# Properties

 Label 2352.4.a.cr.1.4 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: 4.4.145408.2 Defining polynomial: $$x^{4} - 24 x^{2} + 142$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1176) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$3.66254$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +16.6019 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +16.6019 q^{5} +9.00000 q^{9} +49.2481 q^{11} -64.4774 q^{13} +49.8056 q^{15} -132.429 q^{17} -82.0605 q^{19} -82.0466 q^{23} +150.622 q^{25} +27.0000 q^{27} +157.717 q^{29} -185.291 q^{31} +147.744 q^{33} -51.9002 q^{37} -193.432 q^{39} +49.4929 q^{41} -313.788 q^{43} +149.417 q^{45} -553.259 q^{47} -397.287 q^{51} -619.868 q^{53} +817.611 q^{55} -246.181 q^{57} -712.989 q^{59} +287.421 q^{61} -1070.45 q^{65} -226.636 q^{67} -246.140 q^{69} -55.3834 q^{71} -799.071 q^{73} +451.867 q^{75} +120.313 q^{79} +81.0000 q^{81} -857.862 q^{83} -2198.57 q^{85} +473.151 q^{87} +377.167 q^{89} -555.873 q^{93} -1362.36 q^{95} +1265.20 q^{97} +443.233 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{3} + 8q^{5} + 36q^{9} + O(q^{10})$$ $$4q + 12q^{3} + 8q^{5} + 36q^{9} + 40q^{11} + 48q^{13} + 24q^{15} - 152q^{17} - 224q^{19} + 8q^{23} - 28q^{25} + 108q^{27} - 144q^{29} - 400q^{31} + 120q^{33} - 304q^{37} + 144q^{39} - 152q^{41} - 160q^{43} + 72q^{45} - 544q^{47} - 456q^{51} - 1320q^{53} - 16q^{55} - 672q^{57} - 1040q^{59} + 896q^{61} - 648q^{65} + 416q^{67} + 24q^{69} - 248q^{71} - 752q^{73} - 84q^{75} - 864q^{79} + 324q^{81} - 1456q^{83} - 1608q^{85} - 432q^{87} + 2936q^{89} - 1200q^{93} + 80q^{95} + 144q^{97} + 360q^{99} + O(q^{100})$$

## 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
$$5$$ 16.6019 1.48492 0.742458 0.669892i $$-0.233660\pi$$
0.742458 + 0.669892i $$0.233660\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 49.2481 1.34990 0.674949 0.737865i $$-0.264166\pi$$
0.674949 + 0.737865i $$0.264166\pi$$
$$12$$ 0 0
$$13$$ −64.4774 −1.37560 −0.687801 0.725900i $$-0.741424\pi$$
−0.687801 + 0.725900i $$0.741424\pi$$
$$14$$ 0 0
$$15$$ 49.8056 0.857317
$$16$$ 0 0
$$17$$ −132.429 −1.88934 −0.944669 0.328024i $$-0.893617\pi$$
−0.944669 + 0.328024i $$0.893617\pi$$
$$18$$ 0 0
$$19$$ −82.0605 −0.990840 −0.495420 0.868654i $$-0.664986\pi$$
−0.495420 + 0.868654i $$0.664986\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −82.0466 −0.743822 −0.371911 0.928268i $$-0.621297\pi$$
−0.371911 + 0.928268i $$0.621297\pi$$
$$24$$ 0 0
$$25$$ 150.622 1.20498
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 157.717 1.00991 0.504954 0.863146i $$-0.331509\pi$$
0.504954 + 0.863146i $$0.331509\pi$$
$$30$$ 0 0
$$31$$ −185.291 −1.07352 −0.536762 0.843734i $$-0.680353\pi$$
−0.536762 + 0.843734i $$0.680353\pi$$
$$32$$ 0 0
$$33$$ 147.744 0.779363
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −51.9002 −0.230604 −0.115302 0.993331i $$-0.536784\pi$$
−0.115302 + 0.993331i $$0.536784\pi$$
$$38$$ 0 0
$$39$$ −193.432 −0.794204
$$40$$ 0 0
$$41$$ 49.4929 0.188524 0.0942622 0.995547i $$-0.469951\pi$$
0.0942622 + 0.995547i $$0.469951\pi$$
$$42$$ 0 0
$$43$$ −313.788 −1.11284 −0.556421 0.830901i $$-0.687826\pi$$
−0.556421 + 0.830901i $$0.687826\pi$$
$$44$$ 0 0
$$45$$ 149.417 0.494972
$$46$$ 0 0
$$47$$ −553.259 −1.71705 −0.858523 0.512775i $$-0.828618\pi$$
−0.858523 + 0.512775i $$0.828618\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −397.287 −1.09081
$$52$$ 0 0
$$53$$ −619.868 −1.60652 −0.803259 0.595630i $$-0.796903\pi$$
−0.803259 + 0.595630i $$0.796903\pi$$
$$54$$ 0 0
$$55$$ 817.611 2.00448
$$56$$ 0 0
$$57$$ −246.181 −0.572062
$$58$$ 0 0
$$59$$ −712.989 −1.57328 −0.786638 0.617414i $$-0.788180\pi$$
−0.786638 + 0.617414i $$0.788180\pi$$
$$60$$ 0 0
$$61$$ 287.421 0.603287 0.301644 0.953421i $$-0.402465\pi$$
0.301644 + 0.953421i $$0.402465\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −1070.45 −2.04265
$$66$$ 0 0
$$67$$ −226.636 −0.413254 −0.206627 0.978420i $$-0.566249\pi$$
−0.206627 + 0.978420i $$0.566249\pi$$
$$68$$ 0 0
$$69$$ −246.140 −0.429446
$$70$$ 0 0
$$71$$ −55.3834 −0.0925746 −0.0462873 0.998928i $$-0.514739\pi$$
−0.0462873 + 0.998928i $$0.514739\pi$$
$$72$$ 0 0
$$73$$ −799.071 −1.28115 −0.640577 0.767894i $$-0.721305\pi$$
−0.640577 + 0.767894i $$0.721305\pi$$
$$74$$ 0 0
$$75$$ 451.867 0.695694
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 120.313 0.171345 0.0856725 0.996323i $$-0.472696\pi$$
0.0856725 + 0.996323i $$0.472696\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −857.862 −1.13449 −0.567244 0.823550i $$-0.691991\pi$$
−0.567244 + 0.823550i $$0.691991\pi$$
$$84$$ 0 0
$$85$$ −2198.57 −2.80551
$$86$$ 0 0
$$87$$ 473.151 0.583071
$$88$$ 0 0
$$89$$ 377.167 0.449209 0.224604 0.974450i $$-0.427891\pi$$
0.224604 + 0.974450i $$0.427891\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −555.873 −0.619799
$$94$$ 0 0
$$95$$ −1362.36 −1.47132
$$96$$ 0 0
$$97$$ 1265.20 1.32435 0.662174 0.749351i $$-0.269634\pi$$
0.662174 + 0.749351i $$0.269634\pi$$
$$98$$ 0 0
$$99$$ 443.233 0.449966
$$100$$ 0 0
$$101$$ 2011.24 1.98144 0.990722 0.135906i $$-0.0433946\pi$$
0.990722 + 0.135906i $$0.0433946\pi$$
$$102$$ 0 0
$$103$$ −38.9204 −0.0372325 −0.0186162 0.999827i $$-0.505926\pi$$
−0.0186162 + 0.999827i $$0.505926\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1293.26 1.16845 0.584223 0.811593i $$-0.301399\pi$$
0.584223 + 0.811593i $$0.301399\pi$$
$$108$$ 0 0
$$109$$ 1403.08 1.23294 0.616470 0.787378i $$-0.288562\pi$$
0.616470 + 0.787378i $$0.288562\pi$$
$$110$$ 0 0
$$111$$ −155.701 −0.133139
$$112$$ 0 0
$$113$$ −448.950 −0.373749 −0.186875 0.982384i $$-0.559836\pi$$
−0.186875 + 0.982384i $$0.559836\pi$$
$$114$$ 0 0
$$115$$ −1362.13 −1.10451
$$116$$ 0 0
$$117$$ −580.297 −0.458534
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1094.38 0.822222
$$122$$ 0 0
$$123$$ 148.479 0.108845
$$124$$ 0 0
$$125$$ 425.377 0.304375
$$126$$ 0 0
$$127$$ 634.373 0.443240 0.221620 0.975133i $$-0.428866\pi$$
0.221620 + 0.975133i $$0.428866\pi$$
$$128$$ 0 0
$$129$$ −941.363 −0.642499
$$130$$ 0 0
$$131$$ −627.531 −0.418532 −0.209266 0.977859i $$-0.567107\pi$$
−0.209266 + 0.977859i $$0.567107\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 448.251 0.285772
$$136$$ 0 0
$$137$$ 1795.53 1.11973 0.559863 0.828585i $$-0.310854\pi$$
0.559863 + 0.828585i $$0.310854\pi$$
$$138$$ 0 0
$$139$$ 565.754 0.345228 0.172614 0.984990i $$-0.444779\pi$$
0.172614 + 0.984990i $$0.444779\pi$$
$$140$$ 0 0
$$141$$ −1659.78 −0.991337
$$142$$ 0 0
$$143$$ −3175.39 −1.85692
$$144$$ 0 0
$$145$$ 2618.40 1.49963
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −583.578 −0.320863 −0.160432 0.987047i $$-0.551289\pi$$
−0.160432 + 0.987047i $$0.551289\pi$$
$$150$$ 0 0
$$151$$ −874.848 −0.471484 −0.235742 0.971816i $$-0.575752\pi$$
−0.235742 + 0.971816i $$0.575752\pi$$
$$152$$ 0 0
$$153$$ −1191.86 −0.629780
$$154$$ 0 0
$$155$$ −3076.18 −1.59409
$$156$$ 0 0
$$157$$ 203.881 0.103640 0.0518201 0.998656i $$-0.483498\pi$$
0.0518201 + 0.998656i $$0.483498\pi$$
$$158$$ 0 0
$$159$$ −1859.61 −0.927524
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 2329.09 1.11919 0.559596 0.828765i $$-0.310956\pi$$
0.559596 + 0.828765i $$0.310956\pi$$
$$164$$ 0 0
$$165$$ 2452.83 1.15729
$$166$$ 0 0
$$167$$ −2742.47 −1.27077 −0.635385 0.772196i $$-0.719159\pi$$
−0.635385 + 0.772196i $$0.719159\pi$$
$$168$$ 0 0
$$169$$ 1960.34 0.892279
$$170$$ 0 0
$$171$$ −738.544 −0.330280
$$172$$ 0 0
$$173$$ −2797.24 −1.22931 −0.614654 0.788796i $$-0.710705\pi$$
−0.614654 + 0.788796i $$0.710705\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −2138.97 −0.908332
$$178$$ 0 0
$$179$$ 1363.49 0.569340 0.284670 0.958626i $$-0.408116\pi$$
0.284670 + 0.958626i $$0.408116\pi$$
$$180$$ 0 0
$$181$$ 1845.23 0.757763 0.378882 0.925445i $$-0.376309\pi$$
0.378882 + 0.925445i $$0.376309\pi$$
$$182$$ 0 0
$$183$$ 862.264 0.348308
$$184$$ 0 0
$$185$$ −861.640 −0.342427
$$186$$ 0 0
$$187$$ −6521.88 −2.55041
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −496.967 −0.188268 −0.0941342 0.995560i $$-0.530008\pi$$
−0.0941342 + 0.995560i $$0.530008\pi$$
$$192$$ 0 0
$$193$$ 3290.12 1.22709 0.613544 0.789660i $$-0.289743\pi$$
0.613544 + 0.789660i $$0.289743\pi$$
$$194$$ 0 0
$$195$$ −3211.34 −1.17933
$$196$$ 0 0
$$197$$ −2989.83 −1.08130 −0.540651 0.841247i $$-0.681822\pi$$
−0.540651 + 0.841247i $$0.681822\pi$$
$$198$$ 0 0
$$199$$ 4746.57 1.69083 0.845417 0.534108i $$-0.179352\pi$$
0.845417 + 0.534108i $$0.179352\pi$$
$$200$$ 0 0
$$201$$ −679.909 −0.238592
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 821.676 0.279943
$$206$$ 0 0
$$207$$ −738.419 −0.247941
$$208$$ 0 0
$$209$$ −4041.32 −1.33753
$$210$$ 0 0
$$211$$ −5034.43 −1.64258 −0.821289 0.570512i $$-0.806745\pi$$
−0.821289 + 0.570512i $$0.806745\pi$$
$$212$$ 0 0
$$213$$ −166.150 −0.0534480
$$214$$ 0 0
$$215$$ −5209.46 −1.65248
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −2397.21 −0.739674
$$220$$ 0 0
$$221$$ 8538.68 2.59898
$$222$$ 0 0
$$223$$ 2448.32 0.735208 0.367604 0.929982i $$-0.380178\pi$$
0.367604 + 0.929982i $$0.380178\pi$$
$$224$$ 0 0
$$225$$ 1355.60 0.401659
$$226$$ 0 0
$$227$$ 3573.33 1.04480 0.522401 0.852700i $$-0.325036\pi$$
0.522401 + 0.852700i $$0.325036\pi$$
$$228$$ 0 0
$$229$$ 3378.69 0.974978 0.487489 0.873129i $$-0.337913\pi$$
0.487489 + 0.873129i $$0.337913\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 3130.51 0.880199 0.440100 0.897949i $$-0.354943\pi$$
0.440100 + 0.897949i $$0.354943\pi$$
$$234$$ 0 0
$$235$$ −9185.14 −2.54967
$$236$$ 0 0
$$237$$ 360.939 0.0989261
$$238$$ 0 0
$$239$$ 3964.01 1.07285 0.536424 0.843949i $$-0.319775\pi$$
0.536424 + 0.843949i $$0.319775\pi$$
$$240$$ 0 0
$$241$$ 3173.03 0.848102 0.424051 0.905638i $$-0.360608\pi$$
0.424051 + 0.905638i $$0.360608\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5291.05 1.36300
$$248$$ 0 0
$$249$$ −2573.59 −0.654997
$$250$$ 0 0
$$251$$ −2509.92 −0.631174 −0.315587 0.948897i $$-0.602201\pi$$
−0.315587 + 0.948897i $$0.602201\pi$$
$$252$$ 0 0
$$253$$ −4040.64 −1.00408
$$254$$ 0 0
$$255$$ −6595.71 −1.61976
$$256$$ 0 0
$$257$$ 5695.39 1.38237 0.691184 0.722679i $$-0.257089\pi$$
0.691184 + 0.722679i $$0.257089\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 1419.45 0.336636
$$262$$ 0 0
$$263$$ 6420.02 1.50523 0.752614 0.658462i $$-0.228792\pi$$
0.752614 + 0.658462i $$0.228792\pi$$
$$264$$ 0 0
$$265$$ −10291.0 −2.38555
$$266$$ 0 0
$$267$$ 1131.50 0.259351
$$268$$ 0 0
$$269$$ 290.460 0.0658351 0.0329175 0.999458i $$-0.489520\pi$$
0.0329175 + 0.999458i $$0.489520\pi$$
$$270$$ 0 0
$$271$$ 3681.21 0.825157 0.412578 0.910922i $$-0.364628\pi$$
0.412578 + 0.910922i $$0.364628\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 7417.86 1.62660
$$276$$ 0 0
$$277$$ −5427.03 −1.17718 −0.588590 0.808432i $$-0.700317\pi$$
−0.588590 + 0.808432i $$0.700317\pi$$
$$278$$ 0 0
$$279$$ −1667.62 −0.357841
$$280$$ 0 0
$$281$$ −5237.25 −1.11184 −0.555922 0.831235i $$-0.687634\pi$$
−0.555922 + 0.831235i $$0.687634\pi$$
$$282$$ 0 0
$$283$$ −8302.83 −1.74400 −0.872000 0.489506i $$-0.837177\pi$$
−0.872000 + 0.489506i $$0.837177\pi$$
$$284$$ 0 0
$$285$$ −4087.07 −0.849464
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 12624.5 2.56960
$$290$$ 0 0
$$291$$ 3795.60 0.764612
$$292$$ 0 0
$$293$$ 2773.58 0.553018 0.276509 0.961011i $$-0.410822\pi$$
0.276509 + 0.961011i $$0.410822\pi$$
$$294$$ 0 0
$$295$$ −11837.0 −2.33618
$$296$$ 0 0
$$297$$ 1329.70 0.259788
$$298$$ 0 0
$$299$$ 5290.15 1.02320
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6033.72 1.14399
$$304$$ 0 0
$$305$$ 4771.73 0.895832
$$306$$ 0 0
$$307$$ −6528.52 −1.21369 −0.606844 0.794821i $$-0.707565\pi$$
−0.606844 + 0.794821i $$0.707565\pi$$
$$308$$ 0 0
$$309$$ −116.761 −0.0214962
$$310$$ 0 0
$$311$$ −4350.89 −0.793300 −0.396650 0.917970i $$-0.629827\pi$$
−0.396650 + 0.917970i $$0.629827\pi$$
$$312$$ 0 0
$$313$$ 5411.10 0.977168 0.488584 0.872517i $$-0.337514\pi$$
0.488584 + 0.872517i $$0.337514\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −7013.79 −1.24269 −0.621346 0.783536i $$-0.713414\pi$$
−0.621346 + 0.783536i $$0.713414\pi$$
$$318$$ 0 0
$$319$$ 7767.27 1.36327
$$320$$ 0 0
$$321$$ 3879.77 0.674603
$$322$$ 0 0
$$323$$ 10867.2 1.87203
$$324$$ 0 0
$$325$$ −9711.73 −1.65757
$$326$$ 0 0
$$327$$ 4209.23 0.711839
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −142.799 −0.0237128 −0.0118564 0.999930i $$-0.503774\pi$$
−0.0118564 + 0.999930i $$0.503774\pi$$
$$332$$ 0 0
$$333$$ −467.102 −0.0768679
$$334$$ 0 0
$$335$$ −3762.59 −0.613648
$$336$$ 0 0
$$337$$ −1489.10 −0.240702 −0.120351 0.992731i $$-0.538402\pi$$
−0.120351 + 0.992731i $$0.538402\pi$$
$$338$$ 0 0
$$339$$ −1346.85 −0.215784
$$340$$ 0 0
$$341$$ −9125.23 −1.44915
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −4086.38 −0.637691
$$346$$ 0 0
$$347$$ −4017.06 −0.621461 −0.310730 0.950498i $$-0.600574\pi$$
−0.310730 + 0.950498i $$0.600574\pi$$
$$348$$ 0 0
$$349$$ 4235.87 0.649687 0.324844 0.945768i $$-0.394688\pi$$
0.324844 + 0.945768i $$0.394688\pi$$
$$350$$ 0 0
$$351$$ −1740.89 −0.264735
$$352$$ 0 0
$$353$$ −11399.1 −1.71873 −0.859366 0.511361i $$-0.829142\pi$$
−0.859366 + 0.511361i $$0.829142\pi$$
$$354$$ 0 0
$$355$$ −919.468 −0.137466
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 713.980 0.104965 0.0524825 0.998622i $$-0.483287\pi$$
0.0524825 + 0.998622i $$0.483287\pi$$
$$360$$ 0 0
$$361$$ −125.080 −0.0182359
$$362$$ 0 0
$$363$$ 3283.13 0.474710
$$364$$ 0 0
$$365$$ −13266.1 −1.90241
$$366$$ 0 0
$$367$$ −12235.8 −1.74033 −0.870167 0.492757i $$-0.835989\pi$$
−0.870167 + 0.492757i $$0.835989\pi$$
$$368$$ 0 0
$$369$$ 445.437 0.0628415
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −9060.32 −1.25771 −0.628854 0.777523i $$-0.716476\pi$$
−0.628854 + 0.777523i $$0.716476\pi$$
$$374$$ 0 0
$$375$$ 1276.13 0.175731
$$376$$ 0 0
$$377$$ −10169.2 −1.38923
$$378$$ 0 0
$$379$$ −10607.9 −1.43770 −0.718852 0.695163i $$-0.755332\pi$$
−0.718852 + 0.695163i $$0.755332\pi$$
$$380$$ 0 0
$$381$$ 1903.12 0.255905
$$382$$ 0 0
$$383$$ 4281.14 0.571164 0.285582 0.958354i $$-0.407813\pi$$
0.285582 + 0.958354i $$0.407813\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −2824.09 −0.370947
$$388$$ 0 0
$$389$$ −4334.27 −0.564926 −0.282463 0.959278i $$-0.591151\pi$$
−0.282463 + 0.959278i $$0.591151\pi$$
$$390$$ 0 0
$$391$$ 10865.4 1.40533
$$392$$ 0 0
$$393$$ −1882.59 −0.241639
$$394$$ 0 0
$$395$$ 1997.42 0.254433
$$396$$ 0 0
$$397$$ 12347.5 1.56097 0.780485 0.625175i $$-0.214972\pi$$
0.780485 + 0.625175i $$0.214972\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 8321.66 1.03632 0.518159 0.855284i $$-0.326617\pi$$
0.518159 + 0.855284i $$0.326617\pi$$
$$402$$ 0 0
$$403$$ 11947.1 1.47674
$$404$$ 0 0
$$405$$ 1344.75 0.164991
$$406$$ 0 0
$$407$$ −2555.99 −0.311291
$$408$$ 0 0
$$409$$ −15242.4 −1.84276 −0.921382 0.388659i $$-0.872938\pi$$
−0.921382 + 0.388659i $$0.872938\pi$$
$$410$$ 0 0
$$411$$ 5386.59 0.646474
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −14242.1 −1.68462
$$416$$ 0 0
$$417$$ 1697.26 0.199317
$$418$$ 0 0
$$419$$ 8251.84 0.962121 0.481061 0.876687i $$-0.340252\pi$$
0.481061 + 0.876687i $$0.340252\pi$$
$$420$$ 0 0
$$421$$ −1825.97 −0.211383 −0.105691 0.994399i $$-0.533706\pi$$
−0.105691 + 0.994399i $$0.533706\pi$$
$$422$$ 0 0
$$423$$ −4979.34 −0.572349
$$424$$ 0 0
$$425$$ −19946.8 −2.27661
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −9526.18 −1.07209
$$430$$ 0 0
$$431$$ 11399.2 1.27397 0.636986 0.770876i $$-0.280181\pi$$
0.636986 + 0.770876i $$0.280181\pi$$
$$432$$ 0 0
$$433$$ −2484.03 −0.275692 −0.137846 0.990454i $$-0.544018\pi$$
−0.137846 + 0.990454i $$0.544018\pi$$
$$434$$ 0 0
$$435$$ 7855.20 0.865812
$$436$$ 0 0
$$437$$ 6732.78 0.737008
$$438$$ 0 0
$$439$$ −11572.5 −1.25814 −0.629071 0.777348i $$-0.716565\pi$$
−0.629071 + 0.777348i $$0.716565\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −11072.6 −1.18753 −0.593766 0.804638i $$-0.702360\pi$$
−0.593766 + 0.804638i $$0.702360\pi$$
$$444$$ 0 0
$$445$$ 6261.67 0.667038
$$446$$ 0 0
$$447$$ −1750.73 −0.185250
$$448$$ 0 0
$$449$$ 12248.0 1.28735 0.643673 0.765300i $$-0.277410\pi$$
0.643673 + 0.765300i $$0.277410\pi$$
$$450$$ 0 0
$$451$$ 2437.43 0.254489
$$452$$ 0 0
$$453$$ −2624.54 −0.272211
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 12667.5 1.29663 0.648315 0.761372i $$-0.275474\pi$$
0.648315 + 0.761372i $$0.275474\pi$$
$$458$$ 0 0
$$459$$ −3575.58 −0.363603
$$460$$ 0 0
$$461$$ −11444.0 −1.15618 −0.578089 0.815973i $$-0.696202\pi$$
−0.578089 + 0.815973i $$0.696202\pi$$
$$462$$ 0 0
$$463$$ −5577.93 −0.559888 −0.279944 0.960016i $$-0.590316\pi$$
−0.279944 + 0.960016i $$0.590316\pi$$
$$464$$ 0 0
$$465$$ −9228.53 −0.920350
$$466$$ 0 0
$$467$$ 13494.5 1.33716 0.668578 0.743642i $$-0.266903\pi$$
0.668578 + 0.743642i $$0.266903\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 611.644 0.0598366
$$472$$ 0 0
$$473$$ −15453.5 −1.50222
$$474$$ 0 0
$$475$$ −12360.1 −1.19394
$$476$$ 0 0
$$477$$ −5578.82 −0.535506
$$478$$ 0 0
$$479$$ −5321.13 −0.507576 −0.253788 0.967260i $$-0.581677\pi$$
−0.253788 + 0.967260i $$0.581677\pi$$
$$480$$ 0 0
$$481$$ 3346.39 0.317219
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 21004.7 1.96655
$$486$$ 0 0
$$487$$ −5915.38 −0.550414 −0.275207 0.961385i $$-0.588746\pi$$
−0.275207 + 0.961385i $$0.588746\pi$$
$$488$$ 0 0
$$489$$ 6987.27 0.646166
$$490$$ 0 0
$$491$$ −20748.9 −1.90710 −0.953548 0.301240i $$-0.902600\pi$$
−0.953548 + 0.301240i $$0.902600\pi$$
$$492$$ 0 0
$$493$$ −20886.3 −1.90806
$$494$$ 0 0
$$495$$ 7358.50 0.668162
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −10673.6 −0.957546 −0.478773 0.877939i $$-0.658918\pi$$
−0.478773 + 0.877939i $$0.658918\pi$$
$$500$$ 0 0
$$501$$ −8227.41 −0.733679
$$502$$ 0 0
$$503$$ −18494.4 −1.63941 −0.819706 0.572784i $$-0.805863\pi$$
−0.819706 + 0.572784i $$0.805863\pi$$
$$504$$ 0 0
$$505$$ 33390.3 2.94228
$$506$$ 0 0
$$507$$ 5881.01 0.515158
$$508$$ 0 0
$$509$$ 5766.36 0.502140 0.251070 0.967969i $$-0.419218\pi$$
0.251070 + 0.967969i $$0.419218\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −2215.63 −0.190687
$$514$$ 0 0
$$515$$ −646.152 −0.0552871
$$516$$ 0 0
$$517$$ −27247.0 −2.31784
$$518$$ 0 0
$$519$$ −8391.73 −0.709742
$$520$$ 0 0
$$521$$ −4394.77 −0.369555 −0.184778 0.982780i $$-0.559156\pi$$
−0.184778 + 0.982780i $$0.559156\pi$$
$$522$$ 0 0
$$523$$ 1063.00 0.0888752 0.0444376 0.999012i $$-0.485850\pi$$
0.0444376 + 0.999012i $$0.485850\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 24537.9 2.02825
$$528$$ 0 0
$$529$$ −5435.35 −0.446729
$$530$$ 0 0
$$531$$ −6416.90 −0.524425
$$532$$ 0 0
$$533$$ −3191.18 −0.259334
$$534$$ 0 0
$$535$$ 21470.5 1.73505
$$536$$ 0 0
$$537$$ 4090.47 0.328709
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −4002.20 −0.318056 −0.159028 0.987274i $$-0.550836\pi$$
−0.159028 + 0.987274i $$0.550836\pi$$
$$542$$ 0 0
$$543$$ 5535.70 0.437495
$$544$$ 0 0
$$545$$ 23293.7 1.83081
$$546$$ 0 0
$$547$$ −13365.7 −1.04475 −0.522373 0.852717i $$-0.674953\pi$$
−0.522373 + 0.852717i $$0.674953\pi$$
$$548$$ 0 0
$$549$$ 2586.79 0.201096
$$550$$ 0 0
$$551$$ −12942.3 −1.00066
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −2584.92 −0.197700
$$556$$ 0 0
$$557$$ 5675.70 0.431754 0.215877 0.976421i $$-0.430739\pi$$
0.215877 + 0.976421i $$0.430739\pi$$
$$558$$ 0 0
$$559$$ 20232.2 1.53083
$$560$$ 0 0
$$561$$ −19565.6 −1.47248
$$562$$ 0 0
$$563$$ −281.060 −0.0210396 −0.0105198 0.999945i $$-0.503349\pi$$
−0.0105198 + 0.999945i $$0.503349\pi$$
$$564$$ 0 0
$$565$$ −7453.41 −0.554987
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6736.73 0.496342 0.248171 0.968716i $$-0.420171\pi$$
0.248171 + 0.968716i $$0.420171\pi$$
$$570$$ 0 0
$$571$$ 16445.8 1.20532 0.602659 0.797999i $$-0.294108\pi$$
0.602659 + 0.797999i $$0.294108\pi$$
$$572$$ 0 0
$$573$$ −1490.90 −0.108697
$$574$$ 0 0
$$575$$ −12358.0 −0.896289
$$576$$ 0 0
$$577$$ −6097.01 −0.439900 −0.219950 0.975511i $$-0.570589\pi$$
−0.219950 + 0.975511i $$0.570589\pi$$
$$578$$ 0 0
$$579$$ 9870.36 0.708460
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −30527.4 −2.16863
$$584$$ 0 0
$$585$$ −9634.01 −0.680884
$$586$$ 0 0
$$587$$ −19531.2 −1.37332 −0.686659 0.726980i $$-0.740923\pi$$
−0.686659 + 0.726980i $$0.740923\pi$$
$$588$$ 0 0
$$589$$ 15205.1 1.06369
$$590$$ 0 0
$$591$$ −8969.48 −0.624290
$$592$$ 0 0
$$593$$ −17701.7 −1.22583 −0.612917 0.790147i $$-0.710004\pi$$
−0.612917 + 0.790147i $$0.710004\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 14239.7 0.976203
$$598$$ 0 0
$$599$$ 16786.0 1.14500 0.572501 0.819904i $$-0.305973\pi$$
0.572501 + 0.819904i $$0.305973\pi$$
$$600$$ 0 0
$$601$$ −6281.32 −0.426324 −0.213162 0.977017i $$-0.568376\pi$$
−0.213162 + 0.977017i $$0.568376\pi$$
$$602$$ 0 0
$$603$$ −2039.73 −0.137751
$$604$$ 0 0
$$605$$ 18168.7 1.22093
$$606$$ 0 0
$$607$$ −2452.50 −0.163993 −0.0819967 0.996633i $$-0.526130\pi$$
−0.0819967 + 0.996633i $$0.526130\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 35672.7 2.36197
$$612$$ 0 0
$$613$$ 14792.7 0.974670 0.487335 0.873215i $$-0.337969\pi$$
0.487335 + 0.873215i $$0.337969\pi$$
$$614$$ 0 0
$$615$$ 2465.03 0.161625
$$616$$ 0 0
$$617$$ −15044.2 −0.981614 −0.490807 0.871268i $$-0.663298\pi$$
−0.490807 + 0.871268i $$0.663298\pi$$
$$618$$ 0 0
$$619$$ 8948.32 0.581039 0.290520 0.956869i $$-0.406172\pi$$
0.290520 + 0.956869i $$0.406172\pi$$
$$620$$ 0 0
$$621$$ −2215.26 −0.143149
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −11765.7 −0.753006
$$626$$ 0 0
$$627$$ −12124.0 −0.772225
$$628$$ 0 0
$$629$$ 6873.09 0.435688
$$630$$ 0 0
$$631$$ 27769.9 1.75198 0.875991 0.482327i $$-0.160208\pi$$
0.875991 + 0.482327i $$0.160208\pi$$
$$632$$ 0 0
$$633$$ −15103.3 −0.948343
$$634$$ 0 0
$$635$$ 10531.8 0.658174
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −498.451 −0.0308582
$$640$$ 0 0
$$641$$ −15807.9 −0.974063 −0.487031 0.873384i $$-0.661920\pi$$
−0.487031 + 0.873384i $$0.661920\pi$$
$$642$$ 0 0
$$643$$ 21911.4 1.34386 0.671929 0.740616i $$-0.265466\pi$$
0.671929 + 0.740616i $$0.265466\pi$$
$$644$$ 0 0
$$645$$ −15628.4 −0.954058
$$646$$ 0 0
$$647$$ 11245.6 0.683323 0.341662 0.939823i $$-0.389010\pi$$
0.341662 + 0.939823i $$0.389010\pi$$
$$648$$ 0 0
$$649$$ −35113.4 −2.12376
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 8668.43 0.519482 0.259741 0.965678i $$-0.416363\pi$$
0.259741 + 0.965678i $$0.416363\pi$$
$$654$$ 0 0
$$655$$ −10418.2 −0.621484
$$656$$ 0 0
$$657$$ −7191.64 −0.427051
$$658$$ 0 0
$$659$$ 12625.9 0.746334 0.373167 0.927764i $$-0.378272\pi$$
0.373167 + 0.927764i $$0.378272\pi$$
$$660$$ 0 0
$$661$$ 1884.10 0.110867 0.0554334 0.998462i $$-0.482346\pi$$
0.0554334 + 0.998462i $$0.482346\pi$$
$$662$$ 0 0
$$663$$ 25616.0 1.50052
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −12940.2 −0.751192
$$668$$ 0 0
$$669$$ 7344.95 0.424473
$$670$$ 0 0
$$671$$ 14155.0 0.814376
$$672$$ 0 0
$$673$$ −25932.5 −1.48533 −0.742663 0.669665i $$-0.766438\pi$$
−0.742663 + 0.669665i $$0.766438\pi$$
$$674$$ 0 0
$$675$$ 4066.80 0.231898
$$676$$ 0 0
$$677$$ −18226.4 −1.03471 −0.517354 0.855772i $$-0.673083\pi$$
−0.517354 + 0.855772i $$0.673083\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 10720.0 0.603217
$$682$$ 0 0
$$683$$ 10680.6 0.598362 0.299181 0.954196i $$-0.403287\pi$$
0.299181 + 0.954196i $$0.403287\pi$$
$$684$$ 0 0
$$685$$ 29809.2 1.66270
$$686$$ 0 0
$$687$$ 10136.1 0.562904
$$688$$ 0 0
$$689$$ 39967.5 2.20993
$$690$$ 0 0
$$691$$ −20500.4 −1.12862 −0.564308 0.825565i $$-0.690857\pi$$
−0.564308 + 0.825565i $$0.690857\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 9392.58 0.512634
$$696$$ 0 0
$$697$$ −6554.30 −0.356186
$$698$$ 0 0
$$699$$ 9391.53 0.508183
$$700$$ 0 0
$$701$$ 13903.9 0.749137 0.374568 0.927199i $$-0.377791\pi$$
0.374568 + 0.927199i $$0.377791\pi$$
$$702$$ 0 0
$$703$$ 4258.95 0.228491
$$704$$ 0 0
$$705$$ −27555.4 −1.47205
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 18055.3 0.956391 0.478195 0.878253i $$-0.341291\pi$$
0.478195 + 0.878253i $$0.341291\pi$$
$$710$$ 0 0
$$711$$ 1082.82 0.0571150
$$712$$ 0 0
$$713$$ 15202.5 0.798510
$$714$$ 0 0
$$715$$ −52717.5 −2.75737
$$716$$ 0 0
$$717$$ 11892.0 0.619409
$$718$$ 0 0
$$719$$ 9649.29 0.500498 0.250249 0.968182i $$-0.419488\pi$$
0.250249 + 0.968182i $$0.419488\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 9519.08 0.489652
$$724$$ 0 0
$$725$$ 23755.7 1.21692
$$726$$ 0 0
$$727$$ −12451.5 −0.635214 −0.317607 0.948222i $$-0.602879\pi$$
−0.317607 + 0.948222i $$0.602879\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 41554.6 2.10253
$$732$$ 0 0
$$733$$ 21799.1 1.09845 0.549227 0.835673i $$-0.314922\pi$$
0.549227 + 0.835673i $$0.314922\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −11161.4 −0.557850
$$738$$ 0 0
$$739$$ 15158.8 0.754568 0.377284 0.926098i $$-0.376858\pi$$
0.377284 + 0.926098i $$0.376858\pi$$
$$740$$ 0 0
$$741$$ 15873.1 0.786929
$$742$$ 0 0
$$743$$ −5094.25 −0.251534 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$744$$ 0 0
$$745$$ −9688.49 −0.476455
$$746$$ 0 0
$$747$$ −7720.76 −0.378163
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 13213.6 0.642038 0.321019 0.947073i $$-0.395975\pi$$
0.321019 + 0.947073i $$0.395975\pi$$
$$752$$ 0 0
$$753$$ −7529.76 −0.364408
$$754$$ 0 0
$$755$$ −14524.1 −0.700114
$$756$$ 0 0
$$757$$ 32248.9 1.54836 0.774180 0.632966i $$-0.218163\pi$$
0.774180 + 0.632966i $$0.218163\pi$$
$$758$$ 0 0
$$759$$ −12121.9 −0.579707
$$760$$ 0 0
$$761$$ −15715.4 −0.748596 −0.374298 0.927308i $$-0.622116\pi$$
−0.374298 + 0.927308i $$0.622116\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −19787.1 −0.935170
$$766$$ 0 0
$$767$$ 45971.7 2.16420
$$768$$ 0 0
$$769$$ −24907.4 −1.16799 −0.583994 0.811758i $$-0.698511\pi$$
−0.583994 + 0.811758i $$0.698511\pi$$
$$770$$ 0 0
$$771$$ 17086.2 0.798111
$$772$$ 0 0
$$773$$ 22027.2 1.02492 0.512461 0.858710i $$-0.328734\pi$$
0.512461 + 0.858710i $$0.328734\pi$$
$$774$$ 0 0
$$775$$ −27908.9 −1.29357
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −4061.41 −0.186798
$$780$$ 0 0
$$781$$ −2727.53 −0.124966
$$782$$ 0 0
$$783$$ 4258.36 0.194357
$$784$$ 0 0
$$785$$ 3384.81 0.153897
$$786$$ 0 0
$$787$$ −38388.7 −1.73877 −0.869384 0.494137i $$-0.835484\pi$$
−0.869384 + 0.494137i $$0.835484\pi$$
$$788$$ 0 0
$$789$$ 19260.0 0.869044
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −18532.2 −0.829883
$$794$$ 0 0
$$795$$ −30872.9 −1.37730
$$796$$ 0 0
$$797$$ −28628.1 −1.27235 −0.636174 0.771546i $$-0.719484\pi$$
−0.636174 + 0.771546i $$0.719484\pi$$
$$798$$ 0 0
$$799$$ 73267.6 3.24408
$$800$$ 0 0
$$801$$ 3394.50 0.149736
$$802$$ 0 0
$$803$$ −39352.8 −1.72943
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 871.379 0.0380099
$$808$$ 0 0
$$809$$ −11542.2 −0.501608 −0.250804 0.968038i $$-0.580695\pi$$
−0.250804 + 0.968038i $$0.580695\pi$$
$$810$$ 0 0
$$811$$ −29374.0 −1.27184 −0.635919 0.771756i $$-0.719379\pi$$
−0.635919 + 0.771756i $$0.719379\pi$$
$$812$$ 0 0
$$813$$ 11043.6 0.476404
$$814$$ 0 0
$$815$$ 38667.2 1.66191
$$816$$ 0 0
$$817$$ 25749.6 1.10265
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −17945.2 −0.762839 −0.381419 0.924402i $$-0.624565\pi$$
−0.381419 + 0.924402i $$0.624565\pi$$
$$822$$ 0 0
$$823$$ −42524.3 −1.80110 −0.900549 0.434755i $$-0.856835\pi$$
−0.900549 + 0.434755i $$0.856835\pi$$
$$824$$ 0 0
$$825$$ 22253.6 0.939116
$$826$$ 0 0
$$827$$ 43291.0 1.82029 0.910143 0.414293i $$-0.135971\pi$$
0.910143 + 0.414293i $$0.135971\pi$$
$$828$$ 0 0
$$829$$ 28580.9 1.19741 0.598707 0.800968i $$-0.295681\pi$$
0.598707 + 0.800968i $$0.295681\pi$$
$$830$$ 0 0
$$831$$ −16281.1 −0.679645
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −45530.1 −1.88699
$$836$$ 0 0
$$837$$ −5002.85 −0.206600
$$838$$ 0 0
$$839$$ 11526.0 0.474281 0.237141 0.971475i $$-0.423790\pi$$
0.237141 + 0.971475i $$0.423790\pi$$
$$840$$ 0 0
$$841$$ 485.697 0.0199146
$$842$$ 0 0
$$843$$ −15711.7 −0.641923
$$844$$ 0 0
$$845$$ 32545.3 1.32496
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −24908.5 −1.00690
$$850$$ 0 0
$$851$$ 4258.23 0.171528
$$852$$ 0 0
$$853$$ −25692.6 −1.03130 −0.515650 0.856799i $$-0.672449\pi$$
−0.515650 + 0.856799i $$0.672449\pi$$
$$854$$ 0 0
$$855$$ −12261.2 −0.490438
$$856$$ 0 0
$$857$$ 33824.0 1.34820 0.674100 0.738640i $$-0.264532\pi$$
0.674100 + 0.738640i $$0.264532\pi$$
$$858$$ 0 0
$$859$$ 13254.7 0.526479 0.263240 0.964730i $$-0.415209\pi$$
0.263240 + 0.964730i $$0.415209\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −5166.82 −0.203801 −0.101901 0.994795i $$-0.532492\pi$$
−0.101901 + 0.994795i $$0.532492\pi$$
$$864$$ 0 0
$$865$$ −46439.5 −1.82542
$$866$$ 0 0
$$867$$ 37873.4 1.48356
$$868$$ 0 0
$$869$$ 5925.18 0.231298
$$870$$ 0 0
$$871$$ 14612.9 0.568473
$$872$$ 0 0
$$873$$ 11386.8 0.441449
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −25268.0 −0.972907 −0.486453 0.873707i $$-0.661710\pi$$
−0.486453 + 0.873707i $$0.661710\pi$$
$$878$$ 0 0
$$879$$ 8320.74 0.319285
$$880$$ 0 0
$$881$$ −42652.8 −1.63111 −0.815555 0.578680i $$-0.803568\pi$$
−0.815555 + 0.578680i $$0.803568\pi$$
$$882$$ 0 0
$$883$$ −31070.9 −1.18416 −0.592082 0.805877i $$-0.701694\pi$$
−0.592082 + 0.805877i $$0.701694\pi$$
$$884$$ 0 0
$$885$$ −35510.9 −1.34880
$$886$$ 0 0
$$887$$ 36035.6 1.36410 0.682050 0.731306i $$-0.261089\pi$$
0.682050 + 0.731306i $$0.261089\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 3989.10 0.149989
$$892$$ 0 0
$$893$$ 45400.7 1.70132
$$894$$ 0 0
$$895$$ 22636.5 0.845423
$$896$$ 0 0
$$897$$ 15870.5 0.590746
$$898$$ 0 0
$$899$$ −29223.5 −1.08416
$$900$$ 0 0
$$901$$ 82088.6 3.03526
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 30634.3 1.12522
$$906$$ 0 0
$$907$$ −17945.5 −0.656967 −0.328484 0.944510i $$-0.606538\pi$$
−0.328484 + 0.944510i $$0.606538\pi$$
$$908$$ 0 0
$$909$$ 18101.2 0.660481
$$910$$ 0 0
$$911$$ −17672.5 −0.642717 −0.321359 0.946958i $$-0.604139\pi$$
−0.321359 + 0.946958i $$0.604139\pi$$
$$912$$ 0 0
$$913$$ −42248.1 −1.53144
$$914$$ 0 0
$$915$$ 14315.2 0.517209
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −30422.3 −1.09199 −0.545995 0.837788i $$-0.683848\pi$$
−0.545995 + 0.837788i $$0.683848\pi$$
$$920$$ 0 0
$$921$$ −19585.6 −0.700723
$$922$$ 0 0
$$923$$ 3570.98 0.127346
$$924$$ 0 0
$$925$$ −7817.32 −0.277872
$$926$$ 0 0
$$927$$ −350.284 −0.0124108
$$928$$ 0 0
$$929$$ −16017.6 −0.565683 −0.282841 0.959167i $$-0.591277\pi$$
−0.282841 + 0.959167i $$0.591277\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −13052.7 −0.458012
$$934$$ 0 0
$$935$$ −108275. −3.78715
$$936$$ 0 0
$$937$$ 7370.43 0.256970 0.128485 0.991711i $$-0.458989\pi$$
0.128485 + 0.991711i $$0.458989\pi$$
$$938$$ 0 0
$$939$$ 16233.3 0.564168
$$940$$ 0 0
$$941$$ −14999.7 −0.519634 −0.259817 0.965658i $$-0.583662\pi$$
−0.259817 + 0.965658i $$0.583662\pi$$
$$942$$ 0 0
$$943$$ −4060.73 −0.140229
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −12699.6 −0.435776 −0.217888 0.975974i $$-0.569917\pi$$
−0.217888 + 0.975974i $$0.569917\pi$$
$$948$$ 0 0
$$949$$ 51522.0 1.76236
$$950$$ 0 0
$$951$$ −21041.4 −0.717469
$$952$$ 0 0
$$953$$ 5347.41 0.181762 0.0908811 0.995862i $$-0.471032\pi$$
0.0908811 + 0.995862i $$0.471032\pi$$
$$954$$ 0 0
$$955$$ −8250.59 −0.279563
$$956$$ 0 0
$$957$$ 23301.8 0.787086
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 4541.70 0.152452
$$962$$ 0 0
$$963$$ 11639.3 0.389482
$$964$$ 0 0
$$965$$ 54622.2 1.82212
$$966$$ 0 0
$$967$$ −38787.2 −1.28988 −0.644939 0.764234i $$-0.723117\pi$$
−0.644939 + 0.764234i $$0.723117\pi$$
$$968$$ 0 0
$$969$$ 32601.6 1.08082
$$970$$ 0 0
$$971$$ −28064.9 −0.927545 −0.463772 0.885954i $$-0.653504\pi$$
−0.463772 + 0.885954i $$0.653504\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −29135.2 −0.956998
$$976$$ 0 0
$$977$$ 20896.8 0.684287 0.342144 0.939648i $$-0.388847\pi$$
0.342144 + 0.939648i $$0.388847\pi$$
$$978$$ 0 0
$$979$$ 18574.8 0.606386
$$980$$ 0 0
$$981$$ 12627.7 0.410980
$$982$$ 0 0
$$983$$ −1791.08 −0.0581147 −0.0290573 0.999578i $$-0.509251\pi$$
−0.0290573 + 0.999578i $$0.509251\pi$$
$$984$$ 0 0
$$985$$ −49636.7 −1.60564
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 25745.2 0.827755
$$990$$ 0 0
$$991$$ −22843.9 −0.732251 −0.366126 0.930565i $$-0.619316\pi$$
−0.366126 + 0.930565i $$0.619316\pi$$
$$992$$ 0 0
$$993$$ −428.396 −0.0136906
$$994$$ 0 0
$$995$$ 78802.0 2.51075
$$996$$ 0 0
$$997$$ 12875.8 0.409007 0.204504 0.978866i $$-0.434442\pi$$
0.204504 + 0.978866i $$0.434442\pi$$
$$998$$ 0 0
$$999$$ −1401.30 −0.0443797
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.cr.1.4 4
4.3 odd 2 1176.4.a.bb.1.4 4
7.6 odd 2 2352.4.a.ck.1.1 4
28.27 even 2 1176.4.a.bc.1.1 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.bb.1.4 4 4.3 odd 2
1176.4.a.bc.1.1 yes 4 28.27 even 2
2352.4.a.ck.1.1 4 7.6 odd 2
2352.4.a.cr.1.4 4 1.1 even 1 trivial