Properties

 Label 234.4.a.b.1.1 Level $234$ Weight $4$ Character 234.1 Self dual yes Analytic conductor $13.806$ 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] = [234,4,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, 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("234.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 234.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: $$13.8064469413$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} +20.0000 q^{7} -8.00000 q^{8} +O(q^{10})$$ $$q-2.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} +20.0000 q^{7} -8.00000 q^{8} +12.0000 q^{10} -24.0000 q^{11} +13.0000 q^{13} -40.0000 q^{14} +16.0000 q^{16} +30.0000 q^{17} -16.0000 q^{19} -24.0000 q^{20} +48.0000 q^{22} +72.0000 q^{23} -89.0000 q^{25} -26.0000 q^{26} +80.0000 q^{28} +282.000 q^{29} +164.000 q^{31} -32.0000 q^{32} -60.0000 q^{34} -120.000 q^{35} +110.000 q^{37} +32.0000 q^{38} +48.0000 q^{40} +126.000 q^{41} +164.000 q^{43} -96.0000 q^{44} -144.000 q^{46} +204.000 q^{47} +57.0000 q^{49} +178.000 q^{50} +52.0000 q^{52} +738.000 q^{53} +144.000 q^{55} -160.000 q^{56} -564.000 q^{58} -120.000 q^{59} +614.000 q^{61} -328.000 q^{62} +64.0000 q^{64} -78.0000 q^{65} +848.000 q^{67} +120.000 q^{68} +240.000 q^{70} -132.000 q^{71} +218.000 q^{73} -220.000 q^{74} -64.0000 q^{76} -480.000 q^{77} -1096.00 q^{79} -96.0000 q^{80} -252.000 q^{82} -552.000 q^{83} -180.000 q^{85} -328.000 q^{86} +192.000 q^{88} -210.000 q^{89} +260.000 q^{91} +288.000 q^{92} -408.000 q^{94} +96.0000 q^{95} -1726.00 q^{97} -114.000 q^{98} +O(q^{100})$$

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
$$3$$ 0 0
$$4$$ 4.00000 0.500000
$$5$$ −6.00000 −0.536656 −0.268328 0.963328i $$-0.586471\pi$$
−0.268328 + 0.963328i $$0.586471\pi$$
$$6$$ 0 0
$$7$$ 20.0000 1.07990 0.539949 0.841698i $$-0.318443\pi$$
0.539949 + 0.841698i $$0.318443\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ 0 0
$$10$$ 12.0000 0.379473
$$11$$ −24.0000 −0.657843 −0.328921 0.944357i $$-0.606685\pi$$
−0.328921 + 0.944357i $$0.606685\pi$$
$$12$$ 0 0
$$13$$ 13.0000 0.277350
$$14$$ −40.0000 −0.763604
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ 30.0000 0.428004 0.214002 0.976833i $$-0.431350\pi$$
0.214002 + 0.976833i $$0.431350\pi$$
$$18$$ 0 0
$$19$$ −16.0000 −0.193192 −0.0965961 0.995324i $$-0.530796\pi$$
−0.0965961 + 0.995324i $$0.530796\pi$$
$$20$$ −24.0000 −0.268328
$$21$$ 0 0
$$22$$ 48.0000 0.465165
$$23$$ 72.0000 0.652741 0.326370 0.945242i $$-0.394174\pi$$
0.326370 + 0.945242i $$0.394174\pi$$
$$24$$ 0 0
$$25$$ −89.0000 −0.712000
$$26$$ −26.0000 −0.196116
$$27$$ 0 0
$$28$$ 80.0000 0.539949
$$29$$ 282.000 1.80573 0.902864 0.429927i $$-0.141461\pi$$
0.902864 + 0.429927i $$0.141461\pi$$
$$30$$ 0 0
$$31$$ 164.000 0.950170 0.475085 0.879940i $$-0.342417\pi$$
0.475085 + 0.879940i $$0.342417\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 0 0
$$34$$ −60.0000 −0.302645
$$35$$ −120.000 −0.579534
$$36$$ 0 0
$$37$$ 110.000 0.488754 0.244377 0.969680i $$-0.421417\pi$$
0.244377 + 0.969680i $$0.421417\pi$$
$$38$$ 32.0000 0.136608
$$39$$ 0 0
$$40$$ 48.0000 0.189737
$$41$$ 126.000 0.479949 0.239974 0.970779i $$-0.422861\pi$$
0.239974 + 0.970779i $$0.422861\pi$$
$$42$$ 0 0
$$43$$ 164.000 0.581622 0.290811 0.956780i $$-0.406075\pi$$
0.290811 + 0.956780i $$0.406075\pi$$
$$44$$ −96.0000 −0.328921
$$45$$ 0 0
$$46$$ −144.000 −0.461557
$$47$$ 204.000 0.633116 0.316558 0.948573i $$-0.397473\pi$$
0.316558 + 0.948573i $$0.397473\pi$$
$$48$$ 0 0
$$49$$ 57.0000 0.166181
$$50$$ 178.000 0.503460
$$51$$ 0 0
$$52$$ 52.0000 0.138675
$$53$$ 738.000 1.91268 0.956341 0.292255i $$-0.0944055\pi$$
0.956341 + 0.292255i $$0.0944055\pi$$
$$54$$ 0 0
$$55$$ 144.000 0.353036
$$56$$ −160.000 −0.381802
$$57$$ 0 0
$$58$$ −564.000 −1.27684
$$59$$ −120.000 −0.264791 −0.132396 0.991197i $$-0.542267\pi$$
−0.132396 + 0.991197i $$0.542267\pi$$
$$60$$ 0 0
$$61$$ 614.000 1.28876 0.644382 0.764703i $$-0.277115\pi$$
0.644382 + 0.764703i $$0.277115\pi$$
$$62$$ −328.000 −0.671872
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ −78.0000 −0.148842
$$66$$ 0 0
$$67$$ 848.000 1.54626 0.773132 0.634245i $$-0.218689\pi$$
0.773132 + 0.634245i $$0.218689\pi$$
$$68$$ 120.000 0.214002
$$69$$ 0 0
$$70$$ 240.000 0.409793
$$71$$ −132.000 −0.220641 −0.110321 0.993896i $$-0.535188\pi$$
−0.110321 + 0.993896i $$0.535188\pi$$
$$72$$ 0 0
$$73$$ 218.000 0.349520 0.174760 0.984611i $$-0.444085\pi$$
0.174760 + 0.984611i $$0.444085\pi$$
$$74$$ −220.000 −0.345601
$$75$$ 0 0
$$76$$ −64.0000 −0.0965961
$$77$$ −480.000 −0.710404
$$78$$ 0 0
$$79$$ −1096.00 −1.56088 −0.780441 0.625230i $$-0.785005\pi$$
−0.780441 + 0.625230i $$0.785005\pi$$
$$80$$ −96.0000 −0.134164
$$81$$ 0 0
$$82$$ −252.000 −0.339375
$$83$$ −552.000 −0.729998 −0.364999 0.931008i $$-0.618931\pi$$
−0.364999 + 0.931008i $$0.618931\pi$$
$$84$$ 0 0
$$85$$ −180.000 −0.229691
$$86$$ −328.000 −0.411269
$$87$$ 0 0
$$88$$ 192.000 0.232583
$$89$$ −210.000 −0.250112 −0.125056 0.992150i $$-0.539911\pi$$
−0.125056 + 0.992150i $$0.539911\pi$$
$$90$$ 0 0
$$91$$ 260.000 0.299510
$$92$$ 288.000 0.326370
$$93$$ 0 0
$$94$$ −408.000 −0.447681
$$95$$ 96.0000 0.103678
$$96$$ 0 0
$$97$$ −1726.00 −1.80669 −0.903344 0.428917i $$-0.858895\pi$$
−0.903344 + 0.428917i $$0.858895\pi$$
$$98$$ −114.000 −0.117508
$$99$$ 0 0
$$100$$ −356.000 −0.356000
$$101$$ −798.000 −0.786178 −0.393089 0.919500i $$-0.628594\pi$$
−0.393089 + 0.919500i $$0.628594\pi$$
$$102$$ 0 0
$$103$$ −520.000 −0.497448 −0.248724 0.968574i $$-0.580011\pi$$
−0.248724 + 0.968574i $$0.580011\pi$$
$$104$$ −104.000 −0.0980581
$$105$$ 0 0
$$106$$ −1476.00 −1.35247
$$107$$ −12.0000 −0.0108419 −0.00542095 0.999985i $$-0.501726\pi$$
−0.00542095 + 0.999985i $$0.501726\pi$$
$$108$$ 0 0
$$109$$ −1834.00 −1.61161 −0.805804 0.592182i $$-0.798267\pi$$
−0.805804 + 0.592182i $$0.798267\pi$$
$$110$$ −288.000 −0.249634
$$111$$ 0 0
$$112$$ 320.000 0.269975
$$113$$ 366.000 0.304694 0.152347 0.988327i $$-0.451317\pi$$
0.152347 + 0.988327i $$0.451317\pi$$
$$114$$ 0 0
$$115$$ −432.000 −0.350297
$$116$$ 1128.00 0.902864
$$117$$ 0 0
$$118$$ 240.000 0.187236
$$119$$ 600.000 0.462201
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ −1228.00 −0.911294
$$123$$ 0 0
$$124$$ 656.000 0.475085
$$125$$ 1284.00 0.918756
$$126$$ 0 0
$$127$$ 2144.00 1.49803 0.749013 0.662556i $$-0.230528\pi$$
0.749013 + 0.662556i $$0.230528\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 0 0
$$130$$ 156.000 0.105247
$$131$$ 2748.00 1.83278 0.916389 0.400289i $$-0.131090\pi$$
0.916389 + 0.400289i $$0.131090\pi$$
$$132$$ 0 0
$$133$$ −320.000 −0.208628
$$134$$ −1696.00 −1.09337
$$135$$ 0 0
$$136$$ −240.000 −0.151322
$$137$$ −2754.00 −1.71745 −0.858723 0.512440i $$-0.828742\pi$$
−0.858723 + 0.512440i $$0.828742\pi$$
$$138$$ 0 0
$$139$$ 2252.00 1.37419 0.687094 0.726568i $$-0.258886\pi$$
0.687094 + 0.726568i $$0.258886\pi$$
$$140$$ −480.000 −0.289767
$$141$$ 0 0
$$142$$ 264.000 0.156017
$$143$$ −312.000 −0.182453
$$144$$ 0 0
$$145$$ −1692.00 −0.969055
$$146$$ −436.000 −0.247148
$$147$$ 0 0
$$148$$ 440.000 0.244377
$$149$$ 1770.00 0.973182 0.486591 0.873630i $$-0.338240\pi$$
0.486591 + 0.873630i $$0.338240\pi$$
$$150$$ 0 0
$$151$$ −988.000 −0.532466 −0.266233 0.963909i $$-0.585779\pi$$
−0.266233 + 0.963909i $$0.585779\pi$$
$$152$$ 128.000 0.0683038
$$153$$ 0 0
$$154$$ 960.000 0.502331
$$155$$ −984.000 −0.509915
$$156$$ 0 0
$$157$$ 326.000 0.165717 0.0828587 0.996561i $$-0.473595\pi$$
0.0828587 + 0.996561i $$0.473595\pi$$
$$158$$ 2192.00 1.10371
$$159$$ 0 0
$$160$$ 192.000 0.0948683
$$161$$ 1440.00 0.704894
$$162$$ 0 0
$$163$$ 1496.00 0.718870 0.359435 0.933170i $$-0.382969\pi$$
0.359435 + 0.933170i $$0.382969\pi$$
$$164$$ 504.000 0.239974
$$165$$ 0 0
$$166$$ 1104.00 0.516187
$$167$$ −1116.00 −0.517118 −0.258559 0.965995i $$-0.583248\pi$$
−0.258559 + 0.965995i $$0.583248\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 360.000 0.162416
$$171$$ 0 0
$$172$$ 656.000 0.290811
$$173$$ −4374.00 −1.92225 −0.961124 0.276116i $$-0.910953\pi$$
−0.961124 + 0.276116i $$0.910953\pi$$
$$174$$ 0 0
$$175$$ −1780.00 −0.768888
$$176$$ −384.000 −0.164461
$$177$$ 0 0
$$178$$ 420.000 0.176856
$$179$$ −12.0000 −0.00501074 −0.00250537 0.999997i $$-0.500797\pi$$
−0.00250537 + 0.999997i $$0.500797\pi$$
$$180$$ 0 0
$$181$$ 4718.00 1.93749 0.968746 0.248053i $$-0.0797909\pi$$
0.968746 + 0.248053i $$0.0797909\pi$$
$$182$$ −520.000 −0.211786
$$183$$ 0 0
$$184$$ −576.000 −0.230779
$$185$$ −660.000 −0.262293
$$186$$ 0 0
$$187$$ −720.000 −0.281559
$$188$$ 816.000 0.316558
$$189$$ 0 0
$$190$$ −192.000 −0.0733113
$$191$$ 1368.00 0.518246 0.259123 0.965844i $$-0.416566\pi$$
0.259123 + 0.965844i $$0.416566\pi$$
$$192$$ 0 0
$$193$$ −3310.00 −1.23450 −0.617251 0.786766i $$-0.711754\pi$$
−0.617251 + 0.786766i $$0.711754\pi$$
$$194$$ 3452.00 1.27752
$$195$$ 0 0
$$196$$ 228.000 0.0830904
$$197$$ −3126.00 −1.13055 −0.565275 0.824903i $$-0.691230\pi$$
−0.565275 + 0.824903i $$0.691230\pi$$
$$198$$ 0 0
$$199$$ 4664.00 1.66142 0.830709 0.556707i $$-0.187935\pi$$
0.830709 + 0.556707i $$0.187935\pi$$
$$200$$ 712.000 0.251730
$$201$$ 0 0
$$202$$ 1596.00 0.555912
$$203$$ 5640.00 1.95000
$$204$$ 0 0
$$205$$ −756.000 −0.257567
$$206$$ 1040.00 0.351749
$$207$$ 0 0
$$208$$ 208.000 0.0693375
$$209$$ 384.000 0.127090
$$210$$ 0 0
$$211$$ −556.000 −0.181406 −0.0907029 0.995878i $$-0.528911\pi$$
−0.0907029 + 0.995878i $$0.528911\pi$$
$$212$$ 2952.00 0.956341
$$213$$ 0 0
$$214$$ 24.0000 0.00766638
$$215$$ −984.000 −0.312131
$$216$$ 0 0
$$217$$ 3280.00 1.02609
$$218$$ 3668.00 1.13958
$$219$$ 0 0
$$220$$ 576.000 0.176518
$$221$$ 390.000 0.118707
$$222$$ 0 0
$$223$$ −268.000 −0.0804781 −0.0402390 0.999190i $$-0.512812\pi$$
−0.0402390 + 0.999190i $$0.512812\pi$$
$$224$$ −640.000 −0.190901
$$225$$ 0 0
$$226$$ −732.000 −0.215451
$$227$$ −1800.00 −0.526300 −0.263150 0.964755i $$-0.584761\pi$$
−0.263150 + 0.964755i $$0.584761\pi$$
$$228$$ 0 0
$$229$$ 2990.00 0.862816 0.431408 0.902157i $$-0.358017\pi$$
0.431408 + 0.902157i $$0.358017\pi$$
$$230$$ 864.000 0.247698
$$231$$ 0 0
$$232$$ −2256.00 −0.638421
$$233$$ −2826.00 −0.794581 −0.397291 0.917693i $$-0.630049\pi$$
−0.397291 + 0.917693i $$0.630049\pi$$
$$234$$ 0 0
$$235$$ −1224.00 −0.339766
$$236$$ −480.000 −0.132396
$$237$$ 0 0
$$238$$ −1200.00 −0.326825
$$239$$ 1812.00 0.490412 0.245206 0.969471i $$-0.421144\pi$$
0.245206 + 0.969471i $$0.421144\pi$$
$$240$$ 0 0
$$241$$ −1582.00 −0.422845 −0.211422 0.977395i $$-0.567810\pi$$
−0.211422 + 0.977395i $$0.567810\pi$$
$$242$$ 1510.00 0.401101
$$243$$ 0 0
$$244$$ 2456.00 0.644382
$$245$$ −342.000 −0.0891820
$$246$$ 0 0
$$247$$ −208.000 −0.0535819
$$248$$ −1312.00 −0.335936
$$249$$ 0 0
$$250$$ −2568.00 −0.649658
$$251$$ −2148.00 −0.540162 −0.270081 0.962838i $$-0.587050\pi$$
−0.270081 + 0.962838i $$0.587050\pi$$
$$252$$ 0 0
$$253$$ −1728.00 −0.429401
$$254$$ −4288.00 −1.05926
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ 558.000 0.135436 0.0677181 0.997704i $$-0.478428\pi$$
0.0677181 + 0.997704i $$0.478428\pi$$
$$258$$ 0 0
$$259$$ 2200.00 0.527804
$$260$$ −312.000 −0.0744208
$$261$$ 0 0
$$262$$ −5496.00 −1.29597
$$263$$ −2112.00 −0.495177 −0.247588 0.968865i $$-0.579638\pi$$
−0.247588 + 0.968865i $$0.579638\pi$$
$$264$$ 0 0
$$265$$ −4428.00 −1.02645
$$266$$ 640.000 0.147522
$$267$$ 0 0
$$268$$ 3392.00 0.773132
$$269$$ −5046.00 −1.14372 −0.571859 0.820352i $$-0.693777\pi$$
−0.571859 + 0.820352i $$0.693777\pi$$
$$270$$ 0 0
$$271$$ −3796.00 −0.850888 −0.425444 0.904985i $$-0.639882\pi$$
−0.425444 + 0.904985i $$0.639882\pi$$
$$272$$ 480.000 0.107001
$$273$$ 0 0
$$274$$ 5508.00 1.21442
$$275$$ 2136.00 0.468384
$$276$$ 0 0
$$277$$ 5582.00 1.21079 0.605397 0.795924i $$-0.293014\pi$$
0.605397 + 0.795924i $$0.293014\pi$$
$$278$$ −4504.00 −0.971698
$$279$$ 0 0
$$280$$ 960.000 0.204896
$$281$$ 1950.00 0.413976 0.206988 0.978343i $$-0.433634\pi$$
0.206988 + 0.978343i $$0.433634\pi$$
$$282$$ 0 0
$$283$$ −4732.00 −0.993951 −0.496976 0.867765i $$-0.665556\pi$$
−0.496976 + 0.867765i $$0.665556\pi$$
$$284$$ −528.000 −0.110321
$$285$$ 0 0
$$286$$ 624.000 0.129014
$$287$$ 2520.00 0.518296
$$288$$ 0 0
$$289$$ −4013.00 −0.816813
$$290$$ 3384.00 0.685225
$$291$$ 0 0
$$292$$ 872.000 0.174760
$$293$$ −4998.00 −0.996540 −0.498270 0.867022i $$-0.666031\pi$$
−0.498270 + 0.867022i $$0.666031\pi$$
$$294$$ 0 0
$$295$$ 720.000 0.142102
$$296$$ −880.000 −0.172801
$$297$$ 0 0
$$298$$ −3540.00 −0.688143
$$299$$ 936.000 0.181038
$$300$$ 0 0
$$301$$ 3280.00 0.628093
$$302$$ 1976.00 0.376510
$$303$$ 0 0
$$304$$ −256.000 −0.0482980
$$305$$ −3684.00 −0.691624
$$306$$ 0 0
$$307$$ 6824.00 1.26862 0.634310 0.773079i $$-0.281284\pi$$
0.634310 + 0.773079i $$0.281284\pi$$
$$308$$ −1920.00 −0.355202
$$309$$ 0 0
$$310$$ 1968.00 0.360564
$$311$$ 8760.00 1.59722 0.798608 0.601852i $$-0.205570\pi$$
0.798608 + 0.601852i $$0.205570\pi$$
$$312$$ 0 0
$$313$$ 3962.00 0.715481 0.357740 0.933821i $$-0.383547\pi$$
0.357740 + 0.933821i $$0.383547\pi$$
$$314$$ −652.000 −0.117180
$$315$$ 0 0
$$316$$ −4384.00 −0.780441
$$317$$ −7086.00 −1.25549 −0.627744 0.778420i $$-0.716021\pi$$
−0.627744 + 0.778420i $$0.716021\pi$$
$$318$$ 0 0
$$319$$ −6768.00 −1.18788
$$320$$ −384.000 −0.0670820
$$321$$ 0 0
$$322$$ −2880.00 −0.498435
$$323$$ −480.000 −0.0826870
$$324$$ 0 0
$$325$$ −1157.00 −0.197473
$$326$$ −2992.00 −0.508318
$$327$$ 0 0
$$328$$ −1008.00 −0.169687
$$329$$ 4080.00 0.683701
$$330$$ 0 0
$$331$$ −9016.00 −1.49717 −0.748586 0.663037i $$-0.769267\pi$$
−0.748586 + 0.663037i $$0.769267\pi$$
$$332$$ −2208.00 −0.364999
$$333$$ 0 0
$$334$$ 2232.00 0.365658
$$335$$ −5088.00 −0.829812
$$336$$ 0 0
$$337$$ 2306.00 0.372747 0.186374 0.982479i $$-0.440327\pi$$
0.186374 + 0.982479i $$0.440327\pi$$
$$338$$ −338.000 −0.0543928
$$339$$ 0 0
$$340$$ −720.000 −0.114846
$$341$$ −3936.00 −0.625063
$$342$$ 0 0
$$343$$ −5720.00 −0.900440
$$344$$ −1312.00 −0.205635
$$345$$ 0 0
$$346$$ 8748.00 1.35924
$$347$$ 11076.0 1.71352 0.856759 0.515717i $$-0.172474\pi$$
0.856759 + 0.515717i $$0.172474\pi$$
$$348$$ 0 0
$$349$$ 2342.00 0.359210 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$350$$ 3560.00 0.543686
$$351$$ 0 0
$$352$$ 768.000 0.116291
$$353$$ −4650.00 −0.701118 −0.350559 0.936541i $$-0.614008\pi$$
−0.350559 + 0.936541i $$0.614008\pi$$
$$354$$ 0 0
$$355$$ 792.000 0.118408
$$356$$ −840.000 −0.125056
$$357$$ 0 0
$$358$$ 24.0000 0.00354313
$$359$$ 11268.0 1.65655 0.828276 0.560320i $$-0.189322\pi$$
0.828276 + 0.560320i $$0.189322\pi$$
$$360$$ 0 0
$$361$$ −6603.00 −0.962677
$$362$$ −9436.00 −1.37001
$$363$$ 0 0
$$364$$ 1040.00 0.149755
$$365$$ −1308.00 −0.187572
$$366$$ 0 0
$$367$$ −7288.00 −1.03660 −0.518298 0.855200i $$-0.673434\pi$$
−0.518298 + 0.855200i $$0.673434\pi$$
$$368$$ 1152.00 0.163185
$$369$$ 0 0
$$370$$ 1320.00 0.185469
$$371$$ 14760.0 2.06550
$$372$$ 0 0
$$373$$ −9970.00 −1.38399 −0.691993 0.721904i $$-0.743267\pi$$
−0.691993 + 0.721904i $$0.743267\pi$$
$$374$$ 1440.00 0.199093
$$375$$ 0 0
$$376$$ −1632.00 −0.223840
$$377$$ 3666.00 0.500819
$$378$$ 0 0
$$379$$ 13448.0 1.82263 0.911316 0.411708i $$-0.135068\pi$$
0.911316 + 0.411708i $$0.135068\pi$$
$$380$$ 384.000 0.0518389
$$381$$ 0 0
$$382$$ −2736.00 −0.366455
$$383$$ −11820.0 −1.57696 −0.788478 0.615064i $$-0.789130\pi$$
−0.788478 + 0.615064i $$0.789130\pi$$
$$384$$ 0 0
$$385$$ 2880.00 0.381243
$$386$$ 6620.00 0.872925
$$387$$ 0 0
$$388$$ −6904.00 −0.903344
$$389$$ −174.000 −0.0226790 −0.0113395 0.999936i $$-0.503610\pi$$
−0.0113395 + 0.999936i $$0.503610\pi$$
$$390$$ 0 0
$$391$$ 2160.00 0.279376
$$392$$ −456.000 −0.0587538
$$393$$ 0 0
$$394$$ 6252.00 0.799419
$$395$$ 6576.00 0.837657
$$396$$ 0 0
$$397$$ −2986.00 −0.377489 −0.188744 0.982026i $$-0.560442\pi$$
−0.188744 + 0.982026i $$0.560442\pi$$
$$398$$ −9328.00 −1.17480
$$399$$ 0 0
$$400$$ −1424.00 −0.178000
$$401$$ 10566.0 1.31581 0.657906 0.753100i $$-0.271442\pi$$
0.657906 + 0.753100i $$0.271442\pi$$
$$402$$ 0 0
$$403$$ 2132.00 0.263530
$$404$$ −3192.00 −0.393089
$$405$$ 0 0
$$406$$ −11280.0 −1.37886
$$407$$ −2640.00 −0.321523
$$408$$ 0 0
$$409$$ −7270.00 −0.878920 −0.439460 0.898262i $$-0.644830\pi$$
−0.439460 + 0.898262i $$0.644830\pi$$
$$410$$ 1512.00 0.182128
$$411$$ 0 0
$$412$$ −2080.00 −0.248724
$$413$$ −2400.00 −0.285947
$$414$$ 0 0
$$415$$ 3312.00 0.391758
$$416$$ −416.000 −0.0490290
$$417$$ 0 0
$$418$$ −768.000 −0.0898663
$$419$$ 7308.00 0.852074 0.426037 0.904706i $$-0.359909\pi$$
0.426037 + 0.904706i $$0.359909\pi$$
$$420$$ 0 0
$$421$$ −5938.00 −0.687412 −0.343706 0.939077i $$-0.611682\pi$$
−0.343706 + 0.939077i $$0.611682\pi$$
$$422$$ 1112.00 0.128273
$$423$$ 0 0
$$424$$ −5904.00 −0.676235
$$425$$ −2670.00 −0.304739
$$426$$ 0 0
$$427$$ 12280.0 1.39174
$$428$$ −48.0000 −0.00542095
$$429$$ 0 0
$$430$$ 1968.00 0.220710
$$431$$ −11532.0 −1.28881 −0.644405 0.764685i $$-0.722895\pi$$
−0.644405 + 0.764685i $$0.722895\pi$$
$$432$$ 0 0
$$433$$ −718.000 −0.0796879 −0.0398440 0.999206i $$-0.512686\pi$$
−0.0398440 + 0.999206i $$0.512686\pi$$
$$434$$ −6560.00 −0.725553
$$435$$ 0 0
$$436$$ −7336.00 −0.805804
$$437$$ −1152.00 −0.126104
$$438$$ 0 0
$$439$$ 8984.00 0.976726 0.488363 0.872640i $$-0.337594\pi$$
0.488363 + 0.872640i $$0.337594\pi$$
$$440$$ −1152.00 −0.124817
$$441$$ 0 0
$$442$$ −780.000 −0.0839385
$$443$$ −2604.00 −0.279277 −0.139639 0.990203i $$-0.544594\pi$$
−0.139639 + 0.990203i $$0.544594\pi$$
$$444$$ 0 0
$$445$$ 1260.00 0.134224
$$446$$ 536.000 0.0569066
$$447$$ 0 0
$$448$$ 1280.00 0.134987
$$449$$ 13206.0 1.38804 0.694020 0.719956i $$-0.255838\pi$$
0.694020 + 0.719956i $$0.255838\pi$$
$$450$$ 0 0
$$451$$ −3024.00 −0.315731
$$452$$ 1464.00 0.152347
$$453$$ 0 0
$$454$$ 3600.00 0.372151
$$455$$ −1560.00 −0.160734
$$456$$ 0 0
$$457$$ 8426.00 0.862476 0.431238 0.902238i $$-0.358077\pi$$
0.431238 + 0.902238i $$0.358077\pi$$
$$458$$ −5980.00 −0.610103
$$459$$ 0 0
$$460$$ −1728.00 −0.175149
$$461$$ −16686.0 −1.68578 −0.842890 0.538086i $$-0.819148\pi$$
−0.842890 + 0.538086i $$0.819148\pi$$
$$462$$ 0 0
$$463$$ 15932.0 1.59919 0.799593 0.600543i $$-0.205049\pi$$
0.799593 + 0.600543i $$0.205049\pi$$
$$464$$ 4512.00 0.451432
$$465$$ 0 0
$$466$$ 5652.00 0.561854
$$467$$ −18540.0 −1.83711 −0.918553 0.395297i $$-0.870642\pi$$
−0.918553 + 0.395297i $$0.870642\pi$$
$$468$$ 0 0
$$469$$ 16960.0 1.66981
$$470$$ 2448.00 0.240251
$$471$$ 0 0
$$472$$ 960.000 0.0936178
$$473$$ −3936.00 −0.382616
$$474$$ 0 0
$$475$$ 1424.00 0.137553
$$476$$ 2400.00 0.231100
$$477$$ 0 0
$$478$$ −3624.00 −0.346774
$$479$$ −6180.00 −0.589502 −0.294751 0.955574i $$-0.595237\pi$$
−0.294751 + 0.955574i $$0.595237\pi$$
$$480$$ 0 0
$$481$$ 1430.00 0.135556
$$482$$ 3164.00 0.298996
$$483$$ 0 0
$$484$$ −3020.00 −0.283621
$$485$$ 10356.0 0.969571
$$486$$ 0 0
$$487$$ 11756.0 1.09387 0.546936 0.837175i $$-0.315794\pi$$
0.546936 + 0.837175i $$0.315794\pi$$
$$488$$ −4912.00 −0.455647
$$489$$ 0 0
$$490$$ 684.000 0.0630612
$$491$$ −1908.00 −0.175370 −0.0876852 0.996148i $$-0.527947\pi$$
−0.0876852 + 0.996148i $$0.527947\pi$$
$$492$$ 0 0
$$493$$ 8460.00 0.772858
$$494$$ 416.000 0.0378881
$$495$$ 0 0
$$496$$ 2624.00 0.237542
$$497$$ −2640.00 −0.238270
$$498$$ 0 0
$$499$$ −8944.00 −0.802382 −0.401191 0.915995i $$-0.631404\pi$$
−0.401191 + 0.915995i $$0.631404\pi$$
$$500$$ 5136.00 0.459378
$$501$$ 0 0
$$502$$ 4296.00 0.381952
$$503$$ 6528.00 0.578666 0.289333 0.957228i $$-0.406566\pi$$
0.289333 + 0.957228i $$0.406566\pi$$
$$504$$ 0 0
$$505$$ 4788.00 0.421907
$$506$$ 3456.00 0.303632
$$507$$ 0 0
$$508$$ 8576.00 0.749013
$$509$$ 12114.0 1.05490 0.527450 0.849586i $$-0.323148\pi$$
0.527450 + 0.849586i $$0.323148\pi$$
$$510$$ 0 0
$$511$$ 4360.00 0.377446
$$512$$ −512.000 −0.0441942
$$513$$ 0 0
$$514$$ −1116.00 −0.0957678
$$515$$ 3120.00 0.266958
$$516$$ 0 0
$$517$$ −4896.00 −0.416491
$$518$$ −4400.00 −0.373214
$$519$$ 0 0
$$520$$ 624.000 0.0526235
$$521$$ 14310.0 1.20333 0.601663 0.798750i $$-0.294505\pi$$
0.601663 + 0.798750i $$0.294505\pi$$
$$522$$ 0 0
$$523$$ −18340.0 −1.53337 −0.766685 0.642024i $$-0.778095\pi$$
−0.766685 + 0.642024i $$0.778095\pi$$
$$524$$ 10992.0 0.916389
$$525$$ 0 0
$$526$$ 4224.00 0.350143
$$527$$ 4920.00 0.406677
$$528$$ 0 0
$$529$$ −6983.00 −0.573929
$$530$$ 8856.00 0.725811
$$531$$ 0 0
$$532$$ −1280.00 −0.104314
$$533$$ 1638.00 0.133114
$$534$$ 0 0
$$535$$ 72.0000 0.00581838
$$536$$ −6784.00 −0.546687
$$537$$ 0 0
$$538$$ 10092.0 0.808731
$$539$$ −1368.00 −0.109321
$$540$$ 0 0
$$541$$ 9254.00 0.735417 0.367708 0.929941i $$-0.380142\pi$$
0.367708 + 0.929941i $$0.380142\pi$$
$$542$$ 7592.00 0.601668
$$543$$ 0 0
$$544$$ −960.000 −0.0756611
$$545$$ 11004.0 0.864880
$$546$$ 0 0
$$547$$ 17444.0 1.36353 0.681766 0.731571i $$-0.261212\pi$$
0.681766 + 0.731571i $$0.261212\pi$$
$$548$$ −11016.0 −0.858723
$$549$$ 0 0
$$550$$ −4272.00 −0.331198
$$551$$ −4512.00 −0.348852
$$552$$ 0 0
$$553$$ −21920.0 −1.68559
$$554$$ −11164.0 −0.856160
$$555$$ 0 0
$$556$$ 9008.00 0.687094
$$557$$ 3714.00 0.282526 0.141263 0.989972i $$-0.454884\pi$$
0.141263 + 0.989972i $$0.454884\pi$$
$$558$$ 0 0
$$559$$ 2132.00 0.161313
$$560$$ −1920.00 −0.144884
$$561$$ 0 0
$$562$$ −3900.00 −0.292725
$$563$$ 13812.0 1.03394 0.516968 0.856004i $$-0.327060\pi$$
0.516968 + 0.856004i $$0.327060\pi$$
$$564$$ 0 0
$$565$$ −2196.00 −0.163516
$$566$$ 9464.00 0.702830
$$567$$ 0 0
$$568$$ 1056.00 0.0780084
$$569$$ 15942.0 1.17456 0.587279 0.809385i $$-0.300199\pi$$
0.587279 + 0.809385i $$0.300199\pi$$
$$570$$ 0 0
$$571$$ 1604.00 0.117557 0.0587787 0.998271i $$-0.481279\pi$$
0.0587787 + 0.998271i $$0.481279\pi$$
$$572$$ −1248.00 −0.0912264
$$573$$ 0 0
$$574$$ −5040.00 −0.366490
$$575$$ −6408.00 −0.464751
$$576$$ 0 0
$$577$$ −10654.0 −0.768686 −0.384343 0.923190i $$-0.625572\pi$$
−0.384343 + 0.923190i $$0.625572\pi$$
$$578$$ 8026.00 0.577574
$$579$$ 0 0
$$580$$ −6768.00 −0.484527
$$581$$ −11040.0 −0.788324
$$582$$ 0 0
$$583$$ −17712.0 −1.25824
$$584$$ −1744.00 −0.123574
$$585$$ 0 0
$$586$$ 9996.00 0.704660
$$587$$ 9984.00 0.702017 0.351008 0.936372i $$-0.385839\pi$$
0.351008 + 0.936372i $$0.385839\pi$$
$$588$$ 0 0
$$589$$ −2624.00 −0.183565
$$590$$ −1440.00 −0.100481
$$591$$ 0 0
$$592$$ 1760.00 0.122188
$$593$$ −12618.0 −0.873793 −0.436896 0.899512i $$-0.643922\pi$$
−0.436896 + 0.899512i $$0.643922\pi$$
$$594$$ 0 0
$$595$$ −3600.00 −0.248043
$$596$$ 7080.00 0.486591
$$597$$ 0 0
$$598$$ −1872.00 −0.128013
$$599$$ −11184.0 −0.762881 −0.381441 0.924393i $$-0.624572\pi$$
−0.381441 + 0.924393i $$0.624572\pi$$
$$600$$ 0 0
$$601$$ 2810.00 0.190719 0.0953596 0.995443i $$-0.469600\pi$$
0.0953596 + 0.995443i $$0.469600\pi$$
$$602$$ −6560.00 −0.444129
$$603$$ 0 0
$$604$$ −3952.00 −0.266233
$$605$$ 4530.00 0.304414
$$606$$ 0 0
$$607$$ 1064.00 0.0711473 0.0355737 0.999367i $$-0.488674\pi$$
0.0355737 + 0.999367i $$0.488674\pi$$
$$608$$ 512.000 0.0341519
$$609$$ 0 0
$$610$$ 7368.00 0.489052
$$611$$ 2652.00 0.175595
$$612$$ 0 0
$$613$$ −20914.0 −1.37799 −0.688996 0.724766i $$-0.741948\pi$$
−0.688996 + 0.724766i $$0.741948\pi$$
$$614$$ −13648.0 −0.897050
$$615$$ 0 0
$$616$$ 3840.00 0.251166
$$617$$ −9714.00 −0.633826 −0.316913 0.948455i $$-0.602646\pi$$
−0.316913 + 0.948455i $$0.602646\pi$$
$$618$$ 0 0
$$619$$ −14848.0 −0.964122 −0.482061 0.876138i $$-0.660112\pi$$
−0.482061 + 0.876138i $$0.660112\pi$$
$$620$$ −3936.00 −0.254957
$$621$$ 0 0
$$622$$ −17520.0 −1.12940
$$623$$ −4200.00 −0.270095
$$624$$ 0 0
$$625$$ 3421.00 0.218944
$$626$$ −7924.00 −0.505921
$$627$$ 0 0
$$628$$ 1304.00 0.0828587
$$629$$ 3300.00 0.209189
$$630$$ 0 0
$$631$$ 19172.0 1.20955 0.604774 0.796397i $$-0.293263\pi$$
0.604774 + 0.796397i $$0.293263\pi$$
$$632$$ 8768.00 0.551855
$$633$$ 0 0
$$634$$ 14172.0 0.887763
$$635$$ −12864.0 −0.803925
$$636$$ 0 0
$$637$$ 741.000 0.0460902
$$638$$ 13536.0 0.839961
$$639$$ 0 0
$$640$$ 768.000 0.0474342
$$641$$ 11502.0 0.708739 0.354369 0.935105i $$-0.384696\pi$$
0.354369 + 0.935105i $$0.384696\pi$$
$$642$$ 0 0
$$643$$ −15568.0 −0.954809 −0.477404 0.878684i $$-0.658422\pi$$
−0.477404 + 0.878684i $$0.658422\pi$$
$$644$$ 5760.00 0.352447
$$645$$ 0 0
$$646$$ 960.000 0.0584686
$$647$$ −1128.00 −0.0685414 −0.0342707 0.999413i $$-0.510911\pi$$
−0.0342707 + 0.999413i $$0.510911\pi$$
$$648$$ 0 0
$$649$$ 2880.00 0.174191
$$650$$ 2314.00 0.139635
$$651$$ 0 0
$$652$$ 5984.00 0.359435
$$653$$ −8118.00 −0.486496 −0.243248 0.969964i $$-0.578213\pi$$
−0.243248 + 0.969964i $$0.578213\pi$$
$$654$$ 0 0
$$655$$ −16488.0 −0.983572
$$656$$ 2016.00 0.119987
$$657$$ 0 0
$$658$$ −8160.00 −0.483450
$$659$$ −13572.0 −0.802261 −0.401131 0.916021i $$-0.631383\pi$$
−0.401131 + 0.916021i $$0.631383\pi$$
$$660$$ 0 0
$$661$$ −13138.0 −0.773085 −0.386542 0.922272i $$-0.626331\pi$$
−0.386542 + 0.922272i $$0.626331\pi$$
$$662$$ 18032.0 1.05866
$$663$$ 0 0
$$664$$ 4416.00 0.258093
$$665$$ 1920.00 0.111962
$$666$$ 0 0
$$667$$ 20304.0 1.17867
$$668$$ −4464.00 −0.258559
$$669$$ 0 0
$$670$$ 10176.0 0.586766
$$671$$ −14736.0 −0.847805
$$672$$ 0 0
$$673$$ −718.000 −0.0411246 −0.0205623 0.999789i $$-0.506546\pi$$
−0.0205623 + 0.999789i $$0.506546\pi$$
$$674$$ −4612.00 −0.263572
$$675$$ 0 0
$$676$$ 676.000 0.0384615
$$677$$ 2994.00 0.169969 0.0849843 0.996382i $$-0.472916\pi$$
0.0849843 + 0.996382i $$0.472916\pi$$
$$678$$ 0 0
$$679$$ −34520.0 −1.95104
$$680$$ 1440.00 0.0812081
$$681$$ 0 0
$$682$$ 7872.00 0.441986
$$683$$ −27384.0 −1.53414 −0.767071 0.641562i $$-0.778287\pi$$
−0.767071 + 0.641562i $$0.778287\pi$$
$$684$$ 0 0
$$685$$ 16524.0 0.921678
$$686$$ 11440.0 0.636707
$$687$$ 0 0
$$688$$ 2624.00 0.145406
$$689$$ 9594.00 0.530482
$$690$$ 0 0
$$691$$ 27632.0 1.52123 0.760616 0.649202i $$-0.224897\pi$$
0.760616 + 0.649202i $$0.224897\pi$$
$$692$$ −17496.0 −0.961124
$$693$$ 0 0
$$694$$ −22152.0 −1.21164
$$695$$ −13512.0 −0.737467
$$696$$ 0 0
$$697$$ 3780.00 0.205420
$$698$$ −4684.00 −0.254000
$$699$$ 0 0
$$700$$ −7120.00 −0.384444
$$701$$ −19062.0 −1.02705 −0.513525 0.858075i $$-0.671661\pi$$
−0.513525 + 0.858075i $$0.671661\pi$$
$$702$$ 0 0
$$703$$ −1760.00 −0.0944234
$$704$$ −1536.00 −0.0822304
$$705$$ 0 0
$$706$$ 9300.00 0.495765
$$707$$ −15960.0 −0.848992
$$708$$ 0 0
$$709$$ 3854.00 0.204147 0.102073 0.994777i $$-0.467452\pi$$
0.102073 + 0.994777i $$0.467452\pi$$
$$710$$ −1584.00 −0.0837274
$$711$$ 0 0
$$712$$ 1680.00 0.0884279
$$713$$ 11808.0 0.620215
$$714$$ 0 0
$$715$$ 1872.00 0.0979144
$$716$$ −48.0000 −0.00250537
$$717$$ 0 0
$$718$$ −22536.0 −1.17136
$$719$$ −20976.0 −1.08800 −0.544001 0.839085i $$-0.683091\pi$$
−0.544001 + 0.839085i $$0.683091\pi$$
$$720$$ 0 0
$$721$$ −10400.0 −0.537193
$$722$$ 13206.0 0.680715
$$723$$ 0 0
$$724$$ 18872.0 0.968746
$$725$$ −25098.0 −1.28568
$$726$$ 0 0
$$727$$ −29464.0 −1.50311 −0.751554 0.659672i $$-0.770695\pi$$
−0.751554 + 0.659672i $$0.770695\pi$$
$$728$$ −2080.00 −0.105893
$$729$$ 0 0
$$730$$ 2616.00 0.132634
$$731$$ 4920.00 0.248937
$$732$$ 0 0
$$733$$ −2698.00 −0.135952 −0.0679761 0.997687i $$-0.521654\pi$$
−0.0679761 + 0.997687i $$0.521654\pi$$
$$734$$ 14576.0 0.732984
$$735$$ 0 0
$$736$$ −2304.00 −0.115389
$$737$$ −20352.0 −1.01720
$$738$$ 0 0
$$739$$ 632.000 0.0314594 0.0157297 0.999876i $$-0.494993\pi$$
0.0157297 + 0.999876i $$0.494993\pi$$
$$740$$ −2640.00 −0.131146
$$741$$ 0 0
$$742$$ −29520.0 −1.46053
$$743$$ 20844.0 1.02920 0.514598 0.857432i $$-0.327941\pi$$
0.514598 + 0.857432i $$0.327941\pi$$
$$744$$ 0 0
$$745$$ −10620.0 −0.522264
$$746$$ 19940.0 0.978626
$$747$$ 0 0
$$748$$ −2880.00 −0.140780
$$749$$ −240.000 −0.0117082
$$750$$ 0 0
$$751$$ 272.000 0.0132163 0.00660814 0.999978i $$-0.497897\pi$$
0.00660814 + 0.999978i $$0.497897\pi$$
$$752$$ 3264.00 0.158279
$$753$$ 0 0
$$754$$ −7332.00 −0.354132
$$755$$ 5928.00 0.285751
$$756$$ 0 0
$$757$$ 37550.0 1.80288 0.901439 0.432907i $$-0.142512\pi$$
0.901439 + 0.432907i $$0.142512\pi$$
$$758$$ −26896.0 −1.28880
$$759$$ 0 0
$$760$$ −768.000 −0.0366556
$$761$$ −33330.0 −1.58766 −0.793832 0.608138i $$-0.791917\pi$$
−0.793832 + 0.608138i $$0.791917\pi$$
$$762$$ 0 0
$$763$$ −36680.0 −1.74037
$$764$$ 5472.00 0.259123
$$765$$ 0 0
$$766$$ 23640.0 1.11508
$$767$$ −1560.00 −0.0734398
$$768$$ 0 0
$$769$$ −15406.0 −0.722438 −0.361219 0.932481i $$-0.617639\pi$$
−0.361219 + 0.932481i $$0.617639\pi$$
$$770$$ −5760.00 −0.269579
$$771$$ 0 0
$$772$$ −13240.0 −0.617251
$$773$$ 29514.0 1.37328 0.686640 0.726998i $$-0.259085\pi$$
0.686640 + 0.726998i $$0.259085\pi$$
$$774$$ 0 0
$$775$$ −14596.0 −0.676521
$$776$$ 13808.0 0.638761
$$777$$ 0 0
$$778$$ 348.000 0.0160365
$$779$$ −2016.00 −0.0927223
$$780$$ 0 0
$$781$$ 3168.00 0.145147
$$782$$ −4320.00 −0.197548
$$783$$ 0 0
$$784$$ 912.000 0.0415452
$$785$$ −1956.00 −0.0889333
$$786$$ 0 0
$$787$$ 33176.0 1.50266 0.751332 0.659924i $$-0.229412\pi$$
0.751332 + 0.659924i $$0.229412\pi$$
$$788$$ −12504.0 −0.565275
$$789$$ 0 0
$$790$$ −13152.0 −0.592313
$$791$$ 7320.00 0.329038
$$792$$ 0 0
$$793$$ 7982.00 0.357439
$$794$$ 5972.00 0.266925
$$795$$ 0 0
$$796$$ 18656.0 0.830709
$$797$$ 16746.0 0.744258 0.372129 0.928181i $$-0.378628\pi$$
0.372129 + 0.928181i $$0.378628\pi$$
$$798$$ 0 0
$$799$$ 6120.00 0.270976
$$800$$ 2848.00 0.125865
$$801$$ 0 0
$$802$$ −21132.0 −0.930420
$$803$$ −5232.00 −0.229929
$$804$$ 0 0
$$805$$ −8640.00 −0.378286
$$806$$ −4264.00 −0.186344
$$807$$ 0 0
$$808$$ 6384.00 0.277956
$$809$$ 15846.0 0.688647 0.344324 0.938851i $$-0.388108\pi$$
0.344324 + 0.938851i $$0.388108\pi$$
$$810$$ 0 0
$$811$$ 22952.0 0.993778 0.496889 0.867814i $$-0.334476\pi$$
0.496889 + 0.867814i $$0.334476\pi$$
$$812$$ 22560.0 0.975001
$$813$$ 0 0
$$814$$ 5280.00 0.227351
$$815$$ −8976.00 −0.385786
$$816$$ 0 0
$$817$$ −2624.00 −0.112365
$$818$$ 14540.0 0.621490
$$819$$ 0 0
$$820$$ −3024.00 −0.128784
$$821$$ 37146.0 1.57906 0.789528 0.613715i $$-0.210326\pi$$
0.789528 + 0.613715i $$0.210326\pi$$
$$822$$ 0 0
$$823$$ −9592.00 −0.406265 −0.203133 0.979151i $$-0.565112\pi$$
−0.203133 + 0.979151i $$0.565112\pi$$
$$824$$ 4160.00 0.175874
$$825$$ 0 0
$$826$$ 4800.00 0.202195
$$827$$ 39960.0 1.68022 0.840112 0.542413i $$-0.182489\pi$$
0.840112 + 0.542413i $$0.182489\pi$$
$$828$$ 0 0
$$829$$ −3706.00 −0.155265 −0.0776325 0.996982i $$-0.524736\pi$$
−0.0776325 + 0.996982i $$0.524736\pi$$
$$830$$ −6624.00 −0.277015
$$831$$ 0 0
$$832$$ 832.000 0.0346688
$$833$$ 1710.00 0.0711260
$$834$$ 0 0
$$835$$ 6696.00 0.277515
$$836$$ 1536.00 0.0635451
$$837$$ 0 0
$$838$$ −14616.0 −0.602508
$$839$$ −9756.00 −0.401448 −0.200724 0.979648i $$-0.564329\pi$$
−0.200724 + 0.979648i $$0.564329\pi$$
$$840$$ 0 0
$$841$$ 55135.0 2.26065
$$842$$ 11876.0 0.486074
$$843$$ 0 0
$$844$$ −2224.00 −0.0907029
$$845$$ −1014.00 −0.0412813
$$846$$ 0 0
$$847$$ −15100.0 −0.612565
$$848$$ 11808.0 0.478170
$$849$$ 0 0
$$850$$ 5340.00 0.215483
$$851$$ 7920.00 0.319029
$$852$$ 0 0
$$853$$ 11342.0 0.455267 0.227633 0.973747i $$-0.426901\pi$$
0.227633 + 0.973747i $$0.426901\pi$$
$$854$$ −24560.0 −0.984105
$$855$$ 0 0
$$856$$ 96.0000 0.00383319
$$857$$ 16134.0 0.643089 0.321544 0.946895i $$-0.395798\pi$$
0.321544 + 0.946895i $$0.395798\pi$$
$$858$$ 0 0
$$859$$ −20932.0 −0.831421 −0.415710 0.909497i $$-0.636467\pi$$
−0.415710 + 0.909497i $$0.636467\pi$$
$$860$$ −3936.00 −0.156066
$$861$$ 0 0
$$862$$ 23064.0 0.911326
$$863$$ −10044.0 −0.396178 −0.198089 0.980184i $$-0.563474\pi$$
−0.198089 + 0.980184i $$0.563474\pi$$
$$864$$ 0 0
$$865$$ 26244.0 1.03159
$$866$$ 1436.00 0.0563479
$$867$$ 0 0
$$868$$ 13120.0 0.513044
$$869$$ 26304.0 1.02681
$$870$$ 0 0
$$871$$ 11024.0 0.428856
$$872$$ 14672.0 0.569790
$$873$$ 0 0
$$874$$ 2304.00 0.0891693
$$875$$ 25680.0 0.992163
$$876$$ 0 0
$$877$$ −26314.0 −1.01318 −0.506591 0.862186i $$-0.669095\pi$$
−0.506591 + 0.862186i $$0.669095\pi$$
$$878$$ −17968.0 −0.690650
$$879$$ 0 0
$$880$$ 2304.00 0.0882589
$$881$$ −37506.0 −1.43429 −0.717145 0.696924i $$-0.754551\pi$$
−0.717145 + 0.696924i $$0.754551\pi$$
$$882$$ 0 0
$$883$$ −6388.00 −0.243458 −0.121729 0.992563i $$-0.538844\pi$$
−0.121729 + 0.992563i $$0.538844\pi$$
$$884$$ 1560.00 0.0593535
$$885$$ 0 0
$$886$$ 5208.00 0.197479
$$887$$ 5472.00 0.207138 0.103569 0.994622i $$-0.466974\pi$$
0.103569 + 0.994622i $$0.466974\pi$$
$$888$$ 0 0
$$889$$ 42880.0 1.61772
$$890$$ −2520.00 −0.0949108
$$891$$ 0 0
$$892$$ −1072.00 −0.0402390
$$893$$ −3264.00 −0.122313
$$894$$ 0 0
$$895$$ 72.0000 0.00268904
$$896$$ −2560.00 −0.0954504
$$897$$ 0 0
$$898$$ −26412.0 −0.981492
$$899$$ 46248.0 1.71575
$$900$$ 0 0
$$901$$ 22140.0 0.818635
$$902$$ 6048.00 0.223255
$$903$$ 0 0
$$904$$ −2928.00 −0.107725
$$905$$ −28308.0 −1.03977
$$906$$ 0 0
$$907$$ −7180.00 −0.262853 −0.131427 0.991326i $$-0.541956\pi$$
−0.131427 + 0.991326i $$0.541956\pi$$
$$908$$ −7200.00 −0.263150
$$909$$ 0 0
$$910$$ 3120.00 0.113656
$$911$$ −27624.0 −1.00464 −0.502318 0.864683i $$-0.667519\pi$$
−0.502318 + 0.864683i $$0.667519\pi$$
$$912$$ 0 0
$$913$$ 13248.0 0.480224
$$914$$ −16852.0 −0.609863
$$915$$ 0 0
$$916$$ 11960.0 0.431408
$$917$$ 54960.0 1.97921
$$918$$ 0 0
$$919$$ −30256.0 −1.08602 −0.543011 0.839726i $$-0.682716\pi$$
−0.543011 + 0.839726i $$0.682716\pi$$
$$920$$ 3456.00 0.123849
$$921$$ 0 0
$$922$$ 33372.0 1.19203
$$923$$ −1716.00 −0.0611948
$$924$$ 0 0
$$925$$ −9790.00 −0.347993
$$926$$ −31864.0 −1.13079
$$927$$ 0 0
$$928$$ −9024.00 −0.319210
$$929$$ 1926.00 0.0680194 0.0340097 0.999422i $$-0.489172\pi$$
0.0340097 + 0.999422i $$0.489172\pi$$
$$930$$ 0 0
$$931$$ −912.000 −0.0321048
$$932$$ −11304.0 −0.397291
$$933$$ 0 0
$$934$$ 37080.0 1.29903
$$935$$ 4320.00 0.151101
$$936$$ 0 0
$$937$$ 3962.00 0.138135 0.0690677 0.997612i $$-0.477998\pi$$
0.0690677 + 0.997612i $$0.477998\pi$$
$$938$$ −33920.0 −1.18073
$$939$$ 0 0
$$940$$ −4896.00 −0.169883
$$941$$ 1074.00 0.0372066 0.0186033 0.999827i $$-0.494078\pi$$
0.0186033 + 0.999827i $$0.494078\pi$$
$$942$$ 0 0
$$943$$ 9072.00 0.313282
$$944$$ −1920.00 −0.0661978
$$945$$ 0 0
$$946$$ 7872.00 0.270551
$$947$$ −4848.00 −0.166356 −0.0831778 0.996535i $$-0.526507\pi$$
−0.0831778 + 0.996535i $$0.526507\pi$$
$$948$$ 0 0
$$949$$ 2834.00 0.0969394
$$950$$ −2848.00 −0.0972645
$$951$$ 0 0
$$952$$ −4800.00 −0.163413
$$953$$ −762.000 −0.0259009 −0.0129505 0.999916i $$-0.504122\pi$$
−0.0129505 + 0.999916i $$0.504122\pi$$
$$954$$ 0 0
$$955$$ −8208.00 −0.278120
$$956$$ 7248.00 0.245206
$$957$$ 0 0
$$958$$ 12360.0 0.416841
$$959$$ −55080.0 −1.85467
$$960$$ 0 0
$$961$$ −2895.00 −0.0971770
$$962$$ −2860.00 −0.0958525
$$963$$ 0 0
$$964$$ −6328.00 −0.211422
$$965$$ 19860.0 0.662504
$$966$$ 0 0
$$967$$ 35804.0 1.19067 0.595336 0.803477i $$-0.297019\pi$$
0.595336 + 0.803477i $$0.297019\pi$$
$$968$$ 6040.00 0.200551
$$969$$ 0 0
$$970$$ −20712.0 −0.685590
$$971$$ 4260.00 0.140793 0.0703964 0.997519i $$-0.477574\pi$$
0.0703964 + 0.997519i $$0.477574\pi$$
$$972$$ 0 0
$$973$$ 45040.0 1.48398
$$974$$ −23512.0 −0.773484
$$975$$ 0 0
$$976$$ 9824.00 0.322191
$$977$$ 28710.0 0.940137 0.470069 0.882630i $$-0.344229\pi$$
0.470069 + 0.882630i $$0.344229\pi$$
$$978$$ 0 0
$$979$$ 5040.00 0.164534
$$980$$ −1368.00 −0.0445910
$$981$$ 0 0
$$982$$ 3816.00 0.124006
$$983$$ 49524.0 1.60689 0.803444 0.595381i $$-0.202999\pi$$
0.803444 + 0.595381i $$0.202999\pi$$
$$984$$ 0 0
$$985$$ 18756.0 0.606717
$$986$$ −16920.0 −0.546493
$$987$$ 0 0
$$988$$ −832.000 −0.0267909
$$989$$ 11808.0 0.379649
$$990$$ 0 0
$$991$$ 44408.0 1.42348 0.711739 0.702444i $$-0.247908\pi$$
0.711739 + 0.702444i $$0.247908\pi$$
$$992$$ −5248.00 −0.167968
$$993$$ 0 0
$$994$$ 5280.00 0.168482
$$995$$ −27984.0 −0.891610
$$996$$ 0 0
$$997$$ 18398.0 0.584424 0.292212 0.956354i $$-0.405609\pi$$
0.292212 + 0.956354i $$0.405609\pi$$
$$998$$ 17888.0 0.567369
$$999$$ 0 0
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 234.4.a.b.1.1 1
3.2 odd 2 78.4.a.e.1.1 1
4.3 odd 2 1872.4.a.e.1.1 1
12.11 even 2 624.4.a.i.1.1 1
15.14 odd 2 1950.4.a.c.1.1 1
24.5 odd 2 2496.4.a.k.1.1 1
24.11 even 2 2496.4.a.b.1.1 1
39.5 even 4 1014.4.b.c.337.1 2
39.8 even 4 1014.4.b.c.337.2 2
39.38 odd 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 3.2 odd 2
234.4.a.b.1.1 1 1.1 even 1 trivial
624.4.a.i.1.1 1 12.11 even 2
1014.4.a.b.1.1 1 39.38 odd 2
1014.4.b.c.337.1 2 39.5 even 4
1014.4.b.c.337.2 2 39.8 even 4
1872.4.a.e.1.1 1 4.3 odd 2
1950.4.a.c.1.1 1 15.14 odd 2
2496.4.a.b.1.1 1 24.11 even 2
2496.4.a.k.1.1 1 24.5 odd 2