# Properties

 Label 2352.4.a.cl.1.2 Level $2352$ Weight $4$ Character 2352.1 Self dual yes Analytic conductor $138.772$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.136768.1 Defining polynomial: $$x^{4} - 2 x^{3} - 23 x^{2} + 18 x + 119$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 7$$ Twist minimal: no (minimal twist has level 588) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.51732$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -10.6550 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -10.6550 q^{5} +9.00000 q^{9} +6.65399 q^{11} +75.9335 q^{13} +31.9651 q^{15} -104.287 q^{17} -85.4943 q^{19} +68.6733 q^{23} -11.4700 q^{25} -27.0000 q^{27} +87.7843 q^{29} -62.7683 q^{31} -19.9620 q^{33} +42.2093 q^{37} -227.800 q^{39} +313.904 q^{41} -306.591 q^{43} -95.8954 q^{45} -215.081 q^{47} +312.861 q^{51} +525.024 q^{53} -70.8986 q^{55} +256.483 q^{57} -360.491 q^{59} +800.726 q^{61} -809.075 q^{65} +40.2286 q^{67} -206.020 q^{69} +298.781 q^{71} +517.126 q^{73} +34.4099 q^{75} +1222.47 q^{79} +81.0000 q^{81} -1328.55 q^{83} +1111.18 q^{85} -263.353 q^{87} -639.938 q^{89} +188.305 q^{93} +910.946 q^{95} +1425.65 q^{97} +59.8859 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{3} + 36q^{9} + O(q^{10})$$ $$4q - 12q^{3} + 36q^{9} + 48q^{17} - 192q^{19} - 192q^{23} + 324q^{25} - 108q^{27} + 96q^{29} - 48q^{31} + 256q^{37} + 1008q^{41} + 112q^{43} - 864q^{47} - 144q^{51} - 648q^{53} - 2352q^{55} + 576q^{57} - 336q^{59} + 960q^{61} - 360q^{65} - 720q^{67} + 576q^{69} + 1344q^{71} + 672q^{73} - 972q^{75} + 1984q^{79} + 324q^{81} - 3120q^{83} + 680q^{85} - 288q^{87} + 2160q^{89} + 144q^{93} + 3744q^{95} + 2016q^{97} + O(q^{100})$$

## 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
$$5$$ −10.6550 −0.953016 −0.476508 0.879170i $$-0.658098\pi$$
−0.476508 + 0.879170i $$0.658098\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 6.65399 0.182387 0.0911933 0.995833i $$-0.470932\pi$$
0.0911933 + 0.995833i $$0.470932\pi$$
$$12$$ 0 0
$$13$$ 75.9335 1.62001 0.810006 0.586422i $$-0.199464\pi$$
0.810006 + 0.586422i $$0.199464\pi$$
$$14$$ 0 0
$$15$$ 31.9651 0.550224
$$16$$ 0 0
$$17$$ −104.287 −1.48784 −0.743921 0.668268i $$-0.767036\pi$$
−0.743921 + 0.668268i $$0.767036\pi$$
$$18$$ 0 0
$$19$$ −85.4943 −1.03230 −0.516151 0.856498i $$-0.672636\pi$$
−0.516151 + 0.856498i $$0.672636\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 68.6733 0.622581 0.311291 0.950315i $$-0.399239\pi$$
0.311291 + 0.950315i $$0.399239\pi$$
$$24$$ 0 0
$$25$$ −11.4700 −0.0917596
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 87.7843 0.562108 0.281054 0.959692i $$-0.409316\pi$$
0.281054 + 0.959692i $$0.409316\pi$$
$$30$$ 0 0
$$31$$ −62.7683 −0.363662 −0.181831 0.983330i $$-0.558202\pi$$
−0.181831 + 0.983330i $$0.558202\pi$$
$$32$$ 0 0
$$33$$ −19.9620 −0.105301
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 42.2093 0.187545 0.0937726 0.995594i $$-0.470107\pi$$
0.0937726 + 0.995594i $$0.470107\pi$$
$$38$$ 0 0
$$39$$ −227.800 −0.935314
$$40$$ 0 0
$$41$$ 313.904 1.19570 0.597848 0.801609i $$-0.296023\pi$$
0.597848 + 0.801609i $$0.296023\pi$$
$$42$$ 0 0
$$43$$ −306.591 −1.08732 −0.543659 0.839306i $$-0.682962\pi$$
−0.543659 + 0.839306i $$0.682962\pi$$
$$44$$ 0 0
$$45$$ −95.8954 −0.317672
$$46$$ 0 0
$$47$$ −215.081 −0.667508 −0.333754 0.942660i $$-0.608315\pi$$
−0.333754 + 0.942660i $$0.608315\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 312.861 0.859006
$$52$$ 0 0
$$53$$ 525.024 1.36071 0.680354 0.732884i $$-0.261826\pi$$
0.680354 + 0.732884i $$0.261826\pi$$
$$54$$ 0 0
$$55$$ −70.8986 −0.173818
$$56$$ 0 0
$$57$$ 256.483 0.596000
$$58$$ 0 0
$$59$$ −360.491 −0.795457 −0.397729 0.917503i $$-0.630201\pi$$
−0.397729 + 0.917503i $$0.630201\pi$$
$$60$$ 0 0
$$61$$ 800.726 1.68070 0.840348 0.542048i $$-0.182351\pi$$
0.840348 + 0.542048i $$0.182351\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −809.075 −1.54390
$$66$$ 0 0
$$67$$ 40.2286 0.0733538 0.0366769 0.999327i $$-0.488323\pi$$
0.0366769 + 0.999327i $$0.488323\pi$$
$$68$$ 0 0
$$69$$ −206.020 −0.359447
$$70$$ 0 0
$$71$$ 298.781 0.499419 0.249709 0.968321i $$-0.419665\pi$$
0.249709 + 0.968321i $$0.419665\pi$$
$$72$$ 0 0
$$73$$ 517.126 0.829110 0.414555 0.910024i $$-0.363937\pi$$
0.414555 + 0.910024i $$0.363937\pi$$
$$74$$ 0 0
$$75$$ 34.4099 0.0529774
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1222.47 1.74099 0.870494 0.492178i $$-0.163799\pi$$
0.870494 + 0.492178i $$0.163799\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −1328.55 −1.75696 −0.878479 0.477782i $$-0.841441\pi$$
−0.878479 + 0.477782i $$0.841441\pi$$
$$84$$ 0 0
$$85$$ 1111.18 1.41794
$$86$$ 0 0
$$87$$ −263.353 −0.324533
$$88$$ 0 0
$$89$$ −639.938 −0.762172 −0.381086 0.924540i $$-0.624450\pi$$
−0.381086 + 0.924540i $$0.624450\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 188.305 0.209960
$$94$$ 0 0
$$95$$ 910.946 0.983801
$$96$$ 0 0
$$97$$ 1425.65 1.49230 0.746149 0.665779i $$-0.231901\pi$$
0.746149 + 0.665779i $$0.231901\pi$$
$$98$$ 0 0
$$99$$ 59.8859 0.0607956
$$100$$ 0 0
$$101$$ −992.170 −0.977471 −0.488736 0.872432i $$-0.662542\pi$$
−0.488736 + 0.872432i $$0.662542\pi$$
$$102$$ 0 0
$$103$$ −267.572 −0.255967 −0.127984 0.991776i $$-0.540851\pi$$
−0.127984 + 0.991776i $$0.540851\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1536.62 1.38833 0.694164 0.719817i $$-0.255774\pi$$
0.694164 + 0.719817i $$0.255774\pi$$
$$108$$ 0 0
$$109$$ 998.820 0.877703 0.438852 0.898560i $$-0.355385\pi$$
0.438852 + 0.898560i $$0.355385\pi$$
$$110$$ 0 0
$$111$$ −126.628 −0.108279
$$112$$ 0 0
$$113$$ 939.006 0.781719 0.390860 0.920450i $$-0.372178\pi$$
0.390860 + 0.920450i $$0.372178\pi$$
$$114$$ 0 0
$$115$$ −731.717 −0.593330
$$116$$ 0 0
$$117$$ 683.401 0.540004
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1286.72 −0.966735
$$122$$ 0 0
$$123$$ −941.712 −0.690336
$$124$$ 0 0
$$125$$ 1454.09 1.04046
$$126$$ 0 0
$$127$$ 1621.27 1.13279 0.566397 0.824133i $$-0.308337\pi$$
0.566397 + 0.824133i $$0.308337\pi$$
$$128$$ 0 0
$$129$$ 919.773 0.627764
$$130$$ 0 0
$$131$$ −1518.43 −1.01271 −0.506357 0.862324i $$-0.669008\pi$$
−0.506357 + 0.862324i $$0.669008\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 287.686 0.183408
$$136$$ 0 0
$$137$$ −2484.99 −1.54969 −0.774844 0.632152i $$-0.782172\pi$$
−0.774844 + 0.632152i $$0.782172\pi$$
$$138$$ 0 0
$$139$$ 1655.36 1.01011 0.505057 0.863086i $$-0.331471\pi$$
0.505057 + 0.863086i $$0.331471\pi$$
$$140$$ 0 0
$$141$$ 645.244 0.385386
$$142$$ 0 0
$$143$$ 505.260 0.295469
$$144$$ 0 0
$$145$$ −935.346 −0.535698
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 350.191 0.192542 0.0962709 0.995355i $$-0.469308\pi$$
0.0962709 + 0.995355i $$0.469308\pi$$
$$150$$ 0 0
$$151$$ −3338.14 −1.79903 −0.899516 0.436888i $$-0.856081\pi$$
−0.899516 + 0.436888i $$0.856081\pi$$
$$152$$ 0 0
$$153$$ −938.583 −0.495947
$$154$$ 0 0
$$155$$ 668.799 0.346576
$$156$$ 0 0
$$157$$ −1743.67 −0.886371 −0.443185 0.896430i $$-0.646152\pi$$
−0.443185 + 0.896430i $$0.646152\pi$$
$$158$$ 0 0
$$159$$ −1575.07 −0.785605
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −3431.56 −1.64896 −0.824480 0.565891i $$-0.808532\pi$$
−0.824480 + 0.565891i $$0.808532\pi$$
$$164$$ 0 0
$$165$$ 212.696 0.100354
$$166$$ 0 0
$$167$$ −3381.05 −1.56667 −0.783335 0.621600i $$-0.786483\pi$$
−0.783335 + 0.621600i $$0.786483\pi$$
$$168$$ 0 0
$$169$$ 3568.89 1.62444
$$170$$ 0 0
$$171$$ −769.449 −0.344101
$$172$$ 0 0
$$173$$ −1341.00 −0.589332 −0.294666 0.955600i $$-0.595208\pi$$
−0.294666 + 0.955600i $$0.595208\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1081.47 0.459257
$$178$$ 0 0
$$179$$ −1125.15 −0.469818 −0.234909 0.972017i $$-0.575479\pi$$
−0.234909 + 0.972017i $$0.575479\pi$$
$$180$$ 0 0
$$181$$ −3535.04 −1.45170 −0.725848 0.687855i $$-0.758553\pi$$
−0.725848 + 0.687855i $$0.758553\pi$$
$$182$$ 0 0
$$183$$ −2402.18 −0.970350
$$184$$ 0 0
$$185$$ −449.742 −0.178734
$$186$$ 0 0
$$187$$ −693.925 −0.271363
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2639.89 −1.00008 −0.500041 0.866001i $$-0.666682\pi$$
−0.500041 + 0.866001i $$0.666682\pi$$
$$192$$ 0 0
$$193$$ 1047.05 0.390510 0.195255 0.980752i $$-0.437447\pi$$
0.195255 + 0.980752i $$0.437447\pi$$
$$194$$ 0 0
$$195$$ 2427.22 0.891370
$$196$$ 0 0
$$197$$ −4585.45 −1.65837 −0.829187 0.558972i $$-0.811196\pi$$
−0.829187 + 0.558972i $$0.811196\pi$$
$$198$$ 0 0
$$199$$ −830.742 −0.295928 −0.147964 0.988993i $$-0.547272\pi$$
−0.147964 + 0.988993i $$0.547272\pi$$
$$200$$ 0 0
$$201$$ −120.686 −0.0423508
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −3344.66 −1.13952
$$206$$ 0 0
$$207$$ 618.059 0.207527
$$208$$ 0 0
$$209$$ −568.878 −0.188278
$$210$$ 0 0
$$211$$ 2630.08 0.858114 0.429057 0.903277i $$-0.358846\pi$$
0.429057 + 0.903277i $$0.358846\pi$$
$$212$$ 0 0
$$213$$ −896.342 −0.288340
$$214$$ 0 0
$$215$$ 3266.74 1.03623
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −1551.38 −0.478687
$$220$$ 0 0
$$221$$ −7918.87 −2.41032
$$222$$ 0 0
$$223$$ 863.988 0.259448 0.129724 0.991550i $$-0.458591\pi$$
0.129724 + 0.991550i $$0.458591\pi$$
$$224$$ 0 0
$$225$$ −103.230 −0.0305865
$$226$$ 0 0
$$227$$ 4160.36 1.21644 0.608221 0.793767i $$-0.291883\pi$$
0.608221 + 0.793767i $$0.291883\pi$$
$$228$$ 0 0
$$229$$ −181.210 −0.0522914 −0.0261457 0.999658i $$-0.508323\pi$$
−0.0261457 + 0.999658i $$0.508323\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 2150.45 0.604637 0.302319 0.953207i $$-0.402239\pi$$
0.302319 + 0.953207i $$0.402239\pi$$
$$234$$ 0 0
$$235$$ 2291.70 0.636146
$$236$$ 0 0
$$237$$ −3667.40 −1.00516
$$238$$ 0 0
$$239$$ 6939.47 1.87815 0.939073 0.343719i $$-0.111687\pi$$
0.939073 + 0.343719i $$0.111687\pi$$
$$240$$ 0 0
$$241$$ −206.170 −0.0551060 −0.0275530 0.999620i $$-0.508772\pi$$
−0.0275530 + 0.999620i $$0.508772\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6491.88 −1.67234
$$248$$ 0 0
$$249$$ 3985.65 1.01438
$$250$$ 0 0
$$251$$ −5011.93 −1.26036 −0.630180 0.776449i $$-0.717019\pi$$
−0.630180 + 0.776449i $$0.717019\pi$$
$$252$$ 0 0
$$253$$ 456.951 0.113551
$$254$$ 0 0
$$255$$ −3333.55 −0.818647
$$256$$ 0 0
$$257$$ −5863.08 −1.42307 −0.711535 0.702651i $$-0.752000\pi$$
−0.711535 + 0.702651i $$0.752000\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 790.059 0.187369
$$262$$ 0 0
$$263$$ 5639.56 1.32224 0.661122 0.750279i $$-0.270081\pi$$
0.661122 + 0.750279i $$0.270081\pi$$
$$264$$ 0 0
$$265$$ −5594.15 −1.29678
$$266$$ 0 0
$$267$$ 1919.81 0.440040
$$268$$ 0 0
$$269$$ −2976.63 −0.674679 −0.337339 0.941383i $$-0.609527\pi$$
−0.337339 + 0.941383i $$0.609527\pi$$
$$270$$ 0 0
$$271$$ 2807.33 0.629275 0.314637 0.949212i $$-0.398117\pi$$
0.314637 + 0.949212i $$0.398117\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −76.3210 −0.0167357
$$276$$ 0 0
$$277$$ −1918.39 −0.416118 −0.208059 0.978116i $$-0.566715\pi$$
−0.208059 + 0.978116i $$0.566715\pi$$
$$278$$ 0 0
$$279$$ −564.915 −0.121221
$$280$$ 0 0
$$281$$ 5209.50 1.10595 0.552977 0.833197i $$-0.313492\pi$$
0.552977 + 0.833197i $$0.313492\pi$$
$$282$$ 0 0
$$283$$ −7496.70 −1.57467 −0.787337 0.616523i $$-0.788541\pi$$
−0.787337 + 0.616523i $$0.788541\pi$$
$$284$$ 0 0
$$285$$ −2732.84 −0.567998
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 5962.78 1.21367
$$290$$ 0 0
$$291$$ −4276.95 −0.861579
$$292$$ 0 0
$$293$$ −3358.94 −0.669732 −0.334866 0.942266i $$-0.608691\pi$$
−0.334866 + 0.942266i $$0.608691\pi$$
$$294$$ 0 0
$$295$$ 3841.05 0.758084
$$296$$ 0 0
$$297$$ −179.658 −0.0351003
$$298$$ 0 0
$$299$$ 5214.60 1.00859
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 2976.51 0.564343
$$304$$ 0 0
$$305$$ −8531.77 −1.60173
$$306$$ 0 0
$$307$$ 7330.92 1.36286 0.681429 0.731884i $$-0.261359\pi$$
0.681429 + 0.731884i $$0.261359\pi$$
$$308$$ 0 0
$$309$$ 802.716 0.147783
$$310$$ 0 0
$$311$$ 2038.53 0.371687 0.185843 0.982579i $$-0.440498\pi$$
0.185843 + 0.982579i $$0.440498\pi$$
$$312$$ 0 0
$$313$$ 4138.87 0.747421 0.373711 0.927545i $$-0.378085\pi$$
0.373711 + 0.927545i $$0.378085\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −7552.25 −1.33810 −0.669049 0.743219i $$-0.733298\pi$$
−0.669049 + 0.743219i $$0.733298\pi$$
$$318$$ 0 0
$$319$$ 584.116 0.102521
$$320$$ 0 0
$$321$$ −4609.87 −0.801552
$$322$$ 0 0
$$323$$ 8915.94 1.53590
$$324$$ 0 0
$$325$$ −870.953 −0.148652
$$326$$ 0 0
$$327$$ −2996.46 −0.506742
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −5577.70 −0.926218 −0.463109 0.886301i $$-0.653266\pi$$
−0.463109 + 0.886301i $$0.653266\pi$$
$$332$$ 0 0
$$333$$ 379.884 0.0625151
$$334$$ 0 0
$$335$$ −428.637 −0.0699073
$$336$$ 0 0
$$337$$ 4467.16 0.722083 0.361041 0.932550i $$-0.382421\pi$$
0.361041 + 0.932550i $$0.382421\pi$$
$$338$$ 0 0
$$339$$ −2817.02 −0.451326
$$340$$ 0 0
$$341$$ −417.660 −0.0663271
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 2195.15 0.342559
$$346$$ 0 0
$$347$$ −9788.41 −1.51432 −0.757161 0.653229i $$-0.773414\pi$$
−0.757161 + 0.653229i $$0.773414\pi$$
$$348$$ 0 0
$$349$$ −4746.95 −0.728075 −0.364038 0.931384i $$-0.618602\pi$$
−0.364038 + 0.931384i $$0.618602\pi$$
$$350$$ 0 0
$$351$$ −2050.20 −0.311771
$$352$$ 0 0
$$353$$ −9434.66 −1.42254 −0.711269 0.702920i $$-0.751879\pi$$
−0.711269 + 0.702920i $$0.751879\pi$$
$$354$$ 0 0
$$355$$ −3183.52 −0.475954
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −11307.5 −1.66237 −0.831183 0.555999i $$-0.812336\pi$$
−0.831183 + 0.555999i $$0.812336\pi$$
$$360$$ 0 0
$$361$$ 450.275 0.0656474
$$362$$ 0 0
$$363$$ 3860.17 0.558145
$$364$$ 0 0
$$365$$ −5510.00 −0.790155
$$366$$ 0 0
$$367$$ −6089.61 −0.866144 −0.433072 0.901359i $$-0.642570\pi$$
−0.433072 + 0.901359i $$0.642570\pi$$
$$368$$ 0 0
$$369$$ 2825.14 0.398565
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −2601.84 −0.361175 −0.180588 0.983559i $$-0.557800\pi$$
−0.180588 + 0.983559i $$0.557800\pi$$
$$374$$ 0 0
$$375$$ −4362.28 −0.600713
$$376$$ 0 0
$$377$$ 6665.77 0.910622
$$378$$ 0 0
$$379$$ −10416.2 −1.41173 −0.705865 0.708347i $$-0.749441\pi$$
−0.705865 + 0.708347i $$0.749441\pi$$
$$380$$ 0 0
$$381$$ −4863.82 −0.654018
$$382$$ 0 0
$$383$$ 1664.46 0.222063 0.111031 0.993817i $$-0.464585\pi$$
0.111031 + 0.993817i $$0.464585\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −2759.32 −0.362440
$$388$$ 0 0
$$389$$ −1334.16 −0.173893 −0.0869465 0.996213i $$-0.527711\pi$$
−0.0869465 + 0.996213i $$0.527711\pi$$
$$390$$ 0 0
$$391$$ −7161.73 −0.926302
$$392$$ 0 0
$$393$$ 4555.28 0.584691
$$394$$ 0 0
$$395$$ −13025.4 −1.65919
$$396$$ 0 0
$$397$$ 4444.88 0.561920 0.280960 0.959720i $$-0.409347\pi$$
0.280960 + 0.959720i $$0.409347\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2649.82 −0.329989 −0.164995 0.986294i $$-0.552761\pi$$
−0.164995 + 0.986294i $$0.552761\pi$$
$$402$$ 0 0
$$403$$ −4766.21 −0.589137
$$404$$ 0 0
$$405$$ −863.059 −0.105891
$$406$$ 0 0
$$407$$ 280.860 0.0342057
$$408$$ 0 0
$$409$$ −9592.66 −1.15972 −0.579862 0.814715i $$-0.696893\pi$$
−0.579862 + 0.814715i $$0.696893\pi$$
$$410$$ 0 0
$$411$$ 7454.98 0.894713
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 14155.8 1.67441
$$416$$ 0 0
$$417$$ −4966.09 −0.583190
$$418$$ 0 0
$$419$$ −16850.1 −1.96463 −0.982317 0.187224i $$-0.940051\pi$$
−0.982317 + 0.187224i $$0.940051\pi$$
$$420$$ 0 0
$$421$$ 1691.10 0.195770 0.0978849 0.995198i $$-0.468792\pi$$
0.0978849 + 0.995198i $$0.468792\pi$$
$$422$$ 0 0
$$423$$ −1935.73 −0.222503
$$424$$ 0 0
$$425$$ 1196.17 0.136524
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −1515.78 −0.170589
$$430$$ 0 0
$$431$$ 6789.34 0.758773 0.379386 0.925238i $$-0.376135\pi$$
0.379386 + 0.925238i $$0.376135\pi$$
$$432$$ 0 0
$$433$$ 10386.6 1.15276 0.576382 0.817180i $$-0.304464\pi$$
0.576382 + 0.817180i $$0.304464\pi$$
$$434$$ 0 0
$$435$$ 2806.04 0.309286
$$436$$ 0 0
$$437$$ −5871.17 −0.642692
$$438$$ 0 0
$$439$$ 4212.26 0.457950 0.228975 0.973432i $$-0.426463\pi$$
0.228975 + 0.973432i $$0.426463\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 1809.96 0.194117 0.0970585 0.995279i $$-0.469057\pi$$
0.0970585 + 0.995279i $$0.469057\pi$$
$$444$$ 0 0
$$445$$ 6818.57 0.726362
$$446$$ 0 0
$$447$$ −1050.57 −0.111164
$$448$$ 0 0
$$449$$ 1768.29 0.185859 0.0929297 0.995673i $$-0.470377\pi$$
0.0929297 + 0.995673i $$0.470377\pi$$
$$450$$ 0 0
$$451$$ 2088.71 0.218079
$$452$$ 0 0
$$453$$ 10014.4 1.03867
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4372.61 0.447575 0.223788 0.974638i $$-0.428158\pi$$
0.223788 + 0.974638i $$0.428158\pi$$
$$458$$ 0 0
$$459$$ 2815.75 0.286335
$$460$$ 0 0
$$461$$ −1164.34 −0.117633 −0.0588165 0.998269i $$-0.518733\pi$$
−0.0588165 + 0.998269i $$0.518733\pi$$
$$462$$ 0 0
$$463$$ 14893.9 1.49498 0.747491 0.664272i $$-0.231258\pi$$
0.747491 + 0.664272i $$0.231258\pi$$
$$464$$ 0 0
$$465$$ −2006.40 −0.200096
$$466$$ 0 0
$$467$$ −19160.3 −1.89857 −0.949285 0.314418i $$-0.898191\pi$$
−0.949285 + 0.314418i $$0.898191\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 5231.02 0.511746
$$472$$ 0 0
$$473$$ −2040.05 −0.198312
$$474$$ 0 0
$$475$$ 980.616 0.0947237
$$476$$ 0 0
$$477$$ 4725.21 0.453569
$$478$$ 0 0
$$479$$ 9406.28 0.897253 0.448626 0.893719i $$-0.351913\pi$$
0.448626 + 0.893719i $$0.351913\pi$$
$$480$$ 0 0
$$481$$ 3205.10 0.303825
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −15190.4 −1.42218
$$486$$ 0 0
$$487$$ −12460.8 −1.15945 −0.579725 0.814812i $$-0.696840\pi$$
−0.579725 + 0.814812i $$0.696840\pi$$
$$488$$ 0 0
$$489$$ 10294.7 0.952028
$$490$$ 0 0
$$491$$ −12868.0 −1.18274 −0.591369 0.806401i $$-0.701412\pi$$
−0.591369 + 0.806401i $$0.701412\pi$$
$$492$$ 0 0
$$493$$ −9154.76 −0.836328
$$494$$ 0 0
$$495$$ −638.087 −0.0579392
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 13705.0 1.22950 0.614750 0.788722i $$-0.289257\pi$$
0.614750 + 0.788722i $$0.289257\pi$$
$$500$$ 0 0
$$501$$ 10143.2 0.904517
$$502$$ 0 0
$$503$$ −10126.1 −0.897616 −0.448808 0.893628i $$-0.648151\pi$$
−0.448808 + 0.893628i $$0.648151\pi$$
$$504$$ 0 0
$$505$$ 10571.6 0.931546
$$506$$ 0 0
$$507$$ −10706.7 −0.937870
$$508$$ 0 0
$$509$$ −6236.06 −0.543042 −0.271521 0.962432i $$-0.587527\pi$$
−0.271521 + 0.962432i $$0.587527\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 2308.35 0.198667
$$514$$ 0 0
$$515$$ 2850.99 0.243941
$$516$$ 0 0
$$517$$ −1431.15 −0.121744
$$518$$ 0 0
$$519$$ 4023.00 0.340251
$$520$$ 0 0
$$521$$ 17016.8 1.43094 0.715468 0.698645i $$-0.246213\pi$$
0.715468 + 0.698645i $$0.246213\pi$$
$$522$$ 0 0
$$523$$ 5814.62 0.486148 0.243074 0.970008i $$-0.421844\pi$$
0.243074 + 0.970008i $$0.421844\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6545.92 0.541071
$$528$$ 0 0
$$529$$ −7450.98 −0.612393
$$530$$ 0 0
$$531$$ −3244.42 −0.265152
$$532$$ 0 0
$$533$$ 23835.8 1.93704
$$534$$ 0 0
$$535$$ −16372.8 −1.32310
$$536$$ 0 0
$$537$$ 3375.44 0.271250
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −10069.2 −0.800199 −0.400100 0.916472i $$-0.631025\pi$$
−0.400100 + 0.916472i $$0.631025\pi$$
$$542$$ 0 0
$$543$$ 10605.1 0.838137
$$544$$ 0 0
$$545$$ −10642.5 −0.836466
$$546$$ 0 0
$$547$$ 7437.51 0.581362 0.290681 0.956820i $$-0.406118\pi$$
0.290681 + 0.956820i $$0.406118\pi$$
$$548$$ 0 0
$$549$$ 7206.53 0.560232
$$550$$ 0 0
$$551$$ −7505.06 −0.580265
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 1349.23 0.103192
$$556$$ 0 0
$$557$$ 21787.1 1.65736 0.828680 0.559723i $$-0.189092\pi$$
0.828680 + 0.559723i $$0.189092\pi$$
$$558$$ 0 0
$$559$$ −23280.5 −1.76147
$$560$$ 0 0
$$561$$ 2081.77 0.156671
$$562$$ 0 0
$$563$$ −2572.48 −0.192570 −0.0962851 0.995354i $$-0.530696\pi$$
−0.0962851 + 0.995354i $$0.530696\pi$$
$$564$$ 0 0
$$565$$ −10005.2 −0.744991
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −17580.6 −1.29528 −0.647642 0.761945i $$-0.724245\pi$$
−0.647642 + 0.761945i $$0.724245\pi$$
$$570$$ 0 0
$$571$$ −7220.75 −0.529210 −0.264605 0.964357i $$-0.585242\pi$$
−0.264605 + 0.964357i $$0.585242\pi$$
$$572$$ 0 0
$$573$$ 7919.67 0.577398
$$574$$ 0 0
$$575$$ −787.679 −0.0571278
$$576$$ 0 0
$$577$$ 11155.1 0.804839 0.402419 0.915455i $$-0.368169\pi$$
0.402419 + 0.915455i $$0.368169\pi$$
$$578$$ 0 0
$$579$$ −3141.16 −0.225461
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 3493.50 0.248175
$$584$$ 0 0
$$585$$ −7281.67 −0.514633
$$586$$ 0 0
$$587$$ 15216.2 1.06992 0.534958 0.844879i $$-0.320327\pi$$
0.534958 + 0.844879i $$0.320327\pi$$
$$588$$ 0 0
$$589$$ 5366.33 0.375409
$$590$$ 0 0
$$591$$ 13756.3 0.957462
$$592$$ 0 0
$$593$$ 3843.89 0.266188 0.133094 0.991103i $$-0.457509\pi$$
0.133094 + 0.991103i $$0.457509\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 2492.23 0.170854
$$598$$ 0 0
$$599$$ −18747.0 −1.27876 −0.639382 0.768889i $$-0.720810\pi$$
−0.639382 + 0.768889i $$0.720810\pi$$
$$600$$ 0 0
$$601$$ −9864.63 −0.669529 −0.334764 0.942302i $$-0.608657\pi$$
−0.334764 + 0.942302i $$0.608657\pi$$
$$602$$ 0 0
$$603$$ 362.057 0.0244513
$$604$$ 0 0
$$605$$ 13710.1 0.921314
$$606$$ 0 0
$$607$$ −22292.8 −1.49067 −0.745336 0.666689i $$-0.767711\pi$$
−0.745336 + 0.666689i $$0.767711\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −16331.9 −1.08137
$$612$$ 0 0
$$613$$ 6046.78 0.398413 0.199206 0.979958i $$-0.436164\pi$$
0.199206 + 0.979958i $$0.436164\pi$$
$$614$$ 0 0
$$615$$ 10034.0 0.657901
$$616$$ 0 0
$$617$$ −9383.68 −0.612273 −0.306137 0.951988i $$-0.599036\pi$$
−0.306137 + 0.951988i $$0.599036\pi$$
$$618$$ 0 0
$$619$$ 3989.50 0.259049 0.129525 0.991576i $$-0.458655\pi$$
0.129525 + 0.991576i $$0.458655\pi$$
$$620$$ 0 0
$$621$$ −1854.18 −0.119816
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −14059.7 −0.899821
$$626$$ 0 0
$$627$$ 1706.63 0.108702
$$628$$ 0 0
$$629$$ −4401.88 −0.279038
$$630$$ 0 0
$$631$$ −11434.0 −0.721363 −0.360681 0.932689i $$-0.617456\pi$$
−0.360681 + 0.932689i $$0.617456\pi$$
$$632$$ 0 0
$$633$$ −7890.24 −0.495432
$$634$$ 0 0
$$635$$ −17274.7 −1.07957
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 2689.03 0.166473
$$640$$ 0 0
$$641$$ −15097.3 −0.930274 −0.465137 0.885239i $$-0.653995\pi$$
−0.465137 + 0.885239i $$0.653995\pi$$
$$642$$ 0 0
$$643$$ −4170.19 −0.255764 −0.127882 0.991789i $$-0.540818\pi$$
−0.127882 + 0.991789i $$0.540818\pi$$
$$644$$ 0 0
$$645$$ −9800.23 −0.598269
$$646$$ 0 0
$$647$$ 4563.88 0.277318 0.138659 0.990340i $$-0.455721\pi$$
0.138659 + 0.990340i $$0.455721\pi$$
$$648$$ 0 0
$$649$$ −2398.71 −0.145081
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 3819.04 0.228868 0.114434 0.993431i $$-0.463495\pi$$
0.114434 + 0.993431i $$0.463495\pi$$
$$654$$ 0 0
$$655$$ 16178.9 0.965134
$$656$$ 0 0
$$657$$ 4654.13 0.276370
$$658$$ 0 0
$$659$$ 4326.02 0.255717 0.127859 0.991792i $$-0.459190\pi$$
0.127859 + 0.991792i $$0.459190\pi$$
$$660$$ 0 0
$$661$$ −29317.8 −1.72516 −0.862579 0.505923i $$-0.831152\pi$$
−0.862579 + 0.505923i $$0.831152\pi$$
$$662$$ 0 0
$$663$$ 23756.6 1.39160
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6028.43 0.349958
$$668$$ 0 0
$$669$$ −2591.96 −0.149792
$$670$$ 0 0
$$671$$ 5328.02 0.306536
$$672$$ 0 0
$$673$$ −5483.56 −0.314080 −0.157040 0.987592i $$-0.550195\pi$$
−0.157040 + 0.987592i $$0.550195\pi$$
$$674$$ 0 0
$$675$$ 309.689 0.0176591
$$676$$ 0 0
$$677$$ −12069.3 −0.685174 −0.342587 0.939486i $$-0.611303\pi$$
−0.342587 + 0.939486i $$0.611303\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −12481.1 −0.702314
$$682$$ 0 0
$$683$$ −30796.7 −1.72533 −0.862667 0.505772i $$-0.831208\pi$$
−0.862667 + 0.505772i $$0.831208\pi$$
$$684$$ 0 0
$$685$$ 26477.7 1.47688
$$686$$ 0 0
$$687$$ 543.631 0.0301904
$$688$$ 0 0
$$689$$ 39866.9 2.20436
$$690$$ 0 0
$$691$$ 6501.96 0.357954 0.178977 0.983853i $$-0.442721\pi$$
0.178977 + 0.983853i $$0.442721\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −17638.0 −0.962656
$$696$$ 0 0
$$697$$ −32736.1 −1.77901
$$698$$ 0 0
$$699$$ −6451.34 −0.349088
$$700$$ 0 0
$$701$$ −2235.98 −0.120473 −0.0602367 0.998184i $$-0.519186\pi$$
−0.0602367 + 0.998184i $$0.519186\pi$$
$$702$$ 0 0
$$703$$ −3608.66 −0.193603
$$704$$ 0 0
$$705$$ −6875.11 −0.367279
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 35564.5 1.88385 0.941926 0.335820i $$-0.109014\pi$$
0.941926 + 0.335820i $$0.109014\pi$$
$$710$$ 0 0
$$711$$ 11002.2 0.580330
$$712$$ 0 0
$$713$$ −4310.50 −0.226409
$$714$$ 0 0
$$715$$ −5383.57 −0.281586
$$716$$ 0 0
$$717$$ −20818.4 −1.08435
$$718$$ 0 0
$$719$$ 9924.15 0.514754 0.257377 0.966311i $$-0.417142\pi$$
0.257377 + 0.966311i $$0.417142\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 618.509 0.0318155
$$724$$ 0 0
$$725$$ −1006.88 −0.0515788
$$726$$ 0 0
$$727$$ −880.081 −0.0448974 −0.0224487 0.999748i $$-0.507146\pi$$
−0.0224487 + 0.999748i $$0.507146\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 31973.5 1.61776
$$732$$ 0 0
$$733$$ −17336.2 −0.873568 −0.436784 0.899566i $$-0.643883\pi$$
−0.436784 + 0.899566i $$0.643883\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 267.681 0.0133787
$$738$$ 0 0
$$739$$ 24586.4 1.22385 0.611924 0.790917i $$-0.290396\pi$$
0.611924 + 0.790917i $$0.290396\pi$$
$$740$$ 0 0
$$741$$ 19475.6 0.965527
$$742$$ 0 0
$$743$$ −14579.3 −0.719870 −0.359935 0.932977i $$-0.617201\pi$$
−0.359935 + 0.932977i $$0.617201\pi$$
$$744$$ 0 0
$$745$$ −3731.30 −0.183496
$$746$$ 0 0
$$747$$ −11957.0 −0.585652
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −16086.0 −0.781604 −0.390802 0.920475i $$-0.627802\pi$$
−0.390802 + 0.920475i $$0.627802\pi$$
$$752$$ 0 0
$$753$$ 15035.8 0.727669
$$754$$ 0 0
$$755$$ 35568.0 1.71451
$$756$$ 0 0
$$757$$ −37017.3 −1.77730 −0.888651 0.458584i $$-0.848357\pi$$
−0.888651 + 0.458584i $$0.848357\pi$$
$$758$$ 0 0
$$759$$ −1370.85 −0.0655584
$$760$$ 0 0
$$761$$ 24606.5 1.17212 0.586060 0.810267i $$-0.300678\pi$$
0.586060 + 0.810267i $$0.300678\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 10000.6 0.472646
$$766$$ 0 0
$$767$$ −27373.4 −1.28865
$$768$$ 0 0
$$769$$ −12715.6 −0.596275 −0.298137 0.954523i $$-0.596365\pi$$
−0.298137 + 0.954523i $$0.596365\pi$$
$$770$$ 0 0
$$771$$ 17589.2 0.821610
$$772$$ 0 0
$$773$$ 3174.23 0.147696 0.0738480 0.997270i $$-0.476472\pi$$
0.0738480 + 0.997270i $$0.476472\pi$$
$$774$$ 0 0
$$775$$ 719.949 0.0333695
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −26837.0 −1.23432
$$780$$ 0 0
$$781$$ 1988.08 0.0910874
$$782$$ 0 0
$$783$$ −2370.18 −0.108178
$$784$$ 0 0
$$785$$ 18578.9 0.844726
$$786$$ 0 0
$$787$$ 15569.9 0.705218 0.352609 0.935771i $$-0.385295\pi$$
0.352609 + 0.935771i $$0.385295\pi$$
$$788$$ 0 0
$$789$$ −16918.7 −0.763398
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 60801.9 2.72275
$$794$$ 0 0
$$795$$ 16782.5 0.748695
$$796$$ 0 0
$$797$$ −31514.5 −1.40063 −0.700314 0.713835i $$-0.746957\pi$$
−0.700314 + 0.713835i $$0.746957\pi$$
$$798$$ 0 0
$$799$$ 22430.2 0.993146
$$800$$ 0 0
$$801$$ −5759.44 −0.254057
$$802$$ 0 0
$$803$$ 3440.95 0.151219
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 8929.90 0.389526
$$808$$ 0 0
$$809$$ 11029.1 0.479310 0.239655 0.970858i $$-0.422966\pi$$
0.239655 + 0.970858i $$0.422966\pi$$
$$810$$ 0 0
$$811$$ 20830.5 0.901920 0.450960 0.892544i $$-0.351082\pi$$
0.450960 + 0.892544i $$0.351082\pi$$
$$812$$ 0 0
$$813$$ −8422.00 −0.363312
$$814$$ 0 0
$$815$$ 36563.4 1.57149
$$816$$ 0 0
$$817$$ 26211.8 1.12244
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −9349.06 −0.397423 −0.198712 0.980058i $$-0.563676\pi$$
−0.198712 + 0.980058i $$0.563676\pi$$
$$822$$ 0 0
$$823$$ 32890.6 1.39307 0.696533 0.717525i $$-0.254725\pi$$
0.696533 + 0.717525i $$0.254725\pi$$
$$824$$ 0 0
$$825$$ 228.963 0.00966238
$$826$$ 0 0
$$827$$ 13081.2 0.550033 0.275016 0.961440i $$-0.411317\pi$$
0.275016 + 0.961440i $$0.411317\pi$$
$$828$$ 0 0
$$829$$ 28791.2 1.20622 0.603112 0.797657i $$-0.293927\pi$$
0.603112 + 0.797657i $$0.293927\pi$$
$$830$$ 0 0
$$831$$ 5755.17 0.240246
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 36025.3 1.49306
$$836$$ 0 0
$$837$$ 1694.74 0.0699868
$$838$$ 0 0
$$839$$ 36766.8 1.51291 0.756455 0.654046i $$-0.226930\pi$$
0.756455 + 0.654046i $$0.226930\pi$$
$$840$$ 0 0
$$841$$ −16682.9 −0.684034
$$842$$ 0 0
$$843$$ −15628.5 −0.638523
$$844$$ 0 0
$$845$$ −38026.7 −1.54812
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 22490.1 0.909138
$$850$$ 0 0
$$851$$ 2898.65 0.116762
$$852$$ 0 0
$$853$$ 5899.55 0.236808 0.118404 0.992966i $$-0.462222\pi$$
0.118404 + 0.992966i $$0.462222\pi$$
$$854$$ 0 0
$$855$$ 8198.51 0.327934
$$856$$ 0 0
$$857$$ 42277.0 1.68513 0.842564 0.538596i $$-0.181045\pi$$
0.842564 + 0.538596i $$0.181045\pi$$
$$858$$ 0 0
$$859$$ 6343.46 0.251963 0.125981 0.992033i $$-0.459792\pi$$
0.125981 + 0.992033i $$0.459792\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −6929.78 −0.273340 −0.136670 0.990617i $$-0.543640\pi$$
−0.136670 + 0.990617i $$0.543640\pi$$
$$864$$ 0 0
$$865$$ 14288.4 0.561643
$$866$$ 0 0
$$867$$ −17888.3 −0.700715
$$868$$ 0 0
$$869$$ 8134.27 0.317533
$$870$$ 0 0
$$871$$ 3054.69 0.118834
$$872$$ 0 0
$$873$$ 12830.9 0.497433
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 3287.88 0.126595 0.0632976 0.997995i $$-0.479838\pi$$
0.0632976 + 0.997995i $$0.479838\pi$$
$$878$$ 0 0
$$879$$ 10076.8 0.386670
$$880$$ 0 0
$$881$$ −46875.9 −1.79261 −0.896304 0.443439i $$-0.853758\pi$$
−0.896304 + 0.443439i $$0.853758\pi$$
$$882$$ 0 0
$$883$$ −42479.4 −1.61897 −0.809483 0.587144i $$-0.800252\pi$$
−0.809483 + 0.587144i $$0.800252\pi$$
$$884$$ 0 0
$$885$$ −11523.2 −0.437680
$$886$$ 0 0
$$887$$ −3680.15 −0.139309 −0.0696547 0.997571i $$-0.522190\pi$$
−0.0696547 + 0.997571i $$0.522190\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 538.973 0.0202652
$$892$$ 0 0
$$893$$ 18388.2 0.689069
$$894$$ 0 0
$$895$$ 11988.5 0.447745
$$896$$ 0 0
$$897$$ −15643.8 −0.582309
$$898$$ 0 0
$$899$$ −5510.07 −0.204417
$$900$$ 0 0
$$901$$ −54753.1 −2.02452
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 37666.0 1.38349
$$906$$ 0 0
$$907$$ 7102.18 0.260005 0.130002 0.991514i $$-0.458502\pi$$
0.130002 + 0.991514i $$0.458502\pi$$
$$908$$ 0 0
$$909$$ −8929.53 −0.325824
$$910$$ 0 0
$$911$$ −38119.0 −1.38632 −0.693161 0.720783i $$-0.743782\pi$$
−0.693161 + 0.720783i $$0.743782\pi$$
$$912$$ 0 0
$$913$$ −8840.17 −0.320446
$$914$$ 0 0
$$915$$ 25595.3 0.924760
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −17190.6 −0.617045 −0.308522 0.951217i $$-0.599834\pi$$
−0.308522 + 0.951217i $$0.599834\pi$$
$$920$$ 0 0
$$921$$ −21992.8 −0.786847
$$922$$ 0 0
$$923$$ 22687.5 0.809065
$$924$$ 0 0
$$925$$ −484.139 −0.0172091
$$926$$ 0 0
$$927$$ −2408.15 −0.0853225
$$928$$ 0 0
$$929$$ 32256.5 1.13918 0.569591 0.821928i $$-0.307101\pi$$
0.569591 + 0.821928i $$0.307101\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −6115.59 −0.214593
$$934$$ 0 0
$$935$$ 7393.80 0.258613
$$936$$ 0 0
$$937$$ 26213.6 0.913940 0.456970 0.889482i $$-0.348935\pi$$
0.456970 + 0.889482i $$0.348935\pi$$
$$938$$ 0 0
$$939$$ −12416.6 −0.431524
$$940$$ 0 0
$$941$$ 34240.1 1.18618 0.593089 0.805137i $$-0.297908\pi$$
0.593089 + 0.805137i $$0.297908\pi$$
$$942$$ 0 0
$$943$$ 21556.8 0.744418
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −36004.5 −1.23547 −0.617735 0.786386i $$-0.711950\pi$$
−0.617735 + 0.786386i $$0.711950\pi$$
$$948$$ 0 0
$$949$$ 39267.2 1.34317
$$950$$ 0 0
$$951$$ 22656.8 0.772551
$$952$$ 0 0
$$953$$ −29488.1 −1.00232 −0.501161 0.865354i $$-0.667093\pi$$
−0.501161 + 0.865354i $$0.667093\pi$$
$$954$$ 0 0
$$955$$ 28128.2 0.953095
$$956$$ 0 0
$$957$$ −1752.35 −0.0591905
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −25851.1 −0.867750
$$962$$ 0 0
$$963$$ 13829.6 0.462776
$$964$$ 0 0
$$965$$ −11156.4 −0.372163
$$966$$ 0 0
$$967$$ 56009.0 1.86260 0.931298 0.364259i $$-0.118678\pi$$
0.931298 + 0.364259i $$0.118678\pi$$
$$968$$ 0 0
$$969$$ −26747.8 −0.886753
$$970$$ 0 0
$$971$$ −13576.9 −0.448715 −0.224358 0.974507i $$-0.572028\pi$$
−0.224358 + 0.974507i $$0.572028\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 2612.86 0.0858241
$$976$$ 0 0
$$977$$ −31775.1 −1.04051 −0.520254 0.854011i $$-0.674163\pi$$
−0.520254 + 0.854011i $$0.674163\pi$$
$$978$$ 0 0
$$979$$ −4258.14 −0.139010
$$980$$ 0 0
$$981$$ 8989.38 0.292568
$$982$$ 0 0
$$983$$ −32755.4 −1.06280 −0.531401 0.847120i $$-0.678334\pi$$
−0.531401 + 0.847120i $$0.678334\pi$$
$$984$$ 0 0
$$985$$ 48858.2 1.58046
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −21054.6 −0.676944
$$990$$ 0 0
$$991$$ 40797.0 1.30773 0.653865 0.756611i $$-0.273146\pi$$
0.653865 + 0.756611i $$0.273146\pi$$
$$992$$ 0 0
$$993$$ 16733.1 0.534752
$$994$$ 0 0
$$995$$ 8851.60 0.282025
$$996$$ 0 0
$$997$$ −17370.5 −0.551784 −0.275892 0.961189i $$-0.588973\pi$$
−0.275892 + 0.961189i $$0.588973\pi$$
$$998$$ 0 0
$$999$$ −1139.65 −0.0360931
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.cl.1.2 4
4.3 odd 2 588.4.a.k.1.2 yes 4
7.6 odd 2 2352.4.a.cq.1.3 4
12.11 even 2 1764.4.a.ba.1.3 4
28.3 even 6 588.4.i.l.373.2 8
28.11 odd 6 588.4.i.k.373.3 8
28.19 even 6 588.4.i.l.361.2 8
28.23 odd 6 588.4.i.k.361.3 8
28.27 even 2 588.4.a.j.1.3 4
84.11 even 6 1764.4.k.bd.1549.2 8
84.23 even 6 1764.4.k.bd.361.2 8
84.47 odd 6 1764.4.k.bb.361.3 8
84.59 odd 6 1764.4.k.bb.1549.3 8
84.83 odd 2 1764.4.a.bc.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
588.4.a.j.1.3 4 28.27 even 2
588.4.a.k.1.2 yes 4 4.3 odd 2
588.4.i.k.361.3 8 28.23 odd 6
588.4.i.k.373.3 8 28.11 odd 6
588.4.i.l.361.2 8 28.19 even 6
588.4.i.l.373.2 8 28.3 even 6
1764.4.a.ba.1.3 4 12.11 even 2
1764.4.a.bc.1.2 4 84.83 odd 2
1764.4.k.bb.361.3 8 84.47 odd 6
1764.4.k.bb.1549.3 8 84.59 odd 6
1764.4.k.bd.361.2 8 84.23 even 6
1764.4.k.bd.1549.2 8 84.11 even 6
2352.4.a.cl.1.2 4 1.1 even 1 trivial
2352.4.a.cq.1.3 4 7.6 odd 2