# Properties

 Label 960.4.a.k.1.1 Level $960$ Weight $4$ Character 960.1 Self dual yes Analytic conductor $56.642$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(1,960)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("960.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 960.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$56.6418336055$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 960.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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 $$-0.681557\pi$$
−0.539949 + 0.841698i $$0.681557\pi$$
$$8$$ 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 $$-0.130293\pi$$
0.917389 + 0.397992i $$0.130293\pi$$
$$14$$ 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 $$-0.711441\pi$$
−0.616477 + 0.787373i $$0.711441\pi$$
$$24$$ 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 $$-0.270752\pi$$
0.659539 + 0.751671i $$0.270752\pi$$
$$30$$ 0 0
$$31$$ 152.000 0.880645 0.440323 0.897840i $$-0.354864\pi$$
0.440323 + 0.897840i $$0.354864\pi$$
$$32$$ 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 $$-0.715510\pi$$
−0.626493 + 0.779427i $$0.715510\pi$$
$$38$$ 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 $$-0.519770\pi$$
−0.0620702 + 0.998072i $$0.519770\pi$$
$$48$$ 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 $$-0.447793\pi$$
0.163278 + 0.986580i $$0.447793\pi$$
$$54$$ 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 $$-0.499332\pi$$
0.00209897 + 0.999998i $$0.499332\pi$$
$$62$$ 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 $$-0.310174\pi$$
0.561632 + 0.827387i $$0.310174\pi$$
$$72$$ 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 $$-0.593831\pi$$
−0.290529 + 0.956866i $$0.593831\pi$$
$$80$$ 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 $$-0.468905\pi$$
0.0975333 + 0.995232i $$0.468905\pi$$
$$102$$ 0 0
$$103$$ 1532.00 1.46556 0.732779 0.680467i $$-0.238223\pi$$
0.732779 + 0.680467i $$0.238223\pi$$
$$104$$ 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 $$-0.508672\pi$$
−0.0272409 + 0.999629i $$0.508672\pi$$
$$110$$ 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 $$-0.386876\pi$$
0.347956 + 0.937511i $$0.386876\pi$$
$$128$$ 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 $$-0.361429\pi$$
0.421712 + 0.906730i $$0.361429\pi$$
$$150$$ 0 0
$$151$$ 3016.00 1.62542 0.812711 0.582668i $$-0.197991\pi$$
0.812711 + 0.582668i $$0.197991\pi$$
$$152$$ 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 $$-0.347469\pi$$
0.461060 + 0.887369i $$0.347469\pi$$
$$158$$ 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 $$-0.837818\pi$$
−0.872984 + 0.487749i $$0.837818\pi$$
$$168$$ 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 $$-0.566081\pi$$
−0.206112 + 0.978528i $$0.566081\pi$$
$$174$$ 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 $$-0.243440\pi$$
0.721529 + 0.692384i $$0.243440\pi$$
$$182$$ 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 $$-0.409554\pi$$
0.280338 + 0.959901i $$0.409554\pi$$
$$192$$ 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 $$-0.258384\pi$$
0.688239 + 0.725484i $$0.258384\pi$$
$$198$$ 0 0
$$199$$ 856.000 0.304926 0.152463 0.988309i $$-0.451280\pi$$
0.152463 + 0.988309i $$0.451280\pi$$
$$200$$ 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 $$-0.418634\pi$$
0.252845 + 0.967507i $$0.418634\pi$$
$$224$$ 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 $$-0.240693\pi$$
0.727478 + 0.686131i $$0.240693\pi$$
$$230$$ 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 $$-0.593340\pi$$
−0.289051 + 0.957314i $$0.593340\pi$$
$$240$$ 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 $$-0.553084\pi$$
−0.165997 + 0.986126i $$0.553084\pi$$
$$264$$ 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 $$-0.227197\pi$$
0.755905 + 0.654681i $$0.227197\pi$$
$$270$$ 0 0
$$271$$ −48.0000 −0.0107594 −0.00537969 0.999986i $$-0.501712\pi$$
−0.00537969 + 0.999986i $$0.501712\pi$$
$$272$$ 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 $$-0.771134\pi$$
−0.752462 + 0.658636i $$0.771134\pi$$
$$278$$ 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 $$-0.398853\pi$$
0.312441 + 0.949937i $$0.398853\pi$$
$$294$$ 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 $$-0.611863\pi$$
−0.344240 + 0.938882i $$0.611863\pi$$
$$312$$ 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 $$-0.626291\pi$$
−0.386426 + 0.922320i $$0.626291\pi$$
$$318$$ 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 $$-0.731522\pi$$
−0.664892 + 0.746940i $$0.731522\pi$$
$$350$$ 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 $$-0.455493\pi$$
0.139369 + 0.990240i $$0.455493\pi$$
$$360$$ 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 $$-0.533656\pi$$
−0.105537 + 0.994415i $$0.533656\pi$$
$$368$$ 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 $$-0.828648\pi$$
−0.858571 + 0.512694i $$0.828648\pi$$
$$374$$ 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 $$-0.628299\pi$$
−0.392238 + 0.919864i $$0.628299\pi$$
$$384$$ 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 $$-0.543323\pi$$
−0.135683 + 0.990752i $$0.543323\pi$$
$$390$$ 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 $$-0.464879\pi$$
0.110111 + 0.993919i $$0.464879\pi$$
$$398$$ 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 $$-0.237404\pi$$
0.734528 + 0.678578i $$0.237404\pi$$
$$422$$ 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 $$-0.364146\pi$$
0.413957 + 0.910297i $$0.364146\pi$$
$$432$$ 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 $$-0.372747\pi$$
0.389212 + 0.921148i $$0.372747\pi$$
$$440$$ 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 $$-0.461217\pi$$
0.121539 + 0.992587i $$0.461217\pi$$
$$462$$ 0 0
$$463$$ 14316.0 1.43698 0.718489 0.695538i $$-0.244834\pi$$
0.718489 + 0.695538i $$0.244834\pi$$
$$464$$ 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 $$-0.733874\pi$$
−0.670391 + 0.742008i $$0.733874\pi$$
$$480$$ 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 $$-0.325464\pi$$
0.521254 + 0.853401i $$0.325464\pi$$
$$488$$ 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 $$-0.681717\pi$$
−0.540372 + 0.841426i $$0.681717\pi$$
$$504$$ 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 $$-0.523777\pi$$
−0.0746284 + 0.997211i $$0.523777\pi$$
$$510$$ 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 $$-0.311950\pi$$
0.557006 + 0.830508i $$0.311950\pi$$
$$542$$ 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 $$-0.743679\pi$$
−0.692928 + 0.721007i $$0.743679\pi$$
$$558$$ 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 $$-0.471039\pi$$
0.0908582 + 0.995864i $$0.471039\pi$$
$$600$$ 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 $$-0.630712\pi$$
−0.399201 + 0.916864i $$0.630712\pi$$
$$608$$ 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 $$-0.686618\pi$$
−0.553265 + 0.833005i $$0.686618\pi$$
$$614$$ 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 $$-0.442004\pi$$
0.181193 + 0.983448i $$0.442004\pi$$
$$632$$ 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 $$-0.399462\pi$$
0.310624 + 0.950533i $$0.399462\pi$$
$$648$$ 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 $$-0.372823\pi$$
0.388993 + 0.921241i $$0.372823\pi$$
$$654$$ 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 $$-0.663601\pi$$
−0.491637 + 0.870800i $$0.663601\pi$$
$$662$$ 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 $$-0.612811\pi$$
−0.347033 + 0.937853i $$0.612811\pi$$
$$678$$ 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 $$-0.298774\pi$$
0.590896 + 0.806748i $$0.298774\pi$$
$$702$$ 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 $$-0.408629\pi$$
0.283125 + 0.959083i $$0.408629\pi$$
$$710$$ 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 $$-0.616435\pi$$
−0.357688 + 0.933841i $$0.616435\pi$$
$$720$$ 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 $$-0.290251\pi$$
0.612283 + 0.790639i $$0.290251\pi$$
$$728$$ 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 $$-0.569210\pi$$
−0.215719 + 0.976455i $$0.569210\pi$$
$$734$$ 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 $$-0.315135\pi$$
0.548667 + 0.836041i $$0.315135\pi$$
$$744$$ 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 $$-0.590486\pi$$
−0.280457 + 0.959867i $$0.590486\pi$$
$$752$$ 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 $$-0.470815\pi$$
0.0915602 + 0.995800i $$0.470815\pi$$
$$758$$ 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 $$-0.244717\pi$$
0.718745 + 0.695274i $$0.244717\pi$$
$$774$$ 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 $$-0.438460\pi$$
0.192131 + 0.981369i $$0.438460\pi$$
$$798$$ 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 $$-0.614563\pi$$
−0.352191 + 0.935928i $$0.614563\pi$$
$$822$$ 0 0
$$823$$ 4388.00 0.185852 0.0929259 0.995673i $$-0.470378\pi$$
0.0929259 + 0.995673i $$0.470378\pi$$
$$824$$ 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 $$-0.353749\pi$$
0.443465 + 0.896292i $$0.353749\pi$$
$$830$$ 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 $$-0.570412\pi$$
−0.219405 + 0.975634i $$0.570412\pi$$
$$840$$ 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 $$-0.479607\pi$$
0.0640232 + 0.997948i $$0.479607\pi$$
$$854$$ 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 $$-0.573885\pi$$
−0.230039 + 0.973181i $$0.573885\pi$$
$$864$$ 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 $$-0.449254\pi$$
0.158750 + 0.987319i $$0.449254\pi$$
$$878$$ 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 $$-0.290902\pi$$
0.610665 + 0.791889i $$0.290902\pi$$
$$888$$ 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 $$-0.487587\pi$$
0.0389868 + 0.999240i $$0.487587\pi$$
$$912$$ 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 $$-0.705924\pi$$
−0.602739 + 0.797939i $$0.705924\pi$$
$$920$$ 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 $$-0.471434\pi$$
0.0896215 + 0.995976i $$0.471434\pi$$
$$942$$ 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 $$-0.603364\pi$$
−0.319051 + 0.947738i $$0.603364\pi$$
$$968$$ 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 $$-0.590705\pi$$
−0.281118 + 0.959673i $$0.590705\pi$$
$$984$$ 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 $$-0.405994\pi$$
0.291055 + 0.956706i $$0.405994\pi$$
$$992$$ 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 $$-0.453822\pi$$
0.144565 + 0.989495i $$0.453822\pi$$
$$998$$ 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 960.4.a.k.1.1 1
4.3 odd 2 960.4.a.bj.1.1 1
8.3 odd 2 120.4.a.b.1.1 1
8.5 even 2 240.4.a.g.1.1 1
24.5 odd 2 720.4.a.q.1.1 1
24.11 even 2 360.4.a.n.1.1 1
40.3 even 4 600.4.f.a.49.1 2
40.13 odd 4 1200.4.f.t.49.2 2
40.19 odd 2 600.4.a.i.1.1 1
40.27 even 4 600.4.f.a.49.2 2
40.29 even 2 1200.4.a.p.1.1 1
40.37 odd 4 1200.4.f.t.49.1 2
120.59 even 2 1800.4.a.f.1.1 1
120.83 odd 4 1800.4.f.v.649.1 2
120.107 odd 4 1800.4.f.v.649.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.b.1.1 1 8.3 odd 2
240.4.a.g.1.1 1 8.5 even 2
360.4.a.n.1.1 1 24.11 even 2
600.4.a.i.1.1 1 40.19 odd 2
600.4.f.a.49.1 2 40.3 even 4
600.4.f.a.49.2 2 40.27 even 4
720.4.a.q.1.1 1 24.5 odd 2
960.4.a.k.1.1 1 1.1 even 1 trivial
960.4.a.bj.1.1 1 4.3 odd 2
1200.4.a.p.1.1 1 40.29 even 2
1200.4.f.t.49.1 2 40.37 odd 4
1200.4.f.t.49.2 2 40.13 odd 4
1800.4.a.f.1.1 1 120.59 even 2
1800.4.f.v.649.1 2 120.83 odd 4
1800.4.f.v.649.2 2 120.107 odd 4