# Properties

 Label 3822.2.a.u.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 $1$

# Related objects

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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{16} +7.00000 q^{17} +1.00000 q^{18} +7.00000 q^{19} -5.00000 q^{22} +2.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -9.00000 q^{29} +1.00000 q^{32} +5.00000 q^{33} +7.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} +7.00000 q^{38} +1.00000 q^{39} +4.00000 q^{41} +2.00000 q^{43} -5.00000 q^{44} +2.00000 q^{46} -3.00000 q^{47} -1.00000 q^{48} -5.00000 q^{50} -7.00000 q^{51} -1.00000 q^{52} +1.00000 q^{53} -1.00000 q^{54} -7.00000 q^{57} -9.00000 q^{58} +7.00000 q^{59} +13.0000 q^{61} +1.00000 q^{64} +5.00000 q^{66} +3.00000 q^{67} +7.00000 q^{68} -2.00000 q^{69} +9.00000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +4.00000 q^{74} +5.00000 q^{75} +7.00000 q^{76} +1.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} +4.00000 q^{82} +16.0000 q^{83} +2.00000 q^{86} +9.00000 q^{87} -5.00000 q^{88} -12.0000 q^{89} +2.00000 q^{92} -3.00000 q^{94} -1.00000 q^{96} +6.00000 q^{97} -5.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 7.00000 1.69775 0.848875 0.528594i $$-0.177281\pi$$
0.848875 + 0.528594i $$0.177281\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 7.00000 1.60591 0.802955 0.596040i $$-0.203260\pi$$
0.802955 + 0.596040i $$0.203260\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −5.00000 −1.06600
$$23$$ 2.00000 0.417029 0.208514 0.978019i $$-0.433137\pi$$
0.208514 + 0.978019i $$0.433137\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −5.00000 −1.00000
$$26$$ −1.00000 −0.196116
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 5.00000 0.870388
$$34$$ 7.00000 1.20049
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 7.00000 1.13555
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ −5.00000 −0.753778
$$45$$ 0 0
$$46$$ 2.00000 0.294884
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ −5.00000 −0.707107
$$51$$ −7.00000 −0.980196
$$52$$ −1.00000 −0.138675
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −7.00000 −0.927173
$$58$$ −9.00000 −1.18176
$$59$$ 7.00000 0.911322 0.455661 0.890153i $$-0.349403\pi$$
0.455661 + 0.890153i $$0.349403\pi$$
$$60$$ 0 0
$$61$$ 13.0000 1.66448 0.832240 0.554416i $$-0.187058\pi$$
0.832240 + 0.554416i $$0.187058\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 5.00000 0.615457
$$67$$ 3.00000 0.366508 0.183254 0.983066i $$-0.441337\pi$$
0.183254 + 0.983066i $$0.441337\pi$$
$$68$$ 7.00000 0.848875
$$69$$ −2.00000 −0.240772
$$70$$ 0 0
$$71$$ 9.00000 1.06810 0.534052 0.845452i $$-0.320669\pi$$
0.534052 + 0.845452i $$0.320669\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 5.00000 0.577350
$$76$$ 7.00000 0.802955
$$77$$ 0 0
$$78$$ 1.00000 0.113228
$$79$$ 14.0000 1.57512 0.787562 0.616236i $$-0.211343\pi$$
0.787562 + 0.616236i $$0.211343\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 4.00000 0.441726
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 9.00000 0.964901
$$88$$ −5.00000 −0.533002
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 2.00000 0.208514
$$93$$ 0 0
$$94$$ −3.00000 −0.309426
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ −5.00000 −0.502519
$$100$$ −5.00000 −0.500000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ −7.00000 −0.693103
$$103$$ −10.0000 −0.985329 −0.492665 0.870219i $$-0.663977\pi$$
−0.492665 + 0.870219i $$0.663977\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 1.00000 0.0971286
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 8.00000 0.766261 0.383131 0.923694i $$-0.374846\pi$$
0.383131 + 0.923694i $$0.374846\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ 1.00000 0.0940721 0.0470360 0.998893i $$-0.485022\pi$$
0.0470360 + 0.998893i $$0.485022\pi$$
$$114$$ −7.00000 −0.655610
$$115$$ 0 0
$$116$$ −9.00000 −0.835629
$$117$$ −1.00000 −0.0924500
$$118$$ 7.00000 0.644402
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 13.0000 1.17696
$$123$$ −4.00000 −0.360668
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 22.0000 1.95218 0.976092 0.217357i $$-0.0697436\pi$$
0.976092 + 0.217357i $$0.0697436\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ 5.00000 0.435194
$$133$$ 0 0
$$134$$ 3.00000 0.259161
$$135$$ 0 0
$$136$$ 7.00000 0.600245
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ −2.00000 −0.170251
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ 3.00000 0.252646
$$142$$ 9.00000 0.755263
$$143$$ 5.00000 0.418121
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ 0 0
$$148$$ 4.00000 0.328798
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 5.00000 0.408248
$$151$$ −17.0000 −1.38344 −0.691720 0.722166i $$-0.743147\pi$$
−0.691720 + 0.722166i $$0.743147\pi$$
$$152$$ 7.00000 0.567775
$$153$$ 7.00000 0.565916
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 1.00000 0.0800641
$$157$$ 11.0000 0.877896 0.438948 0.898513i $$-0.355351\pi$$
0.438948 + 0.898513i $$0.355351\pi$$
$$158$$ 14.0000 1.11378
$$159$$ −1.00000 −0.0793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 13.0000 1.01824 0.509119 0.860696i $$-0.329971\pi$$
0.509119 + 0.860696i $$0.329971\pi$$
$$164$$ 4.00000 0.312348
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ −5.00000 −0.386912 −0.193456 0.981109i $$-0.561970\pi$$
−0.193456 + 0.981109i $$0.561970\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 7.00000 0.535303
$$172$$ 2.00000 0.152499
$$173$$ −13.0000 −0.988372 −0.494186 0.869356i $$-0.664534\pi$$
−0.494186 + 0.869356i $$0.664534\pi$$
$$174$$ 9.00000 0.682288
$$175$$ 0 0
$$176$$ −5.00000 −0.376889
$$177$$ −7.00000 −0.526152
$$178$$ −12.0000 −0.899438
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ −11.0000 −0.817624 −0.408812 0.912619i $$-0.634057\pi$$
−0.408812 + 0.912619i $$0.634057\pi$$
$$182$$ 0 0
$$183$$ −13.0000 −0.960988
$$184$$ 2.00000 0.147442
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −35.0000 −2.55945
$$188$$ −3.00000 −0.218797
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −8.00000 −0.575853 −0.287926 0.957653i $$-0.592966\pi$$
−0.287926 + 0.957653i $$0.592966\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 22.0000 1.56744 0.783718 0.621117i $$-0.213321\pi$$
0.783718 + 0.621117i $$0.213321\pi$$
$$198$$ −5.00000 −0.355335
$$199$$ 18.0000 1.27599 0.637993 0.770042i $$-0.279765\pi$$
0.637993 + 0.770042i $$0.279765\pi$$
$$200$$ −5.00000 −0.353553
$$201$$ −3.00000 −0.211604
$$202$$ −6.00000 −0.422159
$$203$$ 0 0
$$204$$ −7.00000 −0.490098
$$205$$ 0 0
$$206$$ −10.0000 −0.696733
$$207$$ 2.00000 0.139010
$$208$$ −1.00000 −0.0693375
$$209$$ −35.0000 −2.42100
$$210$$ 0 0
$$211$$ −26.0000 −1.78991 −0.894957 0.446153i $$-0.852794\pi$$
−0.894957 + 0.446153i $$0.852794\pi$$
$$212$$ 1.00000 0.0686803
$$213$$ −9.00000 −0.616670
$$214$$ 0 0
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 8.00000 0.541828
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ −7.00000 −0.470871
$$222$$ −4.00000 −0.268462
$$223$$ −1.00000 −0.0669650 −0.0334825 0.999439i $$-0.510660\pi$$
−0.0334825 + 0.999439i $$0.510660\pi$$
$$224$$ 0 0
$$225$$ −5.00000 −0.333333
$$226$$ 1.00000 0.0665190
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ −7.00000 −0.463586
$$229$$ 18.0000 1.18947 0.594737 0.803921i $$-0.297256\pi$$
0.594737 + 0.803921i $$0.297256\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −9.00000 −0.590879
$$233$$ −3.00000 −0.196537 −0.0982683 0.995160i $$-0.531330\pi$$
−0.0982683 + 0.995160i $$0.531330\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ 7.00000 0.455661
$$237$$ −14.0000 −0.909398
$$238$$ 0 0
$$239$$ −1.00000 −0.0646846 −0.0323423 0.999477i $$-0.510297\pi$$
−0.0323423 + 0.999477i $$0.510297\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 14.0000 0.899954
$$243$$ −1.00000 −0.0641500
$$244$$ 13.0000 0.832240
$$245$$ 0 0
$$246$$ −4.00000 −0.255031
$$247$$ −7.00000 −0.445399
$$248$$ 0 0
$$249$$ −16.0000 −1.01396
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 0 0
$$253$$ −10.0000 −0.628695
$$254$$ 22.0000 1.38040
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ −2.00000 −0.124515
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −9.00000 −0.557086
$$262$$ 18.0000 1.11204
$$263$$ −4.00000 −0.246651 −0.123325 0.992366i $$-0.539356\pi$$
−0.123325 + 0.992366i $$0.539356\pi$$
$$264$$ 5.00000 0.307729
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 12.0000 0.734388
$$268$$ 3.00000 0.183254
$$269$$ 23.0000 1.40233 0.701167 0.712997i $$-0.252663\pi$$
0.701167 + 0.712997i $$0.252663\pi$$
$$270$$ 0 0
$$271$$ 3.00000 0.182237 0.0911185 0.995840i $$-0.470956\pi$$
0.0911185 + 0.995840i $$0.470956\pi$$
$$272$$ 7.00000 0.424437
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ 25.0000 1.50756
$$276$$ −2.00000 −0.120386
$$277$$ 19.0000 1.14160 0.570800 0.821089i $$-0.306633\pi$$
0.570800 + 0.821089i $$0.306633\pi$$
$$278$$ 16.0000 0.959616
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −24.0000 −1.43172 −0.715860 0.698244i $$-0.753965\pi$$
−0.715860 + 0.698244i $$0.753965\pi$$
$$282$$ 3.00000 0.178647
$$283$$ −10.0000 −0.594438 −0.297219 0.954809i $$-0.596059\pi$$
−0.297219 + 0.954809i $$0.596059\pi$$
$$284$$ 9.00000 0.534052
$$285$$ 0 0
$$286$$ 5.00000 0.295656
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ −6.00000 −0.351726
$$292$$ −10.0000 −0.585206
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 4.00000 0.232495
$$297$$ 5.00000 0.290129
$$298$$ 10.0000 0.579284
$$299$$ −2.00000 −0.115663
$$300$$ 5.00000 0.288675
$$301$$ 0 0
$$302$$ −17.0000 −0.978240
$$303$$ 6.00000 0.344691
$$304$$ 7.00000 0.401478
$$305$$ 0 0
$$306$$ 7.00000 0.400163
$$307$$ 19.0000 1.08439 0.542194 0.840254i $$-0.317594\pi$$
0.542194 + 0.840254i $$0.317594\pi$$
$$308$$ 0 0
$$309$$ 10.0000 0.568880
$$310$$ 0 0
$$311$$ 20.0000 1.13410 0.567048 0.823685i $$-0.308085\pi$$
0.567048 + 0.823685i $$0.308085\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 11.0000 0.620766
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ 26.0000 1.46031 0.730153 0.683284i $$-0.239449\pi$$
0.730153 + 0.683284i $$0.239449\pi$$
$$318$$ −1.00000 −0.0560772
$$319$$ 45.0000 2.51952
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 49.0000 2.72643
$$324$$ 1.00000 0.0555556
$$325$$ 5.00000 0.277350
$$326$$ 13.0000 0.720003
$$327$$ −8.00000 −0.442401
$$328$$ 4.00000 0.220863
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 16.0000 0.878114
$$333$$ 4.00000 0.219199
$$334$$ −5.00000 −0.273588
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −33.0000 −1.79762 −0.898812 0.438334i $$-0.855569\pi$$
−0.898812 + 0.438334i $$0.855569\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ −1.00000 −0.0543125
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 7.00000 0.378517
$$343$$ 0 0
$$344$$ 2.00000 0.107833
$$345$$ 0 0
$$346$$ −13.0000 −0.698884
$$347$$ −16.0000 −0.858925 −0.429463 0.903085i $$-0.641297\pi$$
−0.429463 + 0.903085i $$0.641297\pi$$
$$348$$ 9.00000 0.482451
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ −5.00000 −0.266501
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ −7.00000 −0.372046
$$355$$ 0 0
$$356$$ −12.0000 −0.635999
$$357$$ 0 0
$$358$$ −10.0000 −0.528516
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ −11.0000 −0.578147
$$363$$ −14.0000 −0.734809
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −13.0000 −0.679521
$$367$$ 2.00000 0.104399 0.0521996 0.998637i $$-0.483377\pi$$
0.0521996 + 0.998637i $$0.483377\pi$$
$$368$$ 2.00000 0.104257
$$369$$ 4.00000 0.208232
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −21.0000 −1.08734 −0.543669 0.839299i $$-0.682965\pi$$
−0.543669 + 0.839299i $$0.682965\pi$$
$$374$$ −35.0000 −1.80981
$$375$$ 0 0
$$376$$ −3.00000 −0.154713
$$377$$ 9.00000 0.463524
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ −22.0000 −1.12709
$$382$$ −12.0000 −0.613973
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −8.00000 −0.407189
$$387$$ 2.00000 0.101666
$$388$$ 6.00000 0.304604
$$389$$ −23.0000 −1.16615 −0.583073 0.812420i $$-0.698150\pi$$
−0.583073 + 0.812420i $$0.698150\pi$$
$$390$$ 0 0
$$391$$ 14.0000 0.708010
$$392$$ 0 0
$$393$$ −18.0000 −0.907980
$$394$$ 22.0000 1.10834
$$395$$ 0 0
$$396$$ −5.00000 −0.251259
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 18.0000 0.902258
$$399$$ 0 0
$$400$$ −5.00000 −0.250000
$$401$$ −38.0000 −1.89763 −0.948815 0.315833i $$-0.897716\pi$$
−0.948815 + 0.315833i $$0.897716\pi$$
$$402$$ −3.00000 −0.149626
$$403$$ 0 0
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −20.0000 −0.991363
$$408$$ −7.00000 −0.346552
$$409$$ −4.00000 −0.197787 −0.0988936 0.995098i $$-0.531530\pi$$
−0.0988936 + 0.995098i $$0.531530\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ −10.0000 −0.492665
$$413$$ 0 0
$$414$$ 2.00000 0.0982946
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ −16.0000 −0.783523
$$418$$ −35.0000 −1.71191
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ −26.0000 −1.26566
$$423$$ −3.00000 −0.145865
$$424$$ 1.00000 0.0485643
$$425$$ −35.0000 −1.69775
$$426$$ −9.00000 −0.436051
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −5.00000 −0.241402
$$430$$ 0 0
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −11.0000 −0.528626 −0.264313 0.964437i $$-0.585145\pi$$
−0.264313 + 0.964437i $$0.585145\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 8.00000 0.383131
$$437$$ 14.0000 0.669711
$$438$$ 10.0000 0.477818
$$439$$ −22.0000 −1.05000 −0.525001 0.851101i $$-0.675935\pi$$
−0.525001 + 0.851101i $$0.675935\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −7.00000 −0.332956
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ −1.00000 −0.0473514
$$447$$ −10.0000 −0.472984
$$448$$ 0 0
$$449$$ 4.00000 0.188772 0.0943858 0.995536i $$-0.469911\pi$$
0.0943858 + 0.995536i $$0.469911\pi$$
$$450$$ −5.00000 −0.235702
$$451$$ −20.0000 −0.941763
$$452$$ 1.00000 0.0470360
$$453$$ 17.0000 0.798730
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −7.00000 −0.327805
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 18.0000 0.841085
$$459$$ −7.00000 −0.326732
$$460$$ 0 0
$$461$$ −8.00000 −0.372597 −0.186299 0.982493i $$-0.559649\pi$$
−0.186299 + 0.982493i $$0.559649\pi$$
$$462$$ 0 0
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ −9.00000 −0.417815
$$465$$ 0 0
$$466$$ −3.00000 −0.138972
$$467$$ −6.00000 −0.277647 −0.138823 0.990317i $$-0.544332\pi$$
−0.138823 + 0.990317i $$0.544332\pi$$
$$468$$ −1.00000 −0.0462250
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −11.0000 −0.506853
$$472$$ 7.00000 0.322201
$$473$$ −10.0000 −0.459800
$$474$$ −14.0000 −0.643041
$$475$$ −35.0000 −1.60591
$$476$$ 0 0
$$477$$ 1.00000 0.0457869
$$478$$ −1.00000 −0.0457389
$$479$$ −11.0000 −0.502603 −0.251301 0.967909i $$-0.580859\pi$$
−0.251301 + 0.967909i $$0.580859\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 14.0000 0.636364
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −5.00000 −0.226572 −0.113286 0.993562i $$-0.536138\pi$$
−0.113286 + 0.993562i $$0.536138\pi$$
$$488$$ 13.0000 0.588482
$$489$$ −13.0000 −0.587880
$$490$$ 0 0
$$491$$ −22.0000 −0.992846 −0.496423 0.868081i $$-0.665354\pi$$
−0.496423 + 0.868081i $$0.665354\pi$$
$$492$$ −4.00000 −0.180334
$$493$$ −63.0000 −2.83738
$$494$$ −7.00000 −0.314945
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ −16.0000 −0.716977
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ 0 0
$$501$$ 5.00000 0.223384
$$502$$ 18.0000 0.803379
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −10.0000 −0.444554
$$507$$ −1.00000 −0.0444116
$$508$$ 22.0000 0.976092
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −7.00000 −0.309058
$$514$$ −14.0000 −0.617514
$$515$$ 0 0
$$516$$ −2.00000 −0.0880451
$$517$$ 15.0000 0.659699
$$518$$ 0 0
$$519$$ 13.0000 0.570637
$$520$$ 0 0
$$521$$ −38.0000 −1.66481 −0.832405 0.554168i $$-0.813037\pi$$
−0.832405 + 0.554168i $$0.813037\pi$$
$$522$$ −9.00000 −0.393919
$$523$$ 26.0000 1.13690 0.568450 0.822718i $$-0.307543\pi$$
0.568450 + 0.822718i $$0.307543\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 0 0
$$526$$ −4.00000 −0.174408
$$527$$ 0 0
$$528$$ 5.00000 0.217597
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ 7.00000 0.303774
$$532$$ 0 0
$$533$$ −4.00000 −0.173259
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ 3.00000 0.129580
$$537$$ 10.0000 0.431532
$$538$$ 23.0000 0.991600
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 16.0000 0.687894 0.343947 0.938989i $$-0.388236\pi$$
0.343947 + 0.938989i $$0.388236\pi$$
$$542$$ 3.00000 0.128861
$$543$$ 11.0000 0.472055
$$544$$ 7.00000 0.300123
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −38.0000 −1.62476 −0.812381 0.583127i $$-0.801829\pi$$
−0.812381 + 0.583127i $$0.801829\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ 13.0000 0.554826
$$550$$ 25.0000 1.06600
$$551$$ −63.0000 −2.68389
$$552$$ −2.00000 −0.0851257
$$553$$ 0 0
$$554$$ 19.0000 0.807233
$$555$$ 0 0
$$556$$ 16.0000 0.678551
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ −2.00000 −0.0845910
$$560$$ 0 0
$$561$$ 35.0000 1.47770
$$562$$ −24.0000 −1.01238
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 3.00000 0.126323
$$565$$ 0 0
$$566$$ −10.0000 −0.420331
$$567$$ 0 0
$$568$$ 9.00000 0.377632
$$569$$ −31.0000 −1.29959 −0.649794 0.760111i $$-0.725145\pi$$
−0.649794 + 0.760111i $$0.725145\pi$$
$$570$$ 0 0
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 5.00000 0.209061
$$573$$ 12.0000 0.501307
$$574$$ 0 0
$$575$$ −10.0000 −0.417029
$$576$$ 1.00000 0.0416667
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ 32.0000 1.33102
$$579$$ 8.00000 0.332469
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −6.00000 −0.248708
$$583$$ −5.00000 −0.207079
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ −30.0000 −1.23929
$$587$$ −45.0000 −1.85735 −0.928674 0.370896i $$-0.879051\pi$$
−0.928674 + 0.370896i $$0.879051\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −22.0000 −0.904959
$$592$$ 4.00000 0.164399
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 5.00000 0.205152
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ −18.0000 −0.736691
$$598$$ −2.00000 −0.0817861
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 5.00000 0.204124
$$601$$ 25.0000 1.01977 0.509886 0.860242i $$-0.329688\pi$$
0.509886 + 0.860242i $$0.329688\pi$$
$$602$$ 0 0
$$603$$ 3.00000 0.122169
$$604$$ −17.0000 −0.691720
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ −14.0000 −0.568242 −0.284121 0.958788i $$-0.591702\pi$$
−0.284121 + 0.958788i $$0.591702\pi$$
$$608$$ 7.00000 0.283887
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 3.00000 0.121367
$$612$$ 7.00000 0.282958
$$613$$ 8.00000 0.323117 0.161558 0.986863i $$-0.448348\pi$$
0.161558 + 0.986863i $$0.448348\pi$$
$$614$$ 19.0000 0.766778
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 10.0000 0.402259
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ −2.00000 −0.0802572
$$622$$ 20.0000 0.801927
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ 25.0000 1.00000
$$626$$ −6.00000 −0.239808
$$627$$ 35.0000 1.39777
$$628$$ 11.0000 0.438948
$$629$$ 28.0000 1.11643
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 14.0000 0.556890
$$633$$ 26.0000 1.03341
$$634$$ 26.0000 1.03259
$$635$$ 0 0
$$636$$ −1.00000 −0.0396526
$$637$$ 0 0
$$638$$ 45.0000 1.78157
$$639$$ 9.00000 0.356034
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ −9.00000 −0.354925 −0.177463 0.984128i $$-0.556789\pi$$
−0.177463 + 0.984128i $$0.556789\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 49.0000 1.92788
$$647$$ −6.00000 −0.235884 −0.117942 0.993020i $$-0.537630\pi$$
−0.117942 + 0.993020i $$0.537630\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −35.0000 −1.37387
$$650$$ 5.00000 0.196116
$$651$$ 0 0
$$652$$ 13.0000 0.509119
$$653$$ −22.0000 −0.860927 −0.430463 0.902608i $$-0.641650\pi$$
−0.430463 + 0.902608i $$0.641650\pi$$
$$654$$ −8.00000 −0.312825
$$655$$ 0 0
$$656$$ 4.00000 0.156174
$$657$$ −10.0000 −0.390137
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −4.00000 −0.155582 −0.0777910 0.996970i $$-0.524787\pi$$
−0.0777910 + 0.996970i $$0.524787\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 7.00000 0.271857
$$664$$ 16.0000 0.620920
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ −18.0000 −0.696963
$$668$$ −5.00000 −0.193456
$$669$$ 1.00000 0.0386622
$$670$$ 0 0
$$671$$ −65.0000 −2.50930
$$672$$ 0 0
$$673$$ 22.0000 0.848038 0.424019 0.905653i $$-0.360619\pi$$
0.424019 + 0.905653i $$0.360619\pi$$
$$674$$ −33.0000 −1.27111
$$675$$ 5.00000 0.192450
$$676$$ 1.00000 0.0384615
$$677$$ 25.0000 0.960828 0.480414 0.877042i $$-0.340486\pi$$
0.480414 + 0.877042i $$0.340486\pi$$
$$678$$ −1.00000 −0.0384048
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ −40.0000 −1.53056 −0.765279 0.643699i $$-0.777399\pi$$
−0.765279 + 0.643699i $$0.777399\pi$$
$$684$$ 7.00000 0.267652
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −18.0000 −0.686743
$$688$$ 2.00000 0.0762493
$$689$$ −1.00000 −0.0380970
$$690$$ 0 0
$$691$$ 5.00000 0.190209 0.0951045 0.995467i $$-0.469681\pi$$
0.0951045 + 0.995467i $$0.469681\pi$$
$$692$$ −13.0000 −0.494186
$$693$$ 0 0
$$694$$ −16.0000 −0.607352
$$695$$ 0 0
$$696$$ 9.00000 0.341144
$$697$$ 28.0000 1.06058
$$698$$ 2.00000 0.0757011
$$699$$ 3.00000 0.113470
$$700$$ 0 0
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 1.00000 0.0377426
$$703$$ 28.0000 1.05604
$$704$$ −5.00000 −0.188445
$$705$$ 0 0
$$706$$ 2.00000 0.0752710
$$707$$ 0 0
$$708$$ −7.00000 −0.263076
$$709$$ −46.0000 −1.72757 −0.863783 0.503864i $$-0.831911\pi$$
−0.863783 + 0.503864i $$0.831911\pi$$
$$710$$ 0 0
$$711$$ 14.0000 0.525041
$$712$$ −12.0000 −0.449719
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −10.0000 −0.373718
$$717$$ 1.00000 0.0373457
$$718$$ 0 0
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 30.0000 1.11648
$$723$$ 0 0
$$724$$ −11.0000 −0.408812
$$725$$ 45.0000 1.67126
$$726$$ −14.0000 −0.519589
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 14.0000 0.517809
$$732$$ −13.0000 −0.480494
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 2.00000 0.0738213
$$735$$ 0 0
$$736$$ 2.00000 0.0737210
$$737$$ −15.0000 −0.552532
$$738$$ 4.00000 0.147242
$$739$$ −28.0000 −1.03000 −0.514998 0.857191i $$-0.672207\pi$$
−0.514998 + 0.857191i $$0.672207\pi$$
$$740$$ 0 0
$$741$$ 7.00000 0.257151
$$742$$ 0 0
$$743$$ −5.00000 −0.183432 −0.0917161 0.995785i $$-0.529235\pi$$
−0.0917161 + 0.995785i $$0.529235\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −21.0000 −0.768865
$$747$$ 16.0000 0.585409
$$748$$ −35.0000 −1.27973
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ −3.00000 −0.109399
$$753$$ −18.0000 −0.655956
$$754$$ 9.00000 0.327761
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 29.0000 1.05402 0.527011 0.849858i $$-0.323312\pi$$
0.527011 + 0.849858i $$0.323312\pi$$
$$758$$ −4.00000 −0.145287
$$759$$ 10.0000 0.362977
$$760$$ 0 0
$$761$$ −8.00000 −0.290000 −0.145000 0.989432i $$-0.546318\pi$$
−0.145000 + 0.989432i $$0.546318\pi$$
$$762$$ −22.0000 −0.796976
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ −7.00000 −0.252755
$$768$$ −1.00000 −0.0360844
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 14.0000 0.504198
$$772$$ −8.00000 −0.287926
$$773$$ −34.0000 −1.22290 −0.611448 0.791285i $$-0.709412\pi$$
−0.611448 + 0.791285i $$0.709412\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 0 0
$$778$$ −23.0000 −0.824590
$$779$$ 28.0000 1.00320
$$780$$ 0 0
$$781$$ −45.0000 −1.61023
$$782$$ 14.0000 0.500639
$$783$$ 9.00000 0.321634
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −18.0000 −0.642039
$$787$$ 17.0000 0.605985 0.302992 0.952993i $$-0.402014\pi$$
0.302992 + 0.952993i $$0.402014\pi$$
$$788$$ 22.0000 0.783718
$$789$$ 4.00000 0.142404
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −5.00000 −0.177667
$$793$$ −13.0000 −0.461644
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ 18.0000 0.637993
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 0 0
$$799$$ −21.0000 −0.742927
$$800$$ −5.00000 −0.176777
$$801$$ −12.0000 −0.423999
$$802$$ −38.0000 −1.34183
$$803$$ 50.0000 1.76446
$$804$$ −3.00000 −0.105802
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −23.0000 −0.809638
$$808$$ −6.00000 −0.211079
$$809$$ 25.0000 0.878953 0.439477 0.898254i $$-0.355164\pi$$
0.439477 + 0.898254i $$0.355164\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ −3.00000 −0.105215
$$814$$ −20.0000 −0.701000
$$815$$ 0 0
$$816$$ −7.00000 −0.245049
$$817$$ 14.0000 0.489798
$$818$$ −4.00000 −0.139857
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −20.0000 −0.698005 −0.349002 0.937122i $$-0.613479\pi$$
−0.349002 + 0.937122i $$0.613479\pi$$
$$822$$ 12.0000 0.418548
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ −10.0000 −0.348367
$$825$$ −25.0000 −0.870388
$$826$$ 0 0
$$827$$ 21.0000 0.730242 0.365121 0.930960i $$-0.381028\pi$$
0.365121 + 0.930960i $$0.381028\pi$$
$$828$$ 2.00000 0.0695048
$$829$$ −41.0000 −1.42399 −0.711994 0.702185i $$-0.752208\pi$$
−0.711994 + 0.702185i $$0.752208\pi$$
$$830$$ 0 0
$$831$$ −19.0000 −0.659103
$$832$$ −1.00000 −0.0346688
$$833$$ 0 0
$$834$$ −16.0000 −0.554035
$$835$$ 0 0
$$836$$ −35.0000 −1.21050
$$837$$ 0 0
$$838$$ 26.0000 0.898155
$$839$$ 5.00000 0.172619 0.0863096 0.996268i $$-0.472493\pi$$
0.0863096 + 0.996268i $$0.472493\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 38.0000 1.30957
$$843$$ 24.0000 0.826604
$$844$$ −26.0000 −0.894957
$$845$$ 0 0
$$846$$ −3.00000 −0.103142
$$847$$ 0 0
$$848$$ 1.00000 0.0343401
$$849$$ 10.0000 0.343199
$$850$$ −35.0000 −1.20049
$$851$$ 8.00000 0.274236
$$852$$ −9.00000 −0.308335
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 13.0000 0.444072 0.222036 0.975039i $$-0.428730\pi$$
0.222036 + 0.975039i $$0.428730\pi$$
$$858$$ −5.00000 −0.170697
$$859$$ −8.00000 −0.272956 −0.136478 0.990643i $$-0.543578\pi$$
−0.136478 + 0.990643i $$0.543578\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −16.0000 −0.544962
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −11.0000 −0.373795
$$867$$ −32.0000 −1.08678
$$868$$ 0 0
$$869$$ −70.0000 −2.37459
$$870$$ 0 0
$$871$$ −3.00000 −0.101651
$$872$$ 8.00000 0.270914
$$873$$ 6.00000 0.203069
$$874$$ 14.0000 0.473557
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ −12.0000 −0.405211 −0.202606 0.979260i $$-0.564941\pi$$
−0.202606 + 0.979260i $$0.564941\pi$$
$$878$$ −22.0000 −0.742464
$$879$$ 30.0000 1.01187
$$880$$ 0 0
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ 0 0
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ −7.00000 −0.235435
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 42.0000 1.41022 0.705111 0.709097i $$-0.250897\pi$$
0.705111 + 0.709097i $$0.250897\pi$$
$$888$$ −4.00000 −0.134231
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −5.00000 −0.167506
$$892$$ −1.00000 −0.0334825
$$893$$ −21.0000 −0.702738
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 2.00000 0.0667781
$$898$$ 4.00000 0.133482
$$899$$ 0 0
$$900$$ −5.00000 −0.166667
$$901$$ 7.00000 0.233204
$$902$$ −20.0000 −0.665927
$$903$$ 0 0
$$904$$ 1.00000 0.0332595
$$905$$ 0 0
$$906$$ 17.0000 0.564787
$$907$$ 54.0000 1.79304 0.896520 0.443003i $$-0.146087\pi$$
0.896520 + 0.443003i $$0.146087\pi$$
$$908$$ 12.0000 0.398234
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ −7.00000 −0.231793
$$913$$ −80.0000 −2.64761
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 18.0000 0.594737
$$917$$ 0 0
$$918$$ −7.00000 −0.231034
$$919$$ −14.0000 −0.461817 −0.230909 0.972975i $$-0.574170\pi$$
−0.230909 + 0.972975i $$0.574170\pi$$
$$920$$ 0 0
$$921$$ −19.0000 −0.626071
$$922$$ −8.00000 −0.263466
$$923$$ −9.00000 −0.296239
$$924$$ 0 0
$$925$$ −20.0000 −0.657596
$$926$$ 32.0000 1.05159
$$927$$ −10.0000 −0.328443
$$928$$ −9.00000 −0.295439
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −3.00000 −0.0982683
$$933$$ −20.0000 −0.654771
$$934$$ −6.00000 −0.196326
$$935$$ 0 0
$$936$$ −1.00000 −0.0326860
$$937$$ −1.00000 −0.0326686 −0.0163343 0.999867i $$-0.505200\pi$$
−0.0163343 + 0.999867i $$0.505200\pi$$
$$938$$ 0 0
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ −8.00000 −0.260793 −0.130396 0.991462i $$-0.541625\pi$$
−0.130396 + 0.991462i $$0.541625\pi$$
$$942$$ −11.0000 −0.358399
$$943$$ 8.00000 0.260516
$$944$$ 7.00000 0.227831
$$945$$ 0 0
$$946$$ −10.0000 −0.325128
$$947$$ −5.00000 −0.162478 −0.0812391 0.996695i $$-0.525888\pi$$
−0.0812391 + 0.996695i $$0.525888\pi$$
$$948$$ −14.0000 −0.454699
$$949$$ 10.0000 0.324614
$$950$$ −35.0000 −1.13555
$$951$$ −26.0000 −0.843108
$$952$$ 0 0
$$953$$ 9.00000 0.291539 0.145769 0.989319i $$-0.453434\pi$$
0.145769 + 0.989319i $$0.453434\pi$$
$$954$$ 1.00000 0.0323762
$$955$$ 0 0
$$956$$ −1.00000 −0.0323423
$$957$$ −45.0000 −1.45464
$$958$$ −11.0000 −0.355394
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −4.00000 −0.128965
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 1.00000 0.0321578 0.0160789 0.999871i $$-0.494882\pi$$
0.0160789 + 0.999871i $$0.494882\pi$$
$$968$$ 14.0000 0.449977
$$969$$ −49.0000 −1.57411
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ −5.00000 −0.160210
$$975$$ −5.00000 −0.160128
$$976$$ 13.0000 0.416120
$$977$$ 2.00000 0.0639857 0.0319928 0.999488i $$-0.489815\pi$$
0.0319928 + 0.999488i $$0.489815\pi$$
$$978$$ −13.0000 −0.415694
$$979$$ 60.0000 1.91761
$$980$$ 0 0
$$981$$ 8.00000 0.255420
$$982$$ −22.0000 −0.702048
$$983$$ 59.0000 1.88181 0.940904 0.338674i $$-0.109978\pi$$
0.940904 + 0.338674i $$0.109978\pi$$
$$984$$ −4.00000 −0.127515
$$985$$ 0 0
$$986$$ −63.0000 −2.00633
$$987$$ 0 0
$$988$$ −7.00000 −0.222700
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ −4.00000 −0.126936
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −16.0000 −0.506979
$$997$$ 21.0000 0.665077 0.332538 0.943090i $$-0.392095\pi$$
0.332538 + 0.943090i $$0.392095\pi$$
$$998$$ 40.0000 1.26618
$$999$$ −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.u.1.1 1
7.2 even 3 546.2.i.d.235.1 yes 2
7.4 even 3 546.2.i.d.79.1 2
7.6 odd 2 3822.2.a.bf.1.1 1
21.2 odd 6 1638.2.j.i.235.1 2
21.11 odd 6 1638.2.j.i.1171.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.d.79.1 2 7.4 even 3
546.2.i.d.235.1 yes 2 7.2 even 3
1638.2.j.i.235.1 2 21.2 odd 6
1638.2.j.i.1171.1 2 21.11 odd 6
3822.2.a.u.1.1 1 1.1 even 1 trivial
3822.2.a.bf.1.1 1 7.6 odd 2