# Properties

 Label 4275.2.a.o.1.1 Level $4275$ Weight $2$ Character 4275.1 Self dual yes Analytic conductor $34.136$ 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] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4275 = 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4275.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: $$34.1360468641$$ 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 $$\chi$$ $$=$$ 4275.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{7} -3.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{7} -3.00000 q^{8} +2.00000 q^{11} +4.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{19} +2.00000 q^{22} -4.00000 q^{23} +4.00000 q^{26} -2.00000 q^{28} -4.00000 q^{29} +5.00000 q^{32} +2.00000 q^{34} -1.00000 q^{38} +10.0000 q^{43} -2.00000 q^{44} -4.00000 q^{46} +12.0000 q^{47} -3.00000 q^{49} -4.00000 q^{52} -2.00000 q^{53} -6.00000 q^{56} -4.00000 q^{58} -4.00000 q^{59} +2.00000 q^{61} +7.00000 q^{64} +16.0000 q^{67} -2.00000 q^{68} +2.00000 q^{73} +1.00000 q^{76} +4.00000 q^{77} -8.00000 q^{79} -12.0000 q^{83} +10.0000 q^{86} -6.00000 q^{88} +8.00000 q^{91} +4.00000 q^{92} +12.0000 q^{94} +16.0000 q^{97} -3.00000 q^{98} +O(q^{100})$$

## 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 $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ 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 $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 2.00000 0.426401
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 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 $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −4.00000 −0.554700
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −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 $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 16.0000 1.95471 0.977356 0.211604i $$-0.0678686\pi$$
0.977356 + 0.211604i $$0.0678686\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 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
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −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
$$87$$ 0 0
$$88$$ −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
$$96$$ 0 0
$$97$$ 16.0000 1.62455 0.812277 0.583272i $$-0.198228\pi$$
0.812277 + 0.583272i $$0.198228\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 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 $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 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 $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ 0 0
$$118$$ −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
$$126$$ 0 0
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 0 0
$$133$$ −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 $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.00000 0.668994
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 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
$$153$$ 0 0
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ 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 $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −10.0000 −0.762493
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 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
$$183$$ 0 0
$$184$$ 12.0000 0.884652
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.00000 0.292509
$$188$$ −12.0000 −0.875190
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ −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 $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ 12.0000 0.850657 0.425329 0.905039i $$-0.360158\pi$$
0.425329 + 0.905039i $$0.360158\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −14.0000 −0.985037
$$203$$ −8.00000 −0.561490
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ −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
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 10.0000 0.677285
$$219$$ 0 0
$$220$$ 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 $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 12.0000 0.787839
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ 4.00000 0.259281
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 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
$$243$$ 0 0
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4.00000 −0.254514
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −22.0000 −1.38863 −0.694314 0.719672i $$-0.744292\pi$$
−0.694314 + 0.719672i $$0.744292\pi$$
$$252$$ 0 0
$$253$$ −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 $$-0.259294\pi$$
0.686161 + 0.727450i $$0.259294\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 14.0000 0.864923
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.00000 −0.122628
$$267$$ 0 0
$$268$$ −16.0000 −0.977356
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 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
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −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 $$-0.814631\pi$$
−0.835170 + 0.549992i $$0.814631\pi$$
$$282$$ 0 0
$$283$$ −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
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −2.00000 −0.117041
$$293$$ 26.0000 1.51894 0.759468 0.650545i $$-0.225459\pi$$
0.759468 + 0.650545i $$0.225459\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 18.0000 1.04271
$$299$$ −16.0000 −0.925304
$$300$$ 0 0
$$301$$ 20.0000 1.15278
$$302$$ 24.0000 1.38104
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ 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 $$-0.371382\pi$$
0.393159 + 0.919470i $$0.371382\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −8.00000 −0.445823
$$323$$ −2.00000 −0.111283
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 6.00000 0.332309
$$327$$ 0 0
$$328$$ 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
$$336$$ 0 0
$$337$$ −16.0000 −0.871576 −0.435788 0.900049i $$-0.643530\pi$$
−0.435788 + 0.900049i $$0.643530\pi$$
$$338$$ 3.00000 0.163178
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −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 $$-0.770698\pi$$
−0.751559 + 0.659665i $$0.770698\pi$$
$$348$$ 0 0
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 10.0000 0.533002
$$353$$ −2.00000 −0.106449 −0.0532246 0.998583i $$-0.516950\pi$$
−0.0532246 + 0.998583i $$0.516950\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 2.00000 0.105556 0.0527780 0.998606i $$-0.483192\pi$$
0.0527780 + 0.998606i $$0.483192\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 10.0000 0.525588
$$363$$ 0 0
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −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
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 18.0000 0.920960
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −4.00000 −0.203595
$$387$$ 0 0
$$388$$ −16.0000 −0.812277
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 9.00000 0.454569
$$393$$ 0 0
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 12.0000 0.601506
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 14.0000 0.696526
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 20.0000 0.980581
$$417$$ 0 0
$$418$$ −2.00000 −0.0978232
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 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
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.00000 0.193574
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 4.00000 0.192673 0.0963366 0.995349i $$-0.469287\pi$$
0.0963366 + 0.995349i $$0.469287\pi$$
$$432$$ 0 0
$$433$$ 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
$$438$$ 0 0
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 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
$$447$$ 0 0
$$448$$ 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
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 14.0000 0.654892 0.327446 0.944870i $$-0.393812\pi$$
0.327446 + 0.944870i $$0.393812\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ 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 $$-0.379291\pi$$
0.370196 + 0.928954i $$0.379291\pi$$
$$468$$ 0 0
$$469$$ 32.0000 1.47762
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 12.0000 0.552345
$$473$$ 20.0000 0.919601
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ 0 0
$$478$$ 6.00000 0.274434
$$479$$ −38.0000 −1.73626 −0.868132 0.496333i $$-0.834679\pi$$
−0.868132 + 0.496333i $$0.834679\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 26.0000 1.18427
$$483$$ 0 0
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 6.00000 0.270776 0.135388 0.990793i $$-0.456772\pi$$
0.135388 + 0.990793i $$0.456772\pi$$
$$492$$ 0 0
$$493$$ −8.00000 −0.360302
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 32.0000 1.43252 0.716258 0.697835i $$-0.245853\pi$$
0.716258 + 0.697835i $$0.245853\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −22.0000 −0.981908
$$503$$ −8.00000 −0.356702 −0.178351 0.983967i $$-0.557076\pi$$
−0.178351 + 0.983967i $$0.557076\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −8.00000 −0.355643
$$507$$ 0 0
$$508$$ 4.00000 0.177471
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ 22.0000 0.970378
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 24.0000 1.05552
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 12.0000 0.525730 0.262865 0.964833i $$-0.415333\pi$$
0.262865 + 0.964833i $$0.415333\pi$$
$$522$$ 0 0
$$523$$ 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
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.00000 0.0867110
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −48.0000 −2.07328
$$537$$ 0 0
$$538$$ −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
$$543$$ 0 0
$$544$$ 10.0000 0.428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −16.0000 −0.684111 −0.342055 0.939680i $$-0.611123\pi$$
−0.342055 + 0.939680i $$0.611123\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 4.00000 0.170406
$$552$$ 0 0
$$553$$ −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
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −28.0000 −1.18111
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −26.0000 −1.09286
$$567$$ 0 0
$$568$$ 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
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −42.0000 −1.74848 −0.874241 0.485491i $$-0.838641\pi$$
−0.874241 + 0.485491i $$0.838641\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −24.0000 −0.995688
$$582$$ 0 0
$$583$$ −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 $$-0.729609\pi$$
−0.660391 + 0.750922i $$0.729609\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ 0 0
$$598$$ −16.0000 −0.654289
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 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
$$603$$ 0 0
$$604$$ −24.0000 −0.976546
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 16.0000 0.649420 0.324710 0.945814i $$-0.394733\pi$$
0.324710 + 0.945814i $$0.394733\pi$$
$$608$$ −5.00000 −0.202777
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 48.0000 1.94187
$$612$$ 0 0
$$613$$ 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 $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ 16.0000 0.643094 0.321547 0.946894i $$-0.395797\pi$$
0.321547 + 0.946894i $$0.395797\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 18.0000 0.721734
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 14.0000 0.559553
$$627$$ 0 0
$$628$$ −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
$$633$$ 0 0
$$634$$ 14.0000 0.556011
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −12.0000 −0.475457
$$638$$ −8.00000 −0.316723
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 24.0000 0.947943 0.473972 0.880540i $$-0.342820\pi$$
0.473972 + 0.880540i $$0.342820\pi$$
$$642$$ 0 0
$$643$$ −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 $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 0 0
$$649$$ −8.00000 −0.314027
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −6.00000 −0.234978
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 24.0000 0.935617
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ 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
$$663$$ 0 0
$$664$$ 36.0000 1.39707
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.0000 0.619522
$$668$$ 8.00000 0.309529
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ −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 $$-0.910612\pi$$
−0.960828 + 0.277145i $$0.910612\pi$$
$$678$$ 0 0
$$679$$ 32.0000 1.22805
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ 0 0
$$688$$ −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
$$693$$ 0 0
$$694$$ −28.0000 −1.06287
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −18.0000 −0.681310
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 34.0000 1.28416 0.642081 0.766637i $$-0.278071\pi$$
0.642081 + 0.766637i $$0.278071\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 14.0000 0.527645
$$705$$ 0 0
$$706$$ −2.00000 −0.0752710
$$707$$ −28.0000 −1.05305
$$708$$ 0 0
$$709$$ 18.0000 0.676004 0.338002 0.941145i $$-0.390249\pi$$
0.338002 + 0.941145i $$0.390249\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 2.00000 0.0746393
$$719$$ −42.0000 −1.56634 −0.783168 0.621810i $$-0.786397\pi$$
−0.783168 + 0.621810i $$0.786397\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 14.0000 0.519231 0.259616 0.965712i $$-0.416404\pi$$
0.259616 + 0.965712i $$0.416404\pi$$
$$728$$ −24.0000 −0.889499
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 20.0000 0.739727
$$732$$ 0 0
$$733$$ −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
$$741$$ 0 0
$$742$$ −4.00000 −0.146845
$$743$$ −8.00000 −0.293492 −0.146746 0.989174i $$-0.546880\pi$$
−0.146746 + 0.989174i $$0.546880\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −4.00000 −0.146450
$$747$$ 0 0
$$748$$ −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
$$753$$ 0 0
$$754$$ −16.0000 −0.582686
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 0 0
$$763$$ 20.0000 0.724049
$$764$$ −18.0000 −0.651217
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ −16.0000 −0.577727
$$768$$ 0 0
$$769$$ −46.0000 −1.65880 −0.829401 0.558653i $$-0.811318\pi$$
−0.829401 + 0.558653i $$0.811318\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 4.00000 0.143963
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ 0 0
$$775$$ 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
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 28.0000 0.995565
$$792$$ 0 0
$$793$$ 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 $$-0.579758\pi$$
−0.247953 + 0.968772i $$0.579758\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −12.0000 −0.423735
$$803$$ 4.00000 0.141157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 42.0000 1.47755
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ 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
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −10.0000 −0.349856
$$818$$ −26.0000 −0.909069
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ 0 0
$$823$$ 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 $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 0 0
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 28.0000 0.970725
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 2.00000 0.0691714
$$837$$ 0 0
$$838$$ 26.0000 0.898155
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ −34.0000 −1.17172
$$843$$ 0 0
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −14.0000 −0.481046
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ 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 $$-0.445366\pi$$
0.170797 + 0.985306i $$0.445366\pi$$
$$858$$ 0 0
$$859$$ −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 $$-0.261629\pi$$
0.680808 + 0.732462i $$0.261629\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 36.0000 1.22333
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 64.0000 2.16856
$$872$$ −30.0000 −1.01593
$$873$$ 0 0
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −12.0000 −0.405211 −0.202606 0.979260i $$-0.564941\pi$$
−0.202606 + 0.979260i $$0.564941\pi$$
$$878$$ −40.0000 −1.34993
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 10.0000 0.336909 0.168454 0.985709i $$-0.446122\pi$$
0.168454 + 0.985709i $$0.446122\pi$$
$$882$$ 0 0
$$883$$ 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 $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 24.0000 0.803579
$$893$$ −12.0000 −0.401565
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −6.00000 −0.200446
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −4.00000 −0.133259
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −42.0000 −1.39690
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 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
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 28.0000 0.924641
$$918$$ 0 0
$$919$$ 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
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 22.0000 0.722965
$$927$$ 0 0
$$928$$ −20.0000 −0.656532
$$929$$ −10.0000 −0.328089 −0.164045 0.986453i $$-0.552454\pi$$
−0.164045 + 0.986453i $$0.552454\pi$$
$$930$$ 0 0
$$931$$ 3.00000 0.0983210
$$932$$ 18.0000 0.589610
$$933$$ 0 0
$$934$$ 16.0000 0.523536
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ 32.0000 1.04484
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 60.0000 1.95594 0.977972 0.208736i $$-0.0669349\pi$$
0.977972 + 0.208736i $$0.0669349\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 20.0000 0.650256
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ 0 0
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −12.0000 −0.388922
$$953$$ 46.0000 1.49009 0.745043 0.667016i $$-0.232429\pi$$
0.745043 + 0.667016i $$0.232429\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −6.00000 −0.194054
$$957$$ 0 0
$$958$$ −38.0000 −1.22772
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −26.0000 −0.837404
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −34.0000 −1.09337 −0.546683 0.837340i $$-0.684110\pi$$
−0.546683 + 0.837340i $$0.684110\pi$$
$$968$$ 21.0000 0.674966
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ 0 0
$$973$$ 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 $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 6.00000 0.191468
$$983$$ 16.0000 0.510321 0.255160 0.966899i $$-0.417872\pi$$
0.255160 + 0.966899i $$0.417872\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −8.00000 −0.254772
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 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 4275.2.a.o.1.1 1
3.2 odd 2 1425.2.a.d.1.1 1
5.4 even 2 855.2.a.b.1.1 1
15.2 even 4 1425.2.c.d.799.1 2
15.8 even 4 1425.2.c.d.799.2 2
15.14 odd 2 285.2.a.b.1.1 1
60.59 even 2 4560.2.a.v.1.1 1
285.284 even 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 15.14 odd 2
855.2.a.b.1.1 1 5.4 even 2
1425.2.a.d.1.1 1 3.2 odd 2
1425.2.c.d.799.1 2 15.2 even 4
1425.2.c.d.799.2 2 15.8 even 4
4275.2.a.o.1.1 1 1.1 even 1 trivial
4560.2.a.v.1.1 1 60.59 even 2
5415.2.a.c.1.1 1 285.284 even 2