# Properties

 Label 225.2.a.e.1.1 Level $225$ Weight $2$ Character 225.1 Self dual yes Analytic conductor $1.797$ 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] = [225,2,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, 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("225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.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: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{7} +O(q^{10})$$ $$q+2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{7} -2.00000 q^{11} -1.00000 q^{13} +6.00000 q^{14} -4.00000 q^{16} +2.00000 q^{17} -5.00000 q^{19} -4.00000 q^{22} +6.00000 q^{23} -2.00000 q^{26} +6.00000 q^{28} -10.0000 q^{29} -3.00000 q^{31} -8.00000 q^{32} +4.00000 q^{34} -2.00000 q^{37} -10.0000 q^{38} +8.00000 q^{41} -1.00000 q^{43} -4.00000 q^{44} +12.0000 q^{46} +2.00000 q^{47} +2.00000 q^{49} -2.00000 q^{52} -4.00000 q^{53} -20.0000 q^{58} +10.0000 q^{59} +7.00000 q^{61} -6.00000 q^{62} -8.00000 q^{64} +3.00000 q^{67} +4.00000 q^{68} +8.00000 q^{71} +14.0000 q^{73} -4.00000 q^{74} -10.0000 q^{76} -6.00000 q^{77} +16.0000 q^{82} +6.00000 q^{83} -2.00000 q^{86} -3.00000 q^{91} +12.0000 q^{92} +4.00000 q^{94} -17.0000 q^{97} +4.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ 0 0
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 6.00000 1.60357
$$15$$ 0 0
$$16$$ −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
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 6.00000 1.13389
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ −8.00000 −1.41421
$$33$$ 0 0
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −10.0000 −1.62221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 12.0000 1.76930
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ −4.00000 −0.549442 −0.274721 0.961524i $$-0.588586\pi$$
−0.274721 + 0.961524i $$0.588586\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −20.0000 −2.62613
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ −6.00000 −0.762001
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.00000 0.366508 0.183254 0.983066i $$-0.441337\pi$$
0.183254 + 0.983066i $$0.441337\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 0 0
$$76$$ −10.0000 −1.14708
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 16.0000 1.76690
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 12.0000 1.25109
$$93$$ 0 0
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −17.0000 −1.72609 −0.863044 0.505128i $$-0.831445\pi$$
−0.863044 + 0.505128i $$0.831445\pi$$
$$98$$ 4.00000 0.404061
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −8.00000 −0.777029
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −12.0000 −1.13389
$$113$$ −4.00000 −0.376288 −0.188144 0.982141i $$-0.560247\pi$$
−0.188144 + 0.982141i $$0.560247\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −20.0000 −1.85695
$$117$$ 0 0
$$118$$ 20.0000 1.84115
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 14.0000 1.26750
$$123$$ 0 0
$$124$$ −6.00000 −0.538816
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ −15.0000 −1.30066
$$134$$ 6.00000 0.518321
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 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
$$142$$ 16.0000 1.34269
$$143$$ 2.00000 0.167248
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 28.0000 2.31730
$$147$$ 0 0
$$148$$ −4.00000 −0.328798
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 7.00000 0.569652 0.284826 0.958579i $$-0.408064\pi$$
0.284826 + 0.958579i $$0.408064\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −12.0000 −0.966988
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 13.0000 1.03751 0.518756 0.854922i $$-0.326395\pi$$
0.518756 + 0.854922i $$0.326395\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 18.0000 1.41860
$$162$$ 0 0
$$163$$ −11.0000 −0.861586 −0.430793 0.902451i $$-0.641766\pi$$
−0.430793 + 0.902451i $$0.641766\pi$$
$$164$$ 16.0000 1.24939
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −2.00000 −0.152499
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 8.00000 0.603023
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ 17.0000 1.26360 0.631800 0.775131i $$-0.282316\pi$$
0.631800 + 0.775131i $$0.282316\pi$$
$$182$$ −6.00000 −0.444750
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ 4.00000 0.291730
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −22.0000 −1.59186 −0.795932 0.605386i $$-0.793019\pi$$
−0.795932 + 0.605386i $$0.793019\pi$$
$$192$$ 0 0
$$193$$ −11.0000 −0.791797 −0.395899 0.918294i $$-0.629567\pi$$
−0.395899 + 0.918294i $$0.629567\pi$$
$$194$$ −34.0000 −2.44106
$$195$$ 0 0
$$196$$ 4.00000 0.285714
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ −5.00000 −0.354441 −0.177220 0.984171i $$-0.556711\pi$$
−0.177220 + 0.984171i $$0.556711\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −24.0000 −1.68863
$$203$$ −30.0000 −2.10559
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ 4.00000 0.277350
$$209$$ 10.0000 0.691714
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ −8.00000 −0.549442
$$213$$ 0 0
$$214$$ 24.0000 1.64061
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −9.00000 −0.610960
$$218$$ 10.0000 0.677285
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ 19.0000 1.27233 0.636167 0.771551i $$-0.280519\pi$$
0.636167 + 0.771551i $$0.280519\pi$$
$$224$$ −24.0000 −1.60357
$$225$$ 0 0
$$226$$ −8.00000 −0.532152
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 0 0
$$229$$ −15.0000 −0.991228 −0.495614 0.868543i $$-0.665057\pi$$
−0.495614 + 0.868543i $$0.665057\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 20.0000 1.30189
$$237$$ 0 0
$$238$$ 12.0000 0.777844
$$239$$ −20.0000 −1.29369 −0.646846 0.762620i $$-0.723912\pi$$
−0.646846 + 0.762620i $$0.723912\pi$$
$$240$$ 0 0
$$241$$ −23.0000 −1.48156 −0.740780 0.671748i $$-0.765544\pi$$
−0.740780 + 0.671748i $$0.765544\pi$$
$$242$$ −14.0000 −0.899954
$$243$$ 0 0
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.00000 0.318142
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ −12.0000 −0.754434
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −24.0000 −1.48272
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −30.0000 −1.83942
$$267$$ 0 0
$$268$$ 6.00000 0.366508
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ −8.00000 −0.485071
$$273$$ 0 0
$$274$$ −36.0000 −2.17484
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3.00000 0.180253 0.0901263 0.995930i $$-0.471273\pi$$
0.0901263 + 0.995930i $$0.471273\pi$$
$$278$$ 40.0000 2.39904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ 9.00000 0.534994 0.267497 0.963559i $$-0.413803\pi$$
0.267497 + 0.963559i $$0.413803\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 24.0000 1.41668
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 28.0000 1.63858
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ −20.0000 −1.15857
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 14.0000 0.805609
$$303$$ 0 0
$$304$$ 20.0000 1.14708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −7.00000 −0.399511 −0.199756 0.979846i $$-0.564015\pi$$
−0.199756 + 0.979846i $$0.564015\pi$$
$$308$$ −12.0000 −0.683763
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ −11.0000 −0.621757 −0.310878 0.950450i $$-0.600623\pi$$
−0.310878 + 0.950450i $$0.600623\pi$$
$$314$$ 26.0000 1.46726
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −8.00000 −0.449325 −0.224662 0.974437i $$-0.572128\pi$$
−0.224662 + 0.974437i $$0.572128\pi$$
$$318$$ 0 0
$$319$$ 20.0000 1.11979
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 36.0000 2.00620
$$323$$ −10.0000 −0.556415
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −22.0000 −1.21847
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 0 0
$$334$$ 24.0000 1.31322
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ −24.0000 −1.30543
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 12.0000 0.645124
$$347$$ 2.00000 0.107366 0.0536828 0.998558i $$-0.482904\pi$$
0.0536828 + 0.998558i $$0.482904\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 16.0000 0.852803
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 20.0000 1.05703
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 34.0000 1.78700
$$363$$ 0 0
$$364$$ −6.00000 −0.314485
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −27.0000 −1.40939 −0.704694 0.709511i $$-0.748916\pi$$
−0.704694 + 0.709511i $$0.748916\pi$$
$$368$$ −24.0000 −1.25109
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ 29.0000 1.50156 0.750782 0.660551i $$-0.229677\pi$$
0.750782 + 0.660551i $$0.229677\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 10.0000 0.515026
$$378$$ 0 0
$$379$$ 25.0000 1.28416 0.642082 0.766636i $$-0.278071\pi$$
0.642082 + 0.766636i $$0.278071\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −44.0000 −2.25124
$$383$$ 36.0000 1.83951 0.919757 0.392488i $$-0.128386\pi$$
0.919757 + 0.392488i $$0.128386\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −22.0000 −1.11977
$$387$$ 0 0
$$388$$ −34.0000 −1.72609
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −36.0000 −1.81365
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −7.00000 −0.351320 −0.175660 0.984451i $$-0.556206\pi$$
−0.175660 + 0.984451i $$0.556206\pi$$
$$398$$ −10.0000 −0.501255
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ 3.00000 0.149441
$$404$$ −24.0000 −1.19404
$$405$$ 0 0
$$406$$ −60.0000 −2.97775
$$407$$ 4.00000 0.198273
$$408$$ 0 0
$$409$$ 5.00000 0.247234 0.123617 0.992330i $$-0.460551\pi$$
0.123617 + 0.992330i $$0.460551\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 8.00000 0.394132
$$413$$ 30.0000 1.47620
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 8.00000 0.392232
$$417$$ 0 0
$$418$$ 20.0000 0.978232
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ −26.0000 −1.26566
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 21.0000 1.01626
$$428$$ 24.0000 1.16008
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ 29.0000 1.39365 0.696826 0.717241i $$-0.254595\pi$$
0.696826 + 0.717241i $$0.254595\pi$$
$$434$$ −18.0000 −0.864028
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ −30.0000 −1.43509
$$438$$ 0 0
$$439$$ −35.0000 −1.67046 −0.835229 0.549902i $$-0.814665\pi$$
−0.835229 + 0.549902i $$0.814665\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −4.00000 −0.190261
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 38.0000 1.79935
$$447$$ 0 0
$$448$$ −24.0000 −1.13389
$$449$$ −20.0000 −0.943858 −0.471929 0.881636i $$-0.656442\pi$$
−0.471929 + 0.881636i $$0.656442\pi$$
$$450$$ 0 0
$$451$$ −16.0000 −0.753411
$$452$$ −8.00000 −0.376288
$$453$$ 0 0
$$454$$ −16.0000 −0.750917
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ −30.0000 −1.40181
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ 40.0000 1.85695
$$465$$ 0 0
$$466$$ −48.0000 −2.22356
$$467$$ −38.0000 −1.75843 −0.879215 0.476425i $$-0.841932\pi$$
−0.879215 + 0.476425i $$0.841932\pi$$
$$468$$ 0 0
$$469$$ 9.00000 0.415581
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 2.00000 0.0919601
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ 0 0
$$478$$ −40.0000 −1.82956
$$479$$ −30.0000 −1.37073 −0.685367 0.728197i $$-0.740358\pi$$
−0.685367 + 0.728197i $$0.740358\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ −46.0000 −2.09524
$$483$$ 0 0
$$484$$ −14.0000 −0.636364
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 13.0000 0.589086 0.294543 0.955638i $$-0.404833\pi$$
0.294543 + 0.955638i $$0.404833\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 8.00000 0.361035 0.180517 0.983572i $$-0.442223\pi$$
0.180517 + 0.983572i $$0.442223\pi$$
$$492$$ 0 0
$$493$$ −20.0000 −0.900755
$$494$$ 10.0000 0.449921
$$495$$ 0 0
$$496$$ 12.0000 0.538816
$$497$$ 24.0000 1.07655
$$498$$ 0 0
$$499$$ −5.00000 −0.223831 −0.111915 0.993718i $$-0.535699\pi$$
−0.111915 + 0.993718i $$0.535699\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −24.0000 −1.07117
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −24.0000 −1.06693
$$507$$ 0 0
$$508$$ 16.0000 0.709885
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 42.0000 1.85797
$$512$$ 32.0000 1.41421
$$513$$ 0 0
$$514$$ 24.0000 1.05859
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −4.00000 −0.175920
$$518$$ −12.0000 −0.527250
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 0 0
$$523$$ −31.0000 −1.35554 −0.677768 0.735276i $$-0.737052\pi$$
−0.677768 + 0.735276i $$0.737052\pi$$
$$524$$ −24.0000 −1.04844
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ −6.00000 −0.261364
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −30.0000 −1.30066
$$533$$ −8.00000 −0.346518
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 20.0000 0.862261
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −3.00000 −0.128980 −0.0644900 0.997918i $$-0.520542\pi$$
−0.0644900 + 0.997918i $$0.520542\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 0 0
$$544$$ −16.0000 −0.685994
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ −36.0000 −1.53784
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 50.0000 2.13007
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 6.00000 0.254916
$$555$$ 0 0
$$556$$ 40.0000 1.69638
$$557$$ 42.0000 1.77960 0.889799 0.456354i $$-0.150845\pi$$
0.889799 + 0.456354i $$0.150845\pi$$
$$558$$ 0 0
$$559$$ 1.00000 0.0422955
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 36.0000 1.51857
$$563$$ 6.00000 0.252870 0.126435 0.991975i $$-0.459647\pi$$
0.126435 + 0.991975i $$0.459647\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 18.0000 0.756596
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ −13.0000 −0.544033 −0.272017 0.962293i $$-0.587691\pi$$
−0.272017 + 0.962293i $$0.587691\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ 48.0000 2.00348
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 13.0000 0.541197 0.270599 0.962692i $$-0.412778\pi$$
0.270599 + 0.962692i $$0.412778\pi$$
$$578$$ −26.0000 −1.08146
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 18.0000 0.746766
$$582$$ 0 0
$$583$$ 8.00000 0.331326
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 12.0000 0.495715
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 15.0000 0.618064
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 8.00000 0.328798
$$593$$ 16.0000 0.657041 0.328521 0.944497i $$-0.393450\pi$$
0.328521 + 0.944497i $$0.393450\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −20.0000 −0.819232
$$597$$ 0 0
$$598$$ −12.0000 −0.490716
$$599$$ 20.0000 0.817178 0.408589 0.912719i $$-0.366021\pi$$
0.408589 + 0.912719i $$0.366021\pi$$
$$600$$ 0 0
$$601$$ −13.0000 −0.530281 −0.265141 0.964210i $$-0.585418\pi$$
−0.265141 + 0.964210i $$0.585418\pi$$
$$602$$ −6.00000 −0.244542
$$603$$ 0 0
$$604$$ 14.0000 0.569652
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ 40.0000 1.62221
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2.00000 −0.0809113
$$612$$ 0 0
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ −14.0000 −0.564994
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.0000 0.483102 0.241551 0.970388i $$-0.422344\pi$$
0.241551 + 0.970388i $$0.422344\pi$$
$$618$$ 0 0
$$619$$ −25.0000 −1.00483 −0.502417 0.864625i $$-0.667556\pi$$
−0.502417 + 0.864625i $$0.667556\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 36.0000 1.44347
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ 26.0000 1.03751
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ −23.0000 −0.915616 −0.457808 0.889051i $$-0.651365\pi$$
−0.457808 + 0.889051i $$0.651365\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −16.0000 −0.635441
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −2.00000 −0.0792429
$$638$$ 40.0000 1.58362
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ 0 0
$$643$$ −36.0000 −1.41970 −0.709851 0.704352i $$-0.751238\pi$$
−0.709851 + 0.704352i $$0.751238\pi$$
$$644$$ 36.0000 1.41860
$$645$$ 0 0
$$646$$ −20.0000 −0.786889
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ 0 0
$$649$$ −20.0000 −0.785069
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −22.0000 −0.861586
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −32.0000 −1.24939
$$657$$ 0 0
$$658$$ 12.0000 0.467809
$$659$$ 40.0000 1.55818 0.779089 0.626913i $$-0.215682\pi$$
0.779089 + 0.626913i $$0.215682\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 24.0000 0.932786
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −60.0000 −2.32321
$$668$$ 24.0000 0.928588
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −14.0000 −0.540464
$$672$$ 0 0
$$673$$ −6.00000 −0.231283 −0.115642 0.993291i $$-0.536892\pi$$
−0.115642 + 0.993291i $$0.536892\pi$$
$$674$$ 46.0000 1.77185
$$675$$ 0 0
$$676$$ −24.0000 −0.923077
$$677$$ 42.0000 1.61419 0.807096 0.590421i $$-0.201038\pi$$
0.807096 + 0.590421i $$0.201038\pi$$
$$678$$ 0 0
$$679$$ −51.0000 −1.95720
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 12.0000 0.459504
$$683$$ −24.0000 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −30.0000 −1.14541
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 12.0000 0.456172
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 16.0000 0.606043
$$698$$ 20.0000 0.757011
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 0 0
$$703$$ 10.0000 0.377157
$$704$$ 16.0000 0.603023
$$705$$ 0 0
$$706$$ 12.0000 0.451626
$$707$$ −36.0000 −1.35392
$$708$$ 0 0
$$709$$ 25.0000 0.938895 0.469447 0.882960i $$-0.344453\pi$$
0.469447 + 0.882960i $$0.344453\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −18.0000 −0.674105
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 20.0000 0.747435
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −30.0000 −1.11881 −0.559406 0.828894i $$-0.688971\pi$$
−0.559406 + 0.828894i $$0.688971\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ 12.0000 0.446594
$$723$$ 0 0
$$724$$ 34.0000 1.26360
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 43.0000 1.59478 0.797391 0.603463i $$-0.206213\pi$$
0.797391 + 0.603463i $$0.206213\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2.00000 −0.0739727
$$732$$ 0 0
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ −54.0000 −1.99318
$$735$$ 0 0
$$736$$ −48.0000 −1.76930
$$737$$ −6.00000 −0.221013
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −24.0000 −0.881068
$$743$$ −4.00000 −0.146746 −0.0733729 0.997305i $$-0.523376\pi$$
−0.0733729 + 0.997305i $$0.523376\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 58.0000 2.12353
$$747$$ 0 0
$$748$$ −8.00000 −0.292509
$$749$$ 36.0000 1.31541
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ 20.0000 0.728357
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 23.0000 0.835949 0.417975 0.908459i $$-0.362740\pi$$
0.417975 + 0.908459i $$0.362740\pi$$
$$758$$ 50.0000 1.81608
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −12.0000 −0.435000 −0.217500 0.976060i $$-0.569790\pi$$
−0.217500 + 0.976060i $$0.569790\pi$$
$$762$$ 0 0
$$763$$ 15.0000 0.543036
$$764$$ −44.0000 −1.59186
$$765$$ 0 0
$$766$$ 72.0000 2.60147
$$767$$ −10.0000 −0.361079
$$768$$ 0 0
$$769$$ 35.0000 1.26213 0.631066 0.775729i $$-0.282618\pi$$
0.631066 + 0.775729i $$0.282618\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −22.0000 −0.791797
$$773$$ −24.0000 −0.863220 −0.431610 0.902060i $$-0.642054\pi$$
−0.431610 + 0.902060i $$0.642054\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ 24.0000 0.858238
$$783$$ 0 0
$$784$$ −8.00000 −0.285714
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −7.00000 −0.249523 −0.124762 0.992187i $$-0.539817\pi$$
−0.124762 + 0.992187i $$0.539817\pi$$
$$788$$ −36.0000 −1.28245
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ −7.00000 −0.248577
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ 52.0000 1.84193 0.920967 0.389640i $$-0.127401\pi$$
0.920967 + 0.389640i $$0.127401\pi$$
$$798$$ 0 0
$$799$$ 4.00000 0.141510
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −24.0000 −0.847469
$$803$$ −28.0000 −0.988099
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 6.00000 0.211341
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 20.0000 0.703163 0.351581 0.936157i $$-0.385644\pi$$
0.351581 + 0.936157i $$0.385644\pi$$
$$810$$ 0 0
$$811$$ 27.0000 0.948098 0.474049 0.880498i $$-0.342792\pi$$
0.474049 + 0.880498i $$0.342792\pi$$
$$812$$ −60.0000 −2.10559
$$813$$ 0 0
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 5.00000 0.174928
$$818$$ 10.0000 0.349642
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ −41.0000 −1.42917 −0.714585 0.699549i $$-0.753384\pi$$
−0.714585 + 0.699549i $$0.753384\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 60.0000 2.08767
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ 0 0
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 8.00000 0.277350
$$833$$ 4.00000 0.138592
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 20.0000 0.691714
$$837$$ 0 0
$$838$$ 40.0000 1.38178
$$839$$ −10.0000 −0.345238 −0.172619 0.984989i $$-0.555223\pi$$
−0.172619 + 0.984989i $$0.555223\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 44.0000 1.51634
$$843$$ 0 0
$$844$$ −26.0000 −0.894957
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −21.0000 −0.721569
$$848$$ 16.0000 0.549442
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ −51.0000 −1.74621 −0.873103 0.487535i $$-0.837896\pi$$
−0.873103 + 0.487535i $$0.837896\pi$$
$$854$$ 42.0000 1.43721
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −28.0000 −0.956462 −0.478231 0.878234i $$-0.658722\pi$$
−0.478231 + 0.878234i $$0.658722\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 36.0000 1.22616
$$863$$ 16.0000 0.544646 0.272323 0.962206i $$-0.412208\pi$$
0.272323 + 0.962206i $$0.412208\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 58.0000 1.97092
$$867$$ 0 0
$$868$$ −18.0000 −0.610960
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −3.00000 −0.101651
$$872$$ 0 0
$$873$$ 0 0
$$874$$ −60.0000 −2.02953
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −27.0000 −0.911725 −0.455863 0.890050i $$-0.650669\pi$$
−0.455863 + 0.890050i $$0.650669\pi$$
$$878$$ −70.0000 −2.36239
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −32.0000 −1.07811 −0.539054 0.842271i $$-0.681218\pi$$
−0.539054 + 0.842271i $$0.681218\pi$$
$$882$$ 0 0
$$883$$ −41.0000 −1.37976 −0.689880 0.723924i $$-0.742337\pi$$
−0.689880 + 0.723924i $$0.742337\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ −48.0000 −1.61259
$$887$$ −18.0000 −0.604381 −0.302190 0.953248i $$-0.597718\pi$$
−0.302190 + 0.953248i $$0.597718\pi$$
$$888$$ 0 0
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 38.0000 1.27233
$$893$$ −10.0000 −0.334637
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −40.0000 −1.33482
$$899$$ 30.0000 1.00056
$$900$$ 0 0
$$901$$ −8.00000 −0.266519
$$902$$ −32.0000 −1.06548
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ −16.0000 −0.530979
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 58.0000 1.92163 0.960813 0.277198i $$-0.0894057\pi$$
0.960813 + 0.277198i $$0.0894057\pi$$
$$912$$ 0 0
$$913$$ −12.0000 −0.397142
$$914$$ −44.0000 −1.45539
$$915$$ 0 0
$$916$$ −30.0000 −0.991228
$$917$$ −36.0000 −1.18882
$$918$$ 0 0
$$919$$ 55.0000 1.81428 0.907141 0.420826i $$-0.138260\pi$$
0.907141 + 0.420826i $$0.138260\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −24.0000 −0.790398
$$923$$ −8.00000 −0.263323
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 48.0000 1.57738
$$927$$ 0 0
$$928$$ 80.0000 2.62613
$$929$$ 50.0000 1.64045 0.820223 0.572043i $$-0.193849\pi$$
0.820223 + 0.572043i $$0.193849\pi$$
$$930$$ 0 0
$$931$$ −10.0000 −0.327737
$$932$$ −48.0000 −1.57229
$$933$$ 0 0
$$934$$ −76.0000 −2.48680
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 33.0000 1.07806 0.539032 0.842286i $$-0.318790\pi$$
0.539032 + 0.842286i $$0.318790\pi$$
$$938$$ 18.0000 0.587721
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ 0 0
$$943$$ 48.0000 1.56310
$$944$$ −40.0000 −1.30189
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ −18.0000 −0.584921 −0.292461 0.956278i $$-0.594474\pi$$
−0.292461 + 0.956278i $$0.594474\pi$$
$$948$$ 0 0
$$949$$ −14.0000 −0.454459
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 56.0000 1.81402 0.907009 0.421111i $$-0.138360\pi$$
0.907009 + 0.421111i $$0.138360\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −40.0000 −1.29369
$$957$$ 0 0
$$958$$ −60.0000 −1.93851
$$959$$ −54.0000 −1.74375
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 4.00000 0.128965
$$963$$ 0 0
$$964$$ −46.0000 −1.48156
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −42.0000 −1.34784 −0.673922 0.738802i $$-0.735392\pi$$
−0.673922 + 0.738802i $$0.735392\pi$$
$$972$$ 0 0
$$973$$ 60.0000 1.92351
$$974$$ 26.0000 0.833094
$$975$$ 0 0
$$976$$ −28.0000 −0.896258
$$977$$ 2.00000 0.0639857 0.0319928 0.999488i $$-0.489815\pi$$
0.0319928 + 0.999488i $$0.489815\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 16.0000 0.510581
$$983$$ 36.0000 1.14822 0.574111 0.818778i $$-0.305348\pi$$
0.574111 + 0.818778i $$0.305348\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −40.0000 −1.27386
$$987$$ 0 0
$$988$$ 10.0000 0.318142
$$989$$ −6.00000 −0.190789
$$990$$ 0 0
$$991$$ 17.0000 0.540023 0.270011 0.962857i $$-0.412973\pi$$
0.270011 + 0.962857i $$0.412973\pi$$
$$992$$ 24.0000 0.762001
$$993$$ 0 0
$$994$$ 48.0000 1.52247
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −42.0000 −1.33015 −0.665077 0.746775i $$-0.731601\pi$$
−0.665077 + 0.746775i $$0.731601\pi$$
$$998$$ −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 225.2.a.e.1.1 1
3.2 odd 2 75.2.a.a.1.1 1
4.3 odd 2 3600.2.a.j.1.1 1
5.2 odd 4 225.2.b.a.199.2 2
5.3 odd 4 225.2.b.a.199.1 2
5.4 even 2 225.2.a.a.1.1 1
12.11 even 2 1200.2.a.c.1.1 1
15.2 even 4 75.2.b.a.49.1 2
15.8 even 4 75.2.b.a.49.2 2
15.14 odd 2 75.2.a.c.1.1 yes 1
20.3 even 4 3600.2.f.p.2449.2 2
20.7 even 4 3600.2.f.p.2449.1 2
20.19 odd 2 3600.2.a.bk.1.1 1
21.20 even 2 3675.2.a.b.1.1 1
24.5 odd 2 4800.2.a.bb.1.1 1
24.11 even 2 4800.2.a.br.1.1 1
33.32 even 2 9075.2.a.s.1.1 1
60.23 odd 4 1200.2.f.d.49.1 2
60.47 odd 4 1200.2.f.d.49.2 2
60.59 even 2 1200.2.a.p.1.1 1
105.104 even 2 3675.2.a.q.1.1 1
120.29 odd 2 4800.2.a.bq.1.1 1
120.53 even 4 4800.2.f.l.3649.1 2
120.59 even 2 4800.2.a.be.1.1 1
120.77 even 4 4800.2.f.l.3649.2 2
120.83 odd 4 4800.2.f.y.3649.2 2
120.107 odd 4 4800.2.f.y.3649.1 2
165.164 even 2 9075.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.a.a.1.1 1 3.2 odd 2
75.2.a.c.1.1 yes 1 15.14 odd 2
75.2.b.a.49.1 2 15.2 even 4
75.2.b.a.49.2 2 15.8 even 4
225.2.a.a.1.1 1 5.4 even 2
225.2.a.e.1.1 1 1.1 even 1 trivial
225.2.b.a.199.1 2 5.3 odd 4
225.2.b.a.199.2 2 5.2 odd 4
1200.2.a.c.1.1 1 12.11 even 2
1200.2.a.p.1.1 1 60.59 even 2
1200.2.f.d.49.1 2 60.23 odd 4
1200.2.f.d.49.2 2 60.47 odd 4
3600.2.a.j.1.1 1 4.3 odd 2
3600.2.a.bk.1.1 1 20.19 odd 2
3600.2.f.p.2449.1 2 20.7 even 4
3600.2.f.p.2449.2 2 20.3 even 4
3675.2.a.b.1.1 1 21.20 even 2
3675.2.a.q.1.1 1 105.104 even 2
4800.2.a.bb.1.1 1 24.5 odd 2
4800.2.a.be.1.1 1 120.59 even 2
4800.2.a.bq.1.1 1 120.29 odd 2
4800.2.a.br.1.1 1 24.11 even 2
4800.2.f.l.3649.1 2 120.53 even 4
4800.2.f.l.3649.2 2 120.77 even 4
4800.2.f.y.3649.1 2 120.107 odd 4
4800.2.f.y.3649.2 2 120.83 odd 4
9075.2.a.a.1.1 1 165.164 even 2
9075.2.a.s.1.1 1 33.32 even 2