# Properties

 Label 45.2.a.a.1.1 Level $45$ Weight $2$ Character 45.1 Self dual yes Analytic conductor $0.359$ 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] = [45,2,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("45.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 45.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: $$0.359326809096$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 45.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{8} -1.00000 q^{10} +4.00000 q^{11} -2.00000 q^{13} -1.00000 q^{16} -2.00000 q^{17} +4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{25} -2.00000 q^{26} +2.00000 q^{29} +5.00000 q^{32} -2.00000 q^{34} -10.0000 q^{37} +4.00000 q^{38} +3.00000 q^{40} -10.0000 q^{41} +4.00000 q^{43} -4.00000 q^{44} -8.00000 q^{47} -7.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +10.0000 q^{53} -4.00000 q^{55} +2.00000 q^{58} +4.00000 q^{59} -2.00000 q^{61} +7.00000 q^{64} +2.00000 q^{65} +12.0000 q^{67} +2.00000 q^{68} +8.00000 q^{71} +10.0000 q^{73} -10.0000 q^{74} -4.00000 q^{76} +1.00000 q^{80} -10.0000 q^{82} -12.0000 q^{83} +2.00000 q^{85} +4.00000 q^{86} -12.0000 q^{88} +6.00000 q^{89} -8.00000 q^{94} -4.00000 q^{95} +2.00000 q^{97} -7.00000 q^{98} +O(q^{100})$$

## 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
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 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
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 0 0
$$40$$ 3.00000 0.474342
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ 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 $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ −10.0000 −1.10432
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ −12.0000 −1.27920
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ 2.00000 0.175412
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 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
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ −2.00000 −0.166091
$$146$$ 10.0000 0.827606
$$147$$ 0 0
$$148$$ 10.0000 0.821995
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ −12.0000 −0.973329
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ −5.00000 −0.395285
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 2.00000 0.153393
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 10.0000 0.735215
$$186$$ 0 0
$$187$$ −8.00000 −0.585018
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 7.00000 0.500000
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 0 0
$$202$$ −6.00000 −0.422159
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 10.0000 0.698430
$$206$$ −16.0000 −1.11477
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 14.0000 0.948200
$$219$$ 0 0
$$220$$ 4.00000 0.269680
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 7.00000 0.447214
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −2.00000 −0.124035
$$261$$ 0 0
$$262$$ 12.0000 0.741362
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ −10.0000 −0.614295
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 4.00000 0.241209
$$276$$ 0 0
$$277$$ 6.00000 0.360505 0.180253 0.983620i $$-0.442309\pi$$
0.180253 + 0.983620i $$0.442309\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ −2.00000 −0.117444
$$291$$ 0 0
$$292$$ −10.0000 −0.585206
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 30.0000 1.74371
$$297$$ 0 0
$$298$$ −22.0000 −1.27443
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −8.00000 −0.460348
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 2.00000 0.114520
$$306$$ 0 0
$$307$$ 28.0000 1.59804 0.799022 0.601302i $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ −7.00000 −0.391312
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −8.00000 −0.445132
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ 30.0000 1.65647
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −12.0000 −0.655630
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 0 0
$$340$$ −2.00000 −0.108465
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 20.0000 1.06600
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ −20.0000 −1.05703
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −10.0000 −0.525588
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −10.0000 −0.523424
$$366$$ 0 0
$$367$$ −24.0000 −1.25279 −0.626395 0.779506i $$-0.715470\pi$$
−0.626395 + 0.779506i $$0.715470\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 10.0000 0.519875
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −26.0000 −1.34623 −0.673114 0.739538i $$-0.735044\pi$$
−0.673114 + 0.739538i $$0.735044\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 24.0000 1.23771
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 0 0
$$382$$ −16.0000 −0.818631
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 0 0
$$388$$ −2.00000 −0.101535
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 21.0000 1.06066
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −40.0000 −1.98273
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 10.0000 0.493865
$$411$$ 0 0
$$412$$ 16.0000 0.788263
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ −10.0000 −0.490290
$$417$$ 0 0
$$418$$ 16.0000 0.782586
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 20.0000 0.973585
$$423$$ 0 0
$$424$$ −30.0000 −1.45693
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 40.0000 1.90910 0.954548 0.298057i $$-0.0963387\pi$$
0.954548 + 0.298057i $$0.0963387\pi$$
$$440$$ 12.0000 0.572078
$$441$$ 0 0
$$442$$ 4.00000 0.190261
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ −6.00000 −0.284427
$$446$$ 8.00000 0.378811
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ 2.00000 0.0940721
$$453$$ 0 0
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 6.00000 0.280362
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 16.0000 0.731823
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ −14.0000 −0.637683
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ −2.00000 −0.0908153
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 6.00000 0.271607
$$489$$ 0 0
$$490$$ 7.00000 0.316228
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 0 0
$$493$$ −4.00000 −0.180151
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ −12.0000 −0.535586
$$503$$ 32.0000 1.42681 0.713405 0.700752i $$-0.247152\pi$$
0.713405 + 0.700752i $$0.247152\pi$$
$$504$$ 0 0
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ 34.0000 1.50702 0.753512 0.657434i $$-0.228358\pi$$
0.753512 + 0.657434i $$0.228358\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ −18.0000 −0.793946
$$515$$ 16.0000 0.705044
$$516$$ 0 0
$$517$$ −32.0000 −1.40736
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −6.00000 −0.263117
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 0 0
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ −10.0000 −0.434372
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 20.0000 0.866296
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ −36.0000 −1.55496
$$537$$ 0 0
$$538$$ −14.0000 −0.603583
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 0 0
$$544$$ −10.0000 −0.428746
$$545$$ −14.0000 −0.599694
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 0 0
$$550$$ 4.00000 0.170561
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 6.00000 0.254916
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ −12.0000 −0.504398
$$567$$ 0 0
$$568$$ −24.0000 −1.00702
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 8.00000 0.334497
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 2.00000 0.0830455
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 40.0000 1.65663
$$584$$ −30.0000 −1.24141
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ −4.00000 −0.164677
$$591$$ 0 0
$$592$$ 10.0000 0.410997
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 22.0000 0.901155
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 8.00000 0.325515
$$605$$ −5.00000 −0.203279
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 20.0000 0.811107
$$609$$ 0 0
$$610$$ 2.00000 0.0809776
$$611$$ 16.0000 0.647291
$$612$$ 0 0
$$613$$ 22.0000 0.888572 0.444286 0.895885i $$-0.353457\pi$$
0.444286 + 0.895885i $$0.353457\pi$$
$$614$$ 28.0000 1.12999
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 24.0000 0.962312
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ −14.0000 −0.558661
$$629$$ 20.0000 0.797452
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 2.00000 0.0794301
$$635$$ 8.00000 0.317470
$$636$$ 0 0
$$637$$ 14.0000 0.554700
$$638$$ 8.00000 0.316723
$$639$$ 0 0
$$640$$ 3.00000 0.118585
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ −36.0000 −1.41970 −0.709851 0.704352i $$-0.751238\pi$$
−0.709851 + 0.704352i $$0.751238\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ −32.0000 −1.25805 −0.629025 0.777385i $$-0.716546\pi$$
−0.629025 + 0.777385i $$0.716546\pi$$
$$648$$ 0 0
$$649$$ 16.0000 0.628055
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ −46.0000 −1.80012 −0.900060 0.435767i $$-0.856477\pi$$
−0.900060 + 0.435767i $$0.856477\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ 10.0000 0.390434
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 12.0000 0.466393
$$663$$ 0 0
$$664$$ 36.0000 1.39707
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ −12.0000 −0.463600
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ −30.0000 −1.15642 −0.578208 0.815890i $$-0.696248\pi$$
−0.578208 + 0.815890i $$0.696248\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −6.00000 −0.230089
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 0 0
$$685$$ −6.00000 −0.229248
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 0 0
$$694$$ 28.0000 1.06287
$$695$$ 4.00000 0.151729
$$696$$ 0 0
$$697$$ 20.0000 0.757554
$$698$$ −2.00000 −0.0757011
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 0 0
$$703$$ −40.0000 −1.50863
$$704$$ 28.0000 1.05529
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ −8.00000 −0.300235
$$711$$ 0 0
$$712$$ −18.0000 −0.674579
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 8.00000 0.299183
$$716$$ 20.0000 0.747435
$$717$$ 0 0
$$718$$ 24.0000 0.895672
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ 0 0
$$724$$ 10.0000 0.371647
$$725$$ 2.00000 0.0742781
$$726$$ 0 0
$$727$$ −16.0000 −0.593407 −0.296704 0.954970i $$-0.595887\pi$$
−0.296704 + 0.954970i $$0.595887\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −10.0000 −0.370117
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ −24.0000 −0.885856
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 48.0000 1.76810
$$738$$ 0 0
$$739$$ −44.0000 −1.61857 −0.809283 0.587419i $$-0.800144\pi$$
−0.809283 + 0.587419i $$0.800144\pi$$
$$740$$ −10.0000 −0.367607
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 0 0
$$745$$ 22.0000 0.806018
$$746$$ −26.0000 −0.951928
$$747$$ 0 0
$$748$$ 8.00000 0.292509
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ 8.00000 0.291150
$$756$$ 0 0
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ 12.0000 0.435286
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 16.0000 0.578860
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −2.00000 −0.0719816
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ −6.00000 −0.215110
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 7.00000 0.250000
$$785$$ −14.0000 −0.499681
$$786$$ 0 0
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 4.00000 0.142044
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 5.00000 0.176777
$$801$$ 0 0
$$802$$ −18.0000 −0.635602
$$803$$ 40.0000 1.41157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 18.0000 0.633238
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −40.0000 −1.40200
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ −10.0000 −0.349215
$$821$$ −54.0000 −1.88461 −0.942306 0.334751i $$-0.891348\pi$$
−0.942306 + 0.334751i $$0.891348\pi$$
$$822$$ 0 0
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ 48.0000 1.67216
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.0000 0.973655 0.486828 0.873498i $$-0.338154\pi$$
0.486828 + 0.873498i $$0.338154\pi$$
$$828$$ 0 0
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ 12.0000 0.416526
$$831$$ 0 0
$$832$$ −14.0000 −0.485363
$$833$$ 14.0000 0.485071
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ 0 0
$$838$$ −4.00000 −0.138178
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −26.0000 −0.896019
$$843$$ 0 0
$$844$$ −20.0000 −0.688428
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −10.0000 −0.343401
$$849$$ 0 0
$$850$$ −2.00000 −0.0685994
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 6.00000 0.205436 0.102718 0.994711i $$-0.467246\pi$$
0.102718 + 0.994711i $$0.467246\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 4.00000 0.136399
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 56.0000 1.90626 0.953131 0.302558i $$-0.0978405\pi$$
0.953131 + 0.302558i $$0.0978405\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ −14.0000 −0.475739
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ −42.0000 −1.42230
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 30.0000 1.01303 0.506514 0.862232i $$-0.330934\pi$$
0.506514 + 0.862232i $$0.330934\pi$$
$$878$$ 40.0000 1.34993
$$879$$ 0 0
$$880$$ 4.00000 0.134840
$$881$$ 46.0000 1.54978 0.774890 0.632096i $$-0.217805\pi$$
0.774890 + 0.632096i $$0.217805\pi$$
$$882$$ 0 0
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −6.00000 −0.201120
$$891$$ 0 0
$$892$$ −8.00000 −0.267860
$$893$$ −32.0000 −1.07084
$$894$$ 0 0
$$895$$ 20.0000 0.668526
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −2.00000 −0.0667409
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ −40.0000 −1.33185
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 10.0000 0.332411
$$906$$ 0 0
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −32.0000 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 18.0000 0.592798
$$923$$ −16.0000 −0.526646
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ 24.0000 0.788689
$$927$$ 0 0
$$928$$ 10.0000 0.328266
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ −6.00000 −0.196537
$$933$$ 0 0
$$934$$ −28.0000 −0.916188
$$935$$ 8.00000 0.261628
$$936$$ 0 0
$$937$$ −54.0000 −1.76410 −0.882052 0.471153i $$-0.843838\pi$$
−0.882052 + 0.471153i $$0.843838\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −8.00000 −0.260931
$$941$$ 50.0000 1.62995 0.814977 0.579494i $$-0.196750\pi$$
0.814977 + 0.579494i $$0.196750\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ 0 0
$$949$$ −20.0000 −0.649227
$$950$$ 4.00000 0.129777
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ 0 0
$$955$$ 16.0000 0.517748
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 20.0000 0.644826
$$963$$ 0 0
$$964$$ 14.0000 0.450910
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ −15.0000 −0.482118
$$969$$ 0 0
$$970$$ −2.00000 −0.0642161
$$971$$ −60.0000 −1.92549 −0.962746 0.270408i $$-0.912841\pi$$
−0.962746 + 0.270408i $$0.912841\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 32.0000 1.02535
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ −7.00000 −0.223607
$$981$$ 0 0
$$982$$ −28.0000 −0.893516
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ −4.00000 −0.127386
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 8.00000 0.253617
$$996$$ 0 0
$$997$$ 54.0000 1.71020 0.855099 0.518465i $$-0.173497\pi$$
0.855099 + 0.518465i $$0.173497\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 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 45.2.a.a.1.1 1
3.2 odd 2 15.2.a.a.1.1 1
4.3 odd 2 720.2.a.c.1.1 1
5.2 odd 4 225.2.b.b.199.2 2
5.3 odd 4 225.2.b.b.199.1 2
5.4 even 2 225.2.a.b.1.1 1
7.6 odd 2 2205.2.a.i.1.1 1
8.3 odd 2 2880.2.a.bc.1.1 1
8.5 even 2 2880.2.a.y.1.1 1
9.2 odd 6 405.2.e.f.271.1 2
9.4 even 3 405.2.e.c.136.1 2
9.5 odd 6 405.2.e.f.136.1 2
9.7 even 3 405.2.e.c.271.1 2
11.10 odd 2 5445.2.a.c.1.1 1
12.11 even 2 240.2.a.d.1.1 1
13.12 even 2 7605.2.a.g.1.1 1
15.2 even 4 75.2.b.b.49.1 2
15.8 even 4 75.2.b.b.49.2 2
15.14 odd 2 75.2.a.b.1.1 1
20.3 even 4 3600.2.f.e.2449.1 2
20.7 even 4 3600.2.f.e.2449.2 2
20.19 odd 2 3600.2.a.u.1.1 1
21.2 odd 6 735.2.i.e.361.1 2
21.5 even 6 735.2.i.d.361.1 2
21.11 odd 6 735.2.i.e.226.1 2
21.17 even 6 735.2.i.d.226.1 2
21.20 even 2 735.2.a.c.1.1 1
24.5 odd 2 960.2.a.l.1.1 1
24.11 even 2 960.2.a.a.1.1 1
33.32 even 2 1815.2.a.d.1.1 1
39.38 odd 2 2535.2.a.j.1.1 1
48.5 odd 4 3840.2.k.m.1921.2 2
48.11 even 4 3840.2.k.r.1921.1 2
48.29 odd 4 3840.2.k.m.1921.1 2
48.35 even 4 3840.2.k.r.1921.2 2
51.50 odd 2 4335.2.a.c.1.1 1
57.56 even 2 5415.2.a.j.1.1 1
60.23 odd 4 1200.2.f.h.49.2 2
60.47 odd 4 1200.2.f.h.49.1 2
60.59 even 2 1200.2.a.e.1.1 1
69.68 even 2 7935.2.a.d.1.1 1
105.104 even 2 3675.2.a.j.1.1 1
120.29 odd 2 4800.2.a.t.1.1 1
120.53 even 4 4800.2.f.bf.3649.2 2
120.59 even 2 4800.2.a.bz.1.1 1
120.77 even 4 4800.2.f.bf.3649.1 2
120.83 odd 4 4800.2.f.c.3649.1 2
120.107 odd 4 4800.2.f.c.3649.2 2
165.164 even 2 9075.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.2.a.a.1.1 1 3.2 odd 2
45.2.a.a.1.1 1 1.1 even 1 trivial
75.2.a.b.1.1 1 15.14 odd 2
75.2.b.b.49.1 2 15.2 even 4
75.2.b.b.49.2 2 15.8 even 4
225.2.a.b.1.1 1 5.4 even 2
225.2.b.b.199.1 2 5.3 odd 4
225.2.b.b.199.2 2 5.2 odd 4
240.2.a.d.1.1 1 12.11 even 2
405.2.e.c.136.1 2 9.4 even 3
405.2.e.c.271.1 2 9.7 even 3
405.2.e.f.136.1 2 9.5 odd 6
405.2.e.f.271.1 2 9.2 odd 6
720.2.a.c.1.1 1 4.3 odd 2
735.2.a.c.1.1 1 21.20 even 2
735.2.i.d.226.1 2 21.17 even 6
735.2.i.d.361.1 2 21.5 even 6
735.2.i.e.226.1 2 21.11 odd 6
735.2.i.e.361.1 2 21.2 odd 6
960.2.a.a.1.1 1 24.11 even 2
960.2.a.l.1.1 1 24.5 odd 2
1200.2.a.e.1.1 1 60.59 even 2
1200.2.f.h.49.1 2 60.47 odd 4
1200.2.f.h.49.2 2 60.23 odd 4
1815.2.a.d.1.1 1 33.32 even 2
2205.2.a.i.1.1 1 7.6 odd 2
2535.2.a.j.1.1 1 39.38 odd 2
2880.2.a.y.1.1 1 8.5 even 2
2880.2.a.bc.1.1 1 8.3 odd 2
3600.2.a.u.1.1 1 20.19 odd 2
3600.2.f.e.2449.1 2 20.3 even 4
3600.2.f.e.2449.2 2 20.7 even 4
3675.2.a.j.1.1 1 105.104 even 2
3840.2.k.m.1921.1 2 48.29 odd 4
3840.2.k.m.1921.2 2 48.5 odd 4
3840.2.k.r.1921.1 2 48.11 even 4
3840.2.k.r.1921.2 2 48.35 even 4
4335.2.a.c.1.1 1 51.50 odd 2
4800.2.a.t.1.1 1 120.29 odd 2
4800.2.a.bz.1.1 1 120.59 even 2
4800.2.f.c.3649.1 2 120.83 odd 4
4800.2.f.c.3649.2 2 120.107 odd 4
4800.2.f.bf.3649.1 2 120.77 even 4
4800.2.f.bf.3649.2 2 120.53 even 4
5415.2.a.j.1.1 1 57.56 even 2
5445.2.a.c.1.1 1 11.10 odd 2
7605.2.a.g.1.1 1 13.12 even 2
7935.2.a.d.1.1 1 69.68 even 2
9075.2.a.g.1.1 1 165.164 even 2