# Properties

 Label 3234.2.a.b.1.1 Level $3234$ Weight $2$ Character 3234.1 Self dual yes Analytic conductor $25.824$ 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] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, 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("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.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: $$25.8236200137$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3234.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} +6.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} +6.00000 q^{38} +6.00000 q^{39} +4.00000 q^{41} +8.00000 q^{43} -1.00000 q^{44} +4.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +5.00000 q^{50} +4.00000 q^{51} -6.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} +6.00000 q^{57} -6.00000 q^{58} +2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -1.00000 q^{66} -4.00000 q^{67} -4.00000 q^{68} +4.00000 q^{69} +16.0000 q^{71} -1.00000 q^{72} -12.0000 q^{73} -10.0000 q^{74} +5.00000 q^{75} -6.00000 q^{76} -6.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} -4.00000 q^{82} +2.00000 q^{83} -8.00000 q^{86} -6.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} -4.00000 q^{92} -2.00000 q^{93} -6.00000 q^{94} +1.00000 q^{96} +6.00000 q^{97} -1.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 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
$$11$$ −1.00000 −0.301511
$$12$$ −1.00000 −0.288675
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −5.00000 −1.00000
$$26$$ 6.00000 1.17670
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 1.00000 0.174078
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ 5.00000 0.707107
$$51$$ 4.00000 0.560112
$$52$$ −6.00000 −0.832050
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.00000 0.794719
$$58$$ −6.00000 −0.787839
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −12.0000 −1.40449 −0.702247 0.711934i $$-0.747820\pi$$
−0.702247 + 0.711934i $$0.747820\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 5.00000 0.577350
$$76$$ −6.00000 −0.688247
$$77$$ 0 0
$$78$$ −6.00000 −0.679366
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −4.00000 −0.441726
$$83$$ 2.00000 0.219529 0.109764 0.993958i $$-0.464990\pi$$
0.109764 + 0.993958i $$0.464990\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ −6.00000 −0.643268
$$88$$ 1.00000 0.106600
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ −2.00000 −0.207390
$$94$$ −6.00000 −0.618853
$$95$$ 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
$$99$$ −1.00000 −0.100504
$$100$$ −5.00000 −0.500000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ −4.00000 −0.396059
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ −6.00000 −0.561951
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ −6.00000 −0.554700
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −2.00000 −0.181071
$$123$$ −4.00000 −0.360668
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 22.0000 1.92215 0.961074 0.276289i $$-0.0891049\pi$$
0.961074 + 0.276289i $$0.0891049\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 4.00000 0.342997
$$137$$ 22.0000 1.87959 0.939793 0.341743i $$-0.111017\pi$$
0.939793 + 0.341743i $$0.111017\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ −16.0000 −1.34269
$$143$$ 6.00000 0.501745
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 12.0000 0.993127
$$147$$ 0 0
$$148$$ 10.0000 0.821995
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ −5.00000 −0.408248
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 6.00000 0.486664
$$153$$ −4.00000 −0.323381
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ −12.0000 −0.957704 −0.478852 0.877896i $$-0.658947\pi$$
−0.478852 + 0.877896i $$0.658947\pi$$
$$158$$ 16.0000 1.27289
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 4.00000 0.312348
$$165$$ 0 0
$$166$$ −2.00000 −0.155230
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ −6.00000 −0.458831
$$172$$ 8.00000 0.609994
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −8.00000 −0.594635 −0.297318 0.954779i $$-0.596092\pi$$
−0.297318 + 0.954779i $$0.596092\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ 2.00000 0.146647
$$187$$ 4.00000 0.292509
$$188$$ 6.00000 0.437595
$$189$$ 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
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ 5.00000 0.353553
$$201$$ 4.00000 0.282138
$$202$$ −14.0000 −0.985037
$$203$$ 0 0
$$204$$ 4.00000 0.280056
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ −4.00000 −0.278019
$$208$$ −6.00000 −0.416025
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ −16.0000 −1.09630
$$214$$ −8.00000 −0.546869
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 14.0000 0.948200
$$219$$ 12.0000 0.810885
$$220$$ 0 0
$$221$$ 24.0000 1.61441
$$222$$ 10.0000 0.671156
$$223$$ 26.0000 1.74109 0.870544 0.492090i $$-0.163767\pi$$
0.870544 + 0.492090i $$0.163767\pi$$
$$224$$ 0 0
$$225$$ −5.00000 −0.333333
$$226$$ 2.00000 0.133038
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 6.00000 0.397360
$$229$$ −8.00000 −0.528655 −0.264327 0.964433i $$-0.585150\pi$$
−0.264327 + 0.964433i $$0.585150\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 16.0000 1.03931
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 4.00000 0.255031
$$247$$ 36.0000 2.29063
$$248$$ −2.00000 −0.127000
$$249$$ −2.00000 −0.126745
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −10.0000 −0.623783 −0.311891 0.950118i $$-0.600963\pi$$
−0.311891 + 0.950118i $$0.600963\pi$$
$$258$$ 8.00000 0.498058
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ −22.0000 −1.35916
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ −4.00000 −0.244339
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ 0 0
$$274$$ −22.0000 −1.32907
$$275$$ 5.00000 0.301511
$$276$$ 4.00000 0.240772
$$277$$ −6.00000 −0.360505 −0.180253 0.983620i $$-0.557691\pi$$
−0.180253 + 0.983620i $$0.557691\pi$$
$$278$$ −14.0000 −0.839664
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ 14.0000 0.835170 0.417585 0.908638i $$-0.362877\pi$$
0.417585 + 0.908638i $$0.362877\pi$$
$$282$$ 6.00000 0.357295
$$283$$ −18.0000 −1.06999 −0.534994 0.844856i $$-0.679686\pi$$
−0.534994 + 0.844856i $$0.679686\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ −6.00000 −0.354787
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −6.00000 −0.351726
$$292$$ −12.0000 −0.702247
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ 1.00000 0.0580259
$$298$$ 6.00000 0.347571
$$299$$ 24.0000 1.38796
$$300$$ 5.00000 0.288675
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −14.0000 −0.804279
$$304$$ −6.00000 −0.344124
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ 30.0000 1.71219 0.856095 0.516818i $$-0.172884\pi$$
0.856095 + 0.516818i $$0.172884\pi$$
$$308$$ 0 0
$$309$$ −14.0000 −0.796432
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ −6.00000 −0.339683
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 12.0000 0.677199
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ −10.0000 −0.560772
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 0 0
$$323$$ 24.0000 1.33540
$$324$$ 1.00000 0.0555556
$$325$$ 30.0000 1.66410
$$326$$ 4.00000 0.221540
$$327$$ 14.0000 0.774202
$$328$$ −4.00000 −0.220863
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 2.00000 0.109764
$$333$$ 10.0000 0.547997
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ −2.00000 −0.108306
$$342$$ 6.00000 0.324443
$$343$$ 0 0
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ 16.0000 0.858925 0.429463 0.903085i $$-0.358703\pi$$
0.429463 + 0.903085i $$0.358703\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 1.00000 0.0533002
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ −4.00000 −0.211407
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 8.00000 0.420471
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ 2.00000 0.104399 0.0521996 0.998637i $$-0.483377\pi$$
0.0521996 + 0.998637i $$0.483377\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 4.00000 0.208232
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −2.00000 −0.103695
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ −36.0000 −1.85409
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ 12.0000 0.613973
$$383$$ 34.0000 1.73732 0.868659 0.495410i $$-0.164982\pi$$
0.868659 + 0.495410i $$0.164982\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 6.00000 0.305392
$$387$$ 8.00000 0.406663
$$388$$ 6.00000 0.304604
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ −22.0000 −1.10975
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ 4.00000 0.200754 0.100377 0.994949i $$-0.467995\pi$$
0.100377 + 0.994949i $$0.467995\pi$$
$$398$$ −14.0000 −0.701757
$$399$$ 0 0
$$400$$ −5.00000 −0.250000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ −12.0000 −0.597763
$$404$$ 14.0000 0.696526
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −10.0000 −0.495682
$$408$$ −4.00000 −0.198030
$$409$$ −16.0000 −0.791149 −0.395575 0.918434i $$-0.629455\pi$$
−0.395575 + 0.918434i $$0.629455\pi$$
$$410$$ 0 0
$$411$$ −22.0000 −1.08518
$$412$$ 14.0000 0.689730
$$413$$ 0 0
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ −14.0000 −0.685583
$$418$$ −6.00000 −0.293470
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −8.00000 −0.389434
$$423$$ 6.00000 0.291730
$$424$$ 10.0000 0.485643
$$425$$ 20.0000 0.970143
$$426$$ 16.0000 0.775203
$$427$$ 0 0
$$428$$ 8.00000 0.386695
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ 40.0000 1.92673 0.963366 0.268190i $$-0.0864254\pi$$
0.963366 + 0.268190i $$0.0864254\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −26.0000 −1.24948 −0.624740 0.780833i $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ 24.0000 1.14808
$$438$$ −12.0000 −0.573382
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −24.0000 −1.14156
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ 0 0
$$446$$ −26.0000 −1.23114
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ −34.0000 −1.60456 −0.802280 0.596948i $$-0.796380\pi$$
−0.802280 + 0.596948i $$0.796380\pi$$
$$450$$ 5.00000 0.235702
$$451$$ −4.00000 −0.188353
$$452$$ −2.00000 −0.0940721
$$453$$ 0 0
$$454$$ 18.0000 0.844782
$$455$$ 0 0
$$456$$ −6.00000 −0.280976
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 8.00000 0.373815
$$459$$ 4.00000 0.186704
$$460$$ 0 0
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 12.0000 0.552931
$$472$$ 0 0
$$473$$ −8.00000 −0.367840
$$474$$ −16.0000 −0.734904
$$475$$ 30.0000 1.37649
$$476$$ 0 0
$$477$$ −10.0000 −0.457869
$$478$$ −8.00000 −0.365911
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −60.0000 −2.73576
$$482$$ −20.0000 −0.910975
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ −4.00000 −0.180334
$$493$$ −24.0000 −1.08091
$$494$$ −36.0000 −1.61972
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ 0 0
$$498$$ 2.00000 0.0896221
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ −12.0000 −0.535054 −0.267527 0.963550i $$-0.586206\pi$$
−0.267527 + 0.963550i $$0.586206\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −4.00000 −0.177822
$$507$$ −23.0000 −1.02147
$$508$$ −16.0000 −0.709885
$$509$$ 16.0000 0.709188 0.354594 0.935020i $$-0.384619\pi$$
0.354594 + 0.935020i $$0.384619\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 6.00000 0.264906
$$514$$ 10.0000 0.441081
$$515$$ 0 0
$$516$$ −8.00000 −0.352180
$$517$$ −6.00000 −0.263880
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −2.00000 −0.0876216 −0.0438108 0.999040i $$-0.513950\pi$$
−0.0438108 + 0.999040i $$0.513950\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ 6.00000 0.262362 0.131181 0.991358i $$-0.458123\pi$$
0.131181 + 0.991358i $$0.458123\pi$$
$$524$$ 22.0000 0.961074
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8.00000 −0.348485
$$528$$ 1.00000 0.0435194
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −24.0000 −1.03956
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ −4.00000 −0.172613
$$538$$ −24.0000 −1.03471
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 12.0000 0.515444
$$543$$ 8.00000 0.343313
$$544$$ 4.00000 0.171499
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ 22.0000 0.939793
$$549$$ 2.00000 0.0853579
$$550$$ −5.00000 −0.213201
$$551$$ −36.0000 −1.53365
$$552$$ −4.00000 −0.170251
$$553$$ 0 0
$$554$$ 6.00000 0.254916
$$555$$ 0 0
$$556$$ 14.0000 0.593732
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ −2.00000 −0.0846668
$$559$$ −48.0000 −2.03018
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ −14.0000 −0.590554
$$563$$ 22.0000 0.927189 0.463595 0.886047i $$-0.346559\pi$$
0.463595 + 0.886047i $$0.346559\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ 0 0
$$566$$ 18.0000 0.756596
$$567$$ 0 0
$$568$$ −16.0000 −0.671345
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 6.00000 0.250873
$$573$$ 12.0000 0.501307
$$574$$ 0 0
$$575$$ 20.0000 0.834058
$$576$$ 1.00000 0.0416667
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 6.00000 0.249351
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 6.00000 0.248708
$$583$$ 10.0000 0.414158
$$584$$ 12.0000 0.496564
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ −24.0000 −0.990586 −0.495293 0.868726i $$-0.664939\pi$$
−0.495293 + 0.868726i $$0.664939\pi$$
$$588$$ 0 0
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 10.0000 0.410997
$$593$$ −20.0000 −0.821302 −0.410651 0.911793i $$-0.634698\pi$$
−0.410651 + 0.911793i $$0.634698\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ −14.0000 −0.572982
$$598$$ −24.0000 −0.981433
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ −5.00000 −0.204124
$$601$$ 32.0000 1.30531 0.652654 0.757656i $$-0.273656\pi$$
0.652654 + 0.757656i $$0.273656\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 14.0000 0.568711
$$607$$ −24.0000 −0.974130 −0.487065 0.873366i $$-0.661933\pi$$
−0.487065 + 0.873366i $$0.661933\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −36.0000 −1.45640
$$612$$ −4.00000 −0.161690
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ −30.0000 −1.21070
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 14.0000 0.563163
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ −18.0000 −0.721734
$$623$$ 0 0
$$624$$ 6.00000 0.240192
$$625$$ 25.0000 1.00000
$$626$$ 6.00000 0.239808
$$627$$ −6.00000 −0.239617
$$628$$ −12.0000 −0.478852
$$629$$ −40.0000 −1.59490
$$630$$ 0 0
$$631$$ 4.00000 0.159237 0.0796187 0.996825i $$-0.474630\pi$$
0.0796187 + 0.996825i $$0.474630\pi$$
$$632$$ 16.0000 0.636446
$$633$$ −8.00000 −0.317971
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 10.0000 0.396526
$$637$$ 0 0
$$638$$ 6.00000 0.237542
$$639$$ 16.0000 0.632950
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ 8.00000 0.315735
$$643$$ 24.0000 0.946468 0.473234 0.880937i $$-0.343087\pi$$
0.473234 + 0.880937i $$0.343087\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 34.0000 1.33668 0.668339 0.743857i $$-0.267006\pi$$
0.668339 + 0.743857i $$0.267006\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ −30.0000 −1.17670
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ 0 0
$$656$$ 4.00000 0.156174
$$657$$ −12.0000 −0.468165
$$658$$ 0 0
$$659$$ 16.0000 0.623272 0.311636 0.950202i $$-0.399123\pi$$
0.311636 + 0.950202i $$0.399123\pi$$
$$660$$ 0 0
$$661$$ 32.0000 1.24466 0.622328 0.782757i $$-0.286187\pi$$
0.622328 + 0.782757i $$0.286187\pi$$
$$662$$ 4.00000 0.155464
$$663$$ −24.0000 −0.932083
$$664$$ −2.00000 −0.0776151
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ −24.0000 −0.929284
$$668$$ −12.0000 −0.464294
$$669$$ −26.0000 −1.00522
$$670$$ 0 0
$$671$$ −2.00000 −0.0772091
$$672$$ 0 0
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 5.00000 0.192450
$$676$$ 23.0000 0.884615
$$677$$ −30.0000 −1.15299 −0.576497 0.817099i $$-0.695581\pi$$
−0.576497 + 0.817099i $$0.695581\pi$$
$$678$$ −2.00000 −0.0768095
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ 2.00000 0.0765840
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ −6.00000 −0.229416
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 8.00000 0.305219
$$688$$ 8.00000 0.304997
$$689$$ 60.0000 2.28582
$$690$$ 0 0
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ −16.0000 −0.607352
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ −16.0000 −0.606043
$$698$$ 26.0000 0.984115
$$699$$ −22.0000 −0.832116
$$700$$ 0 0
$$701$$ −46.0000 −1.73740 −0.868698 0.495342i $$-0.835043\pi$$
−0.868698 + 0.495342i $$0.835043\pi$$
$$702$$ −6.00000 −0.226455
$$703$$ −60.0000 −2.26294
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ −6.00000 −0.224860
$$713$$ −8.00000 −0.299602
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ −8.00000 −0.298765
$$718$$ −8.00000 −0.298557
$$719$$ 50.0000 1.86469 0.932343 0.361576i $$-0.117761\pi$$
0.932343 + 0.361576i $$0.117761\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −17.0000 −0.632674
$$723$$ −20.0000 −0.743808
$$724$$ −8.00000 −0.297318
$$725$$ −30.0000 −1.11417
$$726$$ 1.00000 0.0371135
$$727$$ −22.0000 −0.815935 −0.407967 0.912996i $$-0.633762\pi$$
−0.407967 + 0.912996i $$0.633762\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −32.0000 −1.18356
$$732$$ −2.00000 −0.0739221
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ −2.00000 −0.0738213
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 4.00000 0.147342
$$738$$ −4.00000 −0.147242
$$739$$ −52.0000 −1.91285 −0.956425 0.291977i $$-0.905687\pi$$
−0.956425 + 0.291977i $$0.905687\pi$$
$$740$$ 0 0
$$741$$ −36.0000 −1.32249
$$742$$ 0 0
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 2.00000 0.0733236
$$745$$ 0 0
$$746$$ 10.0000 0.366126
$$747$$ 2.00000 0.0731762
$$748$$ 4.00000 0.146254
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 28.0000 1.02173 0.510867 0.859660i $$-0.329324\pi$$
0.510867 + 0.859660i $$0.329324\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 0 0
$$754$$ 36.0000 1.31104
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −14.0000 −0.508839 −0.254419 0.967094i $$-0.581884\pi$$
−0.254419 + 0.967094i $$0.581884\pi$$
$$758$$ 20.0000 0.726433
$$759$$ −4.00000 −0.145191
$$760$$ 0 0
$$761$$ −8.00000 −0.290000 −0.145000 0.989432i $$-0.546318\pi$$
−0.145000 + 0.989432i $$0.546318\pi$$
$$762$$ −16.0000 −0.579619
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ −34.0000 −1.22847
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ −16.0000 −0.576975 −0.288487 0.957484i $$-0.593152\pi$$
−0.288487 + 0.957484i $$0.593152\pi$$
$$770$$ 0 0
$$771$$ 10.0000 0.360141
$$772$$ −6.00000 −0.215945
$$773$$ −20.0000 −0.719350 −0.359675 0.933078i $$-0.617112\pi$$
−0.359675 + 0.933078i $$0.617112\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ −10.0000 −0.359211
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ −16.0000 −0.572159
$$783$$ −6.00000 −0.214423
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 22.0000 0.784714
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 1.00000 0.0355335
$$793$$ −12.0000 −0.426132
$$794$$ −4.00000 −0.141955
$$795$$ 0 0
$$796$$ 14.0000 0.496217
$$797$$ 20.0000 0.708436 0.354218 0.935163i $$-0.384747\pi$$
0.354218 + 0.935163i $$0.384747\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 5.00000 0.176777
$$801$$ 6.00000 0.212000
$$802$$ −30.0000 −1.05934
$$803$$ 12.0000 0.423471
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 12.0000 0.422682
$$807$$ −24.0000 −0.844840
$$808$$ −14.0000 −0.492518
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ −18.0000 −0.632065 −0.316033 0.948748i $$-0.602351\pi$$
−0.316033 + 0.948748i $$0.602351\pi$$
$$812$$ 0 0
$$813$$ 12.0000 0.420858
$$814$$ 10.0000 0.350500
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ −48.0000 −1.67931
$$818$$ 16.0000 0.559427
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ 22.0000 0.767338
$$823$$ 28.0000 0.976019 0.488009 0.872838i $$-0.337723\pi$$
0.488009 + 0.872838i $$0.337723\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ −5.00000 −0.174078
$$826$$ 0 0
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ −24.0000 −0.833554 −0.416777 0.909009i $$-0.636840\pi$$
−0.416777 + 0.909009i $$0.636840\pi$$
$$830$$ 0 0
$$831$$ 6.00000 0.208138
$$832$$ −6.00000 −0.208013
$$833$$ 0 0
$$834$$ 14.0000 0.484780
$$835$$ 0 0
$$836$$ 6.00000 0.207514
$$837$$ −2.00000 −0.0691301
$$838$$ 4.00000 0.138178
$$839$$ 42.0000 1.45000 0.725001 0.688748i $$-0.241839\pi$$
0.725001 + 0.688748i $$0.241839\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000 0.344623
$$843$$ −14.0000 −0.482186
$$844$$ 8.00000 0.275371
$$845$$ 0 0
$$846$$ −6.00000 −0.206284
$$847$$ 0 0
$$848$$ −10.0000 −0.343401
$$849$$ 18.0000 0.617758
$$850$$ −20.0000 −0.685994
$$851$$ −40.0000 −1.37118
$$852$$ −16.0000 −0.548151
$$853$$ 42.0000 1.43805 0.719026 0.694983i $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ −24.0000 −0.819824 −0.409912 0.912125i $$-0.634441\pi$$
−0.409912 + 0.912125i $$0.634441\pi$$
$$858$$ 6.00000 0.204837
$$859$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −40.0000 −1.36241
$$863$$ 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
$$866$$ 26.0000 0.883516
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 14.0000 0.474100
$$873$$ 6.00000 0.203069
$$874$$ −24.0000 −0.811812
$$875$$ 0 0
$$876$$ 12.0000 0.405442
$$877$$ 18.0000 0.607817 0.303908 0.952701i $$-0.401708\pi$$
0.303908 + 0.952701i $$0.401708\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ 50.0000 1.68454 0.842271 0.539054i $$-0.181218\pi$$
0.842271 + 0.539054i $$0.181218\pi$$
$$882$$ 0 0
$$883$$ −28.0000 −0.942275 −0.471138 0.882060i $$-0.656156\pi$$
−0.471138 + 0.882060i $$0.656156\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 10.0000 0.335578
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 26.0000 0.870544
$$893$$ −36.0000 −1.20469
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −24.0000 −0.801337
$$898$$ 34.0000 1.13459
$$899$$ 12.0000 0.400222
$$900$$ −5.00000 −0.166667
$$901$$ 40.0000 1.33259
$$902$$ 4.00000 0.133185
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 20.0000 0.664089 0.332045 0.943264i $$-0.392262\pi$$
0.332045 + 0.943264i $$0.392262\pi$$
$$908$$ −18.0000 −0.597351
$$909$$ 14.0000 0.464351
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 6.00000 0.198680
$$913$$ −2.00000 −0.0661903
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ −8.00000 −0.264327
$$917$$ 0 0
$$918$$ −4.00000 −0.132020
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ −30.0000 −0.988534
$$922$$ 2.00000 0.0658665
$$923$$ −96.0000 −3.15988
$$924$$ 0 0
$$925$$ −50.0000 −1.64399
$$926$$ −16.0000 −0.525793
$$927$$ 14.0000 0.459820
$$928$$ −6.00000 −0.196960
$$929$$ −10.0000 −0.328089 −0.164045 0.986453i $$-0.552454\pi$$
−0.164045 + 0.986453i $$0.552454\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 22.0000 0.720634
$$933$$ −18.0000 −0.589294
$$934$$ 28.0000 0.916188
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ 12.0000 0.392023 0.196011 0.980602i $$-0.437201\pi$$
0.196011 + 0.980602i $$0.437201\pi$$
$$938$$ 0 0
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ −12.0000 −0.390981
$$943$$ −16.0000 −0.521032
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 16.0000 0.519656
$$949$$ 72.0000 2.33722
$$950$$ −30.0000 −0.973329
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ 18.0000 0.583077 0.291539 0.956559i $$-0.405833\pi$$
0.291539 + 0.956559i $$0.405833\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ 8.00000 0.258738
$$957$$ 6.00000 0.193952
$$958$$ −24.0000 −0.775405
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 60.0000 1.93448
$$963$$ 8.00000 0.257796
$$964$$ 20.0000 0.644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 24.0000 0.771788 0.385894 0.922543i $$-0.373893\pi$$
0.385894 + 0.922543i $$0.373893\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ −20.0000 −0.640841
$$975$$ −30.0000 −0.960769
$$976$$ 2.00000 0.0640184
$$977$$ −46.0000 −1.47167 −0.735835 0.677161i $$-0.763210\pi$$
−0.735835 + 0.677161i $$0.763210\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ −12.0000 −0.382935
$$983$$ −10.0000 −0.318950 −0.159475 0.987202i $$-0.550980\pi$$
−0.159475 + 0.987202i $$0.550980\pi$$
$$984$$ 4.00000 0.127515
$$985$$ 0 0
$$986$$ 24.0000 0.764316
$$987$$ 0 0
$$988$$ 36.0000 1.14531
$$989$$ −32.0000 −1.01754
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 4.00000 0.126936
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −2.00000 −0.0633724
$$997$$ 46.0000 1.45683 0.728417 0.685134i $$-0.240256\pi$$
0.728417 + 0.685134i $$0.240256\pi$$
$$998$$ −36.0000 −1.13956
$$999$$ −10.0000 −0.316386
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 3234.2.a.b.1.1 1
3.2 odd 2 9702.2.a.bp.1.1 1
7.6 odd 2 462.2.a.d.1.1 1
21.20 even 2 1386.2.a.j.1.1 1
28.27 even 2 3696.2.a.j.1.1 1
77.76 even 2 5082.2.a.ba.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.d.1.1 1 7.6 odd 2
1386.2.a.j.1.1 1 21.20 even 2
3234.2.a.b.1.1 1 1.1 even 1 trivial
3696.2.a.j.1.1 1 28.27 even 2
5082.2.a.ba.1.1 1 77.76 even 2
9702.2.a.bp.1.1 1 3.2 odd 2