# Properties

 Label 2352.4.a.bn.1.1 Level $2352$ Weight $4$ Character 2352.1 Self dual yes Analytic conductor $138.772$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$138.772492334$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ Twist minimal: no (minimal twist has level 294) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -15.8995 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -15.8995 q^{5} +9.00000 q^{9} -57.3970 q^{11} +5.69848 q^{13} +47.6985 q^{15} -51.8995 q^{17} -16.2010 q^{19} +213.397 q^{23} +127.794 q^{25} -27.0000 q^{27} -218.191 q^{29} +251.397 q^{31} +172.191 q^{33} +386.794 q^{37} -17.0955 q^{39} -328.503 q^{41} +37.5879 q^{43} -143.095 q^{45} +254.995 q^{47} +155.698 q^{51} +211.588 q^{53} +912.583 q^{55} +48.6030 q^{57} +412.201 q^{59} -836.693 q^{61} -90.6030 q^{65} +165.588 q^{67} -640.191 q^{69} +465.015 q^{71} +449.658 q^{73} -383.382 q^{75} +343.558 q^{79} +81.0000 q^{81} -1502.33 q^{83} +825.176 q^{85} +654.573 q^{87} -341.085 q^{89} -754.191 q^{93} +257.588 q^{95} -865.437 q^{97} -516.573 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} - 12q^{5} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} - 12q^{5} + 18q^{9} + 4q^{11} - 48q^{13} + 36q^{15} - 84q^{17} - 72q^{19} + 308q^{23} + 18q^{25} - 54q^{27} - 80q^{29} + 384q^{31} - 12q^{33} + 536q^{37} + 144q^{39} - 756q^{41} - 400q^{43} - 108q^{45} + 312q^{47} + 252q^{51} - 52q^{53} + 1152q^{55} + 216q^{57} + 864q^{59} - 1416q^{61} - 300q^{65} - 144q^{67} - 924q^{69} + 1524q^{71} - 744q^{73} - 54q^{75} - 976q^{79} + 162q^{81} - 312q^{83} + 700q^{85} + 240q^{87} - 108q^{89} - 1152q^{93} + 40q^{95} + 744q^{97} + 36q^{99} + O(q^{100})$$

## 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
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ −15.8995 −1.42209 −0.711047 0.703144i $$-0.751779\pi$$
−0.711047 + 0.703144i $$0.751779\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −57.3970 −1.57326 −0.786629 0.617426i $$-0.788176\pi$$
−0.786629 + 0.617426i $$0.788176\pi$$
$$12$$ 0 0
$$13$$ 5.69848 0.121575 0.0607875 0.998151i $$-0.480639\pi$$
0.0607875 + 0.998151i $$0.480639\pi$$
$$14$$ 0 0
$$15$$ 47.6985 0.821046
$$16$$ 0 0
$$17$$ −51.8995 −0.740440 −0.370220 0.928944i $$-0.620718\pi$$
−0.370220 + 0.928944i $$0.620718\pi$$
$$18$$ 0 0
$$19$$ −16.2010 −0.195619 −0.0978096 0.995205i $$-0.531184\pi$$
−0.0978096 + 0.995205i $$0.531184\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 213.397 1.93462 0.967312 0.253590i $$-0.0816114\pi$$
0.967312 + 0.253590i $$0.0816114\pi$$
$$24$$ 0 0
$$25$$ 127.794 1.02235
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −218.191 −1.39714 −0.698570 0.715542i $$-0.746180\pi$$
−0.698570 + 0.715542i $$0.746180\pi$$
$$30$$ 0 0
$$31$$ 251.397 1.45652 0.728262 0.685299i $$-0.240329\pi$$
0.728262 + 0.685299i $$0.240329\pi$$
$$32$$ 0 0
$$33$$ 172.191 0.908321
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 386.794 1.71861 0.859304 0.511464i $$-0.170897\pi$$
0.859304 + 0.511464i $$0.170897\pi$$
$$38$$ 0 0
$$39$$ −17.0955 −0.0701914
$$40$$ 0 0
$$41$$ −328.503 −1.25130 −0.625652 0.780102i $$-0.715167\pi$$
−0.625652 + 0.780102i $$0.715167\pi$$
$$42$$ 0 0
$$43$$ 37.5879 0.133305 0.0666523 0.997776i $$-0.478768\pi$$
0.0666523 + 0.997776i $$0.478768\pi$$
$$44$$ 0 0
$$45$$ −143.095 −0.474031
$$46$$ 0 0
$$47$$ 254.995 0.791379 0.395690 0.918384i $$-0.370506\pi$$
0.395690 + 0.918384i $$0.370506\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 155.698 0.427493
$$52$$ 0 0
$$53$$ 211.588 0.548374 0.274187 0.961676i $$-0.411591\pi$$
0.274187 + 0.961676i $$0.411591\pi$$
$$54$$ 0 0
$$55$$ 912.583 2.23732
$$56$$ 0 0
$$57$$ 48.6030 0.112941
$$58$$ 0 0
$$59$$ 412.201 0.909559 0.454780 0.890604i $$-0.349718\pi$$
0.454780 + 0.890604i $$0.349718\pi$$
$$60$$ 0 0
$$61$$ −836.693 −1.75619 −0.878095 0.478486i $$-0.841186\pi$$
−0.878095 + 0.478486i $$0.841186\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −90.6030 −0.172891
$$66$$ 0 0
$$67$$ 165.588 0.301937 0.150969 0.988539i $$-0.451761\pi$$
0.150969 + 0.988539i $$0.451761\pi$$
$$68$$ 0 0
$$69$$ −640.191 −1.11696
$$70$$ 0 0
$$71$$ 465.015 0.777284 0.388642 0.921389i $$-0.372944\pi$$
0.388642 + 0.921389i $$0.372944\pi$$
$$72$$ 0 0
$$73$$ 449.658 0.720938 0.360469 0.932771i $$-0.382617\pi$$
0.360469 + 0.932771i $$0.382617\pi$$
$$74$$ 0 0
$$75$$ −383.382 −0.590255
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 343.558 0.489282 0.244641 0.969614i $$-0.421330\pi$$
0.244641 + 0.969614i $$0.421330\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −1502.33 −1.98677 −0.993387 0.114812i $$-0.963373\pi$$
−0.993387 + 0.114812i $$0.963373\pi$$
$$84$$ 0 0
$$85$$ 825.176 1.05298
$$86$$ 0 0
$$87$$ 654.573 0.806639
$$88$$ 0 0
$$89$$ −341.085 −0.406236 −0.203118 0.979154i $$-0.565107\pi$$
−0.203118 + 0.979154i $$0.565107\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −754.191 −0.840924
$$94$$ 0 0
$$95$$ 257.588 0.278189
$$96$$ 0 0
$$97$$ −865.437 −0.905895 −0.452947 0.891537i $$-0.649627\pi$$
−0.452947 + 0.891537i $$0.649627\pi$$
$$98$$ 0 0
$$99$$ −516.573 −0.524419
$$100$$ 0 0
$$101$$ −243.256 −0.239652 −0.119826 0.992795i $$-0.538234\pi$$
−0.119826 + 0.992795i $$0.538234\pi$$
$$102$$ 0 0
$$103$$ 953.346 0.912000 0.456000 0.889980i $$-0.349282\pi$$
0.456000 + 0.889980i $$0.349282\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1344.95 1.21516 0.607578 0.794260i $$-0.292141\pi$$
0.607578 + 0.794260i $$0.292141\pi$$
$$108$$ 0 0
$$109$$ 1734.35 1.52404 0.762022 0.647551i $$-0.224207\pi$$
0.762022 + 0.647551i $$0.224207\pi$$
$$110$$ 0 0
$$111$$ −1160.38 −0.992239
$$112$$ 0 0
$$113$$ 1441.18 1.19977 0.599887 0.800085i $$-0.295212\pi$$
0.599887 + 0.800085i $$0.295212\pi$$
$$114$$ 0 0
$$115$$ −3392.90 −2.75122
$$116$$ 0 0
$$117$$ 51.2864 0.0405250
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1963.41 1.47514
$$122$$ 0 0
$$123$$ 985.508 0.722441
$$124$$ 0 0
$$125$$ −44.4222 −0.0317860
$$126$$ 0 0
$$127$$ −1184.70 −0.827759 −0.413880 0.910332i $$-0.635827\pi$$
−0.413880 + 0.910332i $$0.635827\pi$$
$$128$$ 0 0
$$129$$ −112.764 −0.0769634
$$130$$ 0 0
$$131$$ −297.588 −0.198476 −0.0992381 0.995064i $$-0.531641\pi$$
−0.0992381 + 0.995064i $$0.531641\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 429.286 0.273682
$$136$$ 0 0
$$137$$ 620.985 0.387258 0.193629 0.981075i $$-0.437974\pi$$
0.193629 + 0.981075i $$0.437974\pi$$
$$138$$ 0 0
$$139$$ 898.754 0.548426 0.274213 0.961669i $$-0.411583\pi$$
0.274213 + 0.961669i $$0.411583\pi$$
$$140$$ 0 0
$$141$$ −764.985 −0.456903
$$142$$ 0 0
$$143$$ −327.076 −0.191269
$$144$$ 0 0
$$145$$ 3469.13 1.98686
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −3054.70 −1.67954 −0.839769 0.542945i $$-0.817309\pi$$
−0.839769 + 0.542945i $$0.817309\pi$$
$$150$$ 0 0
$$151$$ 65.1455 0.0351090 0.0175545 0.999846i $$-0.494412\pi$$
0.0175545 + 0.999846i $$0.494412\pi$$
$$152$$ 0 0
$$153$$ −467.095 −0.246813
$$154$$ 0 0
$$155$$ −3997.08 −2.07131
$$156$$ 0 0
$$157$$ 1542.22 0.783966 0.391983 0.919973i $$-0.371789\pi$$
0.391983 + 0.919973i $$0.371789\pi$$
$$158$$ 0 0
$$159$$ −634.764 −0.316604
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 2514.73 1.20840 0.604200 0.796833i $$-0.293493\pi$$
0.604200 + 0.796833i $$0.293493\pi$$
$$164$$ 0 0
$$165$$ −2737.75 −1.29172
$$166$$ 0 0
$$167$$ −528.643 −0.244956 −0.122478 0.992471i $$-0.539084\pi$$
−0.122478 + 0.992471i $$0.539084\pi$$
$$168$$ 0 0
$$169$$ −2164.53 −0.985220
$$170$$ 0 0
$$171$$ −145.809 −0.0652064
$$172$$ 0 0
$$173$$ 96.8439 0.0425602 0.0212801 0.999774i $$-0.493226\pi$$
0.0212801 + 0.999774i $$0.493226\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −1236.60 −0.525134
$$178$$ 0 0
$$179$$ 534.542 0.223204 0.111602 0.993753i $$-0.464402\pi$$
0.111602 + 0.993753i $$0.464402\pi$$
$$180$$ 0 0
$$181$$ 2087.00 0.857049 0.428524 0.903530i $$-0.359034\pi$$
0.428524 + 0.903530i $$0.359034\pi$$
$$182$$ 0 0
$$183$$ 2510.08 1.01394
$$184$$ 0 0
$$185$$ −6149.83 −2.44402
$$186$$ 0 0
$$187$$ 2978.87 1.16490
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3387.69 −1.28337 −0.641687 0.766966i $$-0.721765\pi$$
−0.641687 + 0.766966i $$0.721765\pi$$
$$192$$ 0 0
$$193$$ −1908.35 −0.711742 −0.355871 0.934535i $$-0.615816\pi$$
−0.355871 + 0.934535i $$0.615816\pi$$
$$194$$ 0 0
$$195$$ 271.809 0.0998187
$$196$$ 0 0
$$197$$ −2061.88 −0.745699 −0.372850 0.927892i $$-0.621619\pi$$
−0.372850 + 0.927892i $$0.621619\pi$$
$$198$$ 0 0
$$199$$ −3171.50 −1.12976 −0.564878 0.825174i $$-0.691077\pi$$
−0.564878 + 0.825174i $$0.691077\pi$$
$$200$$ 0 0
$$201$$ −496.764 −0.174323
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 5223.02 1.77947
$$206$$ 0 0
$$207$$ 1920.57 0.644875
$$208$$ 0 0
$$209$$ 929.889 0.307760
$$210$$ 0 0
$$211$$ −1349.97 −0.440454 −0.220227 0.975449i $$-0.570680\pi$$
−0.220227 + 0.975449i $$0.570680\pi$$
$$212$$ 0 0
$$213$$ −1395.05 −0.448765
$$214$$ 0 0
$$215$$ −597.628 −0.189572
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −1348.97 −0.416234
$$220$$ 0 0
$$221$$ −295.748 −0.0900190
$$222$$ 0 0
$$223$$ 1361.85 0.408951 0.204476 0.978872i $$-0.434451\pi$$
0.204476 + 0.978872i $$0.434451\pi$$
$$224$$ 0 0
$$225$$ 1150.15 0.340784
$$226$$ 0 0
$$227$$ −1861.81 −0.544373 −0.272186 0.962245i $$-0.587747\pi$$
−0.272186 + 0.962245i $$0.587747\pi$$
$$228$$ 0 0
$$229$$ −5358.78 −1.54637 −0.773184 0.634181i $$-0.781337\pi$$
−0.773184 + 0.634181i $$0.781337\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 5441.12 1.52987 0.764935 0.644107i $$-0.222771\pi$$
0.764935 + 0.644107i $$0.222771\pi$$
$$234$$ 0 0
$$235$$ −4054.29 −1.12542
$$236$$ 0 0
$$237$$ −1030.67 −0.282487
$$238$$ 0 0
$$239$$ 1157.28 0.313213 0.156607 0.987661i $$-0.449945\pi$$
0.156607 + 0.987661i $$0.449945\pi$$
$$240$$ 0 0
$$241$$ −3969.38 −1.06095 −0.530477 0.847699i $$-0.677987\pi$$
−0.530477 + 0.847699i $$0.677987\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −92.3212 −0.0237824
$$248$$ 0 0
$$249$$ 4506.99 1.14706
$$250$$ 0 0
$$251$$ −5978.75 −1.50349 −0.751744 0.659455i $$-0.770787\pi$$
−0.751744 + 0.659455i $$0.770787\pi$$
$$252$$ 0 0
$$253$$ −12248.3 −3.04366
$$254$$ 0 0
$$255$$ −2475.53 −0.607935
$$256$$ 0 0
$$257$$ 4650.15 1.12867 0.564335 0.825546i $$-0.309132\pi$$
0.564335 + 0.825546i $$0.309132\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −1963.72 −0.465713
$$262$$ 0 0
$$263$$ −3695.37 −0.866411 −0.433205 0.901295i $$-0.642618\pi$$
−0.433205 + 0.901295i $$0.642618\pi$$
$$264$$ 0 0
$$265$$ −3364.14 −0.779840
$$266$$ 0 0
$$267$$ 1023.26 0.234540
$$268$$ 0 0
$$269$$ −7157.69 −1.62235 −0.811175 0.584804i $$-0.801171\pi$$
−0.811175 + 0.584804i $$0.801171\pi$$
$$270$$ 0 0
$$271$$ −4038.37 −0.905216 −0.452608 0.891710i $$-0.649506\pi$$
−0.452608 + 0.891710i $$0.649506\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −7334.98 −1.60842
$$276$$ 0 0
$$277$$ −2754.82 −0.597550 −0.298775 0.954324i $$-0.596578\pi$$
−0.298775 + 0.954324i $$0.596578\pi$$
$$278$$ 0 0
$$279$$ 2262.57 0.485508
$$280$$ 0 0
$$281$$ −772.742 −0.164050 −0.0820248 0.996630i $$-0.526139\pi$$
−0.0820248 + 0.996630i $$0.526139\pi$$
$$282$$ 0 0
$$283$$ 6745.49 1.41688 0.708441 0.705770i $$-0.249399\pi$$
0.708441 + 0.705770i $$0.249399\pi$$
$$284$$ 0 0
$$285$$ −772.764 −0.160613
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −2219.44 −0.451749
$$290$$ 0 0
$$291$$ 2596.31 0.523019
$$292$$ 0 0
$$293$$ 1922.69 0.383362 0.191681 0.981457i $$-0.438606\pi$$
0.191681 + 0.981457i $$0.438606\pi$$
$$294$$ 0 0
$$295$$ −6553.79 −1.29348
$$296$$ 0 0
$$297$$ 1549.72 0.302774
$$298$$ 0 0
$$299$$ 1216.04 0.235202
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 729.768 0.138363
$$304$$ 0 0
$$305$$ 13303.0 2.49747
$$306$$ 0 0
$$307$$ −2016.68 −0.374913 −0.187456 0.982273i $$-0.560024\pi$$
−0.187456 + 0.982273i $$0.560024\pi$$
$$308$$ 0 0
$$309$$ −2860.04 −0.526544
$$310$$ 0 0
$$311$$ −7149.99 −1.30366 −0.651831 0.758365i $$-0.725999\pi$$
−0.651831 + 0.758365i $$0.725999\pi$$
$$312$$ 0 0
$$313$$ 8596.49 1.55240 0.776202 0.630484i $$-0.217144\pi$$
0.776202 + 0.630484i $$0.217144\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2853.24 −0.505532 −0.252766 0.967527i $$-0.581340\pi$$
−0.252766 + 0.967527i $$0.581340\pi$$
$$318$$ 0 0
$$319$$ 12523.5 2.19806
$$320$$ 0 0
$$321$$ −4034.86 −0.701570
$$322$$ 0 0
$$323$$ 840.824 0.144844
$$324$$ 0 0
$$325$$ 728.232 0.124292
$$326$$ 0 0
$$327$$ −5203.05 −0.879907
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1619.12 −0.268866 −0.134433 0.990923i $$-0.542921\pi$$
−0.134433 + 0.990923i $$0.542921\pi$$
$$332$$ 0 0
$$333$$ 3481.15 0.572870
$$334$$ 0 0
$$335$$ −2632.76 −0.429383
$$336$$ 0 0
$$337$$ −3278.67 −0.529972 −0.264986 0.964252i $$-0.585367\pi$$
−0.264986 + 0.964252i $$0.585367\pi$$
$$338$$ 0 0
$$339$$ −4323.53 −0.692690
$$340$$ 0 0
$$341$$ −14429.4 −2.29149
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 10178.7 1.58842
$$346$$ 0 0
$$347$$ 2850.30 0.440957 0.220479 0.975392i $$-0.429238\pi$$
0.220479 + 0.975392i $$0.429238\pi$$
$$348$$ 0 0
$$349$$ −4725.32 −0.724758 −0.362379 0.932031i $$-0.618035\pi$$
−0.362379 + 0.932031i $$0.618035\pi$$
$$350$$ 0 0
$$351$$ −153.859 −0.0233971
$$352$$ 0 0
$$353$$ 6727.44 1.01435 0.507175 0.861843i $$-0.330690\pi$$
0.507175 + 0.861843i $$0.330690\pi$$
$$354$$ 0 0
$$355$$ −7393.51 −1.10537
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 7331.89 1.07789 0.538945 0.842341i $$-0.318823\pi$$
0.538945 + 0.842341i $$0.318823\pi$$
$$360$$ 0 0
$$361$$ −6596.53 −0.961733
$$362$$ 0 0
$$363$$ −5890.24 −0.851673
$$364$$ 0 0
$$365$$ −7149.34 −1.02524
$$366$$ 0 0
$$367$$ −2774.43 −0.394616 −0.197308 0.980342i $$-0.563220\pi$$
−0.197308 + 0.980342i $$0.563220\pi$$
$$368$$ 0 0
$$369$$ −2956.52 −0.417101
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 2527.35 0.350834 0.175417 0.984494i $$-0.443873\pi$$
0.175417 + 0.984494i $$0.443873\pi$$
$$374$$ 0 0
$$375$$ 133.267 0.0183516
$$376$$ 0 0
$$377$$ −1243.36 −0.169857
$$378$$ 0 0
$$379$$ −3116.40 −0.422371 −0.211186 0.977446i $$-0.567732\pi$$
−0.211186 + 0.977446i $$0.567732\pi$$
$$380$$ 0 0
$$381$$ 3554.11 0.477907
$$382$$ 0 0
$$383$$ −1518.07 −0.202532 −0.101266 0.994859i $$-0.532289\pi$$
−0.101266 + 0.994859i $$0.532289\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 338.291 0.0444349
$$388$$ 0 0
$$389$$ −3246.77 −0.423182 −0.211591 0.977358i $$-0.567865\pi$$
−0.211591 + 0.977358i $$0.567865\pi$$
$$390$$ 0 0
$$391$$ −11075.2 −1.43247
$$392$$ 0 0
$$393$$ 892.764 0.114590
$$394$$ 0 0
$$395$$ −5462.39 −0.695804
$$396$$ 0 0
$$397$$ −1830.62 −0.231427 −0.115713 0.993283i $$-0.536915\pi$$
−0.115713 + 0.993283i $$0.536915\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −3385.81 −0.421644 −0.210822 0.977524i $$-0.567614\pi$$
−0.210822 + 0.977524i $$0.567614\pi$$
$$402$$ 0 0
$$403$$ 1432.58 0.177077
$$404$$ 0 0
$$405$$ −1287.86 −0.158010
$$406$$ 0 0
$$407$$ −22200.8 −2.70382
$$408$$ 0 0
$$409$$ −9253.17 −1.11868 −0.559340 0.828938i $$-0.688945\pi$$
−0.559340 + 0.828938i $$0.688945\pi$$
$$410$$ 0 0
$$411$$ −1862.95 −0.223583
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 23886.3 2.82538
$$416$$ 0 0
$$417$$ −2696.26 −0.316634
$$418$$ 0 0
$$419$$ 3547.52 0.413622 0.206811 0.978381i $$-0.433692\pi$$
0.206811 + 0.978381i $$0.433692\pi$$
$$420$$ 0 0
$$421$$ 7848.87 0.908624 0.454312 0.890843i $$-0.349885\pi$$
0.454312 + 0.890843i $$0.349885\pi$$
$$422$$ 0 0
$$423$$ 2294.95 0.263793
$$424$$ 0 0
$$425$$ −6632.44 −0.756990
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 981.227 0.110429
$$430$$ 0 0
$$431$$ 4447.32 0.497030 0.248515 0.968628i $$-0.420057\pi$$
0.248515 + 0.968628i $$0.420057\pi$$
$$432$$ 0 0
$$433$$ 6994.82 0.776327 0.388164 0.921590i $$-0.373110\pi$$
0.388164 + 0.921590i $$0.373110\pi$$
$$434$$ 0 0
$$435$$ −10407.4 −1.14712
$$436$$ 0 0
$$437$$ −3457.25 −0.378450
$$438$$ 0 0
$$439$$ 636.182 0.0691647 0.0345823 0.999402i $$-0.488990\pi$$
0.0345823 + 0.999402i $$0.488990\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4474.24 0.479859 0.239929 0.970790i $$-0.422876\pi$$
0.239929 + 0.970790i $$0.422876\pi$$
$$444$$ 0 0
$$445$$ 5423.08 0.577705
$$446$$ 0 0
$$447$$ 9164.11 0.969681
$$448$$ 0 0
$$449$$ −2389.42 −0.251144 −0.125572 0.992085i $$-0.540077\pi$$
−0.125572 + 0.992085i $$0.540077\pi$$
$$450$$ 0 0
$$451$$ 18855.0 1.96862
$$452$$ 0 0
$$453$$ −195.436 −0.0202702
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −4438.34 −0.454304 −0.227152 0.973859i $$-0.572941\pi$$
−0.227152 + 0.973859i $$0.572941\pi$$
$$458$$ 0 0
$$459$$ 1401.29 0.142498
$$460$$ 0 0
$$461$$ 14079.8 1.42248 0.711240 0.702949i $$-0.248134\pi$$
0.711240 + 0.702949i $$0.248134\pi$$
$$462$$ 0 0
$$463$$ −4687.50 −0.470511 −0.235255 0.971934i $$-0.575593\pi$$
−0.235255 + 0.971934i $$0.575593\pi$$
$$464$$ 0 0
$$465$$ 11991.3 1.19587
$$466$$ 0 0
$$467$$ 8447.26 0.837029 0.418514 0.908210i $$-0.362551\pi$$
0.418514 + 0.908210i $$0.362551\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −4626.66 −0.452623
$$472$$ 0 0
$$473$$ −2157.43 −0.209723
$$474$$ 0 0
$$475$$ −2070.39 −0.199992
$$476$$ 0 0
$$477$$ 1904.29 0.182791
$$478$$ 0 0
$$479$$ −4369.41 −0.416792 −0.208396 0.978045i $$-0.566824\pi$$
−0.208396 + 0.978045i $$0.566824\pi$$
$$480$$ 0 0
$$481$$ 2204.14 0.208940
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 13760.0 1.28827
$$486$$ 0 0
$$487$$ 14477.7 1.34712 0.673561 0.739132i $$-0.264764\pi$$
0.673561 + 0.739132i $$0.264764\pi$$
$$488$$ 0 0
$$489$$ −7544.20 −0.697670
$$490$$ 0 0
$$491$$ −9306.12 −0.855355 −0.427677 0.903931i $$-0.640668\pi$$
−0.427677 + 0.903931i $$0.640668\pi$$
$$492$$ 0 0
$$493$$ 11324.0 1.03450
$$494$$ 0 0
$$495$$ 8213.25 0.745774
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 12237.5 1.09785 0.548926 0.835871i $$-0.315037\pi$$
0.548926 + 0.835871i $$0.315037\pi$$
$$500$$ 0 0
$$501$$ 1585.93 0.141425
$$502$$ 0 0
$$503$$ 5524.30 0.489694 0.244847 0.969562i $$-0.421262\pi$$
0.244847 + 0.969562i $$0.421262\pi$$
$$504$$ 0 0
$$505$$ 3867.65 0.340808
$$506$$ 0 0
$$507$$ 6493.58 0.568817
$$508$$ 0 0
$$509$$ 10079.6 0.877743 0.438871 0.898550i $$-0.355378\pi$$
0.438871 + 0.898550i $$0.355378\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 437.427 0.0376470
$$514$$ 0 0
$$515$$ −15157.7 −1.29695
$$516$$ 0 0
$$517$$ −14635.9 −1.24504
$$518$$ 0 0
$$519$$ −290.532 −0.0245721
$$520$$ 0 0
$$521$$ −5706.61 −0.479868 −0.239934 0.970789i $$-0.577126\pi$$
−0.239934 + 0.970789i $$0.577126\pi$$
$$522$$ 0 0
$$523$$ −10657.3 −0.891032 −0.445516 0.895274i $$-0.646980\pi$$
−0.445516 + 0.895274i $$0.646980\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −13047.4 −1.07847
$$528$$ 0 0
$$529$$ 33371.3 2.74277
$$530$$ 0 0
$$531$$ 3709.81 0.303186
$$532$$ 0 0
$$533$$ −1871.97 −0.152127
$$534$$ 0 0
$$535$$ −21384.1 −1.72807
$$536$$ 0 0
$$537$$ −1603.63 −0.128867
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −4010.48 −0.318714 −0.159357 0.987221i $$-0.550942\pi$$
−0.159357 + 0.987221i $$0.550942\pi$$
$$542$$ 0 0
$$543$$ −6261.01 −0.494817
$$544$$ 0 0
$$545$$ −27575.3 −2.16733
$$546$$ 0 0
$$547$$ 17619.8 1.37728 0.688638 0.725105i $$-0.258209\pi$$
0.688638 + 0.725105i $$0.258209\pi$$
$$548$$ 0 0
$$549$$ −7530.24 −0.585397
$$550$$ 0 0
$$551$$ 3534.91 0.273307
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 18449.5 1.41106
$$556$$ 0 0
$$557$$ −10337.7 −0.786395 −0.393198 0.919454i $$-0.628631\pi$$
−0.393198 + 0.919454i $$0.628631\pi$$
$$558$$ 0 0
$$559$$ 214.194 0.0162065
$$560$$ 0 0
$$561$$ −8936.62 −0.672557
$$562$$ 0 0
$$563$$ −24023.7 −1.79836 −0.899180 0.437580i $$-0.855836\pi$$
−0.899180 + 0.437580i $$0.855836\pi$$
$$564$$ 0 0
$$565$$ −22914.0 −1.70619
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −13179.2 −0.971001 −0.485501 0.874236i $$-0.661363\pi$$
−0.485501 + 0.874236i $$0.661363\pi$$
$$570$$ 0 0
$$571$$ −7776.26 −0.569924 −0.284962 0.958539i $$-0.591981\pi$$
−0.284962 + 0.958539i $$0.591981\pi$$
$$572$$ 0 0
$$573$$ 10163.1 0.740956
$$574$$ 0 0
$$575$$ 27270.8 1.97787
$$576$$ 0 0
$$577$$ −20167.0 −1.45505 −0.727525 0.686081i $$-0.759330\pi$$
−0.727525 + 0.686081i $$0.759330\pi$$
$$578$$ 0 0
$$579$$ 5725.05 0.410924
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −12144.5 −0.862734
$$584$$ 0 0
$$585$$ −815.427 −0.0576304
$$586$$ 0 0
$$587$$ −8365.08 −0.588184 −0.294092 0.955777i $$-0.595017\pi$$
−0.294092 + 0.955777i $$0.595017\pi$$
$$588$$ 0 0
$$589$$ −4072.88 −0.284924
$$590$$ 0 0
$$591$$ 6185.64 0.430530
$$592$$ 0 0
$$593$$ 27621.9 1.91281 0.956403 0.292050i $$-0.0943373\pi$$
0.956403 + 0.292050i $$0.0943373\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 9514.49 0.652265
$$598$$ 0 0
$$599$$ −538.318 −0.0367197 −0.0183598 0.999831i $$-0.505844\pi$$
−0.0183598 + 0.999831i $$0.505844\pi$$
$$600$$ 0 0
$$601$$ −6958.64 −0.472294 −0.236147 0.971717i $$-0.575885\pi$$
−0.236147 + 0.971717i $$0.575885\pi$$
$$602$$ 0 0
$$603$$ 1490.29 0.100646
$$604$$ 0 0
$$605$$ −31217.3 −2.09779
$$606$$ 0 0
$$607$$ −17297.6 −1.15665 −0.578326 0.815806i $$-0.696294\pi$$
−0.578326 + 0.815806i $$0.696294\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1453.08 0.0962120
$$612$$ 0 0
$$613$$ −839.158 −0.0552908 −0.0276454 0.999618i $$-0.508801\pi$$
−0.0276454 + 0.999618i $$0.508801\pi$$
$$614$$ 0 0
$$615$$ −15669.1 −1.02738
$$616$$ 0 0
$$617$$ −16040.0 −1.04659 −0.523295 0.852152i $$-0.675297\pi$$
−0.523295 + 0.852152i $$0.675297\pi$$
$$618$$ 0 0
$$619$$ −5429.28 −0.352538 −0.176269 0.984342i $$-0.556403\pi$$
−0.176269 + 0.984342i $$0.556403\pi$$
$$620$$ 0 0
$$621$$ −5761.72 −0.372318
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −15268.0 −0.977149
$$626$$ 0 0
$$627$$ −2789.67 −0.177685
$$628$$ 0 0
$$629$$ −20074.4 −1.27253
$$630$$ 0 0
$$631$$ 1807.86 0.114057 0.0570284 0.998373i $$-0.481837\pi$$
0.0570284 + 0.998373i $$0.481837\pi$$
$$632$$ 0 0
$$633$$ 4049.91 0.254296
$$634$$ 0 0
$$635$$ 18836.2 1.17715
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 4185.14 0.259095
$$640$$ 0 0
$$641$$ −5904.56 −0.363832 −0.181916 0.983314i $$-0.558230\pi$$
−0.181916 + 0.983314i $$0.558230\pi$$
$$642$$ 0 0
$$643$$ 8092.42 0.496320 0.248160 0.968719i $$-0.420174\pi$$
0.248160 + 0.968719i $$0.420174\pi$$
$$644$$ 0 0
$$645$$ 1792.88 0.109449
$$646$$ 0 0
$$647$$ 20192.7 1.22698 0.613490 0.789702i $$-0.289765\pi$$
0.613490 + 0.789702i $$0.289765\pi$$
$$648$$ 0 0
$$649$$ −23659.1 −1.43097
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −21180.8 −1.26933 −0.634664 0.772789i $$-0.718861\pi$$
−0.634664 + 0.772789i $$0.718861\pi$$
$$654$$ 0 0
$$655$$ 4731.50 0.282252
$$656$$ 0 0
$$657$$ 4046.92 0.240313
$$658$$ 0 0
$$659$$ −28411.3 −1.67944 −0.839718 0.543023i $$-0.817280\pi$$
−0.839718 + 0.543023i $$0.817280\pi$$
$$660$$ 0 0
$$661$$ −16704.9 −0.982975 −0.491488 0.870885i $$-0.663547\pi$$
−0.491488 + 0.870885i $$0.663547\pi$$
$$662$$ 0 0
$$663$$ 887.245 0.0519725
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −46561.3 −2.70294
$$668$$ 0 0
$$669$$ −4085.55 −0.236108
$$670$$ 0 0
$$671$$ 48023.7 2.76294
$$672$$ 0 0
$$673$$ −9047.09 −0.518187 −0.259093 0.965852i $$-0.583424\pi$$
−0.259093 + 0.965852i $$0.583424\pi$$
$$674$$ 0 0
$$675$$ −3450.44 −0.196752
$$676$$ 0 0
$$677$$ 7844.26 0.445316 0.222658 0.974897i $$-0.428527\pi$$
0.222658 + 0.974897i $$0.428527\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 5585.43 0.314294
$$682$$ 0 0
$$683$$ −25766.1 −1.44350 −0.721751 0.692153i $$-0.756662\pi$$
−0.721751 + 0.692153i $$0.756662\pi$$
$$684$$ 0 0
$$685$$ −9873.35 −0.550717
$$686$$ 0 0
$$687$$ 16076.3 0.892796
$$688$$ 0 0
$$689$$ 1205.73 0.0666686
$$690$$ 0 0
$$691$$ 24674.1 1.35839 0.679195 0.733958i $$-0.262329\pi$$
0.679195 + 0.733958i $$0.262329\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −14289.7 −0.779914
$$696$$ 0 0
$$697$$ 17049.1 0.926515
$$698$$ 0 0
$$699$$ −16323.4 −0.883271
$$700$$ 0 0
$$701$$ 29377.9 1.58286 0.791431 0.611258i $$-0.209336\pi$$
0.791431 + 0.611258i $$0.209336\pi$$
$$702$$ 0 0
$$703$$ −6266.45 −0.336193
$$704$$ 0 0
$$705$$ 12162.9 0.649759
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 30594.3 1.62058 0.810291 0.586028i $$-0.199309\pi$$
0.810291 + 0.586028i $$0.199309\pi$$
$$710$$ 0 0
$$711$$ 3092.02 0.163094
$$712$$ 0 0
$$713$$ 53647.4 2.81782
$$714$$ 0 0
$$715$$ 5200.34 0.272002
$$716$$ 0 0
$$717$$ −3471.83 −0.180834
$$718$$ 0 0
$$719$$ 1946.94 0.100985 0.0504927 0.998724i $$-0.483921\pi$$
0.0504927 + 0.998724i $$0.483921\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 11908.1 0.612542
$$724$$ 0 0
$$725$$ −27883.5 −1.42837
$$726$$ 0 0
$$727$$ −15750.6 −0.803518 −0.401759 0.915745i $$-0.631601\pi$$
−0.401759 + 0.915745i $$0.631601\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −1950.79 −0.0987040
$$732$$ 0 0
$$733$$ 14349.6 0.723074 0.361537 0.932358i $$-0.382252\pi$$
0.361537 + 0.932358i $$0.382252\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −9504.24 −0.475025
$$738$$ 0 0
$$739$$ 17758.1 0.883953 0.441976 0.897027i $$-0.354278\pi$$
0.441976 + 0.897027i $$0.354278\pi$$
$$740$$ 0 0
$$741$$ 276.964 0.0137308
$$742$$ 0 0
$$743$$ 29187.6 1.44117 0.720586 0.693366i $$-0.243873\pi$$
0.720586 + 0.693366i $$0.243873\pi$$
$$744$$ 0 0
$$745$$ 48568.2 2.38846
$$746$$ 0 0
$$747$$ −13521.0 −0.662258
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −13781.9 −0.669654 −0.334827 0.942280i $$-0.608678\pi$$
−0.334827 + 0.942280i $$0.608678\pi$$
$$752$$ 0 0
$$753$$ 17936.3 0.868039
$$754$$ 0 0
$$755$$ −1035.78 −0.0499283
$$756$$ 0 0
$$757$$ 36952.7 1.77420 0.887099 0.461579i $$-0.152717\pi$$
0.887099 + 0.461579i $$0.152717\pi$$
$$758$$ 0 0
$$759$$ 36745.0 1.75726
$$760$$ 0 0
$$761$$ 28816.5 1.37266 0.686332 0.727288i $$-0.259220\pi$$
0.686332 + 0.727288i $$0.259220\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 7426.58 0.350992
$$766$$ 0 0
$$767$$ 2348.92 0.110580
$$768$$ 0 0
$$769$$ 25285.2 1.18571 0.592854 0.805310i $$-0.298001\pi$$
0.592854 + 0.805310i $$0.298001\pi$$
$$770$$ 0 0
$$771$$ −13950.4 −0.651638
$$772$$ 0 0
$$773$$ −18418.2 −0.856995 −0.428497 0.903543i $$-0.640957\pi$$
−0.428497 + 0.903543i $$0.640957\pi$$
$$774$$ 0 0
$$775$$ 32127.0 1.48908
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 5322.07 0.244779
$$780$$ 0 0
$$781$$ −26690.5 −1.22287
$$782$$ 0 0
$$783$$ 5891.15 0.268880
$$784$$ 0 0
$$785$$ −24520.5 −1.11487
$$786$$ 0 0
$$787$$ 11075.8 0.501664 0.250832 0.968031i $$-0.419296\pi$$
0.250832 + 0.968031i $$0.419296\pi$$
$$788$$ 0 0
$$789$$ 11086.1 0.500223
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −4767.88 −0.213509
$$794$$ 0 0
$$795$$ 10092.4 0.450241
$$796$$ 0 0
$$797$$ −4838.83 −0.215057 −0.107528 0.994202i $$-0.534294\pi$$
−0.107528 + 0.994202i $$0.534294\pi$$
$$798$$ 0 0
$$799$$ −13234.1 −0.585969
$$800$$ 0 0
$$801$$ −3069.77 −0.135412
$$802$$ 0 0
$$803$$ −25809.0 −1.13422
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 21473.1 0.936664
$$808$$ 0 0
$$809$$ −31509.9 −1.36938 −0.684690 0.728834i $$-0.740062\pi$$
−0.684690 + 0.728834i $$0.740062\pi$$
$$810$$ 0 0
$$811$$ −29463.3 −1.27570 −0.637851 0.770160i $$-0.720177\pi$$
−0.637851 + 0.770160i $$0.720177\pi$$
$$812$$ 0 0
$$813$$ 12115.1 0.522627
$$814$$ 0 0
$$815$$ −39983.0 −1.71846
$$816$$ 0 0
$$817$$ −608.962 −0.0260770
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 3502.22 0.148877 0.0744386 0.997226i $$-0.476284\pi$$
0.0744386 + 0.997226i $$0.476284\pi$$
$$822$$ 0 0
$$823$$ −39993.0 −1.69389 −0.846943 0.531684i $$-0.821560\pi$$
−0.846943 + 0.531684i $$0.821560\pi$$
$$824$$ 0 0
$$825$$ 22005.0 0.928623
$$826$$ 0 0
$$827$$ 10733.6 0.451322 0.225661 0.974206i $$-0.427546\pi$$
0.225661 + 0.974206i $$0.427546\pi$$
$$828$$ 0 0
$$829$$ 14537.5 0.609056 0.304528 0.952503i $$-0.401501\pi$$
0.304528 + 0.952503i $$0.401501\pi$$
$$830$$ 0 0
$$831$$ 8264.47 0.344996
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 8405.16 0.348351
$$836$$ 0 0
$$837$$ −6787.72 −0.280308
$$838$$ 0 0
$$839$$ 7353.57 0.302591 0.151295 0.988489i $$-0.451656\pi$$
0.151295 + 0.988489i $$0.451656\pi$$
$$840$$ 0 0
$$841$$ 23218.3 0.951998
$$842$$ 0 0
$$843$$ 2318.23 0.0947141
$$844$$ 0 0
$$845$$ 34414.9 1.40107
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −20236.5 −0.818037
$$850$$ 0 0
$$851$$ 82540.7 3.32486
$$852$$ 0 0
$$853$$ −19293.6 −0.774442 −0.387221 0.921987i $$-0.626565\pi$$
−0.387221 + 0.921987i $$0.626565\pi$$
$$854$$ 0 0
$$855$$ 2318.29 0.0927297
$$856$$ 0 0
$$857$$ 14161.6 0.564468 0.282234 0.959346i $$-0.408924\pi$$
0.282234 + 0.959346i $$0.408924\pi$$
$$858$$ 0 0
$$859$$ 8219.13 0.326465 0.163232 0.986588i $$-0.447808\pi$$
0.163232 + 0.986588i $$0.447808\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −2574.95 −0.101567 −0.0507835 0.998710i $$-0.516172\pi$$
−0.0507835 + 0.998710i $$0.516172\pi$$
$$864$$ 0 0
$$865$$ −1539.77 −0.0605246
$$866$$ 0 0
$$867$$ 6658.33 0.260817
$$868$$ 0 0
$$869$$ −19719.2 −0.769766
$$870$$ 0 0
$$871$$ 943.600 0.0367080
$$872$$ 0 0
$$873$$ −7788.93 −0.301965
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 30981.1 1.19288 0.596442 0.802656i $$-0.296581\pi$$
0.596442 + 0.802656i $$0.296581\pi$$
$$878$$ 0 0
$$879$$ −5768.08 −0.221334
$$880$$ 0 0
$$881$$ 41781.8 1.59780 0.798902 0.601461i $$-0.205415\pi$$
0.798902 + 0.601461i $$0.205415\pi$$
$$882$$ 0 0
$$883$$ 39289.6 1.49740 0.748699 0.662911i $$-0.230679\pi$$
0.748699 + 0.662911i $$0.230679\pi$$
$$884$$ 0 0
$$885$$ 19661.4 0.746790
$$886$$ 0 0
$$887$$ 5832.44 0.220783 0.110391 0.993888i $$-0.464790\pi$$
0.110391 + 0.993888i $$0.464790\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4649.15 −0.174806
$$892$$ 0 0
$$893$$ −4131.18 −0.154809
$$894$$ 0 0
$$895$$ −8498.95 −0.317418
$$896$$ 0 0
$$897$$ −3648.12 −0.135794
$$898$$ 0 0
$$899$$ −54852.5 −2.03497
$$900$$ 0 0
$$901$$ −10981.3 −0.406038
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −33182.3 −1.21880
$$906$$ 0 0
$$907$$ −42387.9 −1.55178 −0.775892 0.630866i $$-0.782700\pi$$
−0.775892 + 0.630866i $$0.782700\pi$$
$$908$$ 0 0
$$909$$ −2189.30 −0.0798841
$$910$$ 0 0
$$911$$ −2275.12 −0.0827423 −0.0413711 0.999144i $$-0.513173\pi$$
−0.0413711 + 0.999144i $$0.513173\pi$$
$$912$$ 0 0
$$913$$ 86229.3 3.12571
$$914$$ 0 0
$$915$$ −39909.0 −1.44191
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 31284.3 1.12293 0.561465 0.827500i $$-0.310238\pi$$
0.561465 + 0.827500i $$0.310238\pi$$
$$920$$ 0 0
$$921$$ 6050.05 0.216456
$$922$$ 0 0
$$923$$ 2649.88 0.0944983
$$924$$ 0 0
$$925$$ 49429.9 1.75702
$$926$$ 0 0
$$927$$ 8580.12 0.304000
$$928$$ 0 0
$$929$$ −32196.6 −1.13707 −0.568535 0.822659i $$-0.692489\pi$$
−0.568535 + 0.822659i $$0.692489\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 21450.0 0.752669
$$934$$ 0 0
$$935$$ −47362.6 −1.65660
$$936$$ 0 0
$$937$$ −22293.6 −0.777269 −0.388635 0.921392i $$-0.627053\pi$$
−0.388635 + 0.921392i $$0.627053\pi$$
$$938$$ 0 0
$$939$$ −25789.5 −0.896281
$$940$$ 0 0
$$941$$ 31809.7 1.10198 0.550991 0.834511i $$-0.314250\pi$$
0.550991 + 0.834511i $$0.314250\pi$$
$$942$$ 0 0
$$943$$ −70101.4 −2.42080
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −18982.6 −0.651376 −0.325688 0.945477i $$-0.605596\pi$$
−0.325688 + 0.945477i $$0.605596\pi$$
$$948$$ 0 0
$$949$$ 2562.37 0.0876481
$$950$$ 0 0
$$951$$ 8559.71 0.291869
$$952$$ 0 0
$$953$$ 9254.58 0.314570 0.157285 0.987553i $$-0.449726\pi$$
0.157285 + 0.987553i $$0.449726\pi$$
$$954$$ 0 0
$$955$$ 53862.5 1.82508
$$956$$ 0 0
$$957$$ −37570.5 −1.26905
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33409.4 1.12146
$$962$$ 0 0
$$963$$ 12104.6 0.405052
$$964$$ 0 0
$$965$$ 30341.8 1.01216
$$966$$ 0 0
$$967$$ −15317.5 −0.509387 −0.254694 0.967022i $$-0.581975\pi$$
−0.254694 + 0.967022i $$0.581975\pi$$
$$968$$ 0 0
$$969$$ −2522.47 −0.0836259
$$970$$ 0 0
$$971$$ 22432.6 0.741398 0.370699 0.928753i $$-0.379118\pi$$
0.370699 + 0.928753i $$0.379118\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −2184.70 −0.0717603
$$976$$ 0 0
$$977$$ −627.864 −0.0205600 −0.0102800 0.999947i $$-0.503272\pi$$
−0.0102800 + 0.999947i $$0.503272\pi$$
$$978$$ 0 0
$$979$$ 19577.3 0.639114
$$980$$ 0 0
$$981$$ 15609.2 0.508015
$$982$$ 0 0
$$983$$ 45032.6 1.46116 0.730579 0.682828i $$-0.239250\pi$$
0.730579 + 0.682828i $$0.239250\pi$$
$$984$$ 0 0
$$985$$ 32782.8 1.06045
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8021.14 0.257894
$$990$$ 0 0
$$991$$ 31981.0 1.02514 0.512569 0.858646i $$-0.328694\pi$$
0.512569 + 0.858646i $$0.328694\pi$$
$$992$$ 0 0
$$993$$ 4857.35 0.155230
$$994$$ 0 0
$$995$$ 50425.2 1.60662
$$996$$ 0 0
$$997$$ −12378.8 −0.393221 −0.196611 0.980482i $$-0.562993\pi$$
−0.196611 + 0.980482i $$0.562993\pi$$
$$998$$ 0 0
$$999$$ −10443.4 −0.330746
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 2352.4.a.bn.1.1 2
4.3 odd 2 294.4.a.k.1.1 yes 2
7.6 odd 2 2352.4.a.cd.1.2 2
12.11 even 2 882.4.a.bi.1.2 2
28.3 even 6 294.4.e.o.79.1 4
28.11 odd 6 294.4.e.n.79.2 4
28.19 even 6 294.4.e.o.67.1 4
28.23 odd 6 294.4.e.n.67.2 4
28.27 even 2 294.4.a.j.1.2 2
84.11 even 6 882.4.g.y.667.1 4
84.23 even 6 882.4.g.y.361.1 4
84.47 odd 6 882.4.g.bd.361.2 4
84.59 odd 6 882.4.g.bd.667.2 4
84.83 odd 2 882.4.a.bc.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.j.1.2 2 28.27 even 2
294.4.a.k.1.1 yes 2 4.3 odd 2
294.4.e.n.67.2 4 28.23 odd 6
294.4.e.n.79.2 4 28.11 odd 6
294.4.e.o.67.1 4 28.19 even 6
294.4.e.o.79.1 4 28.3 even 6
882.4.a.bc.1.1 2 84.83 odd 2
882.4.a.bi.1.2 2 12.11 even 2
882.4.g.y.361.1 4 84.23 even 6
882.4.g.y.667.1 4 84.11 even 6
882.4.g.bd.361.2 4 84.47 odd 6
882.4.g.bd.667.2 4 84.59 odd 6
2352.4.a.bn.1.1 2 1.1 even 1 trivial
2352.4.a.cd.1.2 2 7.6 odd 2