# Properties

 Label 2450.2.a.f.1.1 Level $2450$ Weight $2$ Character 2450.1 Self dual yes Analytic conductor $19.563$ 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] = [2450,2,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.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: $$19.5633484952$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2450.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} -2.00000 q^{12} +5.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -3.00000 q^{22} -3.00000 q^{23} +2.00000 q^{24} -5.00000 q^{26} +4.00000 q^{27} -6.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -11.0000 q^{37} -1.00000 q^{38} -10.0000 q^{39} -3.00000 q^{41} +10.0000 q^{43} +3.00000 q^{44} +3.00000 q^{46} +3.00000 q^{47} -2.00000 q^{48} -12.0000 q^{51} +5.00000 q^{52} -3.00000 q^{53} -4.00000 q^{54} -2.00000 q^{57} +6.00000 q^{58} +4.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} +4.00000 q^{67} +6.00000 q^{68} +6.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} -4.00000 q^{73} +11.0000 q^{74} +1.00000 q^{76} +10.0000 q^{78} -10.0000 q^{79} -11.0000 q^{81} +3.00000 q^{82} -12.0000 q^{83} -10.0000 q^{86} +12.0000 q^{87} -3.00000 q^{88} -6.00000 q^{89} -3.00000 q^{92} -8.00000 q^{93} -3.00000 q^{94} +2.00000 q^{96} +14.0000 q^{97} +3.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ −2.00000 −0.577350
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 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
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 2.00000 0.408248
$$25$$ 0 0
$$26$$ −5.00000 −0.980581
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −6.00000 −1.04447
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −11.0000 −1.80839 −0.904194 0.427121i $$-0.859528\pi$$
−0.904194 + 0.427121i $$0.859528\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ −10.0000 −1.60128
$$40$$ 0 0
$$41$$ −3.00000 −0.468521 −0.234261 0.972174i $$-0.575267\pi$$
−0.234261 + 0.972174i $$0.575267\pi$$
$$42$$ 0 0
$$43$$ 10.0000 1.52499 0.762493 0.646997i $$-0.223975\pi$$
0.762493 + 0.646997i $$0.223975\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ 3.00000 0.442326
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ −2.00000 −0.288675
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −12.0000 −1.68034
$$52$$ 5.00000 0.693375
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ 6.00000 0.787839
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 6.00000 0.738549
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 11.0000 1.27872
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ 10.0000 1.13228
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 3.00000 0.331295
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −10.0000 −1.07833
$$87$$ 12.0000 1.28654
$$88$$ −3.00000 −0.319801
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −3.00000 −0.312772
$$93$$ −8.00000 −0.829561
$$94$$ −3.00000 −0.309426
$$95$$ 0 0
$$96$$ 2.00000 0.204124
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 0 0
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 12.0000 1.18818
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −5.00000 −0.490290
$$105$$ 0 0
$$106$$ 3.00000 0.291386
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 4.00000 0.384900
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 22.0000 2.08815
$$112$$ 0 0
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 5.00000 0.462250
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −4.00000 −0.362143
$$123$$ 6.00000 0.541002
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 19.0000 1.68598 0.842989 0.537931i $$-0.180794\pi$$
0.842989 + 0.537931i $$0.180794\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −20.0000 −1.76090
$$130$$ 0 0
$$131$$ −3.00000 −0.262111 −0.131056 0.991375i $$-0.541837\pi$$
−0.131056 + 0.991375i $$0.541837\pi$$
$$132$$ −6.00000 −0.522233
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ −6.00000 −0.510754
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ −12.0000 −1.00702
$$143$$ 15.0000 1.25436
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 0 0
$$148$$ −11.0000 −0.904194
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 14.0000 1.13930 0.569652 0.821886i $$-0.307078\pi$$
0.569652 + 0.821886i $$0.307078\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −10.0000 −0.800641
$$157$$ 5.00000 0.399043 0.199522 0.979893i $$-0.436061\pi$$
0.199522 + 0.979893i $$0.436061\pi$$
$$158$$ 10.0000 0.795557
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 11.0000 0.864242
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −3.00000 −0.234261
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 9.00000 0.696441 0.348220 0.937413i $$-0.386786\pi$$
0.348220 + 0.937413i $$0.386786\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 10.0000 0.762493
$$173$$ 3.00000 0.228086 0.114043 0.993476i $$-0.463620\pi$$
0.114043 + 0.993476i $$0.463620\pi$$
$$174$$ −12.0000 −0.909718
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ −3.00000 −0.224231 −0.112115 0.993695i $$-0.535763\pi$$
−0.112115 + 0.993695i $$0.535763\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ −8.00000 −0.591377
$$184$$ 3.00000 0.221163
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ 18.0000 1.31629
$$188$$ 3.00000 0.218797
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ −2.00000 −0.144338
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3.00000 0.213741 0.106871 0.994273i $$-0.465917\pi$$
0.106871 + 0.994273i $$0.465917\pi$$
$$198$$ −3.00000 −0.213201
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ −12.0000 −0.844317
$$203$$ 0 0
$$204$$ −12.0000 −0.840168
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ −3.00000 −0.208514
$$208$$ 5.00000 0.346688
$$209$$ 3.00000 0.207514
$$210$$ 0 0
$$211$$ −1.00000 −0.0688428 −0.0344214 0.999407i $$-0.510959\pi$$
−0.0344214 + 0.999407i $$0.510959\pi$$
$$212$$ −3.00000 −0.206041
$$213$$ −24.0000 −1.64445
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ −4.00000 −0.272166
$$217$$ 0 0
$$218$$ 4.00000 0.270914
$$219$$ 8.00000 0.540590
$$220$$ 0 0
$$221$$ 30.0000 2.01802
$$222$$ −22.0000 −1.47654
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 12.0000 0.798228
$$227$$ −24.0000 −1.59294 −0.796468 0.604681i $$-0.793301\pi$$
−0.796468 + 0.604681i $$0.793301\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ 28.0000 1.85029 0.925146 0.379611i $$-0.123942\pi$$
0.925146 + 0.379611i $$0.123942\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ −5.00000 −0.326860
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 20.0000 1.29914
$$238$$ 0 0
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ 25.0000 1.61039 0.805196 0.593009i $$-0.202060\pi$$
0.805196 + 0.593009i $$0.202060\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 10.0000 0.641500
$$244$$ 4.00000 0.256074
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 5.00000 0.318142
$$248$$ −4.00000 −0.254000
$$249$$ 24.0000 1.52094
$$250$$ 0 0
$$251$$ 15.0000 0.946792 0.473396 0.880850i $$-0.343028\pi$$
0.473396 + 0.880850i $$0.343028\pi$$
$$252$$ 0 0
$$253$$ −9.00000 −0.565825
$$254$$ −19.0000 −1.19217
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 20.0000 1.24515
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 3.00000 0.185341
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 6.00000 0.369274
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 12.0000 0.734388
$$268$$ 4.00000 0.244339
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ 6.00000 0.361158
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −3.00000 −0.178965 −0.0894825 0.995988i $$-0.528521\pi$$
−0.0894825 + 0.995988i $$0.528521\pi$$
$$282$$ 6.00000 0.357295
$$283$$ 26.0000 1.54554 0.772770 0.634686i $$-0.218871\pi$$
0.772770 + 0.634686i $$0.218871\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ −15.0000 −0.886969
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ −28.0000 −1.64139
$$292$$ −4.00000 −0.234082
$$293$$ −27.0000 −1.57736 −0.788678 0.614806i $$-0.789234\pi$$
−0.788678 + 0.614806i $$0.789234\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 11.0000 0.639362
$$297$$ 12.0000 0.696311
$$298$$ −18.0000 −1.04271
$$299$$ −15.0000 −0.867472
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −14.0000 −0.805609
$$303$$ −24.0000 −1.37876
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 10.0000 0.566139
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ −5.00000 −0.282166
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ −18.0000 −1.00781
$$320$$ 0 0
$$321$$ −24.0000 −1.33955
$$322$$ 0 0
$$323$$ 6.00000 0.333849
$$324$$ −11.0000 −0.611111
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 8.00000 0.442401
$$328$$ 3.00000 0.165647
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −7.00000 −0.384755 −0.192377 0.981321i $$-0.561620\pi$$
−0.192377 + 0.981321i $$0.561620\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ −11.0000 −0.602796
$$334$$ −9.00000 −0.492458
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ 24.0000 1.30350
$$340$$ 0 0
$$341$$ 12.0000 0.649836
$$342$$ −1.00000 −0.0540738
$$343$$ 0 0
$$344$$ −10.0000 −0.539164
$$345$$ 0 0
$$346$$ −3.00000 −0.161281
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ 12.0000 0.643268
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 20.0000 1.06752
$$352$$ −3.00000 −0.159901
$$353$$ 12.0000 0.638696 0.319348 0.947638i $$-0.396536\pi$$
0.319348 + 0.947638i $$0.396536\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 3.00000 0.158555
$$359$$ 6.00000 0.316668 0.158334 0.987386i $$-0.449388\pi$$
0.158334 + 0.987386i $$0.449388\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 2.00000 0.105118
$$363$$ 4.00000 0.209946
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 8.00000 0.418167
$$367$$ −1.00000 −0.0521996 −0.0260998 0.999659i $$-0.508309\pi$$
−0.0260998 + 0.999659i $$0.508309\pi$$
$$368$$ −3.00000 −0.156386
$$369$$ −3.00000 −0.156174
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −8.00000 −0.414781
$$373$$ 34.0000 1.76045 0.880227 0.474554i $$-0.157390\pi$$
0.880227 + 0.474554i $$0.157390\pi$$
$$374$$ −18.0000 −0.930758
$$375$$ 0 0
$$376$$ −3.00000 −0.154713
$$377$$ −30.0000 −1.54508
$$378$$ 0 0
$$379$$ −25.0000 −1.28416 −0.642082 0.766636i $$-0.721929\pi$$
−0.642082 + 0.766636i $$0.721929\pi$$
$$380$$ 0 0
$$381$$ −38.0000 −1.94680
$$382$$ −12.0000 −0.613973
$$383$$ 15.0000 0.766464 0.383232 0.923652i $$-0.374811\pi$$
0.383232 + 0.923652i $$0.374811\pi$$
$$384$$ 2.00000 0.102062
$$385$$ 0 0
$$386$$ −4.00000 −0.203595
$$387$$ 10.0000 0.508329
$$388$$ 14.0000 0.710742
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ −18.0000 −0.910299
$$392$$ 0 0
$$393$$ 6.00000 0.302660
$$394$$ −3.00000 −0.151138
$$395$$ 0 0
$$396$$ 3.00000 0.150756
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ −4.00000 −0.200502
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 21.0000 1.04869 0.524345 0.851506i $$-0.324310\pi$$
0.524345 + 0.851506i $$0.324310\pi$$
$$402$$ 8.00000 0.399004
$$403$$ 20.0000 0.996271
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −33.0000 −1.63575
$$408$$ 12.0000 0.594089
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 0 0
$$411$$ 24.0000 1.18383
$$412$$ −4.00000 −0.197066
$$413$$ 0 0
$$414$$ 3.00000 0.147442
$$415$$ 0 0
$$416$$ −5.00000 −0.245145
$$417$$ −8.00000 −0.391762
$$418$$ −3.00000 −0.146735
$$419$$ −15.0000 −0.732798 −0.366399 0.930458i $$-0.619409\pi$$
−0.366399 + 0.930458i $$0.619409\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ 1.00000 0.0486792
$$423$$ 3.00000 0.145865
$$424$$ 3.00000 0.145693
$$425$$ 0 0
$$426$$ 24.0000 1.16280
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ −30.0000 −1.44841
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 4.00000 0.192450
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ −3.00000 −0.143509
$$438$$ −8.00000 −0.382255
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −30.0000 −1.42695
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 22.0000 1.04407
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ −36.0000 −1.70274
$$448$$ 0 0
$$449$$ −3.00000 −0.141579 −0.0707894 0.997491i $$-0.522552\pi$$
−0.0707894 + 0.997491i $$0.522552\pi$$
$$450$$ 0 0
$$451$$ −9.00000 −0.423793
$$452$$ −12.0000 −0.564433
$$453$$ −28.0000 −1.31555
$$454$$ 24.0000 1.12638
$$455$$ 0 0
$$456$$ 2.00000 0.0936586
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ −28.0000 −1.30835
$$459$$ 24.0000 1.12022
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ 19.0000 0.883005 0.441502 0.897260i $$-0.354446\pi$$
0.441502 + 0.897260i $$0.354446\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ −18.0000 −0.832941 −0.416470 0.909149i $$-0.636733\pi$$
−0.416470 + 0.909149i $$0.636733\pi$$
$$468$$ 5.00000 0.231125
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −10.0000 −0.460776
$$472$$ 0 0
$$473$$ 30.0000 1.37940
$$474$$ −20.0000 −0.918630
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −3.00000 −0.137361
$$478$$ −6.00000 −0.274434
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −55.0000 −2.50778
$$482$$ −25.0000 −1.13872
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ −10.0000 −0.453609
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ −4.00000 −0.181071
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 6.00000 0.270501
$$493$$ −36.0000 −1.62136
$$494$$ −5.00000 −0.224961
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ −24.0000 −1.07547
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ −18.0000 −0.804181
$$502$$ −15.0000 −0.669483
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 9.00000 0.400099
$$507$$ −24.0000 −1.06588
$$508$$ 19.0000 0.842989
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 4.00000 0.176604
$$514$$ 12.0000 0.529297
$$515$$ 0 0
$$516$$ −20.0000 −0.880451
$$517$$ 9.00000 0.395820
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −33.0000 −1.44576 −0.722878 0.690976i $$-0.757181\pi$$
−0.722878 + 0.690976i $$0.757181\pi$$
$$522$$ 6.00000 0.262613
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ −3.00000 −0.131056
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 24.0000 1.04546
$$528$$ −6.00000 −0.261116
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −15.0000 −0.649722
$$534$$ −12.0000 −0.519291
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 6.00000 0.258919
$$538$$ −12.0000 −0.517357
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 8.00000 0.343947 0.171973 0.985102i $$-0.444986\pi$$
0.171973 + 0.985102i $$0.444986\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 4.00000 0.171656
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ 4.00000 0.170716
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ −6.00000 −0.255377
$$553$$ 0 0
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ 27.0000 1.14403 0.572013 0.820244i $$-0.306163\pi$$
0.572013 + 0.820244i $$0.306163\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ 50.0000 2.11477
$$560$$ 0 0
$$561$$ −36.0000 −1.51992
$$562$$ 3.00000 0.126547
$$563$$ −18.0000 −0.758610 −0.379305 0.925272i $$-0.623837\pi$$
−0.379305 + 0.925272i $$0.623837\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ 0 0
$$566$$ −26.0000 −1.09286
$$567$$ 0 0
$$568$$ −12.0000 −0.503509
$$569$$ −3.00000 −0.125767 −0.0628833 0.998021i $$-0.520030\pi$$
−0.0628833 + 0.998021i $$0.520030\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 15.0000 0.627182
$$573$$ −24.0000 −1.00261
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 20.0000 0.832611 0.416305 0.909225i $$-0.363325\pi$$
0.416305 + 0.909225i $$0.363325\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 28.0000 1.16064
$$583$$ −9.00000 −0.372742
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 27.0000 1.11536
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ −11.0000 −0.452097
$$593$$ −36.0000 −1.47834 −0.739171 0.673517i $$-0.764783\pi$$
−0.739171 + 0.673517i $$0.764783\pi$$
$$594$$ −12.0000 −0.492366
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ −8.00000 −0.327418
$$598$$ 15.0000 0.613396
$$599$$ 42.0000 1.71607 0.858037 0.513588i $$-0.171684\pi$$
0.858037 + 0.513588i $$0.171684\pi$$
$$600$$ 0 0
$$601$$ −2.00000 −0.0815817 −0.0407909 0.999168i $$-0.512988\pi$$
−0.0407909 + 0.999168i $$0.512988\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 14.0000 0.569652
$$605$$ 0 0
$$606$$ 24.0000 0.974933
$$607$$ −19.0000 −0.771186 −0.385593 0.922669i $$-0.626003\pi$$
−0.385593 + 0.922669i $$0.626003\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 15.0000 0.606835
$$612$$ 6.00000 0.242536
$$613$$ −47.0000 −1.89831 −0.949156 0.314806i $$-0.898061\pi$$
−0.949156 + 0.314806i $$0.898061\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ 1.00000 0.0401934 0.0200967 0.999798i $$-0.493603\pi$$
0.0200967 + 0.999798i $$0.493603\pi$$
$$620$$ 0 0
$$621$$ −12.0000 −0.481543
$$622$$ 12.0000 0.481156
$$623$$ 0 0
$$624$$ −10.0000 −0.400320
$$625$$ 0 0
$$626$$ −8.00000 −0.319744
$$627$$ −6.00000 −0.239617
$$628$$ 5.00000 0.199522
$$629$$ −66.0000 −2.63159
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 10.0000 0.397779
$$633$$ 2.00000 0.0794929
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ 0 0
$$638$$ 18.0000 0.712627
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ −45.0000 −1.77739 −0.888697 0.458496i $$-0.848388\pi$$
−0.888697 + 0.458496i $$0.848388\pi$$
$$642$$ 24.0000 0.947204
$$643$$ 38.0000 1.49857 0.749287 0.662246i $$-0.230396\pi$$
0.749287 + 0.662246i $$0.230396\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −6.00000 −0.236067
$$647$$ −21.0000 −0.825595 −0.412798 0.910823i $$-0.635448\pi$$
−0.412798 + 0.910823i $$0.635448\pi$$
$$648$$ 11.0000 0.432121
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ −21.0000 −0.821794 −0.410897 0.911682i $$-0.634784\pi$$
−0.410897 + 0.911682i $$0.634784\pi$$
$$654$$ −8.00000 −0.312825
$$655$$ 0 0
$$656$$ −3.00000 −0.117130
$$657$$ −4.00000 −0.156055
$$658$$ 0 0
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ 0 0
$$661$$ −44.0000 −1.71140 −0.855701 0.517471i $$-0.826874\pi$$
−0.855701 + 0.517471i $$0.826874\pi$$
$$662$$ 7.00000 0.272063
$$663$$ −60.0000 −2.33021
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 11.0000 0.426241
$$667$$ 18.0000 0.696963
$$668$$ 9.00000 0.348220
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ 12.0000 0.463255
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ 12.0000 0.461538
$$677$$ −3.00000 −0.115299 −0.0576497 0.998337i $$-0.518361\pi$$
−0.0576497 + 0.998337i $$0.518361\pi$$
$$678$$ −24.0000 −0.921714
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 48.0000 1.83936
$$682$$ −12.0000 −0.459504
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −56.0000 −2.13653
$$688$$ 10.0000 0.381246
$$689$$ −15.0000 −0.571454
$$690$$ 0 0
$$691$$ −32.0000 −1.21734 −0.608669 0.793424i $$-0.708296\pi$$
−0.608669 + 0.793424i $$0.708296\pi$$
$$692$$ 3.00000 0.114043
$$693$$ 0 0
$$694$$ 24.0000 0.911028
$$695$$ 0 0
$$696$$ −12.0000 −0.454859
$$697$$ −18.0000 −0.681799
$$698$$ −10.0000 −0.378506
$$699$$ −12.0000 −0.453882
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ −20.0000 −0.754851
$$703$$ −11.0000 −0.414873
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ −12.0000 −0.451626
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 14.0000 0.525781 0.262891 0.964826i $$-0.415324\pi$$
0.262891 + 0.964826i $$0.415324\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ 6.00000 0.224860
$$713$$ −12.0000 −0.449404
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −3.00000 −0.112115
$$717$$ −12.0000 −0.448148
$$718$$ −6.00000 −0.223918
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 18.0000 0.669891
$$723$$ −50.0000 −1.85952
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ −4.00000 −0.148454
$$727$$ 29.0000 1.07555 0.537775 0.843088i $$-0.319265\pi$$
0.537775 + 0.843088i $$0.319265\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 60.0000 2.21918
$$732$$ −8.00000 −0.295689
$$733$$ 47.0000 1.73598 0.867992 0.496578i $$-0.165410\pi$$
0.867992 + 0.496578i $$0.165410\pi$$
$$734$$ 1.00000 0.0369107
$$735$$ 0 0
$$736$$ 3.00000 0.110581
$$737$$ 12.0000 0.442026
$$738$$ 3.00000 0.110432
$$739$$ −37.0000 −1.36107 −0.680534 0.732717i $$-0.738252\pi$$
−0.680534 + 0.732717i $$0.738252\pi$$
$$740$$ 0 0
$$741$$ −10.0000 −0.367359
$$742$$ 0 0
$$743$$ −9.00000 −0.330178 −0.165089 0.986279i $$-0.552791\pi$$
−0.165089 + 0.986279i $$0.552791\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ −34.0000 −1.24483
$$747$$ −12.0000 −0.439057
$$748$$ 18.0000 0.658145
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ 3.00000 0.109399
$$753$$ −30.0000 −1.09326
$$754$$ 30.0000 1.09254
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ 25.0000 0.908041
$$759$$ 18.0000 0.653359
$$760$$ 0 0
$$761$$ 51.0000 1.84875 0.924374 0.381487i $$-0.124588\pi$$
0.924374 + 0.381487i $$0.124588\pi$$
$$762$$ 38.0000 1.37659
$$763$$ 0 0
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ −15.0000 −0.541972
$$767$$ 0 0
$$768$$ −2.00000 −0.0721688
$$769$$ 49.0000 1.76699 0.883493 0.468445i $$-0.155186\pi$$
0.883493 + 0.468445i $$0.155186\pi$$
$$770$$ 0 0
$$771$$ 24.0000 0.864339
$$772$$ 4.00000 0.143963
$$773$$ 39.0000 1.40273 0.701366 0.712801i $$-0.252574\pi$$
0.701366 + 0.712801i $$0.252574\pi$$
$$774$$ −10.0000 −0.359443
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ 24.0000 0.860442
$$779$$ −3.00000 −0.107486
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ 18.0000 0.643679
$$783$$ −24.0000 −0.857690
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −6.00000 −0.214013
$$787$$ −34.0000 −1.21197 −0.605985 0.795476i $$-0.707221\pi$$
−0.605985 + 0.795476i $$0.707221\pi$$
$$788$$ 3.00000 0.106871
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −3.00000 −0.106600
$$793$$ 20.0000 0.710221
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ 4.00000 0.141776
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ 0 0
$$799$$ 18.0000 0.636794
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ −21.0000 −0.741536
$$803$$ −12.0000 −0.423471
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ −20.0000 −0.704470
$$807$$ −24.0000 −0.844840
$$808$$ −12.0000 −0.422159
$$809$$ 39.0000 1.37117 0.685583 0.727994i $$-0.259547\pi$$
0.685583 + 0.727994i $$0.259547\pi$$
$$810$$ 0 0
$$811$$ −47.0000 −1.65039 −0.825197 0.564846i $$-0.808936\pi$$
−0.825197 + 0.564846i $$0.808936\pi$$
$$812$$ 0 0
$$813$$ −32.0000 −1.12229
$$814$$ 33.0000 1.15665
$$815$$ 0 0
$$816$$ −12.0000 −0.420084
$$817$$ 10.0000 0.349856
$$818$$ −22.0000 −0.769212
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ −24.0000 −0.837096
$$823$$ −44.0000 −1.53374 −0.766872 0.641800i $$-0.778188\pi$$
−0.766872 + 0.641800i $$0.778188\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 54.0000 1.87776 0.938882 0.344239i $$-0.111863\pi$$
0.938882 + 0.344239i $$0.111863\pi$$
$$828$$ −3.00000 −0.104257
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 4.00000 0.138758
$$832$$ 5.00000 0.173344
$$833$$ 0 0
$$834$$ 8.00000 0.277017
$$835$$ 0 0
$$836$$ 3.00000 0.103757
$$837$$ 16.0000 0.553041
$$838$$ 15.0000 0.518166
$$839$$ 6.00000 0.207143 0.103572 0.994622i $$-0.466973\pi$$
0.103572 + 0.994622i $$0.466973\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 34.0000 1.17172
$$843$$ 6.00000 0.206651
$$844$$ −1.00000 −0.0344214
$$845$$ 0 0
$$846$$ −3.00000 −0.103142
$$847$$ 0 0
$$848$$ −3.00000 −0.103020
$$849$$ −52.0000 −1.78464
$$850$$ 0 0
$$851$$ 33.0000 1.13123
$$852$$ −24.0000 −0.822226
$$853$$ −1.00000 −0.0342393 −0.0171197 0.999853i $$-0.505450\pi$$
−0.0171197 + 0.999853i $$0.505450\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 30.0000 1.02418
$$859$$ −32.0000 −1.09183 −0.545913 0.837842i $$-0.683817\pi$$
−0.545913 + 0.837842i $$0.683817\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −39.0000 −1.32758 −0.663788 0.747921i $$-0.731052\pi$$
−0.663788 + 0.747921i $$0.731052\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 0 0
$$866$$ 16.0000 0.543702
$$867$$ −38.0000 −1.29055
$$868$$ 0 0
$$869$$ −30.0000 −1.01768
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 4.00000 0.135457
$$873$$ 14.0000 0.473828
$$874$$ 3.00000 0.101477
$$875$$ 0 0
$$876$$ 8.00000 0.270295
$$877$$ 7.00000 0.236373 0.118187 0.992991i $$-0.462292\pi$$
0.118187 + 0.992991i $$0.462292\pi$$
$$878$$ −10.0000 −0.337484
$$879$$ 54.0000 1.82137
$$880$$ 0 0
$$881$$ −33.0000 −1.11180 −0.555899 0.831250i $$-0.687626\pi$$
−0.555899 + 0.831250i $$0.687626\pi$$
$$882$$ 0 0
$$883$$ −8.00000 −0.269221 −0.134611 0.990899i $$-0.542978\pi$$
−0.134611 + 0.990899i $$0.542978\pi$$
$$884$$ 30.0000 1.00901
$$885$$ 0 0
$$886$$ −24.0000 −0.806296
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ −22.0000 −0.738272
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −33.0000 −1.10554
$$892$$ 8.00000 0.267860
$$893$$ 3.00000 0.100391
$$894$$ 36.0000 1.20402
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 30.0000 1.00167
$$898$$ 3.00000 0.100111
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ −18.0000 −0.599667
$$902$$ 9.00000 0.299667
$$903$$ 0 0
$$904$$ 12.0000 0.399114
$$905$$ 0 0
$$906$$ 28.0000 0.930238
$$907$$ 10.0000 0.332045 0.166022 0.986122i $$-0.446908\pi$$
0.166022 + 0.986122i $$0.446908\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ 12.0000 0.398015
$$910$$ 0 0
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ −2.00000 −0.0662266
$$913$$ −36.0000 −1.19143
$$914$$ −22.0000 −0.727695
$$915$$ 0 0
$$916$$ 28.0000 0.925146
$$917$$ 0 0
$$918$$ −24.0000 −0.792118
$$919$$ 38.0000 1.25350 0.626752 0.779219i $$-0.284384\pi$$
0.626752 + 0.779219i $$0.284384\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 6.00000 0.197599
$$923$$ 60.0000 1.97492
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −19.0000 −0.624379
$$927$$ −4.00000 −0.131377
$$928$$ 6.00000 0.196960
$$929$$ −33.0000 −1.08269 −0.541347 0.840799i $$-0.682086\pi$$
−0.541347 + 0.840799i $$0.682086\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000 0.196537
$$933$$ 24.0000 0.785725
$$934$$ 18.0000 0.588978
$$935$$ 0 0
$$936$$ −5.00000 −0.163430
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 0 0
$$939$$ −16.0000 −0.522140
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ 10.0000 0.325818
$$943$$ 9.00000 0.293080
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −30.0000 −0.975384
$$947$$ −30.0000 −0.974869 −0.487435 0.873160i $$-0.662067\pi$$
−0.487435 + 0.873160i $$0.662067\pi$$
$$948$$ 20.0000 0.649570
$$949$$ −20.0000 −0.649227
$$950$$ 0 0
$$951$$ 36.0000 1.16738
$$952$$ 0 0
$$953$$ 12.0000 0.388718 0.194359 0.980930i $$-0.437737\pi$$
0.194359 + 0.980930i $$0.437737\pi$$
$$954$$ 3.00000 0.0971286
$$955$$ 0 0
$$956$$ 6.00000 0.194054
$$957$$ 36.0000 1.16371
$$958$$ 24.0000 0.775405
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 55.0000 1.77327
$$963$$ 12.0000 0.386695
$$964$$ 25.0000 0.805196
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ −12.0000 −0.385496
$$970$$ 0 0
$$971$$ −27.0000 −0.866471 −0.433236 0.901281i $$-0.642628\pi$$
−0.433236 + 0.901281i $$0.642628\pi$$
$$972$$ 10.0000 0.320750
$$973$$ 0 0
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ 4.00000 0.128037
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 8.00000 0.255812
$$979$$ −18.0000 −0.575282
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ 12.0000 0.382935
$$983$$ 57.0000 1.81802 0.909009 0.416777i $$-0.136840\pi$$
0.909009 + 0.416777i $$0.136840\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 36.0000 1.14647
$$987$$ 0 0
$$988$$ 5.00000 0.159071
$$989$$ −30.0000 −0.953945
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 14.0000 0.444277
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 24.0000 0.760469
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ 28.0000 0.886325
$$999$$ −44.0000 −1.39210
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 2450.2.a.f.1.1 1
5.2 odd 4 2450.2.c.p.99.1 2
5.3 odd 4 2450.2.c.p.99.2 2
5.4 even 2 490.2.a.j.1.1 1
7.3 odd 6 350.2.e.h.51.1 2
7.5 odd 6 350.2.e.h.151.1 2
7.6 odd 2 2450.2.a.p.1.1 1
15.14 odd 2 4410.2.a.c.1.1 1
20.19 odd 2 3920.2.a.g.1.1 1
35.3 even 12 350.2.j.a.149.1 4
35.4 even 6 490.2.e.a.471.1 2
35.9 even 6 490.2.e.a.361.1 2
35.12 even 12 350.2.j.a.249.1 4
35.13 even 4 2450.2.c.f.99.2 2
35.17 even 12 350.2.j.a.149.2 4
35.19 odd 6 70.2.e.b.11.1 2
35.24 odd 6 70.2.e.b.51.1 yes 2
35.27 even 4 2450.2.c.f.99.1 2
35.33 even 12 350.2.j.a.249.2 4
35.34 odd 2 490.2.a.g.1.1 1
105.59 even 6 630.2.k.e.541.1 2
105.89 even 6 630.2.k.e.361.1 2
105.104 even 2 4410.2.a.m.1.1 1
140.19 even 6 560.2.q.d.81.1 2
140.59 even 6 560.2.q.d.401.1 2
140.139 even 2 3920.2.a.be.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.b.11.1 2 35.19 odd 6
70.2.e.b.51.1 yes 2 35.24 odd 6
350.2.e.h.51.1 2 7.3 odd 6
350.2.e.h.151.1 2 7.5 odd 6
350.2.j.a.149.1 4 35.3 even 12
350.2.j.a.149.2 4 35.17 even 12
350.2.j.a.249.1 4 35.12 even 12
350.2.j.a.249.2 4 35.33 even 12
490.2.a.g.1.1 1 35.34 odd 2
490.2.a.j.1.1 1 5.4 even 2
490.2.e.a.361.1 2 35.9 even 6
490.2.e.a.471.1 2 35.4 even 6
560.2.q.d.81.1 2 140.19 even 6
560.2.q.d.401.1 2 140.59 even 6
630.2.k.e.361.1 2 105.89 even 6
630.2.k.e.541.1 2 105.59 even 6
2450.2.a.f.1.1 1 1.1 even 1 trivial
2450.2.a.p.1.1 1 7.6 odd 2
2450.2.c.f.99.1 2 35.27 even 4
2450.2.c.f.99.2 2 35.13 even 4
2450.2.c.p.99.1 2 5.2 odd 4
2450.2.c.p.99.2 2 5.3 odd 4
3920.2.a.g.1.1 1 20.19 odd 2
3920.2.a.be.1.1 1 140.139 even 2
4410.2.a.c.1.1 1 15.14 odd 2
4410.2.a.m.1.1 1 105.104 even 2