# Properties

 Label 400.8.a.b.1.1 Level $400$ Weight $8$ Character 400.1 Self dual yes Analytic conductor $124.954$ 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] = [400,8,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, 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("400.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 400.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: $$124.954010194$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 400.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-84.0000 q^{3} -456.000 q^{7} +4869.00 q^{9} +O(q^{10})$$ $$q-84.0000 q^{3} -456.000 q^{7} +4869.00 q^{9} +2524.00 q^{11} +10778.0 q^{13} +11150.0 q^{17} -4124.00 q^{19} +38304.0 q^{21} +81704.0 q^{23} -225288. q^{27} +99798.0 q^{29} +40480.0 q^{31} -212016. q^{33} +419442. q^{37} -905352. q^{39} +141402. q^{41} -690428. q^{43} -682032. q^{47} -615607. q^{49} -936600. q^{51} -1.81312e6 q^{53} +346416. q^{57} +966028. q^{59} +1.88767e6 q^{61} -2.22026e6 q^{63} +2.96587e6 q^{67} -6.86314e6 q^{69} +2.54823e6 q^{71} +1.68033e6 q^{73} -1.15094e6 q^{77} -4.03806e6 q^{79} +8.27569e6 q^{81} -5.38576e6 q^{83} -8.38303e6 q^{87} -6.47305e6 q^{89} -4.91477e6 q^{91} -3.40032e6 q^{93} +6.06576e6 q^{97} +1.22894e7 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −84.0000 −1.79620 −0.898100 0.439790i $$-0.855053\pi$$
−0.898100 + 0.439790i $$0.855053\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −456.000 −0.502483 −0.251242 0.967924i $$-0.580839\pi$$
−0.251242 + 0.967924i $$0.580839\pi$$
$$8$$ 0 0
$$9$$ 4869.00 2.22634
$$10$$ 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 $$-0.261845\pi$$
0.680309 + 0.732925i $$0.261845\pi$$
$$14$$ 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 $$-0.521971\pi$$
−0.0689685 + 0.997619i $$0.521971\pi$$
$$20$$ 0 0
$$21$$ 38304.0 0.902561
$$22$$ 0 0
$$23$$ 81704.0 1.40022 0.700109 0.714036i $$-0.253135\pi$$
0.700109 + 0.714036i $$0.253135\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −225288. −2.20275
$$28$$ 0 0
$$29$$ 99798.0 0.759852 0.379926 0.925017i $$-0.375949\pi$$
0.379926 + 0.925017i $$0.375949\pi$$
$$30$$ 0 0
$$31$$ 40480.0 0.244048 0.122024 0.992527i $$-0.461062\pi$$
0.122024 + 0.992527i $$0.461062\pi$$
$$32$$ 0 0
$$33$$ −212016. −1.02700
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 419442. 1.36134 0.680669 0.732591i $$-0.261689\pi$$
0.680669 + 0.732591i $$0.261689\pi$$
$$38$$ 0 0
$$39$$ −905352. −2.44394
$$40$$ 0 0
$$41$$ 141402. 0.320414 0.160207 0.987083i $$-0.448784\pi$$
0.160207 + 0.987083i $$0.448784\pi$$
$$42$$ 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 $$-0.659039\pi$$
−0.479107 + 0.877757i $$0.659039\pi$$
$$48$$ 0 0
$$49$$ −615607. −0.747510
$$50$$ 0 0
$$51$$ −936600. −0.988686
$$52$$ 0 0
$$53$$ −1.81312e6 −1.67286 −0.836432 0.548071i $$-0.815362\pi$$
−0.836432 + 0.548071i $$0.815362\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 346416. 0.247763
$$58$$ 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
$$63$$ −2.22026e6 −1.11870
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.96587e6 1.20473 0.602365 0.798220i $$-0.294225\pi$$
0.602365 + 0.798220i $$0.294225\pi$$
$$68$$ 0 0
$$69$$ −6.86314e6 −2.51507
$$70$$ 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 $$-0.418657\pi$$
0.252775 + 0.967525i $$0.418657\pi$$
$$74$$ 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 $$-0.652413\pi$$
−0.460732 + 0.887539i $$0.652413\pi$$
$$80$$ 0 0
$$81$$ 8.27569e6 1.73024
$$82$$ 0 0
$$83$$ −5.38576e6 −1.03389 −0.516945 0.856019i $$-0.672931\pi$$
−0.516945 + 0.856019i $$0.672931\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −8.38303e6 −1.36485
$$88$$ 0 0
$$89$$ −6.47305e6 −0.973293 −0.486647 0.873599i $$-0.661780\pi$$
−0.486647 + 0.873599i $$0.661780\pi$$
$$90$$ 0 0
$$91$$ −4.91477e6 −0.683688
$$92$$ 0 0
$$93$$ −3.40032e6 −0.438359
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 6.06576e6 0.674814 0.337407 0.941359i $$-0.390450\pi$$
0.337407 + 0.941359i $$0.390450\pi$$
$$98$$ 0 0
$$99$$ 1.22894e7 1.27293
$$100$$ 0 0
$$101$$ 9.70069e6 0.936866 0.468433 0.883499i $$-0.344819\pi$$
0.468433 + 0.883499i $$0.344819\pi$$
$$102$$ 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 $$-0.499084\pi$$
0.00287643 + 0.999996i $$0.499084\pi$$
$$108$$ 0 0
$$109$$ 9.55841e6 0.706957 0.353478 0.935443i $$-0.384999\pi$$
0.353478 + 0.935443i $$0.384999\pi$$
$$110$$ 0 0
$$111$$ −3.52331e7 −2.44524
$$112$$ 0 0
$$113$$ −9.33890e6 −0.608865 −0.304433 0.952534i $$-0.598467\pi$$
−0.304433 + 0.952534i $$0.598467\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 5.24781e7 3.02920
$$118$$ 0 0
$$119$$ −5.08440e6 −0.276583
$$120$$ 0 0
$$121$$ −1.31166e7 −0.673089
$$122$$ 0 0
$$123$$ −1.18778e7 −0.575529
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −3.59794e7 −1.55862 −0.779311 0.626637i $$-0.784431\pi$$
−0.779311 + 0.626637i $$0.784431\pi$$
$$128$$ 0 0
$$129$$ 5.79960e7 2.37867
$$130$$ 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 $$-0.331907\pi$$
0.503876 + 0.863776i $$0.331907\pi$$
$$140$$ 0 0
$$141$$ 5.72907e7 1.72114
$$142$$ 0 0
$$143$$ 2.72037e7 0.777949
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 5.17110e7 1.34268
$$148$$ 0 0
$$149$$ −1.16603e7 −0.288773 −0.144386 0.989521i $$-0.546121\pi$$
−0.144386 + 0.989521i $$0.546121\pi$$
$$150$$ 0 0
$$151$$ 1.76295e7 0.416698 0.208349 0.978055i $$-0.433191\pi$$
0.208349 + 0.978055i $$0.433191\pi$$
$$152$$ 0 0
$$153$$ 5.42894e7 1.22545
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −6.34658e6 −0.130885 −0.0654427 0.997856i $$-0.520846\pi$$
−0.0654427 + 0.997856i $$0.520846\pi$$
$$158$$ 0 0
$$159$$ 1.52302e8 3.00480
$$160$$ 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 $$-0.0975451\pi$$
0.953411 + 0.301673i $$0.0975451\pi$$
$$168$$ 0 0
$$169$$ 5.34168e7 0.851283
$$170$$ 0 0
$$171$$ −2.00798e7 −0.307095
$$172$$ 0 0
$$173$$ 6.33755e7 0.930594 0.465297 0.885155i $$-0.345947\pi$$
0.465297 + 0.885155i $$0.345947\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −8.11464e7 −1.09992
$$178$$ 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
$$183$$ −1.58564e8 −1.91261
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.81426e7 0.314716
$$188$$ 0 0
$$189$$ 1.02731e8 1.10684
$$190$$ 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 $$-0.695460\pi$$
−0.576186 + 0.817319i $$0.695460\pi$$
$$194$$ 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 $$-0.468508\pi$$
0.0987742 + 0.995110i $$0.468508\pi$$
$$200$$ 0 0
$$201$$ −2.49133e8 −2.16394
$$202$$ 0 0
$$203$$ −4.55079e7 −0.381813
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 3.97817e8 3.11736
$$208$$ 0 0
$$209$$ −1.04090e7 −0.0788671
$$210$$ 0 0
$$211$$ 6.10208e7 0.447187 0.223594 0.974682i $$-0.428221\pi$$
0.223594 + 0.974682i $$0.428221\pi$$
$$212$$ 0 0
$$213$$ −2.14051e8 −1.51771
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −1.84589e7 −0.122630
$$218$$ 0 0
$$219$$ −1.41147e8 −0.908068
$$220$$ 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 $$-0.737311\pi$$
−0.678364 + 0.734726i $$0.737311\pi$$
$$228$$ 0 0
$$229$$ −4.67889e7 −0.257465 −0.128733 0.991679i $$-0.541091\pi$$
−0.128733 + 0.991679i $$0.541091\pi$$
$$230$$ 0 0
$$231$$ 9.66793e7 0.516050
$$232$$ 0 0
$$233$$ −3.45225e8 −1.78795 −0.893977 0.448113i $$-0.852096\pi$$
−0.893977 + 0.448113i $$0.852096\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 3.39197e8 1.65513
$$238$$ 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
$$243$$ −2.02453e8 −0.905112
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4.44485e7 −0.187680
$$248$$ 0 0
$$249$$ 4.52404e8 1.85707
$$250$$ 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
$$261$$ 4.85916e8 1.69169
$$262$$ 0 0
$$263$$ 2.19359e8 0.743549 0.371774 0.928323i $$-0.378750\pi$$
0.371774 + 0.928323i $$0.378750\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 5.43736e8 1.74823
$$268$$ 0 0
$$269$$ −1.48033e8 −0.463687 −0.231844 0.972753i $$-0.574476\pi$$
−0.231844 + 0.972753i $$0.574476\pi$$
$$270$$ 0 0
$$271$$ 3.69934e8 1.12910 0.564549 0.825399i $$-0.309050\pi$$
0.564549 + 0.825399i $$0.309050\pi$$
$$272$$ 0 0
$$273$$ 4.12841e8 1.22804
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3.95860e8 1.11908 0.559541 0.828803i $$-0.310977\pi$$
0.559541 + 0.828803i $$0.310977\pi$$
$$278$$ 0 0
$$279$$ 1.97097e8 0.543332
$$280$$ 0 0
$$281$$ −5.97760e8 −1.60714 −0.803572 0.595208i $$-0.797070\pi$$
−0.803572 + 0.595208i $$0.797070\pi$$
$$282$$ 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
$$291$$ −5.09524e8 −1.21210
$$292$$ 0 0
$$293$$ −7.54530e8 −1.75243 −0.876213 0.481924i $$-0.839938\pi$$
−0.876213 + 0.481924i $$0.839938\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −5.68627e8 −1.25945
$$298$$ 0 0
$$299$$ 8.80606e8 1.90516
$$300$$ 0 0
$$301$$ 3.14835e8 0.665427
$$302$$ 0 0
$$303$$ −8.14858e8 −1.68280
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 8.20472e8 1.61838 0.809188 0.587549i $$-0.199907\pi$$
0.809188 + 0.587549i $$0.199907\pi$$
$$308$$ 0 0
$$309$$ −3.44534e8 −0.664320
$$310$$ 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 $$-0.709471\pi$$
−0.611594 + 0.791172i $$0.709471\pi$$
$$314$$ 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
$$321$$ −6.12360e6 −0.0103333
$$322$$ 0 0
$$323$$ −4.59826e7 −0.0759250
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −8.02906e8 −1.26984
$$328$$ 0 0
$$329$$ 3.11007e8 0.481486
$$330$$ 0 0
$$331$$ 3.05543e8 0.463100 0.231550 0.972823i $$-0.425620\pi$$
0.231550 + 0.972823i $$0.425620\pi$$
$$332$$ 0 0
$$333$$ 2.04226e9 3.03080
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −3.54965e7 −0.0505220 −0.0252610 0.999681i $$-0.508042\pi$$
−0.0252610 + 0.999681i $$0.508042\pi$$
$$338$$ 0 0
$$339$$ 7.84467e8 1.09364
$$340$$ 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 $$-0.539072\pi$$
−0.122441 + 0.992476i $$0.539072\pi$$
$$348$$ 0 0
$$349$$ 8.60864e8 1.08404 0.542020 0.840366i $$-0.317660\pi$$
0.542020 + 0.840366i $$0.317660\pi$$
$$350$$ 0 0
$$351$$ −2.42815e9 −2.99710
$$352$$ 0 0
$$353$$ 1.04544e9 1.26500 0.632498 0.774562i $$-0.282030\pi$$
0.632498 + 0.774562i $$0.282030\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 4.27090e8 0.496798
$$358$$ 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
$$363$$ 1.10179e9 1.20900
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −1.38692e9 −1.46460 −0.732302 0.680980i $$-0.761554\pi$$
−0.732302 + 0.680980i $$0.761554\pi$$
$$368$$ 0 0
$$369$$ 6.88486e8 0.713351
$$370$$ 0 0
$$371$$ 8.26782e8 0.840586
$$372$$ 0 0
$$373$$ −4.77105e8 −0.476029 −0.238015 0.971262i $$-0.576497\pi$$
−0.238015 + 0.971262i $$0.576497\pi$$
$$374$$ 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 $$-0.440721\pi$$
0.185156 + 0.982709i $$0.440721\pi$$
$$380$$ 0 0
$$381$$ 3.02227e9 2.79960
$$382$$ 0 0
$$383$$ 2.10409e9 1.91368 0.956839 0.290617i $$-0.0938605\pi$$
0.956839 + 0.290617i $$0.0938605\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −3.36169e9 −2.94829
$$388$$ 0 0
$$389$$ −1.26019e9 −1.08546 −0.542730 0.839907i $$-0.682609\pi$$
−0.542730 + 0.839907i $$0.682609\pi$$
$$390$$ 0 0
$$391$$ 9.11000e8 0.770725
$$392$$ 0 0
$$393$$ −5.67884e7 −0.0471939
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 9.81298e8 0.787107 0.393554 0.919302i $$-0.371246\pi$$
0.393554 + 0.919302i $$0.371246\pi$$
$$398$$ 0 0
$$399$$ −1.57966e8 −0.124497
$$400$$ 0 0
$$401$$ 9.09981e8 0.704737 0.352369 0.935861i $$-0.385376\pi$$
0.352369 + 0.935861i $$0.385376\pi$$
$$402$$ 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
$$411$$ −2.48507e9 −1.76560
$$412$$ 0 0
$$413$$ −4.40509e8 −0.307701
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −2.68031e9 −1.81013
$$418$$ 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
$$423$$ −3.32081e9 −2.13331
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −8.60778e8 −0.535049
$$428$$ 0 0
$$429$$ −2.28511e9 −1.39735
$$430$$ 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 $$-0.432995\pi$$
0.208951 + 0.977926i $$0.432995\pi$$
$$434$$ 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 $$-0.637577\pi$$
−0.418880 + 0.908042i $$0.637577\pi$$
$$440$$ 0 0
$$441$$ −2.99739e9 −1.66421
$$442$$ 0 0
$$443$$ −7.22153e8 −0.394654 −0.197327 0.980338i $$-0.563226\pi$$
−0.197327 + 0.980338i $$0.563226\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 9.79462e8 0.518694
$$448$$ 0 0
$$449$$ −1.22968e9 −0.641109 −0.320554 0.947230i $$-0.603869\pi$$
−0.320554 + 0.947230i $$0.603869\pi$$
$$450$$ 0 0
$$451$$ 3.56899e8 0.183201
$$452$$ 0 0
$$453$$ −1.48088e9 −0.748473
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 8.85551e7 0.0434017 0.0217009 0.999765i $$-0.493092\pi$$
0.0217009 + 0.999765i $$0.493092\pi$$
$$458$$ 0 0
$$459$$ −2.51196e9 −1.21246
$$460$$ 0 0
$$461$$ 2.10937e8 0.100277 0.0501384 0.998742i $$-0.484034\pi$$
0.0501384 + 0.998742i $$0.484034\pi$$
$$462$$ 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 $$-0.493615\pi$$
0.0200567 + 0.999799i $$0.493615\pi$$
$$468$$ 0 0
$$469$$ −1.35244e9 −0.605357
$$470$$ 0 0
$$471$$ 5.33113e8 0.235096
$$472$$ 0 0
$$473$$ −1.74264e9 −0.757171
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −8.82807e9 −3.72436
$$478$$ 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
$$483$$ 3.12959e9 1.26378
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 3.31338e9 1.29993 0.649964 0.759965i $$-0.274784\pi$$
0.649964 + 0.759965i $$0.274784\pi$$
$$488$$ 0 0
$$489$$ −6.75557e9 −2.61265
$$490$$ 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 $$-0.662326\pi$$
−0.488145 + 0.872763i $$0.662326\pi$$
$$500$$ 0 0
$$501$$ −9.64045e9 −3.42504
$$502$$ 0 0
$$503$$ 3.04579e8 0.106712 0.0533558 0.998576i $$-0.483008\pi$$
0.0533558 + 0.998576i $$0.483008\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −4.48701e9 −1.52908
$$508$$ 0 0
$$509$$ −1.88202e8 −0.0632575 −0.0316287 0.999500i $$-0.510069\pi$$
−0.0316287 + 0.999500i $$0.510069\pi$$
$$510$$ 0 0
$$511$$ −7.66229e8 −0.254030
$$512$$ 0 0
$$513$$ 9.29088e8 0.303841
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −1.72145e9 −0.547870
$$518$$ 0 0
$$519$$ −5.32355e9 −1.67153
$$520$$ 0 0
$$521$$ 4.14963e9 1.28552 0.642758 0.766069i $$-0.277790\pi$$
0.642758 + 0.766069i $$0.277790\pi$$
$$522$$ 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
$$531$$ 4.70359e9 1.36332
$$532$$ 0 0
$$533$$ 1.52403e9 0.435962
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −9.51112e8 −0.265046
$$538$$ 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
$$543$$ 4.39026e8 0.117677
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 5.58047e8 0.145786 0.0728929 0.997340i $$-0.476777\pi$$
0.0728929 + 0.997340i $$0.476777\pi$$
$$548$$ 0 0
$$549$$ 9.19107e9 2.37062
$$550$$ 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
$$561$$ −2.36398e9 −0.565293
$$562$$ 0 0
$$563$$ −1.22011e8 −0.0288152 −0.0144076 0.999896i $$-0.504586\pi$$
−0.0144076 + 0.999896i $$0.504586\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −3.77371e9 −0.869417
$$568$$ 0 0
$$569$$ 5.00925e8 0.113993 0.0569967 0.998374i $$-0.481848\pi$$
0.0569967 + 0.998374i $$0.481848\pi$$
$$570$$ 0 0
$$571$$ 6.98702e9 1.57060 0.785300 0.619116i $$-0.212509\pi$$
0.785300 + 0.619116i $$0.212509\pi$$
$$572$$ 0 0
$$573$$ −7.14253e9 −1.58603
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 8.16573e9 1.76962 0.884809 0.465954i $$-0.154289\pi$$
0.884809 + 0.465954i $$0.154289\pi$$
$$578$$ 0 0
$$579$$ 9.66769e9 2.06989
$$580$$ 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 $$-0.163783\pi$$
0.870519 + 0.492135i $$0.163783\pi$$
$$588$$ 0 0
$$589$$ −1.66940e8 −0.0336632
$$590$$ 0 0
$$591$$ −1.16358e10 −2.31868
$$592$$ 0 0
$$593$$ 1.71175e9 0.337092 0.168546 0.985694i $$-0.446093\pi$$
0.168546 + 0.985694i $$0.446093\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −1.84475e9 −0.354836
$$598$$ 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
$$603$$ 1.44408e10 2.68214
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −1.82652e9 −0.331485 −0.165743 0.986169i $$-0.553002\pi$$
−0.165743 + 0.986169i $$0.553002\pi$$
$$608$$ 0 0
$$609$$ 3.82266e9 0.685813
$$610$$ 0 0
$$611$$ −7.35094e9 −1.30376
$$612$$ 0 0
$$613$$ 6.90339e9 1.21046 0.605231 0.796050i $$-0.293081\pi$$
0.605231 + 0.796050i $$0.293081\pi$$
$$614$$ 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 $$-0.381698\pi$$
0.363161 + 0.931727i $$0.381698\pi$$
$$620$$ 0 0
$$621$$ −1.84069e10 −3.08433
$$622$$ 0 0
$$623$$ 2.95171e9 0.489064
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 8.74354e8 0.141661
$$628$$ 0 0
$$629$$ 4.67678e9 0.749324
$$630$$ 0 0
$$631$$ 5.61602e8 0.0889869 0.0444935 0.999010i $$-0.485833\pi$$
0.0444935 + 0.999010i $$0.485833\pi$$
$$632$$ 0 0
$$633$$ −5.12575e9 −0.803238
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −6.63501e9 −1.01708
$$638$$ 0 0
$$639$$ 1.24073e10 1.88116
$$640$$ 0 0
$$641$$ 5.17445e9 0.775998 0.387999 0.921660i $$-0.373166\pi$$
0.387999 + 0.921660i $$0.373166\pi$$
$$642$$ 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 $$-0.522465\pi$$
−0.0705185 + 0.997510i $$0.522465\pi$$
$$648$$ 0 0
$$649$$ 2.43825e9 0.350125
$$650$$ 0 0
$$651$$ 1.55055e9 0.220268
$$652$$ 0 0
$$653$$ 7.25223e9 1.01924 0.509619 0.860400i $$-0.329786\pi$$
0.509619 + 0.860400i $$0.329786\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 8.18151e9 1.12552
$$658$$ 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
$$663$$ −1.00947e10 −1.34523
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 8.15390e9 1.06396
$$668$$ 0 0
$$669$$ 3.54857e9 0.458207
$$670$$ 0 0
$$671$$ 4.76448e9 0.608817
$$672$$ 0 0
$$673$$ 6.34833e9 0.802798 0.401399 0.915903i $$-0.368524\pi$$
0.401399 + 0.915903i $$0.368524\pi$$
$$674$$ 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
$$681$$ 2.00846e10 2.43696
$$682$$ 0 0
$$683$$ 4.92331e9 0.591268 0.295634 0.955301i $$-0.404469\pi$$
0.295634 + 0.955301i $$0.404469\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 3.93027e9 0.462459
$$688$$ 0 0
$$689$$ −1.95418e10 −2.27613
$$690$$ 0 0
$$691$$ 5.68449e9 0.655418 0.327709 0.944779i $$-0.393723\pi$$
0.327709 + 0.944779i $$0.393723\pi$$
$$692$$ 0 0
$$693$$ −5.60395e9 −0.639628
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1.57663e9 0.176366
$$698$$ 0 0
$$699$$ 2.89989e10 3.21152
$$700$$ 0 0
$$701$$ −1.70567e9 −0.187017 −0.0935085 0.995618i $$-0.529808\pi$$
−0.0935085 + 0.995618i $$0.529808\pi$$
$$702$$ 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
$$711$$ −1.96613e10 −2.05149
$$712$$ 0 0
$$713$$ 3.30738e9 0.341720
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 1.96907e10 1.99500
$$718$$ 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
$$723$$ 9.20280e9 0.905599
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 1.44622e10 1.39593 0.697965 0.716132i $$-0.254089\pi$$
0.697965 + 0.716132i $$0.254089\pi$$
$$728$$ 0 0
$$729$$ −1.09288e9 −0.104478
$$730$$ 0 0
$$731$$ −7.69827e9 −0.728924
$$732$$ 0 0
$$733$$ −3.15415e9 −0.295814 −0.147907 0.989001i $$-0.547254\pi$$
−0.147907 + 0.989001i $$0.547254\pi$$
$$734$$ 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 $$-0.748117\pi$$
−0.702912 + 0.711277i $$0.748117\pi$$
$$740$$ 0 0
$$741$$ 3.73367e9 0.337111
$$742$$ 0 0
$$743$$ −1.59520e10 −1.42677 −0.713385 0.700772i $$-0.752839\pi$$
−0.713385 + 0.700772i $$0.752839\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −2.62233e10 −2.30179
$$748$$ 0 0
$$749$$ −3.32424e7 −0.00289072
$$750$$ 0 0
$$751$$ −6.13964e9 −0.528936 −0.264468 0.964395i $$-0.585196\pi$$
−0.264468 + 0.964395i $$0.585196\pi$$
$$752$$ 0 0
$$753$$ −3.30986e10 −2.82506
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1.42818e10 1.19660 0.598299 0.801273i $$-0.295843\pi$$
0.598299 + 0.801273i $$0.295843\pi$$
$$758$$ 0 0
$$759$$ −1.73226e10 −1.43802
$$760$$ 0 0
$$761$$ 1.47536e10 1.21353 0.606767 0.794880i $$-0.292466\pi$$
0.606767 + 0.794880i $$0.292466\pi$$
$$762$$ 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
$$771$$ 2.68014e10 2.10604
$$772$$ 0 0
$$773$$ −1.01370e10 −0.789374 −0.394687 0.918816i $$-0.629147\pi$$
−0.394687 + 0.918816i $$0.629147\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 1.60663e10 1.22869
$$778$$ 0 0
$$779$$ −5.83142e8 −0.0441970
$$780$$ 0 0
$$781$$ 6.43174e9 0.483114
$$782$$ 0 0
$$783$$ −2.24833e10 −1.67376
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −1.27882e10 −0.935188 −0.467594 0.883943i $$-0.654879\pi$$
−0.467594 + 0.883943i $$0.654879\pi$$
$$788$$ 0 0
$$789$$ −1.84261e10 −1.33556
$$790$$ 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
$$801$$ −3.15173e10 −2.16688
$$802$$ 0 0
$$803$$ 4.24114e9 0.289054
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 1.24348e10 0.832875
$$808$$ 0 0
$$809$$ 1.53742e10 1.02087 0.510437 0.859915i $$-0.329484\pi$$
0.510437 + 0.859915i $$0.329484\pi$$
$$810$$ 0 0
$$811$$ −9.77882e9 −0.643744 −0.321872 0.946783i $$-0.604312\pi$$
−0.321872 + 0.946783i $$0.604312\pi$$
$$812$$ 0 0
$$813$$ −3.10745e10 −2.02809
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.84733e9 0.182667
$$818$$ 0 0
$$819$$ −2.39300e10 −1.52212
$$820$$ 0 0
$$821$$ 1.83470e10 1.15708 0.578540 0.815654i $$-0.303623\pi$$
0.578540 + 0.815654i $$0.303623\pi$$
$$822$$ 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 $$-0.560326\pi$$
−0.188387 + 0.982095i $$0.560326\pi$$
$$828$$ 0 0
$$829$$ −1.24652e10 −0.759904 −0.379952 0.925006i $$-0.624060\pi$$
−0.379952 + 0.925006i $$0.624060\pi$$
$$830$$ 0 0
$$831$$ −3.32522e10 −2.01010
$$832$$ 0 0
$$833$$ −6.86402e9 −0.411454
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −9.11966e9 −0.537575
$$838$$ 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
$$843$$ 5.02118e10 2.88675
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.98117e9 0.338216
$$848$$ 0 0
$$849$$ −6.76870e9 −0.379602
$$850$$ 0 0
$$851$$ 3.42701e10 1.90617
$$852$$ 0 0
$$853$$ 2.48619e10 1.37155 0.685777 0.727812i $$-0.259463\pi$$
0.685777 + 0.727812i $$0.259463\pi$$
$$854$$ 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 $$-0.599964\pi$$
−0.308910 + 0.951091i $$0.599964\pi$$
$$860$$ 0 0
$$861$$ 5.41626e9 0.289194
$$862$$ 0 0
$$863$$ 2.13485e10 1.13066 0.565328 0.824866i $$-0.308750\pi$$
0.565328 + 0.824866i $$0.308750\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 2.40254e10 1.25200
$$868$$ 0 0
$$869$$ −1.01921e10 −0.526858
$$870$$ 0 0
$$871$$ 3.19661e10 1.63918
$$872$$ 0 0
$$873$$ 2.95342e10 1.50236
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 7.92753e9 0.396862 0.198431 0.980115i $$-0.436415\pi$$
0.198431 + 0.980115i $$0.436415\pi$$
$$878$$ 0 0
$$879$$ 6.33805e10 3.14771
$$880$$ 0 0
$$881$$ 7.32045e9 0.360680 0.180340 0.983604i $$-0.442280\pi$$
0.180340 + 0.983604i $$0.442280\pi$$
$$882$$ 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 $$-0.455392\pi$$
0.139682 + 0.990196i $$0.455392\pi$$
$$888$$ 0 0
$$889$$ 1.64066e10 0.783182
$$890$$ 0 0
$$891$$ 2.08878e10 0.989285
$$892$$ 0 0
$$893$$ 2.81270e9 0.132173
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −7.39709e10 −3.42206
$$898$$ 0 0
$$899$$ 4.03982e9 0.185440
$$900$$ 0 0
$$901$$ −2.02163e10 −0.920798
$$902$$ 0 0
$$903$$ −2.64462e10 −1.19524
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 1.78240e10 0.793196 0.396598 0.917992i $$-0.370191\pi$$
0.396598 + 0.917992i $$0.370191\pi$$
$$908$$ 0 0
$$909$$ 4.72326e10 2.08578
$$910$$ 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 $$-0.794469\pi$$
−0.798681 + 0.601754i $$0.794469\pi$$
$$920$$ 0 0
$$921$$ −6.89197e10 −2.90693
$$922$$ 0 0
$$923$$ 2.74648e10 1.14966
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 1.99707e10 0.823404
$$928$$ 0 0
$$929$$ −1.92372e10 −0.787205 −0.393602 0.919281i $$-0.628771\pi$$
−0.393602 + 0.919281i $$0.628771\pi$$
$$930$$ 0 0
$$931$$ 2.53876e9 0.103109
$$932$$ 0 0
$$933$$ 5.48943e10 2.21279
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −1.04732e9 −0.0415900 −0.0207950 0.999784i $$-0.506620\pi$$
−0.0207950 + 0.999784i $$0.506620\pi$$
$$938$$ 0 0
$$939$$ 5.57413e10 2.19709
$$940$$ 0 0
$$941$$ −7.97861e9 −0.312150 −0.156075 0.987745i $$-0.549884\pi$$
−0.156075 + 0.987745i $$0.549884\pi$$
$$942$$ 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 $$-0.526029\pi$$
−0.0816799 + 0.996659i $$0.526029\pi$$
$$948$$ 0 0
$$949$$ 1.81106e10 0.687860
$$950$$ 0 0
$$951$$ −2.97963e10 −1.12339
$$952$$ 0 0
$$953$$ 1.06048e10 0.396897 0.198449 0.980111i $$-0.436410\pi$$
0.198449 + 0.980111i $$0.436410\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −2.11588e10 −0.780367
$$958$$ 0 0
$$959$$ −1.34904e10 −0.493922
$$960$$ 0 0
$$961$$ −2.58740e10 −0.940441
$$962$$ 0 0
$$963$$ 3.54950e8 0.0128078
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −1.65090e10 −0.587123 −0.293562 0.955940i $$-0.594841\pi$$
−0.293562 + 0.955940i $$0.594841\pi$$
$$968$$ 0 0
$$969$$ 3.86254e9 0.136377
$$970$$ 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
$$981$$ 4.65399e10 1.57392
$$982$$ 0 0
$$983$$ 1.87585e10 0.629884 0.314942 0.949111i $$-0.398015\pi$$
0.314942 + 0.949111i $$0.398015\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −2.61246e10 −0.864846
$$988$$ 0 0
$$989$$ −5.64107e10 −1.85428
$$990$$ 0 0
$$991$$ 3.59792e9 0.117434 0.0587170 0.998275i $$-0.481299\pi$$
0.0587170 + 0.998275i $$0.481299\pi$$
$$992$$ 0 0
$$993$$ −2.56656e10 −0.831821
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.34287e10 0.429143 0.214571 0.976708i $$-0.431165\pi$$
0.214571 + 0.976708i $$0.431165\pi$$
$$998$$ 0 0
$$999$$ −9.44952e10 −2.99868
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 400.8.a.b.1.1 1
4.3 odd 2 200.8.a.i.1.1 1
5.2 odd 4 400.8.c.b.49.2 2
5.3 odd 4 400.8.c.b.49.1 2
5.4 even 2 16.8.a.c.1.1 1
15.14 odd 2 144.8.a.g.1.1 1
20.3 even 4 200.8.c.a.49.2 2
20.7 even 4 200.8.c.a.49.1 2
20.19 odd 2 8.8.a.a.1.1 1
40.19 odd 2 64.8.a.g.1.1 1
40.29 even 2 64.8.a.a.1.1 1
60.59 even 2 72.8.a.d.1.1 1
80.19 odd 4 256.8.b.e.129.1 2
80.29 even 4 256.8.b.c.129.2 2
80.59 odd 4 256.8.b.e.129.2 2
80.69 even 4 256.8.b.c.129.1 2
120.29 odd 2 576.8.a.k.1.1 1
120.59 even 2 576.8.a.j.1.1 1
140.139 even 2 392.8.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.a.a.1.1 1 20.19 odd 2
16.8.a.c.1.1 1 5.4 even 2
64.8.a.a.1.1 1 40.29 even 2
64.8.a.g.1.1 1 40.19 odd 2
72.8.a.d.1.1 1 60.59 even 2
144.8.a.g.1.1 1 15.14 odd 2
200.8.a.i.1.1 1 4.3 odd 2
200.8.c.a.49.1 2 20.7 even 4
200.8.c.a.49.2 2 20.3 even 4
256.8.b.c.129.1 2 80.69 even 4
256.8.b.c.129.2 2 80.29 even 4
256.8.b.e.129.1 2 80.19 odd 4
256.8.b.e.129.2 2 80.59 odd 4
392.8.a.d.1.1 1 140.139 even 2
400.8.a.b.1.1 1 1.1 even 1 trivial
400.8.c.b.49.1 2 5.3 odd 4
400.8.c.b.49.2 2 5.2 odd 4
576.8.a.j.1.1 1 120.59 even 2
576.8.a.k.1.1 1 120.29 odd 2