $$-0.770800\pi$$
−0.751770 + 0.659425i $$0.770800\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 3.38361e6 0.567658
$$514$$ 0 0
$$515$$ 3.00512e6 0.499280
$$516$$ 0 0
$$517$$ 4.23721e6 0.697194
$$518$$ 0 0
$$519$$ −2.43355e6 −0.396572
$$520$$ 0 0
$$521$$ −162133. −0.0261684 −0.0130842 0.999914i $$-0.504165\pi$$
−0.0130842 + 0.999914i $$0.504165\pi$$
$$522$$ 0 0
$$523$$ −7.14844e6 −1.14277 −0.571383 0.820684i $$-0.693593\pi$$
−0.571383 + 0.820684i $$0.693593\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1.03714e7 1.62671
$$528$$ 0 0
$$529$$ −6.43437e6 −0.999694
$$530$$ 0 0
$$531$$ −1.97267e6 −0.303611
$$532$$ 0 0
$$533$$ −1.39621e7 −2.12879
$$534$$ 0 0
$$535$$ 5.01106e6 0.756912
$$536$$ 0 0
$$537$$ −556604. −0.0832934
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −4.83604e6 −0.710390 −0.355195 0.934792i $$-0.615586\pi$$
−0.355195 + 0.934792i $$0.615586\pi$$
$$542$$ 0 0
$$543$$ 2.48459e6 0.361622
$$544$$ 0 0
$$545$$ 3.98874e6 0.575234
$$546$$ 0 0
$$547$$ −9.98777e6 −1.42725 −0.713626 0.700527i $$-0.752948\pi$$
−0.713626 + 0.700527i $$0.752948\pi$$
$$548$$ 0 0
$$549$$ −7.70833e6 −1.09151
$$550$$ 0 0
$$551$$ −3.88205e6 −0.544732
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −770078. −0.106121
$$556$$ 0 0
$$557$$ 1.74619e6 0.238481 0.119241 0.992865i $$-0.461954\pi$$
0.119241 + 0.992865i $$0.461954\pi$$
$$558$$ 0 0
$$559$$ −105877. −0.0143309
$$560$$ 0 0
$$561$$ 4.26600e6 0.572287
$$562$$ 0 0
$$563$$ −755218. −0.100416 −0.0502078 0.998739i $$-0.515988\pi$$
−0.0502078 + 0.998739i $$0.515988\pi$$
$$564$$ 0 0
$$565$$ −4.66792e6 −0.615181
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −4.39534e6 −0.569131 −0.284565 0.958657i $$-0.591849\pi$$
−0.284565 + 0.958657i $$0.591849\pi$$
$$570$$ 0 0
$$571$$ −1.16104e7 −1.49024 −0.745121 0.666930i $$-0.767608\pi$$
−0.745121 + 0.666930i $$0.767608\pi$$
$$572$$ 0 0
$$573$$ 6.03855e6 0.768327
$$574$$ 0 0
$$575$$ 44821.1 0.00565344
$$576$$ 0 0
$$577$$ −1.06643e7 −1.33350 −0.666748 0.745283i $$-0.732314\pi$$
−0.666748 + 0.745283i $$0.732314\pi$$
$$578$$ 0 0
$$579$$ 3.30745e6 0.410012
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −6.59562e6 −0.803682
$$584$$ 0 0
$$585$$ 1.00700e7 1.21657
$$586$$ 0 0
$$587$$ −1.39482e7 −1.67079 −0.835396 0.549648i $$-0.814762\pi$$
−0.835396 + 0.549648i $$0.814762\pi$$
$$588$$ 0 0
$$589$$ −1.02433e7 −1.21661
$$590$$ 0 0
$$591$$ 1.64686e6 0.193949
$$592$$ 0 0
$$593$$ −1.17933e7 −1.37720 −0.688600 0.725142i $$-0.741774\pi$$
−0.688600 + 0.725142i $$0.741774\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −1.36220e6 −0.156425
$$598$$ 0 0
$$599$$ 4.38057e6 0.498843 0.249421 0.968395i $$-0.419760\pi$$
0.249421 + 0.968395i $$0.419760\pi$$
$$600$$ 0 0
$$601$$ −688570. −0.0777610 −0.0388805 0.999244i $$-0.512379\pi$$
−0.0388805 + 0.999244i $$0.512379\pi$$
$$602$$ 0 0
$$603$$ −1.35181e7 −1.51399
$$604$$ 0 0
$$605$$ 6.59563e6 0.732601
$$606$$ 0 0
$$607$$ −9.37319e6 −1.03256 −0.516281 0.856420i $$-0.672684\pi$$
−0.516281 + 0.856420i $$0.672684\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −8.40174e6 −0.910472
$$612$$ 0 0
$$613$$ −2.16685e6 −0.232904 −0.116452 0.993196i $$-0.537152\pi$$
−0.116452 + 0.993196i $$0.537152\pi$$
$$614$$ 0 0
$$615$$ 3.84306e6 0.409722
$$616$$ 0 0
$$617$$ −5.07951e6 −0.537166 −0.268583 0.963256i $$-0.586555\pi$$
−0.268583 + 0.963256i $$0.586555\pi$$
$$618$$ 0 0
$$619$$ −2.19034e6 −0.229766 −0.114883 0.993379i $$-0.536649\pi$$
−0.114883 + 0.993379i $$0.536649\pi$$
$$620$$ 0 0
$$621$$ 128787. 0.0134012
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −5.59018e6 −0.572435
$$626$$ 0 0
$$627$$ −4.21329e6 −0.428009
$$628$$ 0 0
$$629$$ −3.01923e6 −0.304278
$$630$$ 0 0
$$631$$ 7.18693e6 0.718572 0.359286 0.933228i $$-0.383020\pi$$
0.359286 + 0.933228i $$0.383020\pi$$
$$632$$ 0 0
$$633$$ 4.19451e6 0.416075
$$634$$ 0 0
$$635$$ 142324. 0.0140070
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −1.22825e7 −1.18997
$$640$$ 0 0
$$641$$ −1.76500e7 −1.69668 −0.848340 0.529452i $$-0.822398\pi$$
−0.848340 + 0.529452i $$0.822398\pi$$
$$642$$ 0 0
$$643$$ 898309. 0.0856837 0.0428419 0.999082i $$-0.486359\pi$$
0.0428419 + 0.999082i $$0.486359\pi$$
$$644$$ 0 0
$$645$$ 29142.7 0.00275823
$$646$$ 0 0
$$647$$ −1.38642e6 −0.130207 −0.0651035 0.997879i $$-0.520738\pi$$
−0.0651035 + 0.997879i $$0.520738\pi$$
$$648$$ 0 0
$$649$$ 5.43904e6 0.506886
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1.75425e7 −1.60994 −0.804968 0.593318i $$-0.797818\pi$$
−0.804968 + 0.593318i $$0.797818\pi$$
$$654$$ 0 0
$$655$$ 1.16556e7 1.06152
$$656$$ 0 0
$$657$$ 370308. 0.0334696
$$658$$ 0 0
$$659$$ −9.87522e6 −0.885795 −0.442898 0.896572i $$-0.646050\pi$$
−0.442898 + 0.896572i $$0.646050\pi$$
$$660$$ 0 0
$$661$$ 8.06792e6 0.718221 0.359110 0.933295i $$-0.383080\pi$$
0.359110 + 0.933295i $$0.383080\pi$$
$$662$$ 0 0
$$663$$ −8.45884e6 −0.747355
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −147759. −0.0128599
$$668$$ 0 0
$$669$$ 2.55461e6 0.220678
$$670$$ 0 0
$$671$$ 2.12534e7 1.82231
$$672$$ 0 0
$$673$$ −1.12772e7 −0.959762 −0.479881 0.877334i $$-0.659320\pi$$
−0.479881 + 0.877334i $$0.659320\pi$$
$$674$$ 0 0
$$675$$ 2.93008e6 0.247526
$$676$$ 0 0
$$677$$ −5.20372e6 −0.436357 −0.218179 0.975909i $$-0.570012\pi$$
−0.218179 + 0.975909i $$0.570012\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 1.90778e6 0.157638
$$682$$ 0 0
$$683$$ −6.05915e6 −0.497004 −0.248502 0.968631i $$-0.579938\pi$$
−0.248502 + 0.968631i $$0.579938\pi$$
$$684$$ 0 0
$$685$$ −4.46816e6 −0.363833
$$686$$ 0 0
$$687$$ 8.05671e6 0.651277
$$688$$ 0 0
$$689$$ 1.30781e7 1.04954
$$690$$ 0 0
$$691$$ 7.36498e6 0.586781 0.293391 0.955993i $$-0.405216\pi$$
0.293391 + 0.955993i $$0.405216\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 9.67228e6 0.759569
$$696$$ 0 0
$$697$$ 1.50674e7 1.17478
$$698$$ 0 0
$$699$$ 748298. 0.0579271
$$700$$ 0 0
$$701$$ 7.80919e6 0.600221 0.300110 0.953904i $$-0.402977\pi$$
0.300110 + 0.953904i $$0.402977\pi$$
$$702$$ 0 0
$$703$$ 2.98193e6 0.227567
$$704$$ 0 0
$$705$$ 2.31258e6 0.175236
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1.75650e7 1.31230 0.656150 0.754631i $$-0.272184\pi$$
0.656150 + 0.754631i $$0.272184\pi$$
$$710$$ 0 0
$$711$$ −1706.88 −0.000126628 0
$$712$$ 0 0
$$713$$ −389879. −0.0287214
$$714$$ 0 0
$$715$$ −2.77649e7 −2.03110
$$716$$ 0 0
$$717$$ 7.51597e6 0.545993
$$718$$ 0 0
$$719$$ −8.09220e6 −0.583773 −0.291887 0.956453i $$-0.594283\pi$$
−0.291887 + 0.956453i $$0.594283\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −5.32162e6 −0.378615
$$724$$ 0 0
$$725$$ −3.36172e6 −0.237529
$$726$$ 0 0
$$727$$ 1.51986e7 1.06652 0.533258 0.845952i $$-0.320967\pi$$
0.533258 + 0.845952i $$0.320967\pi$$
$$728$$ 0 0
$$729$$ −1.31285e6 −0.0914944
$$730$$ 0 0
$$731$$ 114259. 0.00790858
$$732$$ 0 0
$$733$$ −5.83402e6 −0.401059 −0.200530 0.979688i $$-0.564266\pi$$
−0.200530 + 0.979688i $$0.564266\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 3.72722e7 2.52765
$$738$$ 0 0
$$739$$ −6.47719e6 −0.436290 −0.218145 0.975916i $$-0.570001\pi$$
−0.218145 + 0.975916i $$0.570001\pi$$
$$740$$ 0 0
$$741$$ 8.35433e6 0.558941
$$742$$ 0 0
$$743$$ 1.50899e7 1.00280 0.501401 0.865215i $$-0.332818\pi$$
0.501401 + 0.865215i $$0.332818\pi$$
$$744$$ 0 0
$$745$$ 6.45744e6 0.426255
$$746$$ 0 0
$$747$$ 1.90196e7 1.24710
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 2.13997e6 0.138455 0.0692273 0.997601i $$-0.477947\pi$$
0.0692273 + 0.997601i $$0.477947\pi$$
$$752$$ 0 0
$$753$$ −2.66355e6 −0.171188
$$754$$ 0 0
$$755$$ −5.50494e6 −0.351467
$$756$$ 0 0
$$757$$ 2.10943e7 1.33791 0.668954 0.743304i $$-0.266742\pi$$
0.668954 + 0.743304i $$0.266742\pi$$
$$758$$ 0 0
$$759$$ −160366. −0.0101044
$$760$$ 0 0
$$761$$ −9.79958e6 −0.613403 −0.306701 0.951806i $$-0.599225\pi$$
−0.306701 + 0.951806i $$0.599225\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −1.08672e7 −0.671371
$$766$$ 0 0
$$767$$ −1.07848e7 −0.661947
$$768$$ 0 0
$$769$$ 3.23493e7 1.97265 0.986323 0.164825i $$-0.0527060\pi$$
0.986323 + 0.164825i $$0.0527060\pi$$
$$770$$ 0 0
$$771$$ 1.11128e7 0.673267
$$772$$ 0 0
$$773$$ −9.19713e6 −0.553609 −0.276805 0.960926i $$-0.589276\pi$$
−0.276805 + 0.960926i $$0.589276\pi$$
$$774$$ 0 0
$$775$$ −8.87030e6 −0.530499
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −1.48812e7 −0.878609
$$780$$ 0 0
$$781$$ 3.38653e7 1.98668
$$782$$ 0 0
$$783$$ −9.65941e6 −0.563049
$$784$$ 0 0
$$785$$ −4.48950e6 −0.260030
$$786$$ 0 0
$$787$$ 1.46950e7 0.845731 0.422866 0.906192i $$-0.361024\pi$$
0.422866 + 0.906192i $$0.361024\pi$$
$$788$$ 0 0
$$789$$ −1.34689e6 −0.0770263
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −4.21423e7 −2.37977
$$794$$ 0 0
$$795$$ −3.59975e6 −0.202001
$$796$$ 0 0
$$797$$ −2.33344e7 −1.30122 −0.650610 0.759412i $$-0.725487\pi$$
−0.650610 + 0.759412i $$0.725487\pi$$
$$798$$ 0 0
$$799$$ 9.06688e6 0.502448
$$800$$ 0 0
$$801$$ 1.07277e7 0.590782
$$802$$ 0 0
$$803$$ −1.02101e6 −0.0558783
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −1.13559e7 −0.613814
$$808$$ 0 0
$$809$$ 1.69301e6 0.0909468 0.0454734 0.998966i $$-0.485520\pi$$
0.0454734 + 0.998966i $$0.485520\pi$$
$$810$$ 0 0
$$811$$ −2.12400e7 −1.13397 −0.566987 0.823727i $$-0.691891\pi$$
−0.566987 + 0.823727i $$0.691891\pi$$
$$812$$ 0 0
$$813$$ 2.41649e6 0.128221
$$814$$ 0 0
$$815$$ −8.40155e6 −0.443063
$$816$$ 0 0
$$817$$ −112848. −0.00591477
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 8.73550e6 0.452304 0.226152 0.974092i $$-0.427385\pi$$
0.226152 + 0.974092i $$0.427385\pi$$
$$822$$ 0 0
$$823$$ 3.27964e7 1.68782 0.843910 0.536485i $$-0.180248\pi$$
0.843910 + 0.536485i $$0.180248\pi$$
$$824$$ 0 0
$$825$$ −3.64856e6 −0.186632
$$826$$ 0 0
$$827$$ −1.31248e7 −0.667311 −0.333656 0.942695i $$-0.608282\pi$$
−0.333656 + 0.942695i $$0.608282\pi$$
$$828$$ 0 0
$$829$$ −2.05402e7 −1.03805 −0.519026 0.854759i $$-0.673705\pi$$
−0.519026 + 0.854759i $$0.673705\pi$$
$$830$$ 0 0
$$831$$ 8.03880e6 0.403821
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −2.07729e7 −1.03106
$$836$$ 0 0
$$837$$ −2.54875e7 −1.25752
$$838$$ 0 0
$$839$$ 2.83736e7 1.39159 0.695793 0.718243i $$-0.255053\pi$$
0.695793 + 0.718243i $$0.255053\pi$$
$$840$$ 0 0
$$841$$ −9.42879e6 −0.459691
$$842$$ 0 0
$$843$$ 1.31400e7 0.636834
$$844$$ 0 0
$$845$$ 3.79774e7 1.82972
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −1.16935e7 −0.556770
$$850$$ 0 0
$$851$$ 113498. 0.00537236
$$852$$ 0 0
$$853$$ −1.02093e7 −0.480424 −0.240212 0.970720i $$-0.577217\pi$$
−0.240212 + 0.970720i $$0.577217\pi$$
$$854$$ 0 0
$$855$$ 1.07329e7 0.502113
$$856$$ 0 0
$$857$$ 8.33206e6 0.387525 0.193763 0.981048i $$-0.437931\pi$$
0.193763 + 0.981048i $$0.437931\pi$$
$$858$$ 0 0
$$859$$ 3.12766e7 1.44623 0.723113 0.690729i $$-0.242710\pi$$
0.723113 + 0.690729i $$0.242710\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 3.73573e7 1.70745 0.853726 0.520722i $$-0.174337\pi$$
0.853726 + 0.520722i $$0.174337\pi$$
$$864$$ 0 0
$$865$$ −1.70924e7 −0.776719
$$866$$ 0 0
$$867$$ −168747. −0.00762411
$$868$$ 0 0
$$869$$ 4706.21 0.000211408 0
$$870$$ 0 0
$$871$$ −7.39051e7 −3.30088
$$872$$ 0 0
$$873$$ 622543. 0.0276461
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 38996.7 0.00171210 0.000856049 1.00000i $$-0.499728\pi$$
0.000856049 1.00000i $$0.499728\pi$$
$$878$$ 0 0
$$879$$ −5.58708e6 −0.243900
$$880$$ 0 0
$$881$$ −3.15554e7 −1.36973 −0.684864 0.728671i $$-0.740138\pi$$
−0.684864 + 0.728671i $$0.740138\pi$$
$$882$$ 0 0
$$883$$ 3.42253e7 1.47722 0.738611 0.674132i $$-0.235482\pi$$
0.738611 + 0.674132i $$0.235482\pi$$
$$884$$ 0 0
$$885$$ 2.96851e6 0.127403
$$886$$ 0 0
$$887$$ −2.69886e7 −1.15178 −0.575892 0.817526i $$-0.695345\pi$$
−0.575892 + 0.817526i $$0.695345\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 1.63495e7 0.689938
$$892$$ 0 0
$$893$$ −8.95486e6 −0.375777
$$894$$ 0 0
$$895$$ −3.90940e6 −0.163137
$$896$$ 0 0
$$897$$ 317982. 0.0131954
$$898$$ 0 0
$$899$$ 2.92421e7 1.20673
$$900$$ 0 0
$$901$$ −1.41135e7 −0.579191
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 1.74509e7 0.708267
$$906$$ 0 0
$$907$$ −1.92103e7 −0.775381 −0.387690 0.921790i $$-0.626727\pi$$
−0.387690 + 0.921790i $$0.626727\pi$$
$$908$$ 0 0
$$909$$ 4.36988e6 0.175412
$$910$$ 0 0
$$911$$ −2.86013e7 −1.14180 −0.570899 0.821020i $$-0.693405\pi$$
−0.570899 + 0.821020i $$0.693405\pi$$
$$912$$ 0 0
$$913$$ −5.24408e7 −2.08206
$$914$$ 0 0
$$915$$ 1.15997e7 0.458029
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 4.21754e7 1.64729 0.823645 0.567106i $$-0.191937\pi$$
0.823645 + 0.567106i $$0.191937\pi$$
$$920$$ 0 0
$$921$$ −1.28386e7 −0.498733
$$922$$ 0 0
$$923$$ −6.71498e7 −2.59442
$$924$$ 0 0
$$925$$ 2.58224e6 0.0992299
$$926$$ 0 0
$$927$$ −1.30764e7 −0.499790
$$928$$ 0 0
$$929$$ 3.01886e7 1.14763 0.573817 0.818983i $$-0.305462\pi$$
0.573817 + 0.818983i $$0.305462\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 5.65489e6 0.212677
$$934$$ 0 0
$$935$$ 2.99630e7 1.12087
$$936$$ 0 0
$$937$$ 3.64068e6 0.135467 0.0677335 0.997703i $$-0.478423\pi$$
0.0677335 + 0.997703i $$0.478423\pi$$
$$938$$ 0 0
$$939$$ −7.20589e6 −0.266701
$$940$$ 0 0
$$941$$ −1.88601e7 −0.694336 −0.347168 0.937803i $$-0.612857\pi$$
−0.347168 + 0.937803i $$0.612857\pi$$
$$942$$ 0 0
$$943$$ −566410. −0.0207420
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1.82172e7 0.660094 0.330047 0.943965i $$-0.392935\pi$$
0.330047 + 0.943965i $$0.392935\pi$$
$$948$$ 0 0
$$949$$ 2.02452e6 0.0729720
$$950$$ 0 0
$$951$$ 9.79522e6 0.351207
$$952$$ 0 0
$$953$$ 2.76898e7 0.987616 0.493808 0.869571i $$-0.335604\pi$$
0.493808 + 0.869571i $$0.335604\pi$$
$$954$$ 0 0
$$955$$ 4.24127e7 1.50483
$$956$$ 0 0
$$957$$ 1.20280e7 0.424534
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 4.85298e7 1.69512
$$962$$ 0 0
$$963$$ −2.18049e7 −0.757685
$$964$$ 0 0
$$965$$ 2.32304e7 0.803042
$$966$$ 0 0
$$967$$ 2.44768e7 0.841761 0.420881 0.907116i $$-0.361721\pi$$
0.420881 + 0.907116i $$0.361721\pi$$
$$968$$ 0 0
$$969$$ −9.01571e6 −0.308454
$$970$$ 0 0
$$971$$ 9.50151e6 0.323403 0.161702 0.986840i $$-0.448302\pi$$
0.161702 + 0.986840i $$0.448302\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 7.23455e6 0.243725
$$976$$ 0 0
$$977$$ 4.69012e7 1.57198 0.785991 0.618238i $$-0.212153\pi$$
0.785991 + 0.618238i $$0.212153\pi$$
$$978$$ 0 0
$$979$$ −2.95785e7 −0.986325
$$980$$ 0 0
$$981$$ −1.73564e7 −0.575822
$$982$$ 0 0
$$983$$ 2.35382e7 0.776945 0.388473 0.921460i $$-0.373003\pi$$
0.388473 + 0.921460i $$0.373003\pi$$
$$984$$ 0 0
$$985$$ 1.15670e7 0.379865
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −4295.21 −0.000139635 0
$$990$$ 0 0
$$991$$ 2.64104e7 0.854261 0.427130 0.904190i $$-0.359525\pi$$
0.427130 + 0.904190i $$0.359525\pi$$
$$992$$ 0 0
$$993$$ 1.79426e7 0.577446
$$994$$ 0 0
$$995$$ −9.56766e6 −0.306371
$$996$$ 0 0
$$997$$ 1.95164e7 0.621815 0.310907 0.950440i $$-0.399367\pi$$
0.310907 + 0.950440i $$0.399367\pi$$
$$998$$ 0 0
$$999$$ 7.41970e6 0.235219
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 784.6.a.bf.1.3 4
4.3 odd 2 49.6.a.g.1.3 4
7.6 odd 2 inner 784.6.a.bf.1.2 4
12.11 even 2 441.6.a.z.1.1 4
28.3 even 6 49.6.c.h.30.1 8
28.11 odd 6 49.6.c.h.30.2 8
28.19 even 6 49.6.c.h.18.1 8
28.23 odd 6 49.6.c.h.18.2 8
28.27 even 2 49.6.a.g.1.4 yes 4
84.83 odd 2 441.6.a.z.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.3 4 4.3 odd 2
49.6.a.g.1.4 yes 4 28.27 even 2
49.6.c.h.18.1 8 28.19 even 6
49.6.c.h.18.2 8 28.23 odd 6
49.6.c.h.30.1 8 28.3 even 6
49.6.c.h.30.2 8 28.11 odd 6
441.6.a.z.1.1 4 12.11 even 2
441.6.a.z.1.2 4 84.83 odd 2
784.6.a.bf.1.2 4 7.6 odd 2 inner
784.6.a.bf.1.3 4 1.1 even 1 trivial