# Properties

 Label 27.9.b.b.26.1 Level $27$ Weight $9$ Character 27.26 Analytic conductor $10.999$ 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] = [27,9,Mod(26,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("27.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$27 = 3^{3}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 27.b (of order $$2$$, degree $$1$$, 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: $$10.9992224717$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 6$$ x^2 + 6 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 26.1 Root $$-2.44949i$$ of defining polynomial Character $$\chi$$ $$=$$ 27.26 Dual form 27.9.b.b.26.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-29.3939i q^{2} -608.000 q^{4} +823.029i q^{5} +1967.00 q^{7} +10346.6i q^{8} +O(q^{10})$$ $$q-29.3939i q^{2} -608.000 q^{4} +823.029i q^{5} +1967.00 q^{7} +10346.6i q^{8} +24192.0 q^{10} +12580.6i q^{11} -45505.0 q^{13} -57817.8i q^{14} +148480. q^{16} +59610.8i q^{17} +152399. q^{19} -500401. i q^{20} +369792. q^{22} +131332. i q^{23} -286751. q^{25} +1.33757e6i q^{26} -1.19594e6 q^{28} +588583. i q^{29} -164350. q^{31} -1.71566e6i q^{32} +1.75219e6 q^{34} +1.61890e6i q^{35} -663937. q^{37} -4.47960e6i q^{38} -8.51558e6 q^{40} +938017. i q^{41} +575330. q^{43} -7.64899e6i q^{44} +3.86035e6 q^{46} +9.23426e6i q^{47} -1.89571e6 q^{49} +8.42872e6i q^{50} +2.76670e7 q^{52} -1.03765e7i q^{53} -1.03542e7 q^{55} +2.03519e7i q^{56} +1.73007e7 q^{58} +5.03987e6i q^{59} -1.92130e7 q^{61} +4.83088e6i q^{62} -1.24191e7 q^{64} -3.74519e7i q^{65} -598033. q^{67} -3.62434e7i q^{68} +4.75857e7 q^{70} -2.92721e7i q^{71} +1.28502e7 q^{73} +1.95157e7i q^{74} -9.26586e7 q^{76} +2.47460e7i q^{77} -2.35847e7 q^{79} +1.22203e8i q^{80} +2.75720e7 q^{82} -3.34509e7i q^{83} -4.90614e7 q^{85} -1.69112e7i q^{86} -1.30167e8 q^{88} +2.82848e7i q^{89} -8.95083e7 q^{91} -7.98498e7i q^{92} +2.71431e8 q^{94} +1.25429e8i q^{95} +1.36490e8 q^{97} +5.57223e7i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 1216 q^{4} + 3934 q^{7}+O(q^{10})$$ 2 * q - 1216 * q^4 + 3934 * q^7 $$2 q - 1216 q^{4} + 3934 q^{7} + 48384 q^{10} - 91010 q^{13} + 296960 q^{16} + 304798 q^{19} + 739584 q^{22} - 573502 q^{25} - 2391872 q^{28} - 328700 q^{31} + 3504384 q^{34} - 1327874 q^{37} - 17031168 q^{40} + 1150660 q^{43} + 7720704 q^{46} - 3791424 q^{49} + 55334080 q^{52} - 20708352 q^{55} + 34601472 q^{58} - 38425922 q^{61} - 24838144 q^{64} - 1196066 q^{67} + 95171328 q^{70} + 25700350 q^{73} - 185317184 q^{76} - 47169314 q^{79} + 55143936 q^{82} - 98122752 q^{85} - 260333568 q^{88} - 179016670 q^{91} + 542861568 q^{94} + 272979262 q^{97}+O(q^{100})$$ 2 * q - 1216 * q^4 + 3934 * q^7 + 48384 * q^10 - 91010 * q^13 + 296960 * q^16 + 304798 * q^19 + 739584 * q^22 - 573502 * q^25 - 2391872 * q^28 - 328700 * q^31 + 3504384 * q^34 - 1327874 * q^37 - 17031168 * q^40 + 1150660 * q^43 + 7720704 * q^46 - 3791424 * q^49 + 55334080 * q^52 - 20708352 * q^55 + 34601472 * q^58 - 38425922 * q^61 - 24838144 * q^64 - 1196066 * q^67 + 95171328 * q^70 + 25700350 * q^73 - 185317184 * q^76 - 47169314 * q^79 + 55143936 * q^82 - 98122752 * q^85 - 260333568 * q^88 - 179016670 * q^91 + 542861568 * q^94 + 272979262 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 29.3939i − 1.83712i −0.395285 0.918559i $$-0.629354\pi$$
0.395285 0.918559i $$-0.370646\pi$$
$$3$$ 0 0
$$4$$ −608.000 −2.37500
$$5$$ 823.029i 1.31685i 0.752648 + 0.658423i $$0.228776\pi$$
−0.752648 + 0.658423i $$0.771224\pi$$
$$6$$ 0 0
$$7$$ 1967.00 0.819242 0.409621 0.912256i $$-0.365661\pi$$
0.409621 + 0.912256i $$0.365661\pi$$
$$8$$ 10346.6i 2.52604i
$$9$$ 0 0
$$10$$ 24192.0 2.41920
$$11$$ 12580.6i 0.859270i 0.903003 + 0.429635i $$0.141358\pi$$
−0.903003 + 0.429635i $$0.858642\pi$$
$$12$$ 0 0
$$13$$ −45505.0 −1.59326 −0.796628 0.604470i $$-0.793385\pi$$
−0.796628 + 0.604470i $$0.793385\pi$$
$$14$$ − 57817.8i − 1.50504i
$$15$$ 0 0
$$16$$ 148480. 2.26562
$$17$$ 59610.8i 0.713722i 0.934157 + 0.356861i $$0.116153\pi$$
−0.934157 + 0.356861i $$0.883847\pi$$
$$18$$ 0 0
$$19$$ 152399. 1.16941 0.584706 0.811245i $$-0.301210\pi$$
0.584706 + 0.811245i $$0.301210\pi$$
$$20$$ − 500401.i − 3.12751i
$$21$$ 0 0
$$22$$ 369792. 1.57858
$$23$$ 131332.i 0.469309i 0.972079 + 0.234654i $$0.0753958\pi$$
−0.972079 + 0.234654i $$0.924604\pi$$
$$24$$ 0 0
$$25$$ −286751. −0.734083
$$26$$ 1.33757e6i 2.92700i
$$27$$ 0 0
$$28$$ −1.19594e6 −1.94570
$$29$$ 588583.i 0.832177i 0.909324 + 0.416089i $$0.136599\pi$$
−0.909324 + 0.416089i $$0.863401\pi$$
$$30$$ 0 0
$$31$$ −164350. −0.177960 −0.0889801 0.996033i $$-0.528361\pi$$
−0.0889801 + 0.996033i $$0.528361\pi$$
$$32$$ − 1.71566e6i − 1.63618i
$$33$$ 0 0
$$34$$ 1.75219e6 1.31119
$$35$$ 1.61890e6i 1.07882i
$$36$$ 0 0
$$37$$ −663937. −0.354258 −0.177129 0.984188i $$-0.556681\pi$$
−0.177129 + 0.984188i $$0.556681\pi$$
$$38$$ − 4.47960e6i − 2.14835i
$$39$$ 0 0
$$40$$ −8.51558e6 −3.32640
$$41$$ 938017.i 0.331952i 0.986130 + 0.165976i $$0.0530774\pi$$
−0.986130 + 0.165976i $$0.946923\pi$$
$$42$$ 0 0
$$43$$ 575330. 0.168284 0.0841421 0.996454i $$-0.473185\pi$$
0.0841421 + 0.996454i $$0.473185\pi$$
$$44$$ − 7.64899e6i − 2.04077i
$$45$$ 0 0
$$46$$ 3.86035e6 0.862175
$$47$$ 9.23426e6i 1.89239i 0.323596 + 0.946195i $$0.395108\pi$$
−0.323596 + 0.946195i $$0.604892\pi$$
$$48$$ 0 0
$$49$$ −1.89571e6 −0.328843
$$50$$ 8.42872e6i 1.34860i
$$51$$ 0 0
$$52$$ 2.76670e7 3.78398
$$53$$ − 1.03765e7i − 1.31507i −0.753425 0.657533i $$-0.771600\pi$$
0.753425 0.657533i $$-0.228400\pi$$
$$54$$ 0 0
$$55$$ −1.03542e7 −1.13153
$$56$$ 2.03519e7i 2.06943i
$$57$$ 0 0
$$58$$ 1.73007e7 1.52881
$$59$$ 5.03987e6i 0.415922i 0.978137 + 0.207961i $$0.0666827\pi$$
−0.978137 + 0.207961i $$0.933317\pi$$
$$60$$ 0 0
$$61$$ −1.92130e7 −1.38763 −0.693817 0.720151i $$-0.744072\pi$$
−0.693817 + 0.720151i $$0.744072\pi$$
$$62$$ 4.83088e6i 0.326934i
$$63$$ 0 0
$$64$$ −1.24191e7 −0.740234
$$65$$ − 3.74519e7i − 2.09807i
$$66$$ 0 0
$$67$$ −598033. −0.0296774 −0.0148387 0.999890i $$-0.504723\pi$$
−0.0148387 + 0.999890i $$0.504723\pi$$
$$68$$ − 3.62434e7i − 1.69509i
$$69$$ 0 0
$$70$$ 4.75857e7 1.98191
$$71$$ − 2.92721e7i − 1.15191i −0.817480 0.575957i $$-0.804630\pi$$
0.817480 0.575957i $$-0.195370\pi$$
$$72$$ 0 0
$$73$$ 1.28502e7 0.452499 0.226249 0.974069i $$-0.427354\pi$$
0.226249 + 0.974069i $$0.427354\pi$$
$$74$$ 1.95157e7i 0.650814i
$$75$$ 0 0
$$76$$ −9.26586e7 −2.77735
$$77$$ 2.47460e7i 0.703950i
$$78$$ 0 0
$$79$$ −2.35847e7 −0.605510 −0.302755 0.953068i $$-0.597906\pi$$
−0.302755 + 0.953068i $$0.597906\pi$$
$$80$$ 1.22203e8i 2.98348i
$$81$$ 0 0
$$82$$ 2.75720e7 0.609835
$$83$$ − 3.34509e7i − 0.704849i −0.935840 0.352424i $$-0.885357\pi$$
0.935840 0.352424i $$-0.114643\pi$$
$$84$$ 0 0
$$85$$ −4.90614e7 −0.939862
$$86$$ − 1.69112e7i − 0.309158i
$$87$$ 0 0
$$88$$ −1.30167e8 −2.17055
$$89$$ 2.82848e7i 0.450809i 0.974265 + 0.225405i $$0.0723704\pi$$
−0.974265 + 0.225405i $$0.927630\pi$$
$$90$$ 0 0
$$91$$ −8.95083e7 −1.30526
$$92$$ − 7.98498e7i − 1.11461i
$$93$$ 0 0
$$94$$ 2.71431e8 3.47654
$$95$$ 1.25429e8i 1.53994i
$$96$$ 0 0
$$97$$ 1.36490e8 1.54175 0.770873 0.636989i $$-0.219820\pi$$
0.770873 + 0.636989i $$0.219820\pi$$
$$98$$ 5.57223e7i 0.604122i
$$99$$ 0 0
$$100$$ 1.74345e8 1.74345
$$101$$ − 4.02494e7i − 0.386789i −0.981121 0.193394i $$-0.938050\pi$$
0.981121 0.193394i $$-0.0619497\pi$$
$$102$$ 0 0
$$103$$ 3.35907e7 0.298449 0.149225 0.988803i $$-0.452322\pi$$
0.149225 + 0.988803i $$0.452322\pi$$
$$104$$ − 4.70824e8i − 4.02462i
$$105$$ 0 0
$$106$$ −3.05006e8 −2.41593
$$107$$ − 1.95483e8i − 1.49133i −0.666322 0.745664i $$-0.732132\pi$$
0.666322 0.745664i $$-0.267868\pi$$
$$108$$ 0 0
$$109$$ −1.01182e8 −0.716800 −0.358400 0.933568i $$-0.616678\pi$$
−0.358400 + 0.933568i $$0.616678\pi$$
$$110$$ 3.04349e8i 2.07875i
$$111$$ 0 0
$$112$$ 2.92060e8 1.85610
$$113$$ 3.14504e8i 1.92891i 0.264238 + 0.964457i $$0.414880\pi$$
−0.264238 + 0.964457i $$0.585120\pi$$
$$114$$ 0 0
$$115$$ −1.08090e8 −0.618007
$$116$$ − 3.57858e8i − 1.97642i
$$117$$ 0 0
$$118$$ 1.48141e8 0.764097
$$119$$ 1.17254e8i 0.584711i
$$120$$ 0 0
$$121$$ 5.60879e7 0.261654
$$122$$ 5.64743e8i 2.54925i
$$123$$ 0 0
$$124$$ 9.99248e7 0.422656
$$125$$ 8.54913e7i 0.350172i
$$126$$ 0 0
$$127$$ 4.30869e8 1.65627 0.828133 0.560532i $$-0.189403\pi$$
0.828133 + 0.560532i $$0.189403\pi$$
$$128$$ − 7.41647e7i − 0.276285i
$$129$$ 0 0
$$130$$ −1.10086e9 −3.85441
$$131$$ 1.88933e8i 0.641538i 0.947158 + 0.320769i $$0.103941\pi$$
−0.947158 + 0.320769i $$0.896059\pi$$
$$132$$ 0 0
$$133$$ 2.99769e8 0.958032
$$134$$ 1.75785e7i 0.0545209i
$$135$$ 0 0
$$136$$ −6.16772e8 −1.80289
$$137$$ 1.22998e8i 0.349152i 0.984644 + 0.174576i $$0.0558555\pi$$
−0.984644 + 0.174576i $$0.944144\pi$$
$$138$$ 0 0
$$139$$ 1.72072e8 0.460947 0.230474 0.973079i $$-0.425972\pi$$
0.230474 + 0.973079i $$0.425972\pi$$
$$140$$ − 9.84289e8i − 2.56219i
$$141$$ 0 0
$$142$$ −8.60420e8 −2.11620
$$143$$ − 5.72479e8i − 1.36904i
$$144$$ 0 0
$$145$$ −4.84421e8 −1.09585
$$146$$ − 3.77716e8i − 0.831294i
$$147$$ 0 0
$$148$$ 4.03674e8 0.841363
$$149$$ − 7.31667e8i − 1.48446i −0.670145 0.742230i $$-0.733768\pi$$
0.670145 0.742230i $$-0.266232\pi$$
$$150$$ 0 0
$$151$$ 1.85952e8 0.357679 0.178840 0.983878i $$-0.442766\pi$$
0.178840 + 0.983878i $$0.442766\pi$$
$$152$$ 1.57682e9i 2.95398i
$$153$$ 0 0
$$154$$ 7.27381e8 1.29324
$$155$$ − 1.35265e8i − 0.234346i
$$156$$ 0 0
$$157$$ 9.74007e8 1.60311 0.801556 0.597920i $$-0.204006\pi$$
0.801556 + 0.597920i $$0.204006\pi$$
$$158$$ 6.93245e8i 1.11239i
$$159$$ 0 0
$$160$$ 1.41204e9 2.15460
$$161$$ 2.58330e8i 0.384477i
$$162$$ 0 0
$$163$$ −1.15499e9 −1.63616 −0.818082 0.575102i $$-0.804962\pi$$
−0.818082 + 0.575102i $$0.804962\pi$$
$$164$$ − 5.70315e8i − 0.788386i
$$165$$ 0 0
$$166$$ −9.83253e8 −1.29489
$$167$$ − 1.14302e8i − 0.146956i −0.997297 0.0734779i $$-0.976590\pi$$
0.997297 0.0734779i $$-0.0234098\pi$$
$$168$$ 0 0
$$169$$ 1.25497e9 1.53847
$$170$$ 1.44210e9i 1.72664i
$$171$$ 0 0
$$172$$ −3.49801e8 −0.399675
$$173$$ − 1.65322e9i − 1.84564i −0.385235 0.922819i $$-0.625879\pi$$
0.385235 0.922819i $$-0.374121\pi$$
$$174$$ 0 0
$$175$$ −5.64039e8 −0.601391
$$176$$ 1.86796e9i 1.94678i
$$177$$ 0 0
$$178$$ 8.31400e8 0.828190
$$179$$ 4.17229e8i 0.406408i 0.979136 + 0.203204i $$0.0651355\pi$$
−0.979136 + 0.203204i $$0.934865\pi$$
$$180$$ 0 0
$$181$$ 1.16424e9 1.08475 0.542374 0.840137i $$-0.317526\pi$$
0.542374 + 0.840137i $$0.317526\pi$$
$$182$$ 2.63100e9i 2.39792i
$$183$$ 0 0
$$184$$ −1.35884e9 −1.18549
$$185$$ − 5.46439e8i − 0.466503i
$$186$$ 0 0
$$187$$ −7.49938e8 −0.613280
$$188$$ − 5.61443e9i − 4.49443i
$$189$$ 0 0
$$190$$ 3.68684e9 2.82904
$$191$$ 1.61055e9i 1.21016i 0.796166 + 0.605078i $$0.206858\pi$$
−0.796166 + 0.605078i $$0.793142\pi$$
$$192$$ 0 0
$$193$$ −7.10279e8 −0.511917 −0.255959 0.966688i $$-0.582391\pi$$
−0.255959 + 0.966688i $$0.582391\pi$$
$$194$$ − 4.01196e9i − 2.83237i
$$195$$ 0 0
$$196$$ 1.15259e9 0.781001
$$197$$ 1.41623e9i 0.940303i 0.882586 + 0.470152i $$0.155801\pi$$
−0.882586 + 0.470152i $$0.844199\pi$$
$$198$$ 0 0
$$199$$ −2.34324e9 −1.49419 −0.747093 0.664720i $$-0.768551\pi$$
−0.747093 + 0.664720i $$0.768551\pi$$
$$200$$ − 2.96691e9i − 1.85432i
$$201$$ 0 0
$$202$$ −1.18309e9 −0.710576
$$203$$ 1.15774e9i 0.681754i
$$204$$ 0 0
$$205$$ −7.72015e8 −0.437130
$$206$$ − 9.87362e8i − 0.548286i
$$207$$ 0 0
$$208$$ −6.75658e9 −3.60972
$$209$$ 1.91727e9i 1.00484i
$$210$$ 0 0
$$211$$ 1.06517e9 0.537389 0.268695 0.963225i $$-0.413408\pi$$
0.268695 + 0.963225i $$0.413408\pi$$
$$212$$ 6.30892e9i 3.12328i
$$213$$ 0 0
$$214$$ −5.74600e9 −2.73975
$$215$$ 4.73513e8i 0.221604i
$$216$$ 0 0
$$217$$ −3.23276e8 −0.145792
$$218$$ 2.97414e9i 1.31685i
$$219$$ 0 0
$$220$$ 6.29534e9 2.68738
$$221$$ − 2.71259e9i − 1.13714i
$$222$$ 0 0
$$223$$ 1.99586e9 0.807068 0.403534 0.914965i $$-0.367782\pi$$
0.403534 + 0.914965i $$0.367782\pi$$
$$224$$ − 3.37471e9i − 1.34043i
$$225$$ 0 0
$$226$$ 9.24451e9 3.54364
$$227$$ 1.79923e9i 0.677615i 0.940856 + 0.338807i $$0.110023\pi$$
−0.940856 + 0.338807i $$0.889977\pi$$
$$228$$ 0 0
$$229$$ 1.59787e9 0.581032 0.290516 0.956870i $$-0.406173\pi$$
0.290516 + 0.956870i $$0.406173\pi$$
$$230$$ 3.17718e9i 1.13535i
$$231$$ 0 0
$$232$$ −6.08986e9 −2.10211
$$233$$ 4.98411e9i 1.69108i 0.533912 + 0.845540i $$0.320721\pi$$
−0.533912 + 0.845540i $$0.679279\pi$$
$$234$$ 0 0
$$235$$ −7.60006e9 −2.49199
$$236$$ − 3.06424e9i − 0.987814i
$$237$$ 0 0
$$238$$ 3.44656e9 1.07418
$$239$$ − 5.89754e9i − 1.80751i −0.428055 0.903753i $$-0.640801\pi$$
0.428055 0.903753i $$-0.359199\pi$$
$$240$$ 0 0
$$241$$ 2.26174e9 0.670463 0.335231 0.942136i $$-0.391185\pi$$
0.335231 + 0.942136i $$0.391185\pi$$
$$242$$ − 1.64864e9i − 0.480689i
$$243$$ 0 0
$$244$$ 1.16815e10 3.29563
$$245$$ − 1.56023e9i − 0.433035i
$$246$$ 0 0
$$247$$ −6.93492e9 −1.86317
$$248$$ − 1.70047e9i − 0.449534i
$$249$$ 0 0
$$250$$ 2.51292e9 0.643307
$$251$$ − 3.31815e9i − 0.835989i −0.908450 0.417994i $$-0.862733\pi$$
0.908450 0.417994i $$-0.137267\pi$$
$$252$$ 0 0
$$253$$ −1.65223e9 −0.403263
$$254$$ − 1.26649e10i − 3.04276i
$$255$$ 0 0
$$256$$ −5.35927e9 −1.24780
$$257$$ 3.05018e9i 0.699185i 0.936902 + 0.349593i $$0.113680\pi$$
−0.936902 + 0.349593i $$0.886320\pi$$
$$258$$ 0 0
$$259$$ −1.30596e9 −0.290223
$$260$$ 2.27708e10i 4.98292i
$$261$$ 0 0
$$262$$ 5.55347e9 1.17858
$$263$$ − 1.97422e9i − 0.412640i −0.978485 0.206320i $$-0.933851\pi$$
0.978485 0.206320i $$-0.0661489\pi$$
$$264$$ 0 0
$$265$$ 8.54016e9 1.73174
$$266$$ − 8.81137e9i − 1.76002i
$$267$$ 0 0
$$268$$ 3.63604e8 0.0704838
$$269$$ − 3.49334e9i − 0.667163i −0.942721 0.333581i $$-0.891743\pi$$
0.942721 0.333581i $$-0.108257\pi$$
$$270$$ 0 0
$$271$$ 1.43226e9 0.265549 0.132774 0.991146i $$-0.457611\pi$$
0.132774 + 0.991146i $$0.457611\pi$$
$$272$$ 8.85101e9i 1.61703i
$$273$$ 0 0
$$274$$ 3.61538e9 0.641434
$$275$$ − 3.60749e9i − 0.630775i
$$276$$ 0 0
$$277$$ 3.35095e9 0.569179 0.284589 0.958650i $$-0.408143\pi$$
0.284589 + 0.958650i $$0.408143\pi$$
$$278$$ − 5.05787e9i − 0.846814i
$$279$$ 0 0
$$280$$ −1.67502e10 −2.72513
$$281$$ 4.29040e9i 0.688133i 0.938945 + 0.344067i $$0.111805\pi$$
−0.938945 + 0.344067i $$0.888195\pi$$
$$282$$ 0 0
$$283$$ −9.91145e7 −0.0154522 −0.00772612 0.999970i $$-0.502459\pi$$
−0.00772612 + 0.999970i $$0.502459\pi$$
$$284$$ 1.77974e10i 2.73580i
$$285$$ 0 0
$$286$$ −1.68274e10 −2.51508
$$287$$ 1.84508e9i 0.271949i
$$288$$ 0 0
$$289$$ 3.42231e9 0.490601
$$290$$ 1.42390e10i 2.01320i
$$291$$ 0 0
$$292$$ −7.81291e9 −1.07469
$$293$$ − 9.02972e8i − 0.122519i −0.998122 0.0612596i $$-0.980488\pi$$
0.998122 0.0612596i $$-0.0195117\pi$$
$$294$$ 0 0
$$295$$ −4.14796e9 −0.547705
$$296$$ − 6.86952e9i − 0.894869i
$$297$$ 0 0
$$298$$ −2.15065e10 −2.72713
$$299$$ − 5.97626e9i − 0.747729i
$$300$$ 0 0
$$301$$ 1.13167e9 0.137865
$$302$$ − 5.46586e9i − 0.657099i
$$303$$ 0 0
$$304$$ 2.26282e10 2.64945
$$305$$ − 1.58128e10i − 1.82730i
$$306$$ 0 0
$$307$$ 3.36397e9 0.378703 0.189352 0.981909i $$-0.439361\pi$$
0.189352 + 0.981909i $$0.439361\pi$$
$$308$$ − 1.50456e10i − 1.67188i
$$309$$ 0 0
$$310$$ −3.97596e9 −0.430521
$$311$$ 1.33023e10i 1.42196i 0.703215 + 0.710978i $$0.251747\pi$$
−0.703215 + 0.710978i $$0.748253\pi$$
$$312$$ 0 0
$$313$$ 1.31249e10 1.36748 0.683738 0.729728i $$-0.260353\pi$$
0.683738 + 0.729728i $$0.260353\pi$$
$$314$$ − 2.86299e10i − 2.94510i
$$315$$ 0 0
$$316$$ 1.43395e10 1.43809
$$317$$ − 6.98520e9i − 0.691739i −0.938283 0.345869i $$-0.887584\pi$$
0.938283 0.345869i $$-0.112416\pi$$
$$318$$ 0 0
$$319$$ −7.40472e9 −0.715065
$$320$$ − 1.02213e10i − 0.974774i
$$321$$ 0 0
$$322$$ 7.59331e9 0.706330
$$323$$ 9.08462e9i 0.834635i
$$324$$ 0 0
$$325$$ 1.30486e10 1.16958
$$326$$ 3.39495e10i 3.00582i
$$327$$ 0 0
$$328$$ −9.70533e9 −0.838523
$$329$$ 1.81638e10i 1.55033i
$$330$$ 0 0
$$331$$ −1.18733e10 −0.989140 −0.494570 0.869138i $$-0.664674\pi$$
−0.494570 + 0.869138i $$0.664674\pi$$
$$332$$ 2.03382e10i 1.67402i
$$333$$ 0 0
$$334$$ −3.35977e9 −0.269975
$$335$$ − 4.92198e8i − 0.0390806i
$$336$$ 0 0
$$337$$ 9.15419e9 0.709741 0.354871 0.934915i $$-0.384525\pi$$
0.354871 + 0.934915i $$0.384525\pi$$
$$338$$ − 3.68886e10i − 2.82634i
$$339$$ 0 0
$$340$$ 2.98293e10 2.23217
$$341$$ − 2.06762e9i − 0.152916i
$$342$$ 0 0
$$343$$ −1.50682e10 −1.08864
$$344$$ 5.95274e9i 0.425092i
$$345$$ 0 0
$$346$$ −4.85946e10 −3.39065
$$347$$ 2.05081e10i 1.41451i 0.706957 + 0.707257i $$0.250067\pi$$
−0.706957 + 0.707257i $$0.749933\pi$$
$$348$$ 0 0
$$349$$ 1.34799e10 0.908625 0.454312 0.890842i $$-0.349885\pi$$
0.454312 + 0.890842i $$0.349885\pi$$
$$350$$ 1.65793e10i 1.10483i
$$351$$ 0 0
$$352$$ 2.15840e10 1.40592
$$353$$ 9.78259e9i 0.630021i 0.949088 + 0.315011i $$0.102008\pi$$
−0.949088 + 0.315011i $$0.897992\pi$$
$$354$$ 0 0
$$355$$ 2.40917e10 1.51689
$$356$$ − 1.71972e10i − 1.07067i
$$357$$ 0 0
$$358$$ 1.22640e10 0.746619
$$359$$ 1.67338e10i 1.00743i 0.863868 + 0.503717i $$0.168035\pi$$
−0.863868 + 0.503717i $$0.831965\pi$$
$$360$$ 0 0
$$361$$ 6.24189e9 0.367525
$$362$$ − 3.42216e10i − 1.99281i
$$363$$ 0 0
$$364$$ 5.44211e10 3.10000
$$365$$ 1.05761e10i 0.595871i
$$366$$ 0 0
$$367$$ −1.49879e10 −0.826184 −0.413092 0.910689i $$-0.635551\pi$$
−0.413092 + 0.910689i $$0.635551\pi$$
$$368$$ 1.95002e10i 1.06328i
$$369$$ 0 0
$$370$$ −1.60620e10 −0.857022
$$371$$ − 2.04106e10i − 1.07736i
$$372$$ 0 0
$$373$$ −1.97964e10 −1.02271 −0.511353 0.859371i $$-0.670855\pi$$
−0.511353 + 0.859371i $$0.670855\pi$$
$$374$$ 2.20436e10i 1.12667i
$$375$$ 0 0
$$376$$ −9.55436e10 −4.78025
$$377$$ − 2.67835e10i − 1.32587i
$$378$$ 0 0
$$379$$ −5.34899e9 −0.259248 −0.129624 0.991563i $$-0.541377\pi$$
−0.129624 + 0.991563i $$0.541377\pi$$
$$380$$ − 7.62607e10i − 3.65735i
$$381$$ 0 0
$$382$$ 4.73404e10 2.22320
$$383$$ − 1.44938e10i − 0.673577i −0.941580 0.336788i $$-0.890659\pi$$
0.941580 0.336788i $$-0.109341\pi$$
$$384$$ 0 0
$$385$$ −2.03667e10 −0.926994
$$386$$ 2.08778e10i 0.940452i
$$387$$ 0 0
$$388$$ −8.29857e10 −3.66165
$$389$$ 1.80507e10i 0.788308i 0.919044 + 0.394154i $$0.128962\pi$$
−0.919044 + 0.394154i $$0.871038\pi$$
$$390$$ 0 0
$$391$$ −7.82879e9 −0.334956
$$392$$ − 1.96143e10i − 0.830668i
$$393$$ 0 0
$$394$$ 4.16284e10 1.72745
$$395$$ − 1.94108e10i − 0.797363i
$$396$$ 0 0
$$397$$ 3.82640e10 1.54038 0.770191 0.637814i $$-0.220161\pi$$
0.770191 + 0.637814i $$0.220161\pi$$
$$398$$ 6.88769e10i 2.74499i
$$399$$ 0 0
$$400$$ −4.25768e10 −1.66316
$$401$$ − 4.26602e10i − 1.64985i −0.565241 0.824926i $$-0.691217\pi$$
0.565241 0.824926i $$-0.308783\pi$$
$$402$$ 0 0
$$403$$ 7.47875e9 0.283536
$$404$$ 2.44716e10i 0.918623i
$$405$$ 0 0
$$406$$ 3.40305e10 1.25246
$$407$$ − 8.35271e9i − 0.304404i
$$408$$ 0 0
$$409$$ −3.53953e10 −1.26489 −0.632444 0.774606i $$-0.717948\pi$$
−0.632444 + 0.774606i $$0.717948\pi$$
$$410$$ 2.26925e10i 0.803059i
$$411$$ 0 0
$$412$$ −2.04232e10 −0.708817
$$413$$ 9.91343e9i 0.340741i
$$414$$ 0 0
$$415$$ 2.75311e10 0.928177
$$416$$ 7.80712e10i 2.60686i
$$417$$ 0 0
$$418$$ 5.63559e10 1.84601
$$419$$ − 1.45926e10i − 0.473453i −0.971576 0.236726i $$-0.923926\pi$$
0.971576 0.236726i $$-0.0760745\pi$$
$$420$$ 0 0
$$421$$ −2.65099e10 −0.843879 −0.421939 0.906624i $$-0.638650\pi$$
−0.421939 + 0.906624i $$0.638650\pi$$
$$422$$ − 3.13095e10i − 0.987247i
$$423$$ 0 0
$$424$$ 1.07362e11 3.32191
$$425$$ − 1.70935e10i − 0.523931i
$$426$$ 0 0
$$427$$ −3.77919e10 −1.13681
$$428$$ 1.18854e11i 3.54191i
$$429$$ 0 0
$$430$$ 1.39184e10 0.407113
$$431$$ − 4.61676e9i − 0.133791i −0.997760 0.0668957i $$-0.978691\pi$$
0.997760 0.0668957i $$-0.0213095\pi$$
$$432$$ 0 0
$$433$$ −1.64494e10 −0.467948 −0.233974 0.972243i $$-0.575173\pi$$
−0.233974 + 0.972243i $$0.575173\pi$$
$$434$$ 9.50235e9i 0.267838i
$$435$$ 0 0
$$436$$ 6.15188e10 1.70240
$$437$$ 2.00148e10i 0.548816i
$$438$$ 0 0
$$439$$ −2.61475e10 −0.703999 −0.351999 0.936000i $$-0.614498\pi$$
−0.351999 + 0.936000i $$0.614498\pi$$
$$440$$ − 1.07131e11i − 2.85828i
$$441$$ 0 0
$$442$$ −7.97335e10 −2.08906
$$443$$ 4.31455e10i 1.12027i 0.828403 + 0.560133i $$0.189250\pi$$
−0.828403 + 0.560133i $$0.810750\pi$$
$$444$$ 0 0
$$445$$ −2.32792e10 −0.593646
$$446$$ − 5.86660e10i − 1.48268i
$$447$$ 0 0
$$448$$ −2.44283e10 −0.606431
$$449$$ 2.18287e10i 0.537084i 0.963268 + 0.268542i $$0.0865418\pi$$
−0.963268 + 0.268542i $$0.913458\pi$$
$$450$$ 0 0
$$451$$ −1.18008e10 −0.285237
$$452$$ − 1.91219e11i − 4.58117i
$$453$$ 0 0
$$454$$ 5.28863e10 1.24486
$$455$$ − 7.36679e10i − 1.71883i
$$456$$ 0 0
$$457$$ 2.46905e10 0.566064 0.283032 0.959110i $$-0.408660\pi$$
0.283032 + 0.959110i $$0.408660\pi$$
$$458$$ − 4.69676e10i − 1.06742i
$$459$$ 0 0
$$460$$ 6.57186e10 1.46777
$$461$$ 2.61732e10i 0.579500i 0.957102 + 0.289750i $$0.0935722\pi$$
−0.957102 + 0.289750i $$0.906428\pi$$
$$462$$ 0 0
$$463$$ 7.04102e10 1.53219 0.766093 0.642729i $$-0.222198\pi$$
0.766093 + 0.642729i $$0.222198\pi$$
$$464$$ 8.73928e10i 1.88540i
$$465$$ 0 0
$$466$$ 1.46502e11 3.10671
$$467$$ − 1.07220e10i − 0.225428i −0.993627 0.112714i $$-0.964046\pi$$
0.993627 0.112714i $$-0.0359543\pi$$
$$468$$ 0 0
$$469$$ −1.17633e9 −0.0243130
$$470$$ 2.23395e11i 4.57807i
$$471$$ 0 0
$$472$$ −5.21458e10 −1.05063
$$473$$ 7.23798e9i 0.144602i
$$474$$ 0 0
$$475$$ −4.37006e10 −0.858445
$$476$$ − 7.12907e10i − 1.38869i
$$477$$ 0 0
$$478$$ −1.73352e11 −3.32060
$$479$$ 1.40034e10i 0.266005i 0.991116 + 0.133003i $$0.0424619\pi$$
−0.991116 + 0.133003i $$0.957538\pi$$
$$480$$ 0 0
$$481$$ 3.02125e10 0.564424
$$482$$ − 6.64814e10i − 1.23172i
$$483$$ 0 0
$$484$$ −3.41014e10 −0.621429
$$485$$ 1.12335e11i 2.03024i
$$486$$ 0 0
$$487$$ 1.72271e9 0.0306264 0.0153132 0.999883i $$-0.495125\pi$$
0.0153132 + 0.999883i $$0.495125\pi$$
$$488$$ − 1.98790e11i − 3.50521i
$$489$$ 0 0
$$490$$ −4.58611e10 −0.795536
$$491$$ 3.70419e10i 0.637334i 0.947867 + 0.318667i $$0.103235\pi$$
−0.947867 + 0.318667i $$0.896765\pi$$
$$492$$ 0 0
$$493$$ −3.50859e10 −0.593943
$$494$$ 2.03844e11i 3.42287i
$$495$$ 0 0
$$496$$ −2.44027e10 −0.403191
$$497$$ − 5.75782e10i − 0.943696i
$$498$$ 0 0
$$499$$ −1.33497e10 −0.215312 −0.107656 0.994188i $$-0.534335\pi$$
−0.107656 + 0.994188i $$0.534335\pi$$
$$500$$ − 5.19787e10i − 0.831659i
$$501$$ 0 0
$$502$$ −9.75332e10 −1.53581
$$503$$ 5.91175e10i 0.923516i 0.887006 + 0.461758i $$0.152781\pi$$
−0.887006 + 0.461758i $$0.847219\pi$$
$$504$$ 0 0
$$505$$ 3.31264e10 0.509341
$$506$$ 4.85655e10i 0.740842i
$$507$$ 0 0
$$508$$ −2.61968e11 −3.93363
$$509$$ − 3.32051e10i − 0.494691i −0.968927 0.247345i $$-0.920442\pi$$
0.968927 0.247345i $$-0.0795583\pi$$
$$510$$ 0 0
$$511$$ 2.52763e10 0.370706
$$512$$ 1.38544e11i 2.01607i
$$513$$ 0 0
$$514$$ 8.96565e10 1.28449
$$515$$ 2.76461e10i 0.393012i
$$516$$ 0 0
$$517$$ −1.16172e11 −1.62608
$$518$$ 3.83873e10i 0.533174i
$$519$$ 0 0
$$520$$ 3.87502e11 5.29981
$$521$$ 2.53746e10i 0.344388i 0.985063 + 0.172194i $$0.0550856\pi$$
−0.985063 + 0.172194i $$0.944914\pi$$
$$522$$ 0 0
$$523$$ 1.27775e11 1.70781 0.853903 0.520432i $$-0.174229\pi$$
0.853903 + 0.520432i $$0.174229\pi$$
$$524$$ − 1.14871e11i − 1.52365i
$$525$$ 0 0
$$526$$ −5.80299e10 −0.758069
$$527$$ − 9.79703e9i − 0.127014i
$$528$$ 0 0
$$529$$ 6.10629e10 0.779749
$$530$$ − 2.51029e11i − 3.18141i
$$531$$ 0 0
$$532$$ −1.82259e11 −2.27533
$$533$$ − 4.26845e10i − 0.528885i
$$534$$ 0 0
$$535$$ 1.60888e11 1.96385
$$536$$ − 6.18763e9i − 0.0749662i
$$537$$ 0 0
$$538$$ −1.02683e11 −1.22566
$$539$$ − 2.38492e10i − 0.282565i
$$540$$ 0 0
$$541$$ 6.16835e9 0.0720078 0.0360039 0.999352i $$-0.488537\pi$$
0.0360039 + 0.999352i $$0.488537\pi$$
$$542$$ − 4.20997e10i − 0.487845i
$$543$$ 0 0
$$544$$ 1.02272e11 1.16778
$$545$$ − 8.32758e10i − 0.943915i
$$546$$ 0 0
$$547$$ 1.02309e11 1.14279 0.571394 0.820676i $$-0.306403\pi$$
0.571394 + 0.820676i $$0.306403\pi$$
$$548$$ − 7.47826e10i − 0.829237i
$$549$$ 0 0
$$550$$ −1.06038e11 −1.15881
$$551$$ 8.96995e10i 0.973158i
$$552$$ 0 0
$$553$$ −4.63910e10 −0.496059
$$554$$ − 9.84974e10i − 1.04565i
$$555$$ 0 0
$$556$$ −1.04620e11 −1.09475
$$557$$ − 1.00553e11i − 1.04466i −0.852743 0.522331i $$-0.825063\pi$$
0.852743 0.522331i $$-0.174937\pi$$
$$558$$ 0 0
$$559$$ −2.61804e10 −0.268120
$$560$$ 2.40374e11i 2.44419i
$$561$$ 0 0
$$562$$ 1.26112e11 1.26418
$$563$$ 1.29004e11i 1.28402i 0.766697 + 0.642009i $$0.221899\pi$$
−0.766697 + 0.642009i $$0.778101\pi$$
$$564$$ 0 0
$$565$$ −2.58846e11 −2.54008
$$566$$ 2.91336e9i 0.0283876i
$$567$$ 0 0
$$568$$ 3.02868e11 2.90978
$$569$$ − 1.38131e10i − 0.131778i −0.997827 0.0658889i $$-0.979012\pi$$
0.997827 0.0658889i $$-0.0209883\pi$$
$$570$$ 0 0
$$571$$ 7.83161e10 0.736727 0.368363 0.929682i $$-0.379918\pi$$
0.368363 + 0.929682i $$0.379918\pi$$
$$572$$ 3.48067e11i 3.25147i
$$573$$ 0 0
$$574$$ 5.42341e10 0.499602
$$575$$ − 3.76595e10i − 0.344511i
$$576$$ 0 0
$$577$$ −6.44834e10 −0.581761 −0.290880 0.956759i $$-0.593948\pi$$
−0.290880 + 0.956759i $$0.593948\pi$$
$$578$$ − 1.00595e11i − 0.901291i
$$579$$ 0 0
$$580$$ 2.94528e11 2.60264
$$581$$ − 6.57980e10i − 0.577442i
$$582$$ 0 0
$$583$$ 1.30542e11 1.13000
$$584$$ 1.32956e11i 1.14303i
$$585$$ 0 0
$$586$$ −2.65418e10 −0.225082
$$587$$ − 1.10003e11i − 0.926512i −0.886225 0.463256i $$-0.846681\pi$$
0.886225 0.463256i $$-0.153319\pi$$
$$588$$ 0 0
$$589$$ −2.50468e10 −0.208109
$$590$$ 1.21925e11i 1.00620i
$$591$$ 0 0
$$592$$ −9.85814e10 −0.802616
$$593$$ 9.33048e9i 0.0754545i 0.999288 + 0.0377273i $$0.0120118\pi$$
−0.999288 + 0.0377273i $$0.987988\pi$$
$$594$$ 0 0
$$595$$ −9.65037e10 −0.769974
$$596$$ 4.44854e11i 3.52559i
$$597$$ 0 0
$$598$$ −1.75665e11 −1.37367
$$599$$ − 8.12693e10i − 0.631276i −0.948880 0.315638i $$-0.897781\pi$$
0.948880 0.315638i $$-0.102219\pi$$
$$600$$ 0 0
$$601$$ 1.80583e11 1.38414 0.692069 0.721831i $$-0.256699\pi$$
0.692069 + 0.721831i $$0.256699\pi$$
$$602$$ − 3.32643e10i − 0.253275i
$$603$$ 0 0
$$604$$ −1.13059e11 −0.849488
$$605$$ 4.61619e10i 0.344558i
$$606$$ 0 0
$$607$$ 1.15750e11 0.852644 0.426322 0.904571i $$-0.359809\pi$$
0.426322 + 0.904571i $$0.359809\pi$$
$$608$$ − 2.61465e11i − 1.91337i
$$609$$ 0 0
$$610$$ −4.64800e11 −3.35696
$$611$$ − 4.20205e11i − 3.01506i
$$612$$ 0 0
$$613$$ 7.01257e10 0.496632 0.248316 0.968679i $$-0.420123\pi$$
0.248316 + 0.968679i $$0.420123\pi$$
$$614$$ − 9.88802e10i − 0.695722i
$$615$$ 0 0
$$616$$ −2.56038e11 −1.77820
$$617$$ 1.79037e10i 0.123538i 0.998090 + 0.0617691i $$0.0196742\pi$$
−0.998090 + 0.0617691i $$0.980326\pi$$
$$618$$ 0 0
$$619$$ 1.35992e11 0.926300 0.463150 0.886280i $$-0.346719\pi$$
0.463150 + 0.886280i $$0.346719\pi$$
$$620$$ 8.22410e10i 0.556572i
$$621$$ 0 0
$$622$$ 3.91007e11 2.61230
$$623$$ 5.56362e10i 0.369322i
$$624$$ 0 0
$$625$$ −1.82374e11 −1.19521
$$626$$ − 3.85793e11i − 2.51221i
$$627$$ 0 0
$$628$$ −5.92197e11 −3.80739
$$629$$ − 3.95778e10i − 0.252842i
$$630$$ 0 0
$$631$$ −2.50988e11 −1.58320 −0.791600 0.611040i $$-0.790752\pi$$
−0.791600 + 0.611040i $$0.790752\pi$$
$$632$$ − 2.44022e11i − 1.52954i
$$633$$ 0 0
$$634$$ −2.05322e11 −1.27080
$$635$$ 3.54617e11i 2.18105i
$$636$$ 0 0
$$637$$ 8.62644e10 0.523931
$$638$$ 2.17653e11i 1.31366i
$$639$$ 0 0
$$640$$ 6.10397e10 0.363825
$$641$$ − 2.57286e10i − 0.152400i −0.997093 0.0761999i $$-0.975721\pi$$
0.997093 0.0761999i $$-0.0242787\pi$$
$$642$$ 0 0
$$643$$ 4.29706e10 0.251378 0.125689 0.992070i $$-0.459886\pi$$
0.125689 + 0.992070i $$0.459886\pi$$
$$644$$ − 1.57064e11i − 0.913134i
$$645$$ 0 0
$$646$$ 2.67032e11 1.53332
$$647$$ 2.20975e11i 1.26103i 0.776177 + 0.630516i $$0.217157\pi$$
−0.776177 + 0.630516i $$0.782843\pi$$
$$648$$ 0 0
$$649$$ −6.34045e10 −0.357389
$$650$$ − 3.83549e11i − 2.14866i
$$651$$ 0 0
$$652$$ 7.02232e11 3.88589
$$653$$ 2.99011e11i 1.64450i 0.569125 + 0.822251i $$0.307282\pi$$
−0.569125 + 0.822251i $$0.692718\pi$$
$$654$$ 0 0
$$655$$ −1.55497e11 −0.844806
$$656$$ 1.39277e11i 0.752079i
$$657$$ 0 0
$$658$$ 5.33904e11 2.84813
$$659$$ − 2.99806e11i − 1.58964i −0.606846 0.794820i $$-0.707565\pi$$
0.606846 0.794820i $$-0.292435\pi$$
$$660$$ 0 0
$$661$$ −2.11448e11 −1.10764 −0.553820 0.832637i $$-0.686830\pi$$
−0.553820 + 0.832637i $$0.686830\pi$$
$$662$$ 3.49001e11i 1.81717i
$$663$$ 0 0
$$664$$ 3.46105e11 1.78047
$$665$$ 2.46718e11i 1.26158i
$$666$$ 0 0
$$667$$ −7.72997e10 −0.390548
$$668$$ 6.94954e10i 0.349020i
$$669$$ 0 0
$$670$$ −1.44676e10 −0.0717956
$$671$$ − 2.41710e11i − 1.19235i
$$672$$ 0 0
$$673$$ −1.90166e11 −0.926985 −0.463493 0.886101i $$-0.653404\pi$$
−0.463493 + 0.886101i $$0.653404\pi$$
$$674$$ − 2.69077e11i − 1.30388i
$$675$$ 0 0
$$676$$ −7.63024e11 −3.65386
$$677$$ 1.14815e11i 0.546570i 0.961933 + 0.273285i $$0.0881102\pi$$
−0.961933 + 0.273285i $$0.911890\pi$$
$$678$$ 0 0
$$679$$ 2.68475e11 1.26306
$$680$$ − 5.07621e11i − 2.37413i
$$681$$ 0 0
$$682$$ −6.07753e10 −0.280925
$$683$$ 3.07050e11i 1.41100i 0.708710 + 0.705500i $$0.249277\pi$$
−0.708710 + 0.705500i $$0.750723\pi$$
$$684$$ 0 0
$$685$$ −1.01231e11 −0.459780
$$686$$ 4.42914e11i 1.99997i
$$687$$ 0 0
$$688$$ 8.54250e10 0.381269
$$689$$ 4.72183e11i 2.09524i
$$690$$ 0 0
$$691$$ −2.20597e11 −0.967582 −0.483791 0.875184i $$-0.660741\pi$$
−0.483791 + 0.875184i $$0.660741\pi$$
$$692$$ 1.00516e12i 4.38339i
$$693$$ 0 0
$$694$$ 6.02812e11 2.59863
$$695$$ 1.41620e11i 0.606996i
$$696$$ 0 0
$$697$$ −5.59160e10 −0.236922
$$698$$ − 3.96226e11i − 1.66925i
$$699$$ 0 0
$$700$$ 3.42936e11 1.42830
$$701$$ − 4.11156e11i − 1.70269i −0.524609 0.851343i $$-0.675788\pi$$
0.524609 0.851343i $$-0.324212\pi$$
$$702$$ 0 0
$$703$$ −1.01183e11 −0.414274
$$704$$ − 1.56239e11i − 0.636062i
$$705$$ 0 0
$$706$$ 2.87548e11 1.15742
$$707$$ − 7.91705e10i − 0.316874i
$$708$$ 0 0
$$709$$ 1.41043e11 0.558169 0.279084 0.960267i $$-0.409969\pi$$
0.279084 + 0.960267i $$0.409969\pi$$
$$710$$ − 7.08150e11i − 2.78671i
$$711$$ 0 0
$$712$$ −2.92653e11 −1.13876
$$713$$ − 2.15844e10i − 0.0835183i
$$714$$ 0 0
$$715$$ 4.71167e11 1.80281
$$716$$ − 2.53675e11i − 0.965219i
$$717$$ 0 0
$$718$$ 4.91871e11 1.85078
$$719$$ − 1.87514e11i − 0.701645i −0.936442 0.350823i $$-0.885902\pi$$
0.936442 0.350823i $$-0.114098\pi$$
$$720$$ 0 0
$$721$$ 6.60730e10 0.244502
$$722$$ − 1.83473e11i − 0.675187i
$$723$$ 0 0
$$724$$ −7.07859e11 −2.57628
$$725$$ − 1.68777e11i − 0.610887i
$$726$$ 0 0
$$727$$ 6.74621e10 0.241503 0.120751 0.992683i $$-0.461470\pi$$
0.120751 + 0.992683i $$0.461470\pi$$
$$728$$ − 9.26111e11i − 3.29714i
$$729$$ 0 0
$$730$$ 3.10871e11 1.09469
$$731$$ 3.42959e10i 0.120108i
$$732$$ 0 0
$$733$$ −5.51348e11 −1.90990 −0.954948 0.296773i $$-0.904090\pi$$
−0.954948 + 0.296773i $$0.904090\pi$$
$$734$$ 4.40553e11i 1.51780i
$$735$$ 0 0
$$736$$ 2.25321e11 0.767875
$$737$$ − 7.52360e9i − 0.0255009i
$$738$$ 0 0
$$739$$ −3.53894e11 −1.18658 −0.593288 0.804990i $$-0.702171\pi$$
−0.593288 + 0.804990i $$0.702171\pi$$
$$740$$ 3.32235e11i 1.10795i
$$741$$ 0 0
$$742$$ −5.99946e11 −1.97923
$$743$$ − 4.61456e10i − 0.151417i −0.997130 0.0757086i $$-0.975878\pi$$
0.997130 0.0757086i $$-0.0241219\pi$$
$$744$$ 0 0
$$745$$ 6.02183e11 1.95480
$$746$$ 5.81893e11i 1.87883i
$$747$$ 0 0
$$748$$ 4.55962e11 1.45654
$$749$$ − 3.84515e11i − 1.22176i
$$750$$ 0 0
$$751$$ −2.78555e11 −0.875690 −0.437845 0.899050i $$-0.644258\pi$$
−0.437845 + 0.899050i $$0.644258\pi$$
$$752$$ 1.37110e12i 4.28745i
$$753$$ 0 0
$$754$$ −7.87270e11 −2.43578
$$755$$ 1.53044e11i 0.471008i
$$756$$ 0 0
$$757$$ −2.54100e11 −0.773786 −0.386893 0.922125i $$-0.626452\pi$$
−0.386893 + 0.922125i $$0.626452\pi$$
$$758$$ 1.57228e11i 0.476268i
$$759$$ 0 0
$$760$$ −1.29777e12 −3.88993
$$761$$ − 5.04366e11i − 1.50386i −0.659243 0.751930i $$-0.729123\pi$$
0.659243 0.751930i $$-0.270877\pi$$
$$762$$ 0 0
$$763$$ −1.99025e11 −0.587233
$$764$$ − 9.79216e11i − 2.87412i
$$765$$ 0 0
$$766$$ −4.26029e11 −1.23744
$$767$$ − 2.29339e11i − 0.662670i
$$768$$ 0 0
$$769$$ 2.16447e11 0.618937 0.309468 0.950910i $$-0.399849\pi$$
0.309468 + 0.950910i $$0.399849\pi$$
$$770$$ 5.98655e11i 1.70300i
$$771$$ 0 0
$$772$$ 4.31850e11 1.21580
$$773$$ − 4.05778e11i − 1.13650i −0.822855 0.568252i $$-0.807620\pi$$
0.822855 0.568252i $$-0.192380\pi$$
$$774$$ 0 0
$$775$$ 4.71275e10 0.130637
$$776$$ 1.41221e12i 3.89451i
$$777$$ 0 0
$$778$$ 5.30581e11 1.44821
$$779$$ 1.42953e11i 0.388189i
$$780$$ 0 0
$$781$$ 3.68260e11 0.989806
$$782$$ 2.30119e11i 0.615354i
$$783$$ 0 0
$$784$$ −2.81475e11 −0.745034
$$785$$ 8.01636e11i 2.11105i
$$786$$ 0 0
$$787$$ 6.28441e11 1.63820 0.819098 0.573653i $$-0.194474\pi$$
0.819098 + 0.573653i $$0.194474\pi$$
$$788$$ − 8.61066e11i − 2.23322i
$$789$$ 0 0
$$790$$ −5.70560e11 −1.46485
$$791$$ 6.18630e11i 1.58025i
$$792$$ 0 0
$$793$$ 8.74286e11 2.21086
$$794$$ − 1.12473e12i − 2.82986i
$$795$$ 0 0
$$796$$ 1.42469e12 3.54869
$$797$$ − 3.23780e11i − 0.802447i −0.915980 0.401224i $$-0.868585\pi$$
0.915980 0.401224i $$-0.131415\pi$$
$$798$$ 0 0
$$799$$ −5.50462e11 −1.35064
$$800$$ 4.91968e11i 1.20109i
$$801$$ 0 0
$$802$$ −1.25395e12 −3.03097
$$803$$ 1.61663e11i 0.388819i
$$804$$ 0 0
$$805$$ −2.12613e11 −0.506297
$$806$$ − 2.19829e11i − 0.520889i
$$807$$ 0 0
$$808$$ 4.16446e11 0.977042
$$809$$ − 4.65514e10i − 0.108677i −0.998523 0.0543386i $$-0.982695\pi$$
0.998523 0.0543386i $$-0.0173050\pi$$
$$810$$ 0 0
$$811$$ 9.90434e9 0.0228951 0.0114475 0.999934i $$-0.496356\pi$$
0.0114475 + 0.999934i $$0.496356\pi$$
$$812$$ − 7.03908e11i − 1.61917i
$$813$$ 0 0
$$814$$ −2.45519e11 −0.559225
$$815$$ − 9.50587e11i − 2.15457i
$$816$$ 0 0
$$817$$ 8.76797e10 0.196794
$$818$$ 1.04040e12i 2.32375i
$$819$$ 0 0
$$820$$ 4.69385e11 1.03818
$$821$$ − 3.00799e11i − 0.662069i −0.943619 0.331034i $$-0.892602\pi$$
0.943619 0.331034i $$-0.107398\pi$$
$$822$$ 0 0
$$823$$ 3.64988e10 0.0795571 0.0397785 0.999209i $$-0.487335\pi$$
0.0397785 + 0.999209i $$0.487335\pi$$
$$824$$ 3.47551e11i 0.753894i
$$825$$ 0 0
$$826$$ 2.91394e11 0.625980
$$827$$ − 3.63233e11i − 0.776540i −0.921546 0.388270i $$-0.873073\pi$$
0.921546 0.388270i $$-0.126927\pi$$
$$828$$ 0 0
$$829$$ −1.99283e11 −0.421941 −0.210971 0.977492i $$-0.567662\pi$$
−0.210971 + 0.977492i $$0.567662\pi$$
$$830$$ − 8.09245e11i − 1.70517i
$$831$$ 0 0
$$832$$ 5.65130e11 1.17938
$$833$$ − 1.13005e11i − 0.234702i
$$834$$ 0 0
$$835$$ 9.40735e10 0.193518
$$836$$ − 1.16570e12i − 2.38650i
$$837$$ 0 0
$$838$$ −4.28933e11 −0.869788
$$839$$ 4.55688e11i 0.919644i 0.888011 + 0.459822i $$0.152087\pi$$
−0.888011 + 0.459822i $$0.847913\pi$$
$$840$$ 0 0
$$841$$ 1.53816e11 0.307481
$$842$$ 7.79229e11i 1.55030i
$$843$$ 0 0
$$844$$ −6.47623e11 −1.27630
$$845$$ 1.03288e12i 2.02592i
$$846$$ 0 0
$$847$$ 1.10325e11 0.214358
$$848$$ − 1.54070e12i − 2.97945i
$$849$$ 0 0
$$850$$ −5.02443e11 −0.962523
$$851$$ − 8.71961e10i − 0.166257i
$$852$$ 0 0
$$853$$ −3.80568e11 −0.718846 −0.359423 0.933175i $$-0.617027\pi$$
−0.359423 + 0.933175i $$0.617027\pi$$
$$854$$ 1.11085e12i 2.08845i
$$855$$ 0 0
$$856$$ 2.02259e12 3.76715
$$857$$ − 2.04675e11i − 0.379438i −0.981838 0.189719i $$-0.939242\pi$$
0.981838 0.189719i $$-0.0607577\pi$$
$$858$$ 0 0
$$859$$ 2.40882e11 0.442417 0.221208 0.975227i $$-0.429000\pi$$
0.221208 + 0.975227i $$0.429000\pi$$
$$860$$ − 2.87896e11i − 0.526310i
$$861$$ 0 0
$$862$$ −1.35704e11 −0.245790
$$863$$ 8.63484e11i 1.55672i 0.627817 + 0.778361i $$0.283949\pi$$
−0.627817 + 0.778361i $$0.716051\pi$$
$$864$$ 0 0
$$865$$ 1.36065e12 2.43042
$$866$$ 4.83510e11i 0.859675i
$$867$$ 0 0
$$868$$ 1.96552e11 0.346257
$$869$$ − 2.96709e11i − 0.520297i
$$870$$ 0 0
$$871$$ 2.72135e10 0.0472837
$$872$$ − 1.04690e12i − 1.81066i
$$873$$ 0 0
$$874$$ 5.88314e11 1.00824
$$875$$ 1.68161e11i 0.286876i
$$876$$ 0 0
$$877$$ −4.79159e11 −0.809993 −0.404997 0.914318i $$-0.632727\pi$$
−0.404997 + 0.914318i $$0.632727\pi$$
$$878$$ 7.68576e11i 1.29333i
$$879$$ 0 0
$$880$$ −1.53739e12 −2.56362
$$881$$ − 6.43291e11i − 1.06783i −0.845537 0.533917i $$-0.820720\pi$$
0.845537 0.533917i $$-0.179280\pi$$
$$882$$ 0 0
$$883$$ −1.08619e12 −1.78674 −0.893371 0.449319i $$-0.851667\pi$$
−0.893371 + 0.449319i $$0.851667\pi$$
$$884$$ 1.64925e12i 2.70071i
$$885$$ 0 0
$$886$$ 1.26821e12 2.05806
$$887$$ 1.68383e11i 0.272022i 0.990707 + 0.136011i $$0.0434283\pi$$
−0.990707 + 0.136011i $$0.956572\pi$$
$$888$$ 0 0
$$889$$ 8.47519e11 1.35688
$$890$$ 6.84266e11i 1.09060i
$$891$$ 0 0
$$892$$ −1.21348e12 −1.91679
$$893$$ 1.40729e12i 2.21299i
$$894$$ 0 0
$$895$$ −3.43391e11 −0.535177
$$896$$ − 1.45882e11i − 0.226344i
$$897$$ 0 0
$$898$$ 6.41630e11 0.986687
$$899$$ − 9.67336e10i − 0.148094i
$$900$$ 0 0
$$901$$ 6.18552e11 0.938592
$$902$$ 3.46871e11i 0.524013i
$$903$$ 0 0
$$904$$ −3.25407e12 −4.87251
$$905$$ 9.58204e11i 1.42845i
$$906$$ 0 0
$$907$$ −6.69645e11 −0.989500 −0.494750 0.869035i $$-0.664740\pi$$
−0.494750 + 0.869035i $$0.664740\pi$$
$$908$$ − 1.09393e12i − 1.60933i
$$909$$ 0 0
$$910$$ −2.16539e12 −3.15769
$$911$$ − 1.19351e12i − 1.73282i −0.499337 0.866408i $$-0.666423\pi$$
0.499337 0.866408i $$-0.333577\pi$$
$$912$$ 0 0
$$913$$ 4.20832e11 0.605656
$$914$$ − 7.25750e11i − 1.03993i
$$915$$ 0 0
$$916$$ −9.71506e11 −1.37995
$$917$$ 3.71631e11i 0.525575i
$$918$$ 0 0
$$919$$ −7.65817e11 −1.07365 −0.536825 0.843693i $$-0.680377\pi$$
−0.536825 + 0.843693i $$0.680377\pi$$
$$920$$ − 1.11837e12i − 1.56111i
$$921$$ 0 0
$$922$$ 7.69333e11 1.06461
$$923$$ 1.33203e12i 1.83529i
$$924$$ 0 0
$$925$$ 1.90385e11 0.260055
$$926$$ − 2.06963e12i − 2.81481i
$$927$$ 0 0
$$928$$ 1.00981e12 1.36159
$$929$$ − 4.26646e11i − 0.572803i −0.958110 0.286401i $$-0.907541\pi$$
0.958110 0.286401i $$-0.0924591\pi$$
$$930$$ 0 0
$$931$$ −2.88905e11 −0.384553
$$932$$ − 3.03034e12i − 4.01631i
$$933$$ 0 0
$$934$$ −3.15160e11 −0.414137
$$935$$ − 6.17221e11i − 0.807596i
$$936$$ 0 0
$$937$$ 1.35431e12 1.75695 0.878477 0.477784i $$-0.158560\pi$$
0.878477 + 0.477784i $$0.158560\pi$$
$$938$$ 3.45769e10i 0.0446658i
$$939$$ 0 0
$$940$$ 4.62084e12 5.91847
$$941$$ − 1.88655e10i − 0.0240608i −0.999928 0.0120304i $$-0.996171\pi$$
0.999928 0.0120304i $$-0.00382948\pi$$
$$942$$ 0 0
$$943$$ −1.23192e11 −0.155788
$$944$$ 7.48321e11i 0.942323i
$$945$$ 0 0
$$946$$ 2.12752e11 0.265650
$$947$$ − 6.03707e11i − 0.750631i −0.926897 0.375316i $$-0.877534\pi$$
0.926897 0.375316i $$-0.122466\pi$$
$$948$$ 0 0
$$949$$ −5.84747e11 −0.720947
$$950$$ 1.28453e12i 1.57706i
$$951$$ 0 0
$$952$$ −1.21319e12 −1.47700
$$953$$ 6.59941e10i 0.0800080i 0.999200 + 0.0400040i $$0.0127371\pi$$
−0.999200 + 0.0400040i $$0.987263\pi$$
$$954$$ 0 0
$$955$$ −1.32553e12 −1.59359
$$956$$ 3.58571e12i 4.29283i
$$957$$ 0 0
$$958$$ 4.11613e11 0.488683
$$959$$ 2.41937e11i 0.286040i
$$960$$ 0 0
$$961$$ −8.25880e11 −0.968330
$$962$$ − 8.88061e11i − 1.03691i
$$963$$ 0 0
$$964$$ −1.37514e12 −1.59235
$$965$$ − 5.84580e11i − 0.674116i
$$966$$ 0 0
$$967$$ 5.39935e11 0.617498 0.308749 0.951144i $$-0.400090\pi$$
0.308749 + 0.951144i $$0.400090\pi$$
$$968$$ 5.80322e11i 0.660948i
$$969$$ 0 0
$$970$$ 3.30196e12 3.72979
$$971$$ − 7.23031e11i − 0.813355i −0.913572 0.406677i $$-0.866687\pi$$
0.913572 0.406677i $$-0.133313\pi$$
$$972$$ 0 0
$$973$$ 3.38466e11 0.377627
$$974$$ − 5.06370e10i − 0.0562642i
$$975$$ 0 0
$$976$$ −2.85274e12 −3.14386
$$977$$ − 4.79457e11i − 0.526225i −0.964765 0.263112i $$-0.915251\pi$$
0.964765 0.263112i $$-0.0847490\pi$$
$$978$$ 0 0
$$979$$ −3.55839e11 −0.387367
$$980$$ 9.48617e11i 1.02846i
$$981$$ 0 0
$$982$$ 1.08880e12 1.17086
$$983$$ 9.92587e11i 1.06305i 0.847042 + 0.531526i $$0.178381\pi$$
−0.847042 + 0.531526i $$0.821619\pi$$
$$984$$ 0 0
$$985$$ −1.16560e12 −1.23823
$$986$$ 1.03131e12i 1.09114i
$$987$$ 0 0
$$988$$ 4.21643e12 4.42504
$$989$$ 7.55591e10i 0.0789772i
$$990$$ 0 0
$$991$$ 9.53660e11 0.988778 0.494389 0.869241i $$-0.335392\pi$$
0.494389 + 0.869241i $$0.335392\pi$$
$$992$$ 2.81969e11i 0.291175i
$$993$$ 0 0
$$994$$ −1.69245e12 −1.73368
$$995$$ − 1.92855e12i − 1.96761i
$$996$$ 0 0
$$997$$ −1.51632e12 −1.53465 −0.767324 0.641259i $$-0.778412\pi$$
−0.767324 + 0.641259i $$0.778412\pi$$
$$998$$ 3.92399e11i 0.395553i
$$999$$ 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 27.9.b.b.26.1 2
3.2 odd 2 inner 27.9.b.b.26.2 yes 2
4.3 odd 2 432.9.e.d.161.2 2
9.2 odd 6 81.9.d.e.53.2 4
9.4 even 3 81.9.d.e.26.2 4
9.5 odd 6 81.9.d.e.26.1 4
9.7 even 3 81.9.d.e.53.1 4
12.11 even 2 432.9.e.d.161.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.b.b.26.1 2 1.1 even 1 trivial
27.9.b.b.26.2 yes 2 3.2 odd 2 inner
81.9.d.e.26.1 4 9.5 odd 6
81.9.d.e.26.2 4 9.4 even 3
81.9.d.e.53.1 4 9.7 even 3
81.9.d.e.53.2 4 9.2 odd 6
432.9.e.d.161.1 2 12.11 even 2
432.9.e.d.161.2 2 4.3 odd 2