# Properties

 Label 72.8.a.d.1.1 Level $72$ Weight $8$ Character 72.1 Self dual yes Analytic conductor $22.492$ Analytic rank $1$ 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] = [72,8,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 72.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: $$22.4917218349$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 72.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+82.0000 q^{5} -456.000 q^{7} +O(q^{10})$$ $$q+82.0000 q^{5} -456.000 q^{7} +2524.00 q^{11} -10778.0 q^{13} +11150.0 q^{17} +4124.00 q^{19} -81704.0 q^{23} -71401.0 q^{25} -99798.0 q^{29} -40480.0 q^{31} -37392.0 q^{35} -419442. q^{37} -141402. q^{41} -690428. q^{43} +682032. q^{47} -615607. q^{49} -1.81312e6 q^{53} +206968. q^{55} +966028. q^{59} +1.88767e6 q^{61} -883796. q^{65} +2.96587e6 q^{67} +2.54823e6 q^{71} -1.68033e6 q^{73} -1.15094e6 q^{77} +4.03806e6 q^{79} +5.38576e6 q^{83} +914300. q^{85} +6.47305e6 q^{89} +4.91477e6 q^{91} +338168. q^{95} -6.06576e6 q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 82.0000 0.293372 0.146686 0.989183i $$-0.453139\pi$$
0.146686 + 0.989183i $$0.453139\pi$$
$$6$$ 0 0
$$7$$ −456.000 −0.502483 −0.251242 0.967924i $$-0.580839\pi$$
−0.251242 + 0.967924i $$0.580839\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2524.00 0.571762 0.285881 0.958265i $$-0.407714\pi$$
0.285881 + 0.958265i $$0.407714\pi$$
$$12$$ 0 0
$$13$$ −10778.0 −1.36062 −0.680309 0.732925i $$-0.738155\pi$$
−0.680309 + 0.732925i $$0.738155\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 11150.0 0.550432 0.275216 0.961382i $$-0.411251\pi$$
0.275216 + 0.961382i $$0.411251\pi$$
$$18$$ 0 0
$$19$$ 4124.00 0.137937 0.0689685 0.997619i $$-0.478029\pi$$
0.0689685 + 0.997619i $$0.478029\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −81704.0 −1.40022 −0.700109 0.714036i $$-0.746865\pi$$
−0.700109 + 0.714036i $$0.746865\pi$$
$$24$$ 0 0
$$25$$ −71401.0 −0.913933
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −99798.0 −0.759852 −0.379926 0.925017i $$-0.624051\pi$$
−0.379926 + 0.925017i $$0.624051\pi$$
$$30$$ 0 0
$$31$$ −40480.0 −0.244048 −0.122024 0.992527i $$-0.538938\pi$$
−0.122024 + 0.992527i $$0.538938\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −37392.0 −0.147415
$$36$$ 0 0
$$37$$ −419442. −1.36134 −0.680669 0.732591i $$-0.738311\pi$$
−0.680669 + 0.732591i $$0.738311\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −141402. −0.320414 −0.160207 0.987083i $$-0.551216\pi$$
−0.160207 + 0.987083i $$0.551216\pi$$
$$42$$ 0 0
$$43$$ −690428. −1.32428 −0.662138 0.749382i $$-0.730351\pi$$
−0.662138 + 0.749382i $$0.730351\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 682032. 0.958213 0.479107 0.877757i $$-0.340961\pi$$
0.479107 + 0.877757i $$0.340961\pi$$
$$48$$ 0 0
$$49$$ −615607. −0.747510
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −1.81312e6 −1.67286 −0.836432 0.548071i $$-0.815362\pi$$
−0.836432 + 0.548071i $$0.815362\pi$$
$$54$$ 0 0
$$55$$ 206968. 0.167739
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 966028. 0.612361 0.306181 0.951973i $$-0.400949\pi$$
0.306181 + 0.951973i $$0.400949\pi$$
$$60$$ 0 0
$$61$$ 1.88767e6 1.06481 0.532404 0.846490i $$-0.321289\pi$$
0.532404 + 0.846490i $$0.321289\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −883796. −0.399168
$$66$$ 0 0
$$67$$ 2.96587e6 1.20473 0.602365 0.798220i $$-0.294225\pi$$
0.602365 + 0.798220i $$0.294225\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 2.54823e6 0.844957 0.422479 0.906373i $$-0.361160\pi$$
0.422479 + 0.906373i $$0.361160\pi$$
$$72$$ 0 0
$$73$$ −1.68033e6 −0.505549 −0.252775 0.967525i $$-0.581343\pi$$
−0.252775 + 0.967525i $$0.581343\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1.15094e6 −0.287301
$$78$$ 0 0
$$79$$ 4.03806e6 0.921464 0.460732 0.887539i $$-0.347587\pi$$
0.460732 + 0.887539i $$0.347587\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 5.38576e6 1.03389 0.516945 0.856019i $$-0.327069\pi$$
0.516945 + 0.856019i $$0.327069\pi$$
$$84$$ 0 0
$$85$$ 914300. 0.161481
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 6.47305e6 0.973293 0.486647 0.873599i $$-0.338220\pi$$
0.486647 + 0.873599i $$0.338220\pi$$
$$90$$ 0 0
$$91$$ 4.91477e6 0.683688
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 338168. 0.0404669
$$96$$ 0 0
$$97$$ −6.06576e6 −0.674814 −0.337407 0.941359i $$-0.609550\pi$$
−0.337407 + 0.941359i $$0.609550\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −9.70069e6 −0.936866 −0.468433 0.883499i $$-0.655181\pi$$
−0.468433 + 0.883499i $$0.655181\pi$$
$$102$$ 0 0
$$103$$ 4.10159e6 0.369847 0.184924 0.982753i $$-0.440796\pi$$
0.184924 + 0.982753i $$0.440796\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −72900.0 −0.00575287 −0.00287643 0.999996i $$-0.500916\pi$$
−0.00287643 + 0.999996i $$0.500916\pi$$
$$108$$ 0 0
$$109$$ 9.55841e6 0.706957 0.353478 0.935443i $$-0.384999\pi$$
0.353478 + 0.935443i $$0.384999\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −9.33890e6 −0.608865 −0.304433 0.952534i $$-0.598467\pi$$
−0.304433 + 0.952534i $$0.598467\pi$$
$$114$$ 0 0
$$115$$ −6.69973e6 −0.410785
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −5.08440e6 −0.276583
$$120$$ 0 0
$$121$$ −1.31166e7 −0.673089
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.22611e7 −0.561495
$$126$$ 0 0
$$127$$ −3.59794e7 −1.55862 −0.779311 0.626637i $$-0.784431\pi$$
−0.779311 + 0.626637i $$0.784431\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 676052. 0.0262743 0.0131371 0.999914i $$-0.495818\pi$$
0.0131371 + 0.999914i $$0.495818\pi$$
$$132$$ 0 0
$$133$$ −1.88054e6 −0.0693111
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.95841e7 0.982962 0.491481 0.870888i $$-0.336456\pi$$
0.491481 + 0.870888i $$0.336456\pi$$
$$138$$ 0 0
$$139$$ −3.19084e7 −1.00775 −0.503876 0.863776i $$-0.668093\pi$$
−0.503876 + 0.863776i $$0.668093\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −2.72037e7 −0.777949
$$144$$ 0 0
$$145$$ −8.18344e6 −0.222919
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 1.16603e7 0.288773 0.144386 0.989521i $$-0.453879\pi$$
0.144386 + 0.989521i $$0.453879\pi$$
$$150$$ 0 0
$$151$$ −1.76295e7 −0.416698 −0.208349 0.978055i $$-0.566809\pi$$
−0.208349 + 0.978055i $$0.566809\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −3.31936e6 −0.0715968
$$156$$ 0 0
$$157$$ 6.34658e6 0.130885 0.0654427 0.997856i $$-0.479154\pi$$
0.0654427 + 0.997856i $$0.479154\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.72570e7 0.703587
$$162$$ 0 0
$$163$$ 8.04234e7 1.45454 0.727271 0.686351i $$-0.240789\pi$$
0.727271 + 0.686351i $$0.240789\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1.14767e8 −1.90682 −0.953411 0.301673i $$-0.902455\pi$$
−0.953411 + 0.301673i $$0.902455\pi$$
$$168$$ 0 0
$$169$$ 5.34168e7 0.851283
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 6.33755e7 0.930594 0.465297 0.885155i $$-0.345947\pi$$
0.465297 + 0.885155i $$0.345947\pi$$
$$174$$ 0 0
$$175$$ 3.25589e7 0.459236
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 1.13228e7 0.147559 0.0737796 0.997275i $$-0.476494\pi$$
0.0737796 + 0.997275i $$0.476494\pi$$
$$180$$ 0 0
$$181$$ −5.22650e6 −0.0655143 −0.0327571 0.999463i $$-0.510429\pi$$
−0.0327571 + 0.999463i $$0.510429\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −3.43942e7 −0.399379
$$186$$ 0 0
$$187$$ 2.81426e7 0.314716
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 8.50301e7 0.882990 0.441495 0.897264i $$-0.354448\pi$$
0.441495 + 0.897264i $$0.354448\pi$$
$$192$$ 0 0
$$193$$ 1.15092e8 1.15237 0.576186 0.817319i $$-0.304540\pi$$
0.576186 + 0.817319i $$0.304540\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1.38522e8 1.29088 0.645441 0.763810i $$-0.276674\pi$$
0.645441 + 0.763810i $$0.276674\pi$$
$$198$$ 0 0
$$199$$ −2.19614e7 −0.197548 −0.0987742 0.995110i $$-0.531492\pi$$
−0.0987742 + 0.995110i $$0.531492\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 4.55079e7 0.381813
$$204$$ 0 0
$$205$$ −1.15950e7 −0.0940007
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1.04090e7 0.0788671
$$210$$ 0 0
$$211$$ −6.10208e7 −0.447187 −0.223594 0.974682i $$-0.571779\pi$$
−0.223594 + 0.974682i $$0.571779\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −5.66151e7 −0.388506
$$216$$ 0 0
$$217$$ 1.84589e7 0.122630
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1.20175e8 −0.748928
$$222$$ 0 0
$$223$$ −4.22448e7 −0.255098 −0.127549 0.991832i $$-0.540711\pi$$
−0.127549 + 0.991832i $$0.540711\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2.39102e8 1.35673 0.678364 0.734726i $$-0.262689\pi$$
0.678364 + 0.734726i $$0.262689\pi$$
$$228$$ 0 0
$$229$$ −4.67889e7 −0.257465 −0.128733 0.991679i $$-0.541091\pi$$
−0.128733 + 0.991679i $$0.541091\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3.45225e8 −1.78795 −0.893977 0.448113i $$-0.852096\pi$$
−0.893977 + 0.448113i $$0.852096\pi$$
$$234$$ 0 0
$$235$$ 5.59266e7 0.281113
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −2.34413e8 −1.11068 −0.555340 0.831624i $$-0.687412\pi$$
−0.555340 + 0.831624i $$0.687412\pi$$
$$240$$ 0 0
$$241$$ −1.09557e8 −0.504175 −0.252087 0.967705i $$-0.581117\pi$$
−0.252087 + 0.967705i $$0.581117\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −5.04798e7 −0.219299
$$246$$ 0 0
$$247$$ −4.44485e7 −0.187680
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 3.94031e8 1.57280 0.786398 0.617720i $$-0.211943\pi$$
0.786398 + 0.617720i $$0.211943\pi$$
$$252$$ 0 0
$$253$$ −2.06221e8 −0.800591
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −3.19064e8 −1.17250 −0.586248 0.810131i $$-0.699396\pi$$
−0.586248 + 0.810131i $$0.699396\pi$$
$$258$$ 0 0
$$259$$ 1.91266e8 0.684050
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −2.19359e8 −0.743549 −0.371774 0.928323i $$-0.621250\pi$$
−0.371774 + 0.928323i $$0.621250\pi$$
$$264$$ 0 0
$$265$$ −1.48676e8 −0.490772
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 1.48033e8 0.463687 0.231844 0.972753i $$-0.425524\pi$$
0.231844 + 0.972753i $$0.425524\pi$$
$$270$$ 0 0
$$271$$ −3.69934e8 −1.12910 −0.564549 0.825399i $$-0.690950\pi$$
−0.564549 + 0.825399i $$0.690950\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.80216e8 −0.522552
$$276$$ 0 0
$$277$$ −3.95860e8 −1.11908 −0.559541 0.828803i $$-0.689023\pi$$
−0.559541 + 0.828803i $$0.689023\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 5.97760e8 1.60714 0.803572 0.595208i $$-0.202930\pi$$
0.803572 + 0.595208i $$0.202930\pi$$
$$282$$ 0 0
$$283$$ 8.05797e7 0.211336 0.105668 0.994401i $$-0.466302\pi$$
0.105668 + 0.994401i $$0.466302\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.44793e7 0.161003
$$288$$ 0 0
$$289$$ −2.86016e8 −0.697025
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −7.54530e8 −1.75243 −0.876213 0.481924i $$-0.839938\pi$$
−0.876213 + 0.481924i $$0.839938\pi$$
$$294$$ 0 0
$$295$$ 7.92143e7 0.179650
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 8.80606e8 1.90516
$$300$$ 0 0
$$301$$ 3.14835e8 0.665427
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 1.54789e8 0.312385
$$306$$ 0 0
$$307$$ 8.20472e8 1.61838 0.809188 0.587549i $$-0.199907\pi$$
0.809188 + 0.587549i $$0.199907\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6.53503e8 −1.23193 −0.615965 0.787773i $$-0.711234\pi$$
−0.615965 + 0.787773i $$0.711234\pi$$
$$312$$ 0 0
$$313$$ 6.63587e8 1.22319 0.611594 0.791172i $$-0.290529\pi$$
0.611594 + 0.791172i $$0.290529\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3.54718e8 0.625426 0.312713 0.949848i $$-0.398762\pi$$
0.312713 + 0.949848i $$0.398762\pi$$
$$318$$ 0 0
$$319$$ −2.51890e8 −0.434454
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4.59826e7 0.0759250
$$324$$ 0 0
$$325$$ 7.69560e8 1.24351
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −3.11007e8 −0.481486
$$330$$ 0 0
$$331$$ −3.05543e8 −0.463100 −0.231550 0.972823i $$-0.574380\pi$$
−0.231550 + 0.972823i $$0.574380\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 2.43201e8 0.353434
$$336$$ 0 0
$$337$$ 3.54965e7 0.0505220 0.0252610 0.999681i $$-0.491958\pi$$
0.0252610 + 0.999681i $$0.491958\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −1.02172e8 −0.139537
$$342$$ 0 0
$$343$$ 6.56252e8 0.878095
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 1.90594e8 0.244882 0.122441 0.992476i $$-0.460928\pi$$
0.122441 + 0.992476i $$0.460928\pi$$
$$348$$ 0 0
$$349$$ 8.60864e8 1.08404 0.542020 0.840366i $$-0.317660\pi$$
0.542020 + 0.840366i $$0.317660\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 1.04544e9 1.26500 0.632498 0.774562i $$-0.282030\pi$$
0.632498 + 0.774562i $$0.282030\pi$$
$$354$$ 0 0
$$355$$ 2.08955e8 0.247887
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −7.63303e8 −0.870696 −0.435348 0.900262i $$-0.643375\pi$$
−0.435348 + 0.900262i $$0.643375\pi$$
$$360$$ 0 0
$$361$$ −8.76864e8 −0.980973
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1.37787e8 −0.148314
$$366$$ 0 0
$$367$$ −1.38692e9 −1.46460 −0.732302 0.680980i $$-0.761554\pi$$
−0.732302 + 0.680980i $$0.761554\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 8.26782e8 0.840586
$$372$$ 0 0
$$373$$ 4.77105e8 0.476029 0.238015 0.971262i $$-0.423503\pi$$
0.238015 + 0.971262i $$0.423503\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1.07562e9 1.03387
$$378$$ 0 0
$$379$$ −3.92468e8 −0.370311 −0.185156 0.982709i $$-0.559279\pi$$
−0.185156 + 0.982709i $$0.559279\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −2.10409e9 −1.91368 −0.956839 0.290617i $$-0.906139\pi$$
−0.956839 + 0.290617i $$0.906139\pi$$
$$384$$ 0 0
$$385$$ −9.43774e7 −0.0842860
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 1.26019e9 1.08546 0.542730 0.839907i $$-0.317391\pi$$
0.542730 + 0.839907i $$0.317391\pi$$
$$390$$ 0 0
$$391$$ −9.11000e8 −0.770725
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 3.31121e8 0.270332
$$396$$ 0 0
$$397$$ −9.81298e8 −0.787107 −0.393554 0.919302i $$-0.628754\pi$$
−0.393554 + 0.919302i $$0.628754\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −9.09981e8 −0.704737 −0.352369 0.935861i $$-0.614624\pi$$
−0.352369 + 0.935861i $$0.614624\pi$$
$$402$$ 0 0
$$403$$ 4.36293e8 0.332056
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1.05867e9 −0.778361
$$408$$ 0 0
$$409$$ −3.55609e7 −0.0257004 −0.0128502 0.999917i $$-0.504090\pi$$
−0.0128502 + 0.999917i $$0.504090\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −4.40509e8 −0.307701
$$414$$ 0 0
$$415$$ 4.41633e8 0.303314
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −2.65360e9 −1.76233 −0.881163 0.472813i $$-0.843239\pi$$
−0.881163 + 0.472813i $$0.843239\pi$$
$$420$$ 0 0
$$421$$ −1.12113e9 −0.732264 −0.366132 0.930563i $$-0.619318\pi$$
−0.366132 + 0.930563i $$0.619318\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −7.96121e8 −0.503058
$$426$$ 0 0
$$427$$ −8.60778e8 −0.535049
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 1.06344e9 0.639799 0.319900 0.947451i $$-0.396351\pi$$
0.319900 + 0.947451i $$0.396351\pi$$
$$432$$ 0 0
$$433$$ −7.05962e8 −0.417901 −0.208951 0.977926i $$-0.567005\pi$$
−0.208951 + 0.977926i $$0.567005\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −3.36947e8 −0.193142
$$438$$ 0 0
$$439$$ 1.48506e9 0.837760 0.418880 0.908042i $$-0.362423\pi$$
0.418880 + 0.908042i $$0.362423\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 7.22153e8 0.394654 0.197327 0.980338i $$-0.436774\pi$$
0.197327 + 0.980338i $$0.436774\pi$$
$$444$$ 0 0
$$445$$ 5.30790e8 0.285537
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 1.22968e9 0.641109 0.320554 0.947230i $$-0.396131\pi$$
0.320554 + 0.947230i $$0.396131\pi$$
$$450$$ 0 0
$$451$$ −3.56899e8 −0.183201
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 4.03011e8 0.200575
$$456$$ 0 0
$$457$$ −8.85551e7 −0.0434017 −0.0217009 0.999765i $$-0.506908\pi$$
−0.0217009 + 0.999765i $$0.506908\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −2.10937e8 −0.100277 −0.0501384 0.998742i $$-0.515966\pi$$
−0.0501384 + 0.998742i $$0.515966\pi$$
$$462$$ 0 0
$$463$$ −3.29775e9 −1.54413 −0.772066 0.635543i $$-0.780776\pi$$
−0.772066 + 0.635543i $$0.780776\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −8.82873e7 −0.0401134 −0.0200567 0.999799i $$-0.506385\pi$$
−0.0200567 + 0.999799i $$0.506385\pi$$
$$468$$ 0 0
$$469$$ −1.35244e9 −0.605357
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −1.74264e9 −0.757171
$$474$$ 0 0
$$475$$ −2.94458e8 −0.126065
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 4.51507e9 1.87711 0.938557 0.345125i $$-0.112164\pi$$
0.938557 + 0.345125i $$0.112164\pi$$
$$480$$ 0 0
$$481$$ 4.52075e9 1.85226
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −4.97392e8 −0.197972
$$486$$ 0 0
$$487$$ 3.31338e9 1.29993 0.649964 0.759965i $$-0.274784\pi$$
0.649964 + 0.759965i $$0.274784\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 4.01694e9 1.53147 0.765737 0.643154i $$-0.222374\pi$$
0.765737 + 0.643154i $$0.222374\pi$$
$$492$$ 0 0
$$493$$ −1.11275e9 −0.418247
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −1.16199e9 −0.424577
$$498$$ 0 0
$$499$$ 2.70976e9 0.976290 0.488145 0.872763i $$-0.337674\pi$$
0.488145 + 0.872763i $$0.337674\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −3.04579e8 −0.106712 −0.0533558 0.998576i $$-0.516992\pi$$
−0.0533558 + 0.998576i $$0.516992\pi$$
$$504$$ 0 0
$$505$$ −7.95456e8 −0.274850
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 1.88202e8 0.0632575 0.0316287 0.999500i $$-0.489931\pi$$
0.0316287 + 0.999500i $$0.489931\pi$$
$$510$$ 0 0
$$511$$ 7.66229e8 0.254030
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 3.36331e8 0.108503
$$516$$ 0 0
$$517$$ 1.72145e9 0.547870
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −4.14963e9 −1.28552 −0.642758 0.766069i $$-0.722210\pi$$
−0.642758 + 0.766069i $$0.722210\pi$$
$$522$$ 0 0
$$523$$ 2.51360e9 0.768318 0.384159 0.923267i $$-0.374491\pi$$
0.384159 + 0.923267i $$0.374491\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −4.51352e8 −0.134332
$$528$$ 0 0
$$529$$ 3.27072e9 0.960613
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 1.52403e9 0.435962
$$534$$ 0 0
$$535$$ −5.97780e6 −0.00168773
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −1.55379e9 −0.427398
$$540$$ 0 0
$$541$$ −1.32416e9 −0.359543 −0.179772 0.983708i $$-0.557536\pi$$
−0.179772 + 0.983708i $$0.557536\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 7.83789e8 0.207401
$$546$$ 0 0
$$547$$ 5.58047e8 0.145786 0.0728929 0.997340i $$-0.476777\pi$$
0.0728929 + 0.997340i $$0.476777\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −4.11567e8 −0.104812
$$552$$ 0 0
$$553$$ −1.84136e9 −0.463020
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 3.30331e9 0.809946 0.404973 0.914329i $$-0.367281\pi$$
0.404973 + 0.914329i $$0.367281\pi$$
$$558$$ 0 0
$$559$$ 7.44143e9 1.80184
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 1.22011e8 0.0288152 0.0144076 0.999896i $$-0.495414\pi$$
0.0144076 + 0.999896i $$0.495414\pi$$
$$564$$ 0 0
$$565$$ −7.65790e8 −0.178624
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −5.00925e8 −0.113993 −0.0569967 0.998374i $$-0.518152\pi$$
−0.0569967 + 0.998374i $$0.518152\pi$$
$$570$$ 0 0
$$571$$ −6.98702e9 −1.57060 −0.785300 0.619116i $$-0.787491\pi$$
−0.785300 + 0.619116i $$0.787491\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 5.83375e9 1.27971
$$576$$ 0 0
$$577$$ −8.16573e9 −1.76962 −0.884809 0.465954i $$-0.845711\pi$$
−0.884809 + 0.465954i $$0.845711\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −2.45591e9 −0.519512
$$582$$ 0 0
$$583$$ −4.57631e9 −0.956479
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −8.53182e9 −1.74104 −0.870519 0.492135i $$-0.836217\pi$$
−0.870519 + 0.492135i $$0.836217\pi$$
$$588$$ 0 0
$$589$$ −1.66940e8 −0.0336632
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 1.71175e9 0.337092 0.168546 0.985694i $$-0.446093\pi$$
0.168546 + 0.985694i $$0.446093\pi$$
$$594$$ 0 0
$$595$$ −4.16921e8 −0.0811417
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 4.77362e9 0.907516 0.453758 0.891125i $$-0.350083\pi$$
0.453758 + 0.891125i $$0.350083\pi$$
$$600$$ 0 0
$$601$$ 7.89998e8 0.148445 0.0742224 0.997242i $$-0.476353\pi$$
0.0742224 + 0.997242i $$0.476353\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −1.07556e9 −0.197465
$$606$$ 0 0
$$607$$ −1.82652e9 −0.331485 −0.165743 0.986169i $$-0.553002\pi$$
−0.165743 + 0.986169i $$0.553002\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −7.35094e9 −1.30376
$$612$$ 0 0
$$613$$ −6.90339e9 −1.21046 −0.605231 0.796050i $$-0.706919\pi$$
−0.605231 + 0.796050i $$0.706919\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 5.69235e9 0.975649 0.487825 0.872942i $$-0.337791\pi$$
0.487825 + 0.872942i $$0.337791\pi$$
$$618$$ 0 0
$$619$$ −4.28594e9 −0.726321 −0.363161 0.931727i $$-0.618302\pi$$
−0.363161 + 0.931727i $$0.618302\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −2.95171e9 −0.489064
$$624$$ 0 0
$$625$$ 4.57279e9 0.749206
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −4.67678e9 −0.749324
$$630$$ 0 0
$$631$$ −5.61602e8 −0.0889869 −0.0444935 0.999010i $$-0.514167\pi$$
−0.0444935 + 0.999010i $$0.514167\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −2.95031e9 −0.457256
$$636$$ 0 0
$$637$$ 6.63501e9 1.01708
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −5.17445e9 −0.775998 −0.387999 0.921660i $$-0.626834\pi$$
−0.387999 + 0.921660i $$0.626834\pi$$
$$642$$ 0 0
$$643$$ −1.04374e10 −1.54830 −0.774148 0.633004i $$-0.781822\pi$$
−0.774148 + 0.633004i $$0.781822\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 9.71623e8 0.141037 0.0705185 0.997510i $$-0.477535\pi$$
0.0705185 + 0.997510i $$0.477535\pi$$
$$648$$ 0 0
$$649$$ 2.43825e9 0.350125
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 7.25223e9 1.01924 0.509619 0.860400i $$-0.329786\pi$$
0.509619 + 0.860400i $$0.329786\pi$$
$$654$$ 0 0
$$655$$ 5.54363e7 0.00770814
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −3.81924e9 −0.519851 −0.259925 0.965629i $$-0.583698\pi$$
−0.259925 + 0.965629i $$0.583698\pi$$
$$660$$ 0 0
$$661$$ 1.07881e10 1.45292 0.726459 0.687210i $$-0.241165\pi$$
0.726459 + 0.687210i $$0.241165\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −1.54205e8 −0.0203339
$$666$$ 0 0
$$667$$ 8.15390e9 1.06396
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 4.76448e9 0.608817
$$672$$ 0 0
$$673$$ −6.34833e9 −0.802798 −0.401399 0.915903i $$-0.631476\pi$$
−0.401399 + 0.915903i $$0.631476\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −8.82566e9 −1.09317 −0.546584 0.837404i $$-0.684072\pi$$
−0.546584 + 0.837404i $$0.684072\pi$$
$$678$$ 0 0
$$679$$ 2.76599e9 0.339083
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −4.92331e9 −0.591268 −0.295634 0.955301i $$-0.595531\pi$$
−0.295634 + 0.955301i $$0.595531\pi$$
$$684$$ 0 0
$$685$$ 2.42590e9 0.288374
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 1.95418e10 2.27613
$$690$$ 0 0
$$691$$ −5.68449e9 −0.655418 −0.327709 0.944779i $$-0.606277\pi$$
−0.327709 + 0.944779i $$0.606277\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −2.61649e9 −0.295646
$$696$$ 0 0
$$697$$ −1.57663e9 −0.176366
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1.70567e9 0.187017 0.0935085 0.995618i $$-0.470192\pi$$
0.0935085 + 0.995618i $$0.470192\pi$$
$$702$$ 0 0
$$703$$ −1.72978e9 −0.187779
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 4.42351e9 0.470760
$$708$$ 0 0
$$709$$ 4.52189e9 0.476495 0.238248 0.971204i $$-0.423427\pi$$
0.238248 + 0.971204i $$0.423427\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 3.30738e9 0.341720
$$714$$ 0 0
$$715$$ −2.23070e9 −0.228229
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 3.09206e9 0.310239 0.155120 0.987896i $$-0.450424\pi$$
0.155120 + 0.987896i $$0.450424\pi$$
$$720$$ 0 0
$$721$$ −1.87033e9 −0.185842
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 7.12568e9 0.694453
$$726$$ 0 0
$$727$$ 1.44622e10 1.39593 0.697965 0.716132i $$-0.254089\pi$$
0.697965 + 0.716132i $$0.254089\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −7.69827e9 −0.728924
$$732$$ 0 0
$$733$$ 3.15415e9 0.295814 0.147907 0.989001i $$-0.452746\pi$$
0.147907 + 0.989001i $$0.452746\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 7.48585e9 0.688819
$$738$$ 0 0
$$739$$ 1.54236e10 1.40582 0.702912 0.711277i $$-0.251883\pi$$
0.702912 + 0.711277i $$0.251883\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 1.59520e10 1.42677 0.713385 0.700772i $$-0.247161\pi$$
0.713385 + 0.700772i $$0.247161\pi$$
$$744$$ 0 0
$$745$$ 9.56141e8 0.0847179
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 3.32424e7 0.00289072
$$750$$ 0 0
$$751$$ 6.13964e9 0.528936 0.264468 0.964395i $$-0.414804\pi$$
0.264468 + 0.964395i $$0.414804\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −1.44562e9 −0.122248
$$756$$ 0 0
$$757$$ −1.42818e10 −1.19660 −0.598299 0.801273i $$-0.704157\pi$$
−0.598299 + 0.801273i $$0.704157\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −1.47536e10 −1.21353 −0.606767 0.794880i $$-0.707534\pi$$
−0.606767 + 0.794880i $$0.707534\pi$$
$$762$$ 0 0
$$763$$ −4.35863e9 −0.355234
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −1.04118e10 −0.833190
$$768$$ 0 0
$$769$$ 1.97592e10 1.56685 0.783424 0.621487i $$-0.213471\pi$$
0.783424 + 0.621487i $$0.213471\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −1.01370e10 −0.789374 −0.394687 0.918816i $$-0.629147\pi$$
−0.394687 + 0.918816i $$0.629147\pi$$
$$774$$ 0 0
$$775$$ 2.89031e9 0.223043
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −5.83142e8 −0.0441970
$$780$$ 0 0
$$781$$ 6.43174e9 0.483114
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 5.20420e8 0.0383981
$$786$$ 0 0
$$787$$ −1.27882e10 −0.935188 −0.467594 0.883943i $$-0.654879\pi$$
−0.467594 + 0.883943i $$0.654879\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 4.25854e9 0.305945
$$792$$ 0 0
$$793$$ −2.03453e10 −1.44880
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 7.38617e9 0.516791 0.258396 0.966039i $$-0.416806\pi$$
0.258396 + 0.966039i $$0.416806\pi$$
$$798$$ 0 0
$$799$$ 7.60466e9 0.527431
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −4.24114e9 −0.289054
$$804$$ 0 0
$$805$$ 3.05508e9 0.206413
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −1.53742e10 −1.02087 −0.510437 0.859915i $$-0.670516\pi$$
−0.510437 + 0.859915i $$0.670516\pi$$
$$810$$ 0 0
$$811$$ 9.77882e9 0.643744 0.321872 0.946783i $$-0.395688\pi$$
0.321872 + 0.946783i $$0.395688\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 6.59472e9 0.426722
$$816$$ 0 0
$$817$$ −2.84733e9 −0.182667
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −1.83470e10 −1.15708 −0.578540 0.815654i $$-0.696377\pi$$
−0.578540 + 0.815654i $$0.696377\pi$$
$$822$$ 0 0
$$823$$ −3.16960e10 −1.98201 −0.991004 0.133829i $$-0.957273\pi$$
−0.991004 + 0.133829i $$0.957273\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 6.12845e9 0.376774 0.188387 0.982095i $$-0.439674\pi$$
0.188387 + 0.982095i $$0.439674\pi$$
$$828$$ 0 0
$$829$$ −1.24652e10 −0.759904 −0.379952 0.925006i $$-0.624060\pi$$
−0.379952 + 0.925006i $$0.624060\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −6.86402e9 −0.411454
$$834$$ 0 0
$$835$$ −9.41091e9 −0.559409
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 1.82237e10 1.06530 0.532648 0.846337i $$-0.321197\pi$$
0.532648 + 0.846337i $$0.321197\pi$$
$$840$$ 0 0
$$841$$ −7.29024e9 −0.422625
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 4.38017e9 0.249743
$$846$$ 0 0
$$847$$ 5.98117e9 0.338216
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 3.42701e10 1.90617
$$852$$ 0 0
$$853$$ −2.48619e10 −1.37155 −0.685777 0.727812i $$-0.740537\pi$$
−0.685777 + 0.727812i $$0.740537\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −2.96761e10 −1.61055 −0.805275 0.592902i $$-0.797982\pi$$
−0.805275 + 0.592902i $$0.797982\pi$$
$$858$$ 0 0
$$859$$ 1.14772e10 0.617819 0.308910 0.951091i $$-0.400036\pi$$
0.308910 + 0.951091i $$0.400036\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −2.13485e10 −1.13066 −0.565328 0.824866i $$-0.691250\pi$$
−0.565328 + 0.824866i $$0.691250\pi$$
$$864$$ 0 0
$$865$$ 5.19679e9 0.273010
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 1.01921e10 0.526858
$$870$$ 0 0
$$871$$ −3.19661e10 −1.63918
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 5.59108e9 0.282142
$$876$$ 0 0
$$877$$ −7.92753e9 −0.396862 −0.198431 0.980115i $$-0.563585\pi$$
−0.198431 + 0.980115i $$0.563585\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −7.32045e9 −0.360680 −0.180340 0.983604i $$-0.557720\pi$$
−0.180340 + 0.983604i $$0.557720\pi$$
$$882$$ 0 0
$$883$$ −3.54988e9 −0.173521 −0.0867604 0.996229i $$-0.527651\pi$$
−0.0867604 + 0.996229i $$0.527651\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −5.80634e9 −0.279364 −0.139682 0.990196i $$-0.544608\pi$$
−0.139682 + 0.990196i $$0.544608\pi$$
$$888$$ 0 0
$$889$$ 1.64066e10 0.783182
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 2.81270e9 0.132173
$$894$$ 0 0
$$895$$ 9.28466e8 0.0432898
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 4.03982e9 0.185440
$$900$$ 0 0
$$901$$ −2.02163e10 −0.920798
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −4.28573e8 −0.0192201
$$906$$ 0 0
$$907$$ 1.78240e10 0.793196 0.396598 0.917992i $$-0.370191\pi$$
0.396598 + 0.917992i $$0.370191\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −1.87703e10 −0.822538 −0.411269 0.911514i $$-0.634914\pi$$
−0.411269 + 0.911514i $$0.634914\pi$$
$$912$$ 0 0
$$913$$ 1.35937e10 0.591138
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −3.08280e8 −0.0132024
$$918$$ 0 0
$$919$$ 3.75844e10 1.59736 0.798681 0.601754i $$-0.205531\pi$$
0.798681 + 0.601754i $$0.205531\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −2.74648e10 −1.14966
$$924$$ 0 0
$$925$$ 2.99486e10 1.24417
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 1.92372e10 0.787205 0.393602 0.919281i $$-0.371229\pi$$
0.393602 + 0.919281i $$0.371229\pi$$
$$930$$ 0 0
$$931$$ −2.53876e9 −0.103109
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 2.30769e9 0.0923289
$$936$$ 0 0
$$937$$ 1.04732e9 0.0415900 0.0207950 0.999784i $$-0.493380\pi$$
0.0207950 + 0.999784i $$0.493380\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 7.97861e9 0.312150 0.156075 0.987745i $$-0.450116\pi$$
0.156075 + 0.987745i $$0.450116\pi$$
$$942$$ 0 0
$$943$$ 1.15531e10 0.448650
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4.26943e9 0.163360 0.0816799 0.996659i $$-0.473971\pi$$
0.0816799 + 0.996659i $$0.473971\pi$$
$$948$$ 0 0
$$949$$ 1.81106e10 0.687860
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 1.06048e10 0.396897 0.198449 0.980111i $$-0.436410\pi$$
0.198449 + 0.980111i $$0.436410\pi$$
$$954$$ 0 0
$$955$$ 6.97247e9 0.259045
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −1.34904e10 −0.493922
$$960$$ 0 0
$$961$$ −2.58740e10 −0.940441
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 9.43750e9 0.338074
$$966$$ 0 0
$$967$$ −1.65090e10 −0.587123 −0.293562 0.955940i $$-0.594841\pi$$
−0.293562 + 0.955940i $$0.594841\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 2.46094e10 0.862649 0.431324 0.902197i $$-0.358046\pi$$
0.431324 + 0.902197i $$0.358046\pi$$
$$972$$ 0 0
$$973$$ 1.45503e10 0.506379
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 2.53886e10 0.870979 0.435489 0.900194i $$-0.356575\pi$$
0.435489 + 0.900194i $$0.356575\pi$$
$$978$$ 0 0
$$979$$ 1.63380e10 0.556492
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −1.87585e10 −0.629884 −0.314942 0.949111i $$-0.601985\pi$$
−0.314942 + 0.949111i $$0.601985\pi$$
$$984$$ 0 0
$$985$$ 1.13588e10 0.378709
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 5.64107e10 1.85428
$$990$$ 0 0
$$991$$ −3.59792e9 −0.117434 −0.0587170 0.998275i $$-0.518701\pi$$
−0.0587170 + 0.998275i $$0.518701\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −1.80083e9 −0.0579552
$$996$$ 0 0
$$997$$ −1.34287e10 −0.429143 −0.214571 0.976708i $$-0.568835\pi$$
−0.214571 + 0.976708i $$0.568835\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 72.8.a.d.1.1 1
3.2 odd 2 8.8.a.a.1.1 1
4.3 odd 2 144.8.a.g.1.1 1
8.3 odd 2 576.8.a.k.1.1 1
8.5 even 2 576.8.a.j.1.1 1
12.11 even 2 16.8.a.c.1.1 1
15.2 even 4 200.8.c.a.49.2 2
15.8 even 4 200.8.c.a.49.1 2
15.14 odd 2 200.8.a.i.1.1 1
21.20 even 2 392.8.a.d.1.1 1
24.5 odd 2 64.8.a.g.1.1 1
24.11 even 2 64.8.a.a.1.1 1
48.5 odd 4 256.8.b.e.129.2 2
48.11 even 4 256.8.b.c.129.1 2
48.29 odd 4 256.8.b.e.129.1 2
48.35 even 4 256.8.b.c.129.2 2
60.23 odd 4 400.8.c.b.49.2 2
60.47 odd 4 400.8.c.b.49.1 2
60.59 even 2 400.8.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.a.a.1.1 1 3.2 odd 2
16.8.a.c.1.1 1 12.11 even 2
64.8.a.a.1.1 1 24.11 even 2
64.8.a.g.1.1 1 24.5 odd 2
72.8.a.d.1.1 1 1.1 even 1 trivial
144.8.a.g.1.1 1 4.3 odd 2
200.8.a.i.1.1 1 15.14 odd 2
200.8.c.a.49.1 2 15.8 even 4
200.8.c.a.49.2 2 15.2 even 4
256.8.b.c.129.1 2 48.11 even 4
256.8.b.c.129.2 2 48.35 even 4
256.8.b.e.129.1 2 48.29 odd 4
256.8.b.e.129.2 2 48.5 odd 4
392.8.a.d.1.1 1 21.20 even 2
400.8.a.b.1.1 1 60.59 even 2
400.8.c.b.49.1 2 60.47 odd 4
400.8.c.b.49.2 2 60.23 odd 4
576.8.a.j.1.1 1 8.5 even 2
576.8.a.k.1.1 1 8.3 odd 2