Properties

 Label 400.8.c.b.49.2 Level $400$ Weight $8$ Character 400.49 Analytic conductor $124.954$ Analytic rank $0$ Dimension $2$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,8,Mod(49,400)]

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, 1]))

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

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$, degree $$1$$, not minimal)

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: no Analytic conductor: $$124.954010194$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 400.49 Dual form 400.8.c.b.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+84.0000i q^{3} -456.000i q^{7} -4869.00 q^{9} +O(q^{10})$$ $$q+84.0000i q^{3} -456.000i q^{7} -4869.00 q^{9} +2524.00 q^{11} -10778.0i q^{13} +11150.0i q^{17} +4124.00 q^{19} +38304.0 q^{21} -81704.0i q^{23} -225288. i q^{27} -99798.0 q^{29} +40480.0 q^{31} +212016. i q^{33} +419442. i q^{37} +905352. q^{39} +141402. q^{41} +690428. i q^{43} -682032. i q^{47} +615607. q^{49} -936600. q^{51} +1.81312e6i q^{53} +346416. i q^{57} -966028. q^{59} +1.88767e6 q^{61} +2.22026e6i q^{63} +2.96587e6i q^{67} +6.86314e6 q^{69} +2.54823e6 q^{71} -1.68033e6i q^{73} -1.15094e6i q^{77} +4.03806e6 q^{79} +8.27569e6 q^{81} +5.38576e6i q^{83} -8.38303e6i q^{87} +6.47305e6 q^{89} -4.91477e6 q^{91} +3.40032e6i q^{93} +6.06576e6i q^{97} -1.22894e7 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 9738 q^{9}+O(q^{10})$$ 2 * q - 9738 * q^9 $$2 q - 9738 q^{9} + 5048 q^{11} + 8248 q^{19} + 76608 q^{21} - 199596 q^{29} + 80960 q^{31} + 1810704 q^{39} + 282804 q^{41} + 1231214 q^{49} - 1873200 q^{51} - 1932056 q^{59} + 3775340 q^{61} + 13726272 q^{69} + 5096464 q^{71} + 8076128 q^{79} + 16551378 q^{81} + 12946092 q^{89} - 9829536 q^{91} - 24578712 q^{99}+O(q^{100})$$ 2 * q - 9738 * q^9 + 5048 * q^11 + 8248 * q^19 + 76608 * q^21 - 199596 * q^29 + 80960 * q^31 + 1810704 * q^39 + 282804 * q^41 + 1231214 * q^49 - 1873200 * q^51 - 1932056 * q^59 + 3775340 * q^61 + 13726272 * q^69 + 5096464 * q^71 + 8076128 * q^79 + 16551378 * q^81 + 12946092 * q^89 - 9829536 * q^91 - 24578712 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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
$$3$$ 84.0000i 1.79620i 0.439790 + 0.898100i $$0.355053\pi$$
−0.439790 + 0.898100i $$0.644947\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 456.000i − 0.502483i −0.967924 0.251242i $$-0.919161\pi$$
0.967924 0.251242i $$-0.0808389\pi$$
$$8$$ 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
$$13$$ − 10778.0i − 1.36062i −0.732925 0.680309i $$-0.761845\pi$$
0.732925 0.680309i $$-0.238155\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 11150.0i 0.550432i 0.961382 + 0.275216i $$0.0887494\pi$$
−0.961382 + 0.275216i $$0.911251\pi$$
$$18$$ 0 0
$$19$$ 4124.00 0.137937 0.0689685 0.997619i $$-0.478029\pi$$
0.0689685 + 0.997619i $$0.478029\pi$$
$$20$$ 0 0
$$21$$ 38304.0 0.902561
$$22$$ 0 0
$$23$$ − 81704.0i − 1.40022i −0.714036 0.700109i $$-0.753135\pi$$
0.714036 0.700109i $$-0.246865\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 225288.i − 2.20275i
$$28$$ 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 $$-0.461062\pi$$
0.122024 + 0.992527i $$0.461062\pi$$
$$32$$ 0 0
$$33$$ 212016.i 1.02700i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 419442.i 1.36134i 0.732591 + 0.680669i $$0.238311\pi$$
−0.732591 + 0.680669i $$0.761689\pi$$
$$38$$ 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.i 1.32428i 0.749382 + 0.662138i $$0.230351\pi$$
−0.749382 + 0.662138i $$0.769649\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 682032.i − 0.958213i −0.877757 0.479107i $$-0.840961\pi$$
0.877757 0.479107i $$-0.159039\pi$$
$$48$$ 0 0
$$49$$ 615607. 0.747510
$$50$$ 0 0
$$51$$ −936600. −0.988686
$$52$$ 0 0
$$53$$ 1.81312e6i 1.67286i 0.548071 + 0.836432i $$0.315362\pi$$
−0.548071 + 0.836432i $$0.684638\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 346416.i 0.247763i
$$58$$ 0 0
$$59$$ −966028. −0.612361 −0.306181 0.951973i $$-0.599051\pi$$
−0.306181 + 0.951973i $$0.599051\pi$$
$$60$$ 0 0
$$61$$ 1.88767e6 1.06481 0.532404 0.846490i $$-0.321289\pi$$
0.532404 + 0.846490i $$0.321289\pi$$
$$62$$ 0 0
$$63$$ 2.22026e6i 1.11870i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.96587e6i 1.20473i 0.798220 + 0.602365i $$0.205775\pi$$
−0.798220 + 0.602365i $$0.794225\pi$$
$$68$$ 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
$$73$$ − 1.68033e6i − 0.505549i −0.967525 0.252775i $$-0.918657\pi$$
0.967525 0.252775i $$-0.0813431\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 1.15094e6i − 0.287301i
$$78$$ 0 0
$$79$$ 4.03806e6 0.921464 0.460732 0.887539i $$-0.347587\pi$$
0.460732 + 0.887539i $$0.347587\pi$$
$$80$$ 0 0
$$81$$ 8.27569e6 1.73024
$$82$$ 0 0
$$83$$ 5.38576e6i 1.03389i 0.856019 + 0.516945i $$0.172931\pi$$
−0.856019 + 0.516945i $$0.827069\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 8.38303e6i − 1.36485i
$$88$$ 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
$$93$$ 3.40032e6i 0.438359i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 6.06576e6i 0.674814i 0.941359 + 0.337407i $$0.109550\pi$$
−0.941359 + 0.337407i $$0.890450\pi$$
$$98$$ 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
$$103$$ − 4.10159e6i − 0.369847i −0.982753 0.184924i $$-0.940796\pi$$
0.982753 0.184924i $$-0.0592037\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 72900.0i 0.00575287i 0.999996 + 0.00287643i $$0.000915598\pi$$
−0.999996 + 0.00287643i $$0.999084\pi$$
$$108$$ 0 0
$$109$$ −9.55841e6 −0.706957 −0.353478 0.935443i $$-0.615001\pi$$
−0.353478 + 0.935443i $$0.615001\pi$$
$$110$$ 0 0
$$111$$ −3.52331e7 −2.44524
$$112$$ 0 0
$$113$$ 9.33890e6i 0.608865i 0.952534 + 0.304433i $$0.0984668\pi$$
−0.952534 + 0.304433i $$0.901533\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 5.24781e7i 3.02920i
$$118$$ 0 0
$$119$$ 5.08440e6 0.276583
$$120$$ 0 0
$$121$$ −1.31166e7 −0.673089
$$122$$ 0 0
$$123$$ 1.18778e7i 0.575529i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 3.59794e7i − 1.55862i −0.626637 0.779311i $$-0.715569\pi$$
0.626637 0.779311i $$-0.284431\pi$$
$$128$$ 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
$$133$$ − 1.88054e6i − 0.0693111i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.95841e7i 0.982962i 0.870888 + 0.491481i $$0.163544\pi$$
−0.870888 + 0.491481i $$0.836456\pi$$
$$138$$ 0 0
$$139$$ −3.19084e7 −1.00775 −0.503876 0.863776i $$-0.668093\pi$$
−0.503876 + 0.863776i $$0.668093\pi$$
$$140$$ 0 0
$$141$$ 5.72907e7 1.72114
$$142$$ 0 0
$$143$$ − 2.72037e7i − 0.777949i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 5.17110e7i 1.34268i
$$148$$ 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 $$-0.433191\pi$$
0.208349 + 0.978055i $$0.433191\pi$$
$$152$$ 0 0
$$153$$ − 5.42894e7i − 1.22545i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 6.34658e6i − 0.130885i −0.997856 0.0654427i $$-0.979154\pi$$
0.997856 0.0654427i $$-0.0208460\pi$$
$$158$$ 0 0
$$159$$ −1.52302e8 −3.00480
$$160$$ 0 0
$$161$$ −3.72570e7 −0.703587
$$162$$ 0 0
$$163$$ − 8.04234e7i − 1.45454i −0.686351 0.727271i $$-0.740789\pi$$
0.686351 0.727271i $$-0.259211\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.14767e8i 1.90682i 0.301673 + 0.953411i $$0.402455\pi$$
−0.301673 + 0.953411i $$0.597545\pi$$
$$168$$ 0 0
$$169$$ −5.34168e7 −0.851283
$$170$$ 0 0
$$171$$ −2.00798e7 −0.307095
$$172$$ 0 0
$$173$$ − 6.33755e7i − 0.930594i −0.885155 0.465297i $$-0.845947\pi$$
0.885155 0.465297i $$-0.154053\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 8.11464e7i − 1.09992i
$$178$$ 0 0
$$179$$ −1.13228e7 −0.147559 −0.0737796 0.997275i $$-0.523506\pi$$
−0.0737796 + 0.997275i $$0.523506\pi$$
$$180$$ 0 0
$$181$$ −5.22650e6 −0.0655143 −0.0327571 0.999463i $$-0.510429\pi$$
−0.0327571 + 0.999463i $$0.510429\pi$$
$$182$$ 0 0
$$183$$ 1.58564e8i 1.91261i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.81426e7i 0.314716i
$$188$$ 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
$$193$$ 1.15092e8i 1.15237i 0.817319 + 0.576186i $$0.195460\pi$$
−0.817319 + 0.576186i $$0.804540\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1.38522e8i 1.29088i 0.763810 + 0.645441i $$0.223326\pi$$
−0.763810 + 0.645441i $$0.776674\pi$$
$$198$$ 0 0
$$199$$ −2.19614e7 −0.197548 −0.0987742 0.995110i $$-0.531492\pi$$
−0.0987742 + 0.995110i $$0.531492\pi$$
$$200$$ 0 0
$$201$$ −2.49133e8 −2.16394
$$202$$ 0 0
$$203$$ 4.55079e7i 0.381813i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 3.97817e8i 3.11736i
$$208$$ 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
$$213$$ 2.14051e8i 1.51771i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 1.84589e7i − 0.122630i
$$218$$ 0 0
$$219$$ 1.41147e8 0.908068
$$220$$ 0 0
$$221$$ 1.20175e8 0.748928
$$222$$ 0 0
$$223$$ 4.22448e7i 0.255098i 0.991832 + 0.127549i $$0.0407110\pi$$
−0.991832 + 0.127549i $$0.959289\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 2.39102e8i − 1.35673i −0.734726 0.678364i $$-0.762689\pi$$
0.734726 0.678364i $$-0.237311\pi$$
$$228$$ 0 0
$$229$$ 4.67889e7 0.257465 0.128733 0.991679i $$-0.458909\pi$$
0.128733 + 0.991679i $$0.458909\pi$$
$$230$$ 0 0
$$231$$ 9.66793e7 0.516050
$$232$$ 0 0
$$233$$ 3.45225e8i 1.78795i 0.448113 + 0.893977i $$0.352096\pi$$
−0.448113 + 0.893977i $$0.647904\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 3.39197e8i 1.65513i
$$238$$ 0 0
$$239$$ 2.34413e8 1.11068 0.555340 0.831624i $$-0.312588\pi$$
0.555340 + 0.831624i $$0.312588\pi$$
$$240$$ 0 0
$$241$$ −1.09557e8 −0.504175 −0.252087 0.967705i $$-0.581117\pi$$
−0.252087 + 0.967705i $$0.581117\pi$$
$$242$$ 0 0
$$243$$ 2.02453e8i 0.905112i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 4.44485e7i − 0.187680i
$$248$$ 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
$$253$$ − 2.06221e8i − 0.800591i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 3.19064e8i − 1.17250i −0.810131 0.586248i $$-0.800604\pi$$
0.810131 0.586248i $$-0.199396\pi$$
$$258$$ 0 0
$$259$$ 1.91266e8 0.684050
$$260$$ 0 0
$$261$$ 4.85916e8 1.69169
$$262$$ 0 0
$$263$$ − 2.19359e8i − 0.743549i −0.928323 0.371774i $$-0.878750\pi$$
0.928323 0.371774i $$-0.121250\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 5.43736e8i 1.74823i
$$268$$ 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 $$-0.309050\pi$$
0.564549 + 0.825399i $$0.309050\pi$$
$$272$$ 0 0
$$273$$ − 4.12841e8i − 1.22804i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3.95860e8i 1.11908i 0.828803 + 0.559541i $$0.189023\pi$$
−0.828803 + 0.559541i $$0.810977\pi$$
$$278$$ 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
$$283$$ − 8.05797e7i − 0.211336i −0.994401 0.105668i $$-0.966302\pi$$
0.994401 0.105668i $$-0.0336981\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 6.44793e7i − 0.161003i
$$288$$ 0 0
$$289$$ 2.86016e8 0.697025
$$290$$ 0 0
$$291$$ −5.09524e8 −1.21210
$$292$$ 0 0
$$293$$ 7.54530e8i 1.75243i 0.481924 + 0.876213i $$0.339938\pi$$
−0.481924 + 0.876213i $$0.660062\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 5.68627e8i − 1.25945i
$$298$$ 0 0
$$299$$ −8.80606e8 −1.90516
$$300$$ 0 0
$$301$$ 3.14835e8 0.665427
$$302$$ 0 0
$$303$$ 8.14858e8i 1.68280i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 8.20472e8i 1.61838i 0.587549 + 0.809188i $$0.300093\pi$$
−0.587549 + 0.809188i $$0.699907\pi$$
$$308$$ 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
$$313$$ 6.63587e8i 1.22319i 0.791172 + 0.611594i $$0.209471\pi$$
−0.791172 + 0.611594i $$0.790529\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3.54718e8i 0.625426i 0.949848 + 0.312713i $$0.101238\pi$$
−0.949848 + 0.312713i $$0.898762\pi$$
$$318$$ 0 0
$$319$$ −2.51890e8 −0.434454
$$320$$ 0 0
$$321$$ −6.12360e6 −0.0103333
$$322$$ 0 0
$$323$$ 4.59826e7i 0.0759250i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 8.02906e8i − 1.26984i
$$328$$ 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
$$333$$ − 2.04226e9i − 3.03080i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 3.54965e7i − 0.0505220i −0.999681 0.0252610i $$-0.991958\pi$$
0.999681 0.0252610i $$-0.00804169\pi$$
$$338$$ 0 0
$$339$$ −7.84467e8 −1.09364
$$340$$ 0 0
$$341$$ 1.02172e8 0.139537
$$342$$ 0 0
$$343$$ − 6.56252e8i − 0.878095i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 1.90594e8i − 0.244882i −0.992476 0.122441i $$-0.960928\pi$$
0.992476 0.122441i $$-0.0390723\pi$$
$$348$$ 0 0
$$349$$ −8.60864e8 −1.08404 −0.542020 0.840366i $$-0.682340\pi$$
−0.542020 + 0.840366i $$0.682340\pi$$
$$350$$ 0 0
$$351$$ −2.42815e9 −2.99710
$$352$$ 0 0
$$353$$ − 1.04544e9i − 1.26500i −0.774562 0.632498i $$-0.782030\pi$$
0.774562 0.632498i $$-0.217970\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 4.27090e8i 0.496798i
$$358$$ 0 0
$$359$$ 7.63303e8 0.870696 0.435348 0.900262i $$-0.356625\pi$$
0.435348 + 0.900262i $$0.356625\pi$$
$$360$$ 0 0
$$361$$ −8.76864e8 −0.980973
$$362$$ 0 0
$$363$$ − 1.10179e9i − 1.20900i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 1.38692e9i − 1.46460i −0.680980 0.732302i $$-0.738446\pi$$
0.680980 0.732302i $$-0.261554\pi$$
$$368$$ 0 0
$$369$$ −6.88486e8 −0.713351
$$370$$ 0 0
$$371$$ 8.26782e8 0.840586
$$372$$ 0 0
$$373$$ 4.77105e8i 0.476029i 0.971262 + 0.238015i $$0.0764966\pi$$
−0.971262 + 0.238015i $$0.923503\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1.07562e9i 1.03387i
$$378$$ 0 0
$$379$$ −3.92468e8 −0.370311 −0.185156 0.982709i $$-0.559279\pi$$
−0.185156 + 0.982709i $$0.559279\pi$$
$$380$$ 0 0
$$381$$ 3.02227e9 2.79960
$$382$$ 0 0
$$383$$ − 2.10409e9i − 1.91368i −0.290617 0.956839i $$-0.593861\pi$$
0.290617 0.956839i $$-0.406139\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 3.36169e9i − 2.94829i
$$388$$ 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
$$393$$ 5.67884e7i 0.0471939i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 9.81298e8i 0.787107i 0.919302 + 0.393554i $$0.128754\pi$$
−0.919302 + 0.393554i $$0.871246\pi$$
$$398$$ 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
$$403$$ − 4.36293e8i − 0.332056i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1.05867e9i 0.778361i
$$408$$ 0 0
$$409$$ 3.55609e7 0.0257004 0.0128502 0.999917i $$-0.495910\pi$$
0.0128502 + 0.999917i $$0.495910\pi$$
$$410$$ 0 0
$$411$$ −2.48507e9 −1.76560
$$412$$ 0 0
$$413$$ 4.40509e8i 0.307701i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 2.68031e9i − 1.81013i
$$418$$ 0 0
$$419$$ 2.65360e9 1.76233 0.881163 0.472813i $$-0.156761\pi$$
0.881163 + 0.472813i $$0.156761\pi$$
$$420$$ 0 0
$$421$$ −1.12113e9 −0.732264 −0.366132 0.930563i $$-0.619318\pi$$
−0.366132 + 0.930563i $$0.619318\pi$$
$$422$$ 0 0
$$423$$ 3.32081e9i 2.13331i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 8.60778e8i − 0.535049i
$$428$$ 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
$$433$$ − 7.05962e8i − 0.417901i −0.977926 0.208951i $$-0.932995\pi$$
0.977926 0.208951i $$-0.0670048\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 3.36947e8i − 0.193142i
$$438$$ 0 0
$$439$$ 1.48506e9 0.837760 0.418880 0.908042i $$-0.362423\pi$$
0.418880 + 0.908042i $$0.362423\pi$$
$$440$$ 0 0
$$441$$ −2.99739e9 −1.66421
$$442$$ 0 0
$$443$$ 7.22153e8i 0.394654i 0.980338 + 0.197327i $$0.0632260\pi$$
−0.980338 + 0.197327i $$0.936774\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 9.79462e8i 0.518694i
$$448$$ 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
$$453$$ 1.48088e9i 0.748473i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 8.85551e7i 0.0434017i 0.999765 + 0.0217009i $$0.00690814\pi$$
−0.999765 + 0.0217009i $$0.993092\pi$$
$$458$$ 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
$$463$$ 3.29775e9i 1.54413i 0.635543 + 0.772066i $$0.280776\pi$$
−0.635543 + 0.772066i $$0.719224\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 8.82873e7i 0.0401134i 0.999799 + 0.0200567i $$0.00638467\pi$$
−0.999799 + 0.0200567i $$0.993615\pi$$
$$468$$ 0 0
$$469$$ 1.35244e9 0.605357
$$470$$ 0 0
$$471$$ 5.33113e8 0.235096
$$472$$ 0 0
$$473$$ 1.74264e9i 0.757171i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 8.82807e9i − 3.72436i
$$478$$ 0 0
$$479$$ −4.51507e9 −1.87711 −0.938557 0.345125i $$-0.887836\pi$$
−0.938557 + 0.345125i $$0.887836\pi$$
$$480$$ 0 0
$$481$$ 4.52075e9 1.85226
$$482$$ 0 0
$$483$$ − 3.12959e9i − 1.26378i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 3.31338e9i 1.29993i 0.759965 + 0.649964i $$0.225216\pi$$
−0.759965 + 0.649964i $$0.774784\pi$$
$$488$$ 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
$$493$$ − 1.11275e9i − 0.418247i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 1.16199e9i − 0.424577i
$$498$$ 0 0
$$499$$ 2.70976e9 0.976290 0.488145 0.872763i $$-0.337674\pi$$
0.488145 + 0.872763i $$0.337674\pi$$
$$500$$ 0 0
$$501$$ −9.64045e9 −3.42504
$$502$$ 0 0
$$503$$ − 3.04579e8i − 0.106712i −0.998576 0.0533558i $$-0.983008\pi$$
0.998576 0.0533558i $$-0.0169918\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 4.48701e9i − 1.52908i
$$508$$ 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
$$513$$ − 9.29088e8i − 0.303841i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 1.72145e9i − 0.547870i
$$518$$ 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
$$523$$ − 2.51360e9i − 0.768318i −0.923267 0.384159i $$-0.874491\pi$$
0.923267 0.384159i $$-0.125509\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4.51352e8i 0.134332i
$$528$$ 0 0
$$529$$ −3.27072e9 −0.960613
$$530$$ 0 0
$$531$$ 4.70359e9 1.36332
$$532$$ 0 0
$$533$$ − 1.52403e9i − 0.435962i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 9.51112e8i − 0.265046i
$$538$$ 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
$$543$$ − 4.39026e8i − 0.117677i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 5.58047e8i 0.145786i 0.997340 + 0.0728929i $$0.0232231\pi$$
−0.997340 + 0.0728929i $$0.976777\pi$$
$$548$$ 0 0
$$549$$ −9.19107e9 −2.37062
$$550$$ 0 0
$$551$$ −4.11567e8 −0.104812
$$552$$ 0 0
$$553$$ − 1.84136e9i − 0.463020i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 3.30331e9i 0.809946i 0.914329 + 0.404973i $$0.132719\pi$$
−0.914329 + 0.404973i $$0.867281\pi$$
$$558$$ 0 0
$$559$$ 7.44143e9 1.80184
$$560$$ 0 0
$$561$$ −2.36398e9 −0.565293
$$562$$ 0 0
$$563$$ 1.22011e8i 0.0288152i 0.999896 + 0.0144076i $$0.00458623\pi$$
−0.999896 + 0.0144076i $$0.995414\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 3.77371e9i − 0.869417i
$$568$$ 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 $$-0.212509\pi$$
0.785300 + 0.619116i $$0.212509\pi$$
$$572$$ 0 0
$$573$$ 7.14253e9i 1.58603i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 8.16573e9i 1.76962i 0.465954 + 0.884809i $$0.345711\pi$$
−0.465954 + 0.884809i $$0.654289\pi$$
$$578$$ 0 0
$$579$$ −9.66769e9 −2.06989
$$580$$ 0 0
$$581$$ 2.45591e9 0.519512
$$582$$ 0 0
$$583$$ 4.57631e9i 0.956479i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 8.53182e9i 1.74104i 0.492135 + 0.870519i $$0.336217\pi$$
−0.492135 + 0.870519i $$0.663783\pi$$
$$588$$ 0 0
$$589$$ 1.66940e8 0.0336632
$$590$$ 0 0
$$591$$ −1.16358e10 −2.31868
$$592$$ 0 0
$$593$$ − 1.71175e9i − 0.337092i −0.985694 0.168546i $$-0.946093\pi$$
0.985694 0.168546i $$-0.0539072\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 1.84475e9i − 0.354836i
$$598$$ 0 0
$$599$$ −4.77362e9 −0.907516 −0.453758 0.891125i $$-0.649917\pi$$
−0.453758 + 0.891125i $$0.649917\pi$$
$$600$$ 0 0
$$601$$ 7.89998e8 0.148445 0.0742224 0.997242i $$-0.476353\pi$$
0.0742224 + 0.997242i $$0.476353\pi$$
$$602$$ 0 0
$$603$$ − 1.44408e10i − 2.68214i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 1.82652e9i − 0.331485i −0.986169 0.165743i $$-0.946998\pi$$
0.986169 0.165743i $$-0.0530021\pi$$
$$608$$ 0 0
$$609$$ −3.82266e9 −0.685813
$$610$$ 0 0
$$611$$ −7.35094e9 −1.30376
$$612$$ 0 0
$$613$$ − 6.90339e9i − 1.21046i −0.796050 0.605231i $$-0.793081\pi$$
0.796050 0.605231i $$-0.206919\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 5.69235e9i 0.975649i 0.872942 + 0.487825i $$0.162209\pi$$
−0.872942 + 0.487825i $$0.837791\pi$$
$$618$$ 0 0
$$619$$ −4.28594e9 −0.726321 −0.363161 0.931727i $$-0.618302\pi$$
−0.363161 + 0.931727i $$0.618302\pi$$
$$620$$ 0 0
$$621$$ −1.84069e10 −3.08433
$$622$$ 0 0
$$623$$ − 2.95171e9i − 0.489064i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 8.74354e8i 0.141661i
$$628$$ 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
$$633$$ 5.12575e9i 0.803238i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 6.63501e9i − 1.01708i
$$638$$ 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
$$643$$ 1.04374e10i 1.54830i 0.633004 + 0.774148i $$0.281822\pi$$
−0.633004 + 0.774148i $$0.718178\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 9.71623e8i − 0.141037i −0.997510 0.0705185i $$-0.977535\pi$$
0.997510 0.0705185i $$-0.0224654\pi$$
$$648$$ 0 0
$$649$$ −2.43825e9 −0.350125
$$650$$ 0 0
$$651$$ 1.55055e9 0.220268
$$652$$ 0 0
$$653$$ − 7.25223e9i − 1.01924i −0.860400 0.509619i $$-0.829786\pi$$
0.860400 0.509619i $$-0.170214\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 8.18151e9i 1.12552i
$$658$$ 0 0
$$659$$ 3.81924e9 0.519851 0.259925 0.965629i $$-0.416302\pi$$
0.259925 + 0.965629i $$0.416302\pi$$
$$660$$ 0 0
$$661$$ 1.07881e10 1.45292 0.726459 0.687210i $$-0.241165\pi$$
0.726459 + 0.687210i $$0.241165\pi$$
$$662$$ 0 0
$$663$$ 1.00947e10i 1.34523i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 8.15390e9i 1.06396i
$$668$$ 0 0
$$669$$ −3.54857e9 −0.458207
$$670$$ 0 0
$$671$$ 4.76448e9 0.608817
$$672$$ 0 0
$$673$$ − 6.34833e9i − 0.802798i −0.915903 0.401399i $$-0.868524\pi$$
0.915903 0.401399i $$-0.131476\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 8.82566e9i − 1.09317i −0.837404 0.546584i $$-0.815928\pi$$
0.837404 0.546584i $$-0.184072\pi$$
$$678$$ 0 0
$$679$$ 2.76599e9 0.339083
$$680$$ 0 0
$$681$$ 2.00846e10 2.43696
$$682$$ 0 0
$$683$$ − 4.92331e9i − 0.591268i −0.955301 0.295634i $$-0.904469\pi$$
0.955301 0.295634i $$-0.0955309\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 3.93027e9i 0.462459i
$$688$$ 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
$$693$$ 5.60395e9i 0.639628i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1.57663e9i 0.176366i
$$698$$ 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
$$703$$ 1.72978e9i 0.187779i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 4.42351e9i − 0.470760i
$$708$$ 0 0
$$709$$ −4.52189e9 −0.476495 −0.238248 0.971204i $$-0.576573\pi$$
−0.238248 + 0.971204i $$0.576573\pi$$
$$710$$ 0 0
$$711$$ −1.96613e10 −2.05149
$$712$$ 0 0
$$713$$ − 3.30738e9i − 0.341720i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 1.96907e10i 1.99500i
$$718$$ 0 0
$$719$$ −3.09206e9 −0.310239 −0.155120 0.987896i $$-0.549576\pi$$
−0.155120 + 0.987896i $$0.549576\pi$$
$$720$$ 0 0
$$721$$ −1.87033e9 −0.185842
$$722$$ 0 0
$$723$$ − 9.20280e9i − 0.905599i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 1.44622e10i 1.39593i 0.716132 + 0.697965i $$0.245911\pi$$
−0.716132 + 0.697965i $$0.754089\pi$$
$$728$$ 0 0
$$729$$ 1.09288e9 0.104478
$$730$$ 0 0
$$731$$ −7.69827e9 −0.728924
$$732$$ 0 0
$$733$$ 3.15415e9i 0.295814i 0.989001 + 0.147907i $$0.0472536\pi$$
−0.989001 + 0.147907i $$0.952746\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 7.48585e9i 0.688819i
$$738$$ 0 0
$$739$$ 1.54236e10 1.40582 0.702912 0.711277i $$-0.251883\pi$$
0.702912 + 0.711277i $$0.251883\pi$$
$$740$$ 0 0
$$741$$ 3.73367e9 0.337111
$$742$$ 0 0
$$743$$ 1.59520e10i 1.42677i 0.700772 + 0.713385i $$0.252839\pi$$
−0.700772 + 0.713385i $$0.747161\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 2.62233e10i − 2.30179i
$$748$$ 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
$$753$$ 3.30986e10i 2.82506i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1.42818e10i 1.19660i 0.801273 + 0.598299i $$0.204157\pi$$
−0.801273 + 0.598299i $$0.795843\pi$$
$$758$$ 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
$$763$$ 4.35863e9i 0.355234i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1.04118e10i 0.833190i
$$768$$ 0 0
$$769$$ −1.97592e10 −1.56685 −0.783424 0.621487i $$-0.786529\pi$$
−0.783424 + 0.621487i $$0.786529\pi$$
$$770$$ 0 0
$$771$$ 2.68014e10 2.10604
$$772$$ 0 0
$$773$$ 1.01370e10i 0.789374i 0.918816 + 0.394687i $$0.129147\pi$$
−0.918816 + 0.394687i $$0.870853\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 1.60663e10i 1.22869i
$$778$$ 0 0
$$779$$ 5.83142e8 0.0441970
$$780$$ 0 0
$$781$$ 6.43174e9 0.483114
$$782$$ 0 0
$$783$$ 2.24833e10i 1.67376i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 1.27882e10i − 0.935188i −0.883943 0.467594i $$-0.845121\pi$$
0.883943 0.467594i $$-0.154879\pi$$
$$788$$ 0 0
$$789$$ 1.84261e10 1.33556
$$790$$ 0 0
$$791$$ 4.25854e9 0.305945
$$792$$ 0 0
$$793$$ − 2.03453e10i − 1.44880i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 7.38617e9i 0.516791i 0.966039 + 0.258396i $$0.0831938\pi$$
−0.966039 + 0.258396i $$0.916806\pi$$
$$798$$ 0 0
$$799$$ 7.60466e9 0.527431
$$800$$ 0 0
$$801$$ −3.15173e10 −2.16688
$$802$$ 0 0
$$803$$ − 4.24114e9i − 0.289054i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 1.24348e10i 0.832875i
$$808$$ 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 $$-0.604312\pi$$
−0.321872 + 0.946783i $$0.604312\pi$$
$$812$$ 0 0
$$813$$ 3.10745e10i 2.02809i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.84733e9i 0.182667i
$$818$$ 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
$$823$$ 3.16960e10i 1.98201i 0.133829 + 0.991004i $$0.457273\pi$$
−0.133829 + 0.991004i $$0.542727\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 6.12845e9i − 0.376774i −0.982095 0.188387i $$-0.939674\pi$$
0.982095 0.188387i $$-0.0603260\pi$$
$$828$$ 0 0
$$829$$ 1.24652e10 0.759904 0.379952 0.925006i $$-0.375940\pi$$
0.379952 + 0.925006i $$0.375940\pi$$
$$830$$ 0 0
$$831$$ −3.32522e10 −2.01010
$$832$$ 0 0
$$833$$ 6.86402e9i 0.411454i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 9.11966e9i − 0.537575i
$$838$$ 0 0
$$839$$ −1.82237e10 −1.06530 −0.532648 0.846337i $$-0.678803\pi$$
−0.532648 + 0.846337i $$0.678803\pi$$
$$840$$ 0 0
$$841$$ −7.29024e9 −0.422625
$$842$$ 0 0
$$843$$ − 5.02118e10i − 2.88675i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.98117e9i 0.338216i
$$848$$ 0 0
$$849$$ 6.76870e9 0.379602
$$850$$ 0 0
$$851$$ 3.42701e10 1.90617
$$852$$ 0 0
$$853$$ − 2.48619e10i − 1.37155i −0.727812 0.685777i $$-0.759463\pi$$
0.727812 0.685777i $$-0.240537\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 2.96761e10i − 1.61055i −0.592902 0.805275i $$-0.702018\pi$$
0.592902 0.805275i $$-0.297982\pi$$
$$858$$ 0 0
$$859$$ 1.14772e10 0.617819 0.308910 0.951091i $$-0.400036\pi$$
0.308910 + 0.951091i $$0.400036\pi$$
$$860$$ 0 0
$$861$$ 5.41626e9 0.289194
$$862$$ 0 0
$$863$$ − 2.13485e10i − 1.13066i −0.824866 0.565328i $$-0.808750\pi$$
0.824866 0.565328i $$-0.191250\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 2.40254e10i 1.25200i
$$868$$ 0 0
$$869$$ 1.01921e10 0.526858
$$870$$ 0 0
$$871$$ 3.19661e10 1.63918
$$872$$ 0 0
$$873$$ − 2.95342e10i − 1.50236i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 7.92753e9i 0.396862i 0.980115 + 0.198431i $$0.0635846\pi$$
−0.980115 + 0.198431i $$0.936415\pi$$
$$878$$ 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
$$883$$ 3.54988e9i 0.173521i 0.996229 + 0.0867604i $$0.0276514\pi$$
−0.996229 + 0.0867604i $$0.972349\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 5.80634e9i 0.279364i 0.990196 + 0.139682i $$0.0446080\pi$$
−0.990196 + 0.139682i $$0.955392\pi$$
$$888$$ 0 0
$$889$$ −1.64066e10 −0.783182
$$890$$ 0 0
$$891$$ 2.08878e10 0.989285
$$892$$ 0 0
$$893$$ − 2.81270e9i − 0.132173i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 7.39709e10i − 3.42206i
$$898$$ 0 0
$$899$$ −4.03982e9 −0.185440
$$900$$ 0 0
$$901$$ −2.02163e10 −0.920798
$$902$$ 0 0
$$903$$ 2.64462e10i 1.19524i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 1.78240e10i 0.793196i 0.917992 + 0.396598i $$0.129809\pi$$
−0.917992 + 0.396598i $$0.870191\pi$$
$$908$$ 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
$$913$$ 1.35937e10i 0.591138i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 3.08280e8i − 0.0132024i
$$918$$ 0 0
$$919$$ 3.75844e10 1.59736 0.798681 0.601754i $$-0.205531\pi$$
0.798681 + 0.601754i $$0.205531\pi$$
$$920$$ 0 0
$$921$$ −6.89197e10 −2.90693
$$922$$ 0 0
$$923$$ − 2.74648e10i − 1.14966i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 1.99707e10i 0.823404i
$$928$$ 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
$$933$$ − 5.48943e10i − 2.21279i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 1.04732e9i − 0.0415900i −0.999784 0.0207950i $$-0.993380\pi$$
0.999784 0.0207950i $$-0.00661973\pi$$
$$938$$ 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
$$943$$ − 1.15531e10i − 0.448650i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 4.26943e9i − 0.163360i −0.996659 0.0816799i $$-0.973971\pi$$
0.996659 0.0816799i $$-0.0260285\pi$$
$$948$$ 0 0
$$949$$ −1.81106e10 −0.687860
$$950$$ 0 0
$$951$$ −2.97963e10 −1.12339
$$952$$ 0 0
$$953$$ − 1.06048e10i − 0.396897i −0.980111 0.198449i $$-0.936410\pi$$
0.980111 0.198449i $$-0.0635903\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 2.11588e10i − 0.780367i
$$958$$ 0 0
$$959$$ 1.34904e10 0.493922
$$960$$ 0 0
$$961$$ −2.58740e10 −0.940441
$$962$$ 0 0
$$963$$ − 3.54950e8i − 0.0128078i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 1.65090e10i − 0.587123i −0.955940 0.293562i $$-0.905159\pi$$
0.955940 0.293562i $$-0.0948406\pi$$
$$968$$ 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
$$973$$ 1.45503e10i 0.506379i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 2.53886e10i 0.870979i 0.900194 + 0.435489i $$0.143425\pi$$
−0.900194 + 0.435489i $$0.856575\pi$$
$$978$$ 0 0
$$979$$ 1.63380e10 0.556492
$$980$$ 0 0
$$981$$ 4.65399e10 1.57392
$$982$$ 0 0
$$983$$ − 1.87585e10i − 0.629884i −0.949111 0.314942i $$-0.898015\pi$$
0.949111 0.314942i $$-0.101985\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 2.61246e10i − 0.864846i
$$988$$ 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
$$993$$ 2.56656e10i 0.831821i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.34287e10i 0.429143i 0.976708 + 0.214571i $$0.0688354\pi$$
−0.976708 + 0.214571i $$0.931165\pi$$
$$998$$ 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.c.b.49.2 2
4.3 odd 2 200.8.c.a.49.1 2
5.2 odd 4 16.8.a.c.1.1 1
5.3 odd 4 400.8.a.b.1.1 1
5.4 even 2 inner 400.8.c.b.49.1 2
15.2 even 4 144.8.a.g.1.1 1
20.3 even 4 200.8.a.i.1.1 1
20.7 even 4 8.8.a.a.1.1 1
20.19 odd 2 200.8.c.a.49.2 2
40.27 even 4 64.8.a.g.1.1 1
40.37 odd 4 64.8.a.a.1.1 1
60.47 odd 4 72.8.a.d.1.1 1
80.27 even 4 256.8.b.e.129.2 2
80.37 odd 4 256.8.b.c.129.1 2
80.67 even 4 256.8.b.e.129.1 2
80.77 odd 4 256.8.b.c.129.2 2
120.77 even 4 576.8.a.k.1.1 1
120.107 odd 4 576.8.a.j.1.1 1
140.27 odd 4 392.8.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.a.a.1.1 1 20.7 even 4
16.8.a.c.1.1 1 5.2 odd 4
64.8.a.a.1.1 1 40.37 odd 4
64.8.a.g.1.1 1 40.27 even 4
72.8.a.d.1.1 1 60.47 odd 4
144.8.a.g.1.1 1 15.2 even 4
200.8.a.i.1.1 1 20.3 even 4
200.8.c.a.49.1 2 4.3 odd 2
200.8.c.a.49.2 2 20.19 odd 2
256.8.b.c.129.1 2 80.37 odd 4
256.8.b.c.129.2 2 80.77 odd 4
256.8.b.e.129.1 2 80.67 even 4
256.8.b.e.129.2 2 80.27 even 4
392.8.a.d.1.1 1 140.27 odd 4
400.8.a.b.1.1 1 5.3 odd 4
400.8.c.b.49.1 2 5.4 even 2 inner
400.8.c.b.49.2 2 1.1 even 1 trivial
576.8.a.j.1.1 1 120.107 odd 4
576.8.a.k.1.1 1 120.77 even 4