# Properties

 Label 1872.4.a.e.1.1 Level $1872$ Weight $4$ Character 1872.1 Self dual yes Analytic conductor $110.452$ 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] = [1872,4,Mod(1,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, 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("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1872 = 2^{4} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1872.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: $$110.451575531$$ 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 $$\chi$$ $$=$$ 1872.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-6.00000 q^{5} -20.0000 q^{7} +O(q^{10})$$ $$q-6.00000 q^{5} -20.0000 q^{7} +24.0000 q^{11} +13.0000 q^{13} +30.0000 q^{17} +16.0000 q^{19} -72.0000 q^{23} -89.0000 q^{25} +282.000 q^{29} -164.000 q^{31} +120.000 q^{35} +110.000 q^{37} +126.000 q^{41} -164.000 q^{43} -204.000 q^{47} +57.0000 q^{49} +738.000 q^{53} -144.000 q^{55} +120.000 q^{59} +614.000 q^{61} -78.0000 q^{65} -848.000 q^{67} +132.000 q^{71} +218.000 q^{73} -480.000 q^{77} +1096.00 q^{79} +552.000 q^{83} -180.000 q^{85} -210.000 q^{89} -260.000 q^{91} -96.0000 q^{95} -1726.00 q^{97} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −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 $$-0.681557\pi$$
−0.539949 + 0.841698i $$0.681557\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 24.0000 0.657843 0.328921 0.944357i $$-0.393315\pi$$
0.328921 + 0.944357i $$0.393315\pi$$
$$12$$ 0 0
$$13$$ 13.0000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 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 $$-0.469204\pi$$
0.0965961 + 0.995324i $$0.469204\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −72.0000 −0.652741 −0.326370 0.945242i $$-0.605826\pi$$
−0.326370 + 0.945242i $$0.605826\pi$$
$$24$$ 0 0
$$25$$ −89.0000 −0.712000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 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 $$-0.657583\pi$$
−0.475085 + 0.879940i $$0.657583\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 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
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 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 $$-0.593925\pi$$
−0.290811 + 0.956780i $$0.593925\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −204.000 −0.633116 −0.316558 0.948573i $$-0.602527\pi$$
−0.316558 + 0.948573i $$0.602527\pi$$
$$48$$ 0 0
$$49$$ 57.0000 0.166181
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 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
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 120.000 0.264791 0.132396 0.991197i $$-0.457733\pi$$
0.132396 + 0.991197i $$0.457733\pi$$
$$60$$ 0 0
$$61$$ 614.000 1.28876 0.644382 0.764703i $$-0.277115\pi$$
0.644382 + 0.764703i $$0.277115\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 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
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 132.000 0.220641 0.110321 0.993896i $$-0.464812\pi$$
0.110321 + 0.993896i $$0.464812\pi$$
$$72$$ 0 0
$$73$$ 218.000 0.349520 0.174760 0.984611i $$-0.444085\pi$$
0.174760 + 0.984611i $$0.444085\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 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
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 552.000 0.729998 0.364999 0.931008i $$-0.381069\pi$$
0.364999 + 0.931008i $$0.381069\pi$$
$$84$$ 0 0
$$85$$ −180.000 −0.229691
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −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
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 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
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −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 $$-0.419989\pi$$
0.248724 + 0.968574i $$0.419989\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000 0.0108419 0.00542095 0.999985i $$-0.498274\pi$$
0.00542095 + 0.999985i $$0.498274\pi$$
$$108$$ 0 0
$$109$$ −1834.00 −1.61161 −0.805804 0.592182i $$-0.798267\pi$$
−0.805804 + 0.592182i $$0.798267\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 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
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −600.000 −0.462201
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 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
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2748.00 −1.83278 −0.916389 0.400289i $$-0.868910\pi$$
−0.916389 + 0.400289i $$0.868910\pi$$
$$132$$ 0 0
$$133$$ −320.000 −0.208628
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −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 $$-0.741114\pi$$
−0.687094 + 0.726568i $$0.741114\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 312.000 0.182453
$$144$$ 0 0
$$145$$ −1692.00 −0.969055
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 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 $$-0.414221\pi$$
0.266233 + 0.963909i $$0.414221\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 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
$$159$$ 0 0
$$160$$ 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
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1116.00 0.517118 0.258559 0.965995i $$-0.416752\pi$$
0.258559 + 0.965995i $$0.416752\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −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
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 12.0000 0.00501074 0.00250537 0.999997i $$-0.499203\pi$$
0.00250537 + 0.999997i $$0.499203\pi$$
$$180$$ 0 0
$$181$$ 4718.00 1.93749 0.968746 0.248053i $$-0.0797909\pi$$
0.968746 + 0.248053i $$0.0797909\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −660.000 −0.262293
$$186$$ 0 0
$$187$$ 720.000 0.281559
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1368.00 −0.518246 −0.259123 0.965844i $$-0.583434\pi$$
−0.259123 + 0.965844i $$0.583434\pi$$
$$192$$ 0 0
$$193$$ −3310.00 −1.23450 −0.617251 0.786766i $$-0.711754\pi$$
−0.617251 + 0.786766i $$0.711754\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −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 $$-0.812065\pi$$
−0.830709 + 0.556707i $$0.812065\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −5640.00 −1.95000
$$204$$ 0 0
$$205$$ −756.000 −0.257567
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 984.000 0.312131
$$216$$ 0 0
$$217$$ 3280.00 1.02609
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 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
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1800.00 0.526300 0.263150 0.964755i $$-0.415239\pi$$
0.263150 + 0.964755i $$0.415239\pi$$
$$228$$ 0 0
$$229$$ 2990.00 0.862816 0.431408 0.902157i $$-0.358017\pi$$
0.431408 + 0.902157i $$0.358017\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −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
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1812.00 −0.490412 −0.245206 0.969471i $$-0.578856\pi$$
−0.245206 + 0.969471i $$0.578856\pi$$
$$240$$ 0 0
$$241$$ −1582.00 −0.422845 −0.211422 0.977395i $$-0.567810\pi$$
−0.211422 + 0.977395i $$0.567810\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −342.000 −0.0891820
$$246$$ 0 0
$$247$$ 208.000 0.0535819
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2148.00 0.540162 0.270081 0.962838i $$-0.412950\pi$$
0.270081 + 0.962838i $$0.412950\pi$$
$$252$$ 0 0
$$253$$ −1728.00 −0.429401
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 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
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 2112.00 0.495177 0.247588 0.968865i $$-0.420362\pi$$
0.247588 + 0.968865i $$0.420362\pi$$
$$264$$ 0 0
$$265$$ −4428.00 −1.02645
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −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 $$-0.360118\pi$$
0.425444 + 0.904985i $$0.360118\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 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
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 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 $$-0.334444\pi$$
0.496976 + 0.867765i $$0.334444\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −2520.00 −0.518296
$$288$$ 0 0
$$289$$ −4013.00 −0.816813
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −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
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −936.000 −0.181038
$$300$$ 0 0
$$301$$ 3280.00 0.628093
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 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
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8760.00 −1.59722 −0.798608 0.601852i $$-0.794430\pi$$
−0.798608 + 0.601852i $$0.794430\pi$$
$$312$$ 0 0
$$313$$ 3962.00 0.715481 0.357740 0.933821i $$-0.383547\pi$$
0.357740 + 0.933821i $$0.383547\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −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
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 480.000 0.0826870
$$324$$ 0 0
$$325$$ −1157.00 −0.197473
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −3936.00 −0.625063
$$342$$ 0 0
$$343$$ 5720.00 0.900440
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −11076.0 −1.71352 −0.856759 0.515717i $$-0.827526\pi$$
−0.856759 + 0.515717i $$0.827526\pi$$
$$348$$ 0 0
$$349$$ 2342.00 0.359210 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −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
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −11268.0 −1.65655 −0.828276 0.560320i $$-0.810678\pi$$
−0.828276 + 0.560320i $$0.810678\pi$$
$$360$$ 0 0
$$361$$ −6603.00 −0.962677
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 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
$$369$$ 0 0
$$370$$ 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
$$375$$ 0 0
$$376$$ 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
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 11820.0 1.57696 0.788478 0.615064i $$-0.210870\pi$$
0.788478 + 0.615064i $$0.210870\pi$$
$$384$$ 0 0
$$385$$ 2880.00 0.381243
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −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
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 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
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 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
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 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
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −2400.00 −0.285947
$$414$$ 0 0
$$415$$ −3312.00 −0.391758
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −7308.00 −0.852074 −0.426037 0.904706i $$-0.640091\pi$$
−0.426037 + 0.904706i $$0.640091\pi$$
$$420$$ 0 0
$$421$$ −5938.00 −0.687412 −0.343706 0.939077i $$-0.611682\pi$$
−0.343706 + 0.939077i $$0.611682\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −2670.00 −0.304739
$$426$$ 0 0
$$427$$ −12280.0 −1.39174
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 11532.0 1.28881 0.644405 0.764685i $$-0.277105\pi$$
0.644405 + 0.764685i $$0.277105\pi$$
$$432$$ 0 0
$$433$$ −718.000 −0.0796879 −0.0398440 0.999206i $$-0.512686\pi$$
−0.0398440 + 0.999206i $$0.512686\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 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
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 2604.00 0.279277 0.139639 0.990203i $$-0.455406\pi$$
0.139639 + 0.990203i $$0.455406\pi$$
$$444$$ 0 0
$$445$$ 1260.00 0.134224
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 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
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 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
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −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 $$-0.794951\pi$$
−0.799593 + 0.600543i $$0.794951\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 18540.0 1.83711 0.918553 0.395297i $$-0.129358\pi$$
0.918553 + 0.395297i $$0.129358\pi$$
$$468$$ 0 0
$$469$$ 16960.0 1.66981
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −3936.00 −0.382616
$$474$$ 0 0
$$475$$ −1424.00 −0.137553
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 6180.00 0.589502 0.294751 0.955574i $$-0.404763\pi$$
0.294751 + 0.955574i $$0.404763\pi$$
$$480$$ 0 0
$$481$$ 1430.00 0.135556
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 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
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1908.00 0.175370 0.0876852 0.996148i $$-0.472053\pi$$
0.0876852 + 0.996148i $$0.472053\pi$$
$$492$$ 0 0
$$493$$ 8460.00 0.772858
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 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
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −6528.00 −0.578666 −0.289333 0.957228i $$-0.593434\pi$$
−0.289333 + 0.957228i $$0.593434\pi$$
$$504$$ 0 0
$$505$$ 4788.00 0.421907
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 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
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −3120.00 −0.266958
$$516$$ 0 0
$$517$$ −4896.00 −0.416491
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 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 $$-0.221905\pi$$
0.766685 + 0.642024i $$0.221905\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −4920.00 −0.406677
$$528$$ 0 0
$$529$$ −6983.00 −0.573929
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 1638.00 0.133114
$$534$$ 0 0
$$535$$ −72.0000 −0.00581838
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 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
$$543$$ 0 0
$$544$$ 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
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 4512.00 0.348852
$$552$$ 0 0
$$553$$ −21920.0 −1.68559
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 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
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −13812.0 −1.03394 −0.516968 0.856004i $$-0.672940\pi$$
−0.516968 + 0.856004i $$0.672940\pi$$
$$564$$ 0 0
$$565$$ −2196.00 −0.163516
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 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 $$-0.518721\pi$$
−0.0587787 + 0.998271i $$0.518721\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 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
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −11040.0 −0.788324
$$582$$ 0 0
$$583$$ 17712.0 1.25824
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −9984.00 −0.702017 −0.351008 0.936372i $$-0.614161\pi$$
−0.351008 + 0.936372i $$0.614161\pi$$
$$588$$ 0 0
$$589$$ −2624.00 −0.183565
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −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
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 11184.0 0.762881 0.381441 0.924393i $$-0.375428\pi$$
0.381441 + 0.924393i $$0.375428\pi$$
$$600$$ 0 0
$$601$$ 2810.00 0.190719 0.0953596 0.995443i $$-0.469600\pi$$
0.0953596 + 0.995443i $$0.469600\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 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
$$609$$ 0 0
$$610$$ 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
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −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 $$-0.339888\pi$$
0.482061 + 0.876138i $$0.339888\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 4200.00 0.270095
$$624$$ 0 0
$$625$$ 3421.00 0.218944
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 12864.0 0.803925
$$636$$ 0 0
$$637$$ 741.000 0.0460902
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 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 $$-0.341578\pi$$
0.477404 + 0.878684i $$0.341578\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 1128.00 0.0685414 0.0342707 0.999413i $$-0.489089\pi$$
0.0342707 + 0.999413i $$0.489089\pi$$
$$648$$ 0 0
$$649$$ 2880.00 0.174191
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −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
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 13572.0 0.802261 0.401131 0.916021i $$-0.368617\pi$$
0.401131 + 0.916021i $$0.368617\pi$$
$$660$$ 0 0
$$661$$ −13138.0 −0.773085 −0.386542 0.922272i $$-0.626331\pi$$
−0.386542 + 0.922272i $$0.626331\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1920.00 0.111962
$$666$$ 0 0
$$667$$ −20304.0 −1.17867
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 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
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 27384.0 1.53414 0.767071 0.641562i $$-0.221713\pi$$
0.767071 + 0.641562i $$0.221713\pi$$
$$684$$ 0 0
$$685$$ 16524.0 0.921678
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 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
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 13512.0 0.737467
$$696$$ 0 0
$$697$$ 3780.00 0.205420
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −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
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 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
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 11808.0 0.620215
$$714$$ 0 0
$$715$$ −1872.00 −0.0979144
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 20976.0 1.08800 0.544001 0.839085i $$-0.316909\pi$$
0.544001 + 0.839085i $$0.316909\pi$$
$$720$$ 0 0
$$721$$ −10400.0 −0.537193
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 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
$$729$$ 0 0
$$730$$ 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
$$735$$ 0 0
$$736$$ 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
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −20844.0 −1.02920 −0.514598 0.857432i $$-0.672059\pi$$
−0.514598 + 0.857432i $$0.672059\pi$$
$$744$$ 0 0
$$745$$ −10620.0 −0.522264
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 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
$$753$$ 0 0
$$754$$ 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
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −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
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 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
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 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
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2016.00 0.0927223
$$780$$ 0 0
$$781$$ 3168.00 0.145147
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 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
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −7320.00 −0.329038
$$792$$ 0 0
$$793$$ 7982.00 0.357439
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 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
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 5232.00 0.229929
$$804$$ 0 0
$$805$$ −8640.00 −0.378286
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 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 $$-0.665524\pi$$
−0.496889 + 0.867814i $$0.665524\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 8976.00 0.385786
$$816$$ 0 0
$$817$$ −2624.00 −0.112365
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 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 $$-0.434888\pi$$
0.203133 + 0.979151i $$0.434888\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −39960.0 −1.68022 −0.840112 0.542413i $$-0.817511\pi$$
−0.840112 + 0.542413i $$0.817511\pi$$
$$828$$ 0 0
$$829$$ −3706.00 −0.155265 −0.0776325 0.996982i $$-0.524736\pi$$
−0.0776325 + 0.996982i $$0.524736\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 1710.00 0.0711260
$$834$$ 0 0
$$835$$ −6696.00 −0.277515
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 9756.00 0.401448 0.200724 0.979648i $$-0.435671\pi$$
0.200724 + 0.979648i $$0.435671\pi$$
$$840$$ 0 0
$$841$$ 55135.0 2.26065
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −1014.00 −0.0412813
$$846$$ 0 0
$$847$$ 15100.0 0.612565
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 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
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 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 $$-0.363533\pi$$
0.415710 + 0.909497i $$0.363533\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 10044.0 0.396178 0.198089 0.980184i $$-0.436526\pi$$
0.198089 + 0.980184i $$0.436526\pi$$
$$864$$ 0 0
$$865$$ 26244.0 1.03159
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 26304.0 1.02681
$$870$$ 0 0
$$871$$ −11024.0 −0.428856
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 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
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −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 $$-0.461156\pi$$
0.121729 + 0.992563i $$0.461156\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −5472.00 −0.207138 −0.103569 0.994622i $$-0.533026\pi$$
−0.103569 + 0.994622i $$0.533026\pi$$
$$888$$ 0 0
$$889$$ 42880.0 1.61772
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −3264.00 −0.122313
$$894$$ 0 0
$$895$$ −72.0000 −0.00268904
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −46248.0 −1.71575
$$900$$ 0 0
$$901$$ 22140.0 0.818635
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 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
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 27624.0 1.00464 0.502318 0.864683i $$-0.332481\pi$$
0.502318 + 0.864683i $$0.332481\pi$$
$$912$$ 0 0
$$913$$ 13248.0 0.480224
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 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
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 1716.00 0.0611948
$$924$$ 0 0
$$925$$ −9790.00 −0.347993
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 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
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 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
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 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
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4848.00 0.166356 0.0831778 0.996535i $$-0.473493\pi$$
0.0831778 + 0.996535i $$0.473493\pi$$
$$948$$ 0 0
$$949$$ 2834.00 0.0969394
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −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
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 55080.0 1.85467
$$960$$ 0 0
$$961$$ −2895.00 −0.0971770
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 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
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −4260.00 −0.140793 −0.0703964 0.997519i $$-0.522426\pi$$
−0.0703964 + 0.997519i $$0.522426\pi$$
$$972$$ 0 0
$$973$$ 45040.0 1.48398
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 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
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −49524.0 −1.60689 −0.803444 0.595381i $$-0.797001\pi$$
−0.803444 + 0.595381i $$0.797001\pi$$
$$984$$ 0 0
$$985$$ 18756.0 0.606717
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ 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
$$999$$ 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 1872.4.a.e.1.1 1
3.2 odd 2 624.4.a.i.1.1 1
4.3 odd 2 234.4.a.b.1.1 1
12.11 even 2 78.4.a.e.1.1 1
24.5 odd 2 2496.4.a.b.1.1 1
24.11 even 2 2496.4.a.k.1.1 1
60.59 even 2 1950.4.a.c.1.1 1
156.47 odd 4 1014.4.b.c.337.2 2
156.83 odd 4 1014.4.b.c.337.1 2
156.155 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 12.11 even 2
234.4.a.b.1.1 1 4.3 odd 2
624.4.a.i.1.1 1 3.2 odd 2
1014.4.a.b.1.1 1 156.155 even 2
1014.4.b.c.337.1 2 156.83 odd 4
1014.4.b.c.337.2 2 156.47 odd 4
1872.4.a.e.1.1 1 1.1 even 1 trivial
1950.4.a.c.1.1 1 60.59 even 2
2496.4.a.b.1.1 1 24.5 odd 2
2496.4.a.k.1.1 1 24.11 even 2