# Properties

 Label 1078.2.a.h.1.1 Level $1078$ Weight $2$ Character 1078.1 Self dual yes Analytic conductor $8.608$ 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] = [1078,2,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, 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("1078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.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: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1078.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} +4.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -2.00000 q^{19} -2.00000 q^{20} +1.00000 q^{22} -2.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} +6.00000 q^{29} +4.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} -4.00000 q^{39} -2.00000 q^{40} +8.00000 q^{41} +12.0000 q^{43} +1.00000 q^{44} -2.00000 q^{45} +12.0000 q^{47} -2.00000 q^{48} -1.00000 q^{50} +2.00000 q^{52} -2.00000 q^{53} +4.00000 q^{54} -2.00000 q^{55} +4.00000 q^{57} +6.00000 q^{58} +10.0000 q^{59} +4.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -2.00000 q^{66} -12.0000 q^{67} +4.00000 q^{71} +1.00000 q^{72} +12.0000 q^{73} +2.00000 q^{74} +2.00000 q^{75} -2.00000 q^{76} -4.00000 q^{78} -2.00000 q^{80} -11.0000 q^{81} +8.00000 q^{82} -18.0000 q^{83} +12.0000 q^{86} -12.0000 q^{87} +1.00000 q^{88} -2.00000 q^{90} -8.00000 q^{93} +12.0000 q^{94} +4.00000 q^{95} -2.00000 q^{96} -12.0000 q^{97} +1.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ −2.00000 −0.816497
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −2.00000 −0.632456
$$11$$ 1.00000 0.301511
$$12$$ −2.00000 −0.577350
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 4.00000 1.03280
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −2.00000 −0.408248
$$25$$ −1.00000 −0.200000
$$26$$ 2.00000 0.392232
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 4.00000 0.730297
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −2.00000 −0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ −4.00000 −0.640513
$$40$$ −2.00000 −0.316228
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ 12.0000 1.82998 0.914991 0.403473i $$-0.132197\pi$$
0.914991 + 0.403473i $$0.132197\pi$$
$$44$$ 1.00000 0.150756
$$45$$ −2.00000 −0.298142
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ −2.00000 −0.288675
$$49$$ 0 0
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 4.00000 0.544331
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 6.00000 0.787839
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 4.00000 0.516398
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −4.00000 −0.496139
$$66$$ −2.00000 −0.246183
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 12.0000 1.40449 0.702247 0.711934i $$-0.252180\pi$$
0.702247 + 0.711934i $$0.252180\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 2.00000 0.230940
$$76$$ −2.00000 −0.229416
$$77$$ 0 0
$$78$$ −4.00000 −0.452911
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ −11.0000 −1.22222
$$82$$ 8.00000 0.883452
$$83$$ −18.0000 −1.97576 −0.987878 0.155230i $$-0.950388\pi$$
−0.987878 + 0.155230i $$0.950388\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ −12.0000 −1.28654
$$88$$ 1.00000 0.106600
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ −2.00000 −0.210819
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −8.00000 −0.829561
$$94$$ 12.0000 1.23771
$$95$$ 4.00000 0.410391
$$96$$ −2.00000 −0.204124
$$97$$ −12.0000 −1.21842 −0.609208 0.793011i $$-0.708512\pi$$
−0.609208 + 0.793011i $$0.708512\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ −1.00000 −0.100000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 12.0000 1.18240 0.591198 0.806527i $$-0.298655\pi$$
0.591198 + 0.806527i $$0.298655\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 4.00000 0.384900
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 2.00000 0.184900
$$118$$ 10.0000 0.920575
$$119$$ 0 0
$$120$$ 4.00000 0.365148
$$121$$ 1.00000 0.0909091
$$122$$ −10.0000 −0.905357
$$123$$ −16.0000 −1.44267
$$124$$ 4.00000 0.359211
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −24.0000 −2.11308
$$130$$ −4.00000 −0.350823
$$131$$ −2.00000 −0.174741 −0.0873704 0.996176i $$-0.527846\pi$$
−0.0873704 + 0.996176i $$0.527846\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ −8.00000 −0.688530
$$136$$ 0 0
$$137$$ 22.0000 1.87959 0.939793 0.341743i $$-0.111017\pi$$
0.939793 + 0.341743i $$0.111017\pi$$
$$138$$ 0 0
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ −24.0000 −2.02116
$$142$$ 4.00000 0.335673
$$143$$ 2.00000 0.167248
$$144$$ 1.00000 0.0833333
$$145$$ −12.0000 −0.996546
$$146$$ 12.0000 0.993127
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 2.00000 0.163299
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ −4.00000 −0.320256
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ 0 0
$$159$$ 4.00000 0.317221
$$160$$ −2.00000 −0.158114
$$161$$ 0 0
$$162$$ −11.0000 −0.864242
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ 8.00000 0.624695
$$165$$ 4.00000 0.311400
$$166$$ −18.0000 −1.39707
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 12.0000 0.914991
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ −12.0000 −0.909718
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ −20.0000 −1.50329
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ −2.00000 −0.149071
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 0 0
$$183$$ 20.0000 1.47844
$$184$$ 0 0
$$185$$ −4.00000 −0.294086
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ 12.0000 0.875190
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ −2.00000 −0.144338
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 8.00000 0.572892
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 24.0000 1.69283
$$202$$ 6.00000 0.422159
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −16.0000 −1.11749
$$206$$ 12.0000 0.836080
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ −8.00000 −0.548151
$$214$$ 12.0000 0.820303
$$215$$ −24.0000 −1.63679
$$216$$ 4.00000 0.272166
$$217$$ 0 0
$$218$$ −10.0000 −0.677285
$$219$$ −24.0000 −1.62177
$$220$$ −2.00000 −0.134840
$$221$$ 0 0
$$222$$ −4.00000 −0.268462
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ −10.0000 −0.665190
$$227$$ 6.00000 0.398234 0.199117 0.979976i $$-0.436193\pi$$
0.199117 + 0.979976i $$0.436193\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −30.0000 −1.98246 −0.991228 0.132164i $$-0.957808\pi$$
−0.991228 + 0.132164i $$0.957808\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 2.00000 0.130744
$$235$$ −24.0000 −1.56559
$$236$$ 10.0000 0.650945
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −4.00000 −0.258738 −0.129369 0.991596i $$-0.541295\pi$$
−0.129369 + 0.991596i $$0.541295\pi$$
$$240$$ 4.00000 0.258199
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 10.0000 0.641500
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ −16.0000 −1.02012
$$247$$ −4.00000 −0.254514
$$248$$ 4.00000 0.254000
$$249$$ 36.0000 2.28141
$$250$$ 12.0000 0.758947
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 16.0000 0.998053 0.499026 0.866587i $$-0.333691\pi$$
0.499026 + 0.866587i $$0.333691\pi$$
$$258$$ −24.0000 −1.49417
$$259$$ 0 0
$$260$$ −4.00000 −0.248069
$$261$$ 6.00000 0.371391
$$262$$ −2.00000 −0.123560
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ 4.00000 0.245718
$$266$$ 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 $$0.558553\pi$$
$$270$$ −8.00000 −0.486864
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 22.0000 1.32907
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ −14.0000 −0.841178 −0.420589 0.907251i $$-0.638177\pi$$
−0.420589 + 0.907251i $$0.638177\pi$$
$$278$$ −14.0000 −0.839664
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ −24.0000 −1.42918
$$283$$ −2.00000 −0.118888 −0.0594438 0.998232i $$-0.518933\pi$$
−0.0594438 + 0.998232i $$0.518933\pi$$
$$284$$ 4.00000 0.237356
$$285$$ −8.00000 −0.473879
$$286$$ 2.00000 0.118262
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ −17.0000 −1.00000
$$290$$ −12.0000 −0.704664
$$291$$ 24.0000 1.40690
$$292$$ 12.0000 0.702247
$$293$$ −26.0000 −1.51894 −0.759468 0.650545i $$-0.774541\pi$$
−0.759468 + 0.650545i $$0.774541\pi$$
$$294$$ 0 0
$$295$$ −20.0000 −1.16445
$$296$$ 2.00000 0.116248
$$297$$ 4.00000 0.232104
$$298$$ −14.0000 −0.810998
$$299$$ 0 0
$$300$$ 2.00000 0.115470
$$301$$ 0 0
$$302$$ −4.00000 −0.230174
$$303$$ −12.0000 −0.689382
$$304$$ −2.00000 −0.114708
$$305$$ 20.0000 1.14520
$$306$$ 0 0
$$307$$ 14.0000 0.799022 0.399511 0.916728i $$-0.369180\pi$$
0.399511 + 0.916728i $$0.369180\pi$$
$$308$$ 0 0
$$309$$ −24.0000 −1.36531
$$310$$ −8.00000 −0.454369
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ −4.00000 −0.226093 −0.113047 0.993590i $$-0.536061\pi$$
−0.113047 + 0.993590i $$0.536061\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 4.00000 0.224309
$$319$$ 6.00000 0.335936
$$320$$ −2.00000 −0.111803
$$321$$ −24.0000 −1.33955
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −11.0000 −0.611111
$$325$$ −2.00000 −0.110940
$$326$$ 12.0000 0.664619
$$327$$ 20.0000 1.10600
$$328$$ 8.00000 0.441726
$$329$$ 0 0
$$330$$ 4.00000 0.220193
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ −18.0000 −0.987878
$$333$$ 2.00000 0.109599
$$334$$ 12.0000 0.656611
$$335$$ 24.0000 1.31126
$$336$$ 0 0
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 20.0000 1.08625
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ −2.00000 −0.108148
$$343$$ 0 0
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ −12.0000 −0.643268
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 8.00000 0.427008
$$352$$ 1.00000 0.0533002
$$353$$ 24.0000 1.27739 0.638696 0.769460i $$-0.279474\pi$$
0.638696 + 0.769460i $$0.279474\pi$$
$$354$$ −20.0000 −1.06299
$$355$$ −8.00000 −0.424596
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ −2.00000 −0.105409
$$361$$ −15.0000 −0.789474
$$362$$ 18.0000 0.946059
$$363$$ −2.00000 −0.104973
$$364$$ 0 0
$$365$$ −24.0000 −1.25622
$$366$$ 20.0000 1.04542
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 8.00000 0.416463
$$370$$ −4.00000 −0.207950
$$371$$ 0 0
$$372$$ −8.00000 −0.414781
$$373$$ −30.0000 −1.55334 −0.776671 0.629907i $$-0.783093\pi$$
−0.776671 + 0.629907i $$0.783093\pi$$
$$374$$ 0 0
$$375$$ −24.0000 −1.23935
$$376$$ 12.0000 0.618853
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 4.00000 0.205196
$$381$$ −8.00000 −0.409852
$$382$$ 12.0000 0.613973
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ −2.00000 −0.102062
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ 12.0000 0.609994
$$388$$ −12.0000 −0.609208
$$389$$ 10.0000 0.507020 0.253510 0.967333i $$-0.418415\pi$$
0.253510 + 0.967333i $$0.418415\pi$$
$$390$$ 8.00000 0.405096
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 4.00000 0.201773
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 1.00000 0.0502519
$$397$$ 34.0000 1.70641 0.853206 0.521575i $$-0.174655\pi$$
0.853206 + 0.521575i $$0.174655\pi$$
$$398$$ 4.00000 0.200502
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 24.0000 1.19701
$$403$$ 8.00000 0.398508
$$404$$ 6.00000 0.298511
$$405$$ 22.0000 1.09319
$$406$$ 0 0
$$407$$ 2.00000 0.0991363
$$408$$ 0 0
$$409$$ 16.0000 0.791149 0.395575 0.918434i $$-0.370545\pi$$
0.395575 + 0.918434i $$0.370545\pi$$
$$410$$ −16.0000 −0.790184
$$411$$ −44.0000 −2.17036
$$412$$ 12.0000 0.591198
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 36.0000 1.76717
$$416$$ 2.00000 0.0980581
$$417$$ 28.0000 1.37117
$$418$$ −2.00000 −0.0978232
$$419$$ −18.0000 −0.879358 −0.439679 0.898155i $$-0.644908\pi$$
−0.439679 + 0.898155i $$0.644908\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ 12.0000 0.583460
$$424$$ −2.00000 −0.0971286
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ −4.00000 −0.193122
$$430$$ −24.0000 −1.15738
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 4.00000 0.192450
$$433$$ 4.00000 0.192228 0.0961139 0.995370i $$-0.469359\pi$$
0.0961139 + 0.995370i $$0.469359\pi$$
$$434$$ 0 0
$$435$$ 24.0000 1.15071
$$436$$ −10.0000 −0.478913
$$437$$ 0 0
$$438$$ −24.0000 −1.14676
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ −2.00000 −0.0953463
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 28.0000 1.32435
$$448$$ 0 0
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 8.00000 0.376705
$$452$$ −10.0000 −0.470360
$$453$$ 8.00000 0.375873
$$454$$ 6.00000 0.281594
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ −30.0000 −1.40181
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 16.0000 0.741982
$$466$$ −10.0000 −0.463241
$$467$$ −22.0000 −1.01804 −0.509019 0.860755i $$-0.669992\pi$$
−0.509019 + 0.860755i $$0.669992\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 0 0
$$470$$ −24.0000 −1.10704
$$471$$ 28.0000 1.29017
$$472$$ 10.0000 0.460287
$$473$$ 12.0000 0.551761
$$474$$ 0 0
$$475$$ 2.00000 0.0917663
$$476$$ 0 0
$$477$$ −2.00000 −0.0915737
$$478$$ −4.00000 −0.182956
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 4.00000 0.182574
$$481$$ 4.00000 0.182384
$$482$$ 8.00000 0.364390
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 24.0000 1.08978
$$486$$ 10.0000 0.453609
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ −24.0000 −1.08532
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ −16.0000 −0.721336
$$493$$ 0 0
$$494$$ −4.00000 −0.179969
$$495$$ −2.00000 −0.0898933
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ 36.0000 1.61320
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 12.0000 0.536656
$$501$$ −24.0000 −1.07224
$$502$$ −18.0000 −0.803379
$$503$$ −8.00000 −0.356702 −0.178351 0.983967i $$-0.557076\pi$$
−0.178351 + 0.983967i $$0.557076\pi$$
$$504$$ 0 0
$$505$$ −12.0000 −0.533993
$$506$$ 0 0
$$507$$ 18.0000 0.799408
$$508$$ 4.00000 0.177471
$$509$$ −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
$$513$$ −8.00000 −0.353209
$$514$$ 16.0000 0.705730
$$515$$ −24.0000 −1.05757
$$516$$ −24.0000 −1.05654
$$517$$ 12.0000 0.527759
$$518$$ 0 0
$$519$$ 12.0000 0.526742
$$520$$ −4.00000 −0.175412
$$521$$ −4.00000 −0.175243 −0.0876216 0.996154i $$-0.527927\pi$$
−0.0876216 + 0.996154i $$0.527927\pi$$
$$522$$ 6.00000 0.262613
$$523$$ −34.0000 −1.48672 −0.743358 0.668894i $$-0.766768\pi$$
−0.743358 + 0.668894i $$0.766768\pi$$
$$524$$ −2.00000 −0.0873704
$$525$$ 0 0
$$526$$ 16.0000 0.697633
$$527$$ 0 0
$$528$$ −2.00000 −0.0870388
$$529$$ −23.0000 −1.00000
$$530$$ 4.00000 0.173749
$$531$$ 10.0000 0.433963
$$532$$ 0 0
$$533$$ 16.0000 0.693037
$$534$$ 0 0
$$535$$ −24.0000 −1.03761
$$536$$ −12.0000 −0.518321
$$537$$ −24.0000 −1.03568
$$538$$ −6.00000 −0.258678
$$539$$ 0 0
$$540$$ −8.00000 −0.344265
$$541$$ 34.0000 1.46177 0.730887 0.682498i $$-0.239107\pi$$
0.730887 + 0.682498i $$0.239107\pi$$
$$542$$ 16.0000 0.687259
$$543$$ −36.0000 −1.54491
$$544$$ 0 0
$$545$$ 20.0000 0.856706
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 22.0000 0.939793
$$549$$ −10.0000 −0.426790
$$550$$ −1.00000 −0.0426401
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −14.0000 −0.594803
$$555$$ 8.00000 0.339581
$$556$$ −14.0000 −0.593732
$$557$$ 34.0000 1.44063 0.720313 0.693649i $$-0.243998\pi$$
0.720313 + 0.693649i $$0.243998\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −10.0000 −0.421825
$$563$$ −22.0000 −0.927189 −0.463595 0.886047i $$-0.653441\pi$$
−0.463595 + 0.886047i $$0.653441\pi$$
$$564$$ −24.0000 −1.01058
$$565$$ 20.0000 0.841406
$$566$$ −2.00000 −0.0840663
$$567$$ 0 0
$$568$$ 4.00000 0.167836
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ −8.00000 −0.335083
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ −24.0000 −1.00261
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −32.0000 −1.33218 −0.666089 0.745873i $$-0.732033\pi$$
−0.666089 + 0.745873i $$0.732033\pi$$
$$578$$ −17.0000 −0.707107
$$579$$ 4.00000 0.166234
$$580$$ −12.0000 −0.498273
$$581$$ 0 0
$$582$$ 24.0000 0.994832
$$583$$ −2.00000 −0.0828315
$$584$$ 12.0000 0.496564
$$585$$ −4.00000 −0.165380
$$586$$ −26.0000 −1.07405
$$587$$ −42.0000 −1.73353 −0.866763 0.498721i $$-0.833803\pi$$
−0.866763 + 0.498721i $$0.833803\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ −20.0000 −0.823387
$$591$$ 12.0000 0.493614
$$592$$ 2.00000 0.0821995
$$593$$ 20.0000 0.821302 0.410651 0.911793i $$-0.365302\pi$$
0.410651 + 0.911793i $$0.365302\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −14.0000 −0.573462
$$597$$ −8.00000 −0.327418
$$598$$ 0 0
$$599$$ 28.0000 1.14405 0.572024 0.820237i $$-0.306158\pi$$
0.572024 + 0.820237i $$0.306158\pi$$
$$600$$ 2.00000 0.0816497
$$601$$ 4.00000 0.163163 0.0815817 0.996667i $$-0.474003\pi$$
0.0815817 + 0.996667i $$0.474003\pi$$
$$602$$ 0 0
$$603$$ −12.0000 −0.488678
$$604$$ −4.00000 −0.162758
$$605$$ −2.00000 −0.0813116
$$606$$ −12.0000 −0.487467
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 0 0
$$610$$ 20.0000 0.809776
$$611$$ 24.0000 0.970936
$$612$$ 0 0
$$613$$ −42.0000 −1.69636 −0.848182 0.529705i $$-0.822303\pi$$
−0.848182 + 0.529705i $$0.822303\pi$$
$$614$$ 14.0000 0.564994
$$615$$ 32.0000 1.29036
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ −24.0000 −0.965422
$$619$$ −14.0000 −0.562708 −0.281354 0.959604i $$-0.590783\pi$$
−0.281354 + 0.959604i $$0.590783\pi$$
$$620$$ −8.00000 −0.321288
$$621$$ 0 0
$$622$$ 16.0000 0.641542
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ −19.0000 −0.760000
$$626$$ −4.00000 −0.159872
$$627$$ 4.00000 0.159745
$$628$$ −14.0000 −0.558661
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −28.0000 −1.11466 −0.557331 0.830290i $$-0.688175\pi$$
−0.557331 + 0.830290i $$0.688175\pi$$
$$632$$ 0 0
$$633$$ 8.00000 0.317971
$$634$$ 30.0000 1.19145
$$635$$ −8.00000 −0.317470
$$636$$ 4.00000 0.158610
$$637$$ 0 0
$$638$$ 6.00000 0.237542
$$639$$ 4.00000 0.158238
$$640$$ −2.00000 −0.0790569
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ −24.0000 −0.947204
$$643$$ −34.0000 −1.34083 −0.670415 0.741987i $$-0.733884\pi$$
−0.670415 + 0.741987i $$0.733884\pi$$
$$644$$ 0 0
$$645$$ 48.0000 1.89000
$$646$$ 0 0
$$647$$ −44.0000 −1.72982 −0.864909 0.501928i $$-0.832624\pi$$
−0.864909 + 0.501928i $$0.832624\pi$$
$$648$$ −11.0000 −0.432121
$$649$$ 10.0000 0.392534
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ 12.0000 0.469956
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 20.0000 0.782062
$$655$$ 4.00000 0.156293
$$656$$ 8.00000 0.312348
$$657$$ 12.0000 0.468165
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 4.00000 0.155700
$$661$$ 6.00000 0.233373 0.116686 0.993169i $$-0.462773\pi$$
0.116686 + 0.993169i $$0.462773\pi$$
$$662$$ 12.0000 0.466393
$$663$$ 0 0
$$664$$ −18.0000 −0.698535
$$665$$ 0 0
$$666$$ 2.00000 0.0774984
$$667$$ 0 0
$$668$$ 12.0000 0.464294
$$669$$ −16.0000 −0.618596
$$670$$ 24.0000 0.927201
$$671$$ −10.0000 −0.386046
$$672$$ 0 0
$$673$$ 6.00000 0.231283 0.115642 0.993291i $$-0.463108\pi$$
0.115642 + 0.993291i $$0.463108\pi$$
$$674$$ 34.0000 1.30963
$$675$$ −4.00000 −0.153960
$$676$$ −9.00000 −0.346154
$$677$$ 42.0000 1.61419 0.807096 0.590421i $$-0.201038\pi$$
0.807096 + 0.590421i $$0.201038\pi$$
$$678$$ 20.0000 0.768095
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 4.00000 0.153168
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ −44.0000 −1.68115
$$686$$ 0 0
$$687$$ 60.0000 2.28914
$$688$$ 12.0000 0.457496
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ 50.0000 1.90209 0.951045 0.309053i $$-0.100012\pi$$
0.951045 + 0.309053i $$0.100012\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ 20.0000 0.759190
$$695$$ 28.0000 1.06210
$$696$$ −12.0000 −0.454859
$$697$$ 0 0
$$698$$ −14.0000 −0.529908
$$699$$ 20.0000 0.756469
$$700$$ 0 0
$$701$$ −34.0000 −1.28416 −0.642081 0.766637i $$-0.721929\pi$$
−0.642081 + 0.766637i $$0.721929\pi$$
$$702$$ 8.00000 0.301941
$$703$$ −4.00000 −0.150863
$$704$$ 1.00000 0.0376889
$$705$$ 48.0000 1.80778
$$706$$ 24.0000 0.903252
$$707$$ 0 0
$$708$$ −20.0000 −0.751646
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ −8.00000 −0.300235
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −4.00000 −0.149592
$$716$$ 12.0000 0.448461
$$717$$ 8.00000 0.298765
$$718$$ 20.0000 0.746393
$$719$$ −28.0000 −1.04422 −0.522112 0.852877i $$-0.674856\pi$$
−0.522112 + 0.852877i $$0.674856\pi$$
$$720$$ −2.00000 −0.0745356
$$721$$ 0 0
$$722$$ −15.0000 −0.558242
$$723$$ −16.0000 −0.595046
$$724$$ 18.0000 0.668965
$$725$$ −6.00000 −0.222834
$$726$$ −2.00000 −0.0742270
$$727$$ 12.0000 0.445055 0.222528 0.974926i $$-0.428569\pi$$
0.222528 + 0.974926i $$0.428569\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ −24.0000 −0.888280
$$731$$ 0 0
$$732$$ 20.0000 0.739221
$$733$$ 6.00000 0.221615 0.110808 0.993842i $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −12.0000 −0.442026
$$738$$ 8.00000 0.294484
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 8.00000 0.293887
$$742$$ 0 0
$$743$$ 36.0000 1.32071 0.660356 0.750953i $$-0.270405\pi$$
0.660356 + 0.750953i $$0.270405\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 28.0000 1.02584
$$746$$ −30.0000 −1.09838
$$747$$ −18.0000 −0.658586
$$748$$ 0 0
$$749$$ 0 0
$$750$$ −24.0000 −0.876356
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ 12.0000 0.437595
$$753$$ 36.0000 1.31191
$$754$$ 12.0000 0.437014
$$755$$ 8.00000 0.291150
$$756$$ 0 0
$$757$$ −30.0000 −1.09037 −0.545184 0.838316i $$-0.683540\pi$$
−0.545184 + 0.838316i $$0.683540\pi$$
$$758$$ 28.0000 1.01701
$$759$$ 0 0
$$760$$ 4.00000 0.145095
$$761$$ 8.00000 0.290000 0.145000 0.989432i $$-0.453682\pi$$
0.145000 + 0.989432i $$0.453682\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ 0 0
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ 20.0000 0.722158
$$768$$ −2.00000 −0.0721688
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ 0 0
$$771$$ −32.0000 −1.15245
$$772$$ −2.00000 −0.0719816
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ 12.0000 0.431331
$$775$$ −4.00000 −0.143684
$$776$$ −12.0000 −0.430775
$$777$$ 0 0
$$778$$ 10.0000 0.358517
$$779$$ −16.0000 −0.573259
$$780$$ 8.00000 0.286446
$$781$$ 4.00000 0.143131
$$782$$ 0 0
$$783$$ 24.0000 0.857690
$$784$$ 0 0
$$785$$ 28.0000 0.999363
$$786$$ 4.00000 0.142675
$$787$$ 14.0000 0.499046 0.249523 0.968369i $$-0.419726\pi$$
0.249523 + 0.968369i $$0.419726\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ −32.0000 −1.13923
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 1.00000 0.0355335
$$793$$ −20.0000 −0.710221
$$794$$ 34.0000 1.20661
$$795$$ −8.00000 −0.283731
$$796$$ 4.00000 0.141776
$$797$$ 18.0000 0.637593 0.318796 0.947823i $$-0.396721\pi$$
0.318796 + 0.947823i $$0.396721\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 18.0000 0.635602
$$803$$ 12.0000 0.423471
$$804$$ 24.0000 0.846415
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 12.0000 0.422420
$$808$$ 6.00000 0.211079
$$809$$ 38.0000 1.33601 0.668004 0.744157i $$-0.267149\pi$$
0.668004 + 0.744157i $$0.267149\pi$$
$$810$$ 22.0000 0.773001
$$811$$ −50.0000 −1.75574 −0.877869 0.478901i $$-0.841035\pi$$
−0.877869 + 0.478901i $$0.841035\pi$$
$$812$$ 0 0
$$813$$ −32.0000 −1.12229
$$814$$ 2.00000 0.0701000
$$815$$ −24.0000 −0.840683
$$816$$ 0 0
$$817$$ −24.0000 −0.839654
$$818$$ 16.0000 0.559427
$$819$$ 0 0
$$820$$ −16.0000 −0.558744
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ −44.0000 −1.53468
$$823$$ −20.0000 −0.697156 −0.348578 0.937280i $$-0.613335\pi$$
−0.348578 + 0.937280i $$0.613335\pi$$
$$824$$ 12.0000 0.418040
$$825$$ 2.00000 0.0696311
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 36.0000 1.24958
$$831$$ 28.0000 0.971309
$$832$$ 2.00000 0.0693375
$$833$$ 0 0
$$834$$ 28.0000 0.969561
$$835$$ −24.0000 −0.830554
$$836$$ −2.00000 −0.0691714
$$837$$ 16.0000 0.553041
$$838$$ −18.0000 −0.621800
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −34.0000 −1.17172
$$843$$ 20.0000 0.688837
$$844$$ −4.00000 −0.137686
$$845$$ 18.0000 0.619219
$$846$$ 12.0000 0.412568
$$847$$ 0 0
$$848$$ −2.00000 −0.0686803
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 0 0
$$852$$ −8.00000 −0.274075
$$853$$ 42.0000 1.43805 0.719026 0.694983i $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ 12.0000 0.410152
$$857$$ 48.0000 1.63965 0.819824 0.572615i $$-0.194071\pi$$
0.819824 + 0.572615i $$0.194071\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ −34.0000 −1.16007 −0.580033 0.814593i $$-0.696960\pi$$
−0.580033 + 0.814593i $$0.696960\pi$$
$$860$$ −24.0000 −0.818393
$$861$$ 0 0
$$862$$ −36.0000 −1.22616
$$863$$ −52.0000 −1.77010 −0.885050 0.465495i $$-0.845876\pi$$
−0.885050 + 0.465495i $$0.845876\pi$$
$$864$$ 4.00000 0.136083
$$865$$ 12.0000 0.408012
$$866$$ 4.00000 0.135926
$$867$$ 34.0000 1.15470
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 24.0000 0.813676
$$871$$ −24.0000 −0.813209
$$872$$ −10.0000 −0.338643
$$873$$ −12.0000 −0.406138
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −24.0000 −0.810885
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ −16.0000 −0.539974
$$879$$ 52.0000 1.75392
$$880$$ −2.00000 −0.0674200
$$881$$ 24.0000 0.808581 0.404290 0.914631i $$-0.367519\pi$$
0.404290 + 0.914631i $$0.367519\pi$$
$$882$$ 0 0
$$883$$ −52.0000 −1.74994 −0.874970 0.484178i $$-0.839119\pi$$
−0.874970 + 0.484178i $$0.839119\pi$$
$$884$$ 0 0
$$885$$ 40.0000 1.34459
$$886$$ 12.0000 0.403148
$$887$$ 20.0000 0.671534 0.335767 0.941945i $$-0.391004\pi$$
0.335767 + 0.941945i $$0.391004\pi$$
$$888$$ −4.00000 −0.134231
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −11.0000 −0.368514
$$892$$ 8.00000 0.267860
$$893$$ −24.0000 −0.803129
$$894$$ 28.0000 0.936460
$$895$$ −24.0000 −0.802232
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ 24.0000 0.800445
$$900$$ −1.00000 −0.0333333
$$901$$ 0 0
$$902$$ 8.00000 0.266371
$$903$$ 0 0
$$904$$ −10.0000 −0.332595
$$905$$ −36.0000 −1.19668
$$906$$ 8.00000 0.265782
$$907$$ 4.00000 0.132818 0.0664089 0.997792i $$-0.478846\pi$$
0.0664089 + 0.997792i $$0.478846\pi$$
$$908$$ 6.00000 0.199117
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ −32.0000 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$912$$ 4.00000 0.132453
$$913$$ −18.0000 −0.595713
$$914$$ 6.00000 0.198462
$$915$$ −40.0000 −1.32236
$$916$$ −30.0000 −0.991228
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ −6.00000 −0.197599
$$923$$ 8.00000 0.263323
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ 4.00000 0.131448
$$927$$ 12.0000 0.394132
$$928$$ 6.00000 0.196960
$$929$$ −4.00000 −0.131236 −0.0656179 0.997845i $$-0.520902\pi$$
−0.0656179 + 0.997845i $$0.520902\pi$$
$$930$$ 16.0000 0.524661
$$931$$ 0 0
$$932$$ −10.0000 −0.327561
$$933$$ −32.0000 −1.04763
$$934$$ −22.0000 −0.719862
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ −20.0000 −0.653372 −0.326686 0.945133i $$-0.605932\pi$$
−0.326686 + 0.945133i $$0.605932\pi$$
$$938$$ 0 0
$$939$$ 8.00000 0.261070
$$940$$ −24.0000 −0.782794
$$941$$ −42.0000 −1.36916 −0.684580 0.728937i $$-0.740015\pi$$
−0.684580 + 0.728937i $$0.740015\pi$$
$$942$$ 28.0000 0.912289
$$943$$ 0 0
$$944$$ 10.0000 0.325472
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 0 0
$$949$$ 24.0000 0.779073
$$950$$ 2.00000 0.0648886
$$951$$ −60.0000 −1.94563
$$952$$ 0 0
$$953$$ 34.0000 1.10137 0.550684 0.834714i $$-0.314367\pi$$
0.550684 + 0.834714i $$0.314367\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ −24.0000 −0.776622
$$956$$ −4.00000 −0.129369
$$957$$ −12.0000 −0.387905
$$958$$ −36.0000 −1.16311
$$959$$ 0 0
$$960$$ 4.00000 0.129099
$$961$$ −15.0000 −0.483871
$$962$$ 4.00000 0.128965
$$963$$ 12.0000 0.386695
$$964$$ 8.00000 0.257663
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ −4.00000 −0.128631 −0.0643157 0.997930i $$-0.520486\pi$$
−0.0643157 + 0.997930i $$0.520486\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 0 0
$$970$$ 24.0000 0.770594
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ 10.0000 0.320750
$$973$$ 0 0
$$974$$ 8.00000 0.256337
$$975$$ 4.00000 0.128103
$$976$$ −10.0000 −0.320092
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ −24.0000 −0.767435
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 28.0000 0.893516
$$983$$ 44.0000 1.40338 0.701691 0.712481i $$-0.252429\pi$$
0.701691 + 0.712481i $$0.252429\pi$$
$$984$$ −16.0000 −0.510061
$$985$$ 12.0000 0.382352
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −4.00000 −0.127257
$$989$$ 0 0
$$990$$ −2.00000 −0.0635642
$$991$$ −52.0000 −1.65183 −0.825917 0.563791i $$-0.809342\pi$$
−0.825917 + 0.563791i $$0.809342\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −24.0000 −0.761617
$$994$$ 0 0
$$995$$ −8.00000 −0.253617
$$996$$ 36.0000 1.14070
$$997$$ 62.0000 1.96356 0.981780 0.190022i $$-0.0608559\pi$$
0.981780 + 0.190022i $$0.0608559\pi$$
$$998$$ 20.0000 0.633089
$$999$$ 8.00000 0.253109
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 1078.2.a.h.1.1 1
3.2 odd 2 9702.2.a.t.1.1 1
4.3 odd 2 8624.2.a.y.1.1 1
7.2 even 3 1078.2.e.e.67.1 2
7.3 odd 6 1078.2.e.a.177.1 2
7.4 even 3 1078.2.e.e.177.1 2
7.5 odd 6 1078.2.e.a.67.1 2
7.6 odd 2 1078.2.a.l.1.1 yes 1
21.20 even 2 9702.2.a.e.1.1 1
28.27 even 2 8624.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.h.1.1 1 1.1 even 1 trivial
1078.2.a.l.1.1 yes 1 7.6 odd 2
1078.2.e.a.67.1 2 7.5 odd 6
1078.2.e.a.177.1 2 7.3 odd 6
1078.2.e.e.67.1 2 7.2 even 3
1078.2.e.e.177.1 2 7.4 even 3
8624.2.a.g.1.1 1 28.27 even 2
8624.2.a.y.1.1 1 4.3 odd 2
9702.2.a.e.1.1 1 21.20 even 2
9702.2.a.t.1.1 1 3.2 odd 2