# Properties

 Label 5415.2.a.c.1.1 Level $5415$ Weight $2$ Character 5415.1 Self dual yes Analytic conductor $43.239$ Analytic rank $1$ 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] = [5415,2,Mod(1,5415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5415, 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("5415.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5415 = 3 \cdot 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5415.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: $$43.2389926945$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 5415.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} +2.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{20} -2.00000 q^{21} +2.00000 q^{22} -4.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -4.00000 q^{29} +1.00000 q^{30} -5.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} +4.00000 q^{39} -3.00000 q^{40} +2.00000 q^{42} -10.0000 q^{43} +2.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} -4.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} -6.00000 q^{56} +4.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} +2.00000 q^{61} -2.00000 q^{63} +7.00000 q^{64} -4.00000 q^{65} +2.00000 q^{66} +16.0000 q^{67} -2.00000 q^{68} -4.00000 q^{69} -2.00000 q^{70} +3.00000 q^{72} -2.00000 q^{73} +1.00000 q^{75} +4.00000 q^{77} -4.00000 q^{78} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} +2.00000 q^{84} -2.00000 q^{85} +10.0000 q^{86} -4.00000 q^{87} -6.00000 q^{88} +1.00000 q^{90} -8.00000 q^{91} +4.00000 q^{92} -12.0000 q^{94} -5.00000 q^{96} +16.0000 q^{97} +3.00000 q^{98} -2.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 3.00000 1.06066
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −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
$$15$$ −1.00000 −0.258199
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 0 0
$$20$$ 1.00000 0.223607
$$21$$ −2.00000 −0.436436
$$22$$ 2.00000 0.426401
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 3.00000 0.612372
$$25$$ 1.00000 0.200000
$$26$$ −4.00000 −0.784465
$$27$$ 1.00000 0.192450
$$28$$ 2.00000 0.377964
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 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
$$35$$ 2.00000 0.338062
$$36$$ −1.00000 −0.166667
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ 4.00000 0.640513
$$40$$ −3.00000 −0.474342
$$41$$ 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 $$-0.776025\pi$$
−0.762493 + 0.646997i $$0.776025\pi$$
$$44$$ 2.00000 0.301511
$$45$$ −1.00000 −0.149071
$$46$$ 4.00000 0.589768
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ −1.00000 −0.141421
$$51$$ 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
$$55$$ 2.00000 0.269680
$$56$$ −6.00000 −0.801784
$$57$$ 0 0
$$58$$ 4.00000 0.525226
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 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
$$65$$ −4.00000 −0.496139
$$66$$ 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
$$70$$ −2.00000 −0.239046
$$71$$ 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 $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 4.00000 0.455842
$$78$$ −4.00000 −0.452911
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 2.00000 0.218218
$$85$$ −2.00000 −0.216930
$$86$$ 10.0000 1.07833
$$87$$ −4.00000 −0.428845
$$88$$ −6.00000 −0.639602
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 1.00000 0.105409
$$91$$ −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
$$100$$ −1.00000 −0.100000
$$101$$ 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
$$105$$ 2.00000 0.195180
$$106$$ −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 $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 0 0
$$112$$ 2.00000 0.188982
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 4.00000 0.371391
$$117$$ 4.00000 0.369800
$$118$$ 4.00000 0.368230
$$119$$ −4.00000 −0.366679
$$120$$ −3.00000 −0.273861
$$121$$ −7.00000 −0.636364
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 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
$$130$$ 4.00000 0.350823
$$131$$ −14.0000 −1.22319 −0.611593 0.791173i $$-0.709471\pi$$
−0.611593 + 0.791173i $$0.709471\pi$$
$$132$$ 2.00000 0.174078
$$133$$ 0 0
$$134$$ −16.0000 −1.38219
$$135$$ −1.00000 −0.0860663
$$136$$ 6.00000 0.514496
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 4.00000 0.340503
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ −1.00000 −0.0833333
$$145$$ 4.00000 0.332182
$$146$$ 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 $$0.763904\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −24.0000 −1.95309 −0.976546 0.215308i $$-0.930924\pi$$
−0.976546 + 0.215308i $$0.930924\pi$$
$$152$$ 0 0
$$153$$ 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 $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 2.00000 0.158610
$$160$$ 5.00000 0.395285
$$161$$ 8.00000 0.630488
$$162$$ −1.00000 −0.0785674
$$163$$ −6.00000 −0.469956 −0.234978 0.972001i $$-0.575502\pi$$
−0.234978 + 0.972001i $$0.575502\pi$$
$$164$$ 0 0
$$165$$ 2.00000 0.155700
$$166$$ 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
$$170$$ 2.00000 0.153393
$$171$$ 0 0
$$172$$ 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
$$175$$ −2.00000 −0.151186
$$176$$ 2.00000 0.150756
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 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
$$195$$ −4.00000 −0.286446
$$196$$ 3.00000 0.214286
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 2.00000 0.142134
$$199$$ 12.0000 0.850657 0.425329 0.905039i $$-0.360158\pi$$
0.425329 + 0.905039i $$0.360158\pi$$
$$200$$ 3.00000 0.212132
$$201$$ 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
$$209$$ 0 0
$$210$$ −2.00000 −0.138013
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 10.0000 0.681994
$$216$$ 3.00000 0.204124
$$217$$ 0 0
$$218$$ 10.0000 0.677285
$$219$$ −2.00000 −0.135147
$$220$$ −2.00000 −0.134840
$$221$$ 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
$$225$$ 1.00000 0.0666667
$$226$$ 14.0000 0.931266
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 4.00000 0.263181
$$232$$ −12.0000 −0.787839
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ −12.0000 −0.782794
$$236$$ 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 $$0.562164\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 1.00000 0.0632456
$$251$$ 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
$$255$$ −2.00000 −0.125245
$$256$$ −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
$$260$$ 4.00000 0.248069
$$261$$ −4.00000 −0.247594
$$262$$ 14.0000 0.864923
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ −6.00000 −0.369274
$$265$$ −2.00000 −0.122859
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −16.0000 −0.977356
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 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
$$275$$ −2.00000 −0.120605
$$276$$ 4.00000 0.240772
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ −8.00000 −0.479808
$$279$$ 0 0
$$280$$ 6.00000 0.358569
$$281$$ −28.0000 −1.67034 −0.835170 0.549992i $$-0.814631\pi$$
−0.835170 + 0.549992i $$0.814631\pi$$
$$282$$ −12.0000 −0.714590
$$283$$ 26.0000 1.54554 0.772770 0.634686i $$-0.218871\pi$$
0.772770 + 0.634686i $$0.218871\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ −5.00000 −0.294628
$$289$$ −13.0000 −0.764706
$$290$$ −4.00000 −0.234888
$$291$$ 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
$$295$$ 4.00000 0.232889
$$296$$ 0 0
$$297$$ −2.00000 −0.116052
$$298$$ 18.0000 1.04271
$$299$$ −16.0000 −0.925304
$$300$$ −1.00000 −0.0577350
$$301$$ 20.0000 1.15278
$$302$$ 24.0000 1.38104
$$303$$ 14.0000 0.804279
$$304$$ 0 0
$$305$$ −2.00000 −0.114520
$$306$$ −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 $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 2.00000 0.112687
$$316$$ −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
$$320$$ −7.00000 −0.391312
$$321$$ 12.0000 0.669775
$$322$$ −8.00000 −0.445823
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 4.00000 0.221880
$$326$$ 6.00000 0.332309
$$327$$ −10.0000 −0.553001
$$328$$ 0 0
$$329$$ −24.0000 −1.32316
$$330$$ −2.00000 −0.110096
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 0 0
$$334$$ −8.00000 −0.437741
$$335$$ −16.0000 −0.874173
$$336$$ 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
$$340$$ 2.00000 0.108465
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ −30.0000 −1.61749
$$345$$ 4.00000 0.215353
$$346$$ 18.0000 0.967686
$$347$$ −28.0000 −1.50312 −0.751559 0.659665i $$-0.770698\pi$$
−0.751559 + 0.659665i $$0.770698\pi$$
$$348$$ 4.00000 0.214423
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 4.00000 0.213504
$$352$$ 10.0000 0.533002
$$353$$ −2.00000 −0.106449 −0.0532246 0.998583i $$-0.516950\pi$$
−0.0532246 + 0.998583i $$0.516950\pi$$
$$354$$ 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 $$0.516808\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ 0 0
$$362$$ 10.0000 0.525588
$$363$$ −7.00000 −0.367405
$$364$$ 8.00000 0.419314
$$365$$ 2.00000 0.104685
$$366$$ −2.00000 −0.104542
$$367$$ 10.0000 0.521996 0.260998 0.965339i $$-0.415948\pi$$
0.260998 + 0.965339i $$0.415948\pi$$
$$368$$ 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
$$375$$ −1.00000 −0.0516398
$$376$$ 36.0000 1.85656
$$377$$ −16.0000 −0.824042
$$378$$ 2.00000 0.102869
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 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
$$385$$ −4.00000 −0.203859
$$386$$ 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 $$0.775061\pi$$
$$390$$ 4.00000 0.202548
$$391$$ −8.00000 −0.404577
$$392$$ −9.00000 −0.454569
$$393$$ −14.0000 −0.706207
$$394$$ 2.00000 0.100759
$$395$$ −8.00000 −0.402524
$$396$$ 2.00000 0.100504
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ −12.0000 −0.601506
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ −16.0000 −0.798007
$$403$$ 0 0
$$404$$ −14.0000 −0.696526
$$405$$ −1.00000 −0.0496904
$$406$$ −8.00000 −0.397033
$$407$$ 0 0
$$408$$ 6.00000 0.297044
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ −8.00000 −0.394132
$$413$$ 8.00000 0.393654
$$414$$ 4.00000 0.196589
$$415$$ 12.0000 0.589057
$$416$$ −20.0000 −0.980581
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 12.0000 0.583460
$$424$$ 6.00000 0.291386
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ −4.00000 −0.193574
$$428$$ −12.0000 −0.580042
$$429$$ −8.00000 −0.386244
$$430$$ −10.0000 −0.482243
$$431$$ 4.00000 0.192673 0.0963366 0.995349i $$-0.469287\pi$$
0.0963366 + 0.995349i $$0.469287\pi$$
$$432$$ −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
$$435$$ 4.00000 0.191785
$$436$$ 10.0000 0.478913
$$437$$ 0 0
$$438$$ 2.00000 0.0955637
$$439$$ 40.0000 1.90910 0.954548 0.298057i $$-0.0963387\pi$$
0.954548 + 0.298057i $$0.0963387\pi$$
$$440$$ 6.00000 0.286039
$$441$$ −3.00000 −0.142857
$$442$$ −8.00000 −0.380521
$$443$$ 16.0000 0.760183 0.380091 0.924949i $$-0.375893\pi$$
0.380091 + 0.924949i $$0.375893\pi$$
$$444$$ 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 $$0.5\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 0 0
$$452$$ 14.0000 0.658505
$$453$$ −24.0000 −1.12762
$$454$$ 12.0000 0.563188
$$455$$ 8.00000 0.375046
$$456$$ 0 0
$$457$$ −14.0000 −0.654892 −0.327446 0.944870i $$-0.606188\pi$$
−0.327446 + 0.944870i $$0.606188\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 2.00000 0.0933520
$$460$$ −4.00000 −0.186501
$$461$$ 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 $$-0.670804\pi$$
−0.511213 + 0.859454i $$0.670804\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ 16.0000 0.740392 0.370196 0.928954i $$-0.379291\pi$$
0.370196 + 0.928954i $$0.379291\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ −32.0000 −1.47762
$$470$$ 12.0000 0.553519
$$471$$ −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 $$0.165321\pi$$
$$480$$ 5.00000 0.228218
$$481$$ 0 0
$$482$$ 26.0000 1.18427
$$483$$ 8.00000 0.364013
$$484$$ 7.00000 0.318182
$$485$$ −16.0000 −0.726523
$$486$$ −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
$$490$$ −3.00000 −0.135526
$$491$$ −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
$$494$$ 0 0
$$495$$ 2.00000 0.0898933
$$496$$ 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 $$0.245853\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 8.00000 0.357414
$$502$$ −22.0000 −0.981908
$$503$$ −8.00000 −0.356702 −0.178351 0.983967i $$-0.557076\pi$$
−0.178351 + 0.983967i $$0.557076\pi$$
$$504$$ −6.00000 −0.267261
$$505$$ −14.0000 −0.622992
$$506$$ −8.00000 −0.355643
$$507$$ 3.00000 0.133235
$$508$$ 4.00000 0.177471
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 2.00000 0.0885615
$$511$$ 4.00000 0.176950
$$512$$ 11.0000 0.486136
$$513$$ 0 0
$$514$$ 22.0000 0.970378
$$515$$ −8.00000 −0.352522
$$516$$ 10.0000 0.440225
$$517$$ −24.0000 −1.05552
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ −12.0000 −0.526235
$$521$$ 12.0000 0.525730 0.262865 0.964833i $$-0.415333\pi$$
0.262865 + 0.964833i $$0.415333\pi$$
$$522$$ 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
$$525$$ −2.00000 −0.0872872
$$526$$ 16.0000 0.697633
$$527$$ 0 0
$$528$$ 2.00000 0.0870388
$$529$$ −7.00000 −0.304348
$$530$$ 2.00000 0.0868744
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ 48.0000 2.07328
$$537$$ 12.0000 0.517838
$$538$$ 4.00000 0.172452
$$539$$ 6.00000 0.258438
$$540$$ 1.00000 0.0430331
$$541$$ −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
$$545$$ 10.0000 0.428353
$$546$$ 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
$$550$$ 2.00000 0.0852803
$$551$$ 0 0
$$552$$ −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 $$-0.797865\pi$$
−0.805056 + 0.593199i $$0.797865\pi$$
$$558$$ 0 0
$$559$$ −40.0000 −1.69182
$$560$$ −2.00000 −0.0845154
$$561$$ −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
$$565$$ 14.0000 0.588984
$$566$$ −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
$$575$$ −4.00000 −0.166812
$$576$$ 7.00000 0.291667
$$577$$ 42.0000 1.74848 0.874241 0.485491i $$-0.161359\pi$$
0.874241 + 0.485491i $$0.161359\pi$$
$$578$$ 13.0000 0.540729
$$579$$ −4.00000 −0.166234
$$580$$ −4.00000 −0.166091
$$581$$ 24.0000 0.995688
$$582$$ −16.0000 −0.663221
$$583$$ −4.00000 −0.165663
$$584$$ −6.00000 −0.248282
$$585$$ −4.00000 −0.165380
$$586$$ 26.0000 1.07405
$$587$$ −32.0000 −1.32078 −0.660391 0.750922i $$-0.729609\pi$$
−0.660391 + 0.750922i $$0.729609\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 0 0
$$590$$ −4.00000 −0.164677
$$591$$ −2.00000 −0.0822690
$$592$$ 0 0
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 4.00000 0.163984
$$596$$ 18.0000 0.737309
$$597$$ 12.0000 0.491127
$$598$$ 16.0000 0.654289
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 3.00000 0.122474
$$601$$ −30.0000 −1.22373 −0.611863 0.790964i $$-0.709580\pi$$
−0.611863 + 0.790964i $$0.709580\pi$$
$$602$$ −20.0000 −0.815139
$$603$$ 16.0000 0.651570
$$604$$ 24.0000 0.976546
$$605$$ 7.00000 0.284590
$$606$$ −14.0000 −0.568711
$$607$$ 16.0000 0.649420 0.324710 0.945814i $$-0.394733\pi$$
0.324710 + 0.945814i $$0.394733\pi$$
$$608$$ 0 0
$$609$$ 8.00000 0.324176
$$610$$ 2.00000 0.0809776
$$611$$ 48.0000 1.94187
$$612$$ −2.00000 −0.0808452
$$613$$ −18.0000 −0.727013 −0.363507 0.931592i $$-0.618421\pi$$
−0.363507 + 0.931592i $$0.618421\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 12.0000 0.483494
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ −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
$$625$$ 1.00000 0.0400000
$$626$$ 14.0000 0.559553
$$627$$ 0 0
$$628$$ 18.0000 0.718278
$$629$$ 0 0
$$630$$ −2.00000 −0.0796819
$$631$$ 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
$$635$$ 4.00000 0.158735
$$636$$ −2.00000 −0.0793052
$$637$$ −12.0000 −0.475457
$$638$$ −8.00000 −0.316723
$$639$$ 0 0
$$640$$ −3.00000 −0.118585
$$641$$ 24.0000 0.947943 0.473972 0.880540i $$-0.342820\pi$$
0.473972 + 0.880540i $$0.342820\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 10.0000 0.393750
$$646$$ 0 0
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 3.00000 0.117851
$$649$$ 8.00000 0.314027
$$650$$ −4.00000 −0.156893
$$651$$ 0 0
$$652$$ 6.00000 0.234978
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 14.0000 0.547025
$$656$$ 0 0
$$657$$ −2.00000 −0.0780274
$$658$$ 24.0000 0.935617
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ −2.00000 −0.0778499
$$661$$ −18.0000 −0.700119 −0.350059 0.936727i $$-0.613839\pi$$
−0.350059 + 0.936727i $$0.613839\pi$$
$$662$$ −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
$$670$$ 16.0000 0.618134
$$671$$ −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
$$675$$ 1.00000 0.0384900
$$676$$ −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
$$680$$ −6.00000 −0.230089
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ −6.00000 −0.229248
$$686$$ −20.0000 −0.763604
$$687$$ −14.0000 −0.534133
$$688$$ 10.0000 0.381246
$$689$$ 8.00000 0.304776
$$690$$ −4.00000 −0.152277
$$691$$ −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
$$695$$ −8.00000 −0.303457
$$696$$ −12.0000 −0.454859
$$697$$ 0 0
$$698$$ 18.0000 0.681310
$$699$$ −18.0000 −0.680823
$$700$$ 2.00000 0.0755929
$$701$$ −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
$$705$$ −12.0000 −0.451946
$$706$$ 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
$$715$$ 8.00000 0.299183
$$716$$ −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 $$0.213603\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ −16.0000 −0.595871
$$722$$ 0 0
$$723$$ −26.0000 −0.966950
$$724$$ 10.0000 0.371647
$$725$$ −4.00000 −0.148556
$$726$$ 7.00000 0.259794
$$727$$ −14.0000 −0.519231 −0.259616 0.965712i $$-0.583596\pi$$
−0.259616 + 0.965712i $$0.583596\pi$$
$$728$$ −24.0000 −0.889499
$$729$$ 1.00000 0.0370370
$$730$$ −2.00000 −0.0740233
$$731$$ −20.0000 −0.739727
$$732$$ −2.00000 −0.0739221
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 3.00000 0.110657
$$736$$ 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
$$741$$ 0 0
$$742$$ 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
$$745$$ 18.0000 0.659469
$$746$$ 4.00000 0.146450
$$747$$ −12.0000 −0.439057
$$748$$ 4.00000 0.146254
$$749$$ −24.0000 −0.876941
$$750$$ 1.00000 0.0365148
$$751$$ 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
$$755$$ 24.0000 0.873449
$$756$$ 2.00000 0.0727393
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ −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
$$765$$ −2.00000 −0.0723102
$$766$$ 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 $$0.811318\pi$$
$$770$$ 4.00000 0.144150
$$771$$ −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
$$780$$ 4.00000 0.143223
$$781$$ 0 0
$$782$$ 8.00000 0.286079
$$783$$ −4.00000 −0.142948
$$784$$ 3.00000 0.107143
$$785$$ 18.0000 0.642448
$$786$$ 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
$$790$$ 8.00000 0.284627
$$791$$ 28.0000 0.995565
$$792$$ −6.00000 −0.213201
$$793$$ 8.00000 0.284088
$$794$$ 2.00000 0.0709773
$$795$$ −2.00000 −0.0709327
$$796$$ −12.0000 −0.425329
$$797$$ 14.0000 0.495905 0.247953 0.968772i $$-0.420242\pi$$
0.247953 + 0.968772i $$0.420242\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ −5.00000 −0.176777
$$801$$ 0 0
$$802$$ 12.0000 0.423735
$$803$$ 4.00000 0.141157
$$804$$ −16.0000 −0.564276
$$805$$ −8.00000 −0.281963
$$806$$ 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 $$0.488807\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ −8.00000 −0.280745
$$813$$ 12.0000 0.420858
$$814$$ 0 0
$$815$$ 6.00000 0.210171
$$816$$ −2.00000 −0.0700140
$$817$$ 0 0
$$818$$ −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 $$-0.578461\pi$$
−0.244005 + 0.969774i $$0.578461\pi$$
$$824$$ 24.0000 0.836080
$$825$$ −2.00000 −0.0696311
$$826$$ −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 $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ −12.0000 −0.416526
$$831$$ 22.0000 0.763172
$$832$$ 28.0000 0.970725
$$833$$ −6.00000 −0.207888
$$834$$ −8.00000 −0.277017
$$835$$ −8.00000 −0.276851
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 26.0000 0.898155
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 6.00000 0.207020
$$841$$ −13.0000 −0.448276
$$842$$ −34.0000 −1.17172
$$843$$ −28.0000 −0.964371
$$844$$ −12.0000 −0.413057
$$845$$ −3.00000 −0.103203
$$846$$ −12.0000 −0.412568
$$847$$ 14.0000 0.481046
$$848$$ −2.00000 −0.0686803
$$849$$ 26.0000 0.892318
$$850$$ −2.00000 −0.0685994
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 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 $$0.565628\pi$$
$$860$$ −10.0000 −0.340997
$$861$$ 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
$$865$$ 18.0000 0.612018
$$866$$ −36.0000 −1.22333
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ −4.00000 −0.135613
$$871$$ 64.0000 2.16856
$$872$$ −30.0000 −1.01593
$$873$$ 16.0000 0.541518
$$874$$ 0 0
$$875$$ 2.00000 0.0676123
$$876$$ 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
$$880$$ −2.00000 −0.0674200
$$881$$ −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 $$-0.781753\pi$$
−0.774012 + 0.633171i $$0.781753\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 4.00000 0.134459
$$886$$ −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
$$893$$ 0 0
$$894$$ 18.0000 0.602010
$$895$$ −12.0000 −0.401116
$$896$$ −6.00000 −0.200446
$$897$$ −16.0000 −0.534224
$$898$$ 0 0
$$899$$ 0 0
$$900$$ −1.00000 −0.0333333
$$901$$ 4.00000 0.133259
$$902$$ 0 0
$$903$$ 20.0000 0.665558
$$904$$ −42.0000 −1.39690
$$905$$ 10.0000 0.332411
$$906$$ 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
$$910$$ −8.00000 −0.265197
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 24.0000 0.794284
$$914$$ 14.0000 0.463079
$$915$$ −2.00000 −0.0661180
$$916$$ 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 $$0.323020\pi$$
$$920$$ 12.0000 0.395628
$$921$$ 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
$$931$$ 0 0
$$932$$ 18.0000 0.589610
$$933$$ −18.0000 −0.589294
$$934$$ −16.0000 −0.523536
$$935$$ 4.00000 0.130814
$$936$$ 12.0000 0.392232
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ 32.0000 1.04484
$$939$$ −14.0000 −0.456873
$$940$$ 12.0000 0.391397
$$941$$ 60.0000 1.95594 0.977972 0.208736i $$-0.0669349\pi$$
0.977972 + 0.208736i $$0.0669349\pi$$
$$942$$ 18.0000 0.586472
$$943$$ 0 0
$$944$$ 4.00000 0.130189
$$945$$ 2.00000 0.0650600
$$946$$ −20.0000 −0.650256
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ −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
$$955$$ 18.0000 0.582466
$$956$$ 6.00000 0.194054
$$957$$ 8.00000 0.258603
$$958$$ −38.0000 −1.22772
$$959$$ −12.0000 −0.387500
$$960$$ −7.00000 −0.225924
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ 26.0000 0.837404
$$965$$ 4.00000 0.128765
$$966$$ −8.00000 −0.257396
$$967$$ 34.0000 1.09337 0.546683 0.837340i $$-0.315890\pi$$
0.546683 + 0.837340i $$0.315890\pi$$
$$968$$ −21.0000 −0.674966
$$969$$ 0 0
$$970$$ 16.0000 0.513729
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −16.0000 −0.512936
$$974$$ 8.00000 0.256337
$$975$$ 4.00000 0.128103
$$976$$ −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
$$980$$ −3.00000 −0.0958315
$$981$$ −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
$$985$$ 2.00000 0.0637253
$$986$$ 8.00000 0.254772
$$987$$ −24.0000 −0.763928
$$988$$ 0 0
$$989$$ 40.0000 1.27193
$$990$$ −2.00000 −0.0635642
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 0 0
$$993$$ 28.0000 0.888553
$$994$$ 0 0
$$995$$ −12.0000 −0.380426
$$996$$ 12.0000 0.380235
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ −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 5415.2.a.c.1.1 1
19.18 odd 2 285.2.a.b.1.1 1
57.56 even 2 855.2.a.b.1.1 1
76.75 even 2 4560.2.a.v.1.1 1
95.18 even 4 1425.2.c.d.799.1 2
95.37 even 4 1425.2.c.d.799.2 2
95.94 odd 2 1425.2.a.d.1.1 1
285.284 even 2 4275.2.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.b.1.1 1 19.18 odd 2
855.2.a.b.1.1 1 57.56 even 2
1425.2.a.d.1.1 1 95.94 odd 2
1425.2.c.d.799.1 2 95.18 even 4
1425.2.c.d.799.2 2 95.37 even 4
4275.2.a.o.1.1 1 285.284 even 2
4560.2.a.v.1.1 1 76.75 even 2
5415.2.a.c.1.1 1 1.1 even 1 trivial