Properties

 Label 120.4.a.b.1.1 Level $120$ Weight $4$ Character 120.1 Self dual yes Analytic conductor $7.080$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$120 = 2^{3} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 120.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$7.08022920069$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 120.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -5.00000 q^{5} +20.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -5.00000 q^{5} +20.0000 q^{7} +9.00000 q^{9} -56.0000 q^{11} -86.0000 q^{13} +15.0000 q^{15} -106.000 q^{17} +4.00000 q^{19} -60.0000 q^{21} +136.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -206.000 q^{29} -152.000 q^{31} +168.000 q^{33} -100.000 q^{35} +282.000 q^{37} +258.000 q^{39} -246.000 q^{41} +412.000 q^{43} -45.0000 q^{45} +40.0000 q^{47} +57.0000 q^{49} +318.000 q^{51} -126.000 q^{53} +280.000 q^{55} -12.0000 q^{57} +56.0000 q^{59} -2.00000 q^{61} +180.000 q^{63} +430.000 q^{65} -388.000 q^{67} -408.000 q^{69} -672.000 q^{71} +1170.00 q^{73} -75.0000 q^{75} -1120.00 q^{77} +408.000 q^{79} +81.0000 q^{81} +668.000 q^{83} +530.000 q^{85} +618.000 q^{87} +66.0000 q^{89} -1720.00 q^{91} +456.000 q^{93} -20.0000 q^{95} -926.000 q^{97} -504.000 q^{99} +O(q^{100})$$

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
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 20.0000 1.07990 0.539949 0.841698i $$-0.318443\pi$$
0.539949 + 0.841698i $$0.318443\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −56.0000 −1.53497 −0.767483 0.641069i $$-0.778491\pi$$
−0.767483 + 0.641069i $$0.778491\pi$$
$$12$$ 0 0
$$13$$ −86.0000 −1.83478 −0.917389 0.397992i $$-0.869707\pi$$
−0.917389 + 0.397992i $$0.869707\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ 0 0
$$17$$ −106.000 −1.51228 −0.756140 0.654409i $$-0.772917\pi$$
−0.756140 + 0.654409i $$0.772917\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.0482980 0.0241490 0.999708i $$-0.492312\pi$$
0.0241490 + 0.999708i $$0.492312\pi$$
$$20$$ 0 0
$$21$$ −60.0000 −0.623480
$$22$$ 0 0
$$23$$ 136.000 1.23295 0.616477 0.787373i $$-0.288559\pi$$
0.616477 + 0.787373i $$0.288559\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −206.000 −1.31908 −0.659539 0.751671i $$-0.729248\pi$$
−0.659539 + 0.751671i $$0.729248\pi$$
$$30$$ 0 0
$$31$$ −152.000 −0.880645 −0.440323 0.897840i $$-0.645136\pi$$
−0.440323 + 0.897840i $$0.645136\pi$$
$$32$$ 0 0
$$33$$ 168.000 0.886214
$$34$$ 0 0
$$35$$ −100.000 −0.482945
$$36$$ 0 0
$$37$$ 282.000 1.25299 0.626493 0.779427i $$-0.284490\pi$$
0.626493 + 0.779427i $$0.284490\pi$$
$$38$$ 0 0
$$39$$ 258.000 1.05931
$$40$$ 0 0
$$41$$ −246.000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 412.000 1.46115 0.730575 0.682833i $$-0.239252\pi$$
0.730575 + 0.682833i $$0.239252\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 0 0
$$47$$ 40.0000 0.124140 0.0620702 0.998072i $$-0.480230\pi$$
0.0620702 + 0.998072i $$0.480230\pi$$
$$48$$ 0 0
$$49$$ 57.0000 0.166181
$$50$$ 0 0
$$51$$ 318.000 0.873116
$$52$$ 0 0
$$53$$ −126.000 −0.326555 −0.163278 0.986580i $$-0.552207\pi$$
−0.163278 + 0.986580i $$0.552207\pi$$
$$54$$ 0 0
$$55$$ 280.000 0.686458
$$56$$ 0 0
$$57$$ −12.0000 −0.0278849
$$58$$ 0 0
$$59$$ 56.0000 0.123569 0.0617846 0.998090i $$-0.480321\pi$$
0.0617846 + 0.998090i $$0.480321\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.00419793 −0.00209897 0.999998i $$-0.500668\pi$$
−0.00209897 + 0.999998i $$0.500668\pi$$
$$62$$ 0 0
$$63$$ 180.000 0.359966
$$64$$ 0 0
$$65$$ 430.000 0.820537
$$66$$ 0 0
$$67$$ −388.000 −0.707489 −0.353744 0.935342i $$-0.615092\pi$$
−0.353744 + 0.935342i $$0.615092\pi$$
$$68$$ 0 0
$$69$$ −408.000 −0.711847
$$70$$ 0 0
$$71$$ −672.000 −1.12326 −0.561632 0.827387i $$-0.689826\pi$$
−0.561632 + 0.827387i $$0.689826\pi$$
$$72$$ 0 0
$$73$$ 1170.00 1.87586 0.937932 0.346818i $$-0.112738\pi$$
0.937932 + 0.346818i $$0.112738\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −0.115470
$$76$$ 0 0
$$77$$ −1120.00 −1.65761
$$78$$ 0 0
$$79$$ 408.000 0.581058 0.290529 0.956866i $$-0.406169\pi$$
0.290529 + 0.956866i $$0.406169\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 668.000 0.883404 0.441702 0.897162i $$-0.354375\pi$$
0.441702 + 0.897162i $$0.354375\pi$$
$$84$$ 0 0
$$85$$ 530.000 0.676313
$$86$$ 0 0
$$87$$ 618.000 0.761570
$$88$$ 0 0
$$89$$ 66.0000 0.0786066 0.0393033 0.999227i $$-0.487486\pi$$
0.0393033 + 0.999227i $$0.487486\pi$$
$$90$$ 0 0
$$91$$ −1720.00 −1.98137
$$92$$ 0 0
$$93$$ 456.000 0.508441
$$94$$ 0 0
$$95$$ −20.0000 −0.0215995
$$96$$ 0 0
$$97$$ −926.000 −0.969289 −0.484645 0.874711i $$-0.661051\pi$$
−0.484645 + 0.874711i $$0.661051\pi$$
$$98$$ 0 0
$$99$$ −504.000 −0.511656
$$100$$ 0 0
$$101$$ −198.000 −0.195067 −0.0975333 0.995232i $$-0.531095\pi$$
−0.0975333 + 0.995232i $$0.531095\pi$$
$$102$$ 0 0
$$103$$ −1532.00 −1.46556 −0.732779 0.680467i $$-0.761777\pi$$
−0.732779 + 0.680467i $$0.761777\pi$$
$$104$$ 0 0
$$105$$ 300.000 0.278829
$$106$$ 0 0
$$107$$ −444.000 −0.401150 −0.200575 0.979678i $$-0.564281\pi$$
−0.200575 + 0.979678i $$0.564281\pi$$
$$108$$ 0 0
$$109$$ 62.0000 0.0544819 0.0272409 0.999629i $$-0.491328\pi$$
0.0272409 + 0.999629i $$0.491328\pi$$
$$110$$ 0 0
$$111$$ −846.000 −0.723412
$$112$$ 0 0
$$113$$ 414.000 0.344653 0.172327 0.985040i $$-0.444872\pi$$
0.172327 + 0.985040i $$0.444872\pi$$
$$114$$ 0 0
$$115$$ −680.000 −0.551394
$$116$$ 0 0
$$117$$ −774.000 −0.611593
$$118$$ 0 0
$$119$$ −2120.00 −1.63311
$$120$$ 0 0
$$121$$ 1805.00 1.35612
$$122$$ 0 0
$$123$$ 738.000 0.541002
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ −996.000 −0.695911 −0.347956 0.937511i $$-0.613124\pi$$
−0.347956 + 0.937511i $$0.613124\pi$$
$$128$$ 0 0
$$129$$ −1236.00 −0.843595
$$130$$ 0 0
$$131$$ −264.000 −0.176075 −0.0880374 0.996117i $$-0.528059\pi$$
−0.0880374 + 0.996117i $$0.528059\pi$$
$$132$$ 0 0
$$133$$ 80.0000 0.0521570
$$134$$ 0 0
$$135$$ 135.000 0.0860663
$$136$$ 0 0
$$137$$ 2278.00 1.42060 0.710302 0.703897i $$-0.248558\pi$$
0.710302 + 0.703897i $$0.248558\pi$$
$$138$$ 0 0
$$139$$ 1812.00 1.10570 0.552848 0.833282i $$-0.313541\pi$$
0.552848 + 0.833282i $$0.313541\pi$$
$$140$$ 0 0
$$141$$ −120.000 −0.0716725
$$142$$ 0 0
$$143$$ 4816.00 2.81632
$$144$$ 0 0
$$145$$ 1030.00 0.589909
$$146$$ 0 0
$$147$$ −171.000 −0.0959445
$$148$$ 0 0
$$149$$ −1534.00 −0.843424 −0.421712 0.906730i $$-0.638571\pi$$
−0.421712 + 0.906730i $$0.638571\pi$$
$$150$$ 0 0
$$151$$ −3016.00 −1.62542 −0.812711 0.582668i $$-0.802009\pi$$
−0.812711 + 0.582668i $$0.802009\pi$$
$$152$$ 0 0
$$153$$ −954.000 −0.504094
$$154$$ 0 0
$$155$$ 760.000 0.393837
$$156$$ 0 0
$$157$$ −1814.00 −0.922121 −0.461060 0.887369i $$-0.652531\pi$$
−0.461060 + 0.887369i $$0.652531\pi$$
$$158$$ 0 0
$$159$$ 378.000 0.188537
$$160$$ 0 0
$$161$$ 2720.00 1.33147
$$162$$ 0 0
$$163$$ −1844.00 −0.886093 −0.443047 0.896499i $$-0.646102\pi$$
−0.443047 + 0.896499i $$0.646102\pi$$
$$164$$ 0 0
$$165$$ −840.000 −0.396327
$$166$$ 0 0
$$167$$ 3768.00 1.74597 0.872984 0.487749i $$-0.162182\pi$$
0.872984 + 0.487749i $$0.162182\pi$$
$$168$$ 0 0
$$169$$ 5199.00 2.36641
$$170$$ 0 0
$$171$$ 36.0000 0.0160993
$$172$$ 0 0
$$173$$ 938.000 0.412224 0.206112 0.978528i $$-0.433919\pi$$
0.206112 + 0.978528i $$0.433919\pi$$
$$174$$ 0 0
$$175$$ 500.000 0.215980
$$176$$ 0 0
$$177$$ −168.000 −0.0713427
$$178$$ 0 0
$$179$$ 3968.00 1.65688 0.828442 0.560075i $$-0.189228\pi$$
0.828442 + 0.560075i $$0.189228\pi$$
$$180$$ 0 0
$$181$$ −3514.00 −1.44306 −0.721529 0.692384i $$-0.756560\pi$$
−0.721529 + 0.692384i $$0.756560\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.00242368
$$184$$ 0 0
$$185$$ −1410.00 −0.560353
$$186$$ 0 0
$$187$$ 5936.00 2.32130
$$188$$ 0 0
$$189$$ −540.000 −0.207827
$$190$$ 0 0
$$191$$ −1480.00 −0.560676 −0.280338 0.959901i $$-0.590446\pi$$
−0.280338 + 0.959901i $$0.590446\pi$$
$$192$$ 0 0
$$193$$ −2774.00 −1.03460 −0.517298 0.855806i $$-0.673062\pi$$
−0.517298 + 0.855806i $$0.673062\pi$$
$$194$$ 0 0
$$195$$ −1290.00 −0.473738
$$196$$ 0 0
$$197$$ −3806.00 −1.37648 −0.688239 0.725484i $$-0.741616\pi$$
−0.688239 + 0.725484i $$0.741616\pi$$
$$198$$ 0 0
$$199$$ −856.000 −0.304926 −0.152463 0.988309i $$-0.548720\pi$$
−0.152463 + 0.988309i $$0.548720\pi$$
$$200$$ 0 0
$$201$$ 1164.00 0.408469
$$202$$ 0 0
$$203$$ −4120.00 −1.42447
$$204$$ 0 0
$$205$$ 1230.00 0.419058
$$206$$ 0 0
$$207$$ 1224.00 0.410985
$$208$$ 0 0
$$209$$ −224.000 −0.0741359
$$210$$ 0 0
$$211$$ 3020.00 0.985334 0.492667 0.870218i $$-0.336022\pi$$
0.492667 + 0.870218i $$0.336022\pi$$
$$212$$ 0 0
$$213$$ 2016.00 0.648517
$$214$$ 0 0
$$215$$ −2060.00 −0.653446
$$216$$ 0 0
$$217$$ −3040.00 −0.951008
$$218$$ 0 0
$$219$$ −3510.00 −1.08303
$$220$$ 0 0
$$221$$ 9116.00 2.77470
$$222$$ 0 0
$$223$$ −1684.00 −0.505690 −0.252845 0.967507i $$-0.581366\pi$$
−0.252845 + 0.967507i $$0.581366\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ 2004.00 0.585948 0.292974 0.956120i $$-0.405355\pi$$
0.292974 + 0.956120i $$0.405355\pi$$
$$228$$ 0 0
$$229$$ −5042.00 −1.45496 −0.727478 0.686131i $$-0.759307\pi$$
−0.727478 + 0.686131i $$0.759307\pi$$
$$230$$ 0 0
$$231$$ 3360.00 0.957021
$$232$$ 0 0
$$233$$ −3090.00 −0.868810 −0.434405 0.900718i $$-0.643041\pi$$
−0.434405 + 0.900718i $$0.643041\pi$$
$$234$$ 0 0
$$235$$ −200.000 −0.0555173
$$236$$ 0 0
$$237$$ −1224.00 −0.335474
$$238$$ 0 0
$$239$$ 2136.00 0.578102 0.289051 0.957314i $$-0.406660\pi$$
0.289051 + 0.957314i $$0.406660\pi$$
$$240$$ 0 0
$$241$$ 98.0000 0.0261939 0.0130970 0.999914i $$-0.495831\pi$$
0.0130970 + 0.999914i $$0.495831\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ −285.000 −0.0743183
$$246$$ 0 0
$$247$$ −344.000 −0.0886162
$$248$$ 0 0
$$249$$ −2004.00 −0.510033
$$250$$ 0 0
$$251$$ −5040.00 −1.26742 −0.633709 0.773571i $$-0.718468\pi$$
−0.633709 + 0.773571i $$0.718468\pi$$
$$252$$ 0 0
$$253$$ −7616.00 −1.89254
$$254$$ 0 0
$$255$$ −1590.00 −0.390469
$$256$$ 0 0
$$257$$ −1986.00 −0.482036 −0.241018 0.970521i $$-0.577481\pi$$
−0.241018 + 0.970521i $$0.577481\pi$$
$$258$$ 0 0
$$259$$ 5640.00 1.35310
$$260$$ 0 0
$$261$$ −1854.00 −0.439692
$$262$$ 0 0
$$263$$ 1416.00 0.331994 0.165997 0.986126i $$-0.446916\pi$$
0.165997 + 0.986126i $$0.446916\pi$$
$$264$$ 0 0
$$265$$ 630.000 0.146040
$$266$$ 0 0
$$267$$ −198.000 −0.0453835
$$268$$ 0 0
$$269$$ −6670.00 −1.51181 −0.755905 0.654681i $$-0.772803\pi$$
−0.755905 + 0.654681i $$0.772803\pi$$
$$270$$ 0 0
$$271$$ 48.0000 0.0107594 0.00537969 0.999986i $$-0.498288\pi$$
0.00537969 + 0.999986i $$0.498288\pi$$
$$272$$ 0 0
$$273$$ 5160.00 1.14395
$$274$$ 0 0
$$275$$ −1400.00 −0.306993
$$276$$ 0 0
$$277$$ 6938.00 1.50492 0.752462 0.658636i $$-0.228866\pi$$
0.752462 + 0.658636i $$0.228866\pi$$
$$278$$ 0 0
$$279$$ −1368.00 −0.293548
$$280$$ 0 0
$$281$$ −1694.00 −0.359628 −0.179814 0.983701i $$-0.557550\pi$$
−0.179814 + 0.983701i $$0.557550\pi$$
$$282$$ 0 0
$$283$$ −6364.00 −1.33675 −0.668376 0.743824i $$-0.733010\pi$$
−0.668376 + 0.743824i $$0.733010\pi$$
$$284$$ 0 0
$$285$$ 60.0000 0.0124705
$$286$$ 0 0
$$287$$ −4920.00 −1.01191
$$288$$ 0 0
$$289$$ 6323.00 1.28699
$$290$$ 0 0
$$291$$ 2778.00 0.559619
$$292$$ 0 0
$$293$$ −3134.00 −0.624881 −0.312441 0.949937i $$-0.601147\pi$$
−0.312441 + 0.949937i $$0.601147\pi$$
$$294$$ 0 0
$$295$$ −280.000 −0.0552618
$$296$$ 0 0
$$297$$ 1512.00 0.295405
$$298$$ 0 0
$$299$$ −11696.0 −2.26220
$$300$$ 0 0
$$301$$ 8240.00 1.57789
$$302$$ 0 0
$$303$$ 594.000 0.112622
$$304$$ 0 0
$$305$$ 10.0000 0.00187737
$$306$$ 0 0
$$307$$ −236.000 −0.0438737 −0.0219369 0.999759i $$-0.506983\pi$$
−0.0219369 + 0.999759i $$0.506983\pi$$
$$308$$ 0 0
$$309$$ 4596.00 0.846140
$$310$$ 0 0
$$311$$ 3776.00 0.688480 0.344240 0.938882i $$-0.388137\pi$$
0.344240 + 0.938882i $$0.388137\pi$$
$$312$$ 0 0
$$313$$ −7918.00 −1.42988 −0.714939 0.699187i $$-0.753546\pi$$
−0.714939 + 0.699187i $$0.753546\pi$$
$$314$$ 0 0
$$315$$ −900.000 −0.160982
$$316$$ 0 0
$$317$$ 4362.00 0.772853 0.386426 0.922320i $$-0.373709\pi$$
0.386426 + 0.922320i $$0.373709\pi$$
$$318$$ 0 0
$$319$$ 11536.0 2.02474
$$320$$ 0 0
$$321$$ 1332.00 0.231604
$$322$$ 0 0
$$323$$ −424.000 −0.0730402
$$324$$ 0 0
$$325$$ −2150.00 −0.366956
$$326$$ 0 0
$$327$$ −186.000 −0.0314551
$$328$$ 0 0
$$329$$ 800.000 0.134059
$$330$$ 0 0
$$331$$ 7980.00 1.32514 0.662569 0.749001i $$-0.269466\pi$$
0.662569 + 0.749001i $$0.269466\pi$$
$$332$$ 0 0
$$333$$ 2538.00 0.417662
$$334$$ 0 0
$$335$$ 1940.00 0.316399
$$336$$ 0 0
$$337$$ −8294.00 −1.34066 −0.670331 0.742062i $$-0.733848\pi$$
−0.670331 + 0.742062i $$0.733848\pi$$
$$338$$ 0 0
$$339$$ −1242.00 −0.198986
$$340$$ 0 0
$$341$$ 8512.00 1.35176
$$342$$ 0 0
$$343$$ −5720.00 −0.900440
$$344$$ 0 0
$$345$$ 2040.00 0.318348
$$346$$ 0 0
$$347$$ −964.000 −0.149136 −0.0745681 0.997216i $$-0.523758\pi$$
−0.0745681 + 0.997216i $$0.523758\pi$$
$$348$$ 0 0
$$349$$ 8670.00 1.32978 0.664892 0.746940i $$-0.268478\pi$$
0.664892 + 0.746940i $$0.268478\pi$$
$$350$$ 0 0
$$351$$ 2322.00 0.353103
$$352$$ 0 0
$$353$$ −2314.00 −0.348900 −0.174450 0.984666i $$-0.555815\pi$$
−0.174450 + 0.984666i $$0.555815\pi$$
$$354$$ 0 0
$$355$$ 3360.00 0.502339
$$356$$ 0 0
$$357$$ 6360.00 0.942876
$$358$$ 0 0
$$359$$ −1896.00 −0.278738 −0.139369 0.990240i $$-0.544507\pi$$
−0.139369 + 0.990240i $$0.544507\pi$$
$$360$$ 0 0
$$361$$ −6843.00 −0.997667
$$362$$ 0 0
$$363$$ −5415.00 −0.782958
$$364$$ 0 0
$$365$$ −5850.00 −0.838912
$$366$$ 0 0
$$367$$ 1484.00 0.211074 0.105537 0.994415i $$-0.466344\pi$$
0.105537 + 0.994415i $$0.466344\pi$$
$$368$$ 0 0
$$369$$ −2214.00 −0.312348
$$370$$ 0 0
$$371$$ −2520.00 −0.352647
$$372$$ 0 0
$$373$$ 12370.0 1.71714 0.858571 0.512694i $$-0.171352\pi$$
0.858571 + 0.512694i $$0.171352\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 0 0
$$377$$ 17716.0 2.42021
$$378$$ 0 0
$$379$$ 5620.00 0.761689 0.380844 0.924639i $$-0.375633\pi$$
0.380844 + 0.924639i $$0.375633\pi$$
$$380$$ 0 0
$$381$$ 2988.00 0.401784
$$382$$ 0 0
$$383$$ 5880.00 0.784475 0.392238 0.919864i $$-0.371701\pi$$
0.392238 + 0.919864i $$0.371701\pi$$
$$384$$ 0 0
$$385$$ 5600.00 0.741305
$$386$$ 0 0
$$387$$ 3708.00 0.487050
$$388$$ 0 0
$$389$$ 2082.00 0.271367 0.135683 0.990752i $$-0.456677\pi$$
0.135683 + 0.990752i $$0.456677\pi$$
$$390$$ 0 0
$$391$$ −14416.0 −1.86457
$$392$$ 0 0
$$393$$ 792.000 0.101657
$$394$$ 0 0
$$395$$ −2040.00 −0.259857
$$396$$ 0 0
$$397$$ −1742.00 −0.220223 −0.110111 0.993919i $$-0.535121\pi$$
−0.110111 + 0.993919i $$0.535121\pi$$
$$398$$ 0 0
$$399$$ −240.000 −0.0301129
$$400$$ 0 0
$$401$$ −3270.00 −0.407222 −0.203611 0.979052i $$-0.565268\pi$$
−0.203611 + 0.979052i $$0.565268\pi$$
$$402$$ 0 0
$$403$$ 13072.0 1.61579
$$404$$ 0 0
$$405$$ −405.000 −0.0496904
$$406$$ 0 0
$$407$$ −15792.0 −1.92329
$$408$$ 0 0
$$409$$ −6134.00 −0.741581 −0.370791 0.928716i $$-0.620913\pi$$
−0.370791 + 0.928716i $$0.620913\pi$$
$$410$$ 0 0
$$411$$ −6834.00 −0.820186
$$412$$ 0 0
$$413$$ 1120.00 0.133442
$$414$$ 0 0
$$415$$ −3340.00 −0.395070
$$416$$ 0 0
$$417$$ −5436.00 −0.638374
$$418$$ 0 0
$$419$$ −10392.0 −1.21165 −0.605826 0.795597i $$-0.707157\pi$$
−0.605826 + 0.795597i $$0.707157\pi$$
$$420$$ 0 0
$$421$$ −12690.0 −1.46906 −0.734528 0.678578i $$-0.762596\pi$$
−0.734528 + 0.678578i $$0.762596\pi$$
$$422$$ 0 0
$$423$$ 360.000 0.0413801
$$424$$ 0 0
$$425$$ −2650.00 −0.302456
$$426$$ 0 0
$$427$$ −40.0000 −0.00453334
$$428$$ 0 0
$$429$$ −14448.0 −1.62600
$$430$$ 0 0
$$431$$ −7408.00 −0.827914 −0.413957 0.910297i $$-0.635854\pi$$
−0.413957 + 0.910297i $$0.635854\pi$$
$$432$$ 0 0
$$433$$ −5062.00 −0.561811 −0.280906 0.959735i $$-0.590635\pi$$
−0.280906 + 0.959735i $$0.590635\pi$$
$$434$$ 0 0
$$435$$ −3090.00 −0.340584
$$436$$ 0 0
$$437$$ 544.000 0.0595493
$$438$$ 0 0
$$439$$ −7160.00 −0.778424 −0.389212 0.921148i $$-0.627253\pi$$
−0.389212 + 0.921148i $$0.627253\pi$$
$$440$$ 0 0
$$441$$ 513.000 0.0553936
$$442$$ 0 0
$$443$$ 17100.0 1.83396 0.916981 0.398930i $$-0.130618\pi$$
0.916981 + 0.398930i $$0.130618\pi$$
$$444$$ 0 0
$$445$$ −330.000 −0.0351539
$$446$$ 0 0
$$447$$ 4602.00 0.486951
$$448$$ 0 0
$$449$$ 8634.00 0.907491 0.453746 0.891131i $$-0.350087\pi$$
0.453746 + 0.891131i $$0.350087\pi$$
$$450$$ 0 0
$$451$$ 13776.0 1.43833
$$452$$ 0 0
$$453$$ 9048.00 0.938437
$$454$$ 0 0
$$455$$ 8600.00 0.886097
$$456$$ 0 0
$$457$$ 2986.00 0.305644 0.152822 0.988254i $$-0.451164\pi$$
0.152822 + 0.988254i $$0.451164\pi$$
$$458$$ 0 0
$$459$$ 2862.00 0.291039
$$460$$ 0 0
$$461$$ −2406.00 −0.243077 −0.121539 0.992587i $$-0.538783\pi$$
−0.121539 + 0.992587i $$0.538783\pi$$
$$462$$ 0 0
$$463$$ −14316.0 −1.43698 −0.718489 0.695538i $$-0.755166\pi$$
−0.718489 + 0.695538i $$0.755166\pi$$
$$464$$ 0 0
$$465$$ −2280.00 −0.227382
$$466$$ 0 0
$$467$$ −292.000 −0.0289339 −0.0144670 0.999895i $$-0.504605\pi$$
−0.0144670 + 0.999895i $$0.504605\pi$$
$$468$$ 0 0
$$469$$ −7760.00 −0.764016
$$470$$ 0 0
$$471$$ 5442.00 0.532387
$$472$$ 0 0
$$473$$ −23072.0 −2.24282
$$474$$ 0 0
$$475$$ 100.000 0.00965961
$$476$$ 0 0
$$477$$ −1134.00 −0.108852
$$478$$ 0 0
$$479$$ 14056.0 1.34078 0.670391 0.742008i $$-0.266126\pi$$
0.670391 + 0.742008i $$0.266126\pi$$
$$480$$ 0 0
$$481$$ −24252.0 −2.29895
$$482$$ 0 0
$$483$$ −8160.00 −0.768722
$$484$$ 0 0
$$485$$ 4630.00 0.433479
$$486$$ 0 0
$$487$$ −11204.0 −1.04251 −0.521254 0.853401i $$-0.674536\pi$$
−0.521254 + 0.853401i $$0.674536\pi$$
$$488$$ 0 0
$$489$$ 5532.00 0.511586
$$490$$ 0 0
$$491$$ 4608.00 0.423536 0.211768 0.977320i $$-0.432078\pi$$
0.211768 + 0.977320i $$0.432078\pi$$
$$492$$ 0 0
$$493$$ 21836.0 1.99482
$$494$$ 0 0
$$495$$ 2520.00 0.228819
$$496$$ 0 0
$$497$$ −13440.0 −1.21301
$$498$$ 0 0
$$499$$ 2468.00 0.221409 0.110704 0.993853i $$-0.464689\pi$$
0.110704 + 0.993853i $$0.464689\pi$$
$$500$$ 0 0
$$501$$ −11304.0 −1.00803
$$502$$ 0 0
$$503$$ 12192.0 1.08074 0.540372 0.841426i $$-0.318283\pi$$
0.540372 + 0.841426i $$0.318283\pi$$
$$504$$ 0 0
$$505$$ 990.000 0.0872365
$$506$$ 0 0
$$507$$ −15597.0 −1.36625
$$508$$ 0 0
$$509$$ 1714.00 0.149257 0.0746284 0.997211i $$-0.476223\pi$$
0.0746284 + 0.997211i $$0.476223\pi$$
$$510$$ 0 0
$$511$$ 23400.0 2.02574
$$512$$ 0 0
$$513$$ −108.000 −0.00929496
$$514$$ 0 0
$$515$$ 7660.00 0.655417
$$516$$ 0 0
$$517$$ −2240.00 −0.190551
$$518$$ 0 0
$$519$$ −2814.00 −0.237998
$$520$$ 0 0
$$521$$ −18014.0 −1.51479 −0.757397 0.652955i $$-0.773529\pi$$
−0.757397 + 0.652955i $$0.773529\pi$$
$$522$$ 0 0
$$523$$ −16748.0 −1.40027 −0.700133 0.714013i $$-0.746876\pi$$
−0.700133 + 0.714013i $$0.746876\pi$$
$$524$$ 0 0
$$525$$ −1500.00 −0.124696
$$526$$ 0 0
$$527$$ 16112.0 1.33178
$$528$$ 0 0
$$529$$ 6329.00 0.520178
$$530$$ 0 0
$$531$$ 504.000 0.0411897
$$532$$ 0 0
$$533$$ 21156.0 1.71926
$$534$$ 0 0
$$535$$ 2220.00 0.179400
$$536$$ 0 0
$$537$$ −11904.0 −0.956602
$$538$$ 0 0
$$539$$ −3192.00 −0.255082
$$540$$ 0 0
$$541$$ −14018.0 −1.11401 −0.557006 0.830508i $$-0.688050\pi$$
−0.557006 + 0.830508i $$0.688050\pi$$
$$542$$ 0 0
$$543$$ 10542.0 0.833150
$$544$$ 0 0
$$545$$ −310.000 −0.0243650
$$546$$ 0 0
$$547$$ −412.000 −0.0322045 −0.0161022 0.999870i $$-0.505126\pi$$
−0.0161022 + 0.999870i $$0.505126\pi$$
$$548$$ 0 0
$$549$$ −18.0000 −0.00139931
$$550$$ 0 0
$$551$$ −824.000 −0.0637089
$$552$$ 0 0
$$553$$ 8160.00 0.627484
$$554$$ 0 0
$$555$$ 4230.00 0.323520
$$556$$ 0 0
$$557$$ 18218.0 1.38586 0.692928 0.721007i $$-0.256321\pi$$
0.692928 + 0.721007i $$0.256321\pi$$
$$558$$ 0 0
$$559$$ −35432.0 −2.68088
$$560$$ 0 0
$$561$$ −17808.0 −1.34020
$$562$$ 0 0
$$563$$ 23524.0 1.76096 0.880478 0.474087i $$-0.157222\pi$$
0.880478 + 0.474087i $$0.157222\pi$$
$$564$$ 0 0
$$565$$ −2070.00 −0.154134
$$566$$ 0 0
$$567$$ 1620.00 0.119989
$$568$$ 0 0
$$569$$ 23330.0 1.71888 0.859442 0.511234i $$-0.170811\pi$$
0.859442 + 0.511234i $$0.170811\pi$$
$$570$$ 0 0
$$571$$ −13124.0 −0.961860 −0.480930 0.876759i $$-0.659701\pi$$
−0.480930 + 0.876759i $$0.659701\pi$$
$$572$$ 0 0
$$573$$ 4440.00 0.323706
$$574$$ 0 0
$$575$$ 3400.00 0.246591
$$576$$ 0 0
$$577$$ 11714.0 0.845165 0.422582 0.906324i $$-0.361124\pi$$
0.422582 + 0.906324i $$0.361124\pi$$
$$578$$ 0 0
$$579$$ 8322.00 0.597324
$$580$$ 0 0
$$581$$ 13360.0 0.953987
$$582$$ 0 0
$$583$$ 7056.00 0.501252
$$584$$ 0 0
$$585$$ 3870.00 0.273512
$$586$$ 0 0
$$587$$ −17628.0 −1.23950 −0.619749 0.784800i $$-0.712766\pi$$
−0.619749 + 0.784800i $$0.712766\pi$$
$$588$$ 0 0
$$589$$ −608.000 −0.0425335
$$590$$ 0 0
$$591$$ 11418.0 0.794710
$$592$$ 0 0
$$593$$ −2802.00 −0.194038 −0.0970188 0.995283i $$-0.530931\pi$$
−0.0970188 + 0.995283i $$0.530931\pi$$
$$594$$ 0 0
$$595$$ 10600.0 0.730349
$$596$$ 0 0
$$597$$ 2568.00 0.176049
$$598$$ 0 0
$$599$$ −2664.00 −0.181716 −0.0908582 0.995864i $$-0.528961\pi$$
−0.0908582 + 0.995864i $$0.528961\pi$$
$$600$$ 0 0
$$601$$ 23962.0 1.62634 0.813170 0.582026i $$-0.197740\pi$$
0.813170 + 0.582026i $$0.197740\pi$$
$$602$$ 0 0
$$603$$ −3492.00 −0.235830
$$604$$ 0 0
$$605$$ −9025.00 −0.606477
$$606$$ 0 0
$$607$$ 11940.0 0.798401 0.399201 0.916864i $$-0.369288\pi$$
0.399201 + 0.916864i $$0.369288\pi$$
$$608$$ 0 0
$$609$$ 12360.0 0.822418
$$610$$ 0 0
$$611$$ −3440.00 −0.227770
$$612$$ 0 0
$$613$$ 16794.0 1.10653 0.553265 0.833005i $$-0.313382\pi$$
0.553265 + 0.833005i $$0.313382\pi$$
$$614$$ 0 0
$$615$$ −3690.00 −0.241943
$$616$$ 0 0
$$617$$ −20706.0 −1.35104 −0.675520 0.737341i $$-0.736081\pi$$
−0.675520 + 0.737341i $$0.736081\pi$$
$$618$$ 0 0
$$619$$ 10724.0 0.696339 0.348170 0.937432i $$-0.386803\pi$$
0.348170 + 0.937432i $$0.386803\pi$$
$$620$$ 0 0
$$621$$ −3672.00 −0.237282
$$622$$ 0 0
$$623$$ 1320.00 0.0848871
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 672.000 0.0428024
$$628$$ 0 0
$$629$$ −29892.0 −1.89487
$$630$$ 0 0
$$631$$ −5744.00 −0.362385 −0.181193 0.983448i $$-0.557996\pi$$
−0.181193 + 0.983448i $$0.557996\pi$$
$$632$$ 0 0
$$633$$ −9060.00 −0.568883
$$634$$ 0 0
$$635$$ 4980.00 0.311221
$$636$$ 0 0
$$637$$ −4902.00 −0.304905
$$638$$ 0 0
$$639$$ −6048.00 −0.374421
$$640$$ 0 0
$$641$$ 27906.0 1.71953 0.859767 0.510687i $$-0.170609\pi$$
0.859767 + 0.510687i $$0.170609\pi$$
$$642$$ 0 0
$$643$$ 20556.0 1.26073 0.630365 0.776299i $$-0.282905\pi$$
0.630365 + 0.776299i $$0.282905\pi$$
$$644$$ 0 0
$$645$$ 6180.00 0.377267
$$646$$ 0 0
$$647$$ −10224.0 −0.621247 −0.310624 0.950533i $$-0.600538\pi$$
−0.310624 + 0.950533i $$0.600538\pi$$
$$648$$ 0 0
$$649$$ −3136.00 −0.189675
$$650$$ 0 0
$$651$$ 9120.00 0.549064
$$652$$ 0 0
$$653$$ −12982.0 −0.777986 −0.388993 0.921241i $$-0.627177\pi$$
−0.388993 + 0.921241i $$0.627177\pi$$
$$654$$ 0 0
$$655$$ 1320.00 0.0787430
$$656$$ 0 0
$$657$$ 10530.0 0.625288
$$658$$ 0 0
$$659$$ −1512.00 −0.0893766 −0.0446883 0.999001i $$-0.514229\pi$$
−0.0446883 + 0.999001i $$0.514229\pi$$
$$660$$ 0 0
$$661$$ 16710.0 0.983273 0.491637 0.870800i $$-0.336399\pi$$
0.491637 + 0.870800i $$0.336399\pi$$
$$662$$ 0 0
$$663$$ −27348.0 −1.60197
$$664$$ 0 0
$$665$$ −400.000 −0.0233253
$$666$$ 0 0
$$667$$ −28016.0 −1.62636
$$668$$ 0 0
$$669$$ 5052.00 0.291961
$$670$$ 0 0
$$671$$ 112.000 0.00644368
$$672$$ 0 0
$$673$$ 7962.00 0.456036 0.228018 0.973657i $$-0.426775\pi$$
0.228018 + 0.973657i $$0.426775\pi$$
$$674$$ 0 0
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ 12226.0 0.694067 0.347033 0.937853i $$-0.387189\pi$$
0.347033 + 0.937853i $$0.387189\pi$$
$$678$$ 0 0
$$679$$ −18520.0 −1.04673
$$680$$ 0 0
$$681$$ −6012.00 −0.338297
$$682$$ 0 0
$$683$$ −8748.00 −0.490092 −0.245046 0.969511i $$-0.578803\pi$$
−0.245046 + 0.969511i $$0.578803\pi$$
$$684$$ 0 0
$$685$$ −11390.0 −0.635313
$$686$$ 0 0
$$687$$ 15126.0 0.840019
$$688$$ 0 0
$$689$$ 10836.0 0.599156
$$690$$ 0 0
$$691$$ −7324.00 −0.403210 −0.201605 0.979467i $$-0.564616\pi$$
−0.201605 + 0.979467i $$0.564616\pi$$
$$692$$ 0 0
$$693$$ −10080.0 −0.552536
$$694$$ 0 0
$$695$$ −9060.00 −0.494483
$$696$$ 0 0
$$697$$ 26076.0 1.41707
$$698$$ 0 0
$$699$$ 9270.00 0.501607
$$700$$ 0 0
$$701$$ −21934.0 −1.18179 −0.590896 0.806748i $$-0.701226\pi$$
−0.590896 + 0.806748i $$0.701226\pi$$
$$702$$ 0 0
$$703$$ 1128.00 0.0605168
$$704$$ 0 0
$$705$$ 600.000 0.0320529
$$706$$ 0 0
$$707$$ −3960.00 −0.210652
$$708$$ 0 0
$$709$$ −10690.0 −0.566250 −0.283125 0.959083i $$-0.591371\pi$$
−0.283125 + 0.959083i $$0.591371\pi$$
$$710$$ 0 0
$$711$$ 3672.00 0.193686
$$712$$ 0 0
$$713$$ −20672.0 −1.08580
$$714$$ 0 0
$$715$$ −24080.0 −1.25950
$$716$$ 0 0
$$717$$ −6408.00 −0.333767
$$718$$ 0 0
$$719$$ 13792.0 0.715375 0.357688 0.933841i $$-0.383565\pi$$
0.357688 + 0.933841i $$0.383565\pi$$
$$720$$ 0 0
$$721$$ −30640.0 −1.58265
$$722$$ 0 0
$$723$$ −294.000 −0.0151231
$$724$$ 0 0
$$725$$ −5150.00 −0.263815
$$726$$ 0 0
$$727$$ −24004.0 −1.22457 −0.612283 0.790639i $$-0.709749\pi$$
−0.612283 + 0.790639i $$0.709749\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −43672.0 −2.20967
$$732$$ 0 0
$$733$$ 8562.00 0.431439 0.215719 0.976455i $$-0.430790\pi$$
0.215719 + 0.976455i $$0.430790\pi$$
$$734$$ 0 0
$$735$$ 855.000 0.0429077
$$736$$ 0 0
$$737$$ 21728.0 1.08597
$$738$$ 0 0
$$739$$ −13836.0 −0.688722 −0.344361 0.938837i $$-0.611904\pi$$
−0.344361 + 0.938837i $$0.611904\pi$$
$$740$$ 0 0
$$741$$ 1032.00 0.0511626
$$742$$ 0 0
$$743$$ −22224.0 −1.09733 −0.548667 0.836041i $$-0.684865\pi$$
−0.548667 + 0.836041i $$0.684865\pi$$
$$744$$ 0 0
$$745$$ 7670.00 0.377191
$$746$$ 0 0
$$747$$ 6012.00 0.294468
$$748$$ 0 0
$$749$$ −8880.00 −0.433202
$$750$$ 0 0
$$751$$ 11544.0 0.560914 0.280457 0.959867i $$-0.409514\pi$$
0.280457 + 0.959867i $$0.409514\pi$$
$$752$$ 0 0
$$753$$ 15120.0 0.731744
$$754$$ 0 0
$$755$$ 15080.0 0.726910
$$756$$ 0 0
$$757$$ −3814.00 −0.183120 −0.0915602 0.995800i $$-0.529185\pi$$
−0.0915602 + 0.995800i $$0.529185\pi$$
$$758$$ 0 0
$$759$$ 22848.0 1.09266
$$760$$ 0 0
$$761$$ −25662.0 −1.22240 −0.611200 0.791476i $$-0.709313\pi$$
−0.611200 + 0.791476i $$0.709313\pi$$
$$762$$ 0 0
$$763$$ 1240.00 0.0588349
$$764$$ 0 0
$$765$$ 4770.00 0.225438
$$766$$ 0 0
$$767$$ −4816.00 −0.226722
$$768$$ 0 0
$$769$$ 30658.0 1.43765 0.718827 0.695189i $$-0.244679\pi$$
0.718827 + 0.695189i $$0.244679\pi$$
$$770$$ 0 0
$$771$$ 5958.00 0.278304
$$772$$ 0 0
$$773$$ −30894.0 −1.43749 −0.718745 0.695274i $$-0.755283\pi$$
−0.718745 + 0.695274i $$0.755283\pi$$
$$774$$ 0 0
$$775$$ −3800.00 −0.176129
$$776$$ 0 0
$$777$$ −16920.0 −0.781212
$$778$$ 0 0
$$779$$ −984.000 −0.0452573
$$780$$ 0 0
$$781$$ 37632.0 1.72417
$$782$$ 0 0
$$783$$ 5562.00 0.253857
$$784$$ 0 0
$$785$$ 9070.00 0.412385
$$786$$ 0 0
$$787$$ 21596.0 0.978163 0.489081 0.872238i $$-0.337332\pi$$
0.489081 + 0.872238i $$0.337332\pi$$
$$788$$ 0 0
$$789$$ −4248.00 −0.191677
$$790$$ 0 0
$$791$$ 8280.00 0.372191
$$792$$ 0 0
$$793$$ 172.000 0.00770227
$$794$$ 0 0
$$795$$ −1890.00 −0.0843162
$$796$$ 0 0
$$797$$ −8646.00 −0.384262 −0.192131 0.981369i $$-0.561540\pi$$
−0.192131 + 0.981369i $$0.561540\pi$$
$$798$$ 0 0
$$799$$ −4240.00 −0.187735
$$800$$ 0 0
$$801$$ 594.000 0.0262022
$$802$$ 0 0
$$803$$ −65520.0 −2.87939
$$804$$ 0 0
$$805$$ −13600.0 −0.595450
$$806$$ 0 0
$$807$$ 20010.0 0.872844
$$808$$ 0 0
$$809$$ 24954.0 1.08447 0.542235 0.840227i $$-0.317578\pi$$
0.542235 + 0.840227i $$0.317578\pi$$
$$810$$ 0 0
$$811$$ 40004.0 1.73210 0.866048 0.499960i $$-0.166652\pi$$
0.866048 + 0.499960i $$0.166652\pi$$
$$812$$ 0 0
$$813$$ −144.000 −0.00621193
$$814$$ 0 0
$$815$$ 9220.00 0.396273
$$816$$ 0 0
$$817$$ 1648.00 0.0705707
$$818$$ 0 0
$$819$$ −15480.0 −0.660458
$$820$$ 0 0
$$821$$ 16570.0 0.704381 0.352191 0.935928i $$-0.385437\pi$$
0.352191 + 0.935928i $$0.385437\pi$$
$$822$$ 0 0
$$823$$ −4388.00 −0.185852 −0.0929259 0.995673i $$-0.529622\pi$$
−0.0929259 + 0.995673i $$0.529622\pi$$
$$824$$ 0 0
$$825$$ 4200.00 0.177243
$$826$$ 0 0
$$827$$ 14364.0 0.603972 0.301986 0.953312i $$-0.402350\pi$$
0.301986 + 0.953312i $$0.402350\pi$$
$$828$$ 0 0
$$829$$ −21170.0 −0.886929 −0.443465 0.896292i $$-0.646251\pi$$
−0.443465 + 0.896292i $$0.646251\pi$$
$$830$$ 0 0
$$831$$ −20814.0 −0.868868
$$832$$ 0 0
$$833$$ −6042.00 −0.251312
$$834$$ 0 0
$$835$$ −18840.0 −0.780820
$$836$$ 0 0
$$837$$ 4104.00 0.169480
$$838$$ 0 0
$$839$$ 10664.0 0.438811 0.219405 0.975634i $$-0.429588\pi$$
0.219405 + 0.975634i $$0.429588\pi$$
$$840$$ 0 0
$$841$$ 18047.0 0.739965
$$842$$ 0 0
$$843$$ 5082.00 0.207632
$$844$$ 0 0
$$845$$ −25995.0 −1.05829
$$846$$ 0 0
$$847$$ 36100.0 1.46448
$$848$$ 0 0
$$849$$ 19092.0 0.771774
$$850$$ 0 0
$$851$$ 38352.0 1.54488
$$852$$ 0 0
$$853$$ −3190.00 −0.128046 −0.0640232 0.997948i $$-0.520393\pi$$
−0.0640232 + 0.997948i $$0.520393\pi$$
$$854$$ 0 0
$$855$$ −180.000 −0.00719985
$$856$$ 0 0
$$857$$ 20814.0 0.829630 0.414815 0.909906i $$-0.363846\pi$$
0.414815 + 0.909906i $$0.363846\pi$$
$$858$$ 0 0
$$859$$ −18988.0 −0.754205 −0.377103 0.926172i $$-0.623080\pi$$
−0.377103 + 0.926172i $$0.623080\pi$$
$$860$$ 0 0
$$861$$ 14760.0 0.584227
$$862$$ 0 0
$$863$$ 11664.0 0.460078 0.230039 0.973181i $$-0.426115\pi$$
0.230039 + 0.973181i $$0.426115\pi$$
$$864$$ 0 0
$$865$$ −4690.00 −0.184352
$$866$$ 0 0
$$867$$ −18969.0 −0.743046
$$868$$ 0 0
$$869$$ −22848.0 −0.891905
$$870$$ 0 0
$$871$$ 33368.0 1.29808
$$872$$ 0 0
$$873$$ −8334.00 −0.323096
$$874$$ 0 0
$$875$$ −2500.00 −0.0965891
$$876$$ 0 0
$$877$$ −8246.00 −0.317500 −0.158750 0.987319i $$-0.550746\pi$$
−0.158750 + 0.987319i $$0.550746\pi$$
$$878$$ 0 0
$$879$$ 9402.00 0.360775
$$880$$ 0 0
$$881$$ 22890.0 0.875350 0.437675 0.899133i $$-0.355802\pi$$
0.437675 + 0.899133i $$0.355802\pi$$
$$882$$ 0 0
$$883$$ 33548.0 1.27857 0.639287 0.768969i $$-0.279230\pi$$
0.639287 + 0.768969i $$0.279230\pi$$
$$884$$ 0 0
$$885$$ 840.000 0.0319054
$$886$$ 0 0
$$887$$ −32264.0 −1.22133 −0.610665 0.791889i $$-0.709098\pi$$
−0.610665 + 0.791889i $$0.709098\pi$$
$$888$$ 0 0
$$889$$ −19920.0 −0.751513
$$890$$ 0 0
$$891$$ −4536.00 −0.170552
$$892$$ 0 0
$$893$$ 160.000 0.00599574
$$894$$ 0 0
$$895$$ −19840.0 −0.740981
$$896$$ 0 0
$$897$$ 35088.0 1.30608
$$898$$ 0 0
$$899$$ 31312.0 1.16164
$$900$$ 0 0
$$901$$ 13356.0 0.493843
$$902$$ 0 0
$$903$$ −24720.0 −0.910997
$$904$$ 0 0
$$905$$ 17570.0 0.645355
$$906$$ 0 0
$$907$$ 51228.0 1.87541 0.937706 0.347431i $$-0.112946\pi$$
0.937706 + 0.347431i $$0.112946\pi$$
$$908$$ 0 0
$$909$$ −1782.00 −0.0650222
$$910$$ 0 0
$$911$$ −2144.00 −0.0779735 −0.0389868 0.999240i $$-0.512413\pi$$
−0.0389868 + 0.999240i $$0.512413\pi$$
$$912$$ 0 0
$$913$$ −37408.0 −1.35600
$$914$$ 0 0
$$915$$ −30.0000 −0.00108390
$$916$$ 0 0
$$917$$ −5280.00 −0.190143
$$918$$ 0 0
$$919$$ 33584.0 1.20548 0.602739 0.797939i $$-0.294076\pi$$
0.602739 + 0.797939i $$0.294076\pi$$
$$920$$ 0 0
$$921$$ 708.000 0.0253305
$$922$$ 0 0
$$923$$ 57792.0 2.06094
$$924$$ 0 0
$$925$$ 7050.00 0.250597
$$926$$ 0 0
$$927$$ −13788.0 −0.488519
$$928$$ 0 0
$$929$$ −3590.00 −0.126786 −0.0633929 0.997989i $$-0.520192\pi$$
−0.0633929 + 0.997989i $$0.520192\pi$$
$$930$$ 0 0
$$931$$ 228.000 0.00802621
$$932$$ 0 0
$$933$$ −11328.0 −0.397494
$$934$$ 0 0
$$935$$ −29680.0 −1.03812
$$936$$ 0 0
$$937$$ −21686.0 −0.756084 −0.378042 0.925788i $$-0.623403\pi$$
−0.378042 + 0.925788i $$0.623403\pi$$
$$938$$ 0 0
$$939$$ 23754.0 0.825540
$$940$$ 0 0
$$941$$ −5174.00 −0.179243 −0.0896215 0.995976i $$-0.528566\pi$$
−0.0896215 + 0.995976i $$0.528566\pi$$
$$942$$ 0 0
$$943$$ −33456.0 −1.15533
$$944$$ 0 0
$$945$$ 2700.00 0.0929429
$$946$$ 0 0
$$947$$ 35524.0 1.21898 0.609490 0.792793i $$-0.291374\pi$$
0.609490 + 0.792793i $$0.291374\pi$$
$$948$$ 0 0
$$949$$ −100620. −3.44179
$$950$$ 0 0
$$951$$ −13086.0 −0.446207
$$952$$ 0 0
$$953$$ −16122.0 −0.547999 −0.273999 0.961730i $$-0.588347\pi$$
−0.273999 + 0.961730i $$0.588347\pi$$
$$954$$ 0 0
$$955$$ 7400.00 0.250742
$$956$$ 0 0
$$957$$ −34608.0 −1.16898
$$958$$ 0 0
$$959$$ 45560.0 1.53411
$$960$$ 0 0
$$961$$ −6687.00 −0.224464
$$962$$ 0 0
$$963$$ −3996.00 −0.133717
$$964$$ 0 0
$$965$$ 13870.0 0.462685
$$966$$ 0 0
$$967$$ 19188.0 0.638102 0.319051 0.947738i $$-0.396636\pi$$
0.319051 + 0.947738i $$0.396636\pi$$
$$968$$ 0 0
$$969$$ 1272.00 0.0421698
$$970$$ 0 0
$$971$$ −38464.0 −1.27123 −0.635617 0.772004i $$-0.719254\pi$$
−0.635617 + 0.772004i $$0.719254\pi$$
$$972$$ 0 0
$$973$$ 36240.0 1.19404
$$974$$ 0 0
$$975$$ 6450.00 0.211862
$$976$$ 0 0
$$977$$ −43930.0 −1.43853 −0.719266 0.694735i $$-0.755522\pi$$
−0.719266 + 0.694735i $$0.755522\pi$$
$$978$$ 0 0
$$979$$ −3696.00 −0.120659
$$980$$ 0 0
$$981$$ 558.000 0.0181606
$$982$$ 0 0
$$983$$ 17328.0 0.562235 0.281118 0.959673i $$-0.409295\pi$$
0.281118 + 0.959673i $$0.409295\pi$$
$$984$$ 0 0
$$985$$ 19030.0 0.615580
$$986$$ 0 0
$$987$$ −2400.00 −0.0773990
$$988$$ 0 0
$$989$$ 56032.0 1.80153
$$990$$ 0 0
$$991$$ −18160.0 −0.582110 −0.291055 0.956706i $$-0.594006\pi$$
−0.291055 + 0.956706i $$0.594006\pi$$
$$992$$ 0 0
$$993$$ −23940.0 −0.765068
$$994$$ 0 0
$$995$$ 4280.00 0.136367
$$996$$ 0 0
$$997$$ −9102.00 −0.289131 −0.144565 0.989495i $$-0.546178\pi$$
−0.144565 + 0.989495i $$0.546178\pi$$
$$998$$ 0 0
$$999$$ −7614.00 −0.241137
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 120.4.a.b.1.1 1
3.2 odd 2 360.4.a.n.1.1 1
4.3 odd 2 240.4.a.g.1.1 1
5.2 odd 4 600.4.f.a.49.2 2
5.3 odd 4 600.4.f.a.49.1 2
5.4 even 2 600.4.a.i.1.1 1
8.3 odd 2 960.4.a.k.1.1 1
8.5 even 2 960.4.a.bj.1.1 1
12.11 even 2 720.4.a.q.1.1 1
15.2 even 4 1800.4.f.v.649.2 2
15.8 even 4 1800.4.f.v.649.1 2
15.14 odd 2 1800.4.a.f.1.1 1
20.3 even 4 1200.4.f.t.49.2 2
20.7 even 4 1200.4.f.t.49.1 2
20.19 odd 2 1200.4.a.p.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.b.1.1 1 1.1 even 1 trivial
240.4.a.g.1.1 1 4.3 odd 2
360.4.a.n.1.1 1 3.2 odd 2
600.4.a.i.1.1 1 5.4 even 2
600.4.f.a.49.1 2 5.3 odd 4
600.4.f.a.49.2 2 5.2 odd 4
720.4.a.q.1.1 1 12.11 even 2
960.4.a.k.1.1 1 8.3 odd 2
960.4.a.bj.1.1 1 8.5 even 2
1200.4.a.p.1.1 1 20.19 odd 2
1200.4.f.t.49.1 2 20.7 even 4
1200.4.f.t.49.2 2 20.3 even 4
1800.4.a.f.1.1 1 15.14 odd 2
1800.4.f.v.649.1 2 15.8 even 4
1800.4.f.v.649.2 2 15.2 even 4