# Properties

 Label 2352.4.a.b.1.1 Level $2352$ Weight $4$ Character 2352.1 Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $1$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -12.0000 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -12.0000 q^{5} +9.00000 q^{9} -20.0000 q^{11} +84.0000 q^{13} +36.0000 q^{15} +96.0000 q^{17} +12.0000 q^{19} +176.000 q^{23} +19.0000 q^{25} -27.0000 q^{27} +58.0000 q^{29} -264.000 q^{31} +60.0000 q^{33} +258.000 q^{37} -252.000 q^{39} -156.000 q^{43} -108.000 q^{45} -408.000 q^{47} -288.000 q^{51} -722.000 q^{53} +240.000 q^{55} -36.0000 q^{57} +492.000 q^{59} +492.000 q^{61} -1008.00 q^{65} -412.000 q^{67} -528.000 q^{69} -296.000 q^{71} -240.000 q^{73} -57.0000 q^{75} -776.000 q^{79} +81.0000 q^{81} +924.000 q^{83} -1152.00 q^{85} -174.000 q^{87} +744.000 q^{89} +792.000 q^{93} -144.000 q^{95} +168.000 q^{97} -180.000 q^{99} +O(q^{100})$$

## 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
$$5$$ −12.0000 −1.07331 −0.536656 0.843801i $$-0.680313\pi$$
−0.536656 + 0.843801i $$0.680313\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −20.0000 −0.548202 −0.274101 0.961701i $$-0.588380\pi$$
−0.274101 + 0.961701i $$0.588380\pi$$
$$12$$ 0 0
$$13$$ 84.0000 1.79211 0.896054 0.443945i $$-0.146421\pi$$
0.896054 + 0.443945i $$0.146421\pi$$
$$14$$ 0 0
$$15$$ 36.0000 0.619677
$$16$$ 0 0
$$17$$ 96.0000 1.36961 0.684806 0.728725i $$-0.259887\pi$$
0.684806 + 0.728725i $$0.259887\pi$$
$$18$$ 0 0
$$19$$ 12.0000 0.144894 0.0724471 0.997372i $$-0.476919\pi$$
0.0724471 + 0.997372i $$0.476919\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 176.000 1.59559 0.797794 0.602930i $$-0.206000\pi$$
0.797794 + 0.602930i $$0.206000\pi$$
$$24$$ 0 0
$$25$$ 19.0000 0.152000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 58.0000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −264.000 −1.52954 −0.764771 0.644302i $$-0.777148\pi$$
−0.764771 + 0.644302i $$0.777148\pi$$
$$32$$ 0 0
$$33$$ 60.0000 0.316505
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 258.000 1.14635 0.573175 0.819433i $$-0.305712\pi$$
0.573175 + 0.819433i $$0.305712\pi$$
$$38$$ 0 0
$$39$$ −252.000 −1.03467
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ −156.000 −0.553251 −0.276625 0.960978i $$-0.589216\pi$$
−0.276625 + 0.960978i $$0.589216\pi$$
$$44$$ 0 0
$$45$$ −108.000 −0.357771
$$46$$ 0 0
$$47$$ −408.000 −1.26623 −0.633116 0.774057i $$-0.718224\pi$$
−0.633116 + 0.774057i $$0.718224\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −288.000 −0.790746
$$52$$ 0 0
$$53$$ −722.000 −1.87121 −0.935607 0.353044i $$-0.885147\pi$$
−0.935607 + 0.353044i $$0.885147\pi$$
$$54$$ 0 0
$$55$$ 240.000 0.588393
$$56$$ 0 0
$$57$$ −36.0000 −0.0836547
$$58$$ 0 0
$$59$$ 492.000 1.08564 0.542822 0.839848i $$-0.317356\pi$$
0.542822 + 0.839848i $$0.317356\pi$$
$$60$$ 0 0
$$61$$ 492.000 1.03269 0.516345 0.856380i $$-0.327292\pi$$
0.516345 + 0.856380i $$0.327292\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −1008.00 −1.92349
$$66$$ 0 0
$$67$$ −412.000 −0.751251 −0.375625 0.926772i $$-0.622572\pi$$
−0.375625 + 0.926772i $$0.622572\pi$$
$$68$$ 0 0
$$69$$ −528.000 −0.921213
$$70$$ 0 0
$$71$$ −296.000 −0.494771 −0.247385 0.968917i $$-0.579571\pi$$
−0.247385 + 0.968917i $$0.579571\pi$$
$$72$$ 0 0
$$73$$ −240.000 −0.384793 −0.192396 0.981317i $$-0.561626\pi$$
−0.192396 + 0.981317i $$0.561626\pi$$
$$74$$ 0 0
$$75$$ −57.0000 −0.0877572
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −776.000 −1.10515 −0.552575 0.833463i $$-0.686355\pi$$
−0.552575 + 0.833463i $$0.686355\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 924.000 1.22195 0.610977 0.791648i $$-0.290777\pi$$
0.610977 + 0.791648i $$0.290777\pi$$
$$84$$ 0 0
$$85$$ −1152.00 −1.47002
$$86$$ 0 0
$$87$$ −174.000 −0.214423
$$88$$ 0 0
$$89$$ 744.000 0.886111 0.443055 0.896494i $$-0.353895\pi$$
0.443055 + 0.896494i $$0.353895\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 792.000 0.883081
$$94$$ 0 0
$$95$$ −144.000 −0.155517
$$96$$ 0 0
$$97$$ 168.000 0.175854 0.0879269 0.996127i $$-0.471976\pi$$
0.0879269 + 0.996127i $$0.471976\pi$$
$$98$$ 0 0
$$99$$ −180.000 −0.182734
$$100$$ 0 0
$$101$$ 1524.00 1.50142 0.750711 0.660630i $$-0.229711\pi$$
0.750711 + 0.660630i $$0.229711\pi$$
$$102$$ 0 0
$$103$$ −408.000 −0.390305 −0.195153 0.980773i $$-0.562520\pi$$
−0.195153 + 0.980773i $$0.562520\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 820.000 0.740863 0.370432 0.928860i $$-0.379210\pi$$
0.370432 + 0.928860i $$0.379210\pi$$
$$108$$ 0 0
$$109$$ −918.000 −0.806683 −0.403342 0.915050i $$-0.632151\pi$$
−0.403342 + 0.915050i $$0.632151\pi$$
$$110$$ 0 0
$$111$$ −774.000 −0.661845
$$112$$ 0 0
$$113$$ −110.000 −0.0915746 −0.0457873 0.998951i $$-0.514580\pi$$
−0.0457873 + 0.998951i $$0.514580\pi$$
$$114$$ 0 0
$$115$$ −2112.00 −1.71257
$$116$$ 0 0
$$117$$ 756.000 0.597369
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −931.000 −0.699474
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1272.00 0.910169
$$126$$ 0 0
$$127$$ −16.0000 −0.0111793 −0.00558965 0.999984i $$-0.501779\pi$$
−0.00558965 + 0.999984i $$0.501779\pi$$
$$128$$ 0 0
$$129$$ 468.000 0.319419
$$130$$ 0 0
$$131$$ 1692.00 1.12848 0.564239 0.825611i $$-0.309169\pi$$
0.564239 + 0.825611i $$0.309169\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 324.000 0.206559
$$136$$ 0 0
$$137$$ 1126.00 0.702195 0.351097 0.936339i $$-0.385809\pi$$
0.351097 + 0.936339i $$0.385809\pi$$
$$138$$ 0 0
$$139$$ 1092.00 0.666347 0.333173 0.942866i $$-0.391881\pi$$
0.333173 + 0.942866i $$0.391881\pi$$
$$140$$ 0 0
$$141$$ 1224.00 0.731060
$$142$$ 0 0
$$143$$ −1680.00 −0.982438
$$144$$ 0 0
$$145$$ −696.000 −0.398618
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 1070.00 0.588307 0.294154 0.955758i $$-0.404962\pi$$
0.294154 + 0.955758i $$0.404962\pi$$
$$150$$ 0 0
$$151$$ 120.000 0.0646719 0.0323360 0.999477i $$-0.489705\pi$$
0.0323360 + 0.999477i $$0.489705\pi$$
$$152$$ 0 0
$$153$$ 864.000 0.456538
$$154$$ 0 0
$$155$$ 3168.00 1.64168
$$156$$ 0 0
$$157$$ −1836.00 −0.933304 −0.466652 0.884441i $$-0.654540\pi$$
−0.466652 + 0.884441i $$0.654540\pi$$
$$158$$ 0 0
$$159$$ 2166.00 1.08035
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −916.000 −0.440164 −0.220082 0.975481i $$-0.570632\pi$$
−0.220082 + 0.975481i $$0.570632\pi$$
$$164$$ 0 0
$$165$$ −720.000 −0.339709
$$166$$ 0 0
$$167$$ 504.000 0.233537 0.116769 0.993159i $$-0.462746\pi$$
0.116769 + 0.993159i $$0.462746\pi$$
$$168$$ 0 0
$$169$$ 4859.00 2.21165
$$170$$ 0 0
$$171$$ 108.000 0.0482980
$$172$$ 0 0
$$173$$ 1836.00 0.806870 0.403435 0.915008i $$-0.367816\pi$$
0.403435 + 0.915008i $$0.367816\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −1476.00 −0.626796
$$178$$ 0 0
$$179$$ −2372.00 −0.990456 −0.495228 0.868763i $$-0.664915\pi$$
−0.495228 + 0.868763i $$0.664915\pi$$
$$180$$ 0 0
$$181$$ 1092.00 0.448440 0.224220 0.974539i $$-0.428017\pi$$
0.224220 + 0.974539i $$0.428017\pi$$
$$182$$ 0 0
$$183$$ −1476.00 −0.596224
$$184$$ 0 0
$$185$$ −3096.00 −1.23039
$$186$$ 0 0
$$187$$ −1920.00 −0.750825
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2512.00 −0.951633 −0.475817 0.879545i $$-0.657847\pi$$
−0.475817 + 0.879545i $$0.657847\pi$$
$$192$$ 0 0
$$193$$ −2430.00 −0.906297 −0.453148 0.891435i $$-0.649699\pi$$
−0.453148 + 0.891435i $$0.649699\pi$$
$$194$$ 0 0
$$195$$ 3024.00 1.11053
$$196$$ 0 0
$$197$$ −1762.00 −0.637245 −0.318623 0.947882i $$-0.603220\pi$$
−0.318623 + 0.947882i $$0.603220\pi$$
$$198$$ 0 0
$$199$$ 3096.00 1.10286 0.551431 0.834220i $$-0.314082\pi$$
0.551431 + 0.834220i $$0.314082\pi$$
$$200$$ 0 0
$$201$$ 1236.00 0.433735
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1584.00 0.531863
$$208$$ 0 0
$$209$$ −240.000 −0.0794313
$$210$$ 0 0
$$211$$ −156.000 −0.0508980 −0.0254490 0.999676i $$-0.508102\pi$$
−0.0254490 + 0.999676i $$0.508102\pi$$
$$212$$ 0 0
$$213$$ 888.000 0.285656
$$214$$ 0 0
$$215$$ 1872.00 0.593811
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 720.000 0.222160
$$220$$ 0 0
$$221$$ 8064.00 2.45449
$$222$$ 0 0
$$223$$ 5040.00 1.51347 0.756734 0.653723i $$-0.226794\pi$$
0.756734 + 0.653723i $$0.226794\pi$$
$$224$$ 0 0
$$225$$ 171.000 0.0506667
$$226$$ 0 0
$$227$$ 2172.00 0.635069 0.317535 0.948247i $$-0.397145\pi$$
0.317535 + 0.948247i $$0.397145\pi$$
$$228$$ 0 0
$$229$$ −2700.00 −0.779131 −0.389566 0.920999i $$-0.627375\pi$$
−0.389566 + 0.920999i $$0.627375\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3802.00 −1.06900 −0.534501 0.845168i $$-0.679500\pi$$
−0.534501 + 0.845168i $$0.679500\pi$$
$$234$$ 0 0
$$235$$ 4896.00 1.35906
$$236$$ 0 0
$$237$$ 2328.00 0.638058
$$238$$ 0 0
$$239$$ 4408.00 1.19301 0.596506 0.802609i $$-0.296555\pi$$
0.596506 + 0.802609i $$0.296555\pi$$
$$240$$ 0 0
$$241$$ −3096.00 −0.827514 −0.413757 0.910387i $$-0.635784\pi$$
−0.413757 + 0.910387i $$0.635784\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1008.00 0.259666
$$248$$ 0 0
$$249$$ −2772.00 −0.705495
$$250$$ 0 0
$$251$$ −924.000 −0.232360 −0.116180 0.993228i $$-0.537065\pi$$
−0.116180 + 0.993228i $$0.537065\pi$$
$$252$$ 0 0
$$253$$ −3520.00 −0.874706
$$254$$ 0 0
$$255$$ 3456.00 0.848718
$$256$$ 0 0
$$257$$ 2760.00 0.669899 0.334950 0.942236i $$-0.391281\pi$$
0.334950 + 0.942236i $$0.391281\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 522.000 0.123797
$$262$$ 0 0
$$263$$ 2360.00 0.553323 0.276661 0.960967i $$-0.410772\pi$$
0.276661 + 0.960967i $$0.410772\pi$$
$$264$$ 0 0
$$265$$ 8664.00 2.00840
$$266$$ 0 0
$$267$$ −2232.00 −0.511596
$$268$$ 0 0
$$269$$ −4020.00 −0.911166 −0.455583 0.890193i $$-0.650569\pi$$
−0.455583 + 0.890193i $$0.650569\pi$$
$$270$$ 0 0
$$271$$ 4800.00 1.07594 0.537969 0.842965i $$-0.319192\pi$$
0.537969 + 0.842965i $$0.319192\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −380.000 −0.0833268
$$276$$ 0 0
$$277$$ 6446.00 1.39820 0.699102 0.715022i $$-0.253583\pi$$
0.699102 + 0.715022i $$0.253583\pi$$
$$278$$ 0 0
$$279$$ −2376.00 −0.509847
$$280$$ 0 0
$$281$$ −2602.00 −0.552393 −0.276196 0.961101i $$-0.589074\pi$$
−0.276196 + 0.961101i $$0.589074\pi$$
$$282$$ 0 0
$$283$$ −6900.00 −1.44934 −0.724669 0.689098i $$-0.758007\pi$$
−0.724669 + 0.689098i $$0.758007\pi$$
$$284$$ 0 0
$$285$$ 432.000 0.0897876
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 4303.00 0.875840
$$290$$ 0 0
$$291$$ −504.000 −0.101529
$$292$$ 0 0
$$293$$ 4452.00 0.887674 0.443837 0.896107i $$-0.353617\pi$$
0.443837 + 0.896107i $$0.353617\pi$$
$$294$$ 0 0
$$295$$ −5904.00 −1.16523
$$296$$ 0 0
$$297$$ 540.000 0.105502
$$298$$ 0 0
$$299$$ 14784.0 2.85947
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −4572.00 −0.866847
$$304$$ 0 0
$$305$$ −5904.00 −1.10840
$$306$$ 0 0
$$307$$ −2436.00 −0.452866 −0.226433 0.974027i $$-0.572706\pi$$
−0.226433 + 0.974027i $$0.572706\pi$$
$$308$$ 0 0
$$309$$ 1224.00 0.225343
$$310$$ 0 0
$$311$$ −7488.00 −1.36529 −0.682646 0.730750i $$-0.739171\pi$$
−0.682646 + 0.730750i $$0.739171\pi$$
$$312$$ 0 0
$$313$$ 1752.00 0.316386 0.158193 0.987408i $$-0.449433\pi$$
0.158193 + 0.987408i $$0.449433\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −1562.00 −0.276753 −0.138376 0.990380i $$-0.544188\pi$$
−0.138376 + 0.990380i $$0.544188\pi$$
$$318$$ 0 0
$$319$$ −1160.00 −0.203597
$$320$$ 0 0
$$321$$ −2460.00 −0.427738
$$322$$ 0 0
$$323$$ 1152.00 0.198449
$$324$$ 0 0
$$325$$ 1596.00 0.272400
$$326$$ 0 0
$$327$$ 2754.00 0.465739
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 7092.00 1.17768 0.588839 0.808250i $$-0.299585\pi$$
0.588839 + 0.808250i $$0.299585\pi$$
$$332$$ 0 0
$$333$$ 2322.00 0.382117
$$334$$ 0 0
$$335$$ 4944.00 0.806327
$$336$$ 0 0
$$337$$ 366.000 0.0591611 0.0295805 0.999562i $$-0.490583\pi$$
0.0295805 + 0.999562i $$0.490583\pi$$
$$338$$ 0 0
$$339$$ 330.000 0.0528706
$$340$$ 0 0
$$341$$ 5280.00 0.838499
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 6336.00 0.988750
$$346$$ 0 0
$$347$$ 6364.00 0.984546 0.492273 0.870441i $$-0.336166\pi$$
0.492273 + 0.870441i $$0.336166\pi$$
$$348$$ 0 0
$$349$$ 10500.0 1.61046 0.805232 0.592960i $$-0.202041\pi$$
0.805232 + 0.592960i $$0.202041\pi$$
$$350$$ 0 0
$$351$$ −2268.00 −0.344891
$$352$$ 0 0
$$353$$ −408.000 −0.0615174 −0.0307587 0.999527i $$-0.509792\pi$$
−0.0307587 + 0.999527i $$0.509792\pi$$
$$354$$ 0 0
$$355$$ 3552.00 0.531044
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 11936.0 1.75476 0.877379 0.479798i $$-0.159290\pi$$
0.877379 + 0.479798i $$0.159290\pi$$
$$360$$ 0 0
$$361$$ −6715.00 −0.979006
$$362$$ 0 0
$$363$$ 2793.00 0.403842
$$364$$ 0 0
$$365$$ 2880.00 0.413003
$$366$$ 0 0
$$367$$ −2448.00 −0.348187 −0.174093 0.984729i $$-0.555699\pi$$
−0.174093 + 0.984729i $$0.555699\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 11374.0 1.57888 0.789442 0.613826i $$-0.210370\pi$$
0.789442 + 0.613826i $$0.210370\pi$$
$$374$$ 0 0
$$375$$ −3816.00 −0.525486
$$376$$ 0 0
$$377$$ 4872.00 0.665572
$$378$$ 0 0
$$379$$ 5892.00 0.798553 0.399277 0.916830i $$-0.369261\pi$$
0.399277 + 0.916830i $$0.369261\pi$$
$$380$$ 0 0
$$381$$ 48.0000 0.00645437
$$382$$ 0 0
$$383$$ −10488.0 −1.39925 −0.699624 0.714511i $$-0.746649\pi$$
−0.699624 + 0.714511i $$0.746649\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1404.00 −0.184417
$$388$$ 0 0
$$389$$ 4514.00 0.588352 0.294176 0.955751i $$-0.404955\pi$$
0.294176 + 0.955751i $$0.404955\pi$$
$$390$$ 0 0
$$391$$ 16896.0 2.18534
$$392$$ 0 0
$$393$$ −5076.00 −0.651528
$$394$$ 0 0
$$395$$ 9312.00 1.18617
$$396$$ 0 0
$$397$$ 6036.00 0.763068 0.381534 0.924355i $$-0.375396\pi$$
0.381534 + 0.924355i $$0.375396\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6770.00 −0.843086 −0.421543 0.906808i $$-0.638511\pi$$
−0.421543 + 0.906808i $$0.638511\pi$$
$$402$$ 0 0
$$403$$ −22176.0 −2.74110
$$404$$ 0 0
$$405$$ −972.000 −0.119257
$$406$$ 0 0
$$407$$ −5160.00 −0.628432
$$408$$ 0 0
$$409$$ −12504.0 −1.51169 −0.755847 0.654748i $$-0.772775\pi$$
−0.755847 + 0.654748i $$0.772775\pi$$
$$410$$ 0 0
$$411$$ −3378.00 −0.405412
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −11088.0 −1.31154
$$416$$ 0 0
$$417$$ −3276.00 −0.384716
$$418$$ 0 0
$$419$$ 9492.00 1.10672 0.553359 0.832943i $$-0.313346\pi$$
0.553359 + 0.832943i $$0.313346\pi$$
$$420$$ 0 0
$$421$$ 5182.00 0.599894 0.299947 0.953956i $$-0.403031\pi$$
0.299947 + 0.953956i $$0.403031\pi$$
$$422$$ 0 0
$$423$$ −3672.00 −0.422077
$$424$$ 0 0
$$425$$ 1824.00 0.208181
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 5040.00 0.567211
$$430$$ 0 0
$$431$$ 5720.00 0.639264 0.319632 0.947542i $$-0.396441\pi$$
0.319632 + 0.947542i $$0.396441\pi$$
$$432$$ 0 0
$$433$$ −13608.0 −1.51030 −0.755149 0.655554i $$-0.772435\pi$$
−0.755149 + 0.655554i $$0.772435\pi$$
$$434$$ 0 0
$$435$$ 2088.00 0.230142
$$436$$ 0 0
$$437$$ 2112.00 0.231191
$$438$$ 0 0
$$439$$ 12864.0 1.39855 0.699277 0.714851i $$-0.253505\pi$$
0.699277 + 0.714851i $$0.253505\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 13252.0 1.42127 0.710634 0.703562i $$-0.248408\pi$$
0.710634 + 0.703562i $$0.248408\pi$$
$$444$$ 0 0
$$445$$ −8928.00 −0.951074
$$446$$ 0 0
$$447$$ −3210.00 −0.339659
$$448$$ 0 0
$$449$$ 226.000 0.0237541 0.0118771 0.999929i $$-0.496219\pi$$
0.0118771 + 0.999929i $$0.496219\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −360.000 −0.0373384
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11334.0 −1.16014 −0.580068 0.814568i $$-0.696974\pi$$
−0.580068 + 0.814568i $$0.696974\pi$$
$$458$$ 0 0
$$459$$ −2592.00 −0.263582
$$460$$ 0 0
$$461$$ 1596.00 0.161243 0.0806216 0.996745i $$-0.474309\pi$$
0.0806216 + 0.996745i $$0.474309\pi$$
$$462$$ 0 0
$$463$$ −12728.0 −1.27758 −0.638791 0.769380i $$-0.720565\pi$$
−0.638791 + 0.769380i $$0.720565\pi$$
$$464$$ 0 0
$$465$$ −9504.00 −0.947822
$$466$$ 0 0
$$467$$ −3012.00 −0.298456 −0.149228 0.988803i $$-0.547679\pi$$
−0.149228 + 0.988803i $$0.547679\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 5508.00 0.538843
$$472$$ 0 0
$$473$$ 3120.00 0.303293
$$474$$ 0 0
$$475$$ 228.000 0.0220239
$$476$$ 0 0
$$477$$ −6498.00 −0.623738
$$478$$ 0 0
$$479$$ −4296.00 −0.409790 −0.204895 0.978784i $$-0.565685\pi$$
−0.204895 + 0.978784i $$0.565685\pi$$
$$480$$ 0 0
$$481$$ 21672.0 2.05438
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −2016.00 −0.188746
$$486$$ 0 0
$$487$$ 8184.00 0.761504 0.380752 0.924677i $$-0.375665\pi$$
0.380752 + 0.924677i $$0.375665\pi$$
$$488$$ 0 0
$$489$$ 2748.00 0.254129
$$490$$ 0 0
$$491$$ 12164.0 1.11803 0.559016 0.829157i $$-0.311179\pi$$
0.559016 + 0.829157i $$0.311179\pi$$
$$492$$ 0 0
$$493$$ 5568.00 0.508661
$$494$$ 0 0
$$495$$ 2160.00 0.196131
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −972.000 −0.0871998 −0.0435999 0.999049i $$-0.513883\pi$$
−0.0435999 + 0.999049i $$0.513883\pi$$
$$500$$ 0 0
$$501$$ −1512.00 −0.134833
$$502$$ 0 0
$$503$$ 7728.00 0.685039 0.342519 0.939511i $$-0.388720\pi$$
0.342519 + 0.939511i $$0.388720\pi$$
$$504$$ 0 0
$$505$$ −18288.0 −1.61150
$$506$$ 0 0
$$507$$ −14577.0 −1.27690
$$508$$ 0 0
$$509$$ −11604.0 −1.01049 −0.505244 0.862977i $$-0.668597\pi$$
−0.505244 + 0.862977i $$0.668597\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −324.000 −0.0278849
$$514$$ 0 0
$$515$$ 4896.00 0.418919
$$516$$ 0 0
$$517$$ 8160.00 0.694152
$$518$$ 0 0
$$519$$ −5508.00 −0.465847
$$520$$ 0 0
$$521$$ 10848.0 0.912206 0.456103 0.889927i $$-0.349245\pi$$
0.456103 + 0.889927i $$0.349245\pi$$
$$522$$ 0 0
$$523$$ −18132.0 −1.51598 −0.757989 0.652267i $$-0.773818\pi$$
−0.757989 + 0.652267i $$0.773818\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −25344.0 −2.09488
$$528$$ 0 0
$$529$$ 18809.0 1.54590
$$530$$ 0 0
$$531$$ 4428.00 0.361881
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −9840.00 −0.795178
$$536$$ 0 0
$$537$$ 7116.00 0.571840
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 6950.00 0.552318 0.276159 0.961112i $$-0.410938\pi$$
0.276159 + 0.961112i $$0.410938\pi$$
$$542$$ 0 0
$$543$$ −3276.00 −0.258907
$$544$$ 0 0
$$545$$ 11016.0 0.865823
$$546$$ 0 0
$$547$$ −17012.0 −1.32976 −0.664882 0.746949i $$-0.731518\pi$$
−0.664882 + 0.746949i $$0.731518\pi$$
$$548$$ 0 0
$$549$$ 4428.00 0.344230
$$550$$ 0 0
$$551$$ 696.000 0.0538123
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 9288.00 0.710367
$$556$$ 0 0
$$557$$ 3926.00 0.298653 0.149327 0.988788i $$-0.452289\pi$$
0.149327 + 0.988788i $$0.452289\pi$$
$$558$$ 0 0
$$559$$ −13104.0 −0.991485
$$560$$ 0 0
$$561$$ 5760.00 0.433489
$$562$$ 0 0
$$563$$ −18828.0 −1.40942 −0.704712 0.709494i $$-0.748924\pi$$
−0.704712 + 0.709494i $$0.748924\pi$$
$$564$$ 0 0
$$565$$ 1320.00 0.0982882
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 11990.0 0.883387 0.441693 0.897166i $$-0.354378\pi$$
0.441693 + 0.897166i $$0.354378\pi$$
$$570$$ 0 0
$$571$$ 15716.0 1.15183 0.575914 0.817510i $$-0.304646\pi$$
0.575914 + 0.817510i $$0.304646\pi$$
$$572$$ 0 0
$$573$$ 7536.00 0.549426
$$574$$ 0 0
$$575$$ 3344.00 0.242529
$$576$$ 0 0
$$577$$ 13872.0 1.00086 0.500432 0.865776i $$-0.333174\pi$$
0.500432 + 0.865776i $$0.333174\pi$$
$$578$$ 0 0
$$579$$ 7290.00 0.523251
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 14440.0 1.02580
$$584$$ 0 0
$$585$$ −9072.00 −0.641164
$$586$$ 0 0
$$587$$ 8820.00 0.620171 0.310085 0.950709i $$-0.399642\pi$$
0.310085 + 0.950709i $$0.399642\pi$$
$$588$$ 0 0
$$589$$ −3168.00 −0.221622
$$590$$ 0 0
$$591$$ 5286.00 0.367914
$$592$$ 0 0
$$593$$ 16872.0 1.16838 0.584191 0.811617i $$-0.301412\pi$$
0.584191 + 0.811617i $$0.301412\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −9288.00 −0.636738
$$598$$ 0 0
$$599$$ 6056.00 0.413091 0.206545 0.978437i $$-0.433778\pi$$
0.206545 + 0.978437i $$0.433778\pi$$
$$600$$ 0 0
$$601$$ −10752.0 −0.729756 −0.364878 0.931055i $$-0.618889\pi$$
−0.364878 + 0.931055i $$0.618889\pi$$
$$602$$ 0 0
$$603$$ −3708.00 −0.250417
$$604$$ 0 0
$$605$$ 11172.0 0.750754
$$606$$ 0 0
$$607$$ 20256.0 1.35447 0.677237 0.735765i $$-0.263177\pi$$
0.677237 + 0.735765i $$0.263177\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −34272.0 −2.26923
$$612$$ 0 0
$$613$$ −28190.0 −1.85740 −0.928698 0.370838i $$-0.879071\pi$$
−0.928698 + 0.370838i $$0.879071\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 29318.0 1.91296 0.956482 0.291793i $$-0.0942518\pi$$
0.956482 + 0.291793i $$0.0942518\pi$$
$$618$$ 0 0
$$619$$ 24348.0 1.58098 0.790492 0.612473i $$-0.209825\pi$$
0.790492 + 0.612473i $$0.209825\pi$$
$$620$$ 0 0
$$621$$ −4752.00 −0.307071
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −17639.0 −1.12890
$$626$$ 0 0
$$627$$ 720.000 0.0458597
$$628$$ 0 0
$$629$$ 24768.0 1.57006
$$630$$ 0 0
$$631$$ 25184.0 1.58884 0.794421 0.607368i $$-0.207774\pi$$
0.794421 + 0.607368i $$0.207774\pi$$
$$632$$ 0 0
$$633$$ 468.000 0.0293860
$$634$$ 0 0
$$635$$ 192.000 0.0119989
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −2664.00 −0.164924
$$640$$ 0 0
$$641$$ 32318.0 1.99140 0.995698 0.0926628i $$-0.0295379\pi$$
0.995698 + 0.0926628i $$0.0295379\pi$$
$$642$$ 0 0
$$643$$ −3948.00 −0.242137 −0.121068 0.992644i $$-0.538632\pi$$
−0.121068 + 0.992644i $$0.538632\pi$$
$$644$$ 0 0
$$645$$ −5616.00 −0.342837
$$646$$ 0 0
$$647$$ 13848.0 0.841454 0.420727 0.907187i $$-0.361775\pi$$
0.420727 + 0.907187i $$0.361775\pi$$
$$648$$ 0 0
$$649$$ −9840.00 −0.595152
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −3158.00 −0.189253 −0.0946264 0.995513i $$-0.530166\pi$$
−0.0946264 + 0.995513i $$0.530166\pi$$
$$654$$ 0 0
$$655$$ −20304.0 −1.21121
$$656$$ 0 0
$$657$$ −2160.00 −0.128264
$$658$$ 0 0
$$659$$ 24596.0 1.45391 0.726953 0.686687i $$-0.240936\pi$$
0.726953 + 0.686687i $$0.240936\pi$$
$$660$$ 0 0
$$661$$ 15468.0 0.910190 0.455095 0.890443i $$-0.349605\pi$$
0.455095 + 0.890443i $$0.349605\pi$$
$$662$$ 0 0
$$663$$ −24192.0 −1.41710
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 10208.0 0.592587
$$668$$ 0 0
$$669$$ −15120.0 −0.873801
$$670$$ 0 0
$$671$$ −9840.00 −0.566124
$$672$$ 0 0
$$673$$ 13470.0 0.771516 0.385758 0.922600i $$-0.373940\pi$$
0.385758 + 0.922600i $$0.373940\pi$$
$$674$$ 0 0
$$675$$ −513.000 −0.0292524
$$676$$ 0 0
$$677$$ 9564.00 0.542946 0.271473 0.962446i $$-0.412489\pi$$
0.271473 + 0.962446i $$0.412489\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −6516.00 −0.366657
$$682$$ 0 0
$$683$$ −13852.0 −0.776035 −0.388018 0.921652i $$-0.626840\pi$$
−0.388018 + 0.921652i $$0.626840\pi$$
$$684$$ 0 0
$$685$$ −13512.0 −0.753674
$$686$$ 0 0
$$687$$ 8100.00 0.449832
$$688$$ 0 0
$$689$$ −60648.0 −3.35342
$$690$$ 0 0
$$691$$ −324.000 −0.0178373 −0.00891863 0.999960i $$-0.502839\pi$$
−0.00891863 + 0.999960i $$0.502839\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −13104.0 −0.715199
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 11406.0 0.617188
$$700$$ 0 0
$$701$$ 24922.0 1.34278 0.671392 0.741103i $$-0.265697\pi$$
0.671392 + 0.741103i $$0.265697\pi$$
$$702$$ 0 0
$$703$$ 3096.00 0.166099
$$704$$ 0 0
$$705$$ −14688.0 −0.784655
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −17886.0 −0.947423 −0.473711 0.880680i $$-0.657086\pi$$
−0.473711 + 0.880680i $$0.657086\pi$$
$$710$$ 0 0
$$711$$ −6984.00 −0.368383
$$712$$ 0 0
$$713$$ −46464.0 −2.44052
$$714$$ 0 0
$$715$$ 20160.0 1.05446
$$716$$ 0 0
$$717$$ −13224.0 −0.688786
$$718$$ 0 0
$$719$$ −6792.00 −0.352293 −0.176147 0.984364i $$-0.556363\pi$$
−0.176147 + 0.984364i $$0.556363\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 9288.00 0.477765
$$724$$ 0 0
$$725$$ 1102.00 0.0564514
$$726$$ 0 0
$$727$$ 1512.00 0.0771348 0.0385674 0.999256i $$-0.487721\pi$$
0.0385674 + 0.999256i $$0.487721\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −14976.0 −0.757739
$$732$$ 0 0
$$733$$ 11244.0 0.566585 0.283292 0.959034i $$-0.408573\pi$$
0.283292 + 0.959034i $$0.408573\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8240.00 0.411838
$$738$$ 0 0
$$739$$ 1996.00 0.0993559 0.0496780 0.998765i $$-0.484180\pi$$
0.0496780 + 0.998765i $$0.484180\pi$$
$$740$$ 0 0
$$741$$ −3024.00 −0.149918
$$742$$ 0 0
$$743$$ 656.000 0.0323907 0.0161954 0.999869i $$-0.494845\pi$$
0.0161954 + 0.999869i $$0.494845\pi$$
$$744$$ 0 0
$$745$$ −12840.0 −0.631438
$$746$$ 0 0
$$747$$ 8316.00 0.407318
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −1056.00 −0.0513102 −0.0256551 0.999671i $$-0.508167\pi$$
−0.0256551 + 0.999671i $$0.508167\pi$$
$$752$$ 0 0
$$753$$ 2772.00 0.134153
$$754$$ 0 0
$$755$$ −1440.00 −0.0694132
$$756$$ 0 0
$$757$$ −18702.0 −0.897934 −0.448967 0.893548i $$-0.648208\pi$$
−0.448967 + 0.893548i $$0.648208\pi$$
$$758$$ 0 0
$$759$$ 10560.0 0.505011
$$760$$ 0 0
$$761$$ −17904.0 −0.852851 −0.426425 0.904523i $$-0.640227\pi$$
−0.426425 + 0.904523i $$0.640227\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −10368.0 −0.490008
$$766$$ 0 0
$$767$$ 41328.0 1.94559
$$768$$ 0 0
$$769$$ 7560.00 0.354513 0.177257 0.984165i $$-0.443278\pi$$
0.177257 + 0.984165i $$0.443278\pi$$
$$770$$ 0 0
$$771$$ −8280.00 −0.386766
$$772$$ 0 0
$$773$$ 14292.0 0.665003 0.332502 0.943103i $$-0.392107\pi$$
0.332502 + 0.943103i $$0.392107\pi$$
$$774$$ 0 0
$$775$$ −5016.00 −0.232490
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 5920.00 0.271235
$$782$$ 0 0
$$783$$ −1566.00 −0.0714742
$$784$$ 0 0
$$785$$ 22032.0 1.00173
$$786$$ 0 0
$$787$$ 26364.0 1.19412 0.597062 0.802195i $$-0.296335\pi$$
0.597062 + 0.802195i $$0.296335\pi$$
$$788$$ 0 0
$$789$$ −7080.00 −0.319461
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 41328.0 1.85069
$$794$$ 0 0
$$795$$ −25992.0 −1.15955
$$796$$ 0 0
$$797$$ −17220.0 −0.765325 −0.382662 0.923888i $$-0.624993\pi$$
−0.382662 + 0.923888i $$0.624993\pi$$
$$798$$ 0 0
$$799$$ −39168.0 −1.73425
$$800$$ 0 0
$$801$$ 6696.00 0.295370
$$802$$ 0 0
$$803$$ 4800.00 0.210944
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 12060.0 0.526062
$$808$$ 0 0
$$809$$ 16442.0 0.714549 0.357274 0.933999i $$-0.383706\pi$$
0.357274 + 0.933999i $$0.383706\pi$$
$$810$$ 0 0
$$811$$ −31332.0 −1.35662 −0.678308 0.734778i $$-0.737286\pi$$
−0.678308 + 0.734778i $$0.737286\pi$$
$$812$$ 0 0
$$813$$ −14400.0 −0.621193
$$814$$ 0 0
$$815$$ 10992.0 0.472433
$$816$$ 0 0
$$817$$ −1872.00 −0.0801628
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −25810.0 −1.09717 −0.548584 0.836095i $$-0.684833\pi$$
−0.548584 + 0.836095i $$0.684833\pi$$
$$822$$ 0 0
$$823$$ −12368.0 −0.523841 −0.261921 0.965089i $$-0.584356\pi$$
−0.261921 + 0.965089i $$0.584356\pi$$
$$824$$ 0 0
$$825$$ 1140.00 0.0481087
$$826$$ 0 0
$$827$$ −6316.00 −0.265573 −0.132786 0.991145i $$-0.542392\pi$$
−0.132786 + 0.991145i $$0.542392\pi$$
$$828$$ 0 0
$$829$$ 23868.0 0.999964 0.499982 0.866036i $$-0.333340\pi$$
0.499982 + 0.866036i $$0.333340\pi$$
$$830$$ 0 0
$$831$$ −19338.0 −0.807254
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −6048.00 −0.250658
$$836$$ 0 0
$$837$$ 7128.00 0.294360
$$838$$ 0 0
$$839$$ −48216.0 −1.98403 −0.992015 0.126120i $$-0.959748\pi$$
−0.992015 + 0.126120i $$0.959748\pi$$
$$840$$ 0 0
$$841$$ −21025.0 −0.862069
$$842$$ 0 0
$$843$$ 7806.00 0.318924
$$844$$ 0 0
$$845$$ −58308.0 −2.37379
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 20700.0 0.836775
$$850$$ 0 0
$$851$$ 45408.0 1.82910
$$852$$ 0 0
$$853$$ 27300.0 1.09582 0.547910 0.836537i $$-0.315424\pi$$
0.547910 + 0.836537i $$0.315424\pi$$
$$854$$ 0 0
$$855$$ −1296.00 −0.0518389
$$856$$ 0 0
$$857$$ −8640.00 −0.344384 −0.172192 0.985063i $$-0.555085\pi$$
−0.172192 + 0.985063i $$0.555085\pi$$
$$858$$ 0 0
$$859$$ 24372.0 0.968058 0.484029 0.875052i $$-0.339173\pi$$
0.484029 + 0.875052i $$0.339173\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −2176.00 −0.0858307 −0.0429154 0.999079i $$-0.513665\pi$$
−0.0429154 + 0.999079i $$0.513665\pi$$
$$864$$ 0 0
$$865$$ −22032.0 −0.866024
$$866$$ 0 0
$$867$$ −12909.0 −0.505666
$$868$$ 0 0
$$869$$ 15520.0 0.605846
$$870$$ 0 0
$$871$$ −34608.0 −1.34632
$$872$$ 0 0
$$873$$ 1512.00 0.0586179
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −27574.0 −1.06170 −0.530848 0.847467i $$-0.678127\pi$$
−0.530848 + 0.847467i $$0.678127\pi$$
$$878$$ 0 0
$$879$$ −13356.0 −0.512499
$$880$$ 0 0
$$881$$ −16968.0 −0.648884 −0.324442 0.945906i $$-0.605176\pi$$
−0.324442 + 0.945906i $$0.605176\pi$$
$$882$$ 0 0
$$883$$ 1860.00 0.0708879 0.0354439 0.999372i $$-0.488715\pi$$
0.0354439 + 0.999372i $$0.488715\pi$$
$$884$$ 0 0
$$885$$ 17712.0 0.672748
$$886$$ 0 0
$$887$$ 2280.00 0.0863077 0.0431538 0.999068i $$-0.486259\pi$$
0.0431538 + 0.999068i $$0.486259\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −1620.00 −0.0609114
$$892$$ 0 0
$$893$$ −4896.00 −0.183470
$$894$$ 0 0
$$895$$ 28464.0 1.06307
$$896$$ 0 0
$$897$$ −44352.0 −1.65091
$$898$$ 0 0
$$899$$ −15312.0 −0.568058
$$900$$ 0 0
$$901$$ −69312.0 −2.56284
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −13104.0 −0.481317
$$906$$ 0 0
$$907$$ −36084.0 −1.32100 −0.660501 0.750825i $$-0.729656\pi$$
−0.660501 + 0.750825i $$0.729656\pi$$
$$908$$ 0 0
$$909$$ 13716.0 0.500474
$$910$$ 0 0
$$911$$ −24152.0 −0.878366 −0.439183 0.898398i $$-0.644732\pi$$
−0.439183 + 0.898398i $$0.644732\pi$$
$$912$$ 0 0
$$913$$ −18480.0 −0.669878
$$914$$ 0 0
$$915$$ 17712.0 0.639935
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −36336.0 −1.30426 −0.652130 0.758108i $$-0.726124\pi$$
−0.652130 + 0.758108i $$0.726124\pi$$
$$920$$ 0 0
$$921$$ 7308.00 0.261462
$$922$$ 0 0
$$923$$ −24864.0 −0.886683
$$924$$ 0 0
$$925$$ 4902.00 0.174245
$$926$$ 0 0
$$927$$ −3672.00 −0.130102
$$928$$ 0 0
$$929$$ −432.000 −0.0152567 −0.00762834 0.999971i $$-0.502428\pi$$
−0.00762834 + 0.999971i $$0.502428\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 22464.0 0.788251
$$934$$ 0 0
$$935$$ 23040.0 0.805870
$$936$$ 0 0
$$937$$ −22176.0 −0.773168 −0.386584 0.922254i $$-0.626345\pi$$
−0.386584 + 0.922254i $$0.626345\pi$$
$$938$$ 0 0
$$939$$ −5256.00 −0.182666
$$940$$ 0 0
$$941$$ 43524.0 1.50780 0.753901 0.656988i $$-0.228170\pi$$
0.753901 + 0.656988i $$0.228170\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1868.00 −0.0640991 −0.0320495 0.999486i $$-0.510203\pi$$
−0.0320495 + 0.999486i $$0.510203\pi$$
$$948$$ 0 0
$$949$$ −20160.0 −0.689590
$$950$$ 0 0
$$951$$ 4686.00 0.159783
$$952$$ 0 0
$$953$$ −9238.00 −0.314006 −0.157003 0.987598i $$-0.550183\pi$$
−0.157003 + 0.987598i $$0.550183\pi$$
$$954$$ 0 0
$$955$$ 30144.0 1.02140
$$956$$ 0 0
$$957$$ 3480.00 0.117547
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 39905.0 1.33950
$$962$$ 0 0
$$963$$ 7380.00 0.246954
$$964$$ 0 0
$$965$$ 29160.0 0.972739
$$966$$ 0 0
$$967$$ 30616.0 1.01814 0.509071 0.860724i $$-0.329989\pi$$
0.509071 + 0.860724i $$0.329989\pi$$
$$968$$ 0 0
$$969$$ −3456.00 −0.114575
$$970$$ 0 0
$$971$$ −27540.0 −0.910196 −0.455098 0.890441i $$-0.650396\pi$$
−0.455098 + 0.890441i $$0.650396\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −4788.00 −0.157270
$$976$$ 0 0
$$977$$ −16402.0 −0.537100 −0.268550 0.963266i $$-0.586544\pi$$
−0.268550 + 0.963266i $$0.586544\pi$$
$$978$$ 0 0
$$979$$ −14880.0 −0.485768
$$980$$ 0 0
$$981$$ −8262.00 −0.268894
$$982$$ 0 0
$$983$$ 55176.0 1.79028 0.895138 0.445789i $$-0.147077\pi$$
0.895138 + 0.445789i $$0.147077\pi$$
$$984$$ 0 0
$$985$$ 21144.0 0.683963
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −27456.0 −0.882760
$$990$$ 0 0
$$991$$ −27096.0 −0.868550 −0.434275 0.900780i $$-0.642995\pi$$
−0.434275 + 0.900780i $$0.642995\pi$$
$$992$$ 0 0
$$993$$ −21276.0 −0.679933
$$994$$ 0 0
$$995$$ −37152.0 −1.18372
$$996$$ 0 0
$$997$$ 16812.0 0.534044 0.267022 0.963691i $$-0.413960\pi$$
0.267022 + 0.963691i $$0.413960\pi$$
$$998$$ 0 0
$$999$$ −6966.00 −0.220615
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.b.1.1 1
4.3 odd 2 147.4.a.e.1.1 yes 1
7.6 odd 2 2352.4.a.bi.1.1 1
12.11 even 2 441.4.a.h.1.1 1
28.3 even 6 147.4.e.f.79.1 2
28.11 odd 6 147.4.e.e.79.1 2
28.19 even 6 147.4.e.f.67.1 2
28.23 odd 6 147.4.e.e.67.1 2
28.27 even 2 147.4.a.d.1.1 1
84.11 even 6 441.4.e.f.226.1 2
84.23 even 6 441.4.e.f.361.1 2
84.47 odd 6 441.4.e.g.361.1 2
84.59 odd 6 441.4.e.g.226.1 2
84.83 odd 2 441.4.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.d.1.1 1 28.27 even 2
147.4.a.e.1.1 yes 1 4.3 odd 2
147.4.e.e.67.1 2 28.23 odd 6
147.4.e.e.79.1 2 28.11 odd 6
147.4.e.f.67.1 2 28.19 even 6
147.4.e.f.79.1 2 28.3 even 6
441.4.a.g.1.1 1 84.83 odd 2
441.4.a.h.1.1 1 12.11 even 2
441.4.e.f.226.1 2 84.11 even 6
441.4.e.f.361.1 2 84.23 even 6
441.4.e.g.226.1 2 84.59 odd 6
441.4.e.g.361.1 2 84.47 odd 6
2352.4.a.b.1.1 1 1.1 even 1 trivial
2352.4.a.bi.1.1 1 7.6 odd 2