# Properties

 Label 825.4.a.c.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 $$-0.677932\pi$$
−0.530330 + 0.847791i $$0.677932\pi$$
$$3$$ −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 $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 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 $$-0.554595\pi$$
−0.170677 + 0.985327i $$0.554595\pi$$
$$14$$ 21.0000 0.400892
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ 21.0000 0.299603 0.149801 0.988716i $$-0.452137\pi$$
0.149801 + 0.988716i $$0.452137\pi$$
$$18$$ −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 $$-0.619672\pi$$
−0.367167 + 0.930155i $$0.619672\pi$$
$$24$$ −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 $$-0.689994\pi$$
−0.562067 + 0.827092i $$0.689994\pi$$
$$38$$ −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 $$-0.556744\pi$$
−0.177324 + 0.984153i $$0.556744\pi$$
$$44$$ 11.0000 0.0376889
$$45$$ 0 0
$$46$$ 243.000 0.778878
$$47$$ −219.000 −0.679669 −0.339834 0.940485i $$-0.610371\pi$$
−0.339834 + 0.940485i $$0.610371\pi$$
$$48$$ 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 $$-0.580038\pi$$
−0.248804 + 0.968554i $$0.580038\pi$$
$$54$$ 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 $$-0.420231\pi$$
0.247986 + 0.968764i $$0.420231\pi$$
$$68$$ 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 $$-0.528618\pi$$
−0.0897850 + 0.995961i $$0.528618\pi$$
$$74$$ 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 $$-0.811194\pi$$
−0.829183 + 0.558977i $$0.811194\pi$$
$$84$$ 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 $$-0.355293\pi$$
0.439111 + 0.898433i $$0.355293\pi$$
$$98$$ 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 $$-0.214421\pi$$
0.781567 + 0.623821i $$0.214421\pi$$
$$104$$ −336.000 −0.316803
$$105$$ 0 0
$$106$$ 576.000 0.527793
$$107$$ 726.000 0.655935 0.327968 0.944689i $$-0.393636\pi$$
0.327968 + 0.944689i $$0.393636\pi$$
$$108$$ −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 $$-0.655576\pi$$
−0.469528 + 0.882918i $$0.655576\pi$$
$$114$$ 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 $$-0.371188\pi$$
0.393721 + 0.919230i $$0.371188\pi$$
$$128$$ −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 $$-0.494640\pi$$
0.0168377 + 0.999858i $$0.494640\pi$$
$$138$$ −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 $$-0.545624\pi$$
−0.142842 + 0.989745i $$0.545624\pi$$
$$158$$ −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 $$-0.301361\pi$$
0.584322 + 0.811522i $$0.301361\pi$$
$$164$$ 63.0000 0.0299968
$$165$$ 0 0
$$166$$ 3762.00 1.75896
$$167$$ 2340.00 1.08428 0.542140 0.840288i $$-0.317614\pi$$
0.542140 + 0.840288i $$0.317614\pi$$
$$168$$ 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 $$-0.192102\pi$$
0.823350 + 0.567534i $$0.192102\pi$$
$$174$$ 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 $$-0.379110\pi$$
0.370724 + 0.928743i $$0.379110\pi$$
$$194$$ −2517.00 −0.931495
$$195$$ 0 0
$$196$$ −294.000 −0.107143
$$197$$ 3327.00 1.20324 0.601622 0.798781i $$-0.294522\pi$$
0.601622 + 0.798781i $$0.294522\pi$$
$$198$$ −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 $$-0.303927\pi$$
0.577760 + 0.816206i $$0.303927\pi$$
$$224$$ −315.000 −0.0939590
$$225$$ 0 0
$$226$$ 3384.00 0.996019
$$227$$ −1386.00 −0.405251 −0.202626 0.979256i $$-0.564947\pi$$
−0.202626 + 0.979256i $$0.564947\pi$$
$$228$$ −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 $$-0.456232\pi$$
0.137069 + 0.990561i $$0.456232\pi$$
$$234$$ 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 $$-0.684724\pi$$
−0.548298 + 0.836283i $$0.684724\pi$$
$$258$$ −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 $$-0.516351\pi$$
−0.0513465 + 0.998681i $$0.516351\pi$$
$$264$$ −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 $$-0.435063\pi$$
0.202594 + 0.979263i $$0.435063\pi$$
$$278$$ −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 $$-0.413942\pi$$
0.267077 + 0.963675i $$0.413942\pi$$
$$284$$ −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 $$-0.652468\pi$$
−0.460885 + 0.887460i $$0.652468\pi$$
$$294$$ −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 $$-0.480934\pi$$
0.0598616 + 0.998207i $$0.480934\pi$$
$$308$$ −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 $$-0.379938\pi$$
0.368305 + 0.929705i $$0.379938\pi$$
$$314$$ 1686.00 0.303014
$$315$$ 0 0
$$316$$ 509.000 0.0906123
$$317$$ 3504.00 0.620834 0.310417 0.950601i $$-0.399531\pi$$
0.310417 + 0.950601i $$0.399531\pi$$
$$318$$ −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 $$-0.844406\pi$$
−0.882891 + 0.469579i $$0.844406\pi$$
$$338$$ 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 $$-0.590135\pi$$
−0.279398 + 0.960175i $$0.590135\pi$$
$$348$$ −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 $$-0.519594\pi$$
−0.0615174 + 0.998106i $$0.519594\pi$$
$$354$$ 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 $$-0.447562\pi$$
0.163995 + 0.986461i $$0.447562\pi$$
$$368$$ 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 $$-0.430202\pi$$
0.217523 + 0.976055i $$0.430202\pi$$
$$374$$ −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 $$-0.222609\pi$$
0.765263 + 0.643717i $$0.222609\pi$$
$$384$$ 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 $$-0.449773\pi$$
0.157140 + 0.987576i $$0.449773\pi$$
$$398$$ 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 $$-0.359818\pi$$
0.426297 + 0.904583i $$0.359818\pi$$
$$434$$ −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 $$-0.526505\pi$$
−0.0831718 + 0.996535i $$0.526505\pi$$
$$444$$ 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 $$-0.328108\pi$$
0.514149 + 0.857701i $$0.328108\pi$$
$$458$$ 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 $$-0.387814\pi$$
0.345192 + 0.938532i $$0.387814\pi$$
$$464$$ −13206.0 −1.32128
$$465$$ 0 0
$$466$$ −2925.00 −0.290768
$$467$$ −16284.0 −1.61356 −0.806781 0.590850i $$-0.798792\pi$$
−0.806781 + 0.590850i $$0.798792\pi$$
$$468$$ −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 $$-0.197244\pi$$
0.814076 + 0.580758i $$0.197244\pi$$
$$488$$ −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 $$-0.527630\pi$$
−0.0866936 + 0.996235i $$0.527630\pi$$
$$504$$ −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 $$-0.334310\pi$$
0.497342 + 0.867555i $$0.334310\pi$$
$$524$$ −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 $$-0.277220\pi$$
0.644129 + 0.764917i $$0.277220\pi$$
$$548$$ 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 $$-0.885822\pi$$
−0.936354 + 0.351058i $$0.885822\pi$$
$$558$$ 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 $$-0.422535\pi$$
0.240967 + 0.970533i $$0.422535\pi$$
$$564$$ 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 $$-0.712408\pi$$
−0.618867 + 0.785496i $$0.712408\pi$$
$$578$$ 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 $$-0.166936\pi$$
0.865603 + 0.500732i $$0.166936\pi$$
$$588$$ 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 $$-0.601639\pi$$
−0.313909 + 0.949453i $$0.601639\pi$$
$$594$$ 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 $$-0.209328\pi$$
0.791447 + 0.611238i $$0.209328\pi$$
$$608$$ 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 $$-0.531806\pi$$
−0.0997551 + 0.995012i $$0.531806\pi$$
$$614$$ −1932.00 −0.126986
$$615$$ 0 0
$$616$$ −1617.00 −0.105764
$$617$$ −1170.00 −0.0763410 −0.0381705 0.999271i $$-0.512153\pi$$
−0.0381705 + 0.999271i $$0.512153\pi$$
$$618$$ 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 $$-0.477824\pi$$
0.0696112 + 0.997574i $$0.477824\pi$$
$$644$$ 567.000 0.0346940
$$645$$ 0 0
$$646$$ −7875.00 −0.479625
$$647$$ 23361.0 1.41950 0.709749 0.704454i $$-0.248808\pi$$
0.709749 + 0.704454i $$0.248808\pi$$
$$648$$ 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 $$-0.379913\pi$$
0.368378 + 0.929676i $$0.379913\pi$$
$$654$$ 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 $$-0.371545\pi$$
0.392689 + 0.919672i $$0.371545\pi$$
$$674$$ 32772.0 1.87289
$$675$$ 0 0
$$676$$ −1941.00 −0.110435
$$677$$ −3714.00 −0.210843 −0.105421 0.994428i $$-0.533619\pi$$
−0.105421 + 0.994428i $$0.533619\pi$$
$$678$$ −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 $$-0.619264\pi$$
−0.365972 + 0.930626i $$0.619264\pi$$
$$684$$ 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 $$-0.588162\pi$$
−0.273441 + 0.961889i $$0.588162\pi$$
$$728$$ 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 $$-0.412113\pi$$
0.272610 + 0.962125i $$0.412113\pi$$
$$734$$ −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 $$-0.547325\pi$$
−0.148128 + 0.988968i $$0.547325\pi$$
$$744$$ 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 $$-0.286904\pi$$
0.620564 + 0.784156i $$0.286904\pi$$
$$758$$ −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 $$-0.688140\pi$$
−0.557240 + 0.830351i $$0.688140\pi$$
$$774$$ 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 $$-0.854631\pi$$
−0.897518 + 0.440979i $$0.854631\pi$$
$$788$$ 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 $$-0.553515\pi$$
−0.167331 + 0.985901i $$0.553515\pi$$
$$798$$ −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 $$-0.430889\pi$$
0.215415 + 0.976523i $$0.430889\pi$$
$$824$$ 34314.0 1.45071
$$825$$ 0 0
$$826$$ 5229.00 0.220267
$$827$$ 4560.00 0.191737 0.0958686 0.995394i $$-0.469437\pi$$
0.0958686 + 0.995394i $$0.469437\pi$$
$$828$$ −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 $$-0.349300\pi$$
0.455949 + 0.890006i $$0.349300\pi$$
$$854$$ −1344.00 −0.0538533
$$855$$ 0 0
$$856$$ 15246.0 0.608759
$$857$$ −27435.0 −1.09354 −0.546769 0.837284i $$-0.684142\pi$$
−0.546769 + 0.837284i $$0.684142\pi$$
$$858$$ −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 $$-0.595439\pi$$
−0.295359 + 0.955386i $$0.595439\pi$$
$$864$$ −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 $$-0.701415\pi$$
−0.591376 + 0.806396i $$0.701415\pi$$
$$878$$ −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 $$-0.772823\pi$$
−0.755947 + 0.654633i $$0.772823\pi$$
$$884$$ −336.000 −0.0127838
$$885$$ 0 0
$$886$$ 4653.00 0.176434
$$887$$ −23724.0 −0.898054 −0.449027 0.893518i $$-0.648229\pi$$
−0.449027 + 0.893518i $$0.648229\pi$$
$$888$$ 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 $$-0.780658\pi$$
−0.771830 + 0.635829i $$0.780658\pi$$
$$908$$ −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 $$-0.462949\pi$$
0.116136 + 0.993233i $$0.462949\pi$$
$$938$$ 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 $$-0.370587\pi$$
0.395455 + 0.918485i $$0.370587\pi$$
$$948$$ −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 $$-0.444425\pi$$
0.173710 + 0.984797i $$0.444425\pi$$
$$954$$ 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 $$-0.505674\pi$$
−0.0178248 + 0.999841i $$0.505674\pi$$
$$968$$ 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 $$-0.197984\pi$$
0.812723 + 0.582651i $$0.197984\pi$$
$$978$$ 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 $$-0.505903\pi$$
−0.0185433 + 0.999828i $$0.505903\pi$$
$$984$$ −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 $$-0.844343\pi$$
−0.882798 + 0.469752i $$0.844343\pi$$
$$998$$ 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.c.1.1 1
3.2 odd 2 2475.4.a.i.1.1 1
5.2 odd 4 825.4.c.d.199.1 2
5.3 odd 4 825.4.c.d.199.2 2
5.4 even 2 825.4.a.g.1.1 yes 1
15.14 odd 2 2475.4.a.d.1.1 1

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