# Properties

 Label 78.4.a.e.1.1 Level $78$ Weight $4$ Character 78.1 Self dual yes Analytic conductor $4.602$ 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] = [78,4,Mod(1,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("78.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 78.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: $$4.60214898045$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 78.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{5} -6.00000 q^{6} +20.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{5} -6.00000 q^{6} +20.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +12.0000 q^{10} +24.0000 q^{11} -12.0000 q^{12} +13.0000 q^{13} +40.0000 q^{14} -18.0000 q^{15} +16.0000 q^{16} -30.0000 q^{17} +18.0000 q^{18} -16.0000 q^{19} +24.0000 q^{20} -60.0000 q^{21} +48.0000 q^{22} -72.0000 q^{23} -24.0000 q^{24} -89.0000 q^{25} +26.0000 q^{26} -27.0000 q^{27} +80.0000 q^{28} -282.000 q^{29} -36.0000 q^{30} +164.000 q^{31} +32.0000 q^{32} -72.0000 q^{33} -60.0000 q^{34} +120.000 q^{35} +36.0000 q^{36} +110.000 q^{37} -32.0000 q^{38} -39.0000 q^{39} +48.0000 q^{40} -126.000 q^{41} -120.000 q^{42} +164.000 q^{43} +96.0000 q^{44} +54.0000 q^{45} -144.000 q^{46} -204.000 q^{47} -48.0000 q^{48} +57.0000 q^{49} -178.000 q^{50} +90.0000 q^{51} +52.0000 q^{52} -738.000 q^{53} -54.0000 q^{54} +144.000 q^{55} +160.000 q^{56} +48.0000 q^{57} -564.000 q^{58} +120.000 q^{59} -72.0000 q^{60} +614.000 q^{61} +328.000 q^{62} +180.000 q^{63} +64.0000 q^{64} +78.0000 q^{65} -144.000 q^{66} +848.000 q^{67} -120.000 q^{68} +216.000 q^{69} +240.000 q^{70} +132.000 q^{71} +72.0000 q^{72} +218.000 q^{73} +220.000 q^{74} +267.000 q^{75} -64.0000 q^{76} +480.000 q^{77} -78.0000 q^{78} -1096.00 q^{79} +96.0000 q^{80} +81.0000 q^{81} -252.000 q^{82} +552.000 q^{83} -240.000 q^{84} -180.000 q^{85} +328.000 q^{86} +846.000 q^{87} +192.000 q^{88} +210.000 q^{89} +108.000 q^{90} +260.000 q^{91} -288.000 q^{92} -492.000 q^{93} -408.000 q^{94} -96.0000 q^{95} -96.0000 q^{96} -1726.00 q^{97} +114.000 q^{98} +216.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.707107
$$3$$ −3.00000 −0.577350
$$4$$ 4.00000 0.500000
$$5$$ 6.00000 0.536656 0.268328 0.963328i $$-0.413529\pi$$
0.268328 + 0.963328i $$0.413529\pi$$
$$6$$ −6.00000 −0.408248
$$7$$ 20.0000 1.07990 0.539949 0.841698i $$-0.318443\pi$$
0.539949 + 0.841698i $$0.318443\pi$$
$$8$$ 8.00000 0.353553
$$9$$ 9.00000 0.333333
$$10$$ 12.0000 0.379473
$$11$$ 24.0000 0.657843 0.328921 0.944357i $$-0.393315\pi$$
0.328921 + 0.944357i $$0.393315\pi$$
$$12$$ −12.0000 −0.288675
$$13$$ 13.0000 0.277350
$$14$$ 40.0000 0.763604
$$15$$ −18.0000 −0.309839
$$16$$ 16.0000 0.250000
$$17$$ −30.0000 −0.428004 −0.214002 0.976833i $$-0.568650\pi$$
−0.214002 + 0.976833i $$0.568650\pi$$
$$18$$ 18.0000 0.235702
$$19$$ −16.0000 −0.193192 −0.0965961 0.995324i $$-0.530796\pi$$
−0.0965961 + 0.995324i $$0.530796\pi$$
$$20$$ 24.0000 0.268328
$$21$$ −60.0000 −0.623480
$$22$$ 48.0000 0.465165
$$23$$ −72.0000 −0.652741 −0.326370 0.945242i $$-0.605826\pi$$
−0.326370 + 0.945242i $$0.605826\pi$$
$$24$$ −24.0000 −0.204124
$$25$$ −89.0000 −0.712000
$$26$$ 26.0000 0.196116
$$27$$ −27.0000 −0.192450
$$28$$ 80.0000 0.539949
$$29$$ −282.000 −1.80573 −0.902864 0.429927i $$-0.858539\pi$$
−0.902864 + 0.429927i $$0.858539\pi$$
$$30$$ −36.0000 −0.219089
$$31$$ 164.000 0.950170 0.475085 0.879940i $$-0.342417\pi$$
0.475085 + 0.879940i $$0.342417\pi$$
$$32$$ 32.0000 0.176777
$$33$$ −72.0000 −0.379806
$$34$$ −60.0000 −0.302645
$$35$$ 120.000 0.579534
$$36$$ 36.0000 0.166667
$$37$$ 110.000 0.488754 0.244377 0.969680i $$-0.421417\pi$$
0.244377 + 0.969680i $$0.421417\pi$$
$$38$$ −32.0000 −0.136608
$$39$$ −39.0000 −0.160128
$$40$$ 48.0000 0.189737
$$41$$ −126.000 −0.479949 −0.239974 0.970779i $$-0.577139\pi$$
−0.239974 + 0.970779i $$0.577139\pi$$
$$42$$ −120.000 −0.440867
$$43$$ 164.000 0.581622 0.290811 0.956780i $$-0.406075\pi$$
0.290811 + 0.956780i $$0.406075\pi$$
$$44$$ 96.0000 0.328921
$$45$$ 54.0000 0.178885
$$46$$ −144.000 −0.461557
$$47$$ −204.000 −0.633116 −0.316558 0.948573i $$-0.602527\pi$$
−0.316558 + 0.948573i $$0.602527\pi$$
$$48$$ −48.0000 −0.144338
$$49$$ 57.0000 0.166181
$$50$$ −178.000 −0.503460
$$51$$ 90.0000 0.247108
$$52$$ 52.0000 0.138675
$$53$$ −738.000 −1.91268 −0.956341 0.292255i $$-0.905595\pi$$
−0.956341 + 0.292255i $$0.905595\pi$$
$$54$$ −54.0000 −0.136083
$$55$$ 144.000 0.353036
$$56$$ 160.000 0.381802
$$57$$ 48.0000 0.111540
$$58$$ −564.000 −1.27684
$$59$$ 120.000 0.264791 0.132396 0.991197i $$-0.457733\pi$$
0.132396 + 0.991197i $$0.457733\pi$$
$$60$$ −72.0000 −0.154919
$$61$$ 614.000 1.28876 0.644382 0.764703i $$-0.277115\pi$$
0.644382 + 0.764703i $$0.277115\pi$$
$$62$$ 328.000 0.671872
$$63$$ 180.000 0.359966
$$64$$ 64.0000 0.125000
$$65$$ 78.0000 0.148842
$$66$$ −144.000 −0.268563
$$67$$ 848.000 1.54626 0.773132 0.634245i $$-0.218689\pi$$
0.773132 + 0.634245i $$0.218689\pi$$
$$68$$ −120.000 −0.214002
$$69$$ 216.000 0.376860
$$70$$ 240.000 0.409793
$$71$$ 132.000 0.220641 0.110321 0.993896i $$-0.464812\pi$$
0.110321 + 0.993896i $$0.464812\pi$$
$$72$$ 72.0000 0.117851
$$73$$ 218.000 0.349520 0.174760 0.984611i $$-0.444085\pi$$
0.174760 + 0.984611i $$0.444085\pi$$
$$74$$ 220.000 0.345601
$$75$$ 267.000 0.411073
$$76$$ −64.0000 −0.0965961
$$77$$ 480.000 0.710404
$$78$$ −78.0000 −0.113228
$$79$$ −1096.00 −1.56088 −0.780441 0.625230i $$-0.785005\pi$$
−0.780441 + 0.625230i $$0.785005\pi$$
$$80$$ 96.0000 0.134164
$$81$$ 81.0000 0.111111
$$82$$ −252.000 −0.339375
$$83$$ 552.000 0.729998 0.364999 0.931008i $$-0.381069\pi$$
0.364999 + 0.931008i $$0.381069\pi$$
$$84$$ −240.000 −0.311740
$$85$$ −180.000 −0.229691
$$86$$ 328.000 0.411269
$$87$$ 846.000 1.04254
$$88$$ 192.000 0.232583
$$89$$ 210.000 0.250112 0.125056 0.992150i $$-0.460089\pi$$
0.125056 + 0.992150i $$0.460089\pi$$
$$90$$ 108.000 0.126491
$$91$$ 260.000 0.299510
$$92$$ −288.000 −0.326370
$$93$$ −492.000 −0.548581
$$94$$ −408.000 −0.447681
$$95$$ −96.0000 −0.103678
$$96$$ −96.0000 −0.102062
$$97$$ −1726.00 −1.80669 −0.903344 0.428917i $$-0.858895\pi$$
−0.903344 + 0.428917i $$0.858895\pi$$
$$98$$ 114.000 0.117508
$$99$$ 216.000 0.219281
$$100$$ −356.000 −0.356000
$$101$$ 798.000 0.786178 0.393089 0.919500i $$-0.371406\pi$$
0.393089 + 0.919500i $$0.371406\pi$$
$$102$$ 180.000 0.174732
$$103$$ −520.000 −0.497448 −0.248724 0.968574i $$-0.580011\pi$$
−0.248724 + 0.968574i $$0.580011\pi$$
$$104$$ 104.000 0.0980581
$$105$$ −360.000 −0.334594
$$106$$ −1476.00 −1.35247
$$107$$ 12.0000 0.0108419 0.00542095 0.999985i $$-0.498274\pi$$
0.00542095 + 0.999985i $$0.498274\pi$$
$$108$$ −108.000 −0.0962250
$$109$$ −1834.00 −1.61161 −0.805804 0.592182i $$-0.798267\pi$$
−0.805804 + 0.592182i $$0.798267\pi$$
$$110$$ 288.000 0.249634
$$111$$ −330.000 −0.282182
$$112$$ 320.000 0.269975
$$113$$ −366.000 −0.304694 −0.152347 0.988327i $$-0.548683\pi$$
−0.152347 + 0.988327i $$0.548683\pi$$
$$114$$ 96.0000 0.0788704
$$115$$ −432.000 −0.350297
$$116$$ −1128.00 −0.902864
$$117$$ 117.000 0.0924500
$$118$$ 240.000 0.187236
$$119$$ −600.000 −0.462201
$$120$$ −144.000 −0.109545
$$121$$ −755.000 −0.567243
$$122$$ 1228.00 0.911294
$$123$$ 378.000 0.277098
$$124$$ 656.000 0.475085
$$125$$ −1284.00 −0.918756
$$126$$ 360.000 0.254535
$$127$$ 2144.00 1.49803 0.749013 0.662556i $$-0.230528\pi$$
0.749013 + 0.662556i $$0.230528\pi$$
$$128$$ 128.000 0.0883883
$$129$$ −492.000 −0.335800
$$130$$ 156.000 0.105247
$$131$$ −2748.00 −1.83278 −0.916389 0.400289i $$-0.868910\pi$$
−0.916389 + 0.400289i $$0.868910\pi$$
$$132$$ −288.000 −0.189903
$$133$$ −320.000 −0.208628
$$134$$ 1696.00 1.09337
$$135$$ −162.000 −0.103280
$$136$$ −240.000 −0.151322
$$137$$ 2754.00 1.71745 0.858723 0.512440i $$-0.171258\pi$$
0.858723 + 0.512440i $$0.171258\pi$$
$$138$$ 432.000 0.266480
$$139$$ 2252.00 1.37419 0.687094 0.726568i $$-0.258886\pi$$
0.687094 + 0.726568i $$0.258886\pi$$
$$140$$ 480.000 0.289767
$$141$$ 612.000 0.365530
$$142$$ 264.000 0.156017
$$143$$ 312.000 0.182453
$$144$$ 144.000 0.0833333
$$145$$ −1692.00 −0.969055
$$146$$ 436.000 0.247148
$$147$$ −171.000 −0.0959445
$$148$$ 440.000 0.244377
$$149$$ −1770.00 −0.973182 −0.486591 0.873630i $$-0.661760\pi$$
−0.486591 + 0.873630i $$0.661760\pi$$
$$150$$ 534.000 0.290673
$$151$$ −988.000 −0.532466 −0.266233 0.963909i $$-0.585779\pi$$
−0.266233 + 0.963909i $$0.585779\pi$$
$$152$$ −128.000 −0.0683038
$$153$$ −270.000 −0.142668
$$154$$ 960.000 0.502331
$$155$$ 984.000 0.509915
$$156$$ −156.000 −0.0800641
$$157$$ 326.000 0.165717 0.0828587 0.996561i $$-0.473595\pi$$
0.0828587 + 0.996561i $$0.473595\pi$$
$$158$$ −2192.00 −1.10371
$$159$$ 2214.00 1.10429
$$160$$ 192.000 0.0948683
$$161$$ −1440.00 −0.704894
$$162$$ 162.000 0.0785674
$$163$$ 1496.00 0.718870 0.359435 0.933170i $$-0.382969\pi$$
0.359435 + 0.933170i $$0.382969\pi$$
$$164$$ −504.000 −0.239974
$$165$$ −432.000 −0.203825
$$166$$ 1104.00 0.516187
$$167$$ 1116.00 0.517118 0.258559 0.965995i $$-0.416752\pi$$
0.258559 + 0.965995i $$0.416752\pi$$
$$168$$ −480.000 −0.220433
$$169$$ 169.000 0.0769231
$$170$$ −360.000 −0.162416
$$171$$ −144.000 −0.0643974
$$172$$ 656.000 0.290811
$$173$$ 4374.00 1.92225 0.961124 0.276116i $$-0.0890472\pi$$
0.961124 + 0.276116i $$0.0890472\pi$$
$$174$$ 1692.00 0.737185
$$175$$ −1780.00 −0.768888
$$176$$ 384.000 0.164461
$$177$$ −360.000 −0.152877
$$178$$ 420.000 0.176856
$$179$$ 12.0000 0.00501074 0.00250537 0.999997i $$-0.499203\pi$$
0.00250537 + 0.999997i $$0.499203\pi$$
$$180$$ 216.000 0.0894427
$$181$$ 4718.00 1.93749 0.968746 0.248053i $$-0.0797909\pi$$
0.968746 + 0.248053i $$0.0797909\pi$$
$$182$$ 520.000 0.211786
$$183$$ −1842.00 −0.744069
$$184$$ −576.000 −0.230779
$$185$$ 660.000 0.262293
$$186$$ −984.000 −0.387905
$$187$$ −720.000 −0.281559
$$188$$ −816.000 −0.316558
$$189$$ −540.000 −0.207827
$$190$$ −192.000 −0.0733113
$$191$$ −1368.00 −0.518246 −0.259123 0.965844i $$-0.583434\pi$$
−0.259123 + 0.965844i $$0.583434\pi$$
$$192$$ −192.000 −0.0721688
$$193$$ −3310.00 −1.23450 −0.617251 0.786766i $$-0.711754\pi$$
−0.617251 + 0.786766i $$0.711754\pi$$
$$194$$ −3452.00 −1.27752
$$195$$ −234.000 −0.0859338
$$196$$ 228.000 0.0830904
$$197$$ 3126.00 1.13055 0.565275 0.824903i $$-0.308770\pi$$
0.565275 + 0.824903i $$0.308770\pi$$
$$198$$ 432.000 0.155055
$$199$$ 4664.00 1.66142 0.830709 0.556707i $$-0.187935\pi$$
0.830709 + 0.556707i $$0.187935\pi$$
$$200$$ −712.000 −0.251730
$$201$$ −2544.00 −0.892736
$$202$$ 1596.00 0.555912
$$203$$ −5640.00 −1.95000
$$204$$ 360.000 0.123554
$$205$$ −756.000 −0.257567
$$206$$ −1040.00 −0.351749
$$207$$ −648.000 −0.217580
$$208$$ 208.000 0.0693375
$$209$$ −384.000 −0.127090
$$210$$ −720.000 −0.236594
$$211$$ −556.000 −0.181406 −0.0907029 0.995878i $$-0.528911\pi$$
−0.0907029 + 0.995878i $$0.528911\pi$$
$$212$$ −2952.00 −0.956341
$$213$$ −396.000 −0.127387
$$214$$ 24.0000 0.00766638
$$215$$ 984.000 0.312131
$$216$$ −216.000 −0.0680414
$$217$$ 3280.00 1.02609
$$218$$ −3668.00 −1.13958
$$219$$ −654.000 −0.201796
$$220$$ 576.000 0.176518
$$221$$ −390.000 −0.118707
$$222$$ −660.000 −0.199533
$$223$$ −268.000 −0.0804781 −0.0402390 0.999190i $$-0.512812\pi$$
−0.0402390 + 0.999190i $$0.512812\pi$$
$$224$$ 640.000 0.190901
$$225$$ −801.000 −0.237333
$$226$$ −732.000 −0.215451
$$227$$ 1800.00 0.526300 0.263150 0.964755i $$-0.415239\pi$$
0.263150 + 0.964755i $$0.415239\pi$$
$$228$$ 192.000 0.0557698
$$229$$ 2990.00 0.862816 0.431408 0.902157i $$-0.358017\pi$$
0.431408 + 0.902157i $$0.358017\pi$$
$$230$$ −864.000 −0.247698
$$231$$ −1440.00 −0.410152
$$232$$ −2256.00 −0.638421
$$233$$ 2826.00 0.794581 0.397291 0.917693i $$-0.369951\pi$$
0.397291 + 0.917693i $$0.369951\pi$$
$$234$$ 234.000 0.0653720
$$235$$ −1224.00 −0.339766
$$236$$ 480.000 0.132396
$$237$$ 3288.00 0.901175
$$238$$ −1200.00 −0.326825
$$239$$ −1812.00 −0.490412 −0.245206 0.969471i $$-0.578856\pi$$
−0.245206 + 0.969471i $$0.578856\pi$$
$$240$$ −288.000 −0.0774597
$$241$$ −1582.00 −0.422845 −0.211422 0.977395i $$-0.567810\pi$$
−0.211422 + 0.977395i $$0.567810\pi$$
$$242$$ −1510.00 −0.401101
$$243$$ −243.000 −0.0641500
$$244$$ 2456.00 0.644382
$$245$$ 342.000 0.0891820
$$246$$ 756.000 0.195938
$$247$$ −208.000 −0.0535819
$$248$$ 1312.00 0.335936
$$249$$ −1656.00 −0.421465
$$250$$ −2568.00 −0.649658
$$251$$ 2148.00 0.540162 0.270081 0.962838i $$-0.412950\pi$$
0.270081 + 0.962838i $$0.412950\pi$$
$$252$$ 720.000 0.179983
$$253$$ −1728.00 −0.429401
$$254$$ 4288.00 1.05926
$$255$$ 540.000 0.132612
$$256$$ 256.000 0.0625000
$$257$$ −558.000 −0.135436 −0.0677181 0.997704i $$-0.521572\pi$$
−0.0677181 + 0.997704i $$0.521572\pi$$
$$258$$ −984.000 −0.237446
$$259$$ 2200.00 0.527804
$$260$$ 312.000 0.0744208
$$261$$ −2538.00 −0.601909
$$262$$ −5496.00 −1.29597
$$263$$ 2112.00 0.495177 0.247588 0.968865i $$-0.420362\pi$$
0.247588 + 0.968865i $$0.420362\pi$$
$$264$$ −576.000 −0.134282
$$265$$ −4428.00 −1.02645
$$266$$ −640.000 −0.147522
$$267$$ −630.000 −0.144402
$$268$$ 3392.00 0.773132
$$269$$ 5046.00 1.14372 0.571859 0.820352i $$-0.306223\pi$$
0.571859 + 0.820352i $$0.306223\pi$$
$$270$$ −324.000 −0.0730297
$$271$$ −3796.00 −0.850888 −0.425444 0.904985i $$-0.639882\pi$$
−0.425444 + 0.904985i $$0.639882\pi$$
$$272$$ −480.000 −0.107001
$$273$$ −780.000 −0.172922
$$274$$ 5508.00 1.21442
$$275$$ −2136.00 −0.468384
$$276$$ 864.000 0.188430
$$277$$ 5582.00 1.21079 0.605397 0.795924i $$-0.293014\pi$$
0.605397 + 0.795924i $$0.293014\pi$$
$$278$$ 4504.00 0.971698
$$279$$ 1476.00 0.316723
$$280$$ 960.000 0.204896
$$281$$ −1950.00 −0.413976 −0.206988 0.978343i $$-0.566366\pi$$
−0.206988 + 0.978343i $$0.566366\pi$$
$$282$$ 1224.00 0.258469
$$283$$ −4732.00 −0.993951 −0.496976 0.867765i $$-0.665556\pi$$
−0.496976 + 0.867765i $$0.665556\pi$$
$$284$$ 528.000 0.110321
$$285$$ 288.000 0.0598584
$$286$$ 624.000 0.129014
$$287$$ −2520.00 −0.518296
$$288$$ 288.000 0.0589256
$$289$$ −4013.00 −0.816813
$$290$$ −3384.00 −0.685225
$$291$$ 5178.00 1.04309
$$292$$ 872.000 0.174760
$$293$$ 4998.00 0.996540 0.498270 0.867022i $$-0.333969\pi$$
0.498270 + 0.867022i $$0.333969\pi$$
$$294$$ −342.000 −0.0678430
$$295$$ 720.000 0.142102
$$296$$ 880.000 0.172801
$$297$$ −648.000 −0.126602
$$298$$ −3540.00 −0.688143
$$299$$ −936.000 −0.181038
$$300$$ 1068.00 0.205537
$$301$$ 3280.00 0.628093
$$302$$ −1976.00 −0.376510
$$303$$ −2394.00 −0.453900
$$304$$ −256.000 −0.0482980
$$305$$ 3684.00 0.691624
$$306$$ −540.000 −0.100882
$$307$$ 6824.00 1.26862 0.634310 0.773079i $$-0.281284\pi$$
0.634310 + 0.773079i $$0.281284\pi$$
$$308$$ 1920.00 0.355202
$$309$$ 1560.00 0.287202
$$310$$ 1968.00 0.360564
$$311$$ −8760.00 −1.59722 −0.798608 0.601852i $$-0.794430\pi$$
−0.798608 + 0.601852i $$0.794430\pi$$
$$312$$ −312.000 −0.0566139
$$313$$ 3962.00 0.715481 0.357740 0.933821i $$-0.383547\pi$$
0.357740 + 0.933821i $$0.383547\pi$$
$$314$$ 652.000 0.117180
$$315$$ 1080.00 0.193178
$$316$$ −4384.00 −0.780441
$$317$$ 7086.00 1.25549 0.627744 0.778420i $$-0.283979\pi$$
0.627744 + 0.778420i $$0.283979\pi$$
$$318$$ 4428.00 0.780849
$$319$$ −6768.00 −1.18788
$$320$$ 384.000 0.0670820
$$321$$ −36.0000 −0.00625958
$$322$$ −2880.00 −0.498435
$$323$$ 480.000 0.0826870
$$324$$ 324.000 0.0555556
$$325$$ −1157.00 −0.197473
$$326$$ 2992.00 0.508318
$$327$$ 5502.00 0.930463
$$328$$ −1008.00 −0.169687
$$329$$ −4080.00 −0.683701
$$330$$ −864.000 −0.144126
$$331$$ −9016.00 −1.49717 −0.748586 0.663037i $$-0.769267\pi$$
−0.748586 + 0.663037i $$0.769267\pi$$
$$332$$ 2208.00 0.364999
$$333$$ 990.000 0.162918
$$334$$ 2232.00 0.365658
$$335$$ 5088.00 0.829812
$$336$$ −960.000 −0.155870
$$337$$ 2306.00 0.372747 0.186374 0.982479i $$-0.440327\pi$$
0.186374 + 0.982479i $$0.440327\pi$$
$$338$$ 338.000 0.0543928
$$339$$ 1098.00 0.175915
$$340$$ −720.000 −0.114846
$$341$$ 3936.00 0.625063
$$342$$ −288.000 −0.0455358
$$343$$ −5720.00 −0.900440
$$344$$ 1312.00 0.205635
$$345$$ 1296.00 0.202244
$$346$$ 8748.00 1.35924
$$347$$ −11076.0 −1.71352 −0.856759 0.515717i $$-0.827526\pi$$
−0.856759 + 0.515717i $$0.827526\pi$$
$$348$$ 3384.00 0.521269
$$349$$ 2342.00 0.359210 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$350$$ −3560.00 −0.543686
$$351$$ −351.000 −0.0533761
$$352$$ 768.000 0.116291
$$353$$ 4650.00 0.701118 0.350559 0.936541i $$-0.385992\pi$$
0.350559 + 0.936541i $$0.385992\pi$$
$$354$$ −720.000 −0.108100
$$355$$ 792.000 0.118408
$$356$$ 840.000 0.125056
$$357$$ 1800.00 0.266852
$$358$$ 24.0000 0.00354313
$$359$$ −11268.0 −1.65655 −0.828276 0.560320i $$-0.810678\pi$$
−0.828276 + 0.560320i $$0.810678\pi$$
$$360$$ 432.000 0.0632456
$$361$$ −6603.00 −0.962677
$$362$$ 9436.00 1.37001
$$363$$ 2265.00 0.327498
$$364$$ 1040.00 0.149755
$$365$$ 1308.00 0.187572
$$366$$ −3684.00 −0.526136
$$367$$ −7288.00 −1.03660 −0.518298 0.855200i $$-0.673434\pi$$
−0.518298 + 0.855200i $$0.673434\pi$$
$$368$$ −1152.00 −0.163185
$$369$$ −1134.00 −0.159983
$$370$$ 1320.00 0.185469
$$371$$ −14760.0 −2.06550
$$372$$ −1968.00 −0.274290
$$373$$ −9970.00 −1.38399 −0.691993 0.721904i $$-0.743267\pi$$
−0.691993 + 0.721904i $$0.743267\pi$$
$$374$$ −1440.00 −0.199093
$$375$$ 3852.00 0.530444
$$376$$ −1632.00 −0.223840
$$377$$ −3666.00 −0.500819
$$378$$ −1080.00 −0.146956
$$379$$ 13448.0 1.82263 0.911316 0.411708i $$-0.135068\pi$$
0.911316 + 0.411708i $$0.135068\pi$$
$$380$$ −384.000 −0.0518389
$$381$$ −6432.00 −0.864885
$$382$$ −2736.00 −0.366455
$$383$$ 11820.0 1.57696 0.788478 0.615064i $$-0.210870\pi$$
0.788478 + 0.615064i $$0.210870\pi$$
$$384$$ −384.000 −0.0510310
$$385$$ 2880.00 0.381243
$$386$$ −6620.00 −0.872925
$$387$$ 1476.00 0.193874
$$388$$ −6904.00 −0.903344
$$389$$ 174.000 0.0226790 0.0113395 0.999936i $$-0.496390\pi$$
0.0113395 + 0.999936i $$0.496390\pi$$
$$390$$ −468.000 −0.0607644
$$391$$ 2160.00 0.279376
$$392$$ 456.000 0.0587538
$$393$$ 8244.00 1.05815
$$394$$ 6252.00 0.799419
$$395$$ −6576.00 −0.837657
$$396$$ 864.000 0.109640
$$397$$ −2986.00 −0.377489 −0.188744 0.982026i $$-0.560442\pi$$
−0.188744 + 0.982026i $$0.560442\pi$$
$$398$$ 9328.00 1.17480
$$399$$ 960.000 0.120451
$$400$$ −1424.00 −0.178000
$$401$$ −10566.0 −1.31581 −0.657906 0.753100i $$-0.728558\pi$$
−0.657906 + 0.753100i $$0.728558\pi$$
$$402$$ −5088.00 −0.631260
$$403$$ 2132.00 0.263530
$$404$$ 3192.00 0.393089
$$405$$ 486.000 0.0596285
$$406$$ −11280.0 −1.37886
$$407$$ 2640.00 0.321523
$$408$$ 720.000 0.0873660
$$409$$ −7270.00 −0.878920 −0.439460 0.898262i $$-0.644830\pi$$
−0.439460 + 0.898262i $$0.644830\pi$$
$$410$$ −1512.00 −0.182128
$$411$$ −8262.00 −0.991568
$$412$$ −2080.00 −0.248724
$$413$$ 2400.00 0.285947
$$414$$ −1296.00 −0.153852
$$415$$ 3312.00 0.391758
$$416$$ 416.000 0.0490290
$$417$$ −6756.00 −0.793388
$$418$$ −768.000 −0.0898663
$$419$$ −7308.00 −0.852074 −0.426037 0.904706i $$-0.640091\pi$$
−0.426037 + 0.904706i $$0.640091\pi$$
$$420$$ −1440.00 −0.167297
$$421$$ −5938.00 −0.687412 −0.343706 0.939077i $$-0.611682\pi$$
−0.343706 + 0.939077i $$0.611682\pi$$
$$422$$ −1112.00 −0.128273
$$423$$ −1836.00 −0.211039
$$424$$ −5904.00 −0.676235
$$425$$ 2670.00 0.304739
$$426$$ −792.000 −0.0900764
$$427$$ 12280.0 1.39174
$$428$$ 48.0000 0.00542095
$$429$$ −936.000 −0.105339
$$430$$ 1968.00 0.220710
$$431$$ 11532.0 1.28881 0.644405 0.764685i $$-0.277105\pi$$
0.644405 + 0.764685i $$0.277105\pi$$
$$432$$ −432.000 −0.0481125
$$433$$ −718.000 −0.0796879 −0.0398440 0.999206i $$-0.512686\pi$$
−0.0398440 + 0.999206i $$0.512686\pi$$
$$434$$ 6560.00 0.725553
$$435$$ 5076.00 0.559484
$$436$$ −7336.00 −0.805804
$$437$$ 1152.00 0.126104
$$438$$ −1308.00 −0.142691
$$439$$ 8984.00 0.976726 0.488363 0.872640i $$-0.337594\pi$$
0.488363 + 0.872640i $$0.337594\pi$$
$$440$$ 1152.00 0.124817
$$441$$ 513.000 0.0553936
$$442$$ −780.000 −0.0839385
$$443$$ 2604.00 0.279277 0.139639 0.990203i $$-0.455406\pi$$
0.139639 + 0.990203i $$0.455406\pi$$
$$444$$ −1320.00 −0.141091
$$445$$ 1260.00 0.134224
$$446$$ −536.000 −0.0569066
$$447$$ 5310.00 0.561867
$$448$$ 1280.00 0.134987
$$449$$ −13206.0 −1.38804 −0.694020 0.719956i $$-0.744162\pi$$
−0.694020 + 0.719956i $$0.744162\pi$$
$$450$$ −1602.00 −0.167820
$$451$$ −3024.00 −0.315731
$$452$$ −1464.00 −0.152347
$$453$$ 2964.00 0.307419
$$454$$ 3600.00 0.372151
$$455$$ 1560.00 0.160734
$$456$$ 384.000 0.0394352
$$457$$ 8426.00 0.862476 0.431238 0.902238i $$-0.358077\pi$$
0.431238 + 0.902238i $$0.358077\pi$$
$$458$$ 5980.00 0.610103
$$459$$ 810.000 0.0823694
$$460$$ −1728.00 −0.175149
$$461$$ 16686.0 1.68578 0.842890 0.538086i $$-0.180852\pi$$
0.842890 + 0.538086i $$0.180852\pi$$
$$462$$ −2880.00 −0.290021
$$463$$ 15932.0 1.59919 0.799593 0.600543i $$-0.205049\pi$$
0.799593 + 0.600543i $$0.205049\pi$$
$$464$$ −4512.00 −0.451432
$$465$$ −2952.00 −0.294399
$$466$$ 5652.00 0.561854
$$467$$ 18540.0 1.83711 0.918553 0.395297i $$-0.129358\pi$$
0.918553 + 0.395297i $$0.129358\pi$$
$$468$$ 468.000 0.0462250
$$469$$ 16960.0 1.66981
$$470$$ −2448.00 −0.240251
$$471$$ −978.000 −0.0956770
$$472$$ 960.000 0.0936178
$$473$$ 3936.00 0.382616
$$474$$ 6576.00 0.637227
$$475$$ 1424.00 0.137553
$$476$$ −2400.00 −0.231100
$$477$$ −6642.00 −0.637560
$$478$$ −3624.00 −0.346774
$$479$$ 6180.00 0.589502 0.294751 0.955574i $$-0.404763\pi$$
0.294751 + 0.955574i $$0.404763\pi$$
$$480$$ −576.000 −0.0547723
$$481$$ 1430.00 0.135556
$$482$$ −3164.00 −0.298996
$$483$$ 4320.00 0.406971
$$484$$ −3020.00 −0.283621
$$485$$ −10356.0 −0.969571
$$486$$ −486.000 −0.0453609
$$487$$ 11756.0 1.09387 0.546936 0.837175i $$-0.315794\pi$$
0.546936 + 0.837175i $$0.315794\pi$$
$$488$$ 4912.00 0.455647
$$489$$ −4488.00 −0.415040
$$490$$ 684.000 0.0630612
$$491$$ 1908.00 0.175370 0.0876852 0.996148i $$-0.472053\pi$$
0.0876852 + 0.996148i $$0.472053\pi$$
$$492$$ 1512.00 0.138549
$$493$$ 8460.00 0.772858
$$494$$ −416.000 −0.0378881
$$495$$ 1296.00 0.117679
$$496$$ 2624.00 0.237542
$$497$$ 2640.00 0.238270
$$498$$ −3312.00 −0.298021
$$499$$ −8944.00 −0.802382 −0.401191 0.915995i $$-0.631404\pi$$
−0.401191 + 0.915995i $$0.631404\pi$$
$$500$$ −5136.00 −0.459378
$$501$$ −3348.00 −0.298558
$$502$$ 4296.00 0.381952
$$503$$ −6528.00 −0.578666 −0.289333 0.957228i $$-0.593434\pi$$
−0.289333 + 0.957228i $$0.593434\pi$$
$$504$$ 1440.00 0.127267
$$505$$ 4788.00 0.421907
$$506$$ −3456.00 −0.303632
$$507$$ −507.000 −0.0444116
$$508$$ 8576.00 0.749013
$$509$$ −12114.0 −1.05490 −0.527450 0.849586i $$-0.676852\pi$$
−0.527450 + 0.849586i $$0.676852\pi$$
$$510$$ 1080.00 0.0937710
$$511$$ 4360.00 0.377446
$$512$$ 512.000 0.0441942
$$513$$ 432.000 0.0371799
$$514$$ −1116.00 −0.0957678
$$515$$ −3120.00 −0.266958
$$516$$ −1968.00 −0.167900
$$517$$ −4896.00 −0.416491
$$518$$ 4400.00 0.373214
$$519$$ −13122.0 −1.10981
$$520$$ 624.000 0.0526235
$$521$$ −14310.0 −1.20333 −0.601663 0.798750i $$-0.705495\pi$$
−0.601663 + 0.798750i $$0.705495\pi$$
$$522$$ −5076.00 −0.425614
$$523$$ −18340.0 −1.53337 −0.766685 0.642024i $$-0.778095\pi$$
−0.766685 + 0.642024i $$0.778095\pi$$
$$524$$ −10992.0 −0.916389
$$525$$ 5340.00 0.443918
$$526$$ 4224.00 0.350143
$$527$$ −4920.00 −0.406677
$$528$$ −1152.00 −0.0949514
$$529$$ −6983.00 −0.573929
$$530$$ −8856.00 −0.725811
$$531$$ 1080.00 0.0882637
$$532$$ −1280.00 −0.104314
$$533$$ −1638.00 −0.133114
$$534$$ −1260.00 −0.102108
$$535$$ 72.0000 0.00581838
$$536$$ 6784.00 0.546687
$$537$$ −36.0000 −0.00289295
$$538$$ 10092.0 0.808731
$$539$$ 1368.00 0.109321
$$540$$ −648.000 −0.0516398
$$541$$ 9254.00 0.735417 0.367708 0.929941i $$-0.380142\pi$$
0.367708 + 0.929941i $$0.380142\pi$$
$$542$$ −7592.00 −0.601668
$$543$$ −14154.0 −1.11861
$$544$$ −960.000 −0.0756611
$$545$$ −11004.0 −0.864880
$$546$$ −1560.00 −0.122274
$$547$$ 17444.0 1.36353 0.681766 0.731571i $$-0.261212\pi$$
0.681766 + 0.731571i $$0.261212\pi$$
$$548$$ 11016.0 0.858723
$$549$$ 5526.00 0.429588
$$550$$ −4272.00 −0.331198
$$551$$ 4512.00 0.348852
$$552$$ 1728.00 0.133240
$$553$$ −21920.0 −1.68559
$$554$$ 11164.0 0.856160
$$555$$ −1980.00 −0.151435
$$556$$ 9008.00 0.687094
$$557$$ −3714.00 −0.282526 −0.141263 0.989972i $$-0.545116\pi$$
−0.141263 + 0.989972i $$0.545116\pi$$
$$558$$ 2952.00 0.223957
$$559$$ 2132.00 0.161313
$$560$$ 1920.00 0.144884
$$561$$ 2160.00 0.162558
$$562$$ −3900.00 −0.292725
$$563$$ −13812.0 −1.03394 −0.516968 0.856004i $$-0.672940\pi$$
−0.516968 + 0.856004i $$0.672940\pi$$
$$564$$ 2448.00 0.182765
$$565$$ −2196.00 −0.163516
$$566$$ −9464.00 −0.702830
$$567$$ 1620.00 0.119989
$$568$$ 1056.00 0.0780084
$$569$$ −15942.0 −1.17456 −0.587279 0.809385i $$-0.699801\pi$$
−0.587279 + 0.809385i $$0.699801\pi$$
$$570$$ 576.000 0.0423263
$$571$$ 1604.00 0.117557 0.0587787 0.998271i $$-0.481279\pi$$
0.0587787 + 0.998271i $$0.481279\pi$$
$$572$$ 1248.00 0.0912264
$$573$$ 4104.00 0.299210
$$574$$ −5040.00 −0.366490
$$575$$ 6408.00 0.464751
$$576$$ 576.000 0.0416667
$$577$$ −10654.0 −0.768686 −0.384343 0.923190i $$-0.625572\pi$$
−0.384343 + 0.923190i $$0.625572\pi$$
$$578$$ −8026.00 −0.577574
$$579$$ 9930.00 0.712740
$$580$$ −6768.00 −0.484527
$$581$$ 11040.0 0.788324
$$582$$ 10356.0 0.737577
$$583$$ −17712.0 −1.25824
$$584$$ 1744.00 0.123574
$$585$$ 702.000 0.0496139
$$586$$ 9996.00 0.704660
$$587$$ −9984.00 −0.702017 −0.351008 0.936372i $$-0.614161\pi$$
−0.351008 + 0.936372i $$0.614161\pi$$
$$588$$ −684.000 −0.0479723
$$589$$ −2624.00 −0.183565
$$590$$ 1440.00 0.100481
$$591$$ −9378.00 −0.652723
$$592$$ 1760.00 0.122188
$$593$$ 12618.0 0.873793 0.436896 0.899512i $$-0.356078\pi$$
0.436896 + 0.899512i $$0.356078\pi$$
$$594$$ −1296.00 −0.0895211
$$595$$ −3600.00 −0.248043
$$596$$ −7080.00 −0.486591
$$597$$ −13992.0 −0.959220
$$598$$ −1872.00 −0.128013
$$599$$ 11184.0 0.762881 0.381441 0.924393i $$-0.375428\pi$$
0.381441 + 0.924393i $$0.375428\pi$$
$$600$$ 2136.00 0.145336
$$601$$ 2810.00 0.190719 0.0953596 0.995443i $$-0.469600\pi$$
0.0953596 + 0.995443i $$0.469600\pi$$
$$602$$ 6560.00 0.444129
$$603$$ 7632.00 0.515421
$$604$$ −3952.00 −0.266233
$$605$$ −4530.00 −0.304414
$$606$$ −4788.00 −0.320956
$$607$$ 1064.00 0.0711473 0.0355737 0.999367i $$-0.488674\pi$$
0.0355737 + 0.999367i $$0.488674\pi$$
$$608$$ −512.000 −0.0341519
$$609$$ 16920.0 1.12583
$$610$$ 7368.00 0.489052
$$611$$ −2652.00 −0.175595
$$612$$ −1080.00 −0.0713340
$$613$$ −20914.0 −1.37799 −0.688996 0.724766i $$-0.741948\pi$$
−0.688996 + 0.724766i $$0.741948\pi$$
$$614$$ 13648.0 0.897050
$$615$$ 2268.00 0.148707
$$616$$ 3840.00 0.251166
$$617$$ 9714.00 0.633826 0.316913 0.948455i $$-0.397354\pi$$
0.316913 + 0.948455i $$0.397354\pi$$
$$618$$ 3120.00 0.203082
$$619$$ −14848.0 −0.964122 −0.482061 0.876138i $$-0.660112\pi$$
−0.482061 + 0.876138i $$0.660112\pi$$
$$620$$ 3936.00 0.254957
$$621$$ 1944.00 0.125620
$$622$$ −17520.0 −1.12940
$$623$$ 4200.00 0.270095
$$624$$ −624.000 −0.0400320
$$625$$ 3421.00 0.218944
$$626$$ 7924.00 0.505921
$$627$$ 1152.00 0.0733755
$$628$$ 1304.00 0.0828587
$$629$$ −3300.00 −0.209189
$$630$$ 2160.00 0.136598
$$631$$ 19172.0 1.20955 0.604774 0.796397i $$-0.293263\pi$$
0.604774 + 0.796397i $$0.293263\pi$$
$$632$$ −8768.00 −0.551855
$$633$$ 1668.00 0.104735
$$634$$ 14172.0 0.887763
$$635$$ 12864.0 0.803925
$$636$$ 8856.00 0.552143
$$637$$ 741.000 0.0460902
$$638$$ −13536.0 −0.839961
$$639$$ 1188.00 0.0735470
$$640$$ 768.000 0.0474342
$$641$$ −11502.0 −0.708739 −0.354369 0.935105i $$-0.615304\pi$$
−0.354369 + 0.935105i $$0.615304\pi$$
$$642$$ −72.0000 −0.00442619
$$643$$ −15568.0 −0.954809 −0.477404 0.878684i $$-0.658422\pi$$
−0.477404 + 0.878684i $$0.658422\pi$$
$$644$$ −5760.00 −0.352447
$$645$$ −2952.00 −0.180209
$$646$$ 960.000 0.0584686
$$647$$ 1128.00 0.0685414 0.0342707 0.999413i $$-0.489089\pi$$
0.0342707 + 0.999413i $$0.489089\pi$$
$$648$$ 648.000 0.0392837
$$649$$ 2880.00 0.174191
$$650$$ −2314.00 −0.139635
$$651$$ −9840.00 −0.592412
$$652$$ 5984.00 0.359435
$$653$$ 8118.00 0.486496 0.243248 0.969964i $$-0.421787\pi$$
0.243248 + 0.969964i $$0.421787\pi$$
$$654$$ 11004.0 0.657936
$$655$$ −16488.0 −0.983572
$$656$$ −2016.00 −0.119987
$$657$$ 1962.00 0.116507
$$658$$ −8160.00 −0.483450
$$659$$ 13572.0 0.802261 0.401131 0.916021i $$-0.368617\pi$$
0.401131 + 0.916021i $$0.368617\pi$$
$$660$$ −1728.00 −0.101913
$$661$$ −13138.0 −0.773085 −0.386542 0.922272i $$-0.626331\pi$$
−0.386542 + 0.922272i $$0.626331\pi$$
$$662$$ −18032.0 −1.05866
$$663$$ 1170.00 0.0685355
$$664$$ 4416.00 0.258093
$$665$$ −1920.00 −0.111962
$$666$$ 1980.00 0.115200
$$667$$ 20304.0 1.17867
$$668$$ 4464.00 0.258559
$$669$$ 804.000 0.0464640
$$670$$ 10176.0 0.586766
$$671$$ 14736.0 0.847805
$$672$$ −1920.00 −0.110217
$$673$$ −718.000 −0.0411246 −0.0205623 0.999789i $$-0.506546\pi$$
−0.0205623 + 0.999789i $$0.506546\pi$$
$$674$$ 4612.00 0.263572
$$675$$ 2403.00 0.137024
$$676$$ 676.000 0.0384615
$$677$$ −2994.00 −0.169969 −0.0849843 0.996382i $$-0.527084\pi$$
−0.0849843 + 0.996382i $$0.527084\pi$$
$$678$$ 2196.00 0.124391
$$679$$ −34520.0 −1.95104
$$680$$ −1440.00 −0.0812081
$$681$$ −5400.00 −0.303860
$$682$$ 7872.00 0.441986
$$683$$ 27384.0 1.53414 0.767071 0.641562i $$-0.221713\pi$$
0.767071 + 0.641562i $$0.221713\pi$$
$$684$$ −576.000 −0.0321987
$$685$$ 16524.0 0.921678
$$686$$ −11440.0 −0.636707
$$687$$ −8970.00 −0.498147
$$688$$ 2624.00 0.145406
$$689$$ −9594.00 −0.530482
$$690$$ 2592.00 0.143008
$$691$$ 27632.0 1.52123 0.760616 0.649202i $$-0.224897\pi$$
0.760616 + 0.649202i $$0.224897\pi$$
$$692$$ 17496.0 0.961124
$$693$$ 4320.00 0.236801
$$694$$ −22152.0 −1.21164
$$695$$ 13512.0 0.737467
$$696$$ 6768.00 0.368592
$$697$$ 3780.00 0.205420
$$698$$ 4684.00 0.254000
$$699$$ −8478.00 −0.458752
$$700$$ −7120.00 −0.384444
$$701$$ 19062.0 1.02705 0.513525 0.858075i $$-0.328339\pi$$
0.513525 + 0.858075i $$0.328339\pi$$
$$702$$ −702.000 −0.0377426
$$703$$ −1760.00 −0.0944234
$$704$$ 1536.00 0.0822304
$$705$$ 3672.00 0.196164
$$706$$ 9300.00 0.495765
$$707$$ 15960.0 0.848992
$$708$$ −1440.00 −0.0764386
$$709$$ 3854.00 0.204147 0.102073 0.994777i $$-0.467452\pi$$
0.102073 + 0.994777i $$0.467452\pi$$
$$710$$ 1584.00 0.0837274
$$711$$ −9864.00 −0.520294
$$712$$ 1680.00 0.0884279
$$713$$ −11808.0 −0.620215
$$714$$ 3600.00 0.188693
$$715$$ 1872.00 0.0979144
$$716$$ 48.0000 0.00250537
$$717$$ 5436.00 0.283140
$$718$$ −22536.0 −1.17136
$$719$$ 20976.0 1.08800 0.544001 0.839085i $$-0.316909\pi$$
0.544001 + 0.839085i $$0.316909\pi$$
$$720$$ 864.000 0.0447214
$$721$$ −10400.0 −0.537193
$$722$$ −13206.0 −0.680715
$$723$$ 4746.00 0.244130
$$724$$ 18872.0 0.968746
$$725$$ 25098.0 1.28568
$$726$$ 4530.00 0.231576
$$727$$ −29464.0 −1.50311 −0.751554 0.659672i $$-0.770695\pi$$
−0.751554 + 0.659672i $$0.770695\pi$$
$$728$$ 2080.00 0.105893
$$729$$ 729.000 0.0370370
$$730$$ 2616.00 0.132634
$$731$$ −4920.00 −0.248937
$$732$$ −7368.00 −0.372034
$$733$$ −2698.00 −0.135952 −0.0679761 0.997687i $$-0.521654\pi$$
−0.0679761 + 0.997687i $$0.521654\pi$$
$$734$$ −14576.0 −0.732984
$$735$$ −1026.00 −0.0514892
$$736$$ −2304.00 −0.115389
$$737$$ 20352.0 1.01720
$$738$$ −2268.00 −0.113125
$$739$$ 632.000 0.0314594 0.0157297 0.999876i $$-0.494993\pi$$
0.0157297 + 0.999876i $$0.494993\pi$$
$$740$$ 2640.00 0.131146
$$741$$ 624.000 0.0309355
$$742$$ −29520.0 −1.46053
$$743$$ −20844.0 −1.02920 −0.514598 0.857432i $$-0.672059\pi$$
−0.514598 + 0.857432i $$0.672059\pi$$
$$744$$ −3936.00 −0.193953
$$745$$ −10620.0 −0.522264
$$746$$ −19940.0 −0.978626
$$747$$ 4968.00 0.243333
$$748$$ −2880.00 −0.140780
$$749$$ 240.000 0.0117082
$$750$$ 7704.00 0.375080
$$751$$ 272.000 0.0132163 0.00660814 0.999978i $$-0.497897\pi$$
0.00660814 + 0.999978i $$0.497897\pi$$
$$752$$ −3264.00 −0.158279
$$753$$ −6444.00 −0.311862
$$754$$ −7332.00 −0.354132
$$755$$ −5928.00 −0.285751
$$756$$ −2160.00 −0.103913
$$757$$ 37550.0 1.80288 0.901439 0.432907i $$-0.142512\pi$$
0.901439 + 0.432907i $$0.142512\pi$$
$$758$$ 26896.0 1.28880
$$759$$ 5184.00 0.247915
$$760$$ −768.000 −0.0366556
$$761$$ 33330.0 1.58766 0.793832 0.608138i $$-0.208083\pi$$
0.793832 + 0.608138i $$0.208083\pi$$
$$762$$ −12864.0 −0.611566
$$763$$ −36680.0 −1.74037
$$764$$ −5472.00 −0.259123
$$765$$ −1620.00 −0.0765637
$$766$$ 23640.0 1.11508
$$767$$ 1560.00 0.0734398
$$768$$ −768.000 −0.0360844
$$769$$ −15406.0 −0.722438 −0.361219 0.932481i $$-0.617639\pi$$
−0.361219 + 0.932481i $$0.617639\pi$$
$$770$$ 5760.00 0.269579
$$771$$ 1674.00 0.0781941
$$772$$ −13240.0 −0.617251
$$773$$ −29514.0 −1.37328 −0.686640 0.726998i $$-0.740915\pi$$
−0.686640 + 0.726998i $$0.740915\pi$$
$$774$$ 2952.00 0.137090
$$775$$ −14596.0 −0.676521
$$776$$ −13808.0 −0.638761
$$777$$ −6600.00 −0.304728
$$778$$ 348.000 0.0160365
$$779$$ 2016.00 0.0927223
$$780$$ −936.000 −0.0429669
$$781$$ 3168.00 0.145147
$$782$$ 4320.00 0.197548
$$783$$ 7614.00 0.347512
$$784$$ 912.000 0.0415452
$$785$$ 1956.00 0.0889333
$$786$$ 16488.0 0.748228
$$787$$ 33176.0 1.50266 0.751332 0.659924i $$-0.229412\pi$$
0.751332 + 0.659924i $$0.229412\pi$$
$$788$$ 12504.0 0.565275
$$789$$ −6336.00 −0.285890
$$790$$ −13152.0 −0.592313
$$791$$ −7320.00 −0.329038
$$792$$ 1728.00 0.0775275
$$793$$ 7982.00 0.357439
$$794$$ −5972.00 −0.266925
$$795$$ 13284.0 0.592623
$$796$$ 18656.0 0.830709
$$797$$ −16746.0 −0.744258 −0.372129 0.928181i $$-0.621372\pi$$
−0.372129 + 0.928181i $$0.621372\pi$$
$$798$$ 1920.00 0.0851720
$$799$$ 6120.00 0.270976
$$800$$ −2848.00 −0.125865
$$801$$ 1890.00 0.0833706
$$802$$ −21132.0 −0.930420
$$803$$ 5232.00 0.229929
$$804$$ −10176.0 −0.446368
$$805$$ −8640.00 −0.378286
$$806$$ 4264.00 0.186344
$$807$$ −15138.0 −0.660326
$$808$$ 6384.00 0.277956
$$809$$ −15846.0 −0.688647 −0.344324 0.938851i $$-0.611892\pi$$
−0.344324 + 0.938851i $$0.611892\pi$$
$$810$$ 972.000 0.0421637
$$811$$ 22952.0 0.993778 0.496889 0.867814i $$-0.334476\pi$$
0.496889 + 0.867814i $$0.334476\pi$$
$$812$$ −22560.0 −0.975001
$$813$$ 11388.0 0.491260
$$814$$ 5280.00 0.227351
$$815$$ 8976.00 0.385786
$$816$$ 1440.00 0.0617771
$$817$$ −2624.00 −0.112365
$$818$$ −14540.0 −0.621490
$$819$$ 2340.00 0.0998367
$$820$$ −3024.00 −0.128784
$$821$$ −37146.0 −1.57906 −0.789528 0.613715i $$-0.789674\pi$$
−0.789528 + 0.613715i $$0.789674\pi$$
$$822$$ −16524.0 −0.701144
$$823$$ −9592.00 −0.406265 −0.203133 0.979151i $$-0.565112\pi$$
−0.203133 + 0.979151i $$0.565112\pi$$
$$824$$ −4160.00 −0.175874
$$825$$ 6408.00 0.270422
$$826$$ 4800.00 0.202195
$$827$$ −39960.0 −1.68022 −0.840112 0.542413i $$-0.817511\pi$$
−0.840112 + 0.542413i $$0.817511\pi$$
$$828$$ −2592.00 −0.108790
$$829$$ −3706.00 −0.155265 −0.0776325 0.996982i $$-0.524736\pi$$
−0.0776325 + 0.996982i $$0.524736\pi$$
$$830$$ 6624.00 0.277015
$$831$$ −16746.0 −0.699052
$$832$$ 832.000 0.0346688
$$833$$ −1710.00 −0.0711260
$$834$$ −13512.0 −0.561010
$$835$$ 6696.00 0.277515
$$836$$ −1536.00 −0.0635451
$$837$$ −4428.00 −0.182860
$$838$$ −14616.0 −0.602508
$$839$$ 9756.00 0.401448 0.200724 0.979648i $$-0.435671\pi$$
0.200724 + 0.979648i $$0.435671\pi$$
$$840$$ −2880.00 −0.118297
$$841$$ 55135.0 2.26065
$$842$$ −11876.0 −0.486074
$$843$$ 5850.00 0.239009
$$844$$ −2224.00 −0.0907029
$$845$$ 1014.00 0.0412813
$$846$$ −3672.00 −0.149227
$$847$$ −15100.0 −0.612565
$$848$$ −11808.0 −0.478170
$$849$$ 14196.0 0.573858
$$850$$ 5340.00 0.215483
$$851$$ −7920.00 −0.319029
$$852$$ −1584.00 −0.0636936
$$853$$ 11342.0 0.455267 0.227633 0.973747i $$-0.426901\pi$$
0.227633 + 0.973747i $$0.426901\pi$$
$$854$$ 24560.0 0.984105
$$855$$ −864.000 −0.0345593
$$856$$ 96.0000 0.00383319
$$857$$ −16134.0 −0.643089 −0.321544 0.946895i $$-0.604202\pi$$
−0.321544 + 0.946895i $$0.604202\pi$$
$$858$$ −1872.00 −0.0744860
$$859$$ −20932.0 −0.831421 −0.415710 0.909497i $$-0.636467\pi$$
−0.415710 + 0.909497i $$0.636467\pi$$
$$860$$ 3936.00 0.156066
$$861$$ 7560.00 0.299238
$$862$$ 23064.0 0.911326
$$863$$ 10044.0 0.396178 0.198089 0.980184i $$-0.436526\pi$$
0.198089 + 0.980184i $$0.436526\pi$$
$$864$$ −864.000 −0.0340207
$$865$$ 26244.0 1.03159
$$866$$ −1436.00 −0.0563479
$$867$$ 12039.0 0.471587
$$868$$ 13120.0 0.513044
$$869$$ −26304.0 −1.02681
$$870$$ 10152.0 0.395615
$$871$$ 11024.0 0.428856
$$872$$ −14672.0 −0.569790
$$873$$ −15534.0 −0.602229
$$874$$ 2304.00 0.0891693
$$875$$ −25680.0 −0.992163
$$876$$ −2616.00 −0.100898
$$877$$ −26314.0 −1.01318 −0.506591 0.862186i $$-0.669095\pi$$
−0.506591 + 0.862186i $$0.669095\pi$$
$$878$$ 17968.0 0.690650
$$879$$ −14994.0 −0.575353
$$880$$ 2304.00 0.0882589
$$881$$ 37506.0 1.43429 0.717145 0.696924i $$-0.245449\pi$$
0.717145 + 0.696924i $$0.245449\pi$$
$$882$$ 1026.00 0.0391692
$$883$$ −6388.00 −0.243458 −0.121729 0.992563i $$-0.538844\pi$$
−0.121729 + 0.992563i $$0.538844\pi$$
$$884$$ −1560.00 −0.0593535
$$885$$ −2160.00 −0.0820425
$$886$$ 5208.00 0.197479
$$887$$ −5472.00 −0.207138 −0.103569 0.994622i $$-0.533026\pi$$
−0.103569 + 0.994622i $$0.533026\pi$$
$$888$$ −2640.00 −0.0997664
$$889$$ 42880.0 1.61772
$$890$$ 2520.00 0.0949108
$$891$$ 1944.00 0.0730937
$$892$$ −1072.00 −0.0402390
$$893$$ 3264.00 0.122313
$$894$$ 10620.0 0.397300
$$895$$ 72.0000 0.00268904
$$896$$ 2560.00 0.0954504
$$897$$ 2808.00 0.104522
$$898$$ −26412.0 −0.981492
$$899$$ −46248.0 −1.71575
$$900$$ −3204.00 −0.118667
$$901$$ 22140.0 0.818635
$$902$$ −6048.00 −0.223255
$$903$$ −9840.00 −0.362630
$$904$$ −2928.00 −0.107725
$$905$$ 28308.0 1.03977
$$906$$ 5928.00 0.217378
$$907$$ −7180.00 −0.262853 −0.131427 0.991326i $$-0.541956\pi$$
−0.131427 + 0.991326i $$0.541956\pi$$
$$908$$ 7200.00 0.263150
$$909$$ 7182.00 0.262059
$$910$$ 3120.00 0.113656
$$911$$ 27624.0 1.00464 0.502318 0.864683i $$-0.332481\pi$$
0.502318 + 0.864683i $$0.332481\pi$$
$$912$$ 768.000 0.0278849
$$913$$ 13248.0 0.480224
$$914$$ 16852.0 0.609863
$$915$$ −11052.0 −0.399309
$$916$$ 11960.0 0.431408
$$917$$ −54960.0 −1.97921
$$918$$ 1620.00 0.0582440
$$919$$ −30256.0 −1.08602 −0.543011 0.839726i $$-0.682716\pi$$
−0.543011 + 0.839726i $$0.682716\pi$$
$$920$$ −3456.00 −0.123849
$$921$$ −20472.0 −0.732438
$$922$$ 33372.0 1.19203
$$923$$ 1716.00 0.0611948
$$924$$ −5760.00 −0.205076
$$925$$ −9790.00 −0.347993
$$926$$ 31864.0 1.13079
$$927$$ −4680.00 −0.165816
$$928$$ −9024.00 −0.319210
$$929$$ −1926.00 −0.0680194 −0.0340097 0.999422i $$-0.510828\pi$$
−0.0340097 + 0.999422i $$0.510828\pi$$
$$930$$ −5904.00 −0.208172
$$931$$ −912.000 −0.0321048
$$932$$ 11304.0 0.397291
$$933$$ 26280.0 0.922153
$$934$$ 37080.0 1.29903
$$935$$ −4320.00 −0.151101
$$936$$ 936.000 0.0326860
$$937$$ 3962.00 0.138135 0.0690677 0.997612i $$-0.477998\pi$$
0.0690677 + 0.997612i $$0.477998\pi$$
$$938$$ 33920.0 1.18073
$$939$$ −11886.0 −0.413083
$$940$$ −4896.00 −0.169883
$$941$$ −1074.00 −0.0372066 −0.0186033 0.999827i $$-0.505922\pi$$
−0.0186033 + 0.999827i $$0.505922\pi$$
$$942$$ −1956.00 −0.0676538
$$943$$ 9072.00 0.313282
$$944$$ 1920.00 0.0661978
$$945$$ −3240.00 −0.111531
$$946$$ 7872.00 0.270551
$$947$$ 4848.00 0.166356 0.0831778 0.996535i $$-0.473493\pi$$
0.0831778 + 0.996535i $$0.473493\pi$$
$$948$$ 13152.0 0.450588
$$949$$ 2834.00 0.0969394
$$950$$ 2848.00 0.0972645
$$951$$ −21258.0 −0.724856
$$952$$ −4800.00 −0.163413
$$953$$ 762.000 0.0259009 0.0129505 0.999916i $$-0.495878\pi$$
0.0129505 + 0.999916i $$0.495878\pi$$
$$954$$ −13284.0 −0.450823
$$955$$ −8208.00 −0.278120
$$956$$ −7248.00 −0.245206
$$957$$ 20304.0 0.685826
$$958$$ 12360.0 0.416841
$$959$$ 55080.0 1.85467
$$960$$ −1152.00 −0.0387298
$$961$$ −2895.00 −0.0971770
$$962$$ 2860.00 0.0958525
$$963$$ 108.000 0.00361397
$$964$$ −6328.00 −0.211422
$$965$$ −19860.0 −0.662504
$$966$$ 8640.00 0.287772
$$967$$ 35804.0 1.19067 0.595336 0.803477i $$-0.297019\pi$$
0.595336 + 0.803477i $$0.297019\pi$$
$$968$$ −6040.00 −0.200551
$$969$$ −1440.00 −0.0477394
$$970$$ −20712.0 −0.685590
$$971$$ −4260.00 −0.140793 −0.0703964 0.997519i $$-0.522426\pi$$
−0.0703964 + 0.997519i $$0.522426\pi$$
$$972$$ −972.000 −0.0320750
$$973$$ 45040.0 1.48398
$$974$$ 23512.0 0.773484
$$975$$ 3471.00 0.114011
$$976$$ 9824.00 0.322191
$$977$$ −28710.0 −0.940137 −0.470069 0.882630i $$-0.655771\pi$$
−0.470069 + 0.882630i $$0.655771\pi$$
$$978$$ −8976.00 −0.293477
$$979$$ 5040.00 0.164534
$$980$$ 1368.00 0.0445910
$$981$$ −16506.0 −0.537203
$$982$$ 3816.00 0.124006
$$983$$ −49524.0 −1.60689 −0.803444 0.595381i $$-0.797001\pi$$
−0.803444 + 0.595381i $$0.797001\pi$$
$$984$$ 3024.00 0.0979691
$$985$$ 18756.0 0.606717
$$986$$ 16920.0 0.546493
$$987$$ 12240.0 0.394735
$$988$$ −832.000 −0.0267909
$$989$$ −11808.0 −0.379649
$$990$$ 2592.00 0.0832113
$$991$$ 44408.0 1.42348 0.711739 0.702444i $$-0.247908\pi$$
0.711739 + 0.702444i $$0.247908\pi$$
$$992$$ 5248.00 0.167968
$$993$$ 27048.0 0.864393
$$994$$ 5280.00 0.168482
$$995$$ 27984.0 0.891610
$$996$$ −6624.00 −0.210732
$$997$$ 18398.0 0.584424 0.292212 0.956354i $$-0.405609\pi$$
0.292212 + 0.956354i $$0.405609\pi$$
$$998$$ −17888.0 −0.567369
$$999$$ −2970.00 −0.0940607
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 78.4.a.e.1.1 1
3.2 odd 2 234.4.a.b.1.1 1
4.3 odd 2 624.4.a.i.1.1 1
5.4 even 2 1950.4.a.c.1.1 1
8.3 odd 2 2496.4.a.b.1.1 1
8.5 even 2 2496.4.a.k.1.1 1
12.11 even 2 1872.4.a.e.1.1 1
13.5 odd 4 1014.4.b.c.337.1 2
13.8 odd 4 1014.4.b.c.337.2 2
13.12 even 2 1014.4.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.e.1.1 1 1.1 even 1 trivial
234.4.a.b.1.1 1 3.2 odd 2
624.4.a.i.1.1 1 4.3 odd 2
1014.4.a.b.1.1 1 13.12 even 2
1014.4.b.c.337.1 2 13.5 odd 4
1014.4.b.c.337.2 2 13.8 odd 4
1872.4.a.e.1.1 1 12.11 even 2
1950.4.a.c.1.1 1 5.4 even 2
2496.4.a.b.1.1 1 8.3 odd 2
2496.4.a.k.1.1 1 8.5 even 2