# Properties

 Label 2352.4.a.cc.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 $1$

# Related objects

## 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: $$\Q(\sqrt{505})$$ Defining polynomial: $$x^{2} - x - 126$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$11.7361$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -6.73610 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -6.73610 q^{5} +9.00000 q^{9} +53.6805 q^{11} +1.73610 q^{13} -20.2083 q^{15} -46.9444 q^{17} +20.1527 q^{19} -118.944 q^{23} -79.6249 q^{25} +27.0000 q^{27} -103.681 q^{29} +157.528 q^{31} +161.042 q^{33} -37.7361 q^{37} +5.20831 q^{39} -287.250 q^{41} -504.875 q^{43} -60.6249 q^{45} +220.500 q^{47} -140.833 q^{51} +292.319 q^{53} -361.597 q^{55} +60.4582 q^{57} +595.875 q^{59} -265.389 q^{61} -11.6946 q^{65} -936.735 q^{67} -356.833 q^{69} +545.944 q^{71} +299.374 q^{73} -238.875 q^{75} +940.333 q^{79} +81.0000 q^{81} -611.319 q^{83} +316.222 q^{85} -311.042 q^{87} -1176.19 q^{89} +472.583 q^{93} -135.751 q^{95} +1482.49 q^{97} +483.125 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} + 9q^{5} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} + 9q^{5} + 18q^{9} - 5q^{11} - 19q^{13} + 27q^{15} - 4q^{17} - 117q^{19} - 148q^{23} + 43q^{25} + 54q^{27} - 95q^{29} + 360q^{31} - 15q^{33} - 53q^{37} - 57q^{39} - 170q^{41} - 403q^{43} + 81q^{45} - 368q^{47} - 12q^{51} + 697q^{53} - 1285q^{55} - 351q^{57} + 585q^{59} - 1160q^{61} - 338q^{65} - 233q^{67} - 444q^{69} - 616q^{71} - 817q^{73} + 129q^{75} + 802q^{79} + 162q^{81} + 283q^{83} + 992q^{85} - 285q^{87} - 1858q^{89} + 1080q^{93} - 2294q^{95} + 1729q^{97} - 45q^{99} + O(q^{100})$$

## 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
$$5$$ −6.73610 −0.602495 −0.301248 0.953546i $$-0.597403\pi$$
−0.301248 + 0.953546i $$0.597403\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 53.6805 1.47139 0.735695 0.677313i $$-0.236856\pi$$
0.735695 + 0.677313i $$0.236856\pi$$
$$12$$ 0 0
$$13$$ 1.73610 0.0370391 0.0185195 0.999828i $$-0.494105\pi$$
0.0185195 + 0.999828i $$0.494105\pi$$
$$14$$ 0 0
$$15$$ −20.2083 −0.347851
$$16$$ 0 0
$$17$$ −46.9444 −0.669747 −0.334873 0.942263i $$-0.608694\pi$$
−0.334873 + 0.942263i $$0.608694\pi$$
$$18$$ 0 0
$$19$$ 20.1527 0.243334 0.121667 0.992571i $$-0.461176\pi$$
0.121667 + 0.992571i $$0.461176\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −118.944 −1.07833 −0.539166 0.842200i $$-0.681260\pi$$
−0.539166 + 0.842200i $$0.681260\pi$$
$$24$$ 0 0
$$25$$ −79.6249 −0.636999
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −103.681 −0.663896 −0.331948 0.943298i $$-0.607706\pi$$
−0.331948 + 0.943298i $$0.607706\pi$$
$$30$$ 0 0
$$31$$ 157.528 0.912672 0.456336 0.889808i $$-0.349162\pi$$
0.456336 + 0.889808i $$0.349162\pi$$
$$32$$ 0 0
$$33$$ 161.042 0.849507
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −37.7361 −0.167670 −0.0838348 0.996480i $$-0.526717\pi$$
−0.0838348 + 0.996480i $$0.526717\pi$$
$$38$$ 0 0
$$39$$ 5.20831 0.0213845
$$40$$ 0 0
$$41$$ −287.250 −1.09417 −0.547084 0.837078i $$-0.684262\pi$$
−0.547084 + 0.837078i $$0.684262\pi$$
$$42$$ 0 0
$$43$$ −504.875 −1.79053 −0.895264 0.445537i $$-0.853013\pi$$
−0.895264 + 0.445537i $$0.853013\pi$$
$$44$$ 0 0
$$45$$ −60.6249 −0.200832
$$46$$ 0 0
$$47$$ 220.500 0.684323 0.342162 0.939641i $$-0.388841\pi$$
0.342162 + 0.939641i $$0.388841\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −140.833 −0.386678
$$52$$ 0 0
$$53$$ 292.319 0.757607 0.378803 0.925477i $$-0.376336\pi$$
0.378803 + 0.925477i $$0.376336\pi$$
$$54$$ 0 0
$$55$$ −361.597 −0.886505
$$56$$ 0 0
$$57$$ 60.4582 0.140489
$$58$$ 0 0
$$59$$ 595.875 1.31485 0.657426 0.753519i $$-0.271645\pi$$
0.657426 + 0.753519i $$0.271645\pi$$
$$60$$ 0 0
$$61$$ −265.389 −0.557043 −0.278521 0.960430i $$-0.589844\pi$$
−0.278521 + 0.960430i $$0.589844\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −11.6946 −0.0223159
$$66$$ 0 0
$$67$$ −936.735 −1.70807 −0.854033 0.520218i $$-0.825851\pi$$
−0.854033 + 0.520218i $$0.825851\pi$$
$$68$$ 0 0
$$69$$ −356.833 −0.622575
$$70$$ 0 0
$$71$$ 545.944 0.912558 0.456279 0.889837i $$-0.349182\pi$$
0.456279 + 0.889837i $$0.349182\pi$$
$$72$$ 0 0
$$73$$ 299.374 0.479988 0.239994 0.970774i $$-0.422855\pi$$
0.239994 + 0.970774i $$0.422855\pi$$
$$74$$ 0 0
$$75$$ −238.875 −0.367772
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 940.333 1.33919 0.669593 0.742728i $$-0.266468\pi$$
0.669593 + 0.742728i $$0.266468\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −611.319 −0.808445 −0.404223 0.914661i $$-0.632458\pi$$
−0.404223 + 0.914661i $$0.632458\pi$$
$$84$$ 0 0
$$85$$ 316.222 0.403519
$$86$$ 0 0
$$87$$ −311.042 −0.383301
$$88$$ 0 0
$$89$$ −1176.19 −1.40086 −0.700429 0.713722i $$-0.747008\pi$$
−0.700429 + 0.713722i $$0.747008\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 472.583 0.526931
$$94$$ 0 0
$$95$$ −135.751 −0.146608
$$96$$ 0 0
$$97$$ 1482.49 1.55179 0.775895 0.630862i $$-0.217299\pi$$
0.775895 + 0.630862i $$0.217299\pi$$
$$98$$ 0 0
$$99$$ 483.125 0.490463
$$100$$ 0 0
$$101$$ 617.944 0.608789 0.304395 0.952546i $$-0.401546\pi$$
0.304395 + 0.952546i $$0.401546\pi$$
$$102$$ 0 0
$$103$$ 816.764 0.781341 0.390670 0.920531i $$-0.372243\pi$$
0.390670 + 0.920531i $$0.372243\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1685.38 −1.52272 −0.761361 0.648328i $$-0.775469\pi$$
−0.761361 + 0.648328i $$0.775469\pi$$
$$108$$ 0 0
$$109$$ 91.5968 0.0804898 0.0402449 0.999190i $$-0.487186\pi$$
0.0402449 + 0.999190i $$0.487186\pi$$
$$110$$ 0 0
$$111$$ −113.208 −0.0968041
$$112$$ 0 0
$$113$$ 614.194 0.511314 0.255657 0.966767i $$-0.417708\pi$$
0.255657 + 0.966767i $$0.417708\pi$$
$$114$$ 0 0
$$115$$ 801.222 0.649690
$$116$$ 0 0
$$117$$ 15.6249 0.0123464
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1550.60 1.16499
$$122$$ 0 0
$$123$$ −861.750 −0.631718
$$124$$ 0 0
$$125$$ 1378.37 0.986284
$$126$$ 0 0
$$127$$ −234.278 −0.163691 −0.0818457 0.996645i $$-0.526081\pi$$
−0.0818457 + 0.996645i $$0.526081\pi$$
$$128$$ 0 0
$$129$$ −1514.62 −1.03376
$$130$$ 0 0
$$131$$ −2664.76 −1.77726 −0.888631 0.458622i $$-0.848343\pi$$
−0.888631 + 0.458622i $$0.848343\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −181.875 −0.115950
$$136$$ 0 0
$$137$$ −1046.31 −0.652496 −0.326248 0.945284i $$-0.605784\pi$$
−0.326248 + 0.945284i $$0.605784\pi$$
$$138$$ 0 0
$$139$$ −647.265 −0.394966 −0.197483 0.980306i $$-0.563277\pi$$
−0.197483 + 0.980306i $$0.563277\pi$$
$$140$$ 0 0
$$141$$ 661.499 0.395094
$$142$$ 0 0
$$143$$ 93.1949 0.0544989
$$144$$ 0 0
$$145$$ 698.403 0.399994
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 1326.81 0.729505 0.364752 0.931105i $$-0.381154\pi$$
0.364752 + 0.931105i $$0.381154\pi$$
$$150$$ 0 0
$$151$$ −3683.24 −1.98502 −0.992508 0.122178i $$-0.961012\pi$$
−0.992508 + 0.122178i $$0.961012\pi$$
$$152$$ 0 0
$$153$$ −422.500 −0.223249
$$154$$ 0 0
$$155$$ −1061.12 −0.549881
$$156$$ 0 0
$$157$$ −2818.25 −1.43262 −0.716308 0.697784i $$-0.754169\pi$$
−0.716308 + 0.697784i $$0.754169\pi$$
$$158$$ 0 0
$$159$$ 876.958 0.437405
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4106.61 −1.97334 −0.986670 0.162733i $$-0.947969\pi$$
−0.986670 + 0.162733i $$0.947969\pi$$
$$164$$ 0 0
$$165$$ −1084.79 −0.511824
$$166$$ 0 0
$$167$$ −1045.56 −0.484476 −0.242238 0.970217i $$-0.577881\pi$$
−0.242238 + 0.970217i $$0.577881\pi$$
$$168$$ 0 0
$$169$$ −2193.99 −0.998628
$$170$$ 0 0
$$171$$ 181.374 0.0811114
$$172$$ 0 0
$$173$$ −1910.31 −0.839525 −0.419763 0.907634i $$-0.637887\pi$$
−0.419763 + 0.907634i $$0.637887\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1787.62 0.759130
$$178$$ 0 0
$$179$$ −231.140 −0.0965151 −0.0482576 0.998835i $$-0.515367\pi$$
−0.0482576 + 0.998835i $$0.515367\pi$$
$$180$$ 0 0
$$181$$ 3308.40 1.35863 0.679314 0.733848i $$-0.262278\pi$$
0.679314 + 0.733848i $$0.262278\pi$$
$$182$$ 0 0
$$183$$ −796.167 −0.321609
$$184$$ 0 0
$$185$$ 254.194 0.101020
$$186$$ 0 0
$$187$$ −2520.00 −0.985458
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1800.75 −0.682187 −0.341093 0.940029i $$-0.610797\pi$$
−0.341093 + 0.940029i $$0.610797\pi$$
$$192$$ 0 0
$$193$$ −1283.36 −0.478644 −0.239322 0.970940i $$-0.576925\pi$$
−0.239322 + 0.970940i $$0.576925\pi$$
$$194$$ 0 0
$$195$$ −35.0837 −0.0128841
$$196$$ 0 0
$$197$$ 60.7514 0.0219714 0.0109857 0.999940i $$-0.496503\pi$$
0.0109857 + 0.999940i $$0.496503\pi$$
$$198$$ 0 0
$$199$$ −3860.39 −1.37515 −0.687577 0.726112i $$-0.741325\pi$$
−0.687577 + 0.726112i $$0.741325\pi$$
$$200$$ 0 0
$$201$$ −2810.21 −0.986153
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 1934.94 0.659231
$$206$$ 0 0
$$207$$ −1070.50 −0.359444
$$208$$ 0 0
$$209$$ 1081.81 0.358039
$$210$$ 0 0
$$211$$ −491.637 −0.160406 −0.0802031 0.996779i $$-0.525557\pi$$
−0.0802031 + 0.996779i $$0.525557\pi$$
$$212$$ 0 0
$$213$$ 1637.83 0.526866
$$214$$ 0 0
$$215$$ 3400.89 1.07878
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 898.123 0.277121
$$220$$ 0 0
$$221$$ −81.5003 −0.0248068
$$222$$ 0 0
$$223$$ −823.376 −0.247253 −0.123626 0.992329i $$-0.539452\pi$$
−0.123626 + 0.992329i $$0.539452\pi$$
$$224$$ 0 0
$$225$$ −716.624 −0.212333
$$226$$ 0 0
$$227$$ 3758.87 1.09905 0.549527 0.835476i $$-0.314808\pi$$
0.549527 + 0.835476i $$0.314808\pi$$
$$228$$ 0 0
$$229$$ −2440.10 −0.704132 −0.352066 0.935975i $$-0.614521\pi$$
−0.352066 + 0.935975i $$0.614521\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1949.86 −0.548239 −0.274120 0.961696i $$-0.588386\pi$$
−0.274120 + 0.961696i $$0.588386\pi$$
$$234$$ 0 0
$$235$$ −1485.31 −0.412301
$$236$$ 0 0
$$237$$ 2821.00 0.773180
$$238$$ 0 0
$$239$$ −2705.64 −0.732273 −0.366137 0.930561i $$-0.619320\pi$$
−0.366137 + 0.930561i $$0.619320\pi$$
$$240$$ 0 0
$$241$$ 604.211 0.161496 0.0807482 0.996735i $$-0.474269\pi$$
0.0807482 + 0.996735i $$0.474269\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 34.9872 0.00901288
$$248$$ 0 0
$$249$$ −1833.96 −0.466756
$$250$$ 0 0
$$251$$ 1631.82 0.410356 0.205178 0.978725i $$-0.434223\pi$$
0.205178 + 0.978725i $$0.434223\pi$$
$$252$$ 0 0
$$253$$ −6385.00 −1.58665
$$254$$ 0 0
$$255$$ 948.667 0.232972
$$256$$ 0 0
$$257$$ −7106.55 −1.72488 −0.862441 0.506158i $$-0.831065\pi$$
−0.862441 + 0.506158i $$0.831065\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −933.125 −0.221299
$$262$$ 0 0
$$263$$ −5570.25 −1.30599 −0.652996 0.757361i $$-0.726488\pi$$
−0.652996 + 0.757361i $$0.726488\pi$$
$$264$$ 0 0
$$265$$ −1969.09 −0.456455
$$266$$ 0 0
$$267$$ −3528.58 −0.808786
$$268$$ 0 0
$$269$$ 3913.18 0.886955 0.443477 0.896286i $$-0.353745\pi$$
0.443477 + 0.896286i $$0.353745\pi$$
$$270$$ 0 0
$$271$$ −6072.51 −1.36118 −0.680588 0.732666i $$-0.738276\pi$$
−0.680588 + 0.732666i $$0.738276\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −4274.31 −0.937274
$$276$$ 0 0
$$277$$ 6140.62 1.33196 0.665982 0.745968i $$-0.268013\pi$$
0.665982 + 0.745968i $$0.268013\pi$$
$$278$$ 0 0
$$279$$ 1417.75 0.304224
$$280$$ 0 0
$$281$$ −365.751 −0.0776472 −0.0388236 0.999246i $$-0.512361\pi$$
−0.0388236 + 0.999246i $$0.512361\pi$$
$$282$$ 0 0
$$283$$ 710.348 0.149208 0.0746039 0.997213i $$-0.476231\pi$$
0.0746039 + 0.997213i $$0.476231\pi$$
$$284$$ 0 0
$$285$$ −407.252 −0.0846440
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −2709.22 −0.551440
$$290$$ 0 0
$$291$$ 4447.46 0.895926
$$292$$ 0 0
$$293$$ 296.373 0.0590932 0.0295466 0.999563i $$-0.490594\pi$$
0.0295466 + 0.999563i $$0.490594\pi$$
$$294$$ 0 0
$$295$$ −4013.87 −0.792192
$$296$$ 0 0
$$297$$ 1449.37 0.283169
$$298$$ 0 0
$$299$$ −206.500 −0.0399404
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 1853.83 0.351485
$$304$$ 0 0
$$305$$ 1787.69 0.335616
$$306$$ 0 0
$$307$$ −8596.87 −1.59821 −0.799103 0.601194i $$-0.794692\pi$$
−0.799103 + 0.601194i $$0.794692\pi$$
$$308$$ 0 0
$$309$$ 2450.29 0.451107
$$310$$ 0 0
$$311$$ 10475.3 1.90997 0.954984 0.296658i $$-0.0958723\pi$$
0.954984 + 0.296658i $$0.0958723\pi$$
$$312$$ 0 0
$$313$$ 5496.52 0.992594 0.496297 0.868153i $$-0.334693\pi$$
0.496297 + 0.868153i $$0.334693\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6556.37 1.16165 0.580824 0.814029i $$-0.302730\pi$$
0.580824 + 0.814029i $$0.302730\pi$$
$$318$$ 0 0
$$319$$ −5565.62 −0.976850
$$320$$ 0 0
$$321$$ −5056.13 −0.879145
$$322$$ 0 0
$$323$$ −946.057 −0.162972
$$324$$ 0 0
$$325$$ −138.237 −0.0235939
$$326$$ 0 0
$$327$$ 274.790 0.0464708
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −102.376 −0.0170003 −0.00850017 0.999964i $$-0.502706\pi$$
−0.00850017 + 0.999964i $$0.502706\pi$$
$$332$$ 0 0
$$333$$ −339.625 −0.0558899
$$334$$ 0 0
$$335$$ 6309.95 1.02910
$$336$$ 0 0
$$337$$ 1333.50 0.215549 0.107775 0.994175i $$-0.465627\pi$$
0.107775 + 0.994175i $$0.465627\pi$$
$$338$$ 0 0
$$339$$ 1842.58 0.295208
$$340$$ 0 0
$$341$$ 8456.17 1.34290
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 2403.67 0.375099
$$346$$ 0 0
$$347$$ 1101.00 0.170331 0.0851655 0.996367i $$-0.472858\pi$$
0.0851655 + 0.996367i $$0.472858\pi$$
$$348$$ 0 0
$$349$$ 9467.89 1.45216 0.726081 0.687609i $$-0.241340\pi$$
0.726081 + 0.687609i $$0.241340\pi$$
$$350$$ 0 0
$$351$$ 46.8748 0.00712818
$$352$$ 0 0
$$353$$ −1684.69 −0.254015 −0.127007 0.991902i $$-0.540537\pi$$
−0.127007 + 0.991902i $$0.540537\pi$$
$$354$$ 0 0
$$355$$ −3677.53 −0.549812
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 6635.25 0.975474 0.487737 0.872991i $$-0.337823\pi$$
0.487737 + 0.872991i $$0.337823\pi$$
$$360$$ 0 0
$$361$$ −6452.87 −0.940788
$$362$$ 0 0
$$363$$ 4651.79 0.672605
$$364$$ 0 0
$$365$$ −2016.62 −0.289191
$$366$$ 0 0
$$367$$ 7121.97 1.01298 0.506490 0.862246i $$-0.330943\pi$$
0.506490 + 0.862246i $$0.330943\pi$$
$$368$$ 0 0
$$369$$ −2585.25 −0.364723
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −13621.3 −1.89085 −0.945424 0.325843i $$-0.894352\pi$$
−0.945424 + 0.325843i $$0.894352\pi$$
$$374$$ 0 0
$$375$$ 4135.12 0.569432
$$376$$ 0 0
$$377$$ −180.000 −0.0245901
$$378$$ 0 0
$$379$$ 3331.51 0.451525 0.225763 0.974182i $$-0.427513\pi$$
0.225763 + 0.974182i $$0.427513\pi$$
$$380$$ 0 0
$$381$$ −702.834 −0.0945073
$$382$$ 0 0
$$383$$ −3275.75 −0.437031 −0.218515 0.975833i $$-0.570121\pi$$
−0.218515 + 0.975833i $$0.570121\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4543.87 −0.596842
$$388$$ 0 0
$$389$$ −14975.4 −1.95188 −0.975941 0.218035i $$-0.930035\pi$$
−0.975941 + 0.218035i $$0.930035\pi$$
$$390$$ 0 0
$$391$$ 5583.78 0.722209
$$392$$ 0 0
$$393$$ −7994.29 −1.02610
$$394$$ 0 0
$$395$$ −6334.18 −0.806853
$$396$$ 0 0
$$397$$ −10758.6 −1.36010 −0.680050 0.733166i $$-0.738042\pi$$
−0.680050 + 0.733166i $$0.738042\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2607.00 −0.324657 −0.162328 0.986737i $$-0.551900\pi$$
−0.162328 + 0.986737i $$0.551900\pi$$
$$402$$ 0 0
$$403$$ 273.484 0.0338045
$$404$$ 0 0
$$405$$ −545.624 −0.0669439
$$406$$ 0 0
$$407$$ −2025.69 −0.246707
$$408$$ 0 0
$$409$$ 6214.64 0.751330 0.375665 0.926756i $$-0.377414\pi$$
0.375665 + 0.926756i $$0.377414\pi$$
$$410$$ 0 0
$$411$$ −3138.92 −0.376719
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 4117.91 0.487085
$$416$$ 0 0
$$417$$ −1941.79 −0.228034
$$418$$ 0 0
$$419$$ 13111.4 1.52872 0.764358 0.644792i $$-0.223056\pi$$
0.764358 + 0.644792i $$0.223056\pi$$
$$420$$ 0 0
$$421$$ −8410.12 −0.973596 −0.486798 0.873514i $$-0.661835\pi$$
−0.486798 + 0.873514i $$0.661835\pi$$
$$422$$ 0 0
$$423$$ 1984.50 0.228108
$$424$$ 0 0
$$425$$ 3737.95 0.426628
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 279.585 0.0314650
$$430$$ 0 0
$$431$$ −7783.31 −0.869858 −0.434929 0.900465i $$-0.643226\pi$$
−0.434929 + 0.900465i $$0.643226\pi$$
$$432$$ 0 0
$$433$$ 13870.1 1.53939 0.769695 0.638412i $$-0.220409\pi$$
0.769695 + 0.638412i $$0.220409\pi$$
$$434$$ 0 0
$$435$$ 2095.21 0.230937
$$436$$ 0 0
$$437$$ −2397.05 −0.262395
$$438$$ 0 0
$$439$$ −5679.63 −0.617480 −0.308740 0.951146i $$-0.599907\pi$$
−0.308740 + 0.951146i $$0.599907\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 2316.93 0.248489 0.124245 0.992252i $$-0.460349\pi$$
0.124245 + 0.992252i $$0.460349\pi$$
$$444$$ 0 0
$$445$$ 7922.97 0.844010
$$446$$ 0 0
$$447$$ 3980.42 0.421180
$$448$$ 0 0
$$449$$ 9526.86 1.00134 0.500669 0.865639i $$-0.333088\pi$$
0.500669 + 0.865639i $$0.333088\pi$$
$$450$$ 0 0
$$451$$ −15419.7 −1.60995
$$452$$ 0 0
$$453$$ −11049.7 −1.14605
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −16627.3 −1.70195 −0.850974 0.525208i $$-0.823987\pi$$
−0.850974 + 0.525208i $$0.823987\pi$$
$$458$$ 0 0
$$459$$ −1267.50 −0.128893
$$460$$ 0 0
$$461$$ −11338.3 −1.14550 −0.572749 0.819730i $$-0.694123\pi$$
−0.572749 + 0.819730i $$0.694123\pi$$
$$462$$ 0 0
$$463$$ −9207.87 −0.924246 −0.462123 0.886816i $$-0.652912\pi$$
−0.462123 + 0.886816i $$0.652912\pi$$
$$464$$ 0 0
$$465$$ −3183.37 −0.317474
$$466$$ 0 0
$$467$$ 17215.7 1.70588 0.852941 0.522007i $$-0.174816\pi$$
0.852941 + 0.522007i $$0.174816\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −8454.74 −0.827121
$$472$$ 0 0
$$473$$ −27101.9 −2.63456
$$474$$ 0 0
$$475$$ −1604.66 −0.155004
$$476$$ 0 0
$$477$$ 2630.88 0.252536
$$478$$ 0 0
$$479$$ −10293.8 −0.981911 −0.490956 0.871185i $$-0.663352\pi$$
−0.490956 + 0.871185i $$0.663352\pi$$
$$480$$ 0 0
$$481$$ −65.5137 −0.00621033
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −9986.18 −0.934946
$$486$$ 0 0
$$487$$ −11109.0 −1.03367 −0.516833 0.856086i $$-0.672889\pi$$
−0.516833 + 0.856086i $$0.672889\pi$$
$$488$$ 0 0
$$489$$ −12319.8 −1.13931
$$490$$ 0 0
$$491$$ 6573.88 0.604226 0.302113 0.953272i $$-0.402308\pi$$
0.302113 + 0.953272i $$0.402308\pi$$
$$492$$ 0 0
$$493$$ 4867.22 0.444642
$$494$$ 0 0
$$495$$ −3254.38 −0.295502
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 16841.4 1.51087 0.755435 0.655223i $$-0.227425\pi$$
0.755435 + 0.655223i $$0.227425\pi$$
$$500$$ 0 0
$$501$$ −3136.67 −0.279712
$$502$$ 0 0
$$503$$ 18436.6 1.63429 0.817146 0.576431i $$-0.195555\pi$$
0.817146 + 0.576431i $$0.195555\pi$$
$$504$$ 0 0
$$505$$ −4162.53 −0.366793
$$506$$ 0 0
$$507$$ −6581.96 −0.576558
$$508$$ 0 0
$$509$$ 20825.2 1.81348 0.906738 0.421694i $$-0.138564\pi$$
0.906738 + 0.421694i $$0.138564\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 544.123 0.0468297
$$514$$ 0 0
$$515$$ −5501.80 −0.470754
$$516$$ 0 0
$$517$$ 11836.5 1.00691
$$518$$ 0 0
$$519$$ −5730.92 −0.484700
$$520$$ 0 0
$$521$$ 14384.7 1.20961 0.604805 0.796373i $$-0.293251\pi$$
0.604805 + 0.796373i $$0.293251\pi$$
$$522$$ 0 0
$$523$$ −6487.49 −0.542405 −0.271203 0.962522i $$-0.587421\pi$$
−0.271203 + 0.962522i $$0.587421\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −7395.05 −0.611259
$$528$$ 0 0
$$529$$ 1980.77 0.162799
$$530$$ 0 0
$$531$$ 5362.87 0.438284
$$532$$ 0 0
$$533$$ −498.695 −0.0405270
$$534$$ 0 0
$$535$$ 11352.9 0.917433
$$536$$ 0 0
$$537$$ −693.420 −0.0557230
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14416.6 1.14569 0.572844 0.819664i $$-0.305840\pi$$
0.572844 + 0.819664i $$0.305840\pi$$
$$542$$ 0 0
$$543$$ 9925.21 0.784404
$$544$$ 0 0
$$545$$ −617.006 −0.0484947
$$546$$ 0 0
$$547$$ 4881.38 0.381559 0.190780 0.981633i $$-0.438898\pi$$
0.190780 + 0.981633i $$0.438898\pi$$
$$548$$ 0 0
$$549$$ −2388.50 −0.185681
$$550$$ 0 0
$$551$$ −2089.44 −0.161549
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 762.583 0.0583240
$$556$$ 0 0
$$557$$ −3153.13 −0.239860 −0.119930 0.992782i $$-0.538267\pi$$
−0.119930 + 0.992782i $$0.538267\pi$$
$$558$$ 0 0
$$559$$ −876.514 −0.0663195
$$560$$ 0 0
$$561$$ −7560.00 −0.568954
$$562$$ 0 0
$$563$$ −6795.87 −0.508724 −0.254362 0.967109i $$-0.581866\pi$$
−0.254362 + 0.967109i $$0.581866\pi$$
$$564$$ 0 0
$$565$$ −4137.28 −0.308065
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −10708.3 −0.788955 −0.394477 0.918906i $$-0.629074\pi$$
−0.394477 + 0.918906i $$0.629074\pi$$
$$570$$ 0 0
$$571$$ −6531.66 −0.478706 −0.239353 0.970933i $$-0.576935\pi$$
−0.239353 + 0.970933i $$0.576935\pi$$
$$572$$ 0 0
$$573$$ −5402.25 −0.393861
$$574$$ 0 0
$$575$$ 9470.94 0.686896
$$576$$ 0 0
$$577$$ 17461.9 1.25987 0.629937 0.776646i $$-0.283081\pi$$
0.629937 + 0.776646i $$0.283081\pi$$
$$578$$ 0 0
$$579$$ −3850.08 −0.276345
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 15691.9 1.11473
$$584$$ 0 0
$$585$$ −105.251 −0.00743863
$$586$$ 0 0
$$587$$ −9254.42 −0.650717 −0.325358 0.945591i $$-0.605485\pi$$
−0.325358 + 0.945591i $$0.605485\pi$$
$$588$$ 0 0
$$589$$ 3174.61 0.222084
$$590$$ 0 0
$$591$$ 182.254 0.0126852
$$592$$ 0 0
$$593$$ −21019.2 −1.45557 −0.727786 0.685804i $$-0.759451\pi$$
−0.727786 + 0.685804i $$0.759451\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −11581.2 −0.793945
$$598$$ 0 0
$$599$$ 26971.7 1.83979 0.919894 0.392167i $$-0.128275\pi$$
0.919894 + 0.392167i $$0.128275\pi$$
$$600$$ 0 0
$$601$$ −26510.3 −1.79930 −0.899648 0.436616i $$-0.856177\pi$$
−0.899648 + 0.436616i $$0.856177\pi$$
$$602$$ 0 0
$$603$$ −8430.62 −0.569355
$$604$$ 0 0
$$605$$ −10445.0 −0.701899
$$606$$ 0 0
$$607$$ −29412.0 −1.96671 −0.983356 0.181687i $$-0.941844\pi$$
−0.983356 + 0.181687i $$0.941844\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 382.810 0.0253467
$$612$$ 0 0
$$613$$ −6068.64 −0.399853 −0.199926 0.979811i $$-0.564070\pi$$
−0.199926 + 0.979811i $$0.564070\pi$$
$$614$$ 0 0
$$615$$ 5804.83 0.380607
$$616$$ 0 0
$$617$$ 1904.56 0.124270 0.0621352 0.998068i $$-0.480209\pi$$
0.0621352 + 0.998068i $$0.480209\pi$$
$$618$$ 0 0
$$619$$ 406.478 0.0263937 0.0131969 0.999913i $$-0.495799\pi$$
0.0131969 + 0.999913i $$0.495799\pi$$
$$620$$ 0 0
$$621$$ −3211.50 −0.207525
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 668.244 0.0427676
$$626$$ 0 0
$$627$$ 3245.42 0.206714
$$628$$ 0 0
$$629$$ 1771.50 0.112296
$$630$$ 0 0
$$631$$ 15294.2 0.964903 0.482452 0.875923i $$-0.339746\pi$$
0.482452 + 0.875923i $$0.339746\pi$$
$$632$$ 0 0
$$633$$ −1474.91 −0.0926105
$$634$$ 0 0
$$635$$ 1578.12 0.0986233
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 4913.49 0.304186
$$640$$ 0 0
$$641$$ −15722.2 −0.968782 −0.484391 0.874852i $$-0.660959\pi$$
−0.484391 + 0.874852i $$0.660959\pi$$
$$642$$ 0 0
$$643$$ −18529.9 −1.13646 −0.568232 0.822868i $$-0.692372\pi$$
−0.568232 + 0.822868i $$0.692372\pi$$
$$644$$ 0 0
$$645$$ 10202.7 0.622836
$$646$$ 0 0
$$647$$ 8638.95 0.524934 0.262467 0.964941i $$-0.415464\pi$$
0.262467 + 0.964941i $$0.415464\pi$$
$$648$$ 0 0
$$649$$ 31986.9 1.93466
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 27884.4 1.67106 0.835528 0.549448i $$-0.185162\pi$$
0.835528 + 0.549448i $$0.185162\pi$$
$$654$$ 0 0
$$655$$ 17950.1 1.07079
$$656$$ 0 0
$$657$$ 2694.37 0.159996
$$658$$ 0 0
$$659$$ 5767.55 0.340928 0.170464 0.985364i $$-0.445473\pi$$
0.170464 + 0.985364i $$0.445473\pi$$
$$660$$ 0 0
$$661$$ 4090.12 0.240677 0.120338 0.992733i $$-0.461602\pi$$
0.120338 + 0.992733i $$0.461602\pi$$
$$662$$ 0 0
$$663$$ −244.501 −0.0143222
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 12332.2 0.715900
$$668$$ 0 0
$$669$$ −2470.13 −0.142751
$$670$$ 0 0
$$671$$ −14246.2 −0.819627
$$672$$ 0 0
$$673$$ −21667.7 −1.24106 −0.620528 0.784185i $$-0.713082\pi$$
−0.620528 + 0.784185i $$0.713082\pi$$
$$674$$ 0 0
$$675$$ −2149.87 −0.122591
$$676$$ 0 0
$$677$$ −15318.9 −0.869648 −0.434824 0.900515i $$-0.643189\pi$$
−0.434824 + 0.900515i $$0.643189\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 11276.6 0.634539
$$682$$ 0 0
$$683$$ −9981.36 −0.559189 −0.279595 0.960118i $$-0.590200\pi$$
−0.279595 + 0.960118i $$0.590200\pi$$
$$684$$ 0 0
$$685$$ 7048.02 0.393126
$$686$$ 0 0
$$687$$ −7320.29 −0.406531
$$688$$ 0 0
$$689$$ 507.497 0.0280611
$$690$$ 0 0
$$691$$ −23243.1 −1.27961 −0.639805 0.768537i $$-0.720985\pi$$
−0.639805 + 0.768537i $$0.720985\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4360.04 0.237965
$$696$$ 0 0
$$697$$ 13484.8 0.732815
$$698$$ 0 0
$$699$$ −5849.59 −0.316526
$$700$$ 0 0
$$701$$ −11295.6 −0.608598 −0.304299 0.952577i $$-0.598422\pi$$
−0.304299 + 0.952577i $$0.598422\pi$$
$$702$$ 0 0
$$703$$ −760.485 −0.0407998
$$704$$ 0 0
$$705$$ −4455.93 −0.238042
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 6868.11 0.363804 0.181902 0.983317i $$-0.441775\pi$$
0.181902 + 0.983317i $$0.441775\pi$$
$$710$$ 0 0
$$711$$ 8463.00 0.446395
$$712$$ 0 0
$$713$$ −18737.1 −0.984163
$$714$$ 0 0
$$715$$ −627.770 −0.0328353
$$716$$ 0 0
$$717$$ −8116.92 −0.422778
$$718$$ 0 0
$$719$$ 31479.5 1.63281 0.816404 0.577481i $$-0.195964\pi$$
0.816404 + 0.577481i $$0.195964\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 1812.63 0.0932400
$$724$$ 0 0
$$725$$ 8255.55 0.422901
$$726$$ 0 0
$$727$$ −26045.8 −1.32873 −0.664363 0.747410i $$-0.731297\pi$$
−0.664363 + 0.747410i $$0.731297\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 23701.0 1.19920
$$732$$ 0 0
$$733$$ −3538.40 −0.178300 −0.0891499 0.996018i $$-0.528415\pi$$
−0.0891499 + 0.996018i $$0.528415\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −50284.4 −2.51323
$$738$$ 0 0
$$739$$ −10426.9 −0.519024 −0.259512 0.965740i $$-0.583562\pi$$
−0.259512 + 0.965740i $$0.583562\pi$$
$$740$$ 0 0
$$741$$ 104.962 0.00520359
$$742$$ 0 0
$$743$$ −2518.88 −0.124372 −0.0621862 0.998065i $$-0.519807\pi$$
−0.0621862 + 0.998065i $$0.519807\pi$$
$$744$$ 0 0
$$745$$ −8937.50 −0.439523
$$746$$ 0 0
$$747$$ −5501.87 −0.269482
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −14820.3 −0.720105 −0.360053 0.932932i $$-0.617241\pi$$
−0.360053 + 0.932932i $$0.617241\pi$$
$$752$$ 0 0
$$753$$ 4895.46 0.236919
$$754$$ 0 0
$$755$$ 24810.7 1.19596
$$756$$ 0 0
$$757$$ 8892.84 0.426969 0.213485 0.976946i $$-0.431519\pi$$
0.213485 + 0.976946i $$0.431519\pi$$
$$758$$ 0 0
$$759$$ −19155.0 −0.916050
$$760$$ 0 0
$$761$$ 22795.0 1.08583 0.542916 0.839787i $$-0.317320\pi$$
0.542916 + 0.839787i $$0.317320\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 2846.00 0.134506
$$766$$ 0 0
$$767$$ 1034.50 0.0487009
$$768$$ 0 0
$$769$$ −13025.5 −0.610809 −0.305405 0.952223i $$-0.598792\pi$$
−0.305405 + 0.952223i $$0.598792\pi$$
$$770$$ 0 0
$$771$$ −21319.7 −0.995861
$$772$$ 0 0
$$773$$ 26467.0 1.23150 0.615751 0.787941i $$-0.288853\pi$$
0.615751 + 0.787941i $$0.288853\pi$$
$$774$$ 0 0
$$775$$ −12543.1 −0.581371
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −5788.87 −0.266249
$$780$$ 0 0
$$781$$ 29306.5 1.34273
$$782$$ 0 0
$$783$$ −2799.37 −0.127767
$$784$$ 0 0
$$785$$ 18984.0 0.863144
$$786$$ 0 0
$$787$$ −33216.6 −1.50450 −0.752252 0.658876i $$-0.771032\pi$$
−0.752252 + 0.658876i $$0.771032\pi$$
$$788$$ 0 0
$$789$$ −16710.7 −0.754015
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −460.743 −0.0206324
$$794$$ 0 0
$$795$$ −5907.28 −0.263534
$$796$$ 0 0
$$797$$ −15363.9 −0.682830 −0.341415 0.939913i $$-0.610906\pi$$
−0.341415 + 0.939913i $$0.610906\pi$$
$$798$$ 0 0
$$799$$ −10351.2 −0.458323
$$800$$ 0 0
$$801$$ −10585.7 −0.466953
$$802$$ 0 0
$$803$$ 16070.6 0.706249
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 11739.5 0.512083
$$808$$ 0 0
$$809$$ 26742.7 1.16220 0.581102 0.813831i $$-0.302622\pi$$
0.581102 + 0.813831i $$0.302622\pi$$
$$810$$ 0 0
$$811$$ 23651.0 1.02404 0.512022 0.858973i $$-0.328897\pi$$
0.512022 + 0.858973i $$0.328897\pi$$
$$812$$ 0 0
$$813$$ −18217.5 −0.785876
$$814$$ 0 0
$$815$$ 27662.5 1.18893
$$816$$ 0 0
$$817$$ −10174.6 −0.435697
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 16974.1 0.721558 0.360779 0.932651i $$-0.382511\pi$$
0.360779 + 0.932651i $$0.382511\pi$$
$$822$$ 0 0
$$823$$ −22328.9 −0.945730 −0.472865 0.881135i $$-0.656780\pi$$
−0.472865 + 0.881135i $$0.656780\pi$$
$$824$$ 0 0
$$825$$ −12822.9 −0.541135
$$826$$ 0 0
$$827$$ −15731.8 −0.661485 −0.330743 0.943721i $$-0.607299\pi$$
−0.330743 + 0.943721i $$0.607299\pi$$
$$828$$ 0 0
$$829$$ −38025.4 −1.59309 −0.796547 0.604576i $$-0.793342\pi$$
−0.796547 + 0.604576i $$0.793342\pi$$
$$830$$ 0 0
$$831$$ 18421.9 0.769010
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 7042.97 0.291895
$$836$$ 0 0
$$837$$ 4253.25 0.175644
$$838$$ 0 0
$$839$$ −16546.5 −0.680868 −0.340434 0.940268i $$-0.610574\pi$$
−0.340434 + 0.940268i $$0.610574\pi$$
$$840$$ 0 0
$$841$$ −13639.4 −0.559242
$$842$$ 0 0
$$843$$ −1097.25 −0.0448296
$$844$$ 0 0
$$845$$ 14778.9 0.601669
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 2131.04 0.0861452
$$850$$ 0 0
$$851$$ 4488.50 0.180803
$$852$$ 0 0
$$853$$ 13511.9 0.542365 0.271182 0.962528i $$-0.412585\pi$$
0.271182 + 0.962528i $$0.412585\pi$$
$$854$$ 0 0
$$855$$ −1221.76 −0.0488692
$$856$$ 0 0
$$857$$ 17178.2 0.684711 0.342355 0.939570i $$-0.388775\pi$$
0.342355 + 0.939570i $$0.388775\pi$$
$$858$$ 0 0
$$859$$ 35914.3 1.42652 0.713260 0.700899i $$-0.247218\pi$$
0.713260 + 0.700899i $$0.247218\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 882.508 0.0348099 0.0174049 0.999849i $$-0.494460\pi$$
0.0174049 + 0.999849i $$0.494460\pi$$
$$864$$ 0 0
$$865$$ 12868.0 0.505810
$$866$$ 0 0
$$867$$ −8127.67 −0.318374
$$868$$ 0 0
$$869$$ 50477.6 1.97046
$$870$$ 0 0
$$871$$ −1626.27 −0.0632652
$$872$$ 0 0
$$873$$ 13342.4 0.517263
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 17420.1 0.670737 0.335368 0.942087i $$-0.391139\pi$$
0.335368 + 0.942087i $$0.391139\pi$$
$$878$$ 0 0
$$879$$ 889.120 0.0341175
$$880$$ 0 0
$$881$$ 38097.8 1.45692 0.728460 0.685088i $$-0.240236\pi$$
0.728460 + 0.685088i $$0.240236\pi$$
$$882$$ 0 0
$$883$$ 13870.6 0.528633 0.264317 0.964436i $$-0.414854\pi$$
0.264317 + 0.964436i $$0.414854\pi$$
$$884$$ 0 0
$$885$$ −12041.6 −0.457372
$$886$$ 0 0
$$887$$ 17467.8 0.661230 0.330615 0.943766i $$-0.392744\pi$$
0.330615 + 0.943766i $$0.392744\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4348.12 0.163488
$$892$$ 0 0
$$893$$ 4443.67 0.166519
$$894$$ 0 0
$$895$$ 1556.98 0.0581499
$$896$$ 0 0
$$897$$ −619.499 −0.0230596
$$898$$ 0 0
$$899$$ −16332.6 −0.605919
$$900$$ 0 0
$$901$$ −13722.8 −0.507405
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −22285.7 −0.818567
$$906$$ 0 0
$$907$$ 28536.3 1.04469 0.522344 0.852735i $$-0.325058\pi$$
0.522344 + 0.852735i $$0.325058\pi$$
$$908$$ 0 0
$$909$$ 5561.49 0.202930
$$910$$ 0 0
$$911$$ 50235.7 1.82698 0.913492 0.406857i $$-0.133375\pi$$
0.913492 + 0.406857i $$0.133375\pi$$
$$912$$ 0 0
$$913$$ −32815.9 −1.18954
$$914$$ 0 0
$$915$$ 5363.07 0.193768
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −5317.08 −0.190854 −0.0954268 0.995436i $$-0.530422\pi$$
−0.0954268 + 0.995436i $$0.530422\pi$$
$$920$$ 0 0
$$921$$ −25790.6 −0.922725
$$922$$ 0 0
$$923$$ 947.814 0.0338003
$$924$$ 0 0
$$925$$ 3004.73 0.106805
$$926$$ 0 0
$$927$$ 7350.87 0.260447
$$928$$ 0 0
$$929$$ 25493.1 0.900325 0.450162 0.892947i $$-0.351366\pi$$
0.450162 + 0.892947i $$0.351366\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 31425.9 1.10272
$$934$$ 0 0
$$935$$ 16975.0 0.593734
$$936$$ 0 0
$$937$$ 12691.4 0.442486 0.221243 0.975219i $$-0.428989\pi$$
0.221243 + 0.975219i $$0.428989\pi$$
$$938$$ 0 0
$$939$$ 16489.6 0.573074
$$940$$ 0 0
$$941$$ −15378.4 −0.532753 −0.266377 0.963869i $$-0.585826\pi$$
−0.266377 + 0.963869i $$0.585826\pi$$
$$942$$ 0 0
$$943$$ 34166.8 1.17988
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −50442.9 −1.73091 −0.865455 0.500986i $$-0.832971\pi$$
−0.865455 + 0.500986i $$0.832971\pi$$
$$948$$ 0 0
$$949$$ 519.745 0.0177783
$$950$$ 0 0
$$951$$ 19669.1 0.670678
$$952$$ 0 0
$$953$$ 31787.3 1.08047 0.540237 0.841513i $$-0.318335\pi$$
0.540237 + 0.841513i $$0.318335\pi$$
$$954$$ 0 0
$$955$$ 12130.0 0.411014
$$956$$ 0 0
$$957$$ −16696.9 −0.563984
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −4975.99 −0.167030
$$962$$ 0 0
$$963$$ −15168.4 −0.507574
$$964$$ 0 0
$$965$$ 8644.84 0.288381
$$966$$ 0 0
$$967$$ −1005.61 −0.0334417 −0.0167209 0.999860i $$-0.505323\pi$$
−0.0167209 + 0.999860i $$0.505323\pi$$
$$968$$ 0 0
$$969$$ −2838.17 −0.0940921
$$970$$ 0 0
$$971$$ −7070.91 −0.233693 −0.116847 0.993150i $$-0.537279\pi$$
−0.116847 + 0.993150i $$0.537279\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −414.711 −0.0136219
$$976$$ 0 0
$$977$$ −11097.1 −0.363384 −0.181692 0.983355i $$-0.558157\pi$$
−0.181692 + 0.983355i $$0.558157\pi$$
$$978$$ 0 0
$$979$$ −63138.7 −2.06121
$$980$$ 0 0
$$981$$ 824.371 0.0268299
$$982$$ 0 0
$$983$$ 17775.5 0.576755 0.288377 0.957517i $$-0.406884\pi$$
0.288377 + 0.957517i $$0.406884\pi$$
$$984$$ 0 0
$$985$$ −409.228 −0.0132376
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 60052.0 1.93078
$$990$$ 0 0
$$991$$ −4691.86 −0.150395 −0.0751977 0.997169i $$-0.523959\pi$$
−0.0751977 + 0.997169i $$0.523959\pi$$
$$992$$ 0 0
$$993$$ −307.129 −0.00981515
$$994$$ 0 0
$$995$$ 26004.0 0.828523
$$996$$ 0 0
$$997$$ −8471.39 −0.269099 −0.134549 0.990907i $$-0.542959\pi$$
−0.134549 + 0.990907i $$0.542959\pi$$
$$998$$ 0 0
$$999$$ −1018.87 −0.0322680
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.cc.1.1 2
4.3 odd 2 1176.4.a.r.1.1 2
7.2 even 3 336.4.q.g.193.2 4
7.4 even 3 336.4.q.g.289.2 4
7.6 odd 2 2352.4.a.bo.1.2 2
28.11 odd 6 168.4.q.d.121.2 yes 4
28.23 odd 6 168.4.q.d.25.2 4
28.27 even 2 1176.4.a.u.1.2 2
84.11 even 6 504.4.s.f.289.1 4
84.23 even 6 504.4.s.f.361.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.d.25.2 4 28.23 odd 6
168.4.q.d.121.2 yes 4 28.11 odd 6
336.4.q.g.193.2 4 7.2 even 3
336.4.q.g.289.2 4 7.4 even 3
504.4.s.f.289.1 4 84.11 even 6
504.4.s.f.361.1 4 84.23 even 6
1176.4.a.r.1.1 2 4.3 odd 2
1176.4.a.u.1.2 2 28.27 even 2
2352.4.a.bo.1.2 2 7.6 odd 2
2352.4.a.cc.1.1 2 1.1 even 1 trivial