# Properties

 Label 77.2.a.a.1.1 Level $77$ Weight $2$ Character 77.1 Self dual yes Analytic conductor $0.615$ 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] = [77,2,Mod(1,77)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(77, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("77.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$77 = 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 77.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: $$0.614848095564$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 77.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} -1.00000 q^{11} +6.00000 q^{12} -4.00000 q^{13} +3.00000 q^{15} +4.00000 q^{16} +2.00000 q^{17} -6.00000 q^{19} +2.00000 q^{20} +3.00000 q^{21} -5.00000 q^{23} -4.00000 q^{25} -9.00000 q^{27} +2.00000 q^{28} +10.0000 q^{29} +1.00000 q^{31} +3.00000 q^{33} +1.00000 q^{35} -12.0000 q^{36} -5.00000 q^{37} +12.0000 q^{39} -2.00000 q^{41} -8.00000 q^{43} +2.00000 q^{44} -6.00000 q^{45} +8.00000 q^{47} -12.0000 q^{48} +1.00000 q^{49} -6.00000 q^{51} +8.00000 q^{52} -6.00000 q^{53} +1.00000 q^{55} +18.0000 q^{57} +3.00000 q^{59} -6.00000 q^{60} -2.00000 q^{61} -6.00000 q^{63} -8.00000 q^{64} +4.00000 q^{65} -3.00000 q^{67} -4.00000 q^{68} +15.0000 q^{69} +1.00000 q^{71} +10.0000 q^{73} +12.0000 q^{75} +12.0000 q^{76} +1.00000 q^{77} +6.00000 q^{79} -4.00000 q^{80} +9.00000 q^{81} +12.0000 q^{83} -6.00000 q^{84} -2.00000 q^{85} -30.0000 q^{87} -15.0000 q^{89} +4.00000 q^{91} +10.0000 q^{92} -3.00000 q^{93} +6.00000 q^{95} -5.00000 q^{97} -6.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ −3.00000 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$4$$ −2.00000 −1.00000
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 6.00000 2.00000
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 6.00000 1.73205
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 4.00000 1.00000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 3.00000 0.654654
$$22$$ 0 0
$$23$$ −5.00000 −1.04257 −0.521286 0.853382i $$-0.674548\pi$$
−0.521286 + 0.853382i $$0.674548\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ −9.00000 −1.73205
$$28$$ 2.00000 0.377964
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605 0.0898027 0.995960i $$-0.471376\pi$$
0.0898027 + 0.995960i $$0.471376\pi$$
$$32$$ 0 0
$$33$$ 3.00000 0.522233
$$34$$ 0 0
$$35$$ 1.00000 0.169031
$$36$$ −12.0000 −2.00000
$$37$$ −5.00000 −0.821995 −0.410997 0.911636i $$-0.634819\pi$$
−0.410997 + 0.911636i $$0.634819\pi$$
$$38$$ 0 0
$$39$$ 12.0000 1.92154
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 2.00000 0.301511
$$45$$ −6.00000 −0.894427
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ −12.0000 −1.73205
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ 8.00000 1.10940
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 1.00000 0.134840
$$56$$ 0 0
$$57$$ 18.0000 2.38416
$$58$$ 0 0
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ −6.00000 −0.774597
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ −6.00000 −0.755929
$$64$$ −8.00000 −1.00000
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ −3.00000 −0.366508 −0.183254 0.983066i $$-0.558663\pi$$
−0.183254 + 0.983066i $$0.558663\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 15.0000 1.80579
$$70$$ 0 0
$$71$$ 1.00000 0.118678 0.0593391 0.998238i $$-0.481101\pi$$
0.0593391 + 0.998238i $$0.481101\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 0 0
$$75$$ 12.0000 1.38564
$$76$$ 12.0000 1.37649
$$77$$ 1.00000 0.113961
$$78$$ 0 0
$$79$$ 6.00000 0.675053 0.337526 0.941316i $$-0.390410\pi$$
0.337526 + 0.941316i $$0.390410\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ −6.00000 −0.654654
$$85$$ −2.00000 −0.216930
$$86$$ 0 0
$$87$$ −30.0000 −3.21634
$$88$$ 0 0
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 10.0000 1.04257
$$93$$ −3.00000 −0.311086
$$94$$ 0 0
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ −5.00000 −0.507673 −0.253837 0.967247i $$-0.581693\pi$$
−0.253837 + 0.967247i $$0.581693\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 8.00000 0.800000
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ 0 0
$$105$$ −3.00000 −0.292770
$$106$$ 0 0
$$107$$ −10.0000 −0.966736 −0.483368 0.875417i $$-0.660587\pi$$
−0.483368 + 0.875417i $$0.660587\pi$$
$$108$$ 18.0000 1.73205
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 15.0000 1.42374
$$112$$ −4.00000 −0.377964
$$113$$ −19.0000 −1.78737 −0.893685 0.448695i $$-0.851889\pi$$
−0.893685 + 0.448695i $$0.851889\pi$$
$$114$$ 0 0
$$115$$ 5.00000 0.466252
$$116$$ −20.0000 −1.85695
$$117$$ −24.0000 −2.21880
$$118$$ 0 0
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 6.00000 0.541002
$$124$$ −2.00000 −0.179605
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ 24.0000 2.11308
$$130$$ 0 0
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ −6.00000 −0.522233
$$133$$ 6.00000 0.520266
$$134$$ 0 0
$$135$$ 9.00000 0.774597
$$136$$ 0 0
$$137$$ −3.00000 −0.256307 −0.128154 0.991754i $$-0.540905\pi$$
−0.128154 + 0.991754i $$0.540905\pi$$
$$138$$ 0 0
$$139$$ −10.0000 −0.848189 −0.424094 0.905618i $$-0.639408\pi$$
−0.424094 + 0.905618i $$0.639408\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ −24.0000 −2.02116
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 24.0000 2.00000
$$145$$ −10.0000 −0.830455
$$146$$ 0 0
$$147$$ −3.00000 −0.247436
$$148$$ 10.0000 0.821995
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 0 0
$$151$$ 6.00000 0.488273 0.244137 0.969741i $$-0.421495\pi$$
0.244137 + 0.969741i $$0.421495\pi$$
$$152$$ 0 0
$$153$$ 12.0000 0.970143
$$154$$ 0 0
$$155$$ −1.00000 −0.0803219
$$156$$ −24.0000 −1.92154
$$157$$ 7.00000 0.558661 0.279330 0.960195i $$-0.409888\pi$$
0.279330 + 0.960195i $$0.409888\pi$$
$$158$$ 0 0
$$159$$ 18.0000 1.42749
$$160$$ 0 0
$$161$$ 5.00000 0.394055
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 4.00000 0.312348
$$165$$ −3.00000 −0.233550
$$166$$ 0 0
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ −36.0000 −2.75299
$$172$$ 16.0000 1.21999
$$173$$ 16.0000 1.21646 0.608229 0.793762i $$-0.291880\pi$$
0.608229 + 0.793762i $$0.291880\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ −4.00000 −0.301511
$$177$$ −9.00000 −0.676481
$$178$$ 0 0
$$179$$ 1.00000 0.0747435 0.0373718 0.999301i $$-0.488101\pi$$
0.0373718 + 0.999301i $$0.488101\pi$$
$$180$$ 12.0000 0.894427
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ 0 0
$$185$$ 5.00000 0.367607
$$186$$ 0 0
$$187$$ −2.00000 −0.146254
$$188$$ −16.0000 −1.16692
$$189$$ 9.00000 0.654654
$$190$$ 0 0
$$191$$ 5.00000 0.361787 0.180894 0.983503i $$-0.442101\pi$$
0.180894 + 0.983503i $$0.442101\pi$$
$$192$$ 24.0000 1.73205
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 0 0
$$195$$ −12.0000 −0.859338
$$196$$ −2.00000 −0.142857
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ 9.00000 0.634811
$$202$$ 0 0
$$203$$ −10.0000 −0.701862
$$204$$ 12.0000 0.840168
$$205$$ 2.00000 0.139686
$$206$$ 0 0
$$207$$ −30.0000 −2.08514
$$208$$ −16.0000 −1.10940
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ −2.00000 −0.137686 −0.0688428 0.997628i $$-0.521931\pi$$
−0.0688428 + 0.997628i $$0.521931\pi$$
$$212$$ 12.0000 0.824163
$$213$$ −3.00000 −0.205557
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ −1.00000 −0.0678844
$$218$$ 0 0
$$219$$ −30.0000 −2.02721
$$220$$ −2.00000 −0.134840
$$221$$ −8.00000 −0.538138
$$222$$ 0 0
$$223$$ 1.00000 0.0669650 0.0334825 0.999439i $$-0.489340\pi$$
0.0334825 + 0.999439i $$0.489340\pi$$
$$224$$ 0 0
$$225$$ −24.0000 −1.60000
$$226$$ 0 0
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ −36.0000 −2.38416
$$229$$ −7.00000 −0.462573 −0.231287 0.972886i $$-0.574293\pi$$
−0.231287 + 0.972886i $$0.574293\pi$$
$$230$$ 0 0
$$231$$ −3.00000 −0.197386
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ −6.00000 −0.390567
$$237$$ −18.0000 −1.16923
$$238$$ 0 0
$$239$$ 4.00000 0.258738 0.129369 0.991596i $$-0.458705\pi$$
0.129369 + 0.991596i $$0.458705\pi$$
$$240$$ 12.0000 0.774597
$$241$$ −12.0000 −0.772988 −0.386494 0.922292i $$-0.626314\pi$$
−0.386494 + 0.922292i $$0.626314\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 4.00000 0.256074
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ 24.0000 1.52708
$$248$$ 0 0
$$249$$ −36.0000 −2.28141
$$250$$ 0 0
$$251$$ −21.0000 −1.32551 −0.662754 0.748837i $$-0.730613\pi$$
−0.662754 + 0.748837i $$0.730613\pi$$
$$252$$ 12.0000 0.755929
$$253$$ 5.00000 0.314347
$$254$$ 0 0
$$255$$ 6.00000 0.375735
$$256$$ 16.0000 1.00000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ 5.00000 0.310685
$$260$$ −8.00000 −0.496139
$$261$$ 60.0000 3.71391
$$262$$ 0 0
$$263$$ 18.0000 1.10993 0.554964 0.831875i $$-0.312732\pi$$
0.554964 + 0.831875i $$0.312732\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ 0 0
$$267$$ 45.0000 2.75396
$$268$$ 6.00000 0.366508
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 8.00000 0.485071
$$273$$ −12.0000 −0.726273
$$274$$ 0 0
$$275$$ 4.00000 0.241209
$$276$$ −30.0000 −1.80579
$$277$$ 24.0000 1.44202 0.721010 0.692925i $$-0.243678\pi$$
0.721010 + 0.692925i $$0.243678\pi$$
$$278$$ 0 0
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ −4.00000 −0.238620 −0.119310 0.992857i $$-0.538068\pi$$
−0.119310 + 0.992857i $$0.538068\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ −18.0000 −1.06623
$$286$$ 0 0
$$287$$ 2.00000 0.118056
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 15.0000 0.879316
$$292$$ −20.0000 −1.17041
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ −3.00000 −0.174667
$$296$$ 0 0
$$297$$ 9.00000 0.522233
$$298$$ 0 0
$$299$$ 20.0000 1.15663
$$300$$ −24.0000 −1.38564
$$301$$ 8.00000 0.461112
$$302$$ 0 0
$$303$$ 36.0000 2.06815
$$304$$ −24.0000 −1.37649
$$305$$ 2.00000 0.114520
$$306$$ 0 0
$$307$$ −28.0000 −1.59804 −0.799022 0.601302i $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ −2.00000 −0.113961
$$309$$ 36.0000 2.04797
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ −23.0000 −1.30004 −0.650018 0.759918i $$-0.725239\pi$$
−0.650018 + 0.759918i $$0.725239\pi$$
$$314$$ 0 0
$$315$$ 6.00000 0.338062
$$316$$ −12.0000 −0.675053
$$317$$ 9.00000 0.505490 0.252745 0.967533i $$-0.418667\pi$$
0.252745 + 0.967533i $$0.418667\pi$$
$$318$$ 0 0
$$319$$ −10.0000 −0.559893
$$320$$ 8.00000 0.447214
$$321$$ 30.0000 1.67444
$$322$$ 0 0
$$323$$ −12.0000 −0.667698
$$324$$ −18.0000 −1.00000
$$325$$ 16.0000 0.887520
$$326$$ 0 0
$$327$$ −12.0000 −0.663602
$$328$$ 0 0
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ −17.0000 −0.934405 −0.467202 0.884150i $$-0.654738\pi$$
−0.467202 + 0.884150i $$0.654738\pi$$
$$332$$ −24.0000 −1.31717
$$333$$ −30.0000 −1.64399
$$334$$ 0 0
$$335$$ 3.00000 0.163908
$$336$$ 12.0000 0.654654
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 0 0
$$339$$ 57.0000 3.09582
$$340$$ 4.00000 0.216930
$$341$$ −1.00000 −0.0541530
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ −15.0000 −0.807573
$$346$$ 0 0
$$347$$ 14.0000 0.751559 0.375780 0.926709i $$-0.377375\pi$$
0.375780 + 0.926709i $$0.377375\pi$$
$$348$$ 60.0000 3.21634
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ 36.0000 1.92154
$$352$$ 0 0
$$353$$ 9.00000 0.479022 0.239511 0.970894i $$-0.423013\pi$$
0.239511 + 0.970894i $$0.423013\pi$$
$$354$$ 0 0
$$355$$ −1.00000 −0.0530745
$$356$$ 30.0000 1.59000
$$357$$ 6.00000 0.317554
$$358$$ 0 0
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ −3.00000 −0.157459
$$364$$ −8.00000 −0.419314
$$365$$ −10.0000 −0.523424
$$366$$ 0 0
$$367$$ −11.0000 −0.574195 −0.287098 0.957901i $$-0.592690\pi$$
−0.287098 + 0.957901i $$0.592690\pi$$
$$368$$ −20.0000 −1.04257
$$369$$ −12.0000 −0.624695
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ 6.00000 0.311086
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 0 0
$$375$$ −27.0000 −1.39427
$$376$$ 0 0
$$377$$ −40.0000 −2.06010
$$378$$ 0 0
$$379$$ −29.0000 −1.48963 −0.744815 0.667271i $$-0.767462\pi$$
−0.744815 + 0.667271i $$0.767462\pi$$
$$380$$ −12.0000 −0.615587
$$381$$ −6.00000 −0.307389
$$382$$ 0 0
$$383$$ 17.0000 0.868659 0.434330 0.900754i $$-0.356985\pi$$
0.434330 + 0.900754i $$0.356985\pi$$
$$384$$ 0 0
$$385$$ −1.00000 −0.0509647
$$386$$ 0 0
$$387$$ −48.0000 −2.43998
$$388$$ 10.0000 0.507673
$$389$$ 9.00000 0.456318 0.228159 0.973624i $$-0.426729\pi$$
0.228159 + 0.973624i $$0.426729\pi$$
$$390$$ 0 0
$$391$$ −10.0000 −0.505722
$$392$$ 0 0
$$393$$ −54.0000 −2.72394
$$394$$ 0 0
$$395$$ −6.00000 −0.301893
$$396$$ 12.0000 0.603023
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ 0 0
$$399$$ −18.0000 −0.901127
$$400$$ −16.0000 −0.800000
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ 24.0000 1.19404
$$405$$ −9.00000 −0.447214
$$406$$ 0 0
$$407$$ 5.00000 0.247841
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 9.00000 0.443937
$$412$$ 24.0000 1.18240
$$413$$ −3.00000 −0.147620
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 30.0000 1.46911
$$418$$ 0 0
$$419$$ 16.0000 0.781651 0.390826 0.920465i $$-0.372190\pi$$
0.390826 + 0.920465i $$0.372190\pi$$
$$420$$ 6.00000 0.292770
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 0 0
$$423$$ 48.0000 2.33384
$$424$$ 0 0
$$425$$ −8.00000 −0.388057
$$426$$ 0 0
$$427$$ 2.00000 0.0967868
$$428$$ 20.0000 0.966736
$$429$$ −12.0000 −0.579365
$$430$$ 0 0
$$431$$ −20.0000 −0.963366 −0.481683 0.876346i $$-0.659974\pi$$
−0.481683 + 0.876346i $$0.659974\pi$$
$$432$$ −36.0000 −1.73205
$$433$$ −25.0000 −1.20142 −0.600712 0.799466i $$-0.705116\pi$$
−0.600712 + 0.799466i $$0.705116\pi$$
$$434$$ 0 0
$$435$$ 30.0000 1.43839
$$436$$ −8.00000 −0.383131
$$437$$ 30.0000 1.43509
$$438$$ 0 0
$$439$$ −14.0000 −0.668184 −0.334092 0.942541i $$-0.608430\pi$$
−0.334092 + 0.942541i $$0.608430\pi$$
$$440$$ 0 0
$$441$$ 6.00000 0.285714
$$442$$ 0 0
$$443$$ −39.0000 −1.85295 −0.926473 0.376361i $$-0.877175\pi$$
−0.926473 + 0.376361i $$0.877175\pi$$
$$444$$ −30.0000 −1.42374
$$445$$ 15.0000 0.711068
$$446$$ 0 0
$$447$$ 66.0000 3.12169
$$448$$ 8.00000 0.377964
$$449$$ 15.0000 0.707894 0.353947 0.935266i $$-0.384839\pi$$
0.353947 + 0.935266i $$0.384839\pi$$
$$450$$ 0 0
$$451$$ 2.00000 0.0941763
$$452$$ 38.0000 1.78737
$$453$$ −18.0000 −0.845714
$$454$$ 0 0
$$455$$ −4.00000 −0.187523
$$456$$ 0 0
$$457$$ 8.00000 0.374224 0.187112 0.982339i $$-0.440087\pi$$
0.187112 + 0.982339i $$0.440087\pi$$
$$458$$ 0 0
$$459$$ −18.0000 −0.840168
$$460$$ −10.0000 −0.466252
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 13.0000 0.604161 0.302081 0.953282i $$-0.402319\pi$$
0.302081 + 0.953282i $$0.402319\pi$$
$$464$$ 40.0000 1.85695
$$465$$ 3.00000 0.139122
$$466$$ 0 0
$$467$$ 3.00000 0.138823 0.0694117 0.997588i $$-0.477888\pi$$
0.0694117 + 0.997588i $$0.477888\pi$$
$$468$$ 48.0000 2.21880
$$469$$ 3.00000 0.138527
$$470$$ 0 0
$$471$$ −21.0000 −0.967629
$$472$$ 0 0
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ 24.0000 1.10120
$$476$$ 4.00000 0.183340
$$477$$ −36.0000 −1.64833
$$478$$ 0 0
$$479$$ −28.0000 −1.27935 −0.639676 0.768644i $$-0.720932\pi$$
−0.639676 + 0.768644i $$0.720932\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 0 0
$$483$$ −15.0000 −0.682524
$$484$$ −2.00000 −0.0909091
$$485$$ 5.00000 0.227038
$$486$$ 0 0
$$487$$ −13.0000 −0.589086 −0.294543 0.955638i $$-0.595167\pi$$
−0.294543 + 0.955638i $$0.595167\pi$$
$$488$$ 0 0
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ −30.0000 −1.35388 −0.676941 0.736038i $$-0.736695\pi$$
−0.676941 + 0.736038i $$0.736695\pi$$
$$492$$ −12.0000 −0.541002
$$493$$ 20.0000 0.900755
$$494$$ 0 0
$$495$$ 6.00000 0.269680
$$496$$ 4.00000 0.179605
$$497$$ −1.00000 −0.0448561
$$498$$ 0 0
$$499$$ 44.0000 1.96971 0.984855 0.173379i $$-0.0554684\pi$$
0.984855 + 0.173379i $$0.0554684\pi$$
$$500$$ −18.0000 −0.804984
$$501$$ 6.00000 0.268060
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 12.0000 0.533993
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ −4.00000 −0.177471
$$509$$ −31.0000 −1.37405 −0.687025 0.726633i $$-0.741084\pi$$
−0.687025 + 0.726633i $$0.741084\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ 0 0
$$513$$ 54.0000 2.38416
$$514$$ 0 0
$$515$$ 12.0000 0.528783
$$516$$ −48.0000 −2.11308
$$517$$ −8.00000 −0.351840
$$518$$ 0 0
$$519$$ −48.0000 −2.10697
$$520$$ 0 0
$$521$$ 7.00000 0.306676 0.153338 0.988174i $$-0.450998\pi$$
0.153338 + 0.988174i $$0.450998\pi$$
$$522$$ 0 0
$$523$$ 32.0000 1.39926 0.699631 0.714504i $$-0.253348\pi$$
0.699631 + 0.714504i $$0.253348\pi$$
$$524$$ −36.0000 −1.57267
$$525$$ −12.0000 −0.523723
$$526$$ 0 0
$$527$$ 2.00000 0.0871214
$$528$$ 12.0000 0.522233
$$529$$ 2.00000 0.0869565
$$530$$ 0 0
$$531$$ 18.0000 0.781133
$$532$$ −12.0000 −0.520266
$$533$$ 8.00000 0.346518
$$534$$ 0 0
$$535$$ 10.0000 0.432338
$$536$$ 0 0
$$537$$ −3.00000 −0.129460
$$538$$ 0 0
$$539$$ −1.00000 −0.0430730
$$540$$ −18.0000 −0.774597
$$541$$ 32.0000 1.37579 0.687894 0.725811i $$-0.258536\pi$$
0.687894 + 0.725811i $$0.258536\pi$$
$$542$$ 0 0
$$543$$ −15.0000 −0.643712
$$544$$ 0 0
$$545$$ −4.00000 −0.171341
$$546$$ 0 0
$$547$$ −24.0000 −1.02617 −0.513083 0.858339i $$-0.671497\pi$$
−0.513083 + 0.858339i $$0.671497\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −12.0000 −0.512148
$$550$$ 0 0
$$551$$ −60.0000 −2.55609
$$552$$ 0 0
$$553$$ −6.00000 −0.255146
$$554$$ 0 0
$$555$$ −15.0000 −0.636715
$$556$$ 20.0000 0.848189
$$557$$ 14.0000 0.593199 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$558$$ 0 0
$$559$$ 32.0000 1.35346
$$560$$ 4.00000 0.169031
$$561$$ 6.00000 0.253320
$$562$$ 0 0
$$563$$ 20.0000 0.842900 0.421450 0.906852i $$-0.361521\pi$$
0.421450 + 0.906852i $$0.361521\pi$$
$$564$$ 48.0000 2.02116
$$565$$ 19.0000 0.799336
$$566$$ 0 0
$$567$$ −9.00000 −0.377964
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ −15.0000 −0.626634
$$574$$ 0 0
$$575$$ 20.0000 0.834058
$$576$$ −48.0000 −2.00000
$$577$$ −25.0000 −1.04076 −0.520382 0.853934i $$-0.674210\pi$$
−0.520382 + 0.853934i $$0.674210\pi$$
$$578$$ 0 0
$$579$$ −42.0000 −1.74546
$$580$$ 20.0000 0.830455
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ 6.00000 0.248495
$$584$$ 0 0
$$585$$ 24.0000 0.992278
$$586$$ 0 0
$$587$$ 36.0000 1.48588 0.742940 0.669359i $$-0.233431\pi$$
0.742940 + 0.669359i $$0.233431\pi$$
$$588$$ 6.00000 0.247436
$$589$$ −6.00000 −0.247226
$$590$$ 0 0
$$591$$ −54.0000 −2.22126
$$592$$ −20.0000 −0.821995
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 0 0
$$595$$ 2.00000 0.0819920
$$596$$ 44.0000 1.80231
$$597$$ 24.0000 0.982255
$$598$$ 0 0
$$599$$ −48.0000 −1.96123 −0.980613 0.195952i $$-0.937220\pi$$
−0.980613 + 0.195952i $$0.937220\pi$$
$$600$$ 0 0
$$601$$ 8.00000 0.326327 0.163163 0.986599i $$-0.447830\pi$$
0.163163 + 0.986599i $$0.447830\pi$$
$$602$$ 0 0
$$603$$ −18.0000 −0.733017
$$604$$ −12.0000 −0.488273
$$605$$ −1.00000 −0.0406558
$$606$$ 0 0
$$607$$ −10.0000 −0.405887 −0.202944 0.979190i $$-0.565051\pi$$
−0.202944 + 0.979190i $$0.565051\pi$$
$$608$$ 0 0
$$609$$ 30.0000 1.21566
$$610$$ 0 0
$$611$$ −32.0000 −1.29458
$$612$$ −24.0000 −0.970143
$$613$$ 16.0000 0.646234 0.323117 0.946359i $$-0.395269\pi$$
0.323117 + 0.946359i $$0.395269\pi$$
$$614$$ 0 0
$$615$$ −6.00000 −0.241943
$$616$$ 0 0
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 0 0
$$619$$ 17.0000 0.683288 0.341644 0.939829i $$-0.389016\pi$$
0.341644 + 0.939829i $$0.389016\pi$$
$$620$$ 2.00000 0.0803219
$$621$$ 45.0000 1.80579
$$622$$ 0 0
$$623$$ 15.0000 0.600962
$$624$$ 48.0000 1.92154
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ −18.0000 −0.718851
$$628$$ −14.0000 −0.558661
$$629$$ −10.0000 −0.398726
$$630$$ 0 0
$$631$$ 27.0000 1.07485 0.537427 0.843311i $$-0.319397\pi$$
0.537427 + 0.843311i $$0.319397\pi$$
$$632$$ 0 0
$$633$$ 6.00000 0.238479
$$634$$ 0 0
$$635$$ −2.00000 −0.0793676
$$636$$ −36.0000 −1.42749
$$637$$ −4.00000 −0.158486
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 15.0000 0.592464 0.296232 0.955116i $$-0.404270\pi$$
0.296232 + 0.955116i $$0.404270\pi$$
$$642$$ 0 0
$$643$$ −29.0000 −1.14365 −0.571824 0.820376i $$-0.693764\pi$$
−0.571824 + 0.820376i $$0.693764\pi$$
$$644$$ −10.0000 −0.394055
$$645$$ −24.0000 −0.944999
$$646$$ 0 0
$$647$$ −21.0000 −0.825595 −0.412798 0.910823i $$-0.635448\pi$$
−0.412798 + 0.910823i $$0.635448\pi$$
$$648$$ 0 0
$$649$$ −3.00000 −0.117760
$$650$$ 0 0
$$651$$ 3.00000 0.117579
$$652$$ −8.00000 −0.313304
$$653$$ −17.0000 −0.665261 −0.332631 0.943057i $$-0.607936\pi$$
−0.332631 + 0.943057i $$0.607936\pi$$
$$654$$ 0 0
$$655$$ −18.0000 −0.703318
$$656$$ −8.00000 −0.312348
$$657$$ 60.0000 2.34082
$$658$$ 0 0
$$659$$ −2.00000 −0.0779089 −0.0389545 0.999241i $$-0.512403\pi$$
−0.0389545 + 0.999241i $$0.512403\pi$$
$$660$$ 6.00000 0.233550
$$661$$ 35.0000 1.36134 0.680671 0.732589i $$-0.261688\pi$$
0.680671 + 0.732589i $$0.261688\pi$$
$$662$$ 0 0
$$663$$ 24.0000 0.932083
$$664$$ 0 0
$$665$$ −6.00000 −0.232670
$$666$$ 0 0
$$667$$ −50.0000 −1.93601
$$668$$ 4.00000 0.154765
$$669$$ −3.00000 −0.115987
$$670$$ 0 0
$$671$$ 2.00000 0.0772091
$$672$$ 0 0
$$673$$ 4.00000 0.154189 0.0770943 0.997024i $$-0.475436\pi$$
0.0770943 + 0.997024i $$0.475436\pi$$
$$674$$ 0 0
$$675$$ 36.0000 1.38564
$$676$$ −6.00000 −0.230769
$$677$$ 38.0000 1.46046 0.730229 0.683202i $$-0.239413\pi$$
0.730229 + 0.683202i $$0.239413\pi$$
$$678$$ 0 0
$$679$$ 5.00000 0.191882
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 72.0000 2.75299
$$685$$ 3.00000 0.114624
$$686$$ 0 0
$$687$$ 21.0000 0.801200
$$688$$ −32.0000 −1.21999
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ 15.0000 0.570627 0.285313 0.958434i $$-0.407902\pi$$
0.285313 + 0.958434i $$0.407902\pi$$
$$692$$ −32.0000 −1.21646
$$693$$ 6.00000 0.227921
$$694$$ 0 0
$$695$$ 10.0000 0.379322
$$696$$ 0 0
$$697$$ −4.00000 −0.151511
$$698$$ 0 0
$$699$$ −18.0000 −0.680823
$$700$$ −8.00000 −0.302372
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 30.0000 1.13147
$$704$$ 8.00000 0.301511
$$705$$ 24.0000 0.903892
$$706$$ 0 0
$$707$$ 12.0000 0.451306
$$708$$ 18.0000 0.676481
$$709$$ 39.0000 1.46468 0.732338 0.680941i $$-0.238429\pi$$
0.732338 + 0.680941i $$0.238429\pi$$
$$710$$ 0 0
$$711$$ 36.0000 1.35011
$$712$$ 0 0
$$713$$ −5.00000 −0.187251
$$714$$ 0 0
$$715$$ −4.00000 −0.149592
$$716$$ −2.00000 −0.0747435
$$717$$ −12.0000 −0.448148
$$718$$ 0 0
$$719$$ −11.0000 −0.410231 −0.205115 0.978738i $$-0.565757\pi$$
−0.205115 + 0.978738i $$0.565757\pi$$
$$720$$ −24.0000 −0.894427
$$721$$ 12.0000 0.446903
$$722$$ 0 0
$$723$$ 36.0000 1.33885
$$724$$ −10.0000 −0.371647
$$725$$ −40.0000 −1.48556
$$726$$ 0 0
$$727$$ −19.0000 −0.704671 −0.352335 0.935874i $$-0.614612\pi$$
−0.352335 + 0.935874i $$0.614612\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −16.0000 −0.591781
$$732$$ −12.0000 −0.443533
$$733$$ −4.00000 −0.147743 −0.0738717 0.997268i $$-0.523536\pi$$
−0.0738717 + 0.997268i $$0.523536\pi$$
$$734$$ 0 0
$$735$$ 3.00000 0.110657
$$736$$ 0 0
$$737$$ 3.00000 0.110506
$$738$$ 0 0
$$739$$ −18.0000 −0.662141 −0.331070 0.943606i $$-0.607410\pi$$
−0.331070 + 0.943606i $$0.607410\pi$$
$$740$$ −10.0000 −0.367607
$$741$$ −72.0000 −2.64499
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 22.0000 0.806018
$$746$$ 0 0
$$747$$ 72.0000 2.63434
$$748$$ 4.00000 0.146254
$$749$$ 10.0000 0.365392
$$750$$ 0 0
$$751$$ −23.0000 −0.839282 −0.419641 0.907690i $$-0.637844\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$752$$ 32.0000 1.16692
$$753$$ 63.0000 2.29585
$$754$$ 0 0
$$755$$ −6.00000 −0.218362
$$756$$ −18.0000 −0.654654
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 0 0
$$759$$ −15.0000 −0.544466
$$760$$ 0 0
$$761$$ −48.0000 −1.74000 −0.869999 0.493053i $$-0.835881\pi$$
−0.869999 + 0.493053i $$0.835881\pi$$
$$762$$ 0 0
$$763$$ −4.00000 −0.144810
$$764$$ −10.0000 −0.361787
$$765$$ −12.0000 −0.433861
$$766$$ 0 0
$$767$$ −12.0000 −0.433295
$$768$$ −48.0000 −1.73205
$$769$$ 40.0000 1.44244 0.721218 0.692708i $$-0.243582\pi$$
0.721218 + 0.692708i $$0.243582\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ −28.0000 −1.00774
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ −15.0000 −0.538122
$$778$$ 0 0
$$779$$ 12.0000 0.429945
$$780$$ 24.0000 0.859338
$$781$$ −1.00000 −0.0357828
$$782$$ 0 0
$$783$$ −90.0000 −3.21634
$$784$$ 4.00000 0.142857
$$785$$ −7.00000 −0.249841
$$786$$ 0 0
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ −36.0000 −1.28245
$$789$$ −54.0000 −1.92245
$$790$$ 0 0
$$791$$ 19.0000 0.675562
$$792$$ 0 0
$$793$$ 8.00000 0.284088
$$794$$ 0 0
$$795$$ −18.0000 −0.638394
$$796$$ 16.0000 0.567105
$$797$$ 23.0000 0.814702 0.407351 0.913272i $$-0.366453\pi$$
0.407351 + 0.913272i $$0.366453\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ −90.0000 −3.17999
$$802$$ 0 0
$$803$$ −10.0000 −0.352892
$$804$$ −18.0000 −0.634811
$$805$$ −5.00000 −0.176227
$$806$$ 0 0
$$807$$ 54.0000 1.90089
$$808$$ 0 0
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ −22.0000 −0.772524 −0.386262 0.922389i $$-0.626234\pi$$
−0.386262 + 0.922389i $$0.626234\pi$$
$$812$$ 20.0000 0.701862
$$813$$ −48.0000 −1.68343
$$814$$ 0 0
$$815$$ −4.00000 −0.140114
$$816$$ −24.0000 −0.840168
$$817$$ 48.0000 1.67931
$$818$$ 0 0
$$819$$ 24.0000 0.838628
$$820$$ −4.00000 −0.139686
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ −25.0000 −0.871445 −0.435723 0.900081i $$-0.643507\pi$$
−0.435723 + 0.900081i $$0.643507\pi$$
$$824$$ 0 0
$$825$$ −12.0000 −0.417786
$$826$$ 0 0
$$827$$ 20.0000 0.695468 0.347734 0.937593i $$-0.386951\pi$$
0.347734 + 0.937593i $$0.386951\pi$$
$$828$$ 60.0000 2.08514
$$829$$ −29.0000 −1.00721 −0.503606 0.863934i $$-0.667994\pi$$
−0.503606 + 0.863934i $$0.667994\pi$$
$$830$$ 0 0
$$831$$ −72.0000 −2.49765
$$832$$ 32.0000 1.10940
$$833$$ 2.00000 0.0692959
$$834$$ 0 0
$$835$$ 2.00000 0.0692129
$$836$$ −12.0000 −0.415029
$$837$$ −9.00000 −0.311086
$$838$$ 0 0
$$839$$ 45.0000 1.55357 0.776786 0.629764i $$-0.216849\pi$$
0.776786 + 0.629764i $$0.216849\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 0 0
$$843$$ 12.0000 0.413302
$$844$$ 4.00000 0.137686
$$845$$ −3.00000 −0.103203
$$846$$ 0 0
$$847$$ −1.00000 −0.0343604
$$848$$ −24.0000 −0.824163
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 25.0000 0.856989
$$852$$ 6.00000 0.205557
$$853$$ −34.0000 −1.16414 −0.582069 0.813139i $$-0.697757\pi$$
−0.582069 + 0.813139i $$0.697757\pi$$
$$854$$ 0 0
$$855$$ 36.0000 1.23117
$$856$$ 0 0
$$857$$ −28.0000 −0.956462 −0.478231 0.878234i $$-0.658722\pi$$
−0.478231 + 0.878234i $$0.658722\pi$$
$$858$$ 0 0
$$859$$ 55.0000 1.87658 0.938288 0.345855i $$-0.112411\pi$$
0.938288 + 0.345855i $$0.112411\pi$$
$$860$$ −16.0000 −0.545595
$$861$$ −6.00000 −0.204479
$$862$$ 0 0
$$863$$ 52.0000 1.77010 0.885050 0.465495i $$-0.154124\pi$$
0.885050 + 0.465495i $$0.154124\pi$$
$$864$$ 0 0
$$865$$ −16.0000 −0.544016
$$866$$ 0 0
$$867$$ 39.0000 1.32451
$$868$$ 2.00000 0.0678844
$$869$$ −6.00000 −0.203536
$$870$$ 0 0
$$871$$ 12.0000 0.406604
$$872$$ 0 0
$$873$$ −30.0000 −1.01535
$$874$$ 0 0
$$875$$ −9.00000 −0.304256
$$876$$ 60.0000 2.02721
$$877$$ −38.0000 −1.28317 −0.641584 0.767052i $$-0.721723\pi$$
−0.641584 + 0.767052i $$0.721723\pi$$
$$878$$ 0 0
$$879$$ 18.0000 0.607125
$$880$$ 4.00000 0.134840
$$881$$ 27.0000 0.909653 0.454827 0.890580i $$-0.349701\pi$$
0.454827 + 0.890580i $$0.349701\pi$$
$$882$$ 0 0
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ 16.0000 0.538138
$$885$$ 9.00000 0.302532
$$886$$ 0 0
$$887$$ −2.00000 −0.0671534 −0.0335767 0.999436i $$-0.510690\pi$$
−0.0335767 + 0.999436i $$0.510690\pi$$
$$888$$ 0 0
$$889$$ −2.00000 −0.0670778
$$890$$ 0 0
$$891$$ −9.00000 −0.301511
$$892$$ −2.00000 −0.0669650
$$893$$ −48.0000 −1.60626
$$894$$ 0 0
$$895$$ −1.00000 −0.0334263
$$896$$ 0 0
$$897$$ −60.0000 −2.00334
$$898$$ 0 0
$$899$$ 10.0000 0.333519
$$900$$ 48.0000 1.60000
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ −24.0000 −0.798670
$$904$$ 0 0
$$905$$ −5.00000 −0.166206
$$906$$ 0 0
$$907$$ −40.0000 −1.32818 −0.664089 0.747653i $$-0.731180\pi$$
−0.664089 + 0.747653i $$0.731180\pi$$
$$908$$ −8.00000 −0.265489
$$909$$ −72.0000 −2.38809
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 72.0000 2.38416
$$913$$ −12.0000 −0.397142
$$914$$ 0 0
$$915$$ −6.00000 −0.198354
$$916$$ 14.0000 0.462573
$$917$$ −18.0000 −0.594412
$$918$$ 0 0
$$919$$ 48.0000 1.58337 0.791687 0.610927i $$-0.209203\pi$$
0.791687 + 0.610927i $$0.209203\pi$$
$$920$$ 0 0
$$921$$ 84.0000 2.76789
$$922$$ 0 0
$$923$$ −4.00000 −0.131662
$$924$$ 6.00000 0.197386
$$925$$ 20.0000 0.657596
$$926$$ 0 0
$$927$$ −72.0000 −2.36479
$$928$$ 0 0
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ −12.0000 −0.393073
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ 2.00000 0.0654070
$$936$$ 0 0
$$937$$ 36.0000 1.17607 0.588034 0.808836i $$-0.299902\pi$$
0.588034 + 0.808836i $$0.299902\pi$$
$$938$$ 0 0
$$939$$ 69.0000 2.25173
$$940$$ 16.0000 0.521862
$$941$$ 58.0000 1.89075 0.945373 0.325991i $$-0.105698\pi$$
0.945373 + 0.325991i $$0.105698\pi$$
$$942$$ 0 0
$$943$$ 10.0000 0.325645
$$944$$ 12.0000 0.390567
$$945$$ −9.00000 −0.292770
$$946$$ 0 0
$$947$$ 5.00000 0.162478 0.0812391 0.996695i $$-0.474112\pi$$
0.0812391 + 0.996695i $$0.474112\pi$$
$$948$$ 36.0000 1.16923
$$949$$ −40.0000 −1.29845
$$950$$ 0 0
$$951$$ −27.0000 −0.875535
$$952$$ 0 0
$$953$$ 44.0000 1.42530 0.712650 0.701520i $$-0.247495\pi$$
0.712650 + 0.701520i $$0.247495\pi$$
$$954$$ 0 0
$$955$$ −5.00000 −0.161796
$$956$$ −8.00000 −0.258738
$$957$$ 30.0000 0.969762
$$958$$ 0 0
$$959$$ 3.00000 0.0968751
$$960$$ −24.0000 −0.774597
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ −60.0000 −1.93347
$$964$$ 24.0000 0.772988
$$965$$ −14.0000 −0.450676
$$966$$ 0 0
$$967$$ −34.0000 −1.09337 −0.546683 0.837340i $$-0.684110\pi$$
−0.546683 + 0.837340i $$0.684110\pi$$
$$968$$ 0 0
$$969$$ 36.0000 1.15649
$$970$$ 0 0
$$971$$ 29.0000 0.930654 0.465327 0.885139i $$-0.345937\pi$$
0.465327 + 0.885139i $$0.345937\pi$$
$$972$$ 0 0
$$973$$ 10.0000 0.320585
$$974$$ 0 0
$$975$$ −48.0000 −1.53723
$$976$$ −8.00000 −0.256074
$$977$$ −31.0000 −0.991778 −0.495889 0.868386i $$-0.665158\pi$$
−0.495889 + 0.868386i $$0.665158\pi$$
$$978$$ 0 0
$$979$$ 15.0000 0.479402
$$980$$ 2.00000 0.0638877
$$981$$ 24.0000 0.766261
$$982$$ 0 0
$$983$$ −27.0000 −0.861166 −0.430583 0.902551i $$-0.641692\pi$$
−0.430583 + 0.902551i $$0.641692\pi$$
$$984$$ 0 0
$$985$$ −18.0000 −0.573528
$$986$$ 0 0
$$987$$ 24.0000 0.763928
$$988$$ −48.0000 −1.52708
$$989$$ 40.0000 1.27193
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ 51.0000 1.61844
$$994$$ 0 0
$$995$$ 8.00000 0.253617
$$996$$ 72.0000 2.28141
$$997$$ −12.0000 −0.380044 −0.190022 0.981780i $$-0.560856\pi$$
−0.190022 + 0.981780i $$0.560856\pi$$
$$998$$ 0 0
$$999$$ 45.0000 1.42374
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 77.2.a.a.1.1 1
3.2 odd 2 693.2.a.c.1.1 1
4.3 odd 2 1232.2.a.l.1.1 1
5.2 odd 4 1925.2.b.e.1849.2 2
5.3 odd 4 1925.2.b.e.1849.1 2
5.4 even 2 1925.2.a.h.1.1 1
7.2 even 3 539.2.e.f.67.1 2
7.3 odd 6 539.2.e.c.177.1 2
7.4 even 3 539.2.e.f.177.1 2
7.5 odd 6 539.2.e.c.67.1 2
7.6 odd 2 539.2.a.c.1.1 1
8.3 odd 2 4928.2.a.a.1.1 1
8.5 even 2 4928.2.a.bj.1.1 1
11.2 odd 10 847.2.f.h.323.1 4
11.3 even 5 847.2.f.i.372.1 4
11.4 even 5 847.2.f.i.148.1 4
11.5 even 5 847.2.f.i.729.1 4
11.6 odd 10 847.2.f.h.729.1 4
11.7 odd 10 847.2.f.h.148.1 4
11.8 odd 10 847.2.f.h.372.1 4
11.9 even 5 847.2.f.i.323.1 4
11.10 odd 2 847.2.a.b.1.1 1
21.20 even 2 4851.2.a.j.1.1 1
28.27 even 2 8624.2.a.a.1.1 1
33.32 even 2 7623.2.a.j.1.1 1
77.76 even 2 5929.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.a.1.1 1 1.1 even 1 trivial
539.2.a.c.1.1 1 7.6 odd 2
539.2.e.c.67.1 2 7.5 odd 6
539.2.e.c.177.1 2 7.3 odd 6
539.2.e.f.67.1 2 7.2 even 3
539.2.e.f.177.1 2 7.4 even 3
693.2.a.c.1.1 1 3.2 odd 2
847.2.a.b.1.1 1 11.10 odd 2
847.2.f.h.148.1 4 11.7 odd 10
847.2.f.h.323.1 4 11.2 odd 10
847.2.f.h.372.1 4 11.8 odd 10
847.2.f.h.729.1 4 11.6 odd 10
847.2.f.i.148.1 4 11.4 even 5
847.2.f.i.323.1 4 11.9 even 5
847.2.f.i.372.1 4 11.3 even 5
847.2.f.i.729.1 4 11.5 even 5
1232.2.a.l.1.1 1 4.3 odd 2
1925.2.a.h.1.1 1 5.4 even 2
1925.2.b.e.1849.1 2 5.3 odd 4
1925.2.b.e.1849.2 2 5.2 odd 4
4851.2.a.j.1.1 1 21.20 even 2
4928.2.a.a.1.1 1 8.3 odd 2
4928.2.a.bj.1.1 1 8.5 even 2
5929.2.a.f.1.1 1 77.76 even 2
7623.2.a.j.1.1 1 33.32 even 2
8624.2.a.a.1.1 1 28.27 even 2