# Properties

 Label 1078.2.a.l.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 $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 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 $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 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 $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 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 $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 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 $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 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 $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 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 $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 4.00000 0.516398
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ −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 $$-0.747820\pi$$
−0.702247 + 0.711934i $$0.747820\pi$$
$$74$$ 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 $$-0.0496119\pi$$
0.987878 + 0.155230i $$0.0496119\pi$$
$$84$$ 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 $$-0.291488\pi$$
0.609208 + 0.793011i $$0.291488\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ −1.00000 −0.100000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ −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 $$-0.472154\pi$$
0.0873704 + 0.996176i $$0.472154\pi$$
$$132$$ 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 $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 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 $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 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 $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 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 $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 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 $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 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 $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ −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 $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ −10.0000 −0.665190
$$227$$ −6.00000 −0.398234 −0.199117 0.979976i $$-0.563807\pi$$
−0.199117 + 0.979976i $$0.563807\pi$$
$$228$$ 4.00000 0.264906
$$229$$ 30.0000 1.98246 0.991228 0.132164i $$-0.0421925\pi$$
0.991228 + 0.132164i $$0.0421925\pi$$
$$230$$ 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 $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ 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 $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 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 $$-0.666309\pi$$
−0.499026 + 0.866587i $$0.666309\pi$$
$$258$$ 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 $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ −8.00000 −0.486864
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 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 $$-0.481067\pi$$
0.0594438 + 0.998232i $$0.481067\pi$$
$$284$$ 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 $$-0.225459\pi$$
0.759468 + 0.650545i $$0.225459\pi$$
$$294$$ 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 $$-0.630820\pi$$
−0.399511 + 0.916728i $$0.630820\pi$$
$$308$$ 0 0
$$309$$ −24.0000 −1.36531
$$310$$ −8.00000 −0.454369
$$311$$ −16.0000 −0.907277 −0.453638 0.891186i $$-0.649874\pi$$
−0.453638 + 0.891186i $$0.649874\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ 4.00000 0.226093 0.113047 0.993590i $$-0.463939\pi$$
0.113047 + 0.993590i $$0.463939\pi$$
$$314$$ 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 $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 8.00000 0.427008
$$352$$ 1.00000 0.0533002
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ −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 $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 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 $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ 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 $$-0.825345\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ −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 $$-0.629455\pi$$
−0.395575 + 0.918434i $$0.629455\pi$$
$$410$$ −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 $$-0.355092\pi$$
0.439679 + 0.898155i $$0.355092\pi$$
$$420$$ 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 $$-0.530641\pi$$
−0.0961139 + 0.995370i $$0.530641\pi$$
$$434$$ 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 $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 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 $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 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 $$-0.330008\pi$$
0.509019 + 0.860755i $$0.330008\pi$$
$$468$$ −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 $$-0.192612\pi$$
0.822441 + 0.568850i $$0.192612\pi$$
$$480$$ 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 $$-0.442924\pi$$
0.178351 + 0.983967i $$0.442924\pi$$
$$504$$ 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 $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 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 $$-0.472073\pi$$
0.0876216 + 0.996154i $$0.472073\pi$$
$$522$$ 6.00000 0.262613
$$523$$ 34.0000 1.48672 0.743358 0.668894i $$-0.233232\pi$$
0.743358 + 0.668894i $$0.233232\pi$$
$$524$$ 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 $$-0.346559\pi$$
0.463595 + 0.886047i $$0.346559\pi$$
$$564$$ −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 $$-0.267967\pi$$
0.666089 + 0.745873i $$0.267967\pi$$
$$578$$ −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 $$-0.166197\pi$$
0.866763 + 0.498721i $$0.166197\pi$$
$$588$$ 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 $$-0.634698\pi$$
−0.410651 + 0.911793i $$0.634698\pi$$
$$594$$ −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 $$-0.525997\pi$$
−0.0815817 + 0.996667i $$0.525997\pi$$
$$602$$ 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 $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 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 $$-0.409217\pi$$
0.281354 + 0.959604i $$0.409217\pi$$
$$620$$ −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 $$-0.266116\pi$$
0.670415 + 0.741987i $$0.266116\pi$$
$$644$$ 0 0
$$645$$ 48.0000 1.89000
$$646$$ 0 0
$$647$$ 44.0000 1.72982 0.864909 0.501928i $$-0.167376\pi$$
0.864909 + 0.501928i $$0.167376\pi$$
$$648$$ −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 $$-0.537227\pi$$
−0.116686 + 0.993169i $$0.537227\pi$$
$$662$$ 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 $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ −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 $$-0.899988\pi$$
−0.951045 + 0.309053i $$0.899988\pi$$
$$692$$ 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 $$-0.325144\pi$$
0.522112 + 0.852877i $$0.325144\pi$$
$$720$$ 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 $$-0.571431\pi$$
−0.222528 + 0.974926i $$0.571431\pi$$
$$728$$ 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 $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 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 $$-0.546318\pi$$
−0.145000 + 0.989432i $$0.546318\pi$$
$$762$$ 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 $$-0.243582\pi$$
0.721218 + 0.692708i $$0.243582\pi$$
$$770$$ 0 0
$$771$$ −32.0000 −1.15245
$$772$$ −2.00000 −0.0719816
$$773$$ −14.0000 −0.503545 −0.251773 0.967786i $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ 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 $$-0.580274\pi$$
−0.249523 + 0.968369i $$0.580274\pi$$
$$788$$ −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 $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 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 $$-0.158965\pi$$
0.877869 + 0.478901i $$0.158965\pi$$
$$812$$ 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 $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 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 $$-0.286552\pi$$
0.621429 + 0.783470i $$0.286552\pi$$
$$840$$ 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 $$-0.755412\pi$$
−0.719026 + 0.694983i $$0.755412\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ 12.0000 0.410152
$$857$$ −48.0000 −1.63965 −0.819824 0.572615i $$-0.805929\pi$$
−0.819824 + 0.572615i $$0.805929\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ 34.0000 1.16007 0.580033 0.814593i $$-0.303040\pi$$
0.580033 + 0.814593i $$0.303040\pi$$
$$860$$ 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 $$-0.632481\pi$$
−0.404290 + 0.914631i $$0.632481\pi$$
$$882$$ 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 $$-0.608996\pi$$
−0.335767 + 0.941945i $$0.608996\pi$$
$$888$$ 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 $$-0.479098\pi$$
0.0656179 + 0.997845i $$0.479098\pi$$
$$930$$ −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 $$-0.394068\pi$$
0.326686 + 0.945133i $$0.394068\pi$$
$$938$$ 0 0
$$939$$ 8.00000 0.261070
$$940$$ −24.0000 −0.782794
$$941$$ 42.0000 1.36916 0.684580 0.728937i $$-0.259985\pi$$
0.684580 + 0.728937i $$0.259985\pi$$
$$942$$ 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 $$-0.469307\pi$$
0.0962746 + 0.995355i $$0.469307\pi$$
$$972$$ −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 $$-0.747571\pi$$
−0.701691 + 0.712481i $$0.747571\pi$$
$$984$$ −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 $$-0.939144\pi$$
−0.981780 + 0.190022i $$0.939144\pi$$
$$998$$ 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.l.1.1 yes 1
3.2 odd 2 9702.2.a.e.1.1 1
4.3 odd 2 8624.2.a.g.1.1 1
7.2 even 3 1078.2.e.a.67.1 2
7.3 odd 6 1078.2.e.e.177.1 2
7.4 even 3 1078.2.e.a.177.1 2
7.5 odd 6 1078.2.e.e.67.1 2
7.6 odd 2 1078.2.a.h.1.1 1
21.20 even 2 9702.2.a.t.1.1 1
28.27 even 2 8624.2.a.y.1.1 1

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