# Properties

 Label 825.4.a.g.1.1 Level $825$ Weight $4$ Character 825.1 Self dual yes Analytic conductor $48.677$ 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] = [825,4,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.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: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +9.00000 q^{6} +7.00000 q^{7} -21.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +9.00000 q^{6} +7.00000 q^{7} -21.0000 q^{8} +9.00000 q^{9} +11.0000 q^{11} +3.00000 q^{12} +16.0000 q^{13} +21.0000 q^{14} -71.0000 q^{16} -21.0000 q^{17} +27.0000 q^{18} +125.000 q^{19} +21.0000 q^{21} +33.0000 q^{22} +81.0000 q^{23} -63.0000 q^{24} +48.0000 q^{26} +27.0000 q^{27} +7.00000 q^{28} +186.000 q^{29} -58.0000 q^{31} -45.0000 q^{32} +33.0000 q^{33} -63.0000 q^{34} +9.00000 q^{36} +253.000 q^{37} +375.000 q^{38} +48.0000 q^{39} +63.0000 q^{41} +63.0000 q^{42} +100.000 q^{43} +11.0000 q^{44} +243.000 q^{46} +219.000 q^{47} -213.000 q^{48} -294.000 q^{49} -63.0000 q^{51} +16.0000 q^{52} +192.000 q^{53} +81.0000 q^{54} -147.000 q^{56} +375.000 q^{57} +558.000 q^{58} +249.000 q^{59} -64.0000 q^{61} -174.000 q^{62} +63.0000 q^{63} +433.000 q^{64} +99.0000 q^{66} -272.000 q^{67} -21.0000 q^{68} +243.000 q^{69} -645.000 q^{71} -189.000 q^{72} +112.000 q^{73} +759.000 q^{74} +125.000 q^{76} +77.0000 q^{77} +144.000 q^{78} +509.000 q^{79} +81.0000 q^{81} +189.000 q^{82} +1254.00 q^{83} +21.0000 q^{84} +300.000 q^{86} +558.000 q^{87} -231.000 q^{88} +756.000 q^{89} +112.000 q^{91} +81.0000 q^{92} -174.000 q^{93} +657.000 q^{94} -135.000 q^{96} -839.000 q^{97} -882.000 q^{98} +99.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.00000 1.06066 0.530330 0.847791i $$-0.322068\pi$$
0.530330 + 0.847791i $$0.322068\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 1.00000 0.125000
$$5$$ 0 0
$$6$$ 9.00000 0.612372
$$7$$ 7.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ −21.0000 −0.928078
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 3.00000 0.0721688
$$13$$ 16.0000 0.341354 0.170677 0.985327i $$-0.445405\pi$$
0.170677 + 0.985327i $$0.445405\pi$$
$$14$$ 21.0000 0.400892
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ −21.0000 −0.299603 −0.149801 0.988716i $$-0.547863\pi$$
−0.149801 + 0.988716i $$0.547863\pi$$
$$18$$ 27.0000 0.353553
$$19$$ 125.000 1.50931 0.754657 0.656119i $$-0.227803\pi$$
0.754657 + 0.656119i $$0.227803\pi$$
$$20$$ 0 0
$$21$$ 21.0000 0.218218
$$22$$ 33.0000 0.319801
$$23$$ 81.0000 0.734333 0.367167 0.930155i $$-0.380328\pi$$
0.367167 + 0.930155i $$0.380328\pi$$
$$24$$ −63.0000 −0.535826
$$25$$ 0 0
$$26$$ 48.0000 0.362061
$$27$$ 27.0000 0.192450
$$28$$ 7.00000 0.0472456
$$29$$ 186.000 1.19101 0.595506 0.803351i $$-0.296952\pi$$
0.595506 + 0.803351i $$0.296952\pi$$
$$30$$ 0 0
$$31$$ −58.0000 −0.336036 −0.168018 0.985784i $$-0.553737\pi$$
−0.168018 + 0.985784i $$0.553737\pi$$
$$32$$ −45.0000 −0.248592
$$33$$ 33.0000 0.174078
$$34$$ −63.0000 −0.317777
$$35$$ 0 0
$$36$$ 9.00000 0.0416667
$$37$$ 253.000 1.12413 0.562067 0.827092i $$-0.310006\pi$$
0.562067 + 0.827092i $$0.310006\pi$$
$$38$$ 375.000 1.60087
$$39$$ 48.0000 0.197081
$$40$$ 0 0
$$41$$ 63.0000 0.239974 0.119987 0.992775i $$-0.461715\pi$$
0.119987 + 0.992775i $$0.461715\pi$$
$$42$$ 63.0000 0.231455
$$43$$ 100.000 0.354648 0.177324 0.984153i $$-0.443256\pi$$
0.177324 + 0.984153i $$0.443256\pi$$
$$44$$ 11.0000 0.0376889
$$45$$ 0 0
$$46$$ 243.000 0.778878
$$47$$ 219.000 0.679669 0.339834 0.940485i $$-0.389629\pi$$
0.339834 + 0.940485i $$0.389629\pi$$
$$48$$ −213.000 −0.640498
$$49$$ −294.000 −0.857143
$$50$$ 0 0
$$51$$ −63.0000 −0.172976
$$52$$ 16.0000 0.0426692
$$53$$ 192.000 0.497608 0.248804 0.968554i $$-0.419962\pi$$
0.248804 + 0.968554i $$0.419962\pi$$
$$54$$ 81.0000 0.204124
$$55$$ 0 0
$$56$$ −147.000 −0.350780
$$57$$ 375.000 0.871403
$$58$$ 558.000 1.26326
$$59$$ 249.000 0.549441 0.274721 0.961524i $$-0.411415\pi$$
0.274721 + 0.961524i $$0.411415\pi$$
$$60$$ 0 0
$$61$$ −64.0000 −0.134334 −0.0671669 0.997742i $$-0.521396\pi$$
−0.0671669 + 0.997742i $$0.521396\pi$$
$$62$$ −174.000 −0.356420
$$63$$ 63.0000 0.125988
$$64$$ 433.000 0.845703
$$65$$ 0 0
$$66$$ 99.0000 0.184637
$$67$$ −272.000 −0.495971 −0.247986 0.968764i $$-0.579769\pi$$
−0.247986 + 0.968764i $$0.579769\pi$$
$$68$$ −21.0000 −0.0374504
$$69$$ 243.000 0.423968
$$70$$ 0 0
$$71$$ −645.000 −1.07813 −0.539066 0.842263i $$-0.681223\pi$$
−0.539066 + 0.842263i $$0.681223\pi$$
$$72$$ −189.000 −0.309359
$$73$$ 112.000 0.179570 0.0897850 0.995961i $$-0.471382\pi$$
0.0897850 + 0.995961i $$0.471382\pi$$
$$74$$ 759.000 1.19232
$$75$$ 0 0
$$76$$ 125.000 0.188664
$$77$$ 77.0000 0.113961
$$78$$ 144.000 0.209036
$$79$$ 509.000 0.724898 0.362449 0.932004i $$-0.381941\pi$$
0.362449 + 0.932004i $$0.381941\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 189.000 0.254531
$$83$$ 1254.00 1.65837 0.829183 0.558977i $$-0.188806\pi$$
0.829183 + 0.558977i $$0.188806\pi$$
$$84$$ 21.0000 0.0272772
$$85$$ 0 0
$$86$$ 300.000 0.376161
$$87$$ 558.000 0.687631
$$88$$ −231.000 −0.279826
$$89$$ 756.000 0.900403 0.450201 0.892927i $$-0.351352\pi$$
0.450201 + 0.892927i $$0.351352\pi$$
$$90$$ 0 0
$$91$$ 112.000 0.129020
$$92$$ 81.0000 0.0917917
$$93$$ −174.000 −0.194010
$$94$$ 657.000 0.720898
$$95$$ 0 0
$$96$$ −135.000 −0.143525
$$97$$ −839.000 −0.878222 −0.439111 0.898433i $$-0.644707\pi$$
−0.439111 + 0.898433i $$0.644707\pi$$
$$98$$ −882.000 −0.909137
$$99$$ 99.0000 0.100504
$$100$$ 0 0
$$101$$ −1413.00 −1.39207 −0.696033 0.718009i $$-0.745053\pi$$
−0.696033 + 0.718009i $$0.745053\pi$$
$$102$$ −189.000 −0.183469
$$103$$ −1634.00 −1.56313 −0.781567 0.623821i $$-0.785579\pi$$
−0.781567 + 0.623821i $$0.785579\pi$$
$$104$$ −336.000 −0.316803
$$105$$ 0 0
$$106$$ 576.000 0.527793
$$107$$ −726.000 −0.655935 −0.327968 0.944689i $$-0.606364\pi$$
−0.327968 + 0.944689i $$0.606364\pi$$
$$108$$ 27.0000 0.0240563
$$109$$ 1712.00 1.50440 0.752201 0.658934i $$-0.228992\pi$$
0.752201 + 0.658934i $$0.228992\pi$$
$$110$$ 0 0
$$111$$ 759.000 0.649019
$$112$$ −497.000 −0.419304
$$113$$ 1128.00 0.939056 0.469528 0.882918i $$-0.344424\pi$$
0.469528 + 0.882918i $$0.344424\pi$$
$$114$$ 1125.00 0.924262
$$115$$ 0 0
$$116$$ 186.000 0.148876
$$117$$ 144.000 0.113785
$$118$$ 747.000 0.582771
$$119$$ −147.000 −0.113239
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ −192.000 −0.142482
$$123$$ 189.000 0.138549
$$124$$ −58.0000 −0.0420045
$$125$$ 0 0
$$126$$ 189.000 0.133631
$$127$$ −1127.00 −0.787442 −0.393721 0.919230i $$-0.628812\pi$$
−0.393721 + 0.919230i $$0.628812\pi$$
$$128$$ 1659.00 1.14560
$$129$$ 300.000 0.204756
$$130$$ 0 0
$$131$$ −1122.00 −0.748318 −0.374159 0.927365i $$-0.622068\pi$$
−0.374159 + 0.927365i $$0.622068\pi$$
$$132$$ 33.0000 0.0217597
$$133$$ 875.000 0.570467
$$134$$ −816.000 −0.526057
$$135$$ 0 0
$$136$$ 441.000 0.278055
$$137$$ −54.0000 −0.0336754 −0.0168377 0.999858i $$-0.505360\pi$$
−0.0168377 + 0.999858i $$0.505360\pi$$
$$138$$ 729.000 0.449686
$$139$$ 1748.00 1.06664 0.533322 0.845913i $$-0.320944\pi$$
0.533322 + 0.845913i $$0.320944\pi$$
$$140$$ 0 0
$$141$$ 657.000 0.392407
$$142$$ −1935.00 −1.14353
$$143$$ 176.000 0.102922
$$144$$ −639.000 −0.369792
$$145$$ 0 0
$$146$$ 336.000 0.190463
$$147$$ −882.000 −0.494872
$$148$$ 253.000 0.140517
$$149$$ 1797.00 0.988027 0.494013 0.869454i $$-0.335529\pi$$
0.494013 + 0.869454i $$0.335529\pi$$
$$150$$ 0 0
$$151$$ 1040.00 0.560490 0.280245 0.959928i $$-0.409584\pi$$
0.280245 + 0.959928i $$0.409584\pi$$
$$152$$ −2625.00 −1.40076
$$153$$ −189.000 −0.0998676
$$154$$ 231.000 0.120873
$$155$$ 0 0
$$156$$ 48.0000 0.0246351
$$157$$ 562.000 0.285685 0.142842 0.989745i $$-0.454376\pi$$
0.142842 + 0.989745i $$0.454376\pi$$
$$158$$ 1527.00 0.768871
$$159$$ 576.000 0.287294
$$160$$ 0 0
$$161$$ 567.000 0.277552
$$162$$ 243.000 0.117851
$$163$$ −2432.00 −1.16864 −0.584322 0.811522i $$-0.698639\pi$$
−0.584322 + 0.811522i $$0.698639\pi$$
$$164$$ 63.0000 0.0299968
$$165$$ 0 0
$$166$$ 3762.00 1.75896
$$167$$ −2340.00 −1.08428 −0.542140 0.840288i $$-0.682386\pi$$
−0.542140 + 0.840288i $$0.682386\pi$$
$$168$$ −441.000 −0.202523
$$169$$ −1941.00 −0.883477
$$170$$ 0 0
$$171$$ 1125.00 0.503105
$$172$$ 100.000 0.0443310
$$173$$ −3747.00 −1.64670 −0.823350 0.567534i $$-0.807898\pi$$
−0.823350 + 0.567534i $$0.807898\pi$$
$$174$$ 1674.00 0.729343
$$175$$ 0 0
$$176$$ −781.000 −0.334489
$$177$$ 747.000 0.317220
$$178$$ 2268.00 0.955021
$$179$$ −267.000 −0.111489 −0.0557445 0.998445i $$-0.517753\pi$$
−0.0557445 + 0.998445i $$0.517753\pi$$
$$180$$ 0 0
$$181$$ 4277.00 1.75639 0.878196 0.478301i $$-0.158747\pi$$
0.878196 + 0.478301i $$0.158747\pi$$
$$182$$ 336.000 0.136846
$$183$$ −192.000 −0.0775576
$$184$$ −1701.00 −0.681518
$$185$$ 0 0
$$186$$ −522.000 −0.205779
$$187$$ −231.000 −0.0903337
$$188$$ 219.000 0.0849586
$$189$$ 189.000 0.0727393
$$190$$ 0 0
$$191$$ −1797.00 −0.680766 −0.340383 0.940287i $$-0.610557\pi$$
−0.340383 + 0.940287i $$0.610557\pi$$
$$192$$ 1299.00 0.488267
$$193$$ −1988.00 −0.741448 −0.370724 0.928743i $$-0.620890\pi$$
−0.370724 + 0.928743i $$0.620890\pi$$
$$194$$ −2517.00 −0.931495
$$195$$ 0 0
$$196$$ −294.000 −0.107143
$$197$$ −3327.00 −1.20324 −0.601622 0.798781i $$-0.705478\pi$$
−0.601622 + 0.798781i $$0.705478\pi$$
$$198$$ 297.000 0.106600
$$199$$ −1780.00 −0.634075 −0.317037 0.948413i $$-0.602688\pi$$
−0.317037 + 0.948413i $$0.602688\pi$$
$$200$$ 0 0
$$201$$ −816.000 −0.286349
$$202$$ −4239.00 −1.47651
$$203$$ 1302.00 0.450160
$$204$$ −63.0000 −0.0216220
$$205$$ 0 0
$$206$$ −4902.00 −1.65795
$$207$$ 729.000 0.244778
$$208$$ −1136.00 −0.378690
$$209$$ 1375.00 0.455075
$$210$$ 0 0
$$211$$ 2180.00 0.711267 0.355634 0.934625i $$-0.384265\pi$$
0.355634 + 0.934625i $$0.384265\pi$$
$$212$$ 192.000 0.0622010
$$213$$ −1935.00 −0.622460
$$214$$ −2178.00 −0.695724
$$215$$ 0 0
$$216$$ −567.000 −0.178609
$$217$$ −406.000 −0.127010
$$218$$ 5136.00 1.59566
$$219$$ 336.000 0.103675
$$220$$ 0 0
$$221$$ −336.000 −0.102271
$$222$$ 2277.00 0.688388
$$223$$ −3848.00 −1.15552 −0.577760 0.816206i $$-0.696073\pi$$
−0.577760 + 0.816206i $$0.696073\pi$$
$$224$$ −315.000 −0.0939590
$$225$$ 0 0
$$226$$ 3384.00 0.996019
$$227$$ 1386.00 0.405251 0.202626 0.979256i $$-0.435053\pi$$
0.202626 + 0.979256i $$0.435053\pi$$
$$228$$ 375.000 0.108925
$$229$$ −991.000 −0.285970 −0.142985 0.989725i $$-0.545670\pi$$
−0.142985 + 0.989725i $$0.545670\pi$$
$$230$$ 0 0
$$231$$ 231.000 0.0657952
$$232$$ −3906.00 −1.10535
$$233$$ −975.000 −0.274139 −0.137069 0.990561i $$-0.543768\pi$$
−0.137069 + 0.990561i $$0.543768\pi$$
$$234$$ 432.000 0.120687
$$235$$ 0 0
$$236$$ 249.000 0.0686802
$$237$$ 1527.00 0.418520
$$238$$ −441.000 −0.120108
$$239$$ −1524.00 −0.412466 −0.206233 0.978503i $$-0.566120\pi$$
−0.206233 + 0.978503i $$0.566120\pi$$
$$240$$ 0 0
$$241$$ −2230.00 −0.596045 −0.298023 0.954559i $$-0.596327\pi$$
−0.298023 + 0.954559i $$0.596327\pi$$
$$242$$ 363.000 0.0964237
$$243$$ 243.000 0.0641500
$$244$$ −64.0000 −0.0167917
$$245$$ 0 0
$$246$$ 567.000 0.146954
$$247$$ 2000.00 0.515210
$$248$$ 1218.00 0.311867
$$249$$ 3762.00 0.957458
$$250$$ 0 0
$$251$$ 3864.00 0.971687 0.485844 0.874046i $$-0.338512\pi$$
0.485844 + 0.874046i $$0.338512\pi$$
$$252$$ 63.0000 0.0157485
$$253$$ 891.000 0.221410
$$254$$ −3381.00 −0.835208
$$255$$ 0 0
$$256$$ 1513.00 0.369385
$$257$$ 4518.00 1.09660 0.548298 0.836283i $$-0.315276\pi$$
0.548298 + 0.836283i $$0.315276\pi$$
$$258$$ 900.000 0.217177
$$259$$ 1771.00 0.424883
$$260$$ 0 0
$$261$$ 1674.00 0.397004
$$262$$ −3366.00 −0.793711
$$263$$ 438.000 0.102693 0.0513465 0.998681i $$-0.483649\pi$$
0.0513465 + 0.998681i $$0.483649\pi$$
$$264$$ −693.000 −0.161558
$$265$$ 0 0
$$266$$ 2625.00 0.605072
$$267$$ 2268.00 0.519848
$$268$$ −272.000 −0.0619964
$$269$$ 4902.00 1.11108 0.555539 0.831490i $$-0.312512\pi$$
0.555539 + 0.831490i $$0.312512\pi$$
$$270$$ 0 0
$$271$$ −2455.00 −0.550298 −0.275149 0.961402i $$-0.588727\pi$$
−0.275149 + 0.961402i $$0.588727\pi$$
$$272$$ 1491.00 0.332372
$$273$$ 336.000 0.0744895
$$274$$ −162.000 −0.0357182
$$275$$ 0 0
$$276$$ 243.000 0.0529959
$$277$$ −1868.00 −0.405189 −0.202594 0.979263i $$-0.564937\pi$$
−0.202594 + 0.979263i $$0.564937\pi$$
$$278$$ 5244.00 1.13135
$$279$$ −522.000 −0.112012
$$280$$ 0 0
$$281$$ −6093.00 −1.29352 −0.646758 0.762695i $$-0.723876\pi$$
−0.646758 + 0.762695i $$0.723876\pi$$
$$282$$ 1971.00 0.416210
$$283$$ −2543.00 −0.534154 −0.267077 0.963675i $$-0.586058\pi$$
−0.267077 + 0.963675i $$0.586058\pi$$
$$284$$ −645.000 −0.134767
$$285$$ 0 0
$$286$$ 528.000 0.109165
$$287$$ 441.000 0.0907018
$$288$$ −405.000 −0.0828641
$$289$$ −4472.00 −0.910238
$$290$$ 0 0
$$291$$ −2517.00 −0.507042
$$292$$ 112.000 0.0224462
$$293$$ 4623.00 0.921770 0.460885 0.887460i $$-0.347532\pi$$
0.460885 + 0.887460i $$0.347532\pi$$
$$294$$ −2646.00 −0.524891
$$295$$ 0 0
$$296$$ −5313.00 −1.04328
$$297$$ 297.000 0.0580259
$$298$$ 5391.00 1.04796
$$299$$ 1296.00 0.250668
$$300$$ 0 0
$$301$$ 700.000 0.134044
$$302$$ 3120.00 0.594489
$$303$$ −4239.00 −0.803710
$$304$$ −8875.00 −1.67440
$$305$$ 0 0
$$306$$ −567.000 −0.105926
$$307$$ −644.000 −0.119723 −0.0598616 0.998207i $$-0.519066\pi$$
−0.0598616 + 0.998207i $$0.519066\pi$$
$$308$$ 77.0000 0.0142451
$$309$$ −4902.00 −0.902476
$$310$$ 0 0
$$311$$ 2616.00 0.476977 0.238488 0.971145i $$-0.423348\pi$$
0.238488 + 0.971145i $$0.423348\pi$$
$$312$$ −1008.00 −0.182906
$$313$$ −4079.00 −0.736609 −0.368305 0.929705i $$-0.620062\pi$$
−0.368305 + 0.929705i $$0.620062\pi$$
$$314$$ 1686.00 0.303014
$$315$$ 0 0
$$316$$ 509.000 0.0906123
$$317$$ −3504.00 −0.620834 −0.310417 0.950601i $$-0.600469\pi$$
−0.310417 + 0.950601i $$0.600469\pi$$
$$318$$ 1728.00 0.304721
$$319$$ 2046.00 0.359103
$$320$$ 0 0
$$321$$ −2178.00 −0.378704
$$322$$ 1701.00 0.294388
$$323$$ −2625.00 −0.452195
$$324$$ 81.0000 0.0138889
$$325$$ 0 0
$$326$$ −7296.00 −1.23953
$$327$$ 5136.00 0.868567
$$328$$ −1323.00 −0.222715
$$329$$ 1533.00 0.256891
$$330$$ 0 0
$$331$$ 4100.00 0.680835 0.340417 0.940274i $$-0.389432\pi$$
0.340417 + 0.940274i $$0.389432\pi$$
$$332$$ 1254.00 0.207296
$$333$$ 2277.00 0.374711
$$334$$ −7020.00 −1.15005
$$335$$ 0 0
$$336$$ −1491.00 −0.242085
$$337$$ 10924.0 1.76578 0.882891 0.469579i $$-0.155594\pi$$
0.882891 + 0.469579i $$0.155594\pi$$
$$338$$ −5823.00 −0.937069
$$339$$ 3384.00 0.542164
$$340$$ 0 0
$$341$$ −638.000 −0.101319
$$342$$ 3375.00 0.533623
$$343$$ −4459.00 −0.701934
$$344$$ −2100.00 −0.329141
$$345$$ 0 0
$$346$$ −11241.0 −1.74659
$$347$$ 3612.00 0.558796 0.279398 0.960175i $$-0.409865\pi$$
0.279398 + 0.960175i $$0.409865\pi$$
$$348$$ 558.000 0.0859538
$$349$$ −406.000 −0.0622713 −0.0311356 0.999515i $$-0.509912\pi$$
−0.0311356 + 0.999515i $$0.509912\pi$$
$$350$$ 0 0
$$351$$ 432.000 0.0656936
$$352$$ −495.000 −0.0749534
$$353$$ 816.000 0.123035 0.0615174 0.998106i $$-0.480406\pi$$
0.0615174 + 0.998106i $$0.480406\pi$$
$$354$$ 2241.00 0.336463
$$355$$ 0 0
$$356$$ 756.000 0.112550
$$357$$ −441.000 −0.0653787
$$358$$ −801.000 −0.118252
$$359$$ 4818.00 0.708313 0.354156 0.935186i $$-0.384768\pi$$
0.354156 + 0.935186i $$0.384768\pi$$
$$360$$ 0 0
$$361$$ 8766.00 1.27803
$$362$$ 12831.0 1.86293
$$363$$ 363.000 0.0524864
$$364$$ 112.000 0.0161275
$$365$$ 0 0
$$366$$ −576.000 −0.0822623
$$367$$ −2306.00 −0.327990 −0.163995 0.986461i $$-0.552438\pi$$
−0.163995 + 0.986461i $$0.552438\pi$$
$$368$$ −5751.00 −0.814651
$$369$$ 567.000 0.0799914
$$370$$ 0 0
$$371$$ 1344.00 0.188078
$$372$$ −174.000 −0.0242513
$$373$$ −3134.00 −0.435047 −0.217523 0.976055i $$-0.569798\pi$$
−0.217523 + 0.976055i $$0.569798\pi$$
$$374$$ −693.000 −0.0958133
$$375$$ 0 0
$$376$$ −4599.00 −0.630785
$$377$$ 2976.00 0.406556
$$378$$ 567.000 0.0771517
$$379$$ 7202.00 0.976100 0.488050 0.872816i $$-0.337708\pi$$
0.488050 + 0.872816i $$0.337708\pi$$
$$380$$ 0 0
$$381$$ −3381.00 −0.454630
$$382$$ −5391.00 −0.722062
$$383$$ −11472.0 −1.53053 −0.765263 0.643717i $$-0.777391\pi$$
−0.765263 + 0.643717i $$0.777391\pi$$
$$384$$ 4977.00 0.661410
$$385$$ 0 0
$$386$$ −5964.00 −0.786424
$$387$$ 900.000 0.118216
$$388$$ −839.000 −0.109778
$$389$$ −3462.00 −0.451235 −0.225617 0.974216i $$-0.572440\pi$$
−0.225617 + 0.974216i $$0.572440\pi$$
$$390$$ 0 0
$$391$$ −1701.00 −0.220008
$$392$$ 6174.00 0.795495
$$393$$ −3366.00 −0.432041
$$394$$ −9981.00 −1.27623
$$395$$ 0 0
$$396$$ 99.0000 0.0125630
$$397$$ −2486.00 −0.314279 −0.157140 0.987576i $$-0.550227\pi$$
−0.157140 + 0.987576i $$0.550227\pi$$
$$398$$ −5340.00 −0.672538
$$399$$ 2625.00 0.329359
$$400$$ 0 0
$$401$$ −9024.00 −1.12378 −0.561892 0.827211i $$-0.689926\pi$$
−0.561892 + 0.827211i $$0.689926\pi$$
$$402$$ −2448.00 −0.303719
$$403$$ −928.000 −0.114707
$$404$$ −1413.00 −0.174008
$$405$$ 0 0
$$406$$ 3906.00 0.477467
$$407$$ 2783.00 0.338939
$$408$$ 1323.00 0.160535
$$409$$ −14488.0 −1.75155 −0.875777 0.482716i $$-0.839650\pi$$
−0.875777 + 0.482716i $$0.839650\pi$$
$$410$$ 0 0
$$411$$ −162.000 −0.0194425
$$412$$ −1634.00 −0.195392
$$413$$ 1743.00 0.207669
$$414$$ 2187.00 0.259626
$$415$$ 0 0
$$416$$ −720.000 −0.0848579
$$417$$ 5244.00 0.615827
$$418$$ 4125.00 0.482680
$$419$$ −3201.00 −0.373220 −0.186610 0.982434i $$-0.559750\pi$$
−0.186610 + 0.982434i $$0.559750\pi$$
$$420$$ 0 0
$$421$$ −6721.00 −0.778056 −0.389028 0.921226i $$-0.627189\pi$$
−0.389028 + 0.921226i $$0.627189\pi$$
$$422$$ 6540.00 0.754413
$$423$$ 1971.00 0.226556
$$424$$ −4032.00 −0.461819
$$425$$ 0 0
$$426$$ −5805.00 −0.660219
$$427$$ −448.000 −0.0507734
$$428$$ −726.000 −0.0819919
$$429$$ 528.000 0.0594221
$$430$$ 0 0
$$431$$ −354.000 −0.0395628 −0.0197814 0.999804i $$-0.506297\pi$$
−0.0197814 + 0.999804i $$0.506297\pi$$
$$432$$ −1917.00 −0.213499
$$433$$ −7682.00 −0.852594 −0.426297 0.904583i $$-0.640182\pi$$
−0.426297 + 0.904583i $$0.640182\pi$$
$$434$$ −1218.00 −0.134714
$$435$$ 0 0
$$436$$ 1712.00 0.188050
$$437$$ 10125.0 1.10834
$$438$$ 1008.00 0.109964
$$439$$ 5465.00 0.594146 0.297073 0.954855i $$-0.403989\pi$$
0.297073 + 0.954855i $$0.403989\pi$$
$$440$$ 0 0
$$441$$ −2646.00 −0.285714
$$442$$ −1008.00 −0.108474
$$443$$ 1551.00 0.166344 0.0831718 0.996535i $$-0.473495\pi$$
0.0831718 + 0.996535i $$0.473495\pi$$
$$444$$ 759.000 0.0811274
$$445$$ 0 0
$$446$$ −11544.0 −1.22561
$$447$$ 5391.00 0.570437
$$448$$ 3031.00 0.319646
$$449$$ −1092.00 −0.114777 −0.0573883 0.998352i $$-0.518277\pi$$
−0.0573883 + 0.998352i $$0.518277\pi$$
$$450$$ 0 0
$$451$$ 693.000 0.0723550
$$452$$ 1128.00 0.117382
$$453$$ 3120.00 0.323599
$$454$$ 4158.00 0.429834
$$455$$ 0 0
$$456$$ −7875.00 −0.808730
$$457$$ −10046.0 −1.02830 −0.514149 0.857701i $$-0.671892\pi$$
−0.514149 + 0.857701i $$0.671892\pi$$
$$458$$ −2973.00 −0.303317
$$459$$ −567.000 −0.0576586
$$460$$ 0 0
$$461$$ 15510.0 1.56697 0.783485 0.621411i $$-0.213440\pi$$
0.783485 + 0.621411i $$0.213440\pi$$
$$462$$ 693.000 0.0697863
$$463$$ −6878.00 −0.690384 −0.345192 0.938532i $$-0.612186\pi$$
−0.345192 + 0.938532i $$0.612186\pi$$
$$464$$ −13206.0 −1.32128
$$465$$ 0 0
$$466$$ −2925.00 −0.290768
$$467$$ 16284.0 1.61356 0.806781 0.590850i $$-0.201208\pi$$
0.806781 + 0.590850i $$0.201208\pi$$
$$468$$ 144.000 0.0142231
$$469$$ −1904.00 −0.187460
$$470$$ 0 0
$$471$$ 1686.00 0.164940
$$472$$ −5229.00 −0.509924
$$473$$ 1100.00 0.106930
$$474$$ 4581.00 0.443908
$$475$$ 0 0
$$476$$ −147.000 −0.0141549
$$477$$ 1728.00 0.165869
$$478$$ −4572.00 −0.437486
$$479$$ 6732.00 0.642156 0.321078 0.947053i $$-0.395955\pi$$
0.321078 + 0.947053i $$0.395955\pi$$
$$480$$ 0 0
$$481$$ 4048.00 0.383727
$$482$$ −6690.00 −0.632202
$$483$$ 1701.00 0.160245
$$484$$ 121.000 0.0113636
$$485$$ 0 0
$$486$$ 729.000 0.0680414
$$487$$ −17498.0 −1.62815 −0.814076 0.580758i $$-0.802756\pi$$
−0.814076 + 0.580758i $$0.802756\pi$$
$$488$$ 1344.00 0.124672
$$489$$ −7296.00 −0.674717
$$490$$ 0 0
$$491$$ 2454.00 0.225555 0.112777 0.993620i $$-0.464025\pi$$
0.112777 + 0.993620i $$0.464025\pi$$
$$492$$ 189.000 0.0173187
$$493$$ −3906.00 −0.356830
$$494$$ 6000.00 0.546463
$$495$$ 0 0
$$496$$ 4118.00 0.372790
$$497$$ −4515.00 −0.407496
$$498$$ 11286.0 1.01554
$$499$$ −20716.0 −1.85847 −0.929234 0.369492i $$-0.879532\pi$$
−0.929234 + 0.369492i $$0.879532\pi$$
$$500$$ 0 0
$$501$$ −7020.00 −0.626009
$$502$$ 11592.0 1.03063
$$503$$ 1956.00 0.173387 0.0866936 0.996235i $$-0.472370\pi$$
0.0866936 + 0.996235i $$0.472370\pi$$
$$504$$ −1323.00 −0.116927
$$505$$ 0 0
$$506$$ 2673.00 0.234841
$$507$$ −5823.00 −0.510076
$$508$$ −1127.00 −0.0984302
$$509$$ 18240.0 1.58836 0.794179 0.607684i $$-0.207901\pi$$
0.794179 + 0.607684i $$0.207901\pi$$
$$510$$ 0 0
$$511$$ 784.000 0.0678711
$$512$$ −8733.00 −0.753804
$$513$$ 3375.00 0.290468
$$514$$ 13554.0 1.16312
$$515$$ 0 0
$$516$$ 300.000 0.0255945
$$517$$ 2409.00 0.204928
$$518$$ 5313.00 0.450656
$$519$$ −11241.0 −0.950723
$$520$$ 0 0
$$521$$ 20790.0 1.74823 0.874114 0.485721i $$-0.161443\pi$$
0.874114 + 0.485721i $$0.161443\pi$$
$$522$$ 5022.00 0.421086
$$523$$ −11897.0 −0.994684 −0.497342 0.867555i $$-0.665690\pi$$
−0.497342 + 0.867555i $$0.665690\pi$$
$$524$$ −1122.00 −0.0935397
$$525$$ 0 0
$$526$$ 1314.00 0.108922
$$527$$ 1218.00 0.100677
$$528$$ −2343.00 −0.193117
$$529$$ −5606.00 −0.460754
$$530$$ 0 0
$$531$$ 2241.00 0.183147
$$532$$ 875.000 0.0713084
$$533$$ 1008.00 0.0819162
$$534$$ 6804.00 0.551382
$$535$$ 0 0
$$536$$ 5712.00 0.460300
$$537$$ −801.000 −0.0643682
$$538$$ 14706.0 1.17848
$$539$$ −3234.00 −0.258438
$$540$$ 0 0
$$541$$ 20336.0 1.61611 0.808053 0.589110i $$-0.200522\pi$$
0.808053 + 0.589110i $$0.200522\pi$$
$$542$$ −7365.00 −0.583679
$$543$$ 12831.0 1.01405
$$544$$ 945.000 0.0744789
$$545$$ 0 0
$$546$$ 1008.00 0.0790081
$$547$$ −16481.0 −1.28826 −0.644129 0.764917i $$-0.722780\pi$$
−0.644129 + 0.764917i $$0.722780\pi$$
$$548$$ −54.0000 −0.00420943
$$549$$ −576.000 −0.0447779
$$550$$ 0 0
$$551$$ 23250.0 1.79761
$$552$$ −5103.00 −0.393475
$$553$$ 3563.00 0.273986
$$554$$ −5604.00 −0.429767
$$555$$ 0 0
$$556$$ 1748.00 0.133330
$$557$$ 24618.0 1.87271 0.936354 0.351058i $$-0.114178\pi$$
0.936354 + 0.351058i $$0.114178\pi$$
$$558$$ −1566.00 −0.118807
$$559$$ 1600.00 0.121060
$$560$$ 0 0
$$561$$ −693.000 −0.0521542
$$562$$ −18279.0 −1.37198
$$563$$ −6438.00 −0.481935 −0.240967 0.970533i $$-0.577465\pi$$
−0.240967 + 0.970533i $$0.577465\pi$$
$$564$$ 657.000 0.0490509
$$565$$ 0 0
$$566$$ −7629.00 −0.566556
$$567$$ 567.000 0.0419961
$$568$$ 13545.0 1.00059
$$569$$ −18183.0 −1.33967 −0.669834 0.742511i $$-0.733635\pi$$
−0.669834 + 0.742511i $$0.733635\pi$$
$$570$$ 0 0
$$571$$ −17656.0 −1.29401 −0.647006 0.762485i $$-0.723979\pi$$
−0.647006 + 0.762485i $$0.723979\pi$$
$$572$$ 176.000 0.0128653
$$573$$ −5391.00 −0.393041
$$574$$ 1323.00 0.0962038
$$575$$ 0 0
$$576$$ 3897.00 0.281901
$$577$$ 17155.0 1.23773 0.618867 0.785496i $$-0.287592\pi$$
0.618867 + 0.785496i $$0.287592\pi$$
$$578$$ −13416.0 −0.965453
$$579$$ −5964.00 −0.428075
$$580$$ 0 0
$$581$$ 8778.00 0.626803
$$582$$ −7551.00 −0.537799
$$583$$ 2112.00 0.150034
$$584$$ −2352.00 −0.166655
$$585$$ 0 0
$$586$$ 13869.0 0.977684
$$587$$ −24621.0 −1.73121 −0.865603 0.500732i $$-0.833064\pi$$
−0.865603 + 0.500732i $$0.833064\pi$$
$$588$$ −882.000 −0.0618590
$$589$$ −7250.00 −0.507183
$$590$$ 0 0
$$591$$ −9981.00 −0.694693
$$592$$ −17963.0 −1.24709
$$593$$ 9066.00 0.627818 0.313909 0.949453i $$-0.398361\pi$$
0.313909 + 0.949453i $$0.398361\pi$$
$$594$$ 891.000 0.0615457
$$595$$ 0 0
$$596$$ 1797.00 0.123503
$$597$$ −5340.00 −0.366083
$$598$$ 3888.00 0.265873
$$599$$ 22353.0 1.52474 0.762370 0.647142i $$-0.224036\pi$$
0.762370 + 0.647142i $$0.224036\pi$$
$$600$$ 0 0
$$601$$ −6304.00 −0.427863 −0.213931 0.976849i $$-0.568627\pi$$
−0.213931 + 0.976849i $$0.568627\pi$$
$$602$$ 2100.00 0.142175
$$603$$ −2448.00 −0.165324
$$604$$ 1040.00 0.0700613
$$605$$ 0 0
$$606$$ −12717.0 −0.852463
$$607$$ −23672.0 −1.58289 −0.791447 0.611238i $$-0.790672\pi$$
−0.791447 + 0.611238i $$0.790672\pi$$
$$608$$ −5625.00 −0.375204
$$609$$ 3906.00 0.259900
$$610$$ 0 0
$$611$$ 3504.00 0.232008
$$612$$ −189.000 −0.0124835
$$613$$ 3028.00 0.199510 0.0997551 0.995012i $$-0.468194\pi$$
0.0997551 + 0.995012i $$0.468194\pi$$
$$614$$ −1932.00 −0.126986
$$615$$ 0 0
$$616$$ −1617.00 −0.105764
$$617$$ 1170.00 0.0763410 0.0381705 0.999271i $$-0.487847\pi$$
0.0381705 + 0.999271i $$0.487847\pi$$
$$618$$ −14706.0 −0.957220
$$619$$ 9626.00 0.625043 0.312521 0.949911i $$-0.398826\pi$$
0.312521 + 0.949911i $$0.398826\pi$$
$$620$$ 0 0
$$621$$ 2187.00 0.141323
$$622$$ 7848.00 0.505910
$$623$$ 5292.00 0.340320
$$624$$ −3408.00 −0.218637
$$625$$ 0 0
$$626$$ −12237.0 −0.781292
$$627$$ 4125.00 0.262738
$$628$$ 562.000 0.0357106
$$629$$ −5313.00 −0.336794
$$630$$ 0 0
$$631$$ −5794.00 −0.365540 −0.182770 0.983156i $$-0.558506\pi$$
−0.182770 + 0.983156i $$0.558506\pi$$
$$632$$ −10689.0 −0.672762
$$633$$ 6540.00 0.410650
$$634$$ −10512.0 −0.658493
$$635$$ 0 0
$$636$$ 576.000 0.0359118
$$637$$ −4704.00 −0.292589
$$638$$ 6138.00 0.380887
$$639$$ −5805.00 −0.359378
$$640$$ 0 0
$$641$$ 28308.0 1.74430 0.872152 0.489235i $$-0.162724\pi$$
0.872152 + 0.489235i $$0.162724\pi$$
$$642$$ −6534.00 −0.401677
$$643$$ −2270.00 −0.139222 −0.0696112 0.997574i $$-0.522176\pi$$
−0.0696112 + 0.997574i $$0.522176\pi$$
$$644$$ 567.000 0.0346940
$$645$$ 0 0
$$646$$ −7875.00 −0.479625
$$647$$ −23361.0 −1.41950 −0.709749 0.704454i $$-0.751192\pi$$
−0.709749 + 0.704454i $$0.751192\pi$$
$$648$$ −1701.00 −0.103120
$$649$$ 2739.00 0.165663
$$650$$ 0 0
$$651$$ −1218.00 −0.0733290
$$652$$ −2432.00 −0.146080
$$653$$ −12294.0 −0.736756 −0.368378 0.929676i $$-0.620087\pi$$
−0.368378 + 0.929676i $$0.620087\pi$$
$$654$$ 15408.0 0.921255
$$655$$ 0 0
$$656$$ −4473.00 −0.266222
$$657$$ 1008.00 0.0598567
$$658$$ 4599.00 0.272474
$$659$$ 31896.0 1.88542 0.942710 0.333613i $$-0.108268\pi$$
0.942710 + 0.333613i $$0.108268\pi$$
$$660$$ 0 0
$$661$$ 13469.0 0.792562 0.396281 0.918129i $$-0.370301\pi$$
0.396281 + 0.918129i $$0.370301\pi$$
$$662$$ 12300.0 0.722135
$$663$$ −1008.00 −0.0590460
$$664$$ −26334.0 −1.53909
$$665$$ 0 0
$$666$$ 6831.00 0.397441
$$667$$ 15066.0 0.874599
$$668$$ −2340.00 −0.135535
$$669$$ −11544.0 −0.667140
$$670$$ 0 0
$$671$$ −704.000 −0.0405032
$$672$$ −945.000 −0.0542473
$$673$$ −13712.0 −0.785377 −0.392689 0.919672i $$-0.628455\pi$$
−0.392689 + 0.919672i $$0.628455\pi$$
$$674$$ 32772.0 1.87289
$$675$$ 0 0
$$676$$ −1941.00 −0.110435
$$677$$ 3714.00 0.210843 0.105421 0.994428i $$-0.466381\pi$$
0.105421 + 0.994428i $$0.466381\pi$$
$$678$$ 10152.0 0.575052
$$679$$ −5873.00 −0.331937
$$680$$ 0 0
$$681$$ 4158.00 0.233972
$$682$$ −1914.00 −0.107465
$$683$$ 13065.0 0.731945 0.365972 0.930626i $$-0.380736\pi$$
0.365972 + 0.930626i $$0.380736\pi$$
$$684$$ 1125.00 0.0628881
$$685$$ 0 0
$$686$$ −13377.0 −0.744513
$$687$$ −2973.00 −0.165105
$$688$$ −7100.00 −0.393437
$$689$$ 3072.00 0.169860
$$690$$ 0 0
$$691$$ 12512.0 0.688826 0.344413 0.938818i $$-0.388078\pi$$
0.344413 + 0.938818i $$0.388078\pi$$
$$692$$ −3747.00 −0.205838
$$693$$ 693.000 0.0379869
$$694$$ 10836.0 0.592693
$$695$$ 0 0
$$696$$ −11718.0 −0.638175
$$697$$ −1323.00 −0.0718970
$$698$$ −1218.00 −0.0660487
$$699$$ −2925.00 −0.158274
$$700$$ 0 0
$$701$$ 33405.0 1.79984 0.899921 0.436053i $$-0.143624\pi$$
0.899921 + 0.436053i $$0.143624\pi$$
$$702$$ 1296.00 0.0696786
$$703$$ 31625.0 1.69667
$$704$$ 4763.00 0.254989
$$705$$ 0 0
$$706$$ 2448.00 0.130498
$$707$$ −9891.00 −0.526152
$$708$$ 747.000 0.0396525
$$709$$ 22217.0 1.17684 0.588418 0.808557i $$-0.299751\pi$$
0.588418 + 0.808557i $$0.299751\pi$$
$$710$$ 0 0
$$711$$ 4581.00 0.241633
$$712$$ −15876.0 −0.835644
$$713$$ −4698.00 −0.246762
$$714$$ −1323.00 −0.0693446
$$715$$ 0 0
$$716$$ −267.000 −0.0139361
$$717$$ −4572.00 −0.238137
$$718$$ 14454.0 0.751279
$$719$$ 12336.0 0.639854 0.319927 0.947442i $$-0.396342\pi$$
0.319927 + 0.947442i $$0.396342\pi$$
$$720$$ 0 0
$$721$$ −11438.0 −0.590809
$$722$$ 26298.0 1.35555
$$723$$ −6690.00 −0.344127
$$724$$ 4277.00 0.219549
$$725$$ 0 0
$$726$$ 1089.00 0.0556702
$$727$$ 10720.0 0.546881 0.273441 0.961889i $$-0.411838\pi$$
0.273441 + 0.961889i $$0.411838\pi$$
$$728$$ −2352.00 −0.119740
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −2100.00 −0.106253
$$732$$ −192.000 −0.00969471
$$733$$ −10820.0 −0.545219 −0.272610 0.962125i $$-0.587887\pi$$
−0.272610 + 0.962125i $$0.587887\pi$$
$$734$$ −6918.00 −0.347886
$$735$$ 0 0
$$736$$ −3645.00 −0.182550
$$737$$ −2992.00 −0.149541
$$738$$ 1701.00 0.0848437
$$739$$ 10127.0 0.504097 0.252049 0.967715i $$-0.418896\pi$$
0.252049 + 0.967715i $$0.418896\pi$$
$$740$$ 0 0
$$741$$ 6000.00 0.297457
$$742$$ 4032.00 0.199487
$$743$$ 6000.00 0.296257 0.148128 0.988968i $$-0.452675\pi$$
0.148128 + 0.988968i $$0.452675\pi$$
$$744$$ 3654.00 0.180057
$$745$$ 0 0
$$746$$ −9402.00 −0.461437
$$747$$ 11286.0 0.552789
$$748$$ −231.000 −0.0112917
$$749$$ −5082.00 −0.247920
$$750$$ 0 0
$$751$$ 26132.0 1.26973 0.634867 0.772621i $$-0.281055\pi$$
0.634867 + 0.772621i $$0.281055\pi$$
$$752$$ −15549.0 −0.754008
$$753$$ 11592.0 0.561004
$$754$$ 8928.00 0.431218
$$755$$ 0 0
$$756$$ 189.000 0.00909241
$$757$$ −25850.0 −1.24113 −0.620564 0.784156i $$-0.713096\pi$$
−0.620564 + 0.784156i $$0.713096\pi$$
$$758$$ 21606.0 1.03531
$$759$$ 2673.00 0.127831
$$760$$ 0 0
$$761$$ −3654.00 −0.174057 −0.0870285 0.996206i $$-0.527737\pi$$
−0.0870285 + 0.996206i $$0.527737\pi$$
$$762$$ −10143.0 −0.482208
$$763$$ 11984.0 0.568611
$$764$$ −1797.00 −0.0850958
$$765$$ 0 0
$$766$$ −34416.0 −1.62337
$$767$$ 3984.00 0.187554
$$768$$ 4539.00 0.213264
$$769$$ 6248.00 0.292989 0.146495 0.989211i $$-0.453201\pi$$
0.146495 + 0.989211i $$0.453201\pi$$
$$770$$ 0 0
$$771$$ 13554.0 0.633120
$$772$$ −1988.00 −0.0926809
$$773$$ 23952.0 1.11448 0.557240 0.830351i $$-0.311860\pi$$
0.557240 + 0.830351i $$0.311860\pi$$
$$774$$ 2700.00 0.125387
$$775$$ 0 0
$$776$$ 17619.0 0.815058
$$777$$ 5313.00 0.245306
$$778$$ −10386.0 −0.478607
$$779$$ 7875.00 0.362197
$$780$$ 0 0
$$781$$ −7095.00 −0.325069
$$782$$ −5103.00 −0.233354
$$783$$ 5022.00 0.229210
$$784$$ 20874.0 0.950893
$$785$$ 0 0
$$786$$ −10098.0 −0.458249
$$787$$ 39631.0 1.79504 0.897518 0.440979i $$-0.145369\pi$$
0.897518 + 0.440979i $$0.145369\pi$$
$$788$$ −3327.00 −0.150405
$$789$$ 1314.00 0.0592898
$$790$$ 0 0
$$791$$ 7896.00 0.354930
$$792$$ −2079.00 −0.0932753
$$793$$ −1024.00 −0.0458554
$$794$$ −7458.00 −0.333343
$$795$$ 0 0
$$796$$ −1780.00 −0.0792593
$$797$$ 7530.00 0.334663 0.167331 0.985901i $$-0.446485\pi$$
0.167331 + 0.985901i $$0.446485\pi$$
$$798$$ 7875.00 0.349338
$$799$$ −4599.00 −0.203631
$$800$$ 0 0
$$801$$ 6804.00 0.300134
$$802$$ −27072.0 −1.19195
$$803$$ 1232.00 0.0541424
$$804$$ −816.000 −0.0357937
$$805$$ 0 0
$$806$$ −2784.00 −0.121665
$$807$$ 14706.0 0.641482
$$808$$ 29673.0 1.29195
$$809$$ −7239.00 −0.314598 −0.157299 0.987551i $$-0.550279\pi$$
−0.157299 + 0.987551i $$0.550279\pi$$
$$810$$ 0 0
$$811$$ −35611.0 −1.54189 −0.770944 0.636903i $$-0.780215\pi$$
−0.770944 + 0.636903i $$0.780215\pi$$
$$812$$ 1302.00 0.0562700
$$813$$ −7365.00 −0.317714
$$814$$ 8349.00 0.359499
$$815$$ 0 0
$$816$$ 4473.00 0.191895
$$817$$ 12500.0 0.535275
$$818$$ −43464.0 −1.85780
$$819$$ 1008.00 0.0430066
$$820$$ 0 0
$$821$$ −42054.0 −1.78769 −0.893846 0.448375i $$-0.852003\pi$$
−0.893846 + 0.448375i $$0.852003\pi$$
$$822$$ −486.000 −0.0206219
$$823$$ −10172.0 −0.430831 −0.215415 0.976523i $$-0.569111\pi$$
−0.215415 + 0.976523i $$0.569111\pi$$
$$824$$ 34314.0 1.45071
$$825$$ 0 0
$$826$$ 5229.00 0.220267
$$827$$ −4560.00 −0.191737 −0.0958686 0.995394i $$-0.530563\pi$$
−0.0958686 + 0.995394i $$0.530563\pi$$
$$828$$ 729.000 0.0305972
$$829$$ 13202.0 0.553105 0.276553 0.960999i $$-0.410808\pi$$
0.276553 + 0.960999i $$0.410808\pi$$
$$830$$ 0 0
$$831$$ −5604.00 −0.233936
$$832$$ 6928.00 0.288684
$$833$$ 6174.00 0.256802
$$834$$ 15732.0 0.653183
$$835$$ 0 0
$$836$$ 1375.00 0.0568844
$$837$$ −1566.00 −0.0646701
$$838$$ −9603.00 −0.395859
$$839$$ 6216.00 0.255781 0.127890 0.991788i $$-0.459179\pi$$
0.127890 + 0.991788i $$0.459179\pi$$
$$840$$ 0 0
$$841$$ 10207.0 0.418508
$$842$$ −20163.0 −0.825253
$$843$$ −18279.0 −0.746812
$$844$$ 2180.00 0.0889084
$$845$$ 0 0
$$846$$ 5913.00 0.240299
$$847$$ 847.000 0.0343604
$$848$$ −13632.0 −0.552034
$$849$$ −7629.00 −0.308394
$$850$$ 0 0
$$851$$ 20493.0 0.825489
$$852$$ −1935.00 −0.0778075
$$853$$ −22718.0 −0.911899 −0.455949 0.890006i $$-0.650700\pi$$
−0.455949 + 0.890006i $$0.650700\pi$$
$$854$$ −1344.00 −0.0538533
$$855$$ 0 0
$$856$$ 15246.0 0.608759
$$857$$ 27435.0 1.09354 0.546769 0.837284i $$-0.315858\pi$$
0.546769 + 0.837284i $$0.315858\pi$$
$$858$$ 1584.00 0.0630267
$$859$$ 37556.0 1.49173 0.745864 0.666098i $$-0.232037\pi$$
0.745864 + 0.666098i $$0.232037\pi$$
$$860$$ 0 0
$$861$$ 1323.00 0.0523667
$$862$$ −1062.00 −0.0419627
$$863$$ 14976.0 0.590717 0.295359 0.955386i $$-0.404561\pi$$
0.295359 + 0.955386i $$0.404561\pi$$
$$864$$ −1215.00 −0.0478416
$$865$$ 0 0
$$866$$ −23046.0 −0.904313
$$867$$ −13416.0 −0.525526
$$868$$ −406.000 −0.0158762
$$869$$ 5599.00 0.218565
$$870$$ 0 0
$$871$$ −4352.00 −0.169302
$$872$$ −35952.0 −1.39620
$$873$$ −7551.00 −0.292741
$$874$$ 30375.0 1.17557
$$875$$ 0 0
$$876$$ 336.000 0.0129593
$$877$$ 30718.0 1.18275 0.591376 0.806396i $$-0.298585\pi$$
0.591376 + 0.806396i $$0.298585\pi$$
$$878$$ 16395.0 0.630187
$$879$$ 13869.0 0.532184
$$880$$ 0 0
$$881$$ 2916.00 0.111513 0.0557563 0.998444i $$-0.482243\pi$$
0.0557563 + 0.998444i $$0.482243\pi$$
$$882$$ −7938.00 −0.303046
$$883$$ 39670.0 1.51189 0.755947 0.654633i $$-0.227177\pi$$
0.755947 + 0.654633i $$0.227177\pi$$
$$884$$ −336.000 −0.0127838
$$885$$ 0 0
$$886$$ 4653.00 0.176434
$$887$$ 23724.0 0.898054 0.449027 0.893518i $$-0.351771\pi$$
0.449027 + 0.893518i $$0.351771\pi$$
$$888$$ −15939.0 −0.602340
$$889$$ −7889.00 −0.297625
$$890$$ 0 0
$$891$$ 891.000 0.0335013
$$892$$ −3848.00 −0.144440
$$893$$ 27375.0 1.02583
$$894$$ 16173.0 0.605040
$$895$$ 0 0
$$896$$ 11613.0 0.432995
$$897$$ 3888.00 0.144723
$$898$$ −3276.00 −0.121739
$$899$$ −10788.0 −0.400222
$$900$$ 0 0
$$901$$ −4032.00 −0.149085
$$902$$ 2079.00 0.0767440
$$903$$ 2100.00 0.0773905
$$904$$ −23688.0 −0.871517
$$905$$ 0 0
$$906$$ 9360.00 0.343229
$$907$$ 42166.0 1.54366 0.771830 0.635829i $$-0.219342\pi$$
0.771830 + 0.635829i $$0.219342\pi$$
$$908$$ 1386.00 0.0506564
$$909$$ −12717.0 −0.464022
$$910$$ 0 0
$$911$$ 32139.0 1.16884 0.584420 0.811452i $$-0.301322\pi$$
0.584420 + 0.811452i $$0.301322\pi$$
$$912$$ −26625.0 −0.966713
$$913$$ 13794.0 0.500016
$$914$$ −30138.0 −1.09067
$$915$$ 0 0
$$916$$ −991.000 −0.0357462
$$917$$ −7854.00 −0.282837
$$918$$ −1701.00 −0.0611562
$$919$$ 44525.0 1.59820 0.799099 0.601199i $$-0.205310\pi$$
0.799099 + 0.601199i $$0.205310\pi$$
$$920$$ 0 0
$$921$$ −1932.00 −0.0691222
$$922$$ 46530.0 1.66202
$$923$$ −10320.0 −0.368025
$$924$$ 231.000 0.00822440
$$925$$ 0 0
$$926$$ −20634.0 −0.732263
$$927$$ −14706.0 −0.521045
$$928$$ −8370.00 −0.296076
$$929$$ −5964.00 −0.210627 −0.105314 0.994439i $$-0.533585\pi$$
−0.105314 + 0.994439i $$0.533585\pi$$
$$930$$ 0 0
$$931$$ −36750.0 −1.29370
$$932$$ −975.000 −0.0342674
$$933$$ 7848.00 0.275383
$$934$$ 48852.0 1.71144
$$935$$ 0 0
$$936$$ −3024.00 −0.105601
$$937$$ −6662.00 −0.232271 −0.116136 0.993233i $$-0.537051\pi$$
−0.116136 + 0.993233i $$0.537051\pi$$
$$938$$ −5712.00 −0.198831
$$939$$ −12237.0 −0.425282
$$940$$ 0 0
$$941$$ −42129.0 −1.45948 −0.729738 0.683727i $$-0.760358\pi$$
−0.729738 + 0.683727i $$0.760358\pi$$
$$942$$ 5058.00 0.174945
$$943$$ 5103.00 0.176221
$$944$$ −17679.0 −0.609536
$$945$$ 0 0
$$946$$ 3300.00 0.113417
$$947$$ −23049.0 −0.790910 −0.395455 0.918485i $$-0.629413\pi$$
−0.395455 + 0.918485i $$0.629413\pi$$
$$948$$ 1527.00 0.0523150
$$949$$ 1792.00 0.0612969
$$950$$ 0 0
$$951$$ −10512.0 −0.358438
$$952$$ 3087.00 0.105095
$$953$$ −10221.0 −0.347419 −0.173710 0.984797i $$-0.555575\pi$$
−0.173710 + 0.984797i $$0.555575\pi$$
$$954$$ 5184.00 0.175931
$$955$$ 0 0
$$956$$ −1524.00 −0.0515582
$$957$$ 6138.00 0.207328
$$958$$ 20196.0 0.681110
$$959$$ −378.000 −0.0127281
$$960$$ 0 0
$$961$$ −26427.0 −0.887080
$$962$$ 12144.0 0.407004
$$963$$ −6534.00 −0.218645
$$964$$ −2230.00 −0.0745057
$$965$$ 0 0
$$966$$ 5103.00 0.169965
$$967$$ 1072.00 0.0356496 0.0178248 0.999841i $$-0.494326\pi$$
0.0178248 + 0.999841i $$0.494326\pi$$
$$968$$ −2541.00 −0.0843707
$$969$$ −7875.00 −0.261075
$$970$$ 0 0
$$971$$ −50337.0 −1.66364 −0.831818 0.555048i $$-0.812700\pi$$
−0.831818 + 0.555048i $$0.812700\pi$$
$$972$$ 243.000 0.00801875
$$973$$ 12236.0 0.403153
$$974$$ −52494.0 −1.72692
$$975$$ 0 0
$$976$$ 4544.00 0.149027
$$977$$ −49638.0 −1.62545 −0.812723 0.582651i $$-0.802016\pi$$
−0.812723 + 0.582651i $$0.802016\pi$$
$$978$$ −21888.0 −0.715645
$$979$$ 8316.00 0.271482
$$980$$ 0 0
$$981$$ 15408.0 0.501467
$$982$$ 7362.00 0.239237
$$983$$ 1143.00 0.0370865 0.0185433 0.999828i $$-0.494097\pi$$
0.0185433 + 0.999828i $$0.494097\pi$$
$$984$$ −3969.00 −0.128584
$$985$$ 0 0
$$986$$ −11718.0 −0.378476
$$987$$ 4599.00 0.148316
$$988$$ 2000.00 0.0644013
$$989$$ 8100.00 0.260430
$$990$$ 0 0
$$991$$ 35060.0 1.12383 0.561916 0.827194i $$-0.310064\pi$$
0.561916 + 0.827194i $$0.310064\pi$$
$$992$$ 2610.00 0.0835359
$$993$$ 12300.0 0.393080
$$994$$ −13545.0 −0.432215
$$995$$ 0 0
$$996$$ 3762.00 0.119682
$$997$$ 55582.0 1.76560 0.882798 0.469752i $$-0.155657\pi$$
0.882798 + 0.469752i $$0.155657\pi$$
$$998$$ −62148.0 −1.97120
$$999$$ 6831.00 0.216340
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 825.4.a.g.1.1 yes 1
3.2 odd 2 2475.4.a.d.1.1 1
5.2 odd 4 825.4.c.d.199.2 2
5.3 odd 4 825.4.c.d.199.1 2
5.4 even 2 825.4.a.c.1.1 1
15.14 odd 2 2475.4.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.c.1.1 1 5.4 even 2
825.4.a.g.1.1 yes 1 1.1 even 1 trivial
825.4.c.d.199.1 2 5.3 odd 4
825.4.c.d.199.2 2 5.2 odd 4
2475.4.a.d.1.1 1 3.2 odd 2
2475.4.a.i.1.1 1 15.14 odd 2