# Properties

 Label 3822.2.a.v.1.1 Level $3822$ Weight $2$ Character 3822.1 Self dual yes Analytic conductor $30.519$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, 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("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3822.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: $$30.5188236525$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3822.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} -1.00000 q^{30} +11.0000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} +4.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} -12.0000 q^{41} -8.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} -6.00000 q^{51} +1.00000 q^{52} -5.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} -4.00000 q^{57} +3.00000 q^{58} +5.00000 q^{59} -1.00000 q^{60} -12.0000 q^{61} +11.0000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +1.00000 q^{66} +16.0000 q^{67} +6.00000 q^{68} +6.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} +4.00000 q^{74} +4.00000 q^{75} +4.00000 q^{76} -1.00000 q^{78} +7.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} +17.0000 q^{83} +6.00000 q^{85} -8.00000 q^{86} -3.00000 q^{87} -1.00000 q^{88} +12.0000 q^{89} +1.00000 q^{90} -6.00000 q^{92} -11.0000 q^{93} +8.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} -13.0000 q^{97} -1.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −4.00000 −0.800000
$$26$$ 1.00000 0.196116
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 11.0000 1.97566 0.987829 0.155543i $$-0.0497126\pi$$
0.987829 + 0.155543i $$0.0497126\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.00000 0.174078
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 4.00000 0.648886
$$39$$ −1.00000 −0.160128
$$40$$ 1.00000 0.158114
$$41$$ −12.0000 −1.87409 −0.937043 0.349215i $$-0.886448\pi$$
−0.937043 + 0.349215i $$0.886448\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 1.00000 0.149071
$$46$$ −6.00000 −0.884652
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ −4.00000 −0.565685
$$51$$ −6.00000 −0.840168
$$52$$ 1.00000 0.138675
$$53$$ −5.00000 −0.686803 −0.343401 0.939189i $$-0.611579\pi$$
−0.343401 + 0.939189i $$0.611579\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 3.00000 0.393919
$$59$$ 5.00000 0.650945 0.325472 0.945552i $$-0.394477\pi$$
0.325472 + 0.945552i $$0.394477\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −12.0000 −1.53644 −0.768221 0.640184i $$-0.778858\pi$$
−0.768221 + 0.640184i $$0.778858\pi$$
$$62$$ 11.0000 1.39700
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 1.00000 0.124035
$$66$$ 1.00000 0.123091
$$67$$ 16.0000 1.95471 0.977356 0.211604i $$-0.0678686\pi$$
0.977356 + 0.211604i $$0.0678686\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 4.00000 0.461880
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ −1.00000 −0.113228
$$79$$ 7.00000 0.787562 0.393781 0.919204i $$-0.371167\pi$$
0.393781 + 0.919204i $$0.371167\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ −12.0000 −1.32518
$$83$$ 17.0000 1.86599 0.932996 0.359886i $$-0.117184\pi$$
0.932996 + 0.359886i $$0.117184\pi$$
$$84$$ 0 0
$$85$$ 6.00000 0.650791
$$86$$ −8.00000 −0.862662
$$87$$ −3.00000 −0.321634
$$88$$ −1.00000 −0.106600
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ −6.00000 −0.625543
$$93$$ −11.0000 −1.14065
$$94$$ 8.00000 0.825137
$$95$$ 4.00000 0.410391
$$96$$ −1.00000 −0.102062
$$97$$ −13.0000 −1.31995 −0.659975 0.751288i $$-0.729433\pi$$
−0.659975 + 0.751288i $$0.729433\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ −4.00000 −0.400000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ −6.00000 −0.594089
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −5.00000 −0.485643
$$107$$ 7.00000 0.676716 0.338358 0.941018i $$-0.390129\pi$$
0.338358 + 0.941018i $$0.390129\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 12.0000 1.14939 0.574696 0.818367i $$-0.305120\pi$$
0.574696 + 0.818367i $$0.305120\pi$$
$$110$$ −1.00000 −0.0953463
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ −4.00000 −0.376288 −0.188144 0.982141i $$-0.560247\pi$$
−0.188144 + 0.982141i $$0.560247\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ −6.00000 −0.559503
$$116$$ 3.00000 0.278543
$$117$$ 1.00000 0.0924500
$$118$$ 5.00000 0.460287
$$119$$ 0 0
$$120$$ −1.00000 −0.0912871
$$121$$ −10.0000 −0.909091
$$122$$ −12.0000 −1.08643
$$123$$ 12.0000 1.08200
$$124$$ 11.0000 0.987829
$$125$$ −9.00000 −0.804984
$$126$$ 0 0
$$127$$ −7.00000 −0.621150 −0.310575 0.950549i $$-0.600522\pi$$
−0.310575 + 0.950549i $$0.600522\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 1.00000 0.0877058
$$131$$ −1.00000 −0.0873704 −0.0436852 0.999045i $$-0.513910\pi$$
−0.0436852 + 0.999045i $$0.513910\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 0 0
$$134$$ 16.0000 1.38219
$$135$$ −1.00000 −0.0860663
$$136$$ 6.00000 0.514496
$$137$$ −8.00000 −0.683486 −0.341743 0.939793i $$-0.611017\pi$$
−0.341743 + 0.939793i $$0.611017\pi$$
$$138$$ 6.00000 0.510754
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 6.00000 0.503509
$$143$$ −1.00000 −0.0836242
$$144$$ 1.00000 0.0833333
$$145$$ 3.00000 0.249136
$$146$$ 10.0000 0.827606
$$147$$ 0 0
$$148$$ 4.00000 0.328798
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 4.00000 0.326599
$$151$$ 13.0000 1.05792 0.528962 0.848645i $$-0.322581\pi$$
0.528962 + 0.848645i $$0.322581\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 11.0000 0.883541
$$156$$ −1.00000 −0.0800641
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 7.00000 0.556890
$$159$$ 5.00000 0.396526
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −8.00000 −0.626608 −0.313304 0.949653i $$-0.601436\pi$$
−0.313304 + 0.949653i $$0.601436\pi$$
$$164$$ −12.0000 −0.937043
$$165$$ 1.00000 0.0778499
$$166$$ 17.0000 1.31946
$$167$$ 24.0000 1.85718 0.928588 0.371113i $$-0.121024\pi$$
0.928588 + 0.371113i $$0.121024\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 6.00000 0.460179
$$171$$ 4.00000 0.305888
$$172$$ −8.00000 −0.609994
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ −3.00000 −0.227429
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ −5.00000 −0.375823
$$178$$ 12.0000 0.899438
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 16.0000 1.18927 0.594635 0.803996i $$-0.297296\pi$$
0.594635 + 0.803996i $$0.297296\pi$$
$$182$$ 0 0
$$183$$ 12.0000 0.887066
$$184$$ −6.00000 −0.442326
$$185$$ 4.00000 0.294086
$$186$$ −11.0000 −0.806559
$$187$$ −6.00000 −0.438763
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ −6.00000 −0.434145 −0.217072 0.976156i $$-0.569651\pi$$
−0.217072 + 0.976156i $$0.569651\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 15.0000 1.07972 0.539862 0.841754i $$-0.318476\pi$$
0.539862 + 0.841754i $$0.318476\pi$$
$$194$$ −13.0000 −0.933346
$$195$$ −1.00000 −0.0716115
$$196$$ 0 0
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ −4.00000 −0.282843
$$201$$ −16.0000 −1.12855
$$202$$ −10.0000 −0.703598
$$203$$ 0 0
$$204$$ −6.00000 −0.420084
$$205$$ −12.0000 −0.838116
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 1.00000 0.0693375
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ −5.00000 −0.343401
$$213$$ −6.00000 −0.411113
$$214$$ 7.00000 0.478510
$$215$$ −8.00000 −0.545595
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 12.0000 0.812743
$$219$$ −10.0000 −0.675737
$$220$$ −1.00000 −0.0674200
$$221$$ 6.00000 0.403604
$$222$$ −4.00000 −0.268462
$$223$$ −7.00000 −0.468755 −0.234377 0.972146i $$-0.575305\pi$$
−0.234377 + 0.972146i $$0.575305\pi$$
$$224$$ 0 0
$$225$$ −4.00000 −0.266667
$$226$$ −4.00000 −0.266076
$$227$$ 15.0000 0.995585 0.497792 0.867296i $$-0.334144\pi$$
0.497792 + 0.867296i $$0.334144\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ −6.00000 −0.395628
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 8.00000 0.521862
$$236$$ 5.00000 0.325472
$$237$$ −7.00000 −0.454699
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 15.0000 0.966235 0.483117 0.875556i $$-0.339504\pi$$
0.483117 + 0.875556i $$0.339504\pi$$
$$242$$ −10.0000 −0.642824
$$243$$ −1.00000 −0.0641500
$$244$$ −12.0000 −0.768221
$$245$$ 0 0
$$246$$ 12.0000 0.765092
$$247$$ 4.00000 0.254514
$$248$$ 11.0000 0.698501
$$249$$ −17.0000 −1.07733
$$250$$ −9.00000 −0.569210
$$251$$ −17.0000 −1.07303 −0.536515 0.843891i $$-0.680260\pi$$
−0.536515 + 0.843891i $$0.680260\pi$$
$$252$$ 0 0
$$253$$ 6.00000 0.377217
$$254$$ −7.00000 −0.439219
$$255$$ −6.00000 −0.375735
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 8.00000 0.498058
$$259$$ 0 0
$$260$$ 1.00000 0.0620174
$$261$$ 3.00000 0.185695
$$262$$ −1.00000 −0.0617802
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ −5.00000 −0.307148
$$266$$ 0 0
$$267$$ −12.0000 −0.734388
$$268$$ 16.0000 0.977356
$$269$$ 9.00000 0.548740 0.274370 0.961624i $$-0.411531\pi$$
0.274370 + 0.961624i $$0.411531\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 7.00000 0.425220 0.212610 0.977137i $$-0.431804\pi$$
0.212610 + 0.977137i $$0.431804\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ −8.00000 −0.483298
$$275$$ 4.00000 0.241209
$$276$$ 6.00000 0.361158
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 0 0
$$279$$ 11.0000 0.658553
$$280$$ 0 0
$$281$$ 4.00000 0.238620 0.119310 0.992857i $$-0.461932\pi$$
0.119310 + 0.992857i $$0.461932\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ −10.0000 −0.594438 −0.297219 0.954809i $$-0.596059\pi$$
−0.297219 + 0.954809i $$0.596059\pi$$
$$284$$ 6.00000 0.356034
$$285$$ −4.00000 −0.236940
$$286$$ −1.00000 −0.0591312
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 3.00000 0.176166
$$291$$ 13.0000 0.762073
$$292$$ 10.0000 0.585206
$$293$$ 7.00000 0.408944 0.204472 0.978872i $$-0.434452\pi$$
0.204472 + 0.978872i $$0.434452\pi$$
$$294$$ 0 0
$$295$$ 5.00000 0.291111
$$296$$ 4.00000 0.232495
$$297$$ 1.00000 0.0580259
$$298$$ −10.0000 −0.579284
$$299$$ −6.00000 −0.346989
$$300$$ 4.00000 0.230940
$$301$$ 0 0
$$302$$ 13.0000 0.748066
$$303$$ 10.0000 0.574485
$$304$$ 4.00000 0.229416
$$305$$ −12.0000 −0.687118
$$306$$ 6.00000 0.342997
$$307$$ −6.00000 −0.342438 −0.171219 0.985233i $$-0.554771\pi$$
−0.171219 + 0.985233i $$0.554771\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 11.0000 0.624758
$$311$$ −2.00000 −0.113410 −0.0567048 0.998391i $$-0.518059\pi$$
−0.0567048 + 0.998391i $$0.518059\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ −13.0000 −0.734803 −0.367402 0.930062i $$-0.619753\pi$$
−0.367402 + 0.930062i $$0.619753\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 0 0
$$316$$ 7.00000 0.393781
$$317$$ −3.00000 −0.168497 −0.0842484 0.996445i $$-0.526849\pi$$
−0.0842484 + 0.996445i $$0.526849\pi$$
$$318$$ 5.00000 0.280386
$$319$$ −3.00000 −0.167968
$$320$$ 1.00000 0.0559017
$$321$$ −7.00000 −0.390702
$$322$$ 0 0
$$323$$ 24.0000 1.33540
$$324$$ 1.00000 0.0555556
$$325$$ −4.00000 −0.221880
$$326$$ −8.00000 −0.443079
$$327$$ −12.0000 −0.663602
$$328$$ −12.0000 −0.662589
$$329$$ 0 0
$$330$$ 1.00000 0.0550482
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ 17.0000 0.932996
$$333$$ 4.00000 0.219199
$$334$$ 24.0000 1.31322
$$335$$ 16.0000 0.874173
$$336$$ 0 0
$$337$$ 25.0000 1.36184 0.680918 0.732359i $$-0.261581\pi$$
0.680918 + 0.732359i $$0.261581\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 4.00000 0.217250
$$340$$ 6.00000 0.325396
$$341$$ −11.0000 −0.595683
$$342$$ 4.00000 0.216295
$$343$$ 0 0
$$344$$ −8.00000 −0.431331
$$345$$ 6.00000 0.323029
$$346$$ −6.00000 −0.322562
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ −3.00000 −0.160817
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ −1.00000 −0.0533002
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ −5.00000 −0.265747
$$355$$ 6.00000 0.318447
$$356$$ 12.0000 0.635999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ −22.0000 −1.16112 −0.580558 0.814219i $$-0.697165\pi$$
−0.580558 + 0.814219i $$0.697165\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −3.00000 −0.157895
$$362$$ 16.0000 0.840941
$$363$$ 10.0000 0.524864
$$364$$ 0 0
$$365$$ 10.0000 0.523424
$$366$$ 12.0000 0.627250
$$367$$ 7.00000 0.365397 0.182699 0.983169i $$-0.441517\pi$$
0.182699 + 0.983169i $$0.441517\pi$$
$$368$$ −6.00000 −0.312772
$$369$$ −12.0000 −0.624695
$$370$$ 4.00000 0.207950
$$371$$ 0 0
$$372$$ −11.0000 −0.570323
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 9.00000 0.464758
$$376$$ 8.00000 0.412568
$$377$$ 3.00000 0.154508
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 7.00000 0.358621
$$382$$ −6.00000 −0.306987
$$383$$ 14.0000 0.715367 0.357683 0.933843i $$-0.383567\pi$$
0.357683 + 0.933843i $$0.383567\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 15.0000 0.763480
$$387$$ −8.00000 −0.406663
$$388$$ −13.0000 −0.659975
$$389$$ 22.0000 1.11544 0.557722 0.830028i $$-0.311675\pi$$
0.557722 + 0.830028i $$0.311675\pi$$
$$390$$ −1.00000 −0.0506370
$$391$$ −36.0000 −1.82060
$$392$$ 0 0
$$393$$ 1.00000 0.0504433
$$394$$ −18.0000 −0.906827
$$395$$ 7.00000 0.352208
$$396$$ −1.00000 −0.0502519
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ −16.0000 −0.798007
$$403$$ 11.0000 0.547949
$$404$$ −10.0000 −0.497519
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −4.00000 −0.198273
$$408$$ −6.00000 −0.297044
$$409$$ −25.0000 −1.23617 −0.618085 0.786111i $$-0.712091\pi$$
−0.618085 + 0.786111i $$0.712091\pi$$
$$410$$ −12.0000 −0.592638
$$411$$ 8.00000 0.394611
$$412$$ 0 0
$$413$$ 0 0
$$414$$ −6.00000 −0.294884
$$415$$ 17.0000 0.834497
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ −4.00000 −0.195646
$$419$$ −16.0000 −0.781651 −0.390826 0.920465i $$-0.627810\pi$$
−0.390826 + 0.920465i $$0.627810\pi$$
$$420$$ 0 0
$$421$$ −36.0000 −1.75453 −0.877266 0.480004i $$-0.840635\pi$$
−0.877266 + 0.480004i $$0.840635\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 8.00000 0.388973
$$424$$ −5.00000 −0.242821
$$425$$ −24.0000 −1.16417
$$426$$ −6.00000 −0.290701
$$427$$ 0 0
$$428$$ 7.00000 0.338358
$$429$$ 1.00000 0.0482805
$$430$$ −8.00000 −0.385794
$$431$$ −38.0000 −1.83040 −0.915198 0.403005i $$-0.867966\pi$$
−0.915198 + 0.403005i $$0.867966\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ −3.00000 −0.143839
$$436$$ 12.0000 0.574696
$$437$$ −24.0000 −1.14808
$$438$$ −10.0000 −0.477818
$$439$$ 17.0000 0.811366 0.405683 0.914014i $$-0.367034\pi$$
0.405683 + 0.914014i $$0.367034\pi$$
$$440$$ −1.00000 −0.0476731
$$441$$ 0 0
$$442$$ 6.00000 0.285391
$$443$$ −39.0000 −1.85295 −0.926473 0.376361i $$-0.877175\pi$$
−0.926473 + 0.376361i $$0.877175\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 12.0000 0.568855
$$446$$ −7.00000 −0.331460
$$447$$ 10.0000 0.472984
$$448$$ 0 0
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ −4.00000 −0.188562
$$451$$ 12.0000 0.565058
$$452$$ −4.00000 −0.188144
$$453$$ −13.0000 −0.610793
$$454$$ 15.0000 0.703985
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ −11.0000 −0.514558 −0.257279 0.966337i $$-0.582826\pi$$
−0.257279 + 0.966337i $$0.582826\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ −6.00000 −0.280056
$$460$$ −6.00000 −0.279751
$$461$$ −34.0000 −1.58354 −0.791769 0.610821i $$-0.790840\pi$$
−0.791769 + 0.610821i $$0.790840\pi$$
$$462$$ 0 0
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ 3.00000 0.139272
$$465$$ −11.0000 −0.510113
$$466$$ 6.00000 0.277945
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 0 0
$$470$$ 8.00000 0.369012
$$471$$ −18.0000 −0.829396
$$472$$ 5.00000 0.230144
$$473$$ 8.00000 0.367840
$$474$$ −7.00000 −0.321521
$$475$$ −16.0000 −0.734130
$$476$$ 0 0
$$477$$ −5.00000 −0.228934
$$478$$ −6.00000 −0.274434
$$479$$ −14.0000 −0.639676 −0.319838 0.947472i $$-0.603629\pi$$
−0.319838 + 0.947472i $$0.603629\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 4.00000 0.182384
$$482$$ 15.0000 0.683231
$$483$$ 0 0
$$484$$ −10.0000 −0.454545
$$485$$ −13.0000 −0.590300
$$486$$ −1.00000 −0.0453609
$$487$$ 23.0000 1.04223 0.521115 0.853487i $$-0.325516\pi$$
0.521115 + 0.853487i $$0.325516\pi$$
$$488$$ −12.0000 −0.543214
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ −35.0000 −1.57953 −0.789764 0.613411i $$-0.789797\pi$$
−0.789764 + 0.613411i $$0.789797\pi$$
$$492$$ 12.0000 0.541002
$$493$$ 18.0000 0.810679
$$494$$ 4.00000 0.179969
$$495$$ −1.00000 −0.0449467
$$496$$ 11.0000 0.493915
$$497$$ 0 0
$$498$$ −17.0000 −0.761788
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ −9.00000 −0.402492
$$501$$ −24.0000 −1.07224
$$502$$ −17.0000 −0.758747
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ 0 0
$$505$$ −10.0000 −0.444994
$$506$$ 6.00000 0.266733
$$507$$ −1.00000 −0.0444116
$$508$$ −7.00000 −0.310575
$$509$$ −13.0000 −0.576215 −0.288107 0.957598i $$-0.593026\pi$$
−0.288107 + 0.957598i $$0.593026\pi$$
$$510$$ −6.00000 −0.265684
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ −8.00000 −0.351840
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 1.00000 0.0438529
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ 3.00000 0.131306
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ −1.00000 −0.0436852
$$525$$ 0 0
$$526$$ −18.0000 −0.784837
$$527$$ 66.0000 2.87501
$$528$$ 1.00000 0.0435194
$$529$$ 13.0000 0.565217
$$530$$ −5.00000 −0.217186
$$531$$ 5.00000 0.216982
$$532$$ 0 0
$$533$$ −12.0000 −0.519778
$$534$$ −12.0000 −0.519291
$$535$$ 7.00000 0.302636
$$536$$ 16.0000 0.691095
$$537$$ −12.0000 −0.517838
$$538$$ 9.00000 0.388018
$$539$$ 0 0
$$540$$ −1.00000 −0.0430331
$$541$$ 32.0000 1.37579 0.687894 0.725811i $$-0.258536\pi$$
0.687894 + 0.725811i $$0.258536\pi$$
$$542$$ 7.00000 0.300676
$$543$$ −16.0000 −0.686626
$$544$$ 6.00000 0.257248
$$545$$ 12.0000 0.514024
$$546$$ 0 0
$$547$$ 36.0000 1.53925 0.769624 0.638497i $$-0.220443\pi$$
0.769624 + 0.638497i $$0.220443\pi$$
$$548$$ −8.00000 −0.341743
$$549$$ −12.0000 −0.512148
$$550$$ 4.00000 0.170561
$$551$$ 12.0000 0.511217
$$552$$ 6.00000 0.255377
$$553$$ 0 0
$$554$$ −10.0000 −0.424859
$$555$$ −4.00000 −0.169791
$$556$$ 0 0
$$557$$ −17.0000 −0.720313 −0.360157 0.932892i $$-0.617277\pi$$
−0.360157 + 0.932892i $$0.617277\pi$$
$$558$$ 11.0000 0.465667
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 6.00000 0.253320
$$562$$ 4.00000 0.168730
$$563$$ −21.0000 −0.885044 −0.442522 0.896758i $$-0.645916\pi$$
−0.442522 + 0.896758i $$0.645916\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ −4.00000 −0.168281
$$566$$ −10.0000 −0.420331
$$567$$ 0 0
$$568$$ 6.00000 0.251754
$$569$$ −4.00000 −0.167689 −0.0838444 0.996479i $$-0.526720\pi$$
−0.0838444 + 0.996479i $$0.526720\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ −1.00000 −0.0418121
$$573$$ 6.00000 0.250654
$$574$$ 0 0
$$575$$ 24.0000 1.00087
$$576$$ 1.00000 0.0416667
$$577$$ −1.00000 −0.0416305 −0.0208153 0.999783i $$-0.506626\pi$$
−0.0208153 + 0.999783i $$0.506626\pi$$
$$578$$ 19.0000 0.790296
$$579$$ −15.0000 −0.623379
$$580$$ 3.00000 0.124568
$$581$$ 0 0
$$582$$ 13.0000 0.538867
$$583$$ 5.00000 0.207079
$$584$$ 10.0000 0.413803
$$585$$ 1.00000 0.0413449
$$586$$ 7.00000 0.289167
$$587$$ −5.00000 −0.206372 −0.103186 0.994662i $$-0.532904\pi$$
−0.103186 + 0.994662i $$0.532904\pi$$
$$588$$ 0 0
$$589$$ 44.0000 1.81299
$$590$$ 5.00000 0.205847
$$591$$ 18.0000 0.740421
$$592$$ 4.00000 0.164399
$$593$$ 36.0000 1.47834 0.739171 0.673517i $$-0.235217\pi$$
0.739171 + 0.673517i $$0.235217\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 8.00000 0.327418
$$598$$ −6.00000 −0.245358
$$599$$ −10.0000 −0.408589 −0.204294 0.978909i $$-0.565490\pi$$
−0.204294 + 0.978909i $$0.565490\pi$$
$$600$$ 4.00000 0.163299
$$601$$ −11.0000 −0.448699 −0.224350 0.974509i $$-0.572026\pi$$
−0.224350 + 0.974509i $$0.572026\pi$$
$$602$$ 0 0
$$603$$ 16.0000 0.651570
$$604$$ 13.0000 0.528962
$$605$$ −10.0000 −0.406558
$$606$$ 10.0000 0.406222
$$607$$ −29.0000 −1.17707 −0.588537 0.808470i $$-0.700296\pi$$
−0.588537 + 0.808470i $$0.700296\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ −12.0000 −0.485866
$$611$$ 8.00000 0.323645
$$612$$ 6.00000 0.242536
$$613$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$614$$ −6.00000 −0.242140
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ −36.0000 −1.44931 −0.724653 0.689114i $$-0.758000\pi$$
−0.724653 + 0.689114i $$0.758000\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 11.0000 0.441771
$$621$$ 6.00000 0.240772
$$622$$ −2.00000 −0.0801927
$$623$$ 0 0
$$624$$ −1.00000 −0.0400320
$$625$$ 11.0000 0.440000
$$626$$ −13.0000 −0.519584
$$627$$ 4.00000 0.159745
$$628$$ 18.0000 0.718278
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ −29.0000 −1.15447 −0.577236 0.816577i $$-0.695869\pi$$
−0.577236 + 0.816577i $$0.695869\pi$$
$$632$$ 7.00000 0.278445
$$633$$ −12.0000 −0.476957
$$634$$ −3.00000 −0.119145
$$635$$ −7.00000 −0.277787
$$636$$ 5.00000 0.198263
$$637$$ 0 0
$$638$$ −3.00000 −0.118771
$$639$$ 6.00000 0.237356
$$640$$ 1.00000 0.0395285
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ −7.00000 −0.276268
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ 0 0
$$645$$ 8.00000 0.315000
$$646$$ 24.0000 0.944267
$$647$$ 4.00000 0.157256 0.0786281 0.996904i $$-0.474946\pi$$
0.0786281 + 0.996904i $$0.474946\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −5.00000 −0.196267
$$650$$ −4.00000 −0.156893
$$651$$ 0 0
$$652$$ −8.00000 −0.313304
$$653$$ −11.0000 −0.430463 −0.215232 0.976563i $$-0.569051\pi$$
−0.215232 + 0.976563i $$0.569051\pi$$
$$654$$ −12.0000 −0.469237
$$655$$ −1.00000 −0.0390732
$$656$$ −12.0000 −0.468521
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 1.00000 0.0389249
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ −8.00000 −0.310929
$$663$$ −6.00000 −0.233021
$$664$$ 17.0000 0.659728
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ −18.0000 −0.696963
$$668$$ 24.0000 0.928588
$$669$$ 7.00000 0.270636
$$670$$ 16.0000 0.618134
$$671$$ 12.0000 0.463255
$$672$$ 0 0
$$673$$ −39.0000 −1.50334 −0.751670 0.659540i $$-0.770751\pi$$
−0.751670 + 0.659540i $$0.770751\pi$$
$$674$$ 25.0000 0.962964
$$675$$ 4.00000 0.153960
$$676$$ 1.00000 0.0384615
$$677$$ 35.0000 1.34516 0.672580 0.740025i $$-0.265186\pi$$
0.672580 + 0.740025i $$0.265186\pi$$
$$678$$ 4.00000 0.153619
$$679$$ 0 0
$$680$$ 6.00000 0.230089
$$681$$ −15.0000 −0.574801
$$682$$ −11.0000 −0.421212
$$683$$ −27.0000 −1.03313 −0.516563 0.856249i $$-0.672789\pi$$
−0.516563 + 0.856249i $$0.672789\pi$$
$$684$$ 4.00000 0.152944
$$685$$ −8.00000 −0.305664
$$686$$ 0 0
$$687$$ 6.00000 0.228914
$$688$$ −8.00000 −0.304997
$$689$$ −5.00000 −0.190485
$$690$$ 6.00000 0.228416
$$691$$ 40.0000 1.52167 0.760836 0.648944i $$-0.224789\pi$$
0.760836 + 0.648944i $$0.224789\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ 20.0000 0.759190
$$695$$ 0 0
$$696$$ −3.00000 −0.113715
$$697$$ −72.0000 −2.72719
$$698$$ −14.0000 −0.529908
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −13.0000 −0.491003 −0.245502 0.969396i $$-0.578953\pi$$
−0.245502 + 0.969396i $$0.578953\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ 16.0000 0.603451
$$704$$ −1.00000 −0.0376889
$$705$$ −8.00000 −0.301297
$$706$$ −30.0000 −1.12906
$$707$$ 0 0
$$708$$ −5.00000 −0.187912
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 6.00000 0.225176
$$711$$ 7.00000 0.262521
$$712$$ 12.0000 0.449719
$$713$$ −66.0000 −2.47172
$$714$$ 0 0
$$715$$ −1.00000 −0.0373979
$$716$$ 12.0000 0.448461
$$717$$ 6.00000 0.224074
$$718$$ −22.0000 −0.821033
$$719$$ −46.0000 −1.71551 −0.857755 0.514058i $$-0.828142\pi$$
−0.857755 + 0.514058i $$0.828142\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ −15.0000 −0.557856
$$724$$ 16.0000 0.594635
$$725$$ −12.0000 −0.445669
$$726$$ 10.0000 0.371135
$$727$$ −19.0000 −0.704671 −0.352335 0.935874i $$-0.614612\pi$$
−0.352335 + 0.935874i $$0.614612\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 10.0000 0.370117
$$731$$ −48.0000 −1.77534
$$732$$ 12.0000 0.443533
$$733$$ −2.00000 −0.0738717 −0.0369358 0.999318i $$-0.511760\pi$$
−0.0369358 + 0.999318i $$0.511760\pi$$
$$734$$ 7.00000 0.258375
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ −16.0000 −0.589368
$$738$$ −12.0000 −0.441726
$$739$$ −26.0000 −0.956425 −0.478213 0.878244i $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ 4.00000 0.147043
$$741$$ −4.00000 −0.146944
$$742$$ 0 0
$$743$$ −32.0000 −1.17397 −0.586983 0.809599i $$-0.699684\pi$$
−0.586983 + 0.809599i $$0.699684\pi$$
$$744$$ −11.0000 −0.403280
$$745$$ −10.0000 −0.366372
$$746$$ 26.0000 0.951928
$$747$$ 17.0000 0.621997
$$748$$ −6.00000 −0.219382
$$749$$ 0 0
$$750$$ 9.00000 0.328634
$$751$$ 25.0000 0.912263 0.456131 0.889912i $$-0.349235\pi$$
0.456131 + 0.889912i $$0.349235\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 17.0000 0.619514
$$754$$ 3.00000 0.109254
$$755$$ 13.0000 0.473118
$$756$$ 0 0
$$757$$ −38.0000 −1.38113 −0.690567 0.723269i $$-0.742639\pi$$
−0.690567 + 0.723269i $$0.742639\pi$$
$$758$$ 16.0000 0.581146
$$759$$ −6.00000 −0.217786
$$760$$ 4.00000 0.145095
$$761$$ 28.0000 1.01500 0.507500 0.861652i $$-0.330570\pi$$
0.507500 + 0.861652i $$0.330570\pi$$
$$762$$ 7.00000 0.253583
$$763$$ 0 0
$$764$$ −6.00000 −0.217072
$$765$$ 6.00000 0.216930
$$766$$ 14.0000 0.505841
$$767$$ 5.00000 0.180540
$$768$$ −1.00000 −0.0360844
$$769$$ −31.0000 −1.11789 −0.558944 0.829205i $$-0.688793\pi$$
−0.558944 + 0.829205i $$0.688793\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ 15.0000 0.539862
$$773$$ 42.0000 1.51064 0.755318 0.655359i $$-0.227483\pi$$
0.755318 + 0.655359i $$0.227483\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ −44.0000 −1.58053
$$776$$ −13.0000 −0.466673
$$777$$ 0 0
$$778$$ 22.0000 0.788738
$$779$$ −48.0000 −1.71978
$$780$$ −1.00000 −0.0358057
$$781$$ −6.00000 −0.214697
$$782$$ −36.0000 −1.28736
$$783$$ −3.00000 −0.107211
$$784$$ 0 0
$$785$$ 18.0000 0.642448
$$786$$ 1.00000 0.0356688
$$787$$ 12.0000 0.427754 0.213877 0.976861i $$-0.431391\pi$$
0.213877 + 0.976861i $$0.431391\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ 18.0000 0.640817
$$790$$ 7.00000 0.249049
$$791$$ 0 0
$$792$$ −1.00000 −0.0355335
$$793$$ −12.0000 −0.426132
$$794$$ 18.0000 0.638796
$$795$$ 5.00000 0.177332
$$796$$ −8.00000 −0.283552
$$797$$ 3.00000 0.106265 0.0531327 0.998587i $$-0.483079\pi$$
0.0531327 + 0.998587i $$0.483079\pi$$
$$798$$ 0 0
$$799$$ 48.0000 1.69812
$$800$$ −4.00000 −0.141421
$$801$$ 12.0000 0.423999
$$802$$ −18.0000 −0.635602
$$803$$ −10.0000 −0.352892
$$804$$ −16.0000 −0.564276
$$805$$ 0 0
$$806$$ 11.0000 0.387458
$$807$$ −9.00000 −0.316815
$$808$$ −10.0000 −0.351799
$$809$$ 12.0000 0.421898 0.210949 0.977497i $$-0.432345\pi$$
0.210949 + 0.977497i $$0.432345\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 40.0000 1.40459 0.702295 0.711886i $$-0.252159\pi$$
0.702295 + 0.711886i $$0.252159\pi$$
$$812$$ 0 0
$$813$$ −7.00000 −0.245501
$$814$$ −4.00000 −0.140200
$$815$$ −8.00000 −0.280228
$$816$$ −6.00000 −0.210042
$$817$$ −32.0000 −1.11954
$$818$$ −25.0000 −0.874105
$$819$$ 0 0
$$820$$ −12.0000 −0.419058
$$821$$ −11.0000 −0.383903 −0.191951 0.981404i $$-0.561482\pi$$
−0.191951 + 0.981404i $$0.561482\pi$$
$$822$$ 8.00000 0.279032
$$823$$ 44.0000 1.53374 0.766872 0.641800i $$-0.221812\pi$$
0.766872 + 0.641800i $$0.221812\pi$$
$$824$$ 0 0
$$825$$ −4.00000 −0.139262
$$826$$ 0 0
$$827$$ −25.0000 −0.869335 −0.434668 0.900591i $$-0.643134\pi$$
−0.434668 + 0.900591i $$0.643134\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 17.0000 0.590079
$$831$$ 10.0000 0.346896
$$832$$ 1.00000 0.0346688
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 24.0000 0.830554
$$836$$ −4.00000 −0.138343
$$837$$ −11.0000 −0.380216
$$838$$ −16.0000 −0.552711
$$839$$ 16.0000 0.552381 0.276191 0.961103i $$-0.410928\pi$$
0.276191 + 0.961103i $$0.410928\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ −36.0000 −1.24064
$$843$$ −4.00000 −0.137767
$$844$$ 12.0000 0.413057
$$845$$ 1.00000 0.0344010
$$846$$ 8.00000 0.275046
$$847$$ 0 0
$$848$$ −5.00000 −0.171701
$$849$$ 10.0000 0.343199
$$850$$ −24.0000 −0.823193
$$851$$ −24.0000 −0.822709
$$852$$ −6.00000 −0.205557
$$853$$ −26.0000 −0.890223 −0.445112 0.895475i $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ 7.00000 0.239255
$$857$$ −54.0000 −1.84460 −0.922302 0.386469i $$-0.873695\pi$$
−0.922302 + 0.386469i $$0.873695\pi$$
$$858$$ 1.00000 0.0341394
$$859$$ 8.00000 0.272956 0.136478 0.990643i $$-0.456422\pi$$
0.136478 + 0.990643i $$0.456422\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ −38.0000 −1.29429
$$863$$ −4.00000 −0.136162 −0.0680808 0.997680i $$-0.521688\pi$$
−0.0680808 + 0.997680i $$0.521688\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −6.00000 −0.204006
$$866$$ 14.0000 0.475739
$$867$$ −19.0000 −0.645274
$$868$$ 0 0
$$869$$ −7.00000 −0.237459
$$870$$ −3.00000 −0.101710
$$871$$ 16.0000 0.542139
$$872$$ 12.0000 0.406371
$$873$$ −13.0000 −0.439983
$$874$$ −24.0000 −0.811812
$$875$$ 0 0
$$876$$ −10.0000 −0.337869
$$877$$ 18.0000 0.607817 0.303908 0.952701i $$-0.401708\pi$$
0.303908 + 0.952701i $$0.401708\pi$$
$$878$$ 17.0000 0.573722
$$879$$ −7.00000 −0.236104
$$880$$ −1.00000 −0.0337100
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 0 0
$$883$$ 30.0000 1.00958 0.504790 0.863242i $$-0.331570\pi$$
0.504790 + 0.863242i $$0.331570\pi$$
$$884$$ 6.00000 0.201802
$$885$$ −5.00000 −0.168073
$$886$$ −39.0000 −1.31023
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ −4.00000 −0.134231
$$889$$ 0 0
$$890$$ 12.0000 0.402241
$$891$$ −1.00000 −0.0335013
$$892$$ −7.00000 −0.234377
$$893$$ 32.0000 1.07084
$$894$$ 10.0000 0.334450
$$895$$ 12.0000 0.401116
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ 20.0000 0.667409
$$899$$ 33.0000 1.10061
$$900$$ −4.00000 −0.133333
$$901$$ −30.0000 −0.999445
$$902$$ 12.0000 0.399556
$$903$$ 0 0
$$904$$ −4.00000 −0.133038
$$905$$ 16.0000 0.531858
$$906$$ −13.0000 −0.431896
$$907$$ −36.0000 −1.19536 −0.597680 0.801735i $$-0.703911\pi$$
−0.597680 + 0.801735i $$0.703911\pi$$
$$908$$ 15.0000 0.497792
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ −10.0000 −0.331315 −0.165657 0.986183i $$-0.552975\pi$$
−0.165657 + 0.986183i $$0.552975\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ −17.0000 −0.562618
$$914$$ −11.0000 −0.363848
$$915$$ 12.0000 0.396708
$$916$$ −6.00000 −0.198246
$$917$$ 0 0
$$918$$ −6.00000 −0.198030
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ −6.00000 −0.197814
$$921$$ 6.00000 0.197707
$$922$$ −34.0000 −1.11973
$$923$$ 6.00000 0.197492
$$924$$ 0 0
$$925$$ −16.0000 −0.526077
$$926$$ −24.0000 −0.788689
$$927$$ 0 0
$$928$$ 3.00000 0.0984798
$$929$$ 20.0000 0.656179 0.328089 0.944647i $$-0.393595\pi$$
0.328089 + 0.944647i $$0.393595\pi$$
$$930$$ −11.0000 −0.360704
$$931$$ 0 0
$$932$$ 6.00000 0.196537
$$933$$ 2.00000 0.0654771
$$934$$ −12.0000 −0.392652
$$935$$ −6.00000 −0.196221
$$936$$ 1.00000 0.0326860
$$937$$ 13.0000 0.424691 0.212346 0.977195i $$-0.431890\pi$$
0.212346 + 0.977195i $$0.431890\pi$$
$$938$$ 0 0
$$939$$ 13.0000 0.424239
$$940$$ 8.00000 0.260931
$$941$$ −15.0000 −0.488986 −0.244493 0.969651i $$-0.578622\pi$$
−0.244493 + 0.969651i $$0.578622\pi$$
$$942$$ −18.0000 −0.586472
$$943$$ 72.0000 2.34464
$$944$$ 5.00000 0.162736
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ −7.00000 −0.227349
$$949$$ 10.0000 0.324614
$$950$$ −16.0000 −0.519109
$$951$$ 3.00000 0.0972817
$$952$$ 0 0
$$953$$ −44.0000 −1.42530 −0.712650 0.701520i $$-0.752505\pi$$
−0.712650 + 0.701520i $$0.752505\pi$$
$$954$$ −5.00000 −0.161881
$$955$$ −6.00000 −0.194155
$$956$$ −6.00000 −0.194054
$$957$$ 3.00000 0.0969762
$$958$$ −14.0000 −0.452319
$$959$$ 0 0
$$960$$ −1.00000 −0.0322749
$$961$$ 90.0000 2.90323
$$962$$ 4.00000 0.128965
$$963$$ 7.00000 0.225572
$$964$$ 15.0000 0.483117
$$965$$ 15.0000 0.482867
$$966$$ 0 0
$$967$$ −11.0000 −0.353736 −0.176868 0.984235i $$-0.556597\pi$$
−0.176868 + 0.984235i $$0.556597\pi$$
$$968$$ −10.0000 −0.321412
$$969$$ −24.0000 −0.770991
$$970$$ −13.0000 −0.417405
$$971$$ 29.0000 0.930654 0.465327 0.885139i $$-0.345937\pi$$
0.465327 + 0.885139i $$0.345937\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ 23.0000 0.736968
$$975$$ 4.00000 0.128103
$$976$$ −12.0000 −0.384111
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 8.00000 0.255812
$$979$$ −12.0000 −0.383522
$$980$$ 0 0
$$981$$ 12.0000 0.383131
$$982$$ −35.0000 −1.11689
$$983$$ 26.0000 0.829271 0.414636 0.909988i $$-0.363909\pi$$
0.414636 + 0.909988i $$0.363909\pi$$
$$984$$ 12.0000 0.382546
$$985$$ −18.0000 −0.573528
$$986$$ 18.0000 0.573237
$$987$$ 0 0
$$988$$ 4.00000 0.127257
$$989$$ 48.0000 1.52631
$$990$$ −1.00000 −0.0317821
$$991$$ −5.00000 −0.158830 −0.0794151 0.996842i $$-0.525305\pi$$
−0.0794151 + 0.996842i $$0.525305\pi$$
$$992$$ 11.0000 0.349250
$$993$$ 8.00000 0.253872
$$994$$ 0 0
$$995$$ −8.00000 −0.253617
$$996$$ −17.0000 −0.538666
$$997$$ 28.0000 0.886769 0.443384 0.896332i $$-0.353778\pi$$
0.443384 + 0.896332i $$0.353778\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ −4.00000 −0.126554
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 3822.2.a.v.1.1 1
7.3 odd 6 546.2.i.b.79.1 2
7.5 odd 6 546.2.i.b.235.1 yes 2
7.6 odd 2 3822.2.a.be.1.1 1
21.5 even 6 1638.2.j.h.235.1 2
21.17 even 6 1638.2.j.h.1171.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.b.79.1 2 7.3 odd 6
546.2.i.b.235.1 yes 2 7.5 odd 6
1638.2.j.h.235.1 2 21.5 even 6
1638.2.j.h.1171.1 2 21.17 even 6
3822.2.a.v.1.1 1 1.1 even 1 trivial
3822.2.a.be.1.1 1 7.6 odd 2