# Properties

 Label 2312.4.a.k.1.1 Level $2312$ Weight $4$ Character 2312.1 Self dual yes Analytic conductor $136.412$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2312 = 2^{3} \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2312.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$136.412415933$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 95x^{6} + 756x^{4} - 1780x^{2} + 1152$$ x^8 - 95*x^6 + 756*x^4 - 1780*x^2 + 1152 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{13}$$ Twist minimal: no (minimal twist has level 136) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.20783$$ of defining polynomial Character $$\chi$$ $$=$$ 2312.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-9.26065 q^{3} +16.4090 q^{5} -34.5010 q^{7} +58.7597 q^{9} +O(q^{10})$$ $$q-9.26065 q^{3} +16.4090 q^{5} -34.5010 q^{7} +58.7597 q^{9} -7.42843 q^{11} -42.0242 q^{13} -151.958 q^{15} -59.9095 q^{19} +319.502 q^{21} -49.4728 q^{23} +144.255 q^{25} -294.116 q^{27} -259.369 q^{29} -92.2215 q^{31} +68.7922 q^{33} -566.126 q^{35} +207.564 q^{37} +389.172 q^{39} -176.800 q^{41} +19.0809 q^{43} +964.188 q^{45} +80.1389 q^{47} +847.318 q^{49} -319.869 q^{53} -121.893 q^{55} +554.801 q^{57} -11.0809 q^{59} -712.648 q^{61} -2027.27 q^{63} -689.575 q^{65} +484.980 q^{67} +458.150 q^{69} -443.968 q^{71} +337.468 q^{73} -1335.89 q^{75} +256.288 q^{77} -840.905 q^{79} +1137.19 q^{81} -456.004 q^{83} +2401.93 q^{87} -1205.43 q^{89} +1449.88 q^{91} +854.032 q^{93} -983.055 q^{95} -638.484 q^{97} -436.493 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 132 q^{9}+O(q^{10})$$ 8 * q + 132 * q^9 $$8 q + 132 q^{9} + 44 q^{13} - 24 q^{15} - 48 q^{19} + 308 q^{21} + 520 q^{25} + 812 q^{33} - 1064 q^{35} - 8 q^{43} + 312 q^{47} + 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 180 q^{69} - 1660 q^{77} + 3156 q^{81} - 2472 q^{83} + 6664 q^{87} + 68 q^{89} + 4036 q^{93}+O(q^{100})$$ 8 * q + 132 * q^9 + 44 * q^13 - 24 * q^15 - 48 * q^19 + 308 * q^21 + 520 * q^25 + 812 * q^33 - 1064 * q^35 - 8 * q^43 + 312 * q^47 + 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 180 * q^69 - 1660 * q^77 + 3156 * q^81 - 2472 * q^83 + 6664 * q^87 + 68 * q^89 + 4036 * q^93

## 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
$$3$$ −9.26065 −1.78221 −0.891107 0.453794i $$-0.850070\pi$$
−0.891107 + 0.453794i $$0.850070\pi$$
$$4$$ 0 0
$$5$$ 16.4090 1.46766 0.733832 0.679331i $$-0.237730\pi$$
0.733832 + 0.679331i $$0.237730\pi$$
$$6$$ 0 0
$$7$$ −34.5010 −1.86288 −0.931439 0.363898i $$-0.881446\pi$$
−0.931439 + 0.363898i $$0.881446\pi$$
$$8$$ 0 0
$$9$$ 58.7597 2.17629
$$10$$ 0 0
$$11$$ −7.42843 −0.203614 −0.101807 0.994804i $$-0.532462\pi$$
−0.101807 + 0.994804i $$0.532462\pi$$
$$12$$ 0 0
$$13$$ −42.0242 −0.896571 −0.448285 0.893890i $$-0.647965\pi$$
−0.448285 + 0.893890i $$0.647965\pi$$
$$14$$ 0 0
$$15$$ −151.958 −2.61569
$$16$$ 0 0
$$17$$ 0 0
$$18$$ 0 0
$$19$$ −59.9095 −0.723378 −0.361689 0.932299i $$-0.617800\pi$$
−0.361689 + 0.932299i $$0.617800\pi$$
$$20$$ 0 0
$$21$$ 319.502 3.32005
$$22$$ 0 0
$$23$$ −49.4728 −0.448513 −0.224256 0.974530i $$-0.571995\pi$$
−0.224256 + 0.974530i $$0.571995\pi$$
$$24$$ 0 0
$$25$$ 144.255 1.15404
$$26$$ 0 0
$$27$$ −294.116 −2.09639
$$28$$ 0 0
$$29$$ −259.369 −1.66081 −0.830407 0.557157i $$-0.811892\pi$$
−0.830407 + 0.557157i $$0.811892\pi$$
$$30$$ 0 0
$$31$$ −92.2215 −0.534306 −0.267153 0.963654i $$-0.586083\pi$$
−0.267153 + 0.963654i $$0.586083\pi$$
$$32$$ 0 0
$$33$$ 68.7922 0.362884
$$34$$ 0 0
$$35$$ −566.126 −2.73408
$$36$$ 0 0
$$37$$ 207.564 0.922252 0.461126 0.887335i $$-0.347446\pi$$
0.461126 + 0.887335i $$0.347446\pi$$
$$38$$ 0 0
$$39$$ 389.172 1.59788
$$40$$ 0 0
$$41$$ −176.800 −0.673454 −0.336727 0.941602i $$-0.609320\pi$$
−0.336727 + 0.941602i $$0.609320\pi$$
$$42$$ 0 0
$$43$$ 19.0809 0.0676700 0.0338350 0.999427i $$-0.489228\pi$$
0.0338350 + 0.999427i $$0.489228\pi$$
$$44$$ 0 0
$$45$$ 964.188 3.19406
$$46$$ 0 0
$$47$$ 80.1389 0.248712 0.124356 0.992238i $$-0.460314\pi$$
0.124356 + 0.992238i $$0.460314\pi$$
$$48$$ 0 0
$$49$$ 847.318 2.47031
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −319.869 −0.829006 −0.414503 0.910048i $$-0.636045\pi$$
−0.414503 + 0.910048i $$0.636045\pi$$
$$54$$ 0 0
$$55$$ −121.893 −0.298837
$$56$$ 0 0
$$57$$ 554.801 1.28921
$$58$$ 0 0
$$59$$ −11.0809 −0.0244510 −0.0122255 0.999925i $$-0.503892\pi$$
−0.0122255 + 0.999925i $$0.503892\pi$$
$$60$$ 0 0
$$61$$ −712.648 −1.49582 −0.747912 0.663798i $$-0.768943\pi$$
−0.747912 + 0.663798i $$0.768943\pi$$
$$62$$ 0 0
$$63$$ −2027.27 −4.05415
$$64$$ 0 0
$$65$$ −689.575 −1.31587
$$66$$ 0 0
$$67$$ 484.980 0.884325 0.442163 0.896935i $$-0.354211\pi$$
0.442163 + 0.896935i $$0.354211\pi$$
$$68$$ 0 0
$$69$$ 458.150 0.799345
$$70$$ 0 0
$$71$$ −443.968 −0.742103 −0.371051 0.928612i $$-0.621003\pi$$
−0.371051 + 0.928612i $$0.621003\pi$$
$$72$$ 0 0
$$73$$ 337.468 0.541063 0.270531 0.962711i $$-0.412801\pi$$
0.270531 + 0.962711i $$0.412801\pi$$
$$74$$ 0 0
$$75$$ −1335.89 −2.05674
$$76$$ 0 0
$$77$$ 256.288 0.379309
$$78$$ 0 0
$$79$$ −840.905 −1.19759 −0.598793 0.800904i $$-0.704353\pi$$
−0.598793 + 0.800904i $$0.704353\pi$$
$$80$$ 0 0
$$81$$ 1137.19 1.55993
$$82$$ 0 0
$$83$$ −456.004 −0.603047 −0.301524 0.953459i $$-0.597495\pi$$
−0.301524 + 0.953459i $$0.597495\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2401.93 2.95993
$$88$$ 0 0
$$89$$ −1205.43 −1.43568 −0.717841 0.696207i $$-0.754870\pi$$
−0.717841 + 0.696207i $$0.754870\pi$$
$$90$$ 0 0
$$91$$ 1449.88 1.67020
$$92$$ 0 0
$$93$$ 854.032 0.952247
$$94$$ 0 0
$$95$$ −983.055 −1.06168
$$96$$ 0 0
$$97$$ −638.484 −0.668332 −0.334166 0.942514i $$-0.608455\pi$$
−0.334166 + 0.942514i $$0.608455\pi$$
$$98$$ 0 0
$$99$$ −436.493 −0.443123
$$100$$ 0 0
$$101$$ 739.395 0.728441 0.364220 0.931313i $$-0.381335\pi$$
0.364220 + 0.931313i $$0.381335\pi$$
$$102$$ 0 0
$$103$$ 1588.54 1.51965 0.759824 0.650129i $$-0.225285\pi$$
0.759824 + 0.650129i $$0.225285\pi$$
$$104$$ 0 0
$$105$$ 5242.70 4.87271
$$106$$ 0 0
$$107$$ 202.298 0.182775 0.0913875 0.995815i $$-0.470870\pi$$
0.0913875 + 0.995815i $$0.470870\pi$$
$$108$$ 0 0
$$109$$ −1244.71 −1.09378 −0.546890 0.837204i $$-0.684188\pi$$
−0.546890 + 0.837204i $$0.684188\pi$$
$$110$$ 0 0
$$111$$ −1922.18 −1.64365
$$112$$ 0 0
$$113$$ 1492.28 1.24232 0.621159 0.783685i $$-0.286662\pi$$
0.621159 + 0.783685i $$0.286662\pi$$
$$114$$ 0 0
$$115$$ −811.798 −0.658266
$$116$$ 0 0
$$117$$ −2469.33 −1.95119
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1275.82 −0.958541
$$122$$ 0 0
$$123$$ 1637.29 1.20024
$$124$$ 0 0
$$125$$ 315.954 0.226078
$$126$$ 0 0
$$127$$ −2054.49 −1.43548 −0.717741 0.696311i $$-0.754824\pi$$
−0.717741 + 0.696311i $$0.754824\pi$$
$$128$$ 0 0
$$129$$ −176.702 −0.120602
$$130$$ 0 0
$$131$$ −1892.72 −1.26235 −0.631175 0.775640i $$-0.717427\pi$$
−0.631175 + 0.775640i $$0.717427\pi$$
$$132$$ 0 0
$$133$$ 2066.94 1.34757
$$134$$ 0 0
$$135$$ −4826.14 −3.07680
$$136$$ 0 0
$$137$$ −848.618 −0.529214 −0.264607 0.964356i $$-0.585242\pi$$
−0.264607 + 0.964356i $$0.585242\pi$$
$$138$$ 0 0
$$139$$ −20.6414 −0.0125955 −0.00629777 0.999980i $$-0.502005\pi$$
−0.00629777 + 0.999980i $$0.502005\pi$$
$$140$$ 0 0
$$141$$ −742.139 −0.443258
$$142$$ 0 0
$$143$$ 312.174 0.182555
$$144$$ 0 0
$$145$$ −4255.98 −2.43752
$$146$$ 0 0
$$147$$ −7846.71 −4.40263
$$148$$ 0 0
$$149$$ −1202.59 −0.661206 −0.330603 0.943770i $$-0.607252\pi$$
−0.330603 + 0.943770i $$0.607252\pi$$
$$150$$ 0 0
$$151$$ 2080.77 1.12140 0.560698 0.828020i $$-0.310533\pi$$
0.560698 + 0.828020i $$0.310533\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1513.26 −0.784182
$$156$$ 0 0
$$157$$ 1153.74 0.586486 0.293243 0.956038i $$-0.405266\pi$$
0.293243 + 0.956038i $$0.405266\pi$$
$$158$$ 0 0
$$159$$ 2962.19 1.47747
$$160$$ 0 0
$$161$$ 1706.86 0.835524
$$162$$ 0 0
$$163$$ −1461.35 −0.702221 −0.351110 0.936334i $$-0.614196\pi$$
−0.351110 + 0.936334i $$0.614196\pi$$
$$164$$ 0 0
$$165$$ 1128.81 0.532592
$$166$$ 0 0
$$167$$ −4157.98 −1.92667 −0.963337 0.268296i $$-0.913539\pi$$
−0.963337 + 0.268296i $$0.913539\pi$$
$$168$$ 0 0
$$169$$ −430.965 −0.196160
$$170$$ 0 0
$$171$$ −3520.27 −1.57428
$$172$$ 0 0
$$173$$ 2504.08 1.10047 0.550235 0.835010i $$-0.314538\pi$$
0.550235 + 0.835010i $$0.314538\pi$$
$$174$$ 0 0
$$175$$ −4976.94 −2.14983
$$176$$ 0 0
$$177$$ 102.616 0.0435769
$$178$$ 0 0
$$179$$ −839.685 −0.350620 −0.175310 0.984513i $$-0.556093\pi$$
−0.175310 + 0.984513i $$0.556093\pi$$
$$180$$ 0 0
$$181$$ 500.491 0.205532 0.102766 0.994706i $$-0.467231\pi$$
0.102766 + 0.994706i $$0.467231\pi$$
$$182$$ 0 0
$$183$$ 6599.59 2.66588
$$184$$ 0 0
$$185$$ 3405.92 1.35356
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 10147.3 3.90532
$$190$$ 0 0
$$191$$ 199.937 0.0757430 0.0378715 0.999283i $$-0.487942\pi$$
0.0378715 + 0.999283i $$0.487942\pi$$
$$192$$ 0 0
$$193$$ −2758.52 −1.02882 −0.514411 0.857544i $$-0.671989\pi$$
−0.514411 + 0.857544i $$0.671989\pi$$
$$194$$ 0 0
$$195$$ 6385.92 2.34515
$$196$$ 0 0
$$197$$ 3379.87 1.22236 0.611182 0.791490i $$-0.290694\pi$$
0.611182 + 0.791490i $$0.290694\pi$$
$$198$$ 0 0
$$199$$ 2729.46 0.972293 0.486147 0.873877i $$-0.338402\pi$$
0.486147 + 0.873877i $$0.338402\pi$$
$$200$$ 0 0
$$201$$ −4491.24 −1.57606
$$202$$ 0 0
$$203$$ 8948.48 3.09389
$$204$$ 0 0
$$205$$ −2901.12 −0.988404
$$206$$ 0 0
$$207$$ −2907.01 −0.976092
$$208$$ 0 0
$$209$$ 445.034 0.147290
$$210$$ 0 0
$$211$$ −2149.01 −0.701157 −0.350579 0.936533i $$-0.614015\pi$$
−0.350579 + 0.936533i $$0.614015\pi$$
$$212$$ 0 0
$$213$$ 4111.43 1.32259
$$214$$ 0 0
$$215$$ 313.098 0.0993168
$$216$$ 0 0
$$217$$ 3181.73 0.995346
$$218$$ 0 0
$$219$$ −3125.17 −0.964290
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 115.665 0.0347332 0.0173666 0.999849i $$-0.494472\pi$$
0.0173666 + 0.999849i $$0.494472\pi$$
$$224$$ 0 0
$$225$$ 8476.38 2.51152
$$226$$ 0 0
$$227$$ 150.739 0.0440743 0.0220372 0.999757i $$-0.492985\pi$$
0.0220372 + 0.999757i $$0.492985\pi$$
$$228$$ 0 0
$$229$$ 252.262 0.0727946 0.0363973 0.999337i $$-0.488412\pi$$
0.0363973 + 0.999337i $$0.488412\pi$$
$$230$$ 0 0
$$231$$ −2373.40 −0.676009
$$232$$ 0 0
$$233$$ −1025.45 −0.288323 −0.144161 0.989554i $$-0.546048\pi$$
−0.144161 + 0.989554i $$0.546048\pi$$
$$234$$ 0 0
$$235$$ 1315.00 0.365026
$$236$$ 0 0
$$237$$ 7787.33 2.13435
$$238$$ 0 0
$$239$$ 3803.45 1.02939 0.514696 0.857373i $$-0.327905\pi$$
0.514696 + 0.857373i $$0.327905\pi$$
$$240$$ 0 0
$$241$$ −2073.99 −0.554346 −0.277173 0.960820i $$-0.589397\pi$$
−0.277173 + 0.960820i $$0.589397\pi$$
$$242$$ 0 0
$$243$$ −2590.02 −0.683743
$$244$$ 0 0
$$245$$ 13903.6 3.62559
$$246$$ 0 0
$$247$$ 2517.65 0.648560
$$248$$ 0 0
$$249$$ 4222.89 1.07476
$$250$$ 0 0
$$251$$ −1242.05 −0.312341 −0.156170 0.987730i $$-0.549915\pi$$
−0.156170 + 0.987730i $$0.549915\pi$$
$$252$$ 0 0
$$253$$ 367.505 0.0913236
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1469.82 −0.356751 −0.178375 0.983963i $$-0.557084\pi$$
−0.178375 + 0.983963i $$0.557084\pi$$
$$258$$ 0 0
$$259$$ −7161.16 −1.71804
$$260$$ 0 0
$$261$$ −15240.4 −3.61441
$$262$$ 0 0
$$263$$ 4308.70 1.01021 0.505106 0.863057i $$-0.331453\pi$$
0.505106 + 0.863057i $$0.331453\pi$$
$$264$$ 0 0
$$265$$ −5248.72 −1.21670
$$266$$ 0 0
$$267$$ 11163.1 2.55869
$$268$$ 0 0
$$269$$ 3812.72 0.864184 0.432092 0.901830i $$-0.357776\pi$$
0.432092 + 0.901830i $$0.357776\pi$$
$$270$$ 0 0
$$271$$ 2859.05 0.640868 0.320434 0.947271i $$-0.396171\pi$$
0.320434 + 0.947271i $$0.396171\pi$$
$$272$$ 0 0
$$273$$ −13426.8 −2.97666
$$274$$ 0 0
$$275$$ −1071.59 −0.234979
$$276$$ 0 0
$$277$$ 5377.92 1.16653 0.583263 0.812283i $$-0.301776\pi$$
0.583263 + 0.812283i $$0.301776\pi$$
$$278$$ 0 0
$$279$$ −5418.91 −1.16280
$$280$$ 0 0
$$281$$ −1867.29 −0.396418 −0.198209 0.980160i $$-0.563512\pi$$
−0.198209 + 0.980160i $$0.563512\pi$$
$$282$$ 0 0
$$283$$ 4296.76 0.902529 0.451265 0.892390i $$-0.350973\pi$$
0.451265 + 0.892390i $$0.350973\pi$$
$$284$$ 0 0
$$285$$ 9103.73 1.89213
$$286$$ 0 0
$$287$$ 6099.79 1.25456
$$288$$ 0 0
$$289$$ 0 0
$$290$$ 0 0
$$291$$ 5912.78 1.19111
$$292$$ 0 0
$$293$$ 8816.10 1.75782 0.878911 0.476985i $$-0.158270\pi$$
0.878911 + 0.476985i $$0.158270\pi$$
$$294$$ 0 0
$$295$$ −181.826 −0.0358859
$$296$$ 0 0
$$297$$ 2184.82 0.426855
$$298$$ 0 0
$$299$$ 2079.06 0.402123
$$300$$ 0 0
$$301$$ −658.309 −0.126061
$$302$$ 0 0
$$303$$ −6847.28 −1.29824
$$304$$ 0 0
$$305$$ −11693.8 −2.19537
$$306$$ 0 0
$$307$$ 6143.84 1.14217 0.571087 0.820890i $$-0.306522\pi$$
0.571087 + 0.820890i $$0.306522\pi$$
$$308$$ 0 0
$$309$$ −14710.9 −2.70834
$$310$$ 0 0
$$311$$ 7409.80 1.35103 0.675516 0.737345i $$-0.263921\pi$$
0.675516 + 0.737345i $$0.263921\pi$$
$$312$$ 0 0
$$313$$ −9455.98 −1.70761 −0.853807 0.520589i $$-0.825712\pi$$
−0.853807 + 0.520589i $$0.825712\pi$$
$$314$$ 0 0
$$315$$ −33265.4 −5.95014
$$316$$ 0 0
$$317$$ −1045.23 −0.185191 −0.0925957 0.995704i $$-0.529516\pi$$
−0.0925957 + 0.995704i $$0.529516\pi$$
$$318$$ 0 0
$$319$$ 1926.71 0.338165
$$320$$ 0 0
$$321$$ −1873.42 −0.325744
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −6062.20 −1.03468
$$326$$ 0 0
$$327$$ 11526.9 1.94935
$$328$$ 0 0
$$329$$ −2764.87 −0.463320
$$330$$ 0 0
$$331$$ 6370.04 1.05779 0.528896 0.848687i $$-0.322606\pi$$
0.528896 + 0.848687i $$0.322606\pi$$
$$332$$ 0 0
$$333$$ 12196.4 2.00708
$$334$$ 0 0
$$335$$ 7958.04 1.29789
$$336$$ 0 0
$$337$$ 9216.37 1.48976 0.744878 0.667201i $$-0.232508\pi$$
0.744878 + 0.667201i $$0.232508\pi$$
$$338$$ 0 0
$$339$$ −13819.5 −2.21408
$$340$$ 0 0
$$341$$ 685.062 0.108792
$$342$$ 0 0
$$343$$ −17399.4 −2.73901
$$344$$ 0 0
$$345$$ 7517.79 1.17317
$$346$$ 0 0
$$347$$ 7810.63 1.20835 0.604174 0.796852i $$-0.293503\pi$$
0.604174 + 0.796852i $$0.293503\pi$$
$$348$$ 0 0
$$349$$ −8377.96 −1.28499 −0.642495 0.766290i $$-0.722101\pi$$
−0.642495 + 0.766290i $$0.722101\pi$$
$$350$$ 0 0
$$351$$ 12360.0 1.87956
$$352$$ 0 0
$$353$$ 500.551 0.0754721 0.0377360 0.999288i $$-0.487985\pi$$
0.0377360 + 0.999288i $$0.487985\pi$$
$$354$$ 0 0
$$355$$ −7285.06 −1.08916
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 12643.0 1.85870 0.929351 0.369198i $$-0.120368\pi$$
0.929351 + 0.369198i $$0.120368\pi$$
$$360$$ 0 0
$$361$$ −3269.85 −0.476724
$$362$$ 0 0
$$363$$ 11814.9 1.70833
$$364$$ 0 0
$$365$$ 5537.50 0.794099
$$366$$ 0 0
$$367$$ 640.446 0.0910927 0.0455464 0.998962i $$-0.485497\pi$$
0.0455464 + 0.998962i $$0.485497\pi$$
$$368$$ 0 0
$$369$$ −10388.7 −1.46563
$$370$$ 0 0
$$371$$ 11035.8 1.54434
$$372$$ 0 0
$$373$$ −1138.01 −0.157972 −0.0789861 0.996876i $$-0.525168\pi$$
−0.0789861 + 0.996876i $$0.525168\pi$$
$$374$$ 0 0
$$375$$ −2925.94 −0.402920
$$376$$ 0 0
$$377$$ 10899.8 1.48904
$$378$$ 0 0
$$379$$ −12320.8 −1.66987 −0.834933 0.550352i $$-0.814493\pi$$
−0.834933 + 0.550352i $$0.814493\pi$$
$$380$$ 0 0
$$381$$ 19025.9 2.55833
$$382$$ 0 0
$$383$$ 9934.77 1.32544 0.662719 0.748868i $$-0.269402\pi$$
0.662719 + 0.748868i $$0.269402\pi$$
$$384$$ 0 0
$$385$$ 4205.43 0.556698
$$386$$ 0 0
$$387$$ 1121.19 0.147269
$$388$$ 0 0
$$389$$ −3323.46 −0.433178 −0.216589 0.976263i $$-0.569493\pi$$
−0.216589 + 0.976263i $$0.569493\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 17527.8 2.24978
$$394$$ 0 0
$$395$$ −13798.4 −1.75765
$$396$$ 0 0
$$397$$ 3094.45 0.391199 0.195599 0.980684i $$-0.437335\pi$$
0.195599 + 0.980684i $$0.437335\pi$$
$$398$$ 0 0
$$399$$ −19141.2 −2.40165
$$400$$ 0 0
$$401$$ 6261.50 0.779761 0.389881 0.920865i $$-0.372516\pi$$
0.389881 + 0.920865i $$0.372516\pi$$
$$402$$ 0 0
$$403$$ 3875.54 0.479043
$$404$$ 0 0
$$405$$ 18660.2 2.28946
$$406$$ 0 0
$$407$$ −1541.88 −0.187784
$$408$$ 0 0
$$409$$ 13966.7 1.68853 0.844264 0.535927i $$-0.180038\pi$$
0.844264 + 0.535927i $$0.180038\pi$$
$$410$$ 0 0
$$411$$ 7858.76 0.943172
$$412$$ 0 0
$$413$$ 382.302 0.0455492
$$414$$ 0 0
$$415$$ −7482.56 −0.885071
$$416$$ 0 0
$$417$$ 191.153 0.0224479
$$418$$ 0 0
$$419$$ 6308.88 0.735582 0.367791 0.929908i $$-0.380114\pi$$
0.367791 + 0.929908i $$0.380114\pi$$
$$420$$ 0 0
$$421$$ −3142.63 −0.363806 −0.181903 0.983317i $$-0.558226\pi$$
−0.181903 + 0.983317i $$0.558226\pi$$
$$422$$ 0 0
$$423$$ 4708.94 0.541268
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 24587.1 2.78654
$$428$$ 0 0
$$429$$ −2890.94 −0.325351
$$430$$ 0 0
$$431$$ −6900.34 −0.771178 −0.385589 0.922671i $$-0.626002\pi$$
−0.385589 + 0.922671i $$0.626002\pi$$
$$432$$ 0 0
$$433$$ 15236.0 1.69098 0.845490 0.533991i $$-0.179308\pi$$
0.845490 + 0.533991i $$0.179308\pi$$
$$434$$ 0 0
$$435$$ 39413.2 4.34418
$$436$$ 0 0
$$437$$ 2963.89 0.324444
$$438$$ 0 0
$$439$$ 9594.64 1.04311 0.521557 0.853216i $$-0.325351\pi$$
0.521557 + 0.853216i $$0.325351\pi$$
$$440$$ 0 0
$$441$$ 49788.1 5.37611
$$442$$ 0 0
$$443$$ −3375.02 −0.361968 −0.180984 0.983486i $$-0.557928\pi$$
−0.180984 + 0.983486i $$0.557928\pi$$
$$444$$ 0 0
$$445$$ −19780.0 −2.10710
$$446$$ 0 0
$$447$$ 11136.7 1.17841
$$448$$ 0 0
$$449$$ 2918.05 0.306707 0.153354 0.988171i $$-0.450993\pi$$
0.153354 + 0.988171i $$0.450993\pi$$
$$450$$ 0 0
$$451$$ 1313.35 0.137125
$$452$$ 0 0
$$453$$ −19269.3 −1.99857
$$454$$ 0 0
$$455$$ 23791.0 2.45130
$$456$$ 0 0
$$457$$ 3829.50 0.391983 0.195992 0.980606i $$-0.437207\pi$$
0.195992 + 0.980606i $$0.437207\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6988.84 −0.706080 −0.353040 0.935608i $$-0.614852\pi$$
−0.353040 + 0.935608i $$0.614852\pi$$
$$462$$ 0 0
$$463$$ 8669.67 0.870224 0.435112 0.900376i $$-0.356709\pi$$
0.435112 + 0.900376i $$0.356709\pi$$
$$464$$ 0 0
$$465$$ 14013.8 1.39758
$$466$$ 0 0
$$467$$ −12728.0 −1.26120 −0.630600 0.776108i $$-0.717191\pi$$
−0.630600 + 0.776108i $$0.717191\pi$$
$$468$$ 0 0
$$469$$ −16732.3 −1.64739
$$470$$ 0 0
$$471$$ −10684.4 −1.04524
$$472$$ 0 0
$$473$$ −141.741 −0.0137786
$$474$$ 0 0
$$475$$ −8642.24 −0.834807
$$476$$ 0 0
$$477$$ −18795.4 −1.80415
$$478$$ 0 0
$$479$$ 13282.8 1.26703 0.633516 0.773730i $$-0.281611\pi$$
0.633516 + 0.773730i $$0.281611\pi$$
$$480$$ 0 0
$$481$$ −8722.72 −0.826864
$$482$$ 0 0
$$483$$ −15806.6 −1.48908
$$484$$ 0 0
$$485$$ −10476.9 −0.980888
$$486$$ 0 0
$$487$$ −5014.37 −0.466577 −0.233288 0.972408i $$-0.574949\pi$$
−0.233288 + 0.972408i $$0.574949\pi$$
$$488$$ 0 0
$$489$$ 13533.1 1.25151
$$490$$ 0 0
$$491$$ −14518.4 −1.33443 −0.667217 0.744863i $$-0.732515\pi$$
−0.667217 + 0.744863i $$0.732515\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −7162.40 −0.650356
$$496$$ 0 0
$$497$$ 15317.3 1.38245
$$498$$ 0 0
$$499$$ 18158.2 1.62901 0.814504 0.580158i $$-0.197009\pi$$
0.814504 + 0.580158i $$0.197009\pi$$
$$500$$ 0 0
$$501$$ 38505.6 3.43374
$$502$$ 0 0
$$503$$ −12331.2 −1.09309 −0.546543 0.837431i $$-0.684057\pi$$
−0.546543 + 0.837431i $$0.684057\pi$$
$$504$$ 0 0
$$505$$ 12132.7 1.06911
$$506$$ 0 0
$$507$$ 3991.01 0.349600
$$508$$ 0 0
$$509$$ −4051.78 −0.352833 −0.176417 0.984316i $$-0.556451\pi$$
−0.176417 + 0.984316i $$0.556451\pi$$
$$510$$ 0 0
$$511$$ −11643.0 −1.00793
$$512$$ 0 0
$$513$$ 17620.3 1.51648
$$514$$ 0 0
$$515$$ 26066.4 2.23033
$$516$$ 0 0
$$517$$ −595.307 −0.0506413
$$518$$ 0 0
$$519$$ −23189.4 −1.96127
$$520$$ 0 0
$$521$$ −6101.23 −0.513052 −0.256526 0.966537i $$-0.582578\pi$$
−0.256526 + 0.966537i $$0.582578\pi$$
$$522$$ 0 0
$$523$$ −1233.70 −0.103147 −0.0515734 0.998669i $$-0.516424\pi$$
−0.0515734 + 0.998669i $$0.516424\pi$$
$$524$$ 0 0
$$525$$ 46089.7 3.83146
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −9719.44 −0.798836
$$530$$ 0 0
$$531$$ −651.110 −0.0532124
$$532$$ 0 0
$$533$$ 7429.90 0.603799
$$534$$ 0 0
$$535$$ 3319.51 0.268252
$$536$$ 0 0
$$537$$ 7776.03 0.624880
$$538$$ 0 0
$$539$$ −6294.24 −0.502991
$$540$$ 0 0
$$541$$ 8968.57 0.712734 0.356367 0.934346i $$-0.384015\pi$$
0.356367 + 0.934346i $$0.384015\pi$$
$$542$$ 0 0
$$543$$ −4634.87 −0.366301
$$544$$ 0 0
$$545$$ −20424.5 −1.60530
$$546$$ 0 0
$$547$$ −7187.71 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$548$$ 0 0
$$549$$ −41875.0 −3.25534
$$550$$ 0 0
$$551$$ 15538.7 1.20140
$$552$$ 0 0
$$553$$ 29012.0 2.23095
$$554$$ 0 0
$$555$$ −31541.0 −2.41233
$$556$$ 0 0
$$557$$ 12502.2 0.951047 0.475524 0.879703i $$-0.342259\pi$$
0.475524 + 0.879703i $$0.342259\pi$$
$$558$$ 0 0
$$559$$ −801.860 −0.0606709
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −20391.1 −1.52643 −0.763216 0.646143i $$-0.776381\pi$$
−0.763216 + 0.646143i $$0.776381\pi$$
$$564$$ 0 0
$$565$$ 24486.8 1.82331
$$566$$ 0 0
$$567$$ −39234.2 −2.90597
$$568$$ 0 0
$$569$$ −3407.53 −0.251057 −0.125528 0.992090i $$-0.540063\pi$$
−0.125528 + 0.992090i $$0.540063\pi$$
$$570$$ 0 0
$$571$$ −9682.41 −0.709626 −0.354813 0.934937i $$-0.615455\pi$$
−0.354813 + 0.934937i $$0.615455\pi$$
$$572$$ 0 0
$$573$$ −1851.54 −0.134990
$$574$$ 0 0
$$575$$ −7136.69 −0.517601
$$576$$ 0 0
$$577$$ −10357.7 −0.747309 −0.373654 0.927568i $$-0.621895\pi$$
−0.373654 + 0.927568i $$0.621895\pi$$
$$578$$ 0 0
$$579$$ 25545.7 1.83358
$$580$$ 0 0
$$581$$ 15732.6 1.12340
$$582$$ 0 0
$$583$$ 2376.12 0.168798
$$584$$ 0 0
$$585$$ −40519.2 −2.86370
$$586$$ 0 0
$$587$$ 760.786 0.0534940 0.0267470 0.999642i $$-0.491485\pi$$
0.0267470 + 0.999642i $$0.491485\pi$$
$$588$$ 0 0
$$589$$ 5524.95 0.386505
$$590$$ 0 0
$$591$$ −31299.8 −2.17851
$$592$$ 0 0
$$593$$ −23861.5 −1.65241 −0.826203 0.563373i $$-0.809503\pi$$
−0.826203 + 0.563373i $$0.809503\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −25276.6 −1.73283
$$598$$ 0 0
$$599$$ 40.2985 0.00274884 0.00137442 0.999999i $$-0.499563\pi$$
0.00137442 + 0.999999i $$0.499563\pi$$
$$600$$ 0 0
$$601$$ 28351.4 1.92426 0.962129 0.272594i $$-0.0878815\pi$$
0.962129 + 0.272594i $$0.0878815\pi$$
$$602$$ 0 0
$$603$$ 28497.3 1.92454
$$604$$ 0 0
$$605$$ −20934.9 −1.40682
$$606$$ 0 0
$$607$$ 380.879 0.0254686 0.0127343 0.999919i $$-0.495946\pi$$
0.0127343 + 0.999919i $$0.495946\pi$$
$$608$$ 0 0
$$609$$ −82868.8 −5.51398
$$610$$ 0 0
$$611$$ −3367.78 −0.222988
$$612$$ 0 0
$$613$$ −18449.5 −1.21561 −0.607804 0.794087i $$-0.707949\pi$$
−0.607804 + 0.794087i $$0.707949\pi$$
$$614$$ 0 0
$$615$$ 26866.2 1.76155
$$616$$ 0 0
$$617$$ 13692.2 0.893397 0.446698 0.894685i $$-0.352600\pi$$
0.446698 + 0.894685i $$0.352600\pi$$
$$618$$ 0 0
$$619$$ 15000.2 0.974004 0.487002 0.873401i $$-0.338090\pi$$
0.487002 + 0.873401i $$0.338090\pi$$
$$620$$ 0 0
$$621$$ 14550.7 0.940259
$$622$$ 0 0
$$623$$ 41588.7 2.67450
$$624$$ 0 0
$$625$$ −12847.4 −0.822232
$$626$$ 0 0
$$627$$ −4121.31 −0.262503
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 25659.3 1.61883 0.809413 0.587240i $$-0.199785\pi$$
0.809413 + 0.587240i $$0.199785\pi$$
$$632$$ 0 0
$$633$$ 19901.3 1.24961
$$634$$ 0 0
$$635$$ −33712.0 −2.10680
$$636$$ 0 0
$$637$$ −35607.9 −2.21481
$$638$$ 0 0
$$639$$ −26087.4 −1.61503
$$640$$ 0 0
$$641$$ 6709.60 0.413437 0.206719 0.978400i $$-0.433722\pi$$
0.206719 + 0.978400i $$0.433722\pi$$
$$642$$ 0 0
$$643$$ −11976.4 −0.734528 −0.367264 0.930117i $$-0.619705\pi$$
−0.367264 + 0.930117i $$0.619705\pi$$
$$644$$ 0 0
$$645$$ −2899.49 −0.177004
$$646$$ 0 0
$$647$$ 21265.3 1.29215 0.646077 0.763272i $$-0.276408\pi$$
0.646077 + 0.763272i $$0.276408\pi$$
$$648$$ 0 0
$$649$$ 82.3137 0.00497857
$$650$$ 0 0
$$651$$ −29464.9 −1.77392
$$652$$ 0 0
$$653$$ 19671.9 1.17890 0.589449 0.807806i $$-0.299345\pi$$
0.589449 + 0.807806i $$0.299345\pi$$
$$654$$ 0 0
$$655$$ −31057.6 −1.85271
$$656$$ 0 0
$$657$$ 19829.5 1.17751
$$658$$ 0 0
$$659$$ −14782.6 −0.873824 −0.436912 0.899504i $$-0.643928\pi$$
−0.436912 + 0.899504i $$0.643928\pi$$
$$660$$ 0 0
$$661$$ −27770.8 −1.63413 −0.817063 0.576548i $$-0.804399\pi$$
−0.817063 + 0.576548i $$0.804399\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 33916.4 1.97777
$$666$$ 0 0
$$667$$ 12831.7 0.744896
$$668$$ 0 0
$$669$$ −1071.14 −0.0619021
$$670$$ 0 0
$$671$$ 5293.86 0.304571
$$672$$ 0 0
$$673$$ −21891.3 −1.25386 −0.626929 0.779077i $$-0.715688\pi$$
−0.626929 + 0.779077i $$0.715688\pi$$
$$674$$ 0 0
$$675$$ −42427.6 −2.41932
$$676$$ 0 0
$$677$$ −1946.42 −0.110498 −0.0552488 0.998473i $$-0.517595\pi$$
−0.0552488 + 0.998473i $$0.517595\pi$$
$$678$$ 0 0
$$679$$ 22028.3 1.24502
$$680$$ 0 0
$$681$$ −1395.94 −0.0785498
$$682$$ 0 0
$$683$$ 10516.5 0.589169 0.294584 0.955625i $$-0.404819\pi$$
0.294584 + 0.955625i $$0.404819\pi$$
$$684$$ 0 0
$$685$$ −13925.0 −0.776709
$$686$$ 0 0
$$687$$ −2336.12 −0.129736
$$688$$ 0 0
$$689$$ 13442.2 0.743263
$$690$$ 0 0
$$691$$ −26695.9 −1.46970 −0.734848 0.678232i $$-0.762747\pi$$
−0.734848 + 0.678232i $$0.762747\pi$$
$$692$$ 0 0
$$693$$ 15059.4 0.825484
$$694$$ 0 0
$$695$$ −338.704 −0.0184860
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 9496.30 0.513853
$$700$$ 0 0
$$701$$ −22317.0 −1.20243 −0.601214 0.799088i $$-0.705316\pi$$
−0.601214 + 0.799088i $$0.705316\pi$$
$$702$$ 0 0
$$703$$ −12435.1 −0.667137
$$704$$ 0 0
$$705$$ −12177.7 −0.650554
$$706$$ 0 0
$$707$$ −25509.8 −1.35700
$$708$$ 0 0
$$709$$ −28249.7 −1.49639 −0.748196 0.663478i $$-0.769080\pi$$
−0.748196 + 0.663478i $$0.769080\pi$$
$$710$$ 0 0
$$711$$ −49411.3 −2.60629
$$712$$ 0 0
$$713$$ 4562.46 0.239643
$$714$$ 0 0
$$715$$ 5122.46 0.267929
$$716$$ 0 0
$$717$$ −35222.4 −1.83460
$$718$$ 0 0
$$719$$ 30991.2 1.60748 0.803739 0.594982i $$-0.202841\pi$$
0.803739 + 0.594982i $$0.202841\pi$$
$$720$$ 0 0
$$721$$ −54806.3 −2.83092
$$722$$ 0 0
$$723$$ 19206.5 0.987963
$$724$$ 0 0
$$725$$ −37415.2 −1.91664
$$726$$ 0 0
$$727$$ 13328.0 0.679929 0.339965 0.940438i $$-0.389585\pi$$
0.339965 + 0.940438i $$0.389585\pi$$
$$728$$ 0 0
$$729$$ −6718.94 −0.341358
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 27976.7 1.40974 0.704872 0.709334i $$-0.251004\pi$$
0.704872 + 0.709334i $$0.251004\pi$$
$$734$$ 0 0
$$735$$ −128757. −6.46158
$$736$$ 0 0
$$737$$ −3602.64 −0.180061
$$738$$ 0 0
$$739$$ −33971.2 −1.69100 −0.845501 0.533974i $$-0.820698\pi$$
−0.845501 + 0.533974i $$0.820698\pi$$
$$740$$ 0 0
$$741$$ −23315.1 −1.15587
$$742$$ 0 0
$$743$$ −25922.4 −1.27995 −0.639974 0.768397i $$-0.721055\pi$$
−0.639974 + 0.768397i $$0.721055\pi$$
$$744$$ 0 0
$$745$$ −19733.2 −0.970428
$$746$$ 0 0
$$747$$ −26794.6 −1.31240
$$748$$ 0 0
$$749$$ −6979.49 −0.340488
$$750$$ 0 0
$$751$$ −18950.5 −0.920792 −0.460396 0.887714i $$-0.652293\pi$$
−0.460396 + 0.887714i $$0.652293\pi$$
$$752$$ 0 0
$$753$$ 11502.2 0.556658
$$754$$ 0 0
$$755$$ 34143.4 1.64583
$$756$$ 0 0
$$757$$ 8184.07 0.392940 0.196470 0.980510i $$-0.437052\pi$$
0.196470 + 0.980510i $$0.437052\pi$$
$$758$$ 0 0
$$759$$ −3403.34 −0.162758
$$760$$ 0 0
$$761$$ −25803.9 −1.22916 −0.614579 0.788855i $$-0.710674\pi$$
−0.614579 + 0.788855i $$0.710674\pi$$
$$762$$ 0 0
$$763$$ 42943.9 2.03758
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 465.666 0.0219221
$$768$$ 0 0
$$769$$ 5222.53 0.244901 0.122451 0.992475i $$-0.460925\pi$$
0.122451 + 0.992475i $$0.460925\pi$$
$$770$$ 0 0
$$771$$ 13611.5 0.635806
$$772$$ 0 0
$$773$$ −32636.6 −1.51857 −0.759286 0.650757i $$-0.774452\pi$$
−0.759286 + 0.650757i $$0.774452\pi$$
$$774$$ 0 0
$$775$$ −13303.4 −0.616610
$$776$$ 0 0
$$777$$ 66317.0 3.06192
$$778$$ 0 0
$$779$$ 10592.0 0.487162
$$780$$ 0 0
$$781$$ 3297.99 0.151103
$$782$$ 0 0
$$783$$ 76284.5 3.48172
$$784$$ 0 0
$$785$$ 18931.7 0.860764
$$786$$ 0 0
$$787$$ −17014.8 −0.770664 −0.385332 0.922778i $$-0.625913\pi$$
−0.385332 + 0.922778i $$0.625913\pi$$
$$788$$ 0 0
$$789$$ −39901.4 −1.80042
$$790$$ 0 0
$$791$$ −51485.1 −2.31429
$$792$$ 0 0
$$793$$ 29948.5 1.34111
$$794$$ 0 0
$$795$$ 48606.6 2.16842
$$796$$ 0 0
$$797$$ 20826.6 0.925616 0.462808 0.886458i $$-0.346842\pi$$
0.462808 + 0.886458i $$0.346842\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −70831.0 −3.12446
$$802$$ 0 0
$$803$$ −2506.86 −0.110168
$$804$$ 0 0
$$805$$ 28007.8 1.22627
$$806$$ 0 0
$$807$$ −35308.2 −1.54016
$$808$$ 0 0
$$809$$ 8639.25 0.375451 0.187726 0.982222i $$-0.439888\pi$$
0.187726 + 0.982222i $$0.439888\pi$$
$$810$$ 0 0
$$811$$ 22319.6 0.966397 0.483199 0.875511i $$-0.339475\pi$$
0.483199 + 0.875511i $$0.339475\pi$$
$$812$$ 0 0
$$813$$ −26476.7 −1.14216
$$814$$ 0 0
$$815$$ −23979.3 −1.03062
$$816$$ 0 0
$$817$$ −1143.13 −0.0489510
$$818$$ 0 0
$$819$$ 85194.4 3.63484
$$820$$ 0 0
$$821$$ 9361.94 0.397971 0.198985 0.980002i $$-0.436235\pi$$
0.198985 + 0.980002i $$0.436235\pi$$
$$822$$ 0 0
$$823$$ 19128.8 0.810192 0.405096 0.914274i $$-0.367238\pi$$
0.405096 + 0.914274i $$0.367238\pi$$
$$824$$ 0 0
$$825$$ 9923.61 0.418783
$$826$$ 0 0
$$827$$ 40640.9 1.70886 0.854428 0.519570i $$-0.173908\pi$$
0.854428 + 0.519570i $$0.173908\pi$$
$$828$$ 0 0
$$829$$ −25220.3 −1.05662 −0.528310 0.849052i $$-0.677174\pi$$
−0.528310 + 0.849052i $$0.677174\pi$$
$$830$$ 0 0
$$831$$ −49803.0 −2.07900
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −68228.3 −2.82771
$$836$$ 0 0
$$837$$ 27123.8 1.12011
$$838$$ 0 0
$$839$$ 4544.50 0.187001 0.0935004 0.995619i $$-0.470194\pi$$
0.0935004 + 0.995619i $$0.470194\pi$$
$$840$$ 0 0
$$841$$ 42883.3 1.75830
$$842$$ 0 0
$$843$$ 17292.4 0.706501
$$844$$ 0 0
$$845$$ −7071.69 −0.287898
$$846$$ 0 0
$$847$$ 44017.0 1.78565
$$848$$ 0 0
$$849$$ −39790.8 −1.60850
$$850$$ 0 0
$$851$$ −10268.8 −0.413641
$$852$$ 0 0
$$853$$ 22395.8 0.898964 0.449482 0.893289i $$-0.351609\pi$$
0.449482 + 0.893289i $$0.351609\pi$$
$$854$$ 0 0
$$855$$ −57764.0 −2.31051
$$856$$ 0 0
$$857$$ −5735.42 −0.228610 −0.114305 0.993446i $$-0.536464\pi$$
−0.114305 + 0.993446i $$0.536464\pi$$
$$858$$ 0 0
$$859$$ 34718.6 1.37903 0.689513 0.724273i $$-0.257825\pi$$
0.689513 + 0.724273i $$0.257825\pi$$
$$860$$ 0 0
$$861$$ −56488.0 −2.23590
$$862$$ 0 0
$$863$$ −22239.9 −0.877237 −0.438619 0.898673i $$-0.644532\pi$$
−0.438619 + 0.898673i $$0.644532\pi$$
$$864$$ 0 0
$$865$$ 41089.4 1.61512
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 6246.61 0.243845
$$870$$ 0 0
$$871$$ −20380.9 −0.792860
$$872$$ 0 0
$$873$$ −37517.1 −1.45448
$$874$$ 0 0
$$875$$ −10900.7 −0.421156
$$876$$ 0 0
$$877$$ −10379.6 −0.399650 −0.199825 0.979832i $$-0.564037\pi$$
−0.199825 + 0.979832i $$0.564037\pi$$
$$878$$ 0 0
$$879$$ −81642.9 −3.13282
$$880$$ 0 0
$$881$$ 44934.3 1.71836 0.859181 0.511672i $$-0.170974\pi$$
0.859181 + 0.511672i $$0.170974\pi$$
$$882$$ 0 0
$$883$$ −15846.0 −0.603918 −0.301959 0.953321i $$-0.597640\pi$$
−0.301959 + 0.953321i $$0.597640\pi$$
$$884$$ 0 0
$$885$$ 1683.83 0.0639563
$$886$$ 0 0
$$887$$ −30415.0 −1.15134 −0.575669 0.817683i $$-0.695258\pi$$
−0.575669 + 0.817683i $$0.695258\pi$$
$$888$$ 0 0
$$889$$ 70881.8 2.67413
$$890$$ 0 0
$$891$$ −8447.56 −0.317625
$$892$$ 0 0
$$893$$ −4801.08 −0.179913
$$894$$ 0 0
$$895$$ −13778.4 −0.514593
$$896$$ 0 0
$$897$$ −19253.4 −0.716670
$$898$$ 0 0
$$899$$ 23919.4 0.887382
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 6096.38 0.224667
$$904$$ 0 0
$$905$$ 8212.55 0.301651
$$906$$ 0 0
$$907$$ −7288.75 −0.266835 −0.133417 0.991060i $$-0.542595\pi$$
−0.133417 + 0.991060i $$0.542595\pi$$
$$908$$ 0 0
$$909$$ 43446.6 1.58530
$$910$$ 0 0
$$911$$ −10232.6 −0.372140 −0.186070 0.982536i $$-0.559575\pi$$
−0.186070 + 0.982536i $$0.559575\pi$$
$$912$$ 0 0
$$913$$ 3387.39 0.122789
$$914$$ 0 0
$$915$$ 108293. 3.91261
$$916$$ 0 0
$$917$$ 65300.8 2.35160
$$918$$ 0 0
$$919$$ −26533.8 −0.952415 −0.476207 0.879333i $$-0.657989\pi$$
−0.476207 + 0.879333i $$0.657989\pi$$
$$920$$ 0 0
$$921$$ −56895.9 −2.03560
$$922$$ 0 0
$$923$$ 18657.4 0.665348
$$924$$ 0 0
$$925$$ 29942.1 1.06431
$$926$$ 0 0
$$927$$ 93342.3 3.30719
$$928$$ 0 0
$$929$$ 7037.69 0.248546 0.124273 0.992248i $$-0.460340\pi$$
0.124273 + 0.992248i $$0.460340\pi$$
$$930$$ 0 0
$$931$$ −50762.4 −1.78697
$$932$$ 0 0
$$933$$ −68619.6 −2.40783
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −16895.0 −0.589047 −0.294523 0.955644i $$-0.595161\pi$$
−0.294523 + 0.955644i $$0.595161\pi$$
$$938$$ 0 0
$$939$$ 87568.5 3.04333
$$940$$ 0 0
$$941$$ 51103.6 1.77038 0.885190 0.465229i $$-0.154028\pi$$
0.885190 + 0.465229i $$0.154028\pi$$
$$942$$ 0 0
$$943$$ 8746.81 0.302052
$$944$$ 0 0
$$945$$ 166507. 5.73170
$$946$$ 0 0
$$947$$ 22510.4 0.772429 0.386215 0.922409i $$-0.373782\pi$$
0.386215 + 0.922409i $$0.373782\pi$$
$$948$$ 0 0
$$949$$ −14181.8 −0.485101
$$950$$ 0 0
$$951$$ 9679.47 0.330051
$$952$$ 0 0
$$953$$ −4260.98 −0.144834 −0.0724169 0.997374i $$-0.523071\pi$$
−0.0724169 + 0.997374i $$0.523071\pi$$
$$954$$ 0 0
$$955$$ 3280.76 0.111165
$$956$$ 0 0
$$957$$ −17842.5 −0.602683
$$958$$ 0 0
$$959$$ 29278.1 0.985861
$$960$$ 0 0
$$961$$ −21286.2 −0.714517
$$962$$ 0 0
$$963$$ 11887.0 0.397771
$$964$$ 0 0
$$965$$ −45264.5 −1.50996
$$966$$ 0 0
$$967$$ −5646.95 −0.187791 −0.0938953 0.995582i $$-0.529932\pi$$
−0.0938953 + 0.995582i $$0.529932\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −3320.02 −0.109726 −0.0548632 0.998494i $$-0.517472\pi$$
−0.0548632 + 0.998494i $$0.517472\pi$$
$$972$$ 0 0
$$973$$ 712.148 0.0234639
$$974$$ 0 0
$$975$$ 56140.0 1.84402
$$976$$ 0 0
$$977$$ 5821.02 0.190615 0.0953076 0.995448i $$-0.469617\pi$$
0.0953076 + 0.995448i $$0.469617\pi$$
$$978$$ 0 0
$$979$$ 8954.49 0.292326
$$980$$ 0 0
$$981$$ −73139.1 −2.38038
$$982$$ 0 0
$$983$$ 234.506 0.00760894 0.00380447 0.999993i $$-0.498789\pi$$
0.00380447 + 0.999993i $$0.498789\pi$$
$$984$$ 0 0
$$985$$ 55460.3 1.79402
$$986$$ 0 0
$$987$$ 25604.5 0.825735
$$988$$ 0 0
$$989$$ −943.985 −0.0303508
$$990$$ 0 0
$$991$$ 16708.2 0.535573 0.267787 0.963478i $$-0.413708\pi$$
0.267787 + 0.963478i $$0.413708\pi$$
$$992$$ 0 0
$$993$$ −58990.8 −1.88521
$$994$$ 0 0
$$995$$ 44787.7 1.42700
$$996$$ 0 0
$$997$$ −10281.9 −0.326611 −0.163305 0.986576i $$-0.552216\pi$$
−0.163305 + 0.986576i $$0.552216\pi$$
$$998$$ 0 0
$$999$$ −61047.8 −1.93340
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 2312.4.a.k.1.1 8
17.4 even 4 136.4.b.b.33.8 yes 8
17.13 even 4 136.4.b.b.33.1 8
17.16 even 2 inner 2312.4.a.k.1.8 8
51.38 odd 4 1224.4.c.e.577.7 8
51.47 odd 4 1224.4.c.e.577.2 8
68.47 odd 4 272.4.b.f.33.8 8
68.55 odd 4 272.4.b.f.33.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
136.4.b.b.33.1 8 17.13 even 4
136.4.b.b.33.8 yes 8 17.4 even 4
272.4.b.f.33.1 8 68.55 odd 4
272.4.b.f.33.8 8 68.47 odd 4
1224.4.c.e.577.2 8 51.47 odd 4
1224.4.c.e.577.7 8 51.38 odd 4
2312.4.a.k.1.1 8 1.1 even 1 trivial
2312.4.a.k.1.8 8 17.16 even 2 inner