# Properties

 Label 855.2.a.c.1.1 Level $855$ Weight $2$ Character 855.1 Self dual yes Analytic conductor $6.827$ 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] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, 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("855.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$855 = 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 855.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: $$6.82720937282$$ 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 $$\chi$$ $$=$$ 855.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -3.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{10} +6.00000 q^{11} -2.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{19} -1.00000 q^{20} +6.00000 q^{22} +8.00000 q^{23} +1.00000 q^{25} +2.00000 q^{28} -4.00000 q^{29} +5.00000 q^{32} +6.00000 q^{34} -2.00000 q^{35} +4.00000 q^{37} +1.00000 q^{38} -3.00000 q^{40} -2.00000 q^{43} -6.00000 q^{44} +8.00000 q^{46} +8.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{53} +6.00000 q^{55} +6.00000 q^{56} -4.00000 q^{58} -12.0000 q^{59} +2.00000 q^{61} +7.00000 q^{64} -8.00000 q^{67} -6.00000 q^{68} -2.00000 q^{70} -16.0000 q^{71} +14.0000 q^{73} +4.00000 q^{74} -1.00000 q^{76} -12.0000 q^{77} +8.00000 q^{79} -1.00000 q^{80} +6.00000 q^{85} -2.00000 q^{86} -18.0000 q^{88} -8.00000 q^{92} +8.00000 q^{94} +1.00000 q^{95} -12.0000 q^{97} -3.00000 q^{98} +O(q^{100})$$

## 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
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 6.00000 1.27920
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 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
$$34$$ 6.00000 1.02899
$$35$$ −2.00000 −0.338062
$$36$$ 0 0
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ −3.00000 −0.474342
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ 6.00000 0.809040
$$56$$ 6.00000 0.801784
$$57$$ 0 0
$$58$$ −4.00000 −0.525226
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 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
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 0 0
$$70$$ −2.00000 −0.239046
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ 0 0
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −12.0000 −1.36753
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 6.00000 0.650791
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ −18.0000 −1.91881
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −8.00000 −0.834058
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ −12.0000 −1.21842 −0.609208 0.793011i $$-0.708512\pi$$
−0.609208 + 0.793011i $$0.708512\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 6.00000 0.572078
$$111$$ 0 0
$$112$$ 2.00000 0.188982
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 8.00000 0.746004
$$116$$ 4.00000 0.371391
$$117$$ 0 0
$$118$$ −12.0000 −1.10469
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 2.00000 0.181071
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2.00000 0.174741 0.0873704 0.996176i $$-0.472154\pi$$
0.0873704 + 0.996176i $$0.472154\pi$$
$$132$$ 0 0
$$133$$ −2.00000 −0.173422
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ −18.0000 −1.54349
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 0 0
$$142$$ −16.0000 −1.34269
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −4.00000 −0.332182
$$146$$ 14.0000 1.15865
$$147$$ 0 0
$$148$$ −4.00000 −0.328798
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ −3.00000 −0.243332
$$153$$ 0 0
$$154$$ −12.0000 −0.966988
$$155$$ 0 0
$$156$$ 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
$$160$$ 5.00000 0.395285
$$161$$ −16.0000 −1.26098
$$162$$ 0 0
$$163$$ −22.0000 −1.72317 −0.861586 0.507611i $$-0.830529\pi$$
−0.861586 + 0.507611i $$0.830529\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 6.00000 0.460179
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ −2.00000 −0.151186
$$176$$ −6.00000 −0.452267
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −24.0000 −1.76930
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ 36.0000 2.63258
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ 1.00000 0.0725476
$$191$$ −10.0000 −0.723575 −0.361787 0.932261i $$-0.617833\pi$$
−0.361787 + 0.932261i $$0.617833\pi$$
$$192$$ 0 0
$$193$$ 24.0000 1.72756 0.863779 0.503871i $$-0.168091\pi$$
0.863779 + 0.503871i $$0.168091\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 0 0
$$202$$ 18.0000 1.26648
$$203$$ 8.00000 0.561490
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ −2.00000 −0.136399
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −6.00000 −0.406371
$$219$$ 0 0
$$220$$ −6.00000 −0.404520
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ −10.0000 −0.668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 28.0000 1.85843 0.929213 0.369546i $$-0.120487\pi$$
0.929213 + 0.369546i $$0.120487\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 8.00000 0.527504
$$231$$ 0 0
$$232$$ 12.0000 0.787839
$$233$$ −22.0000 −1.44127 −0.720634 0.693316i $$-0.756149\pi$$
−0.720634 + 0.693316i $$0.756149\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ −12.0000 −0.777844
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 25.0000 1.60706
$$243$$ 0 0
$$244$$ −2.00000 −0.128037
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ 0 0
$$253$$ 48.0000 3.01773
$$254$$ 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
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 2.00000 0.123560
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ −2.00000 −0.122859
$$266$$ −2.00000 −0.122628
$$267$$ 0 0
$$268$$ 8.00000 0.488678
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ 10.0000 0.604122
$$275$$ 6.00000 0.361814
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ −16.0000 −0.959616
$$279$$ 0 0
$$280$$ 6.00000 0.358569
$$281$$ 4.00000 0.238620 0.119310 0.992857i $$-0.461932\pi$$
0.119310 + 0.992857i $$0.461932\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ −4.00000 −0.234888
$$291$$ 0 0
$$292$$ −14.0000 −0.819288
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 0 0
$$295$$ −12.0000 −0.698667
$$296$$ −12.0000 −0.697486
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ −16.0000 −0.920697
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 2.00000 0.114520
$$306$$ 0 0
$$307$$ −8.00000 −0.456584 −0.228292 0.973593i $$-0.573314\pi$$
−0.228292 + 0.973593i $$0.573314\pi$$
$$308$$ 12.0000 0.683763
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −34.0000 −1.92796 −0.963982 0.265969i $$-0.914308\pi$$
−0.963982 + 0.265969i $$0.914308\pi$$
$$312$$ 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 $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ −24.0000 −1.34374
$$320$$ 7.00000 0.391312
$$321$$ 0 0
$$322$$ −16.0000 −0.891645
$$323$$ 6.00000 0.333849
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −22.0000 −1.21847
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ −16.0000 −0.875481
$$335$$ −8.00000 −0.437087
$$336$$ 0 0
$$337$$ −12.0000 −0.653682 −0.326841 0.945079i $$-0.605984\pi$$
−0.326841 + 0.945079i $$0.605984\pi$$
$$338$$ −13.0000 −0.707107
$$339$$ 0 0
$$340$$ −6.00000 −0.325396
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 6.00000 0.323498
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ −8.00000 −0.429463 −0.214731 0.976673i $$-0.568888\pi$$
−0.214731 + 0.976673i $$0.568888\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 0 0
$$352$$ 30.0000 1.59901
$$353$$ 34.0000 1.80964 0.904819 0.425797i $$-0.140006\pi$$
0.904819 + 0.425797i $$0.140006\pi$$
$$354$$ 0 0
$$355$$ −16.0000 −0.849192
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 20.0000 1.05703
$$359$$ −10.0000 −0.527780 −0.263890 0.964553i $$-0.585006\pi$$
−0.263890 + 0.964553i $$0.585006\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 10.0000 0.525588
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 14.0000 0.732793
$$366$$ 0 0
$$367$$ −14.0000 −0.730794 −0.365397 0.930852i $$-0.619067\pi$$
−0.365397 + 0.930852i $$0.619067\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ 0 0
$$370$$ 4.00000 0.207950
$$371$$ 4.00000 0.207670
$$372$$ 0 0
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 36.0000 1.86152
$$375$$ 0 0
$$376$$ −24.0000 −1.23771
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ −1.00000 −0.0512989
$$381$$ 0 0
$$382$$ −10.0000 −0.511645
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 0 0
$$385$$ −12.0000 −0.611577
$$386$$ 24.0000 1.22157
$$387$$ 0 0
$$388$$ 12.0000 0.609208
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ 48.0000 2.42746
$$392$$ 9.00000 0.454569
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ −20.0000 −1.00251
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −4.00000 −0.199750 −0.0998752 0.995000i $$-0.531844\pi$$
−0.0998752 + 0.995000i $$0.531844\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ 8.00000 0.397033
$$407$$ 24.0000 1.18964
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 8.00000 0.394132
$$413$$ 24.0000 1.18096
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 6.00000 0.293470
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ −4.00000 −0.193574
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ −2.00000 −0.0964486
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 0 0
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 6.00000 0.287348
$$437$$ 8.00000 0.382692
$$438$$ 0 0
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ −18.0000 −0.858116
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 0 0
$$448$$ −14.0000 −0.661438
$$449$$ 8.00000 0.377543 0.188772 0.982021i $$-0.439549\pi$$
0.188772 + 0.982021i $$0.439549\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ 0 0
$$454$$ 28.0000 1.31411
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ 2.00000 0.0934539
$$459$$ 0 0
$$460$$ −8.00000 −0.373002
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ −14.0000 −0.650635 −0.325318 0.945605i $$-0.605471\pi$$
−0.325318 + 0.945605i $$0.605471\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ 36.0000 1.65703
$$473$$ −12.0000 −0.551761
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 12.0000 0.550019
$$477$$ 0 0
$$478$$ −6.00000 −0.274434
$$479$$ −18.0000 −0.822441 −0.411220 0.911536i $$-0.634897\pi$$
−0.411220 + 0.911536i $$0.634897\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 18.0000 0.819878
$$483$$ 0 0
$$484$$ −25.0000 −1.13636
$$485$$ −12.0000 −0.544892
$$486$$ 0 0
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ 0 0
$$490$$ −3.00000 −0.135526
$$491$$ 26.0000 1.17336 0.586682 0.809818i $$-0.300434\pi$$
0.586682 + 0.809818i $$0.300434\pi$$
$$492$$ 0 0
$$493$$ −24.0000 −1.08091
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 32.0000 1.43540
$$498$$ 0 0
$$499$$ −24.0000 −1.07439 −0.537194 0.843459i $$-0.680516\pi$$
−0.537194 + 0.843459i $$0.680516\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 6.00000 0.267793
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 48.0000 2.13386
$$507$$ 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
$$511$$ −28.0000 −1.23865
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ 22.0000 0.970378
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ 48.0000 2.11104
$$518$$ −8.00000 −0.351500
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 4.00000 0.175243 0.0876216 0.996154i $$-0.472073\pi$$
0.0876216 + 0.996154i $$0.472073\pi$$
$$522$$ 0 0
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −2.00000 −0.0873704
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ −2.00000 −0.0868744
$$531$$ 0 0
$$532$$ 2.00000 0.0867110
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 24.0000 1.03664
$$537$$ 0 0
$$538$$ −12.0000 −0.517357
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ −42.0000 −1.80572 −0.902861 0.429934i $$-0.858537\pi$$
−0.902861 + 0.429934i $$0.858537\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ 0 0
$$544$$ 30.0000 1.28624
$$545$$ −6.00000 −0.257012
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ −10.0000 −0.427179
$$549$$ 0 0
$$550$$ 6.00000 0.255841
$$551$$ −4.00000 −0.170406
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ 16.0000 0.678551
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 2.00000 0.0845154
$$561$$ 0 0
$$562$$ 4.00000 0.168730
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ 0 0
$$565$$ 6.00000 0.252422
$$566$$ −14.0000 −0.588464
$$567$$ 0 0
$$568$$ 48.0000 2.01404
$$569$$ 8.00000 0.335377 0.167689 0.985840i $$-0.446370\pi$$
0.167689 + 0.985840i $$0.446370\pi$$
$$570$$ 0 0
$$571$$ −24.0000 −1.00437 −0.502184 0.864761i $$-0.667470\pi$$
−0.502184 + 0.864761i $$0.667470\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 8.00000 0.333623
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 0 0
$$580$$ 4.00000 0.166091
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −12.0000 −0.496989
$$584$$ −42.0000 −1.73797
$$585$$ 0 0
$$586$$ −30.0000 −1.23929
$$587$$ 20.0000 0.825488 0.412744 0.910847i $$-0.364570\pi$$
0.412744 + 0.910847i $$0.364570\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ −12.0000 −0.494032
$$591$$ 0 0
$$592$$ −4.00000 −0.164399
$$593$$ 10.0000 0.410651 0.205325 0.978694i $$-0.434175\pi$$
0.205325 + 0.978694i $$0.434175\pi$$
$$594$$ 0 0
$$595$$ −12.0000 −0.491952
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 20.0000 0.817178 0.408589 0.912719i $$-0.366021\pi$$
0.408589 + 0.912719i $$0.366021\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 4.00000 0.163028
$$603$$ 0 0
$$604$$ 16.0000 0.651031
$$605$$ 25.0000 1.01639
$$606$$ 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
$$610$$ 2.00000 0.0809776
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 0 0
$$616$$ 36.0000 1.45048
$$617$$ −34.0000 −1.36879 −0.684394 0.729112i $$-0.739933\pi$$
−0.684394 + 0.729112i $$0.739933\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −34.0000 −1.36328
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ 2.00000 0.0798087
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ −24.0000 −0.954669
$$633$$ 0 0
$$634$$ 6.00000 0.238290
$$635$$ 4.00000 0.158735
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −24.0000 −0.950169
$$639$$ 0 0
$$640$$ −3.00000 −0.118585
$$641$$ −40.0000 −1.57991 −0.789953 0.613168i $$-0.789895\pi$$
−0.789953 + 0.613168i $$0.789895\pi$$
$$642$$ 0 0
$$643$$ 2.00000 0.0788723 0.0394362 0.999222i $$-0.487444\pi$$
0.0394362 + 0.999222i $$0.487444\pi$$
$$644$$ 16.0000 0.630488
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ −20.0000 −0.786281 −0.393141 0.919478i $$-0.628611\pi$$
−0.393141 + 0.919478i $$0.628611\pi$$
$$648$$ 0 0
$$649$$ −72.0000 −2.82625
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 22.0000 0.861586
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ 2.00000 0.0781465
$$656$$ 0 0
$$657$$ 0 0
$$658$$ −16.0000 −0.623745
$$659$$ −48.0000 −1.86981 −0.934907 0.354892i $$-0.884518\pi$$
−0.934907 + 0.354892i $$0.884518\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 28.0000 1.08825
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −2.00000 −0.0775567
$$666$$ 0 0
$$667$$ −32.0000 −1.23904
$$668$$ 16.0000 0.619059
$$669$$ 0 0
$$670$$ −8.00000 −0.309067
$$671$$ 12.0000 0.463255
$$672$$ 0 0
$$673$$ 36.0000 1.38770 0.693849 0.720121i $$-0.255914\pi$$
0.693849 + 0.720121i $$0.255914\pi$$
$$674$$ −12.0000 −0.462223
$$675$$ 0 0
$$676$$ 13.0000 0.500000
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 0 0
$$679$$ 24.0000 0.921035
$$680$$ −18.0000 −0.690268
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 0 0
$$685$$ 10.0000 0.382080
$$686$$ 20.0000 0.763604
$$687$$ 0 0
$$688$$ 2.00000 0.0762493
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ −8.00000 −0.303676
$$695$$ −16.0000 −0.606915
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −2.00000 −0.0757011
$$699$$ 0 0
$$700$$ 2.00000 0.0755929
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 42.0000 1.58293
$$705$$ 0 0
$$706$$ 34.0000 1.27961
$$707$$ −36.0000 −1.35392
$$708$$ 0 0
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ −16.0000 −0.600469
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ 0 0
$$718$$ −10.0000 −0.373197
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ −10.0000 −0.371647
$$725$$ −4.00000 −0.148556
$$726$$ 0 0
$$727$$ 26.0000 0.964287 0.482143 0.876092i $$-0.339858\pi$$
0.482143 + 0.876092i $$0.339858\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 14.0000 0.518163
$$731$$ −12.0000 −0.443836
$$732$$ 0 0
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ −14.0000 −0.516749
$$735$$ 0 0
$$736$$ 40.0000 1.47442
$$737$$ −48.0000 −1.76810
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 0 0
$$742$$ 4.00000 0.146845
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 0 0
$$747$$ 0 0
$$748$$ −36.0000 −1.31629
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ −38.0000 −1.38113 −0.690567 0.723269i $$-0.742639\pi$$
−0.690567 + 0.723269i $$0.742639\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ −3.00000 −0.108821
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ 12.0000 0.434429
$$764$$ 10.0000 0.361787
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ −12.0000 −0.432450
$$771$$ 0 0
$$772$$ −24.0000 −0.863779
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 36.0000 1.29232
$$777$$ 0 0
$$778$$ 14.0000 0.501924
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −96.0000 −3.43515
$$782$$ 48.0000 1.71648
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ −2.00000 −0.0713831
$$786$$ 0 0
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ 8.00000 0.284627
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 6.00000 0.212932
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ 26.0000 0.920967 0.460484 0.887668i $$-0.347676\pi$$
0.460484 + 0.887668i $$0.347676\pi$$
$$798$$ 0 0
$$799$$ 48.0000 1.69812
$$800$$ 5.00000 0.176777
$$801$$ 0 0
$$802$$ −4.00000 −0.141245
$$803$$ 84.0000 2.96430
$$804$$ 0 0
$$805$$ −16.0000 −0.563926
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −54.0000 −1.89971
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −8.00000 −0.280745
$$813$$ 0 0
$$814$$ 24.0000 0.841200
$$815$$ −22.0000 −0.770626
$$816$$ 0 0
$$817$$ −2.00000 −0.0699711
$$818$$ −10.0000 −0.349642
$$819$$ 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 $$-0.578461\pi$$
−0.244005 + 0.969774i $$0.578461\pi$$
$$824$$ 24.0000 0.836080
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 0 0
$$829$$ −6.00000 −0.208389 −0.104194 0.994557i $$-0.533226\pi$$
−0.104194 + 0.994557i $$0.533226\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ −16.0000 −0.553703
$$836$$ −6.00000 −0.207514
$$837$$ 0 0
$$838$$ 30.0000 1.03633
$$839$$ −8.00000 −0.276191 −0.138095 0.990419i $$-0.544098\pi$$
−0.138095 + 0.990419i $$0.544098\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 6.00000 0.206774
$$843$$ 0 0
$$844$$ 20.0000 0.688428
$$845$$ −13.0000 −0.447214
$$846$$ 0 0
$$847$$ −50.0000 −1.71802
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ 6.00000 0.205798
$$851$$ 32.0000 1.09695
$$852$$ 0 0
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ −46.0000 −1.57133 −0.785665 0.618652i $$-0.787679\pi$$
−0.785665 + 0.618652i $$0.787679\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 2.00000 0.0681994
$$861$$ 0 0
$$862$$ −36.0000 −1.22616
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ −16.0000 −0.543702
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 48.0000 1.62829
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 18.0000 0.609557
$$873$$ 0 0
$$874$$ 8.00000 0.270604
$$875$$ −2.00000 −0.0676123
$$876$$ 0 0
$$877$$ −24.0000 −0.810422 −0.405211 0.914223i $$-0.632802\pi$$
−0.405211 + 0.914223i $$0.632802\pi$$
$$878$$ 24.0000 0.809961
$$879$$ 0 0
$$880$$ −6.00000 −0.202260
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 0 0
$$883$$ 2.00000 0.0673054 0.0336527 0.999434i $$-0.489286\pi$$
0.0336527 + 0.999434i $$0.489286\pi$$
$$884$$ 0 0
$$885$$ 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
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −8.00000 −0.267860
$$893$$ 8.00000 0.267710
$$894$$ 0 0
$$895$$ 20.0000 0.668526
$$896$$ 6.00000 0.200446
$$897$$ 0 0
$$898$$ 8.00000 0.266963
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −18.0000 −0.598671
$$905$$ 10.0000 0.332411
$$906$$ 0 0
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ −28.0000 −0.929213
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −6.00000 −0.198462
$$915$$ 0 0
$$916$$ −2.00000 −0.0660819
$$917$$ −4.00000 −0.132092
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ −24.0000 −0.791257
$$921$$ 0 0
$$922$$ −30.0000 −0.987997
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 4.00000 0.131519
$$926$$ −14.0000 −0.460069
$$927$$ 0 0
$$928$$ −20.0000 −0.656532
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ 22.0000 0.720634
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 36.0000 1.17733
$$936$$ 0 0
$$937$$ 30.0000 0.980057 0.490029 0.871706i $$-0.336986\pi$$
0.490029 + 0.871706i $$0.336986\pi$$
$$938$$ 16.0000 0.522419
$$939$$ 0 0
$$940$$ −8.00000 −0.260931
$$941$$ −44.0000 −1.43436 −0.717180 0.696888i $$-0.754567\pi$$
−0.717180 + 0.696888i $$0.754567\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ −48.0000 −1.55979 −0.779895 0.625910i $$-0.784728\pi$$
−0.779895 + 0.625910i $$0.784728\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 1.00000 0.0324443
$$951$$ 0 0
$$952$$ 36.0000 1.16677
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ 0 0
$$955$$ −10.0000 −0.323592
$$956$$ 6.00000 0.194054
$$957$$ 0 0
$$958$$ −18.0000 −0.581554
$$959$$ −20.0000 −0.645834
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −18.0000 −0.579741
$$965$$ 24.0000 0.772587
$$966$$ 0 0
$$967$$ −14.0000 −0.450210 −0.225105 0.974335i $$-0.572272\pi$$
−0.225105 + 0.974335i $$0.572272\pi$$
$$968$$ −75.0000 −2.41059
$$969$$ 0 0
$$970$$ −12.0000 −0.385297
$$971$$ 8.00000 0.256732 0.128366 0.991727i $$-0.459027\pi$$
0.128366 + 0.991727i $$0.459027\pi$$
$$972$$ 0 0
$$973$$ 32.0000 1.02587
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 3.00000 0.0958315
$$981$$ 0 0
$$982$$ 26.0000 0.829693
$$983$$ −8.00000 −0.255160 −0.127580 0.991828i $$-0.540721\pi$$
−0.127580 + 0.991828i $$0.540721\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ −24.0000 −0.764316
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 32.0000 1.01498
$$995$$ −20.0000 −0.634043
$$996$$ 0 0
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ −24.0000 −0.759707
$$999$$ 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 855.2.a.c.1.1 1
3.2 odd 2 285.2.a.a.1.1 1
5.4 even 2 4275.2.a.h.1.1 1
12.11 even 2 4560.2.a.h.1.1 1
15.2 even 4 1425.2.c.c.799.1 2
15.8 even 4 1425.2.c.c.799.2 2
15.14 odd 2 1425.2.a.g.1.1 1
57.56 even 2 5415.2.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.a.1.1 1 3.2 odd 2
855.2.a.c.1.1 1 1.1 even 1 trivial
1425.2.a.g.1.1 1 15.14 odd 2
1425.2.c.c.799.1 2 15.2 even 4
1425.2.c.c.799.2 2 15.8 even 4
4275.2.a.h.1.1 1 5.4 even 2
4560.2.a.h.1.1 1 12.11 even 2
5415.2.a.h.1.1 1 57.56 even 2