# Properties

 Label 624.4.a.i.1.1 Level $624$ Weight $4$ Character 624.1 Self dual yes Analytic conductor $36.817$ Analytic rank $1$ 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] = [624,4,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, 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("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 624.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: $$36.8171918436$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 624.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +6.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +6.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} -24.0000 q^{11} +13.0000 q^{13} +18.0000 q^{15} -30.0000 q^{17} +16.0000 q^{19} -60.0000 q^{21} +72.0000 q^{23} -89.0000 q^{25} +27.0000 q^{27} -282.000 q^{29} -164.000 q^{31} -72.0000 q^{33} -120.000 q^{35} +110.000 q^{37} +39.0000 q^{39} -126.000 q^{41} -164.000 q^{43} +54.0000 q^{45} +204.000 q^{47} +57.0000 q^{49} -90.0000 q^{51} -738.000 q^{53} -144.000 q^{55} +48.0000 q^{57} -120.000 q^{59} +614.000 q^{61} -180.000 q^{63} +78.0000 q^{65} -848.000 q^{67} +216.000 q^{69} -132.000 q^{71} +218.000 q^{73} -267.000 q^{75} +480.000 q^{77} +1096.00 q^{79} +81.0000 q^{81} -552.000 q^{83} -180.000 q^{85} -846.000 q^{87} +210.000 q^{89} -260.000 q^{91} -492.000 q^{93} +96.0000 q^{95} -1726.00 q^{97} -216.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 6.00000 0.536656 0.268328 0.963328i $$-0.413529\pi$$
0.268328 + 0.963328i $$0.413529\pi$$
$$6$$ 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
$$11$$ −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
$$14$$ 0 0
$$15$$ 18.0000 0.309839
$$16$$ 0 0
$$17$$ −30.0000 −0.428004 −0.214002 0.976833i $$-0.568650\pi$$
−0.214002 + 0.976833i $$0.568650\pi$$
$$18$$ 0 0
$$19$$ 16.0000 0.193192 0.0965961 0.995324i $$-0.469204\pi$$
0.0965961 + 0.995324i $$0.469204\pi$$
$$20$$ 0 0
$$21$$ −60.0000 −0.623480
$$22$$ 0 0
$$23$$ 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
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −282.000 −1.80573 −0.902864 0.429927i $$-0.858539\pi$$
−0.902864 + 0.429927i $$0.858539\pi$$
$$30$$ 0 0
$$31$$ −164.000 −0.950170 −0.475085 0.879940i $$-0.657583\pi$$
−0.475085 + 0.879940i $$0.657583\pi$$
$$32$$ 0 0
$$33$$ −72.0000 −0.379806
$$34$$ 0 0
$$35$$ −120.000 −0.579534
$$36$$ 0 0
$$37$$ 110.000 0.488754 0.244377 0.969680i $$-0.421417\pi$$
0.244377 + 0.969680i $$0.421417\pi$$
$$38$$ 0 0
$$39$$ 39.0000 0.160128
$$40$$ 0 0
$$41$$ −126.000 −0.479949 −0.239974 0.970779i $$-0.577139\pi$$
−0.239974 + 0.970779i $$0.577139\pi$$
$$42$$ 0 0
$$43$$ −164.000 −0.581622 −0.290811 0.956780i $$-0.593925\pi$$
−0.290811 + 0.956780i $$0.593925\pi$$
$$44$$ 0 0
$$45$$ 54.0000 0.178885
$$46$$ 0 0
$$47$$ 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
$$50$$ 0 0
$$51$$ −90.0000 −0.247108
$$52$$ 0 0
$$53$$ −738.000 −1.91268 −0.956341 0.292255i $$-0.905595\pi$$
−0.956341 + 0.292255i $$0.905595\pi$$
$$54$$ 0 0
$$55$$ −144.000 −0.353036
$$56$$ 0 0
$$57$$ 48.0000 0.111540
$$58$$ 0 0
$$59$$ −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 $$0.277115\pi$$
$$62$$ 0 0
$$63$$ −180.000 −0.359966
$$64$$ 0 0
$$65$$ 78.0000 0.148842
$$66$$ 0 0
$$67$$ −848.000 −1.54626 −0.773132 0.634245i $$-0.781311\pi$$
−0.773132 + 0.634245i $$0.781311\pi$$
$$68$$ 0 0
$$69$$ 216.000 0.376860
$$70$$ 0 0
$$71$$ −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 $$0.444085\pi$$
$$74$$ 0 0
$$75$$ −267.000 −0.411073
$$76$$ 0 0
$$77$$ 480.000 0.710404
$$78$$ 0 0
$$79$$ 1096.00 1.56088 0.780441 0.625230i $$-0.214995\pi$$
0.780441 + 0.625230i $$0.214995\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −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
$$86$$ 0 0
$$87$$ −846.000 −1.04254
$$88$$ 0 0
$$89$$ 210.000 0.250112 0.125056 0.992150i $$-0.460089\pi$$
0.125056 + 0.992150i $$0.460089\pi$$
$$90$$ 0 0
$$91$$ −260.000 −0.299510
$$92$$ 0 0
$$93$$ −492.000 −0.548581
$$94$$ 0 0
$$95$$ 96.0000 0.103678
$$96$$ 0 0
$$97$$ −1726.00 −1.80669 −0.903344 0.428917i $$-0.858895\pi$$
−0.903344 + 0.428917i $$0.858895\pi$$
$$98$$ 0 0
$$99$$ −216.000 −0.219281
$$100$$ 0 0
$$101$$ 798.000 0.786178 0.393089 0.919500i $$-0.371406\pi$$
0.393089 + 0.919500i $$0.371406\pi$$
$$102$$ 0 0
$$103$$ 520.000 0.497448 0.248724 0.968574i $$-0.419989\pi$$
0.248724 + 0.968574i $$0.419989\pi$$
$$104$$ 0 0
$$105$$ −360.000 −0.334594
$$106$$ 0 0
$$107$$ −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 $$0.798267\pi$$
$$110$$ 0 0
$$111$$ 330.000 0.282182
$$112$$ 0 0
$$113$$ −366.000 −0.304694 −0.152347 0.988327i $$-0.548683\pi$$
−0.152347 + 0.988327i $$0.548683\pi$$
$$114$$ 0 0
$$115$$ 432.000 0.350297
$$116$$ 0 0
$$117$$ 117.000 0.0924500
$$118$$ 0 0
$$119$$ 600.000 0.462201
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ 0 0
$$123$$ −378.000 −0.277098
$$124$$ 0 0
$$125$$ −1284.00 −0.918756
$$126$$ 0 0
$$127$$ −2144.00 −1.49803 −0.749013 0.662556i $$-0.769472\pi$$
−0.749013 + 0.662556i $$0.769472\pi$$
$$128$$ 0 0
$$129$$ −492.000 −0.335800
$$130$$ 0 0
$$131$$ 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
$$134$$ 0 0
$$135$$ 162.000 0.103280
$$136$$ 0 0
$$137$$ 2754.00 1.71745 0.858723 0.512440i $$-0.171258\pi$$
0.858723 + 0.512440i $$0.171258\pi$$
$$138$$ 0 0
$$139$$ −2252.00 −1.37419 −0.687094 0.726568i $$-0.741114\pi$$
−0.687094 + 0.726568i $$0.741114\pi$$
$$140$$ 0 0
$$141$$ 612.000 0.365530
$$142$$ 0 0
$$143$$ −312.000 −0.182453
$$144$$ 0 0
$$145$$ −1692.00 −0.969055
$$146$$ 0 0
$$147$$ 171.000 0.0959445
$$148$$ 0 0
$$149$$ −1770.00 −0.973182 −0.486591 0.873630i $$-0.661760\pi$$
−0.486591 + 0.873630i $$0.661760\pi$$
$$150$$ 0 0
$$151$$ 988.000 0.532466 0.266233 0.963909i $$-0.414221\pi$$
0.266233 + 0.963909i $$0.414221\pi$$
$$152$$ 0 0
$$153$$ −270.000 −0.142668
$$154$$ 0 0
$$155$$ −984.000 −0.509915
$$156$$ 0 0
$$157$$ 326.000 0.165717 0.0828587 0.996561i $$-0.473595\pi$$
0.0828587 + 0.996561i $$0.473595\pi$$
$$158$$ 0 0
$$159$$ −2214.00 −1.10429
$$160$$ 0 0
$$161$$ −1440.00 −0.704894
$$162$$ 0 0
$$163$$ −1496.00 −0.718870 −0.359435 0.933170i $$-0.617031\pi$$
−0.359435 + 0.933170i $$0.617031\pi$$
$$164$$ 0 0
$$165$$ −432.000 −0.203825
$$166$$ 0 0
$$167$$ −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
$$170$$ 0 0
$$171$$ 144.000 0.0643974
$$172$$ 0 0
$$173$$ 4374.00 1.92225 0.961124 0.276116i $$-0.0890472\pi$$
0.961124 + 0.276116i $$0.0890472\pi$$
$$174$$ 0 0
$$175$$ 1780.00 0.768888
$$176$$ 0 0
$$177$$ −360.000 −0.152877
$$178$$ 0 0
$$179$$ −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 $$0.0797909\pi$$
$$182$$ 0 0
$$183$$ 1842.00 0.744069
$$184$$ 0 0
$$185$$ 660.000 0.262293
$$186$$ 0 0
$$187$$ 720.000 0.281559
$$188$$ 0 0
$$189$$ −540.000 −0.207827
$$190$$ 0 0
$$191$$ 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 $$0.711754\pi$$
$$194$$ 0 0
$$195$$ 234.000 0.0859338
$$196$$ 0 0
$$197$$ 3126.00 1.13055 0.565275 0.824903i $$-0.308770\pi$$
0.565275 + 0.824903i $$0.308770\pi$$
$$198$$ 0 0
$$199$$ −4664.00 −1.66142 −0.830709 0.556707i $$-0.812065\pi$$
−0.830709 + 0.556707i $$0.812065\pi$$
$$200$$ 0 0
$$201$$ −2544.00 −0.892736
$$202$$ 0 0
$$203$$ 5640.00 1.95000
$$204$$ 0 0
$$205$$ −756.000 −0.257567
$$206$$ 0 0
$$207$$ 648.000 0.217580
$$208$$ 0 0
$$209$$ −384.000 −0.127090
$$210$$ 0 0
$$211$$ 556.000 0.181406 0.0907029 0.995878i $$-0.471089\pi$$
0.0907029 + 0.995878i $$0.471089\pi$$
$$212$$ 0 0
$$213$$ −396.000 −0.127387
$$214$$ 0 0
$$215$$ −984.000 −0.312131
$$216$$ 0 0
$$217$$ 3280.00 1.02609
$$218$$ 0 0
$$219$$ 654.000 0.201796
$$220$$ 0 0
$$221$$ −390.000 −0.118707
$$222$$ 0 0
$$223$$ 268.000 0.0804781 0.0402390 0.999190i $$-0.487188\pi$$
0.0402390 + 0.999190i $$0.487188\pi$$
$$224$$ 0 0
$$225$$ −801.000 −0.237333
$$226$$ 0 0
$$227$$ −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 $$0.358017\pi$$
$$230$$ 0 0
$$231$$ 1440.00 0.410152
$$232$$ 0 0
$$233$$ 2826.00 0.794581 0.397291 0.917693i $$-0.369951\pi$$
0.397291 + 0.917693i $$0.369951\pi$$
$$234$$ 0 0
$$235$$ 1224.00 0.339766
$$236$$ 0 0
$$237$$ 3288.00 0.901175
$$238$$ 0 0
$$239$$ 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 $$0.567810\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 342.000 0.0891820
$$246$$ 0 0
$$247$$ 208.000 0.0535819
$$248$$ 0 0
$$249$$ −1656.00 −0.421465
$$250$$ 0 0
$$251$$ −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
$$254$$ 0 0
$$255$$ −540.000 −0.132612
$$256$$ 0 0
$$257$$ −558.000 −0.135436 −0.0677181 0.997704i $$-0.521572\pi$$
−0.0677181 + 0.997704i $$0.521572\pi$$
$$258$$ 0 0
$$259$$ −2200.00 −0.527804
$$260$$ 0 0
$$261$$ −2538.00 −0.601909
$$262$$ 0 0
$$263$$ −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
$$266$$ 0 0
$$267$$ 630.000 0.144402
$$268$$ 0 0
$$269$$ 5046.00 1.14372 0.571859 0.820352i $$-0.306223\pi$$
0.571859 + 0.820352i $$0.306223\pi$$
$$270$$ 0 0
$$271$$ 3796.00 0.850888 0.425444 0.904985i $$-0.360118\pi$$
0.425444 + 0.904985i $$0.360118\pi$$
$$272$$ 0 0
$$273$$ −780.000 −0.172922
$$274$$ 0 0
$$275$$ 2136.00 0.468384
$$276$$ 0 0
$$277$$ 5582.00 1.21079 0.605397 0.795924i $$-0.293014\pi$$
0.605397 + 0.795924i $$0.293014\pi$$
$$278$$ 0 0
$$279$$ −1476.00 −0.316723
$$280$$ 0 0
$$281$$ −1950.00 −0.413976 −0.206988 0.978343i $$-0.566366\pi$$
−0.206988 + 0.978343i $$0.566366\pi$$
$$282$$ 0 0
$$283$$ 4732.00 0.993951 0.496976 0.867765i $$-0.334444\pi$$
0.496976 + 0.867765i $$0.334444\pi$$
$$284$$ 0 0
$$285$$ 288.000 0.0598584
$$286$$ 0 0
$$287$$ 2520.00 0.518296
$$288$$ 0 0
$$289$$ −4013.00 −0.816813
$$290$$ 0 0
$$291$$ −5178.00 −1.04309
$$292$$ 0 0
$$293$$ 4998.00 0.996540 0.498270 0.867022i $$-0.333969\pi$$
0.498270 + 0.867022i $$0.333969\pi$$
$$294$$ 0 0
$$295$$ −720.000 −0.142102
$$296$$ 0 0
$$297$$ −648.000 −0.126602
$$298$$ 0 0
$$299$$ 936.000 0.181038
$$300$$ 0 0
$$301$$ 3280.00 0.628093
$$302$$ 0 0
$$303$$ 2394.00 0.453900
$$304$$ 0 0
$$305$$ 3684.00 0.691624
$$306$$ 0 0
$$307$$ −6824.00 −1.26862 −0.634310 0.773079i $$-0.718716\pi$$
−0.634310 + 0.773079i $$0.718716\pi$$
$$308$$ 0 0
$$309$$ 1560.00 0.287202
$$310$$ 0 0
$$311$$ 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 $$0.383547\pi$$
$$314$$ 0 0
$$315$$ −1080.00 −0.193178
$$316$$ 0 0
$$317$$ 7086.00 1.25549 0.627744 0.778420i $$-0.283979\pi$$
0.627744 + 0.778420i $$0.283979\pi$$
$$318$$ 0 0
$$319$$ 6768.00 1.18788
$$320$$ 0 0
$$321$$ −36.0000 −0.00625958
$$322$$ 0 0
$$323$$ −480.000 −0.0826870
$$324$$ 0 0
$$325$$ −1157.00 −0.197473
$$326$$ 0 0
$$327$$ −5502.00 −0.930463
$$328$$ 0 0
$$329$$ −4080.00 −0.683701
$$330$$ 0 0
$$331$$ 9016.00 1.49717 0.748586 0.663037i $$-0.230733\pi$$
0.748586 + 0.663037i $$0.230733\pi$$
$$332$$ 0 0
$$333$$ 990.000 0.162918
$$334$$ 0 0
$$335$$ −5088.00 −0.829812
$$336$$ 0 0
$$337$$ 2306.00 0.372747 0.186374 0.982479i $$-0.440327\pi$$
0.186374 + 0.982479i $$0.440327\pi$$
$$338$$ 0 0
$$339$$ −1098.00 −0.175915
$$340$$ 0 0
$$341$$ 3936.00 0.625063
$$342$$ 0 0
$$343$$ 5720.00 0.900440
$$344$$ 0 0
$$345$$ 1296.00 0.202244
$$346$$ 0 0
$$347$$ 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 $$0.442518\pi$$
$$350$$ 0 0
$$351$$ 351.000 0.0533761
$$352$$ 0 0
$$353$$ 4650.00 0.701118 0.350559 0.936541i $$-0.385992\pi$$
0.350559 + 0.936541i $$0.385992\pi$$
$$354$$ 0 0
$$355$$ −792.000 −0.118408
$$356$$ 0 0
$$357$$ 1800.00 0.266852
$$358$$ 0 0
$$359$$ 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
$$362$$ 0 0
$$363$$ −2265.00 −0.327498
$$364$$ 0 0
$$365$$ 1308.00 0.187572
$$366$$ 0 0
$$367$$ 7288.00 1.03660 0.518298 0.855200i $$-0.326566\pi$$
0.518298 + 0.855200i $$0.326566\pi$$
$$368$$ 0 0
$$369$$ −1134.00 −0.159983
$$370$$ 0 0
$$371$$ 14760.0 2.06550
$$372$$ 0 0
$$373$$ −9970.00 −1.38399 −0.691993 0.721904i $$-0.743267\pi$$
−0.691993 + 0.721904i $$0.743267\pi$$
$$374$$ 0 0
$$375$$ −3852.00 −0.530444
$$376$$ 0 0
$$377$$ −3666.00 −0.500819
$$378$$ 0 0
$$379$$ −13448.0 −1.82263 −0.911316 0.411708i $$-0.864932\pi$$
−0.911316 + 0.411708i $$0.864932\pi$$
$$380$$ 0 0
$$381$$ −6432.00 −0.864885
$$382$$ 0 0
$$383$$ −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
$$386$$ 0 0
$$387$$ −1476.00 −0.193874
$$388$$ 0 0
$$389$$ 174.000 0.0226790 0.0113395 0.999936i $$-0.496390\pi$$
0.0113395 + 0.999936i $$0.496390\pi$$
$$390$$ 0 0
$$391$$ −2160.00 −0.279376
$$392$$ 0 0
$$393$$ 8244.00 1.05815
$$394$$ 0 0
$$395$$ 6576.00 0.837657
$$396$$ 0 0
$$397$$ −2986.00 −0.377489 −0.188744 0.982026i $$-0.560442\pi$$
−0.188744 + 0.982026i $$0.560442\pi$$
$$398$$ 0 0
$$399$$ −960.000 −0.120451
$$400$$ 0 0
$$401$$ −10566.0 −1.31581 −0.657906 0.753100i $$-0.728558\pi$$
−0.657906 + 0.753100i $$0.728558\pi$$
$$402$$ 0 0
$$403$$ −2132.00 −0.263530
$$404$$ 0 0
$$405$$ 486.000 0.0596285
$$406$$ 0 0
$$407$$ −2640.00 −0.321523
$$408$$ 0 0
$$409$$ −7270.00 −0.878920 −0.439460 0.898262i $$-0.644830\pi$$
−0.439460 + 0.898262i $$0.644830\pi$$
$$410$$ 0 0
$$411$$ 8262.00 0.991568
$$412$$ 0 0
$$413$$ 2400.00 0.285947
$$414$$ 0 0
$$415$$ −3312.00 −0.391758
$$416$$ 0 0
$$417$$ −6756.00 −0.793388
$$418$$ 0 0
$$419$$ 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 $$0.611682\pi$$
$$422$$ 0 0
$$423$$ 1836.00 0.211039
$$424$$ 0 0
$$425$$ 2670.00 0.304739
$$426$$ 0 0
$$427$$ −12280.0 −1.39174
$$428$$ 0 0
$$429$$ −936.000 −0.105339
$$430$$ 0 0
$$431$$ −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 $$0.512686\pi$$
$$434$$ 0 0
$$435$$ −5076.00 −0.559484
$$436$$ 0 0
$$437$$ 1152.00 0.126104
$$438$$ 0 0
$$439$$ −8984.00 −0.976726 −0.488363 0.872640i $$-0.662406\pi$$
−0.488363 + 0.872640i $$0.662406\pi$$
$$440$$ 0 0
$$441$$ 513.000 0.0553936
$$442$$ 0 0
$$443$$ −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
$$446$$ 0 0
$$447$$ −5310.00 −0.561867
$$448$$ 0 0
$$449$$ −13206.0 −1.38804 −0.694020 0.719956i $$-0.744162\pi$$
−0.694020 + 0.719956i $$0.744162\pi$$
$$450$$ 0 0
$$451$$ 3024.00 0.315731
$$452$$ 0 0
$$453$$ 2964.00 0.307419
$$454$$ 0 0
$$455$$ −1560.00 −0.160734
$$456$$ 0 0
$$457$$ 8426.00 0.862476 0.431238 0.902238i $$-0.358077\pi$$
0.431238 + 0.902238i $$0.358077\pi$$
$$458$$ 0 0
$$459$$ −810.000 −0.0823694
$$460$$ 0 0
$$461$$ 16686.0 1.68578 0.842890 0.538086i $$-0.180852\pi$$
0.842890 + 0.538086i $$0.180852\pi$$
$$462$$ 0 0
$$463$$ −15932.0 −1.59919 −0.799593 0.600543i $$-0.794951\pi$$
−0.799593 + 0.600543i $$0.794951\pi$$
$$464$$ 0 0
$$465$$ −2952.00 −0.294399
$$466$$ 0 0
$$467$$ −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
$$470$$ 0 0
$$471$$ 978.000 0.0956770
$$472$$ 0 0
$$473$$ 3936.00 0.382616
$$474$$ 0 0
$$475$$ −1424.00 −0.137553
$$476$$ 0 0
$$477$$ −6642.00 −0.637560
$$478$$ 0 0
$$479$$ −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
$$482$$ 0 0
$$483$$ −4320.00 −0.406971
$$484$$ 0 0
$$485$$ −10356.0 −0.969571
$$486$$ 0 0
$$487$$ −11756.0 −1.09387 −0.546936 0.837175i $$-0.684206\pi$$
−0.546936 + 0.837175i $$0.684206\pi$$
$$488$$ 0 0
$$489$$ −4488.00 −0.415040
$$490$$ 0 0
$$491$$ −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
$$494$$ 0 0
$$495$$ −1296.00 −0.117679
$$496$$ 0 0
$$497$$ 2640.00 0.238270
$$498$$ 0 0
$$499$$ 8944.00 0.802382 0.401191 0.915995i $$-0.368596\pi$$
0.401191 + 0.915995i $$0.368596\pi$$
$$500$$ 0 0
$$501$$ −3348.00 −0.298558
$$502$$ 0 0
$$503$$ 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
$$506$$ 0 0
$$507$$ 507.000 0.0444116
$$508$$ 0 0
$$509$$ −12114.0 −1.05490 −0.527450 0.849586i $$-0.676852\pi$$
−0.527450 + 0.849586i $$0.676852\pi$$
$$510$$ 0 0
$$511$$ −4360.00 −0.377446
$$512$$ 0 0
$$513$$ 432.000 0.0371799
$$514$$ 0 0
$$515$$ 3120.00 0.266958
$$516$$ 0 0
$$517$$ −4896.00 −0.416491
$$518$$ 0 0
$$519$$ 13122.0 1.10981
$$520$$ 0 0
$$521$$ −14310.0 −1.20333 −0.601663 0.798750i $$-0.705495\pi$$
−0.601663 + 0.798750i $$0.705495\pi$$
$$522$$ 0 0
$$523$$ 18340.0 1.53337 0.766685 0.642024i $$-0.221905\pi$$
0.766685 + 0.642024i $$0.221905\pi$$
$$524$$ 0 0
$$525$$ 5340.00 0.443918
$$526$$ 0 0
$$527$$ 4920.00 0.406677
$$528$$ 0 0
$$529$$ −6983.00 −0.573929
$$530$$ 0 0
$$531$$ −1080.00 −0.0882637
$$532$$ 0 0
$$533$$ −1638.00 −0.133114
$$534$$ 0 0
$$535$$ −72.0000 −0.00581838
$$536$$ 0 0
$$537$$ −36.0000 −0.00289295
$$538$$ 0 0
$$539$$ −1368.00 −0.109321
$$540$$ 0 0
$$541$$ 9254.00 0.735417 0.367708 0.929941i $$-0.380142\pi$$
0.367708 + 0.929941i $$0.380142\pi$$
$$542$$ 0 0
$$543$$ 14154.0 1.11861
$$544$$ 0 0
$$545$$ −11004.0 −0.864880
$$546$$ 0 0
$$547$$ −17444.0 −1.36353 −0.681766 0.731571i $$-0.738788\pi$$
−0.681766 + 0.731571i $$0.738788\pi$$
$$548$$ 0 0
$$549$$ 5526.00 0.429588
$$550$$ 0 0
$$551$$ −4512.00 −0.348852
$$552$$ 0 0
$$553$$ −21920.0 −1.68559
$$554$$ 0 0
$$555$$ 1980.00 0.151435
$$556$$ 0 0
$$557$$ −3714.00 −0.282526 −0.141263 0.989972i $$-0.545116\pi$$
−0.141263 + 0.989972i $$0.545116\pi$$
$$558$$ 0 0
$$559$$ −2132.00 −0.161313
$$560$$ 0 0
$$561$$ 2160.00 0.162558
$$562$$ 0 0
$$563$$ 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
$$566$$ 0 0
$$567$$ −1620.00 −0.119989
$$568$$ 0 0
$$569$$ −15942.0 −1.17456 −0.587279 0.809385i $$-0.699801\pi$$
−0.587279 + 0.809385i $$0.699801\pi$$
$$570$$ 0 0
$$571$$ −1604.00 −0.117557 −0.0587787 0.998271i $$-0.518721\pi$$
−0.0587787 + 0.998271i $$0.518721\pi$$
$$572$$ 0 0
$$573$$ 4104.00 0.299210
$$574$$ 0 0
$$575$$ −6408.00 −0.464751
$$576$$ 0 0
$$577$$ −10654.0 −0.768686 −0.384343 0.923190i $$-0.625572\pi$$
−0.384343 + 0.923190i $$0.625572\pi$$
$$578$$ 0 0
$$579$$ −9930.00 −0.712740
$$580$$ 0 0
$$581$$ 11040.0 0.788324
$$582$$ 0 0
$$583$$ 17712.0 1.25824
$$584$$ 0 0
$$585$$ 702.000 0.0496139
$$586$$ 0 0
$$587$$ 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
$$590$$ 0 0
$$591$$ 9378.00 0.652723
$$592$$ 0 0
$$593$$ 12618.0 0.873793 0.436896 0.899512i $$-0.356078\pi$$
0.436896 + 0.899512i $$0.356078\pi$$
$$594$$ 0 0
$$595$$ 3600.00 0.248043
$$596$$ 0 0
$$597$$ −13992.0 −0.959220
$$598$$ 0 0
$$599$$ −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 $$0.469600\pi$$
$$602$$ 0 0
$$603$$ −7632.00 −0.515421
$$604$$ 0 0
$$605$$ −4530.00 −0.304414
$$606$$ 0 0
$$607$$ −1064.00 −0.0711473 −0.0355737 0.999367i $$-0.511326\pi$$
−0.0355737 + 0.999367i $$0.511326\pi$$
$$608$$ 0 0
$$609$$ 16920.0 1.12583
$$610$$ 0 0
$$611$$ 2652.00 0.175595
$$612$$ 0 0
$$613$$ −20914.0 −1.37799 −0.688996 0.724766i $$-0.741948\pi$$
−0.688996 + 0.724766i $$0.741948\pi$$
$$614$$ 0 0
$$615$$ −2268.00 −0.148707
$$616$$ 0 0
$$617$$ 9714.00 0.633826 0.316913 0.948455i $$-0.397354\pi$$
0.316913 + 0.948455i $$0.397354\pi$$
$$618$$ 0 0
$$619$$ 14848.0 0.964122 0.482061 0.876138i $$-0.339888\pi$$
0.482061 + 0.876138i $$0.339888\pi$$
$$620$$ 0 0
$$621$$ 1944.00 0.125620
$$622$$ 0 0
$$623$$ −4200.00 −0.270095
$$624$$ 0 0
$$625$$ 3421.00 0.218944
$$626$$ 0 0
$$627$$ −1152.00 −0.0733755
$$628$$ 0 0
$$629$$ −3300.00 −0.209189
$$630$$ 0 0
$$631$$ −19172.0 −1.20955 −0.604774 0.796397i $$-0.706737\pi$$
−0.604774 + 0.796397i $$0.706737\pi$$
$$632$$ 0 0
$$633$$ 1668.00 0.104735
$$634$$ 0 0
$$635$$ −12864.0 −0.803925
$$636$$ 0 0
$$637$$ 741.000 0.0460902
$$638$$ 0 0
$$639$$ −1188.00 −0.0735470
$$640$$ 0 0
$$641$$ −11502.0 −0.708739 −0.354369 0.935105i $$-0.615304\pi$$
−0.354369 + 0.935105i $$0.615304\pi$$
$$642$$ 0 0
$$643$$ 15568.0 0.954809 0.477404 0.878684i $$-0.341578\pi$$
0.477404 + 0.878684i $$0.341578\pi$$
$$644$$ 0 0
$$645$$ −2952.00 −0.180209
$$646$$ 0 0
$$647$$ −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
$$650$$ 0 0
$$651$$ 9840.00 0.592412
$$652$$ 0 0
$$653$$ 8118.00 0.486496 0.243248 0.969964i $$-0.421787\pi$$
0.243248 + 0.969964i $$0.421787\pi$$
$$654$$ 0 0
$$655$$ 16488.0 0.983572
$$656$$ 0 0
$$657$$ 1962.00 0.116507
$$658$$ 0 0
$$659$$ −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 $$0.626331\pi$$
$$662$$ 0 0
$$663$$ −1170.00 −0.0685355
$$664$$ 0 0
$$665$$ −1920.00 −0.111962
$$666$$ 0 0
$$667$$ −20304.0 −1.17867
$$668$$ 0 0
$$669$$ 804.000 0.0464640
$$670$$ 0 0
$$671$$ −14736.0 −0.847805
$$672$$ 0 0
$$673$$ −718.000 −0.0411246 −0.0205623 0.999789i $$-0.506546\pi$$
−0.0205623 + 0.999789i $$0.506546\pi$$
$$674$$ 0 0
$$675$$ −2403.00 −0.137024
$$676$$ 0 0
$$677$$ −2994.00 −0.169969 −0.0849843 0.996382i $$-0.527084\pi$$
−0.0849843 + 0.996382i $$0.527084\pi$$
$$678$$ 0 0
$$679$$ 34520.0 1.95104
$$680$$ 0 0
$$681$$ −5400.00 −0.303860
$$682$$ 0 0
$$683$$ −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
$$686$$ 0 0
$$687$$ 8970.00 0.498147
$$688$$ 0 0
$$689$$ −9594.00 −0.530482
$$690$$ 0 0
$$691$$ −27632.0 −1.52123 −0.760616 0.649202i $$-0.775103\pi$$
−0.760616 + 0.649202i $$0.775103\pi$$
$$692$$ 0 0
$$693$$ 4320.00 0.236801
$$694$$ 0 0
$$695$$ −13512.0 −0.737467
$$696$$ 0 0
$$697$$ 3780.00 0.205420
$$698$$ 0 0
$$699$$ 8478.00 0.458752
$$700$$ 0 0
$$701$$ 19062.0 1.02705 0.513525 0.858075i $$-0.328339\pi$$
0.513525 + 0.858075i $$0.328339\pi$$
$$702$$ 0 0
$$703$$ 1760.00 0.0944234
$$704$$ 0 0
$$705$$ 3672.00 0.196164
$$706$$ 0 0
$$707$$ −15960.0 −0.848992
$$708$$ 0 0
$$709$$ 3854.00 0.204147 0.102073 0.994777i $$-0.467452\pi$$
0.102073 + 0.994777i $$0.467452\pi$$
$$710$$ 0 0
$$711$$ 9864.00 0.520294
$$712$$ 0 0
$$713$$ −11808.0 −0.620215
$$714$$ 0 0
$$715$$ −1872.00 −0.0979144
$$716$$ 0 0
$$717$$ 5436.00 0.283140
$$718$$ 0 0
$$719$$ −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
$$722$$ 0 0
$$723$$ −4746.00 −0.244130
$$724$$ 0 0
$$725$$ 25098.0 1.28568
$$726$$ 0 0
$$727$$ 29464.0 1.50311 0.751554 0.659672i $$-0.229305\pi$$
0.751554 + 0.659672i $$0.229305\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 4920.00 0.248937
$$732$$ 0 0
$$733$$ −2698.00 −0.135952 −0.0679761 0.997687i $$-0.521654\pi$$
−0.0679761 + 0.997687i $$0.521654\pi$$
$$734$$ 0 0
$$735$$ 1026.00 0.0514892
$$736$$ 0 0
$$737$$ 20352.0 1.01720
$$738$$ 0 0
$$739$$ −632.000 −0.0314594 −0.0157297 0.999876i $$-0.505007\pi$$
−0.0157297 + 0.999876i $$0.505007\pi$$
$$740$$ 0 0
$$741$$ 624.000 0.0309355
$$742$$ 0 0
$$743$$ 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
$$746$$ 0 0
$$747$$ −4968.00 −0.243333
$$748$$ 0 0
$$749$$ 240.000 0.0117082
$$750$$ 0 0
$$751$$ −272.000 −0.0132163 −0.00660814 0.999978i $$-0.502103\pi$$
−0.00660814 + 0.999978i $$0.502103\pi$$
$$752$$ 0 0
$$753$$ −6444.00 −0.311862
$$754$$ 0 0
$$755$$ 5928.00 0.285751
$$756$$ 0 0
$$757$$ 37550.0 1.80288 0.901439 0.432907i $$-0.142512\pi$$
0.901439 + 0.432907i $$0.142512\pi$$
$$758$$ 0 0
$$759$$ −5184.00 −0.247915
$$760$$ 0 0
$$761$$ 33330.0 1.58766 0.793832 0.608138i $$-0.208083\pi$$
0.793832 + 0.608138i $$0.208083\pi$$
$$762$$ 0 0
$$763$$ 36680.0 1.74037
$$764$$ 0 0
$$765$$ −1620.00 −0.0765637
$$766$$ 0 0
$$767$$ −1560.00 −0.0734398
$$768$$ 0 0
$$769$$ −15406.0 −0.722438 −0.361219 0.932481i $$-0.617639\pi$$
−0.361219 + 0.932481i $$0.617639\pi$$
$$770$$ 0 0
$$771$$ −1674.00 −0.0781941
$$772$$ 0 0
$$773$$ −29514.0 −1.37328 −0.686640 0.726998i $$-0.740915\pi$$
−0.686640 + 0.726998i $$0.740915\pi$$
$$774$$ 0 0
$$775$$ 14596.0 0.676521
$$776$$ 0 0
$$777$$ −6600.00 −0.304728
$$778$$ 0 0
$$779$$ −2016.00 −0.0927223
$$780$$ 0 0
$$781$$ 3168.00 0.145147
$$782$$ 0 0
$$783$$ −7614.00 −0.347512
$$784$$ 0 0
$$785$$ 1956.00 0.0889333
$$786$$ 0 0
$$787$$ −33176.0 −1.50266 −0.751332 0.659924i $$-0.770588\pi$$
−0.751332 + 0.659924i $$0.770588\pi$$
$$788$$ 0 0
$$789$$ −6336.00 −0.285890
$$790$$ 0 0
$$791$$ 7320.00 0.329038
$$792$$ 0 0
$$793$$ 7982.00 0.357439
$$794$$ 0 0
$$795$$ −13284.0 −0.592623
$$796$$ 0 0
$$797$$ −16746.0 −0.744258 −0.372129 0.928181i $$-0.621372\pi$$
−0.372129 + 0.928181i $$0.621372\pi$$
$$798$$ 0 0
$$799$$ −6120.00 −0.270976
$$800$$ 0 0
$$801$$ 1890.00 0.0833706
$$802$$ 0 0
$$803$$ −5232.00 −0.229929
$$804$$ 0 0
$$805$$ −8640.00 −0.378286
$$806$$ 0 0
$$807$$ 15138.0 0.660326
$$808$$ 0 0
$$809$$ −15846.0 −0.688647 −0.344324 0.938851i $$-0.611892\pi$$
−0.344324 + 0.938851i $$0.611892\pi$$
$$810$$ 0 0
$$811$$ −22952.0 −0.993778 −0.496889 0.867814i $$-0.665524\pi$$
−0.496889 + 0.867814i $$0.665524\pi$$
$$812$$ 0 0
$$813$$ 11388.0 0.491260
$$814$$ 0 0
$$815$$ −8976.00 −0.385786
$$816$$ 0 0
$$817$$ −2624.00 −0.112365
$$818$$ 0 0
$$819$$ −2340.00 −0.0998367
$$820$$ 0 0
$$821$$ −37146.0 −1.57906 −0.789528 0.613715i $$-0.789674\pi$$
−0.789528 + 0.613715i $$0.789674\pi$$
$$822$$ 0 0
$$823$$ 9592.00 0.406265 0.203133 0.979151i $$-0.434888\pi$$
0.203133 + 0.979151i $$0.434888\pi$$
$$824$$ 0 0
$$825$$ 6408.00 0.270422
$$826$$ 0 0
$$827$$ 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 $$0.524736\pi$$
$$830$$ 0 0
$$831$$ 16746.0 0.699052
$$832$$ 0 0
$$833$$ −1710.00 −0.0711260
$$834$$ 0 0
$$835$$ −6696.00 −0.277515
$$836$$ 0 0
$$837$$ −4428.00 −0.182860
$$838$$ 0 0
$$839$$ −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
$$842$$ 0 0
$$843$$ −5850.00 −0.239009
$$844$$ 0 0
$$845$$ 1014.00 0.0412813
$$846$$ 0 0
$$847$$ 15100.0 0.612565
$$848$$ 0 0
$$849$$ 14196.0 0.573858
$$850$$ 0 0
$$851$$ 7920.00 0.319029
$$852$$ 0 0
$$853$$ 11342.0 0.455267 0.227633 0.973747i $$-0.426901\pi$$
0.227633 + 0.973747i $$0.426901\pi$$
$$854$$ 0 0
$$855$$ 864.000 0.0345593
$$856$$ 0 0
$$857$$ −16134.0 −0.643089 −0.321544 0.946895i $$-0.604202\pi$$
−0.321544 + 0.946895i $$0.604202\pi$$
$$858$$ 0 0
$$859$$ 20932.0 0.831421 0.415710 0.909497i $$-0.363533\pi$$
0.415710 + 0.909497i $$0.363533\pi$$
$$860$$ 0 0
$$861$$ 7560.00 0.299238
$$862$$ 0 0
$$863$$ −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
$$866$$ 0 0
$$867$$ −12039.0 −0.471587
$$868$$ 0 0
$$869$$ −26304.0 −1.02681
$$870$$ 0 0
$$871$$ −11024.0 −0.428856
$$872$$ 0 0
$$873$$ −15534.0 −0.602229
$$874$$ 0 0
$$875$$ 25680.0 0.992163
$$876$$ 0 0
$$877$$ −26314.0 −1.01318 −0.506591 0.862186i $$-0.669095\pi$$
−0.506591 + 0.862186i $$0.669095\pi$$
$$878$$ 0 0
$$879$$ 14994.0 0.575353
$$880$$ 0 0
$$881$$ 37506.0 1.43429 0.717145 0.696924i $$-0.245449\pi$$
0.717145 + 0.696924i $$0.245449\pi$$
$$882$$ 0 0
$$883$$ 6388.00 0.243458 0.121729 0.992563i $$-0.461156\pi$$
0.121729 + 0.992563i $$0.461156\pi$$
$$884$$ 0 0
$$885$$ −2160.00 −0.0820425
$$886$$ 0 0
$$887$$ 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
$$890$$ 0 0
$$891$$ −1944.00 −0.0730937
$$892$$ 0 0
$$893$$ 3264.00 0.122313
$$894$$ 0 0
$$895$$ −72.0000 −0.00268904
$$896$$ 0 0
$$897$$ 2808.00 0.104522
$$898$$ 0 0
$$899$$ 46248.0 1.71575
$$900$$ 0 0
$$901$$ 22140.0 0.818635
$$902$$ 0 0
$$903$$ 9840.00 0.362630
$$904$$ 0 0
$$905$$ 28308.0 1.03977
$$906$$ 0 0
$$907$$ 7180.00 0.262853 0.131427 0.991326i $$-0.458044\pi$$
0.131427 + 0.991326i $$0.458044\pi$$
$$908$$ 0 0
$$909$$ 7182.00 0.262059
$$910$$ 0 0
$$911$$ −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
$$914$$ 0 0
$$915$$ 11052.0 0.399309
$$916$$ 0 0
$$917$$ −54960.0 −1.97921
$$918$$ 0 0
$$919$$ 30256.0 1.08602 0.543011 0.839726i $$-0.317284\pi$$
0.543011 + 0.839726i $$0.317284\pi$$
$$920$$ 0 0
$$921$$ −20472.0 −0.732438
$$922$$ 0 0
$$923$$ −1716.00 −0.0611948
$$924$$ 0 0
$$925$$ −9790.00 −0.347993
$$926$$ 0 0
$$927$$ 4680.00 0.165816
$$928$$ 0 0
$$929$$ −1926.00 −0.0680194 −0.0340097 0.999422i $$-0.510828\pi$$
−0.0340097 + 0.999422i $$0.510828\pi$$
$$930$$ 0 0
$$931$$ 912.000 0.0321048
$$932$$ 0 0
$$933$$ 26280.0 0.922153
$$934$$ 0 0
$$935$$ 4320.00 0.151101
$$936$$ 0 0
$$937$$ 3962.00 0.138135 0.0690677 0.997612i $$-0.477998\pi$$
0.0690677 + 0.997612i $$0.477998\pi$$
$$938$$ 0 0
$$939$$ 11886.0 0.413083
$$940$$ 0 0
$$941$$ −1074.00 −0.0372066 −0.0186033 0.999827i $$-0.505922\pi$$
−0.0186033 + 0.999827i $$0.505922\pi$$
$$942$$ 0 0
$$943$$ −9072.00 −0.313282
$$944$$ 0 0
$$945$$ −3240.00 −0.111531
$$946$$ 0 0
$$947$$ −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
$$950$$ 0 0
$$951$$ 21258.0 0.724856
$$952$$ 0 0
$$953$$ 762.000 0.0259009 0.0129505 0.999916i $$-0.495878\pi$$
0.0129505 + 0.999916i $$0.495878\pi$$
$$954$$ 0 0
$$955$$ 8208.00 0.278120
$$956$$ 0 0
$$957$$ 20304.0 0.685826
$$958$$ 0 0
$$959$$ −55080.0 −1.85467
$$960$$ 0 0
$$961$$ −2895.00 −0.0971770
$$962$$ 0 0
$$963$$ −108.000 −0.00361397
$$964$$ 0 0
$$965$$ −19860.0 −0.662504
$$966$$ 0 0
$$967$$ −35804.0 −1.19067 −0.595336 0.803477i $$-0.702981\pi$$
−0.595336 + 0.803477i $$0.702981\pi$$
$$968$$ 0 0
$$969$$ −1440.00 −0.0477394
$$970$$ 0 0
$$971$$ 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
$$974$$ 0 0
$$975$$ −3471.00 −0.114011
$$976$$ 0 0
$$977$$ −28710.0 −0.940137 −0.470069 0.882630i $$-0.655771\pi$$
−0.470069 + 0.882630i $$0.655771\pi$$
$$978$$ 0 0
$$979$$ −5040.00 −0.164534
$$980$$ 0 0
$$981$$ −16506.0 −0.537203
$$982$$ 0 0
$$983$$ 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
$$986$$ 0 0
$$987$$ −12240.0 −0.394735
$$988$$ 0 0
$$989$$ −11808.0 −0.379649
$$990$$ 0 0
$$991$$ −44408.0 −1.42348 −0.711739 0.702444i $$-0.752092\pi$$
−0.711739 + 0.702444i $$0.752092\pi$$
$$992$$ 0 0
$$993$$ 27048.0 0.864393
$$994$$ 0 0
$$995$$ −27984.0 −0.891610
$$996$$ 0 0
$$997$$ 18398.0 0.584424 0.292212 0.956354i $$-0.405609\pi$$
0.292212 + 0.956354i $$0.405609\pi$$
$$998$$ 0 0
$$999$$ 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 624.4.a.i.1.1 1
3.2 odd 2 1872.4.a.e.1.1 1
4.3 odd 2 78.4.a.e.1.1 1
8.3 odd 2 2496.4.a.k.1.1 1
8.5 even 2 2496.4.a.b.1.1 1
12.11 even 2 234.4.a.b.1.1 1
20.19 odd 2 1950.4.a.c.1.1 1
52.31 even 4 1014.4.b.c.337.1 2
52.47 even 4 1014.4.b.c.337.2 2
52.51 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 4.3 odd 2
234.4.a.b.1.1 1 12.11 even 2
624.4.a.i.1.1 1 1.1 even 1 trivial
1014.4.a.b.1.1 1 52.51 odd 2
1014.4.b.c.337.1 2 52.31 even 4
1014.4.b.c.337.2 2 52.47 even 4
1872.4.a.e.1.1 1 3.2 odd 2
1950.4.a.c.1.1 1 20.19 odd 2
2496.4.a.b.1.1 1 8.5 even 2
2496.4.a.k.1.1 1 8.3 odd 2