# Properties

 Label 1425.2.a.d.1.1 Level $1425$ Weight $2$ Character 1425.1 Self dual yes Analytic conductor $11.379$ 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] = [1425,2,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, 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("1425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.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: $$11.3786822880$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1425.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +2.00000 q^{21} +2.00000 q^{22} +4.00000 q^{23} +3.00000 q^{24} -4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} +4.00000 q^{29} -5.00000 q^{32} -2.00000 q^{33} +2.00000 q^{34} -1.00000 q^{36} +1.00000 q^{38} +4.00000 q^{39} -2.00000 q^{42} +10.0000 q^{43} +2.00000 q^{44} -4.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -2.00000 q^{51} -4.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +6.00000 q^{56} -1.00000 q^{57} -4.00000 q^{58} +4.00000 q^{59} +2.00000 q^{61} +2.00000 q^{63} +7.00000 q^{64} +2.00000 q^{66} +16.0000 q^{67} +2.00000 q^{68} +4.00000 q^{69} +3.00000 q^{72} +2.00000 q^{73} +1.00000 q^{76} -4.00000 q^{77} -4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -2.00000 q^{84} -10.0000 q^{86} +4.00000 q^{87} -6.00000 q^{88} +8.00000 q^{91} -4.00000 q^{92} +12.0000 q^{94} -5.00000 q^{96} +16.0000 q^{97} +3.00000 q^{98} -2.00000 q^{99} +O(q^{100})$$

## 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 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 3.00000 1.06066
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 2.00000 0.426401
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 3.00000 0.612372
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ 1.00000 0.192450
$$28$$ −2.00000 −0.377964
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ −2.00000 −0.348155
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 10.0000 1.52499 0.762493 0.646997i $$-0.223975\pi$$
0.762493 + 0.646997i $$0.223975\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ −4.00000 −0.554700
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 6.00000 0.801784
$$57$$ −1.00000 −0.132453
$$58$$ −4.00000 −0.525226
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 2.00000 0.251976
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ 16.0000 1.95471 0.977356 0.211604i $$-0.0678686\pi$$
0.977356 + 0.211604i $$0.0678686\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 3.00000 0.353553
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ −4.00000 −0.455842
$$78$$ −4.00000 −0.452911
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ −10.0000 −1.07833
$$87$$ 4.00000 0.428845
$$88$$ −6.00000 −0.639602
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ −5.00000 −0.510310
$$97$$ 16.0000 1.62455 0.812277 0.583272i $$-0.198228\pi$$
0.812277 + 0.583272i $$0.198228\pi$$
$$98$$ 3.00000 0.303046
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 2.00000 0.198030
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 12.0000 1.17670
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −2.00000 −0.188982
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ 4.00000 0.369800
$$118$$ −4.00000 −0.368230
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 3.00000 0.265165
$$129$$ 10.0000 0.880451
$$130$$ 0 0
$$131$$ −14.0000 −1.22319 −0.611593 0.791173i $$-0.709471\pi$$
−0.611593 + 0.791173i $$0.709471\pi$$
$$132$$ 2.00000 0.174078
$$133$$ −2.00000 −0.173422
$$134$$ −16.0000 −1.38219
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ −3.00000 −0.247436
$$148$$ 0 0
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ −3.00000 −0.243332
$$153$$ −2.00000 −0.161690
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ −1.00000 −0.0785674
$$163$$ 6.00000 0.469956 0.234978 0.972001i $$-0.424498\pi$$
0.234978 + 0.972001i $$0.424498\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 6.00000 0.462910
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ −10.0000 −0.762493
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 4.00000 0.300658
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ −8.00000 −0.592999
$$183$$ 2.00000 0.147844
$$184$$ 12.0000 0.884652
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.00000 0.292509
$$188$$ 12.0000 0.875190
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 7.00000 0.505181
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ −16.0000 −1.14873
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 2.00000 0.142134
$$199$$ 12.0000 0.850657 0.425329 0.905039i $$-0.360158\pi$$
0.425329 + 0.905039i $$0.360158\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ −14.0000 −0.985037
$$203$$ 8.00000 0.561490
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 4.00000 0.278019
$$208$$ −4.00000 −0.277350
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 3.00000 0.204124
$$217$$ 0 0
$$218$$ −10.0000 −0.677285
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ 0 0
$$223$$ −24.0000 −1.60716 −0.803579 0.595198i $$-0.797074\pi$$
−0.803579 + 0.595198i $$0.797074\pi$$
$$224$$ −10.0000 −0.668153
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 12.0000 0.787839
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ −8.00000 −0.519656
$$238$$ 4.00000 0.259281
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4.00000 −0.254514
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 22.0000 1.38863 0.694314 0.719672i $$-0.255708\pi$$
0.694314 + 0.719672i $$0.255708\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ −8.00000 −0.502956
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ −10.0000 −0.622573
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ 14.0000 0.864923
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ −6.00000 −0.369274
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ 0 0
$$268$$ −16.0000 −0.977356
$$269$$ 4.00000 0.243884 0.121942 0.992537i $$-0.461088\pi$$
0.121942 + 0.992537i $$0.461088\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 8.00000 0.484182
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ −8.00000 −0.479808
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 28.0000 1.67034 0.835170 0.549992i $$-0.185369\pi$$
0.835170 + 0.549992i $$0.185369\pi$$
$$282$$ 12.0000 0.714590
$$283$$ −26.0000 −1.54554 −0.772770 0.634686i $$-0.781129\pi$$
−0.772770 + 0.634686i $$0.781129\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ −5.00000 −0.294628
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 16.0000 0.937937
$$292$$ −2.00000 −0.117041
$$293$$ −26.0000 −1.51894 −0.759468 0.650545i $$-0.774541\pi$$
−0.759468 + 0.650545i $$0.774541\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −2.00000 −0.116052
$$298$$ 18.0000 1.04271
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ 20.0000 1.15278
$$302$$ −24.0000 −1.38104
$$303$$ 14.0000 0.804279
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 4.00000 0.227921
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 12.0000 0.679366
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ −14.0000 −0.786318 −0.393159 0.919470i $$-0.628618\pi$$
−0.393159 + 0.919470i $$0.628618\pi$$
$$318$$ −2.00000 −0.112154
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ −8.00000 −0.445823
$$323$$ 2.00000 0.111283
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −6.00000 −0.332309
$$327$$ 10.0000 0.553001
$$328$$ 0 0
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 0 0
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ −16.0000 −0.871576 −0.435788 0.900049i $$-0.643530\pi$$
−0.435788 + 0.900049i $$0.643530\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ −14.0000 −0.760376
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 1.00000 0.0540738
$$343$$ −20.0000 −1.07990
$$344$$ 30.0000 1.61749
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ −4.00000 −0.214423
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ 10.0000 0.533002
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −4.00000 −0.211702
$$358$$ 12.0000 0.634220
$$359$$ −2.00000 −0.105556 −0.0527780 0.998606i $$-0.516808\pi$$
−0.0527780 + 0.998606i $$0.516808\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −10.0000 −0.525588
$$363$$ −7.00000 −0.367405
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ −10.0000 −0.521996 −0.260998 0.965339i $$-0.584052\pi$$
−0.260998 + 0.965339i $$0.584052\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 4.00000 0.207670
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ −36.0000 −1.85656
$$377$$ 16.0000 0.824042
$$378$$ −2.00000 −0.102869
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 18.0000 0.920960
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 3.00000 0.153093
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ 10.0000 0.508329
$$388$$ −16.0000 −0.812277
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ −9.00000 −0.454569
$$393$$ −14.0000 −0.706207
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ −12.0000 −0.601506
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ −16.0000 −0.798007
$$403$$ 0 0
$$404$$ −14.0000 −0.696526
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ 0 0
$$408$$ −6.00000 −0.297044
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ −8.00000 −0.394132
$$413$$ 8.00000 0.393654
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ −20.0000 −0.980581
$$417$$ 8.00000 0.391762
$$418$$ −2.00000 −0.0978232
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ 12.0000 0.584151
$$423$$ −12.0000 −0.583460
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.00000 0.193574
$$428$$ −12.0000 −0.580042
$$429$$ −8.00000 −0.386244
$$430$$ 0 0
$$431$$ −4.00000 −0.192673 −0.0963366 0.995349i $$-0.530713\pi$$
−0.0963366 + 0.995349i $$0.530713\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 36.0000 1.73005 0.865025 0.501729i $$-0.167303\pi$$
0.865025 + 0.501729i $$0.167303\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ −4.00000 −0.191346
$$438$$ −2.00000 −0.0955637
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 8.00000 0.380521
$$443$$ −16.0000 −0.760183 −0.380091 0.924949i $$-0.624107\pi$$
−0.380091 + 0.924949i $$0.624107\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 24.0000 1.13643
$$447$$ −18.0000 −0.851371
$$448$$ 14.0000 0.661438
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 14.0000 0.658505
$$453$$ 24.0000 1.12762
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −3.00000 −0.140488
$$457$$ 14.0000 0.654892 0.327446 0.944870i $$-0.393812\pi$$
0.327446 + 0.944870i $$0.393812\pi$$
$$458$$ 14.0000 0.654177
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 4.00000 0.186097
$$463$$ 22.0000 1.02243 0.511213 0.859454i $$-0.329196\pi$$
0.511213 + 0.859454i $$0.329196\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ −16.0000 −0.740392 −0.370196 0.928954i $$-0.620709\pi$$
−0.370196 + 0.928954i $$0.620709\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ 32.0000 1.47762
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 12.0000 0.552345
$$473$$ −20.0000 −0.919601
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ 2.00000 0.0915737
$$478$$ 6.00000 0.274434
$$479$$ 38.0000 1.73626 0.868132 0.496333i $$-0.165321\pi$$
0.868132 + 0.496333i $$0.165321\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −26.0000 −1.18427
$$483$$ 8.00000 0.364013
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 6.00000 0.271607
$$489$$ 6.00000 0.271329
$$490$$ 0 0
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ 0 0
$$493$$ −8.00000 −0.360302
$$494$$ 4.00000 0.179969
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ 32.0000 1.43252 0.716258 0.697835i $$-0.245853\pi$$
0.716258 + 0.697835i $$0.245853\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ −22.0000 −0.981908
$$503$$ 8.00000 0.356702 0.178351 0.983967i $$-0.442924\pi$$
0.178351 + 0.983967i $$0.442924\pi$$
$$504$$ 6.00000 0.267261
$$505$$ 0 0
$$506$$ 8.00000 0.355643
$$507$$ 3.00000 0.133235
$$508$$ 4.00000 0.177471
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ 11.0000 0.486136
$$513$$ −1.00000 −0.0441511
$$514$$ 22.0000 0.970378
$$515$$ 0 0
$$516$$ −10.0000 −0.440225
$$517$$ 24.0000 1.05552
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ −4.00000 −0.175075
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 14.0000 0.611593
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ 2.00000 0.0870388
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 2.00000 0.0867110
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 48.0000 2.07328
$$537$$ −12.0000 −0.517838
$$538$$ −4.00000 −0.172452
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −42.0000 −1.80572 −0.902861 0.429934i $$-0.858537\pi$$
−0.902861 + 0.429934i $$0.858537\pi$$
$$542$$ −12.0000 −0.515444
$$543$$ 10.0000 0.429141
$$544$$ 10.0000 0.428746
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ −16.0000 −0.684111 −0.342055 0.939680i $$-0.611123\pi$$
−0.342055 + 0.939680i $$0.611123\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −4.00000 −0.170406
$$552$$ 12.0000 0.510754
$$553$$ −16.0000 −0.680389
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ −8.00000 −0.339276
$$557$$ 38.0000 1.61011 0.805056 0.593199i $$-0.202135\pi$$
0.805056 + 0.593199i $$0.202135\pi$$
$$558$$ 0 0
$$559$$ 40.0000 1.69182
$$560$$ 0 0
$$561$$ 4.00000 0.168880
$$562$$ −28.0000 −1.18111
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ 26.0000 1.09286
$$567$$ 2.00000 0.0839921
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −24.0000 −1.00437 −0.502184 0.864761i $$-0.667470\pi$$
−0.502184 + 0.864761i $$0.667470\pi$$
$$572$$ 8.00000 0.334497
$$573$$ −18.0000 −0.751961
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ −42.0000 −1.74848 −0.874241 0.485491i $$-0.838641\pi$$
−0.874241 + 0.485491i $$0.838641\pi$$
$$578$$ 13.0000 0.540729
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ −16.0000 −0.663221
$$583$$ −4.00000 −0.165663
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ 26.0000 1.07405
$$587$$ 32.0000 1.32078 0.660391 0.750922i $$-0.270391\pi$$
0.660391 + 0.750922i $$0.270391\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ 0 0
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ 12.0000 0.491127
$$598$$ −16.0000 −0.654289
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ −20.0000 −0.815139
$$603$$ 16.0000 0.651570
$$604$$ −24.0000 −0.976546
$$605$$ 0 0
$$606$$ −14.0000 −0.568711
$$607$$ 16.0000 0.649420 0.324710 0.945814i $$-0.394733\pi$$
0.324710 + 0.945814i $$0.394733\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ −48.0000 −1.94187
$$612$$ 2.00000 0.0808452
$$613$$ 18.0000 0.727013 0.363507 0.931592i $$-0.381579\pi$$
0.363507 + 0.931592i $$0.381579\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −12.0000 −0.483494
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ 16.0000 0.643094 0.321547 0.946894i $$-0.395797\pi$$
0.321547 + 0.946894i $$0.395797\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 18.0000 0.721734
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ −14.0000 −0.559553
$$627$$ 2.00000 0.0798723
$$628$$ −18.0000 −0.718278
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 48.0000 1.91085 0.955425 0.295234i $$-0.0953977\pi$$
0.955425 + 0.295234i $$0.0953977\pi$$
$$632$$ −24.0000 −0.954669
$$633$$ −12.0000 −0.476957
$$634$$ 14.0000 0.556011
$$635$$ 0 0
$$636$$ −2.00000 −0.0793052
$$637$$ −12.0000 −0.475457
$$638$$ 8.00000 0.316723
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ −26.0000 −1.02534 −0.512670 0.858586i $$-0.671344\pi$$
−0.512670 + 0.858586i $$0.671344\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ −2.00000 −0.0786889
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 3.00000 0.117851
$$649$$ −8.00000 −0.314027
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −6.00000 −0.234978
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 24.0000 0.935617
$$659$$ 16.0000 0.623272 0.311636 0.950202i $$-0.399123\pi$$
0.311636 + 0.950202i $$0.399123\pi$$
$$660$$ 0 0
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ 28.0000 1.08825
$$663$$ −8.00000 −0.310694
$$664$$ 36.0000 1.39707
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.0000 0.619522
$$668$$ −8.00000 −0.309529
$$669$$ −24.0000 −0.927894
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ −10.0000 −0.385758
$$673$$ −24.0000 −0.925132 −0.462566 0.886585i $$-0.653071\pi$$
−0.462566 + 0.886585i $$0.653071\pi$$
$$674$$ 16.0000 0.616297
$$675$$ 0 0
$$676$$ −3.00000 −0.115385
$$677$$ 50.0000 1.92166 0.960828 0.277145i $$-0.0893883\pi$$
0.960828 + 0.277145i $$0.0893883\pi$$
$$678$$ 14.0000 0.537667
$$679$$ 32.0000 1.22805
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ −14.0000 −0.534133
$$688$$ −10.0000 −0.381246
$$689$$ 8.00000 0.304776
$$690$$ 0 0
$$691$$ −16.0000 −0.608669 −0.304334 0.952565i $$-0.598434\pi$$
−0.304334 + 0.952565i $$0.598434\pi$$
$$692$$ 18.0000 0.684257
$$693$$ −4.00000 −0.151947
$$694$$ −28.0000 −1.06287
$$695$$ 0 0
$$696$$ 12.0000 0.454859
$$697$$ 0 0
$$698$$ 18.0000 0.681310
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ −34.0000 −1.28416 −0.642081 0.766637i $$-0.721929\pi$$
−0.642081 + 0.766637i $$0.721929\pi$$
$$702$$ −4.00000 −0.150970
$$703$$ 0 0
$$704$$ −14.0000 −0.527645
$$705$$ 0 0
$$706$$ −2.00000 −0.0752710
$$707$$ 28.0000 1.05305
$$708$$ −4.00000 −0.150329
$$709$$ 18.0000 0.676004 0.338002 0.941145i $$-0.390249\pi$$
0.338002 + 0.941145i $$0.390249\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 4.00000 0.149696
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ −6.00000 −0.224074
$$718$$ 2.00000 0.0746393
$$719$$ 42.0000 1.56634 0.783168 0.621810i $$-0.213603\pi$$
0.783168 + 0.621810i $$0.213603\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ −1.00000 −0.0372161
$$723$$ 26.0000 0.966950
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ 7.00000 0.259794
$$727$$ 14.0000 0.519231 0.259616 0.965712i $$-0.416404\pi$$
0.259616 + 0.965712i $$0.416404\pi$$
$$728$$ 24.0000 0.889499
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −20.0000 −0.739727
$$732$$ −2.00000 −0.0739221
$$733$$ −34.0000 −1.25582 −0.627909 0.778287i $$-0.716089\pi$$
−0.627909 + 0.778287i $$0.716089\pi$$
$$734$$ 10.0000 0.369107
$$735$$ 0 0
$$736$$ −20.0000 −0.737210
$$737$$ −32.0000 −1.17874
$$738$$ 0 0
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ −4.00000 −0.146944
$$742$$ −4.00000 −0.146845
$$743$$ 8.00000 0.293492 0.146746 0.989174i $$-0.453120\pi$$
0.146746 + 0.989174i $$0.453120\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 4.00000 0.146450
$$747$$ 12.0000 0.439057
$$748$$ −4.00000 −0.146254
$$749$$ 24.0000 0.876941
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ 12.0000 0.437595
$$753$$ 22.0000 0.801725
$$754$$ −16.0000 −0.582686
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ 20.0000 0.726433
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 4.00000 0.144905
$$763$$ 20.0000 0.724049
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ 16.0000 0.577727
$$768$$ −17.0000 −0.613435
$$769$$ −46.0000 −1.65880 −0.829401 0.558653i $$-0.811318\pi$$
−0.829401 + 0.558653i $$0.811318\pi$$
$$770$$ 0 0
$$771$$ −22.0000 −0.792311
$$772$$ 4.00000 0.143963
$$773$$ −14.0000 −0.503545 −0.251773 0.967786i $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ −10.0000 −0.359443
$$775$$ 0 0
$$776$$ 48.0000 1.72310
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 8.00000 0.286079
$$783$$ 4.00000 0.142948
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 14.0000 0.499363
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ 16.0000 0.569615
$$790$$ 0 0
$$791$$ −28.0000 −0.995565
$$792$$ −6.00000 −0.213201
$$793$$ 8.00000 0.284088
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ −12.0000 −0.425329
$$797$$ 14.0000 0.495905 0.247953 0.968772i $$-0.420242\pi$$
0.247953 + 0.968772i $$0.420242\pi$$
$$798$$ 2.00000 0.0707992
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −12.0000 −0.423735
$$803$$ −4.00000 −0.141157
$$804$$ −16.0000 −0.564276
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 4.00000 0.140807
$$808$$ 42.0000 1.47755
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 0 0
$$811$$ 4.00000 0.140459 0.0702295 0.997531i $$-0.477627\pi$$
0.0702295 + 0.997531i $$0.477627\pi$$
$$812$$ −8.00000 −0.280745
$$813$$ 12.0000 0.420858
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ −10.0000 −0.349856
$$818$$ 26.0000 0.909069
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 6.00000 0.209274
$$823$$ 14.0000 0.488009 0.244005 0.969774i $$-0.421539\pi$$
0.244005 + 0.969774i $$0.421539\pi$$
$$824$$ 24.0000 0.836080
$$825$$ 0 0
$$826$$ −8.00000 −0.278356
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ 28.0000 0.970725
$$833$$ 6.00000 0.207888
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ −2.00000 −0.0691714
$$837$$ 0 0
$$838$$ 26.0000 0.898155
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 34.0000 1.17172
$$843$$ 28.0000 0.964371
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ −14.0000 −0.481046
$$848$$ −2.00000 −0.0686803
$$849$$ −26.0000 −0.892318
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ 36.0000 1.23045
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ 8.00000 0.273115
$$859$$ −12.0000 −0.409435 −0.204717 0.978821i $$-0.565628\pi$$
−0.204717 + 0.978821i $$0.565628\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 4.00000 0.136241
$$863$$ −40.0000 −1.36162 −0.680808 0.732462i $$-0.738371\pi$$
−0.680808 + 0.732462i $$0.738371\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ −36.0000 −1.22333
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 64.0000 2.16856
$$872$$ 30.0000 1.01593
$$873$$ 16.0000 0.541518
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ −2.00000 −0.0675737
$$877$$ −12.0000 −0.405211 −0.202606 0.979260i $$-0.564941\pi$$
−0.202606 + 0.979260i $$0.564941\pi$$
$$878$$ 40.0000 1.34993
$$879$$ −26.0000 −0.876958
$$880$$ 0 0
$$881$$ −10.0000 −0.336909 −0.168454 0.985709i $$-0.553878\pi$$
−0.168454 + 0.985709i $$0.553878\pi$$
$$882$$ 3.00000 0.101015
$$883$$ 46.0000 1.54802 0.774012 0.633171i $$-0.218247\pi$$
0.774012 + 0.633171i $$0.218247\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ 16.0000 0.537531
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 24.0000 0.803579
$$893$$ 12.0000 0.401565
$$894$$ 18.0000 0.602010
$$895$$ 0 0
$$896$$ 6.00000 0.200446
$$897$$ 16.0000 0.534224
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −4.00000 −0.133259
$$902$$ 0 0
$$903$$ 20.0000 0.665558
$$904$$ −42.0000 −1.39690
$$905$$ 0 0
$$906$$ −24.0000 −0.797347
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 14.0000 0.464351
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ −24.0000 −0.794284
$$914$$ −14.0000 −0.463079
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ −28.0000 −0.924641
$$918$$ 2.00000 0.0660098
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −6.00000 −0.197599
$$923$$ 0 0
$$924$$ 4.00000 0.131590
$$925$$ 0 0
$$926$$ −22.0000 −0.722965
$$927$$ 8.00000 0.262754
$$928$$ −20.0000 −0.656532
$$929$$ 10.0000 0.328089 0.164045 0.986453i $$-0.447546\pi$$
0.164045 + 0.986453i $$0.447546\pi$$
$$930$$ 0 0
$$931$$ 3.00000 0.0983210
$$932$$ −18.0000 −0.589610
$$933$$ −18.0000 −0.589294
$$934$$ 16.0000 0.523536
$$935$$ 0 0
$$936$$ 12.0000 0.392232
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ −32.0000 −1.04484
$$939$$ 14.0000 0.456873
$$940$$ 0 0
$$941$$ −60.0000 −1.95594 −0.977972 0.208736i $$-0.933065\pi$$
−0.977972 + 0.208736i $$0.933065\pi$$
$$942$$ −18.0000 −0.586472
$$943$$ 0 0
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 20.0000 0.650256
$$947$$ 52.0000 1.68977 0.844886 0.534946i $$-0.179668\pi$$
0.844886 + 0.534946i $$0.179668\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ −14.0000 −0.453981
$$952$$ −12.0000 −0.388922
$$953$$ −46.0000 −1.49009 −0.745043 0.667016i $$-0.767571\pi$$
−0.745043 + 0.667016i $$0.767571\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ 6.00000 0.194054
$$957$$ −8.00000 −0.258603
$$958$$ −38.0000 −1.22772
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ −26.0000 −0.837404
$$965$$ 0 0
$$966$$ −8.00000 −0.257396
$$967$$ −34.0000 −1.09337 −0.546683 0.837340i $$-0.684110\pi$$
−0.546683 + 0.837340i $$0.684110\pi$$
$$968$$ −21.0000 −0.674966
$$969$$ 2.00000 0.0642493
$$970$$ 0 0
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 16.0000 0.512936
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ −6.00000 −0.191859
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ 6.00000 0.191468
$$983$$ −16.0000 −0.510321 −0.255160 0.966899i $$-0.582128\pi$$
−0.255160 + 0.966899i $$0.582128\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 8.00000 0.254772
$$987$$ −24.0000 −0.763928
$$988$$ 4.00000 0.127257
$$989$$ 40.0000 1.27193
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 0 0
$$993$$ −28.0000 −0.888553
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ 2.00000 0.0633406 0.0316703 0.999498i $$-0.489917\pi$$
0.0316703 + 0.999498i $$0.489917\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1425.2.a.d.1.1 1
3.2 odd 2 4275.2.a.o.1.1 1
5.2 odd 4 1425.2.c.d.799.1 2
5.3 odd 4 1425.2.c.d.799.2 2
5.4 even 2 285.2.a.b.1.1 1
15.14 odd 2 855.2.a.b.1.1 1
20.19 odd 2 4560.2.a.v.1.1 1
95.94 odd 2 5415.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.b.1.1 1 5.4 even 2
855.2.a.b.1.1 1 15.14 odd 2
1425.2.a.d.1.1 1 1.1 even 1 trivial
1425.2.c.d.799.1 2 5.2 odd 4
1425.2.c.d.799.2 2 5.3 odd 4
4275.2.a.o.1.1 1 3.2 odd 2
4560.2.a.v.1.1 1 20.19 odd 2
5415.2.a.c.1.1 1 95.94 odd 2