# Properties

 Label 8624.2.a.bs.1.2 Level $8624$ Weight $2$ Character 8624.1 Self dual yes Analytic conductor $68.863$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8624 = 2^{4} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8624.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: $$68.8629867032$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1078) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 8624.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.41421 q^{3} -4.24264 q^{5} -1.00000 q^{9} +O(q^{10})$$ $$q+1.41421 q^{3} -4.24264 q^{5} -1.00000 q^{9} +1.00000 q^{11} -6.00000 q^{15} +5.65685 q^{17} -6.00000 q^{23} +13.0000 q^{25} -5.65685 q^{27} +2.00000 q^{29} -1.41421 q^{31} +1.41421 q^{33} -10.0000 q^{37} +11.3137 q^{41} +8.00000 q^{43} +4.24264 q^{45} +4.24264 q^{47} +8.00000 q^{51} +8.00000 q^{53} -4.24264 q^{55} +1.41421 q^{59} +2.82843 q^{61} -2.00000 q^{67} -8.48528 q^{69} +2.00000 q^{71} -8.48528 q^{73} +18.3848 q^{75} -16.0000 q^{79} -5.00000 q^{81} -16.9706 q^{83} -24.0000 q^{85} +2.82843 q^{87} +7.07107 q^{89} -2.00000 q^{93} +9.89949 q^{97} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 2 q^{11} - 12 q^{15} - 12 q^{23} + 26 q^{25} + 4 q^{29} - 20 q^{37} + 16 q^{43} + 16 q^{51} + 16 q^{53} - 4 q^{67} + 4 q^{71} - 32 q^{79} - 10 q^{81} - 48 q^{85} - 4 q^{93} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 + 2 * q^11 - 12 * q^15 - 12 * q^23 + 26 * q^25 + 4 * q^29 - 20 * q^37 + 16 * q^43 + 16 * q^51 + 16 * q^53 - 4 * q^67 + 4 * q^71 - 32 * q^79 - 10 * q^81 - 48 * q^85 - 4 * q^93 - 2 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.41421 0.816497 0.408248 0.912871i $$-0.366140\pi$$
0.408248 + 0.912871i $$0.366140\pi$$
$$4$$ 0 0
$$5$$ −4.24264 −1.89737 −0.948683 0.316228i $$-0.897584\pi$$
−0.948683 + 0.316228i $$0.897584\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ −6.00000 −1.54919
$$16$$ 0 0
$$17$$ 5.65685 1.37199 0.685994 0.727607i $$-0.259367\pi$$
0.685994 + 0.727607i $$0.259367\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 13.0000 2.60000
$$26$$ 0 0
$$27$$ −5.65685 −1.08866
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −1.41421 −0.254000 −0.127000 0.991903i $$-0.540535\pi$$
−0.127000 + 0.991903i $$0.540535\pi$$
$$32$$ 0 0
$$33$$ 1.41421 0.246183
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.3137 1.76690 0.883452 0.468521i $$-0.155213\pi$$
0.883452 + 0.468521i $$0.155213\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 4.24264 0.632456
$$46$$ 0 0
$$47$$ 4.24264 0.618853 0.309426 0.950923i $$-0.399863\pi$$
0.309426 + 0.950923i $$0.399863\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 8.00000 1.12022
$$52$$ 0 0
$$53$$ 8.00000 1.09888 0.549442 0.835532i $$-0.314840\pi$$
0.549442 + 0.835532i $$0.314840\pi$$
$$54$$ 0 0
$$55$$ −4.24264 −0.572078
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 1.41421 0.184115 0.0920575 0.995754i $$-0.470656\pi$$
0.0920575 + 0.995754i $$0.470656\pi$$
$$60$$ 0 0
$$61$$ 2.82843 0.362143 0.181071 0.983470i $$-0.442043\pi$$
0.181071 + 0.983470i $$0.442043\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ 0 0
$$69$$ −8.48528 −1.02151
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 0 0
$$73$$ −8.48528 −0.993127 −0.496564 0.868000i $$-0.665405\pi$$
−0.496564 + 0.868000i $$0.665405\pi$$
$$74$$ 0 0
$$75$$ 18.3848 2.12289
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 0 0
$$81$$ −5.00000 −0.555556
$$82$$ 0 0
$$83$$ −16.9706 −1.86276 −0.931381 0.364047i $$-0.881395\pi$$
−0.931381 + 0.364047i $$0.881395\pi$$
$$84$$ 0 0
$$85$$ −24.0000 −2.60317
$$86$$ 0 0
$$87$$ 2.82843 0.303239
$$88$$ 0 0
$$89$$ 7.07107 0.749532 0.374766 0.927119i $$-0.377723\pi$$
0.374766 + 0.927119i $$0.377723\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −2.00000 −0.207390
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 9.89949 1.00514 0.502571 0.864536i $$-0.332388\pi$$
0.502571 + 0.864536i $$0.332388\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −5.65685 −0.562878 −0.281439 0.959579i $$-0.590812\pi$$
−0.281439 + 0.959579i $$0.590812\pi$$
$$102$$ 0 0
$$103$$ 18.3848 1.81151 0.905753 0.423806i $$-0.139306\pi$$
0.905753 + 0.423806i $$0.139306\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −16.0000 −1.54678 −0.773389 0.633932i $$-0.781440\pi$$
−0.773389 + 0.633932i $$0.781440\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −14.1421 −1.34231
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 25.4558 2.37377
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 16.0000 1.44267
$$124$$ 0 0
$$125$$ −33.9411 −3.03579
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 0 0
$$129$$ 11.3137 0.996116
$$130$$ 0 0
$$131$$ −19.7990 −1.72985 −0.864923 0.501905i $$-0.832633\pi$$
−0.864923 + 0.501905i $$0.832633\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 24.0000 2.06559
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ 11.3137 0.959616 0.479808 0.877373i $$-0.340706\pi$$
0.479808 + 0.877373i $$0.340706\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −8.48528 −0.704664
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 0 0
$$153$$ −5.65685 −0.457330
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ −4.24264 −0.338600 −0.169300 0.985565i $$-0.554151\pi$$
−0.169300 + 0.985565i $$0.554151\pi$$
$$158$$ 0 0
$$159$$ 11.3137 0.897235
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 10.0000 0.783260 0.391630 0.920123i $$-0.371911\pi$$
0.391630 + 0.920123i $$0.371911\pi$$
$$164$$ 0 0
$$165$$ −6.00000 −0.467099
$$166$$ 0 0
$$167$$ 5.65685 0.437741 0.218870 0.975754i $$-0.429763\pi$$
0.218870 + 0.975754i $$0.429763\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 11.3137 0.860165 0.430083 0.902790i $$-0.358484\pi$$
0.430083 + 0.902790i $$0.358484\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 2.00000 0.150329
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 7.07107 0.525588 0.262794 0.964852i $$-0.415356\pi$$
0.262794 + 0.964852i $$0.415356\pi$$
$$182$$ 0 0
$$183$$ 4.00000 0.295689
$$184$$ 0 0
$$185$$ 42.4264 3.11925
$$186$$ 0 0
$$187$$ 5.65685 0.413670
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 22.0000 1.56744 0.783718 0.621117i $$-0.213321\pi$$
0.783718 + 0.621117i $$0.213321\pi$$
$$198$$ 0 0
$$199$$ −1.41421 −0.100251 −0.0501255 0.998743i $$-0.515962\pi$$
−0.0501255 + 0.998743i $$0.515962\pi$$
$$200$$ 0 0
$$201$$ −2.82843 −0.199502
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −48.0000 −3.35247
$$206$$ 0 0
$$207$$ 6.00000 0.417029
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 0 0
$$213$$ 2.82843 0.193801
$$214$$ 0 0
$$215$$ −33.9411 −2.31477
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −12.0000 −0.810885
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −21.2132 −1.42054 −0.710271 0.703929i $$-0.751427\pi$$
−0.710271 + 0.703929i $$0.751427\pi$$
$$224$$ 0 0
$$225$$ −13.0000 −0.866667
$$226$$ 0 0
$$227$$ −14.1421 −0.938647 −0.469323 0.883026i $$-0.655502\pi$$
−0.469323 + 0.883026i $$0.655502\pi$$
$$228$$ 0 0
$$229$$ −9.89949 −0.654177 −0.327089 0.944994i $$-0.606068\pi$$
−0.327089 + 0.944994i $$0.606068\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 0 0
$$235$$ −18.0000 −1.17419
$$236$$ 0 0
$$237$$ −22.6274 −1.46981
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ 9.89949 0.635053
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −24.0000 −1.52094
$$250$$ 0 0
$$251$$ 18.3848 1.16044 0.580218 0.814461i $$-0.302967\pi$$
0.580218 + 0.814461i $$0.302967\pi$$
$$252$$ 0 0
$$253$$ −6.00000 −0.377217
$$254$$ 0 0
$$255$$ −33.9411 −2.12548
$$256$$ 0 0
$$257$$ −1.41421 −0.0882162 −0.0441081 0.999027i $$-0.514045\pi$$
−0.0441081 + 0.999027i $$0.514045\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 0 0
$$265$$ −33.9411 −2.08499
$$266$$ 0 0
$$267$$ 10.0000 0.611990
$$268$$ 0 0
$$269$$ −18.3848 −1.12094 −0.560470 0.828175i $$-0.689379\pi$$
−0.560470 + 0.828175i $$0.689379\pi$$
$$270$$ 0 0
$$271$$ −8.48528 −0.515444 −0.257722 0.966219i $$-0.582972\pi$$
−0.257722 + 0.966219i $$0.582972\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 13.0000 0.783929
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 0 0
$$279$$ 1.41421 0.0846668
$$280$$ 0 0
$$281$$ −14.0000 −0.835170 −0.417585 0.908638i $$-0.637123\pi$$
−0.417585 + 0.908638i $$0.637123\pi$$
$$282$$ 0 0
$$283$$ −19.7990 −1.17693 −0.588464 0.808523i $$-0.700267\pi$$
−0.588464 + 0.808523i $$0.700267\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 15.0000 0.882353
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 0 0
$$293$$ 8.48528 0.495715 0.247858 0.968796i $$-0.420273\pi$$
0.247858 + 0.968796i $$0.420273\pi$$
$$294$$ 0 0
$$295$$ −6.00000 −0.349334
$$296$$ 0 0
$$297$$ −5.65685 −0.328244
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −8.00000 −0.459588
$$304$$ 0 0
$$305$$ −12.0000 −0.687118
$$306$$ 0 0
$$307$$ −25.4558 −1.45284 −0.726421 0.687250i $$-0.758818\pi$$
−0.726421 + 0.687250i $$0.758818\pi$$
$$308$$ 0 0
$$309$$ 26.0000 1.47909
$$310$$ 0 0
$$311$$ 18.3848 1.04251 0.521253 0.853402i $$-0.325465\pi$$
0.521253 + 0.853402i $$0.325465\pi$$
$$312$$ 0 0
$$313$$ 12.7279 0.719425 0.359712 0.933063i $$-0.382875\pi$$
0.359712 + 0.933063i $$0.382875\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 0 0
$$319$$ 2.00000 0.111979
$$320$$ 0 0
$$321$$ −22.6274 −1.26294
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −2.82843 −0.156412
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ 10.0000 0.547997
$$334$$ 0 0
$$335$$ 8.48528 0.463600
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 0 0
$$339$$ −2.82843 −0.153619
$$340$$ 0 0
$$341$$ −1.41421 −0.0765840
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 36.0000 1.93817
$$346$$ 0 0
$$347$$ −20.0000 −1.07366 −0.536828 0.843692i $$-0.680378\pi$$
−0.536828 + 0.843692i $$0.680378\pi$$
$$348$$ 0 0
$$349$$ −14.1421 −0.757011 −0.378506 0.925599i $$-0.623562\pi$$
−0.378506 + 0.925599i $$0.623562\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −1.41421 −0.0752710 −0.0376355 0.999292i $$-0.511983\pi$$
−0.0376355 + 0.999292i $$0.511983\pi$$
$$354$$ 0 0
$$355$$ −8.48528 −0.450352
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ 1.41421 0.0742270
$$364$$ 0 0
$$365$$ 36.0000 1.88433
$$366$$ 0 0
$$367$$ −21.2132 −1.10732 −0.553660 0.832743i $$-0.686769\pi$$
−0.553660 + 0.832743i $$0.686769\pi$$
$$368$$ 0 0
$$369$$ −11.3137 −0.588968
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 0 0
$$375$$ −48.0000 −2.47871
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 6.00000 0.308199 0.154100 0.988055i $$-0.450752\pi$$
0.154100 + 0.988055i $$0.450752\pi$$
$$380$$ 0 0
$$381$$ −22.6274 −1.15924
$$382$$ 0 0
$$383$$ 15.5563 0.794892 0.397446 0.917625i $$-0.369897\pi$$
0.397446 + 0.917625i $$0.369897\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −8.00000 −0.406663
$$388$$ 0 0
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ −33.9411 −1.71648
$$392$$ 0 0
$$393$$ −28.0000 −1.41241
$$394$$ 0 0
$$395$$ 67.8823 3.41553
$$396$$ 0 0
$$397$$ −12.7279 −0.638796 −0.319398 0.947621i $$-0.603481\pi$$
−0.319398 + 0.947621i $$0.603481\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 21.2132 1.05409
$$406$$ 0 0
$$407$$ −10.0000 −0.495682
$$408$$ 0 0
$$409$$ 2.82843 0.139857 0.0699284 0.997552i $$-0.477723\pi$$
0.0699284 + 0.997552i $$0.477723\pi$$
$$410$$ 0 0
$$411$$ −25.4558 −1.25564
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 72.0000 3.53434
$$416$$ 0 0
$$417$$ 16.0000 0.783523
$$418$$ 0 0
$$419$$ 24.0416 1.17451 0.587255 0.809402i $$-0.300208\pi$$
0.587255 + 0.809402i $$0.300208\pi$$
$$420$$ 0 0
$$421$$ −20.0000 −0.974740 −0.487370 0.873195i $$-0.662044\pi$$
−0.487370 + 0.873195i $$0.662044\pi$$
$$422$$ 0 0
$$423$$ −4.24264 −0.206284
$$424$$ 0 0
$$425$$ 73.5391 3.56717
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −32.0000 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$432$$ 0 0
$$433$$ 12.7279 0.611665 0.305832 0.952085i $$-0.401065\pi$$
0.305832 + 0.952085i $$0.401065\pi$$
$$434$$ 0 0
$$435$$ −12.0000 −0.575356
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −25.4558 −1.21494 −0.607471 0.794342i $$-0.707816\pi$$
−0.607471 + 0.794342i $$0.707816\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 0 0
$$445$$ −30.0000 −1.42214
$$446$$ 0 0
$$447$$ −14.1421 −0.668900
$$448$$ 0 0
$$449$$ −20.0000 −0.943858 −0.471929 0.881636i $$-0.656442\pi$$
−0.471929 + 0.881636i $$0.656442\pi$$
$$450$$ 0 0
$$451$$ 11.3137 0.532742
$$452$$ 0 0
$$453$$ −5.65685 −0.265782
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 14.0000 0.654892 0.327446 0.944870i $$-0.393812\pi$$
0.327446 + 0.944870i $$0.393812\pi$$
$$458$$ 0 0
$$459$$ −32.0000 −1.49363
$$460$$ 0 0
$$461$$ −2.82843 −0.131733 −0.0658665 0.997828i $$-0.520981\pi$$
−0.0658665 + 0.997828i $$0.520981\pi$$
$$462$$ 0 0
$$463$$ −26.0000 −1.20832 −0.604161 0.796862i $$-0.706492\pi$$
−0.604161 + 0.796862i $$0.706492\pi$$
$$464$$ 0 0
$$465$$ 8.48528 0.393496
$$466$$ 0 0
$$467$$ −4.24264 −0.196326 −0.0981630 0.995170i $$-0.531297\pi$$
−0.0981630 + 0.995170i $$0.531297\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −6.00000 −0.276465
$$472$$ 0 0
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −8.00000 −0.366295
$$478$$ 0 0
$$479$$ 2.82843 0.129234 0.0646171 0.997910i $$-0.479417\pi$$
0.0646171 + 0.997910i $$0.479417\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −42.0000 −1.90712
$$486$$ 0 0
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ 0 0
$$489$$ 14.1421 0.639529
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 11.3137 0.509544
$$494$$ 0 0
$$495$$ 4.24264 0.190693
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −6.00000 −0.268597 −0.134298 0.990941i $$-0.542878\pi$$
−0.134298 + 0.990941i $$0.542878\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ 0 0
$$503$$ 31.1127 1.38725 0.693623 0.720338i $$-0.256013\pi$$
0.693623 + 0.720338i $$0.256013\pi$$
$$504$$ 0 0
$$505$$ 24.0000 1.06799
$$506$$ 0 0
$$507$$ −18.3848 −0.816497
$$508$$ 0 0
$$509$$ −12.7279 −0.564155 −0.282078 0.959392i $$-0.591024\pi$$
−0.282078 + 0.959392i $$0.591024\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −78.0000 −3.43709
$$516$$ 0 0
$$517$$ 4.24264 0.186591
$$518$$ 0 0
$$519$$ 16.0000 0.702322
$$520$$ 0 0
$$521$$ −1.41421 −0.0619578 −0.0309789 0.999520i $$-0.509862\pi$$
−0.0309789 + 0.999520i $$0.509862\pi$$
$$522$$ 0 0
$$523$$ 8.48528 0.371035 0.185518 0.982641i $$-0.440604\pi$$
0.185518 + 0.982641i $$0.440604\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −1.41421 −0.0613716
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 67.8823 2.93481
$$536$$ 0 0
$$537$$ −16.9706 −0.732334
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 0 0
$$543$$ 10.0000 0.429141
$$544$$ 0 0
$$545$$ 8.48528 0.363470
$$546$$ 0 0
$$547$$ −44.0000 −1.88130 −0.940652 0.339372i $$-0.889785\pi$$
−0.940652 + 0.339372i $$0.889785\pi$$
$$548$$ 0 0
$$549$$ −2.82843 −0.120714
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 60.0000 2.54686
$$556$$ 0 0
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ 22.6274 0.953632 0.476816 0.879003i $$-0.341791\pi$$
0.476816 + 0.879003i $$0.341791\pi$$
$$564$$ 0 0
$$565$$ 8.48528 0.356978
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ −8.00000 −0.334790 −0.167395 0.985890i $$-0.553535\pi$$
−0.167395 + 0.985890i $$0.553535\pi$$
$$572$$ 0 0
$$573$$ −22.6274 −0.945274
$$574$$ 0 0
$$575$$ −78.0000 −3.25282
$$576$$ 0 0
$$577$$ 7.07107 0.294372 0.147186 0.989109i $$-0.452978\pi$$
0.147186 + 0.989109i $$0.452978\pi$$
$$578$$ 0 0
$$579$$ −8.48528 −0.352636
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 8.00000 0.331326
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −26.8701 −1.10905 −0.554523 0.832168i $$-0.687099\pi$$
−0.554523 + 0.832168i $$0.687099\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 31.1127 1.27981
$$592$$ 0 0
$$593$$ −42.4264 −1.74224 −0.871122 0.491067i $$-0.836607\pi$$
−0.871122 + 0.491067i $$0.836607\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −2.00000 −0.0818546
$$598$$ 0 0
$$599$$ 6.00000 0.245153 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$600$$ 0 0
$$601$$ −5.65685 −0.230748 −0.115374 0.993322i $$-0.536807\pi$$
−0.115374 + 0.993322i $$0.536807\pi$$
$$602$$ 0 0
$$603$$ 2.00000 0.0814463
$$604$$ 0 0
$$605$$ −4.24264 −0.172488
$$606$$ 0 0
$$607$$ 16.9706 0.688814 0.344407 0.938820i $$-0.388080\pi$$
0.344407 + 0.938820i $$0.388080\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −10.0000 −0.403896 −0.201948 0.979396i $$-0.564727\pi$$
−0.201948 + 0.979396i $$0.564727\pi$$
$$614$$ 0 0
$$615$$ −67.8823 −2.73728
$$616$$ 0 0
$$617$$ 34.0000 1.36879 0.684394 0.729112i $$-0.260067\pi$$
0.684394 + 0.729112i $$0.260067\pi$$
$$618$$ 0 0
$$619$$ 9.89949 0.397894 0.198947 0.980010i $$-0.436248\pi$$
0.198947 + 0.980010i $$0.436248\pi$$
$$620$$ 0 0
$$621$$ 33.9411 1.36201
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 79.0000 3.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −56.5685 −2.25554
$$630$$ 0 0
$$631$$ −24.0000 −0.955425 −0.477712 0.878516i $$-0.658534\pi$$
−0.477712 + 0.878516i $$0.658534\pi$$
$$632$$ 0 0
$$633$$ −11.3137 −0.449680
$$634$$ 0 0
$$635$$ 67.8823 2.69382
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −2.00000 −0.0791188
$$640$$ 0 0
$$641$$ −34.0000 −1.34292 −0.671460 0.741041i $$-0.734332\pi$$
−0.671460 + 0.741041i $$0.734332\pi$$
$$642$$ 0 0
$$643$$ −38.1838 −1.50582 −0.752910 0.658123i $$-0.771351\pi$$
−0.752910 + 0.658123i $$0.771351\pi$$
$$644$$ 0 0
$$645$$ −48.0000 −1.89000
$$646$$ 0 0
$$647$$ −15.5563 −0.611583 −0.305792 0.952098i $$-0.598921\pi$$
−0.305792 + 0.952098i $$0.598921\pi$$
$$648$$ 0 0
$$649$$ 1.41421 0.0555127
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 12.0000 0.469596 0.234798 0.972044i $$-0.424557\pi$$
0.234798 + 0.972044i $$0.424557\pi$$
$$654$$ 0 0
$$655$$ 84.0000 3.28215
$$656$$ 0 0
$$657$$ 8.48528 0.331042
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −12.7279 −0.495059 −0.247529 0.968880i $$-0.579619\pi$$
−0.247529 + 0.968880i $$0.579619\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −12.0000 −0.464642
$$668$$ 0 0
$$669$$ −30.0000 −1.15987
$$670$$ 0 0
$$671$$ 2.82843 0.109190
$$672$$ 0 0
$$673$$ −22.0000 −0.848038 −0.424019 0.905653i $$-0.639381\pi$$
−0.424019 + 0.905653i $$0.639381\pi$$
$$674$$ 0 0
$$675$$ −73.5391 −2.83052
$$676$$ 0 0
$$677$$ 39.5980 1.52187 0.760937 0.648826i $$-0.224740\pi$$
0.760937 + 0.648826i $$0.224740\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 0 0
$$683$$ 28.0000 1.07139 0.535695 0.844411i $$-0.320050\pi$$
0.535695 + 0.844411i $$0.320050\pi$$
$$684$$ 0 0
$$685$$ 76.3675 2.91785
$$686$$ 0 0
$$687$$ −14.0000 −0.534133
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 15.5563 0.591791 0.295896 0.955220i $$-0.404382\pi$$
0.295896 + 0.955220i $$0.404382\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −48.0000 −1.82074
$$696$$ 0 0
$$697$$ 64.0000 2.42417
$$698$$ 0 0
$$699$$ 19.7990 0.748867
$$700$$ 0 0
$$701$$ 50.0000 1.88847 0.944237 0.329267i $$-0.106802\pi$$
0.944237 + 0.329267i $$0.106802\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −25.4558 −0.958723
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −20.0000 −0.751116 −0.375558 0.926799i $$-0.622549\pi$$
−0.375558 + 0.926799i $$0.622549\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ 0 0
$$713$$ 8.48528 0.317776
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 16.9706 0.633777
$$718$$ 0 0
$$719$$ 18.3848 0.685636 0.342818 0.939402i $$-0.388619\pi$$
0.342818 + 0.939402i $$0.388619\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 26.0000 0.965616
$$726$$ 0 0
$$727$$ 12.7279 0.472052 0.236026 0.971747i $$-0.424155\pi$$
0.236026 + 0.971747i $$0.424155\pi$$
$$728$$ 0 0
$$729$$ 29.0000 1.07407
$$730$$ 0 0
$$731$$ 45.2548 1.67381
$$732$$ 0 0
$$733$$ −14.1421 −0.522352 −0.261176 0.965291i $$-0.584110\pi$$
−0.261176 + 0.965291i $$0.584110\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −2.00000 −0.0736709
$$738$$ 0 0
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 44.0000 1.61420 0.807102 0.590412i $$-0.201035\pi$$
0.807102 + 0.590412i $$0.201035\pi$$
$$744$$ 0 0
$$745$$ 42.4264 1.55438
$$746$$ 0 0
$$747$$ 16.9706 0.620920
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 14.0000 0.510867 0.255434 0.966827i $$-0.417782\pi$$
0.255434 + 0.966827i $$0.417782\pi$$
$$752$$ 0 0
$$753$$ 26.0000 0.947493
$$754$$ 0 0
$$755$$ 16.9706 0.617622
$$756$$ 0 0
$$757$$ −8.00000 −0.290765 −0.145382 0.989376i $$-0.546441\pi$$
−0.145382 + 0.989376i $$0.546441\pi$$
$$758$$ 0 0
$$759$$ −8.48528 −0.307996
$$760$$ 0 0
$$761$$ 42.4264 1.53796 0.768978 0.639275i $$-0.220766\pi$$
0.768978 + 0.639275i $$0.220766\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 24.0000 0.867722
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −19.7990 −0.713970 −0.356985 0.934110i $$-0.616195\pi$$
−0.356985 + 0.934110i $$0.616195\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ 0 0
$$773$$ −1.41421 −0.0508657 −0.0254329 0.999677i $$-0.508096\pi$$
−0.0254329 + 0.999677i $$0.508096\pi$$
$$774$$ 0 0
$$775$$ −18.3848 −0.660401
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 2.00000 0.0715656
$$782$$ 0 0
$$783$$ −11.3137 −0.404319
$$784$$ 0 0
$$785$$ 18.0000 0.642448
$$786$$ 0 0
$$787$$ −5.65685 −0.201645 −0.100823 0.994904i $$-0.532147\pi$$
−0.100823 + 0.994904i $$0.532147\pi$$
$$788$$ 0 0
$$789$$ 11.3137 0.402779
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −48.0000 −1.70238
$$796$$ 0 0
$$797$$ −7.07107 −0.250470 −0.125235 0.992127i $$-0.539968\pi$$
−0.125235 + 0.992127i $$0.539968\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ −7.07107 −0.249844
$$802$$ 0 0
$$803$$ −8.48528 −0.299439
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −26.0000 −0.915243
$$808$$ 0 0
$$809$$ 54.0000 1.89854 0.949269 0.314464i $$-0.101825\pi$$
0.949269 + 0.314464i $$0.101825\pi$$
$$810$$ 0 0
$$811$$ 36.7696 1.29115 0.645577 0.763695i $$-0.276617\pi$$
0.645577 + 0.763695i $$0.276617\pi$$
$$812$$ 0 0
$$813$$ −12.0000 −0.420858
$$814$$ 0 0
$$815$$ −42.4264 −1.48613
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ 0 0
$$823$$ −40.0000 −1.39431 −0.697156 0.716919i $$-0.745552\pi$$
−0.697156 + 0.716919i $$0.745552\pi$$
$$824$$ 0 0
$$825$$ 18.3848 0.640076
$$826$$ 0 0
$$827$$ 32.0000 1.11275 0.556375 0.830932i $$-0.312192\pi$$
0.556375 + 0.830932i $$0.312192\pi$$
$$828$$ 0 0
$$829$$ 4.24264 0.147353 0.0736765 0.997282i $$-0.476527\pi$$
0.0736765 + 0.997282i $$0.476527\pi$$
$$830$$ 0 0
$$831$$ −2.82843 −0.0981170
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −24.0000 −0.830554
$$836$$ 0 0
$$837$$ 8.00000 0.276520
$$838$$ 0 0
$$839$$ 1.41421 0.0488241 0.0244120 0.999702i $$-0.492229\pi$$
0.0244120 + 0.999702i $$0.492229\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −19.7990 −0.681913
$$844$$ 0 0
$$845$$ 55.1543 1.89737
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −28.0000 −0.960958
$$850$$ 0 0
$$851$$ 60.0000 2.05677
$$852$$ 0 0
$$853$$ −28.2843 −0.968435 −0.484218 0.874948i $$-0.660896\pi$$
−0.484218 + 0.874948i $$0.660896\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 25.4558 0.869555 0.434778 0.900538i $$-0.356827\pi$$
0.434778 + 0.900538i $$0.356827\pi$$
$$858$$ 0 0
$$859$$ 12.7279 0.434271 0.217136 0.976141i $$-0.430329\pi$$
0.217136 + 0.976141i $$0.430329\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ −48.0000 −1.63205
$$866$$ 0 0
$$867$$ 21.2132 0.720438
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −9.89949 −0.335047
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 34.0000 1.14810 0.574049 0.818821i $$-0.305372\pi$$
0.574049 + 0.818821i $$0.305372\pi$$
$$878$$ 0 0
$$879$$ 12.0000 0.404750
$$880$$ 0 0
$$881$$ 29.6985 1.00057 0.500284 0.865862i $$-0.333229\pi$$
0.500284 + 0.865862i $$0.333229\pi$$
$$882$$ 0 0
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ 0 0
$$885$$ −8.48528 −0.285230
$$886$$ 0 0
$$887$$ 36.7696 1.23460 0.617300 0.786728i $$-0.288226\pi$$
0.617300 + 0.786728i $$0.288226\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −5.00000 −0.167506
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 50.9117 1.70179
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −2.82843 −0.0943333
$$900$$ 0 0
$$901$$ 45.2548 1.50766
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −30.0000 −0.997234
$$906$$ 0 0
$$907$$ 14.0000 0.464862 0.232431 0.972613i $$-0.425332\pi$$
0.232431 + 0.972613i $$0.425332\pi$$
$$908$$ 0 0
$$909$$ 5.65685 0.187626
$$910$$ 0 0
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ 0 0
$$913$$ −16.9706 −0.561644
$$914$$ 0 0
$$915$$ −16.9706 −0.561029
$$916$$ 0 0
$$917$$ 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
$$921$$ −36.0000 −1.18624
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −130.000 −4.27437
$$926$$ 0 0
$$927$$ −18.3848 −0.603835
$$928$$ 0 0
$$929$$ 32.5269 1.06717 0.533587 0.845745i $$-0.320844\pi$$
0.533587 + 0.845745i $$0.320844\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 26.0000 0.851202
$$934$$ 0 0
$$935$$ −24.0000 −0.784884
$$936$$ 0 0
$$937$$ −25.4558 −0.831606 −0.415803 0.909455i $$-0.636499\pi$$
−0.415803 + 0.909455i $$0.636499\pi$$
$$938$$ 0 0
$$939$$ 18.0000 0.587408
$$940$$ 0 0
$$941$$ −31.1127 −1.01424 −0.507122 0.861874i $$-0.669291\pi$$
−0.507122 + 0.861874i $$0.669291\pi$$
$$942$$ 0 0
$$943$$ −67.8823 −2.21055
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 42.4264 1.37577
$$952$$ 0 0
$$953$$ 14.0000 0.453504 0.226752 0.973952i $$-0.427189\pi$$
0.226752 + 0.973952i $$0.427189\pi$$
$$954$$ 0 0
$$955$$ 67.8823 2.19662
$$956$$ 0 0
$$957$$ 2.82843 0.0914301
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −29.0000 −0.935484
$$962$$ 0 0
$$963$$ 16.0000 0.515593
$$964$$ 0 0
$$965$$ 25.4558 0.819453
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 52.3259 1.67922 0.839609 0.543191i $$-0.182784\pi$$
0.839609 + 0.543191i $$0.182784\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 52.0000 1.66363 0.831814 0.555055i $$-0.187303\pi$$
0.831814 + 0.555055i $$0.187303\pi$$
$$978$$ 0 0
$$979$$ 7.07107 0.225992
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ 1.41421 0.0451064 0.0225532 0.999746i $$-0.492820\pi$$
0.0225532 + 0.999746i $$0.492820\pi$$
$$984$$ 0 0
$$985$$ −93.3381 −2.97400
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −48.0000 −1.52631
$$990$$ 0 0
$$991$$ 46.0000 1.46124 0.730619 0.682785i $$-0.239232\pi$$
0.730619 + 0.682785i $$0.239232\pi$$
$$992$$ 0 0
$$993$$ −28.2843 −0.897574
$$994$$ 0 0
$$995$$ 6.00000 0.190213
$$996$$ 0 0
$$997$$ −16.9706 −0.537463 −0.268732 0.963215i $$-0.586604\pi$$
−0.268732 + 0.963215i $$0.586604\pi$$
$$998$$ 0 0
$$999$$ 56.5685 1.78975
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 8624.2.a.bs.1.2 2
4.3 odd 2 1078.2.a.u.1.1 2
7.6 odd 2 inner 8624.2.a.bs.1.1 2
12.11 even 2 9702.2.a.cp.1.2 2
28.3 even 6 1078.2.e.p.177.1 4
28.11 odd 6 1078.2.e.p.177.2 4
28.19 even 6 1078.2.e.p.67.1 4
28.23 odd 6 1078.2.e.p.67.2 4
28.27 even 2 1078.2.a.u.1.2 yes 2
84.83 odd 2 9702.2.a.cp.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.u.1.1 2 4.3 odd 2
1078.2.a.u.1.2 yes 2 28.27 even 2
1078.2.e.p.67.1 4 28.19 even 6
1078.2.e.p.67.2 4 28.23 odd 6
1078.2.e.p.177.1 4 28.3 even 6
1078.2.e.p.177.2 4 28.11 odd 6
8624.2.a.bs.1.1 2 7.6 odd 2 inner
8624.2.a.bs.1.2 2 1.1 even 1 trivial
9702.2.a.cp.1.1 2 84.83 odd 2
9702.2.a.cp.1.2 2 12.11 even 2