# Properties

 Label 306.2.a.b.1.1 Level $306$ Weight $2$ Character 306.1 Self dual yes Analytic conductor $2.443$ 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] = [306,2,Mod(1,306)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(306, 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("306.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$306 = 2 \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 306.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: $$2.44342230185$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 306.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} -2.00000 q^{10} +4.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} -1.00000 q^{17} +4.00000 q^{19} +2.00000 q^{20} -4.00000 q^{22} -1.00000 q^{25} +2.00000 q^{26} +10.0000 q^{29} +8.00000 q^{31} -1.00000 q^{32} +1.00000 q^{34} -2.00000 q^{37} -4.00000 q^{38} -2.00000 q^{40} -10.0000 q^{41} +12.0000 q^{43} +4.00000 q^{44} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} +8.00000 q^{55} -10.0000 q^{58} -12.0000 q^{59} -10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -12.0000 q^{67} -1.00000 q^{68} +10.0000 q^{73} +2.00000 q^{74} +4.00000 q^{76} -8.00000 q^{79} +2.00000 q^{80} +10.0000 q^{82} -4.00000 q^{83} -2.00000 q^{85} -12.0000 q^{86} -4.00000 q^{88} +6.00000 q^{89} +8.00000 q^{95} -14.0000 q^{97} +7.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −2.00000 −0.632456
$$11$$ 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
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 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
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ −2.00000 −0.316228
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 12.0000 1.82998 0.914991 0.403473i $$-0.132197\pi$$
0.914991 + 0.403473i $$0.132197\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 8.00000 1.07872
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −10.0000 −1.31306
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 0 0
$$82$$ 10.0000 1.10432
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ −12.0000 −1.29399
$$87$$ 0 0
$$88$$ −4.00000 −0.426401
$$89$$ 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
$$94$$ 0 0
$$95$$ 8.00000 0.820783
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 7.00000 0.707107
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ −8.00000 −0.762770
$$111$$ 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
$$116$$ 10.0000 0.928477
$$117$$ 0 0
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000 0.905357
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 4.00000 0.350823
$$131$$ 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
$$136$$ 1.00000 0.0857493
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 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
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ 20.0000 1.66091
$$146$$ −10.0000 −0.827606
$$147$$ 0 0
$$148$$ −2.00000 −0.164399
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 16.0000 1.28515
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 0 0
$$160$$ −2.00000 −0.158114
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 2.00000 0.153393
$$171$$ 0 0
$$172$$ 12.0000 0.914991
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −4.00000 −0.294086
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ 0 0
$$189$$ 0 0
$$190$$ −8.00000 −0.580381
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 0 0
$$193$$ 18.0000 1.29567 0.647834 0.761781i $$-0.275675\pi$$
0.647834 + 0.761781i $$0.275675\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ −14.0000 −0.997459 −0.498729 0.866758i $$-0.666200\pi$$
−0.498729 + 0.866758i $$0.666200\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ −10.0000 −0.703598
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −20.0000 −1.39686
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ −28.0000 −1.92760 −0.963800 0.266627i $$-0.914091\pi$$
−0.963800 + 0.266627i $$0.914091\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ 24.0000 1.63679
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 10.0000 0.677285
$$219$$ 0 0
$$220$$ 8.00000 0.539360
$$221$$ 2.00000 0.134535
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 0 0
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −10.0000 −0.656532
$$233$$ −26.0000 −1.70332 −0.851658 0.524097i $$-0.824403\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 0 0
$$244$$ −10.0000 −0.640184
$$245$$ −14.0000 −0.894427
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ −8.00000 −0.508001
$$249$$ 0 0
$$250$$ 12.0000 0.758947
$$251$$ −28.0000 −1.76734 −0.883672 0.468106i $$-0.844936\pi$$
−0.883672 + 0.468106i $$0.844936\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −4.00000 −0.248069
$$261$$ 0 0
$$262$$ −12.0000 −0.741362
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 0 0
$$274$$ 10.0000 0.604122
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ 30.0000 1.80253 0.901263 0.433273i $$-0.142641\pi$$
0.901263 + 0.433273i $$0.142641\pi$$
$$278$$ 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 $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ −20.0000 −1.17444
$$291$$ 0 0
$$292$$ 10.0000 0.585206
$$293$$ 26.0000 1.51894 0.759468 0.650545i $$-0.225459\pi$$
0.759468 + 0.650545i $$0.225459\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −24.0000 −1.38104
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ −20.0000 −1.14520
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −16.0000 −0.908739
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 40.0000 2.23957
$$320$$ 2.00000 0.111803
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −4.00000 −0.222566
$$324$$ 0 0
$$325$$ 2.00000 0.110940
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ 10.0000 0.552158
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ 0 0
$$334$$ 16.0000 0.875481
$$335$$ −24.0000 −1.31126
$$336$$ 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
$$341$$ 32.0000 1.73290
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ −28.0000 −1.50312 −0.751559 0.659665i $$-0.770698\pi$$
−0.751559 + 0.659665i $$0.770698\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −4.00000 −0.213201
$$353$$ 30.0000 1.59674 0.798369 0.602168i $$-0.205696\pi$$
0.798369 + 0.602168i $$0.205696\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ 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
$$362$$ −14.0000 −0.735824
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 20.0000 1.04685
$$366$$ 0 0
$$367$$ 24.0000 1.25279 0.626395 0.779506i $$-0.284530\pi$$
0.626395 + 0.779506i $$0.284530\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 4.00000 0.207950
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −20.0000 −1.03005
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 8.00000 0.410391
$$381$$ 0 0
$$382$$ −16.0000 −0.818631
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −18.0000 −0.916176
$$387$$ 0 0
$$388$$ −14.0000 −0.710742
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 7.00000 0.353553
$$393$$ 0 0
$$394$$ 14.0000 0.705310
$$395$$ −16.0000 −0.805047
$$396$$ 0 0
$$397$$ −26.0000 −1.30490 −0.652451 0.757831i $$-0.726259\pi$$
−0.652451 + 0.757831i $$0.726259\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 14.0000 0.699127 0.349563 0.936913i $$-0.386330\pi$$
0.349563 + 0.936913i $$0.386330\pi$$
$$402$$ 0 0
$$403$$ −16.0000 −0.797017
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 20.0000 0.987730
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −8.00000 −0.392705
$$416$$ 2.00000 0.0980581
$$417$$ 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
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 28.0000 1.36302
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 1.00000 0.0485071
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ −24.0000 −1.15738
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 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
$$436$$ −10.0000 −0.478913
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ −8.00000 −0.381385
$$441$$ 0 0
$$442$$ −2.00000 −0.0951303
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 0 0
$$445$$ 12.0000 0.568855
$$446$$ −16.0000 −0.757622
$$447$$ 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
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 26.0000 1.21490
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ 10.0000 0.464238
$$465$$ 0 0
$$466$$ 26.0000 1.20443
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 12.0000 0.552345
$$473$$ 48.0000 2.20704
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ −2.00000 −0.0910975
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ −28.0000 −1.27141
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 0 0
$$490$$ 14.0000 0.632456
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ −10.0000 −0.450377
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 0 0
$$502$$ 28.0000 1.24970
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 20.0000 0.889988
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 2.00000 0.0882162
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 4.00000 0.175412
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 12.0000 0.521247
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 20.0000 0.866296
$$534$$ 0 0
$$535$$ 8.00000 0.345870
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ 6.00000 0.258678
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 0 0
$$544$$ 1.00000 0.0428746
$$545$$ −20.0000 −0.856706
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ −10.0000 −0.427179
$$549$$ 0 0
$$550$$ 4.00000 0.170561
$$551$$ 40.0000 1.70406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −30.0000 −1.27458
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 34.0000 1.44063 0.720313 0.693649i $$-0.243998\pi$$
0.720313 + 0.693649i $$0.243998\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6.00000 −0.253095
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ 0 0
$$565$$ −4.00000 −0.168281
$$566$$ −12.0000 −0.504398
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ −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 $$0.486745\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 0 0
$$580$$ 20.0000 0.830455
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −24.0000 −0.993978
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ 20.0000 0.825488 0.412744 0.910847i $$-0.364570\pi$$
0.412744 + 0.910847i $$0.364570\pi$$
$$588$$ 0 0
$$589$$ 32.0000 1.31854
$$590$$ 24.0000 0.988064
$$591$$ 0 0
$$592$$ −2.00000 −0.0821995
$$593$$ 14.0000 0.574911 0.287456 0.957794i $$-0.407191\pi$$
0.287456 + 0.957794i $$0.407191\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 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
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 24.0000 0.976546
$$605$$ 10.0000 0.406558
$$606$$ 0 0
$$607$$ −24.0000 −0.974130 −0.487065 0.873366i $$-0.661933\pi$$
−0.487065 + 0.873366i $$0.661933\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 0 0
$$610$$ 20.0000 0.809776
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 38.0000 1.53481 0.767403 0.641165i $$-0.221549\pi$$
0.767403 + 0.641165i $$0.221549\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 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
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 16.0000 0.642575
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ −10.0000 −0.399680
$$627$$ 0 0
$$628$$ −2.00000 −0.0798087
$$629$$ 2.00000 0.0797452
$$630$$ 0 0
$$631$$ −24.0000 −0.955425 −0.477712 0.878516i $$-0.658534\pi$$
−0.477712 + 0.878516i $$0.658534\pi$$
$$632$$ 8.00000 0.318223
$$633$$ 0 0
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 14.0000 0.554700
$$638$$ −40.0000 −1.58362
$$639$$ 0 0
$$640$$ −2.00000 −0.0790569
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 4.00000 0.157378
$$647$$ 40.0000 1.57256 0.786281 0.617869i $$-0.212004\pi$$
0.786281 + 0.617869i $$0.212004\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 0 0
$$655$$ 24.0000 0.937758
$$656$$ −10.0000 −0.390434
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 6.00000 0.233373 0.116686 0.993169i $$-0.462773\pi$$
0.116686 + 0.993169i $$0.462773\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 0 0
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −16.0000 −0.619059
$$669$$ 0 0
$$670$$ 24.0000 0.927201
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ −46.0000 −1.77317 −0.886585 0.462566i $$-0.846929\pi$$
−0.886585 + 0.462566i $$0.846929\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −46.0000 −1.76792 −0.883962 0.467559i $$-0.845134\pi$$
−0.883962 + 0.467559i $$0.845134\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 2.00000 0.0766965
$$681$$ 0 0
$$682$$ −32.0000 −1.22534
$$683$$ 20.0000 0.765279 0.382639 0.923898i $$-0.375015\pi$$
0.382639 + 0.923898i $$0.375015\pi$$
$$684$$ 0 0
$$685$$ −20.0000 −0.764161
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 12.0000 0.457496
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ 28.0000 1.06287
$$695$$ −8.00000 −0.303457
$$696$$ 0 0
$$697$$ 10.0000 0.378777
$$698$$ −14.0000 −0.529908
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −46.0000 −1.73740 −0.868698 0.495342i $$-0.835043\pi$$
−0.868698 + 0.495342i $$0.835043\pi$$
$$702$$ 0 0
$$703$$ −8.00000 −0.301726
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 46.0000 1.72757 0.863783 0.503864i $$-0.168089\pi$$
0.863783 + 0.503864i $$0.168089\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −6.00000 −0.224860
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −16.0000 −0.598366
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ −24.0000 −0.895672
$$719$$ −40.0000 −1.49175 −0.745874 0.666087i $$-0.767968\pi$$
−0.745874 + 0.666087i $$0.767968\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ 0 0
$$724$$ 14.0000 0.520306
$$725$$ −10.0000 −0.371391
$$726$$ 0 0
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −20.0000 −0.740233
$$731$$ −12.0000 −0.443836
$$732$$ 0 0
$$733$$ 46.0000 1.69905 0.849524 0.527549i $$-0.176889\pi$$
0.849524 + 0.527549i $$0.176889\pi$$
$$734$$ −24.0000 −0.885856
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −48.0000 −1.76810
$$738$$ 0 0
$$739$$ 52.0000 1.91285 0.956425 0.291977i $$-0.0943129\pi$$
0.956425 + 0.291977i $$0.0943129\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 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
$$745$$ 20.0000 0.732743
$$746$$ −6.00000 −0.219676
$$747$$ 0 0
$$748$$ −4.00000 −0.146254
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 20.0000 0.728357
$$755$$ 48.0000 1.74690
$$756$$ 0 0
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ 4.00000 0.145287
$$759$$ 0 0
$$760$$ −8.00000 −0.290191
$$761$$ 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
$$766$$ −16.0000 −0.578103
$$767$$ 24.0000 0.866590
$$768$$ 0 0
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 18.0000 0.647834
$$773$$ −38.0000 −1.36677 −0.683383 0.730061i $$-0.739492\pi$$
−0.683383 + 0.730061i $$0.739492\pi$$
$$774$$ 0 0
$$775$$ −8.00000 −0.287368
$$776$$ 14.0000 0.502571
$$777$$ 0 0
$$778$$ −26.0000 −0.932145
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −7.00000 −0.250000
$$785$$ −4.00000 −0.142766
$$786$$ 0 0
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ −14.0000 −0.498729
$$789$$ 0 0
$$790$$ 16.0000 0.569254
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 20.0000 0.710221
$$794$$ 26.0000 0.922705
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −14.0000 −0.495905 −0.247953 0.968772i $$-0.579758\pi$$
−0.247953 + 0.968772i $$0.579758\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ −14.0000 −0.494357
$$803$$ 40.0000 1.41157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 16.0000 0.563576
$$807$$ 0 0
$$808$$ −10.0000 −0.351799
$$809$$ 22.0000 0.773479 0.386739 0.922189i $$-0.373601\pi$$
0.386739 + 0.922189i $$0.373601\pi$$
$$810$$ 0 0
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 8.00000 0.280400
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ 48.0000 1.67931
$$818$$ −26.0000 −0.909069
$$819$$ 0 0
$$820$$ −20.0000 −0.698430
$$821$$ 34.0000 1.18661 0.593304 0.804978i $$-0.297823\pi$$
0.593304 + 0.804978i $$0.297823\pi$$
$$822$$ 0 0
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 20.0000 0.695468 0.347734 0.937593i $$-0.386951\pi$$
0.347734 + 0.937593i $$0.386951\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 8.00000 0.277684
$$831$$ 0 0
$$832$$ −2.00000 −0.0693375
$$833$$ 7.00000 0.242536
$$834$$ 0 0
$$835$$ −32.0000 −1.10741
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ 4.00000 0.138178
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ −22.0000 −0.758170
$$843$$ 0 0
$$844$$ −28.0000 −0.963800
$$845$$ −18.0000 −0.619219
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ 0 0
$$850$$ −1.00000 −0.0342997
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −2.00000 −0.0684787 −0.0342393 0.999414i $$-0.510901\pi$$
−0.0342393 + 0.999414i $$0.510901\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 0 0
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ 24.0000 0.818393
$$861$$ 0 0
$$862$$ −8.00000 −0.272481
$$863$$ 32.0000 1.08929 0.544646 0.838666i $$-0.316664\pi$$
0.544646 + 0.838666i $$0.316664\pi$$
$$864$$ 0 0
$$865$$ −12.0000 −0.408012
$$866$$ 14.0000 0.475739
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −32.0000 −1.08553
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 10.0000 0.338643
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ −16.0000 −0.539974
$$879$$ 0 0
$$880$$ 8.00000 0.269680
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ 0 0
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 48.0000 1.61168 0.805841 0.592132i $$-0.201714\pi$$
0.805841 + 0.592132i $$0.201714\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −12.0000 −0.402241
$$891$$ 0 0
$$892$$ 16.0000 0.535720
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 2.00000 0.0667409
$$899$$ 80.0000 2.66815
$$900$$ 0 0
$$901$$ 6.00000 0.199889
$$902$$ 40.0000 1.33185
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ 28.0000 0.930751
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ −4.00000 −0.132745
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −8.00000 −0.265052 −0.132526 0.991180i $$-0.542309\pi$$
−0.132526 + 0.991180i $$0.542309\pi$$
$$912$$ 0 0
$$913$$ −16.0000 −0.529523
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ −26.0000 −0.859064
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 30.0000 0.987997
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ −32.0000 −1.05159
$$927$$ 0 0
$$928$$ −10.0000 −0.328266
$$929$$ −2.00000 −0.0656179 −0.0328089 0.999462i $$-0.510445\pi$$
−0.0328089 + 0.999462i $$0.510445\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ −26.0000 −0.851658
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ −8.00000 −0.261628
$$936$$ 0 0
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ −48.0000 −1.56061
$$947$$ 28.0000 0.909878 0.454939 0.890523i $$-0.349661\pi$$
0.454939 + 0.890523i $$0.349661\pi$$
$$948$$ 0 0
$$949$$ −20.0000 −0.649227
$$950$$ 4.00000 0.129777
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 0 0
$$955$$ 32.0000 1.03550
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −4.00000 −0.128965
$$963$$ 0 0
$$964$$ 2.00000 0.0644157
$$965$$ 36.0000 1.15888
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 0 0
$$970$$ 28.0000 0.899026
$$971$$ 4.00000 0.128366 0.0641831 0.997938i $$-0.479556\pi$$
0.0641831 + 0.997938i $$0.479556\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 46.0000 1.47167 0.735835 0.677161i $$-0.236790\pi$$
0.735835 + 0.677161i $$0.236790\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ −14.0000 −0.447214
$$981$$ 0 0
$$982$$ 12.0000 0.382935
$$983$$ −48.0000 −1.53096 −0.765481 0.643458i $$-0.777499\pi$$
−0.765481 + 0.643458i $$0.777499\pi$$
$$984$$ 0 0
$$985$$ −28.0000 −0.892154
$$986$$ 10.0000 0.318465
$$987$$ 0 0
$$988$$ −8.00000 −0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ −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 306.2.a.b.1.1 1
3.2 odd 2 102.2.a.c.1.1 1
4.3 odd 2 2448.2.a.p.1.1 1
5.4 even 2 7650.2.a.ca.1.1 1
8.3 odd 2 9792.2.a.l.1.1 1
8.5 even 2 9792.2.a.k.1.1 1
12.11 even 2 816.2.a.b.1.1 1
15.2 even 4 2550.2.d.m.2449.2 2
15.8 even 4 2550.2.d.m.2449.1 2
15.14 odd 2 2550.2.a.c.1.1 1
17.16 even 2 5202.2.a.c.1.1 1
21.20 even 2 4998.2.a.be.1.1 1
24.5 odd 2 3264.2.a.m.1.1 1
24.11 even 2 3264.2.a.bc.1.1 1
51.2 odd 8 1734.2.f.e.1483.2 4
51.8 odd 8 1734.2.f.e.829.1 4
51.26 odd 8 1734.2.f.e.829.2 4
51.32 odd 8 1734.2.f.e.1483.1 4
51.38 odd 4 1734.2.b.b.577.1 2
51.47 odd 4 1734.2.b.b.577.2 2
51.50 odd 2 1734.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.c.1.1 1 3.2 odd 2
306.2.a.b.1.1 1 1.1 even 1 trivial
816.2.a.b.1.1 1 12.11 even 2
1734.2.a.j.1.1 1 51.50 odd 2
1734.2.b.b.577.1 2 51.38 odd 4
1734.2.b.b.577.2 2 51.47 odd 4
1734.2.f.e.829.1 4 51.8 odd 8
1734.2.f.e.829.2 4 51.26 odd 8
1734.2.f.e.1483.1 4 51.32 odd 8
1734.2.f.e.1483.2 4 51.2 odd 8
2448.2.a.p.1.1 1 4.3 odd 2
2550.2.a.c.1.1 1 15.14 odd 2
2550.2.d.m.2449.1 2 15.8 even 4
2550.2.d.m.2449.2 2 15.2 even 4
3264.2.a.m.1.1 1 24.5 odd 2
3264.2.a.bc.1.1 1 24.11 even 2
4998.2.a.be.1.1 1 21.20 even 2
5202.2.a.c.1.1 1 17.16 even 2
7650.2.a.ca.1.1 1 5.4 even 2
9792.2.a.k.1.1 1 8.5 even 2
9792.2.a.l.1.1 1 8.3 odd 2