# Properties

 Label 5175.2.a.e.1.1 Level $5175$ Weight $2$ Character 5175.1 Self dual yes Analytic conductor $41.323$ 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] = [5175,2,Mod(1,5175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5175 = 3^{2} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5175.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: $$41.3225830460$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 575) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{7} +3.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} -5.00000 q^{19} -1.00000 q^{22} +1.00000 q^{23} +1.00000 q^{26} +1.00000 q^{28} +5.00000 q^{29} -2.00000 q^{31} -5.00000 q^{32} +4.00000 q^{37} +5.00000 q^{38} +5.00000 q^{41} +9.00000 q^{43} -1.00000 q^{44} -1.00000 q^{46} -6.00000 q^{47} -6.00000 q^{49} +1.00000 q^{52} +2.00000 q^{53} -3.00000 q^{56} -5.00000 q^{58} -8.00000 q^{59} -8.00000 q^{61} +2.00000 q^{62} +7.00000 q^{64} -8.00000 q^{67} +10.0000 q^{71} +3.00000 q^{73} -4.00000 q^{74} +5.00000 q^{76} -1.00000 q^{77} -3.00000 q^{79} -5.00000 q^{82} +3.00000 q^{83} -9.00000 q^{86} +3.00000 q^{88} -10.0000 q^{89} +1.00000 q^{91} -1.00000 q^{92} +6.00000 q^{94} +2.00000 q^{97} +6.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 3.00000 1.06066
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 5.00000 0.811107
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 5.00000 0.780869 0.390434 0.920631i $$-0.372325\pi$$
0.390434 + 0.920631i $$0.372325\pi$$
$$42$$ 0 0
$$43$$ 9.00000 1.37249 0.686244 0.727372i $$-0.259258\pi$$
0.686244 + 0.727372i $$0.259258\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −3.00000 −0.400892
$$57$$ 0 0
$$58$$ −5.00000 −0.656532
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 2.00000 0.254000
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10.0000 1.18678 0.593391 0.804914i $$-0.297789\pi$$
0.593391 + 0.804914i $$0.297789\pi$$
$$72$$ 0 0
$$73$$ 3.00000 0.351123 0.175562 0.984468i $$-0.443826\pi$$
0.175562 + 0.984468i $$0.443826\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 0 0
$$76$$ 5.00000 0.573539
$$77$$ −1.00000 −0.113961
$$78$$ 0 0
$$79$$ −3.00000 −0.337526 −0.168763 0.985657i $$-0.553977\pi$$
−0.168763 + 0.985657i $$0.553977\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −5.00000 −0.552158
$$83$$ 3.00000 0.329293 0.164646 0.986353i $$-0.447352\pi$$
0.164646 + 0.986353i $$0.447352\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −9.00000 −0.970495
$$87$$ 0 0
$$88$$ 3.00000 0.319801
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ −1.00000 −0.104257
$$93$$ 0 0
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 6.00000 0.606092
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 17.0000 1.67506 0.837530 0.546392i $$-0.183999\pi$$
0.837530 + 0.546392i $$0.183999\pi$$
$$104$$ −3.00000 −0.294174
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −5.00000 −0.464238
$$117$$ 0 0
$$118$$ 8.00000 0.736460
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 8.00000 0.724286
$$123$$ 0 0
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 3.00000 0.265165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 5.00000 0.433555
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −10.0000 −0.839181
$$143$$ −1.00000 −0.0836242
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −3.00000 −0.248282
$$147$$ 0 0
$$148$$ −4.00000 −0.328798
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ 6.00000 0.488273 0.244137 0.969741i $$-0.421495\pi$$
0.244137 + 0.969741i $$0.421495\pi$$
$$152$$ −15.0000 −1.21666
$$153$$ 0 0
$$154$$ 1.00000 0.0805823
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ 3.00000 0.238667
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −1.00000 −0.0788110
$$162$$ 0 0
$$163$$ 6.00000 0.469956 0.234978 0.972001i $$-0.424498\pi$$
0.234978 + 0.972001i $$0.424498\pi$$
$$164$$ −5.00000 −0.390434
$$165$$ 0 0
$$166$$ −3.00000 −0.232845
$$167$$ 18.0000 1.39288 0.696441 0.717614i $$-0.254766\pi$$
0.696441 + 0.717614i $$0.254766\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −9.00000 −0.686244
$$173$$ 15.0000 1.14043 0.570214 0.821496i $$-0.306860\pi$$
0.570214 + 0.821496i $$0.306860\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ −1.00000 −0.0741249
$$183$$ 0 0
$$184$$ 3.00000 0.221163
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 6.00000 0.437595
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −15.0000 −1.08536 −0.542681 0.839939i $$-0.682591\pi$$
−0.542681 + 0.839939i $$0.682591\pi$$
$$192$$ 0 0
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 6.00000 0.428571
$$197$$ −17.0000 −1.21120 −0.605600 0.795769i $$-0.707067\pi$$
−0.605600 + 0.795769i $$0.707067\pi$$
$$198$$ 0 0
$$199$$ −7.00000 −0.496217 −0.248108 0.968732i $$-0.579809\pi$$
−0.248108 + 0.968732i $$0.579809\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 10.0000 0.703598
$$203$$ −5.00000 −0.350931
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −17.0000 −1.18445
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ −5.00000 −0.345857
$$210$$ 0 0
$$211$$ −6.00000 −0.413057 −0.206529 0.978441i $$-0.566217\pi$$
−0.206529 + 0.978441i $$0.566217\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 2.00000 0.135769
$$218$$ 4.00000 0.270914
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 5.00000 0.334077
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 15.0000 0.984798
$$233$$ −5.00000 −0.327561 −0.163780 0.986497i $$-0.552369\pi$$
−0.163780 + 0.986497i $$0.552369\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −10.0000 −0.646846 −0.323423 0.946254i $$-0.604834\pi$$
−0.323423 + 0.946254i $$0.604834\pi$$
$$240$$ 0 0
$$241$$ −30.0000 −1.93247 −0.966235 0.257663i $$-0.917048\pi$$
−0.966235 + 0.257663i $$0.917048\pi$$
$$242$$ 10.0000 0.642824
$$243$$ 0 0
$$244$$ 8.00000 0.512148
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.00000 0.318142
$$248$$ −6.00000 −0.381000
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 8.00000 0.504956 0.252478 0.967603i $$-0.418755\pi$$
0.252478 + 0.967603i $$0.418755\pi$$
$$252$$ 0 0
$$253$$ 1.00000 0.0628695
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 0 0
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −6.00000 −0.370681
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −5.00000 −0.306570
$$267$$ 0 0
$$268$$ 8.00000 0.488678
$$269$$ −3.00000 −0.182913 −0.0914566 0.995809i $$-0.529152\pi$$
−0.0914566 + 0.995809i $$0.529152\pi$$
$$270$$ 0 0
$$271$$ −14.0000 −0.850439 −0.425220 0.905090i $$-0.639803\pi$$
−0.425220 + 0.905090i $$0.639803\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 33.0000 1.98278 0.991389 0.130950i $$-0.0418029\pi$$
0.991389 + 0.130950i $$0.0418029\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 0 0
$$283$$ −32.0000 −1.90220 −0.951101 0.308879i $$-0.900046\pi$$
−0.951101 + 0.308879i $$0.900046\pi$$
$$284$$ −10.0000 −0.593391
$$285$$ 0 0
$$286$$ 1.00000 0.0591312
$$287$$ −5.00000 −0.295141
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −3.00000 −0.175562
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 12.0000 0.697486
$$297$$ 0 0
$$298$$ −2.00000 −0.115857
$$299$$ −1.00000 −0.0578315
$$300$$ 0 0
$$301$$ −9.00000 −0.518751
$$302$$ −6.00000 −0.345261
$$303$$ 0 0
$$304$$ 5.00000 0.286770
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 1.00000 0.0569803
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −28.0000 −1.58773 −0.793867 0.608091i $$-0.791935\pi$$
−0.793867 + 0.608091i $$0.791935\pi$$
$$312$$ 0 0
$$313$$ 12.0000 0.678280 0.339140 0.940736i $$-0.389864\pi$$
0.339140 + 0.940736i $$0.389864\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 3.00000 0.168763
$$317$$ −33.0000 −1.85346 −0.926732 0.375722i $$-0.877395\pi$$
−0.926732 + 0.375722i $$0.877395\pi$$
$$318$$ 0 0
$$319$$ 5.00000 0.279946
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 1.00000 0.0557278
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −6.00000 −0.332309
$$327$$ 0 0
$$328$$ 15.0000 0.828236
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ 14.0000 0.769510 0.384755 0.923019i $$-0.374286\pi$$
0.384755 + 0.923019i $$0.374286\pi$$
$$332$$ −3.00000 −0.164646
$$333$$ 0 0
$$334$$ −18.0000 −0.984916
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 12.0000 0.652714
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2.00000 −0.108306
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ 27.0000 1.45574
$$345$$ 0 0
$$346$$ −15.0000 −0.806405
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ 0 0
$$349$$ −25.0000 −1.33822 −0.669110 0.743164i $$-0.733324\pi$$
−0.669110 + 0.743164i $$0.733324\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −5.00000 −0.266501
$$353$$ −11.0000 −0.585471 −0.292735 0.956193i $$-0.594566\pi$$
−0.292735 + 0.956193i $$0.594566\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ 6.00000 0.317110
$$359$$ −33.0000 −1.74167 −0.870837 0.491572i $$-0.836422\pi$$
−0.870837 + 0.491572i $$0.836422\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 20.0000 1.05118
$$363$$ 0 0
$$364$$ −1.00000 −0.0524142
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −17.0000 −0.887393 −0.443696 0.896177i $$-0.646333\pi$$
−0.443696 + 0.896177i $$0.646333\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2.00000 −0.103835
$$372$$ 0 0
$$373$$ −20.0000 −1.03556 −0.517780 0.855514i $$-0.673242\pi$$
−0.517780 + 0.855514i $$0.673242\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −18.0000 −0.928279
$$377$$ −5.00000 −0.257513
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 15.0000 0.767467
$$383$$ −19.0000 −0.970855 −0.485427 0.874277i $$-0.661336\pi$$
−0.485427 + 0.874277i $$0.661336\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ 0 0
$$388$$ −2.00000 −0.101535
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −18.0000 −0.909137
$$393$$ 0 0
$$394$$ 17.0000 0.856448
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ 7.00000 0.350878
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −32.0000 −1.59800 −0.799002 0.601329i $$-0.794638\pi$$
−0.799002 + 0.601329i $$0.794638\pi$$
$$402$$ 0 0
$$403$$ 2.00000 0.0996271
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ 5.00000 0.248146
$$407$$ 4.00000 0.198273
$$408$$ 0 0
$$409$$ 1.00000 0.0494468 0.0247234 0.999694i $$-0.492129\pi$$
0.0247234 + 0.999694i $$0.492129\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −17.0000 −0.837530
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 5.00000 0.245145
$$417$$ 0 0
$$418$$ 5.00000 0.244558
$$419$$ −9.00000 −0.439679 −0.219839 0.975536i $$-0.570553\pi$$
−0.219839 + 0.975536i $$0.570553\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 6.00000 0.292075
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 8.00000 0.387147
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ −32.0000 −1.53782 −0.768911 0.639356i $$-0.779201\pi$$
−0.768911 + 0.639356i $$0.779201\pi$$
$$434$$ −2.00000 −0.0960031
$$435$$ 0 0
$$436$$ 4.00000 0.191565
$$437$$ −5.00000 −0.239182
$$438$$ 0 0
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −14.0000 −0.665160 −0.332580 0.943075i $$-0.607919\pi$$
−0.332580 + 0.943075i $$0.607919\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 2.00000 0.0947027
$$447$$ 0 0
$$448$$ −7.00000 −0.330719
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 5.00000 0.235441
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.0000 1.77757 0.888783 0.458329i $$-0.151552\pi$$
0.888783 + 0.458329i $$0.151552\pi$$
$$458$$ 22.0000 1.02799
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −7.00000 −0.326023 −0.163011 0.986624i $$-0.552121\pi$$
−0.163011 + 0.986624i $$0.552121\pi$$
$$462$$ 0 0
$$463$$ 28.0000 1.30127 0.650635 0.759390i $$-0.274503\pi$$
0.650635 + 0.759390i $$0.274503\pi$$
$$464$$ −5.00000 −0.232119
$$465$$ 0 0
$$466$$ 5.00000 0.231621
$$467$$ 33.0000 1.52706 0.763529 0.645774i $$-0.223465\pi$$
0.763529 + 0.645774i $$0.223465\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −24.0000 −1.10469
$$473$$ 9.00000 0.413820
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 10.0000 0.457389
$$479$$ 9.00000 0.411220 0.205610 0.978634i $$-0.434082\pi$$
0.205610 + 0.978634i $$0.434082\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 30.0000 1.36646
$$483$$ 0 0
$$484$$ 10.0000 0.454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −14.0000 −0.634401 −0.317200 0.948359i $$-0.602743\pi$$
−0.317200 + 0.948359i $$0.602743\pi$$
$$488$$ −24.0000 −1.08643
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −24.0000 −1.08310 −0.541552 0.840667i $$-0.682163\pi$$
−0.541552 + 0.840667i $$0.682163\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −5.00000 −0.224961
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ −10.0000 −0.448561
$$498$$ 0 0
$$499$$ 10.0000 0.447661 0.223831 0.974628i $$-0.428144\pi$$
0.223831 + 0.974628i $$0.428144\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −8.00000 −0.357057
$$503$$ −21.0000 −0.936344 −0.468172 0.883637i $$-0.655087\pi$$
−0.468172 + 0.883637i $$0.655087\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −1.00000 −0.0444554
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ −3.00000 −0.132712
$$512$$ 11.0000 0.486136
$$513$$ 0 0
$$514$$ 2.00000 0.0882162
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −6.00000 −0.263880
$$518$$ 4.00000 0.175750
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ −5.00000 −0.218635 −0.109317 0.994007i $$-0.534866\pi$$
−0.109317 + 0.994007i $$0.534866\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −5.00000 −0.216777
$$533$$ −5.00000 −0.216574
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −24.0000 −1.03664
$$537$$ 0 0
$$538$$ 3.00000 0.129339
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ 31.0000 1.33279 0.666397 0.745597i $$-0.267836\pi$$
0.666397 + 0.745597i $$0.267836\pi$$
$$542$$ 14.0000 0.601351
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 18.0000 0.769624 0.384812 0.922995i $$-0.374266\pi$$
0.384812 + 0.922995i $$0.374266\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −25.0000 −1.06504
$$552$$ 0 0
$$553$$ 3.00000 0.127573
$$554$$ −33.0000 −1.40204
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ −16.0000 −0.677942 −0.338971 0.940797i $$-0.610079\pi$$
−0.338971 + 0.940797i $$0.610079\pi$$
$$558$$ 0 0
$$559$$ −9.00000 −0.380659
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −30.0000 −1.26547
$$563$$ −41.0000 −1.72794 −0.863972 0.503540i $$-0.832031\pi$$
−0.863972 + 0.503540i $$0.832031\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 32.0000 1.34506
$$567$$ 0 0
$$568$$ 30.0000 1.25877
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 1.00000 0.0418121
$$573$$ 0 0
$$574$$ 5.00000 0.208696
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 31.0000 1.29055 0.645273 0.763952i $$-0.276743\pi$$
0.645273 + 0.763952i $$0.276743\pi$$
$$578$$ 17.0000 0.707107
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3.00000 −0.124461
$$582$$ 0 0
$$583$$ 2.00000 0.0828315
$$584$$ 9.00000 0.372423
$$585$$ 0 0
$$586$$ −24.0000 −0.991431
$$587$$ 8.00000 0.330195 0.165098 0.986277i $$-0.447206\pi$$
0.165098 + 0.986277i $$0.447206\pi$$
$$588$$ 0 0
$$589$$ 10.0000 0.412043
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −4.00000 −0.164399
$$593$$ −25.0000 −1.02663 −0.513313 0.858201i $$-0.671582\pi$$
−0.513313 + 0.858201i $$0.671582\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −2.00000 −0.0819232
$$597$$ 0 0
$$598$$ 1.00000 0.0408930
$$599$$ −30.0000 −1.22577 −0.612883 0.790173i $$-0.709990\pi$$
−0.612883 + 0.790173i $$0.709990\pi$$
$$600$$ 0 0
$$601$$ −30.0000 −1.22373 −0.611863 0.790964i $$-0.709580\pi$$
−0.611863 + 0.790964i $$0.709580\pi$$
$$602$$ 9.00000 0.366813
$$603$$ 0 0
$$604$$ −6.00000 −0.244137
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −2.00000 −0.0811775 −0.0405887 0.999176i $$-0.512923\pi$$
−0.0405887 + 0.999176i $$0.512923\pi$$
$$608$$ 25.0000 1.01388
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 6.00000 0.242734
$$612$$ 0 0
$$613$$ −28.0000 −1.13091 −0.565455 0.824779i $$-0.691299\pi$$
−0.565455 + 0.824779i $$0.691299\pi$$
$$614$$ 16.0000 0.645707
$$615$$ 0 0
$$616$$ −3.00000 −0.120873
$$617$$ 28.0000 1.12724 0.563619 0.826035i $$-0.309409\pi$$
0.563619 + 0.826035i $$0.309409\pi$$
$$618$$ 0 0
$$619$$ 36.0000 1.44696 0.723481 0.690344i $$-0.242541\pi$$
0.723481 + 0.690344i $$0.242541\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 28.0000 1.12270
$$623$$ 10.0000 0.400642
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −12.0000 −0.479616
$$627$$ 0 0
$$628$$ 14.0000 0.558661
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 7.00000 0.278666 0.139333 0.990246i $$-0.455504\pi$$
0.139333 + 0.990246i $$0.455504\pi$$
$$632$$ −9.00000 −0.358001
$$633$$ 0 0
$$634$$ 33.0000 1.31060
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6.00000 0.237729
$$638$$ −5.00000 −0.197952
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 16.0000 0.631962 0.315981 0.948766i $$-0.397666\pi$$
0.315981 + 0.948766i $$0.397666\pi$$
$$642$$ 0 0
$$643$$ −31.0000 −1.22252 −0.611260 0.791430i $$-0.709337\pi$$
−0.611260 + 0.791430i $$0.709337\pi$$
$$644$$ 1.00000 0.0394055
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −38.0000 −1.49393 −0.746967 0.664861i $$-0.768491\pi$$
−0.746967 + 0.664861i $$0.768491\pi$$
$$648$$ 0 0
$$649$$ −8.00000 −0.314027
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −6.00000 −0.234978
$$653$$ −27.0000 −1.05659 −0.528296 0.849060i $$-0.677169\pi$$
−0.528296 + 0.849060i $$0.677169\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −5.00000 −0.195217
$$657$$ 0 0
$$658$$ −6.00000 −0.233904
$$659$$ 3.00000 0.116863 0.0584317 0.998291i $$-0.481390\pi$$
0.0584317 + 0.998291i $$0.481390\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ −14.0000 −0.544125
$$663$$ 0 0
$$664$$ 9.00000 0.349268
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 5.00000 0.193601
$$668$$ −18.0000 −0.696441
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ −19.0000 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ 12.0000 0.461538
$$677$$ −12.0000 −0.461197 −0.230599 0.973049i $$-0.574068\pi$$
−0.230599 + 0.973049i $$0.574068\pi$$
$$678$$ 0 0
$$679$$ −2.00000 −0.0767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 2.00000 0.0765840
$$683$$ 2.00000 0.0765279 0.0382639 0.999268i $$-0.487817\pi$$
0.0382639 + 0.999268i $$0.487817\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −13.0000 −0.496342
$$687$$ 0 0
$$688$$ −9.00000 −0.343122
$$689$$ −2.00000 −0.0761939
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ −15.0000 −0.570214
$$693$$ 0 0
$$694$$ 24.0000 0.911028
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 25.0000 0.946264
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −36.0000 −1.35970 −0.679851 0.733351i $$-0.737955\pi$$
−0.679851 + 0.733351i $$0.737955\pi$$
$$702$$ 0 0
$$703$$ −20.0000 −0.754314
$$704$$ 7.00000 0.263822
$$705$$ 0 0
$$706$$ 11.0000 0.413990
$$707$$ 10.0000 0.376089
$$708$$ 0 0
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −30.0000 −1.12430
$$713$$ −2.00000 −0.0749006
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 6.00000 0.224231
$$717$$ 0 0
$$718$$ 33.0000 1.23155
$$719$$ 40.0000 1.49175 0.745874 0.666087i $$-0.232032\pi$$
0.745874 + 0.666087i $$0.232032\pi$$
$$720$$ 0 0
$$721$$ −17.0000 −0.633113
$$722$$ −6.00000 −0.223297
$$723$$ 0 0
$$724$$ 20.0000 0.743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 3.00000 0.111187
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 22.0000 0.812589 0.406294 0.913742i $$-0.366821\pi$$
0.406294 + 0.913742i $$0.366821\pi$$
$$734$$ 17.0000 0.627481
$$735$$ 0 0
$$736$$ −5.00000 −0.184302
$$737$$ −8.00000 −0.294684
$$738$$ 0 0
$$739$$ −22.0000 −0.809283 −0.404642 0.914475i $$-0.632604\pi$$
−0.404642 + 0.914475i $$0.632604\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 2.00000 0.0734223
$$743$$ −11.0000 −0.403551 −0.201775 0.979432i $$-0.564671\pi$$
−0.201775 + 0.979432i $$0.564671\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 20.0000 0.732252
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 23.0000 0.839282 0.419641 0.907690i $$-0.362156\pi$$
0.419641 + 0.907690i $$0.362156\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 0 0
$$754$$ 5.00000 0.182089
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −40.0000 −1.45382 −0.726912 0.686730i $$-0.759045\pi$$
−0.726912 + 0.686730i $$0.759045\pi$$
$$758$$ 12.0000 0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 39.0000 1.41375 0.706874 0.707339i $$-0.250105\pi$$
0.706874 + 0.707339i $$0.250105\pi$$
$$762$$ 0 0
$$763$$ 4.00000 0.144810
$$764$$ 15.0000 0.542681
$$765$$ 0 0
$$766$$ 19.0000 0.686498
$$767$$ 8.00000 0.288863
$$768$$ 0 0
$$769$$ −52.0000 −1.87517 −0.937584 0.347759i $$-0.886943\pi$$
−0.937584 + 0.347759i $$0.886943\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 10.0000 0.359908
$$773$$ 20.0000 0.719350 0.359675 0.933078i $$-0.382888\pi$$
0.359675 + 0.933078i $$0.382888\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 0 0
$$778$$ −14.0000 −0.501924
$$779$$ −25.0000 −0.895718
$$780$$ 0 0
$$781$$ 10.0000 0.357828
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 6.00000 0.214286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 43.0000 1.53278 0.766392 0.642373i $$-0.222050\pi$$
0.766392 + 0.642373i $$0.222050\pi$$
$$788$$ 17.0000 0.605600
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ 8.00000 0.284088
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ 7.00000 0.248108
$$797$$ 8.00000 0.283375 0.141687 0.989911i $$-0.454747\pi$$
0.141687 + 0.989911i $$0.454747\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 32.0000 1.12996
$$803$$ 3.00000 0.105868
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −2.00000 −0.0704470
$$807$$ 0 0
$$808$$ −30.0000 −1.05540
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 5.00000 0.175466
$$813$$ 0 0
$$814$$ −4.00000 −0.140200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −45.0000 −1.57435
$$818$$ −1.00000 −0.0349642
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −1.00000 −0.0349002 −0.0174501 0.999848i $$-0.505555\pi$$
−0.0174501 + 0.999848i $$0.505555\pi$$
$$822$$ 0 0
$$823$$ −20.0000 −0.697156 −0.348578 0.937280i $$-0.613335\pi$$
−0.348578 + 0.937280i $$0.613335\pi$$
$$824$$ 51.0000 1.77667
$$825$$ 0 0
$$826$$ −8.00000 −0.278356
$$827$$ 37.0000 1.28662 0.643308 0.765607i $$-0.277561\pi$$
0.643308 + 0.765607i $$0.277561\pi$$
$$828$$ 0 0
$$829$$ 15.0000 0.520972 0.260486 0.965478i $$-0.416117\pi$$
0.260486 + 0.965478i $$0.416117\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −7.00000 −0.242681
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 5.00000 0.172929
$$837$$ 0 0
$$838$$ 9.00000 0.310900
$$839$$ 9.00000 0.310715 0.155357 0.987858i $$-0.450347\pi$$
0.155357 + 0.987858i $$0.450347\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ −22.0000 −0.758170
$$843$$ 0 0
$$844$$ 6.00000 0.206529
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ −2.00000 −0.0686803
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 4.00000 0.137118
$$852$$ 0 0
$$853$$ −37.0000 −1.26686 −0.633428 0.773802i $$-0.718353\pi$$
−0.633428 + 0.773802i $$0.718353\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 0 0
$$856$$ 36.0000 1.23045
$$857$$ −26.0000 −0.888143 −0.444072 0.895991i $$-0.646466\pi$$
−0.444072 + 0.895991i $$0.646466\pi$$
$$858$$ 0 0
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −8.00000 −0.272481
$$863$$ 6.00000 0.204242 0.102121 0.994772i $$-0.467437\pi$$
0.102121 + 0.994772i $$0.467437\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 32.0000 1.08740
$$867$$ 0 0
$$868$$ −2.00000 −0.0678844
$$869$$ −3.00000 −0.101768
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ −12.0000 −0.406371
$$873$$ 0 0
$$874$$ 5.00000 0.169128
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 38.0000 1.28317 0.641584 0.767052i $$-0.278277\pi$$
0.641584 + 0.767052i $$0.278277\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 20.0000 0.673817 0.336909 0.941537i $$-0.390619\pi$$
0.336909 + 0.941537i $$0.390619\pi$$
$$882$$ 0 0
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 14.0000 0.470339
$$887$$ 28.0000 0.940148 0.470074 0.882627i $$-0.344227\pi$$
0.470074 + 0.882627i $$0.344227\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 2.00000 0.0669650
$$893$$ 30.0000 1.00391
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −3.00000 −0.100223
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ −10.0000 −0.333519
$$900$$ 0 0
$$901$$ 0 0
$$902$$ −5.00000 −0.166482
$$903$$ 0 0
$$904$$ −18.0000 −0.598671
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −17.0000 −0.564476 −0.282238 0.959344i $$-0.591077\pi$$
−0.282238 + 0.959344i $$0.591077\pi$$
$$908$$ 20.0000 0.663723
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 55.0000 1.82223 0.911116 0.412151i $$-0.135222\pi$$
0.911116 + 0.412151i $$0.135222\pi$$
$$912$$ 0 0
$$913$$ 3.00000 0.0992855
$$914$$ −38.0000 −1.25693
$$915$$ 0 0
$$916$$ 22.0000 0.726900
$$917$$ −6.00000 −0.198137
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 7.00000 0.230533
$$923$$ −10.0000 −0.329154
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −28.0000 −0.920137
$$927$$ 0 0
$$928$$ −25.0000 −0.820665
$$929$$ 45.0000 1.47640 0.738201 0.674581i $$-0.235676\pi$$
0.738201 + 0.674581i $$0.235676\pi$$
$$930$$ 0 0
$$931$$ 30.0000 0.983210
$$932$$ 5.00000 0.163780
$$933$$ 0 0
$$934$$ −33.0000 −1.07979
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −26.0000 −0.849383 −0.424691 0.905338i $$-0.639617\pi$$
−0.424691 + 0.905338i $$0.639617\pi$$
$$938$$ −8.00000 −0.261209
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 0 0
$$943$$ 5.00000 0.162822
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ −9.00000 −0.292615
$$947$$ −22.0000 −0.714904 −0.357452 0.933932i $$-0.616354\pi$$
−0.357452 + 0.933932i $$0.616354\pi$$
$$948$$ 0 0
$$949$$ −3.00000 −0.0973841
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −28.0000 −0.907009 −0.453504 0.891254i $$-0.649826\pi$$
−0.453504 + 0.891254i $$0.649826\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 10.0000 0.323423
$$957$$ 0 0
$$958$$ −9.00000 −0.290777
$$959$$ −18.0000 −0.581250
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 4.00000 0.128965
$$963$$ 0 0
$$964$$ 30.0000 0.966235
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 4.00000 0.128631 0.0643157 0.997930i $$-0.479514\pi$$
0.0643157 + 0.997930i $$0.479514\pi$$
$$968$$ −30.0000 −0.964237
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 15.0000 0.481373 0.240686 0.970603i $$-0.422627\pi$$
0.240686 + 0.970603i $$0.422627\pi$$
$$972$$ 0 0
$$973$$ −20.0000 −0.641171
$$974$$ 14.0000 0.448589
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ −10.0000 −0.319601
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 24.0000 0.765871
$$983$$ 39.0000 1.24391 0.621953 0.783054i $$-0.286339\pi$$
0.621953 + 0.783054i $$0.286339\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −5.00000 −0.159071
$$989$$ 9.00000 0.286183
$$990$$ 0 0
$$991$$ 2.00000 0.0635321 0.0317660 0.999495i $$-0.489887\pi$$
0.0317660 + 0.999495i $$0.489887\pi$$
$$992$$ 10.0000 0.317500
$$993$$ 0 0
$$994$$ 10.0000 0.317181
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 37.0000 1.17180 0.585901 0.810383i $$-0.300741\pi$$
0.585901 + 0.810383i $$0.300741\pi$$
$$998$$ −10.0000 −0.316544
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5175.2.a.e.1.1 1
3.2 odd 2 575.2.a.d.1.1 yes 1
5.4 even 2 5175.2.a.u.1.1 1
12.11 even 2 9200.2.a.u.1.1 1
15.2 even 4 575.2.b.b.24.2 2
15.8 even 4 575.2.b.b.24.1 2
15.14 odd 2 575.2.a.c.1.1 1
60.59 even 2 9200.2.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
575.2.a.c.1.1 1 15.14 odd 2
575.2.a.d.1.1 yes 1 3.2 odd 2
575.2.b.b.24.1 2 15.8 even 4
575.2.b.b.24.2 2 15.2 even 4
5175.2.a.e.1.1 1 1.1 even 1 trivial
5175.2.a.u.1.1 1 5.4 even 2
9200.2.a.r.1.1 1 60.59 even 2
9200.2.a.u.1.1 1 12.11 even 2