# Properties

 Label 18.9.b.a.17.2 Level $18$ Weight $9$ Character 18.17 Analytic conductor $7.333$ 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] = [18,9,Mod(17,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, 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("18.17");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 18.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: $$7.33281498110$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 17.2 Root $$1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 18.17 Dual form 18.9.b.a.17.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+11.3137i q^{2} -128.000 q^{4} +233.345i q^{5} -3532.00 q^{7} -1448.15i q^{8} +O(q^{10})$$ $$q+11.3137i q^{2} -128.000 q^{4} +233.345i q^{5} -3532.00 q^{7} -1448.15i q^{8} -2640.00 q^{10} -20178.0i q^{11} -41824.0 q^{13} -39960.0i q^{14} +16384.0 q^{16} +94784.8i q^{17} -36304.0 q^{19} -29868.2i q^{20} +228288. q^{22} +413624. i q^{23} +336175. q^{25} -473185. i q^{26} +452096. q^{28} +269191. i q^{29} -471196. q^{31} +185364. i q^{32} -1.07237e6 q^{34} -824175. i q^{35} -3.00740e6 q^{37} -410733. i q^{38} +337920. q^{40} -1.71534e6i q^{41} +3.62372e6 q^{43} +2.58278e6i q^{44} -4.67962e6 q^{46} -6.01462e6i q^{47} +6.71022e6 q^{49} +3.80339e6i q^{50} +5.35347e6 q^{52} +1.02767e7i q^{53} +4.70844e6 q^{55} +5.11488e6i q^{56} -3.04555e6 q^{58} +2.68810e6i q^{59} -5.44063e6 q^{61} -5.33097e6i q^{62} -2.09715e6 q^{64} -9.75943e6i q^{65} -6.12158e6 q^{67} -1.21325e7i q^{68} +9.32448e6 q^{70} -2.11941e7i q^{71} -4.90312e7 q^{73} -3.40249e7i q^{74} +4.64691e6 q^{76} +7.12687e7i q^{77} +8.35776e6 q^{79} +3.82313e6i q^{80} +1.94068e7 q^{82} -5.13918e7i q^{83} -2.21176e7 q^{85} +4.09977e7i q^{86} -2.92209e7 q^{88} +1.07337e8i q^{89} +1.47722e8 q^{91} -5.29438e7i q^{92} +6.80477e7 q^{94} -8.47137e6i q^{95} +2.04313e7 q^{97} +7.59175e7i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 256 q^{4} - 7064 q^{7}+O(q^{10})$$ 2 * q - 256 * q^4 - 7064 * q^7 $$2 q - 256 q^{4} - 7064 q^{7} - 5280 q^{10} - 83648 q^{13} + 32768 q^{16} - 72608 q^{19} + 456576 q^{22} + 672350 q^{25} + 904192 q^{28} - 942392 q^{31} - 2144736 q^{34} - 6014804 q^{37} + 675840 q^{40} + 7247440 q^{43} - 9359232 q^{46} + 13420446 q^{49} + 10706944 q^{52} + 9416880 q^{55} - 6091104 q^{58} - 10881260 q^{61} - 4194304 q^{64} - 12243152 q^{67} + 18648960 q^{70} - 98062304 q^{73} + 9293824 q^{76} + 16715512 q^{79} + 38813664 q^{82} - 44235180 q^{85} - 58441728 q^{88} + 295444736 q^{91} + 136095360 q^{94} + 40862656 q^{97}+O(q^{100})$$ 2 * q - 256 * q^4 - 7064 * q^7 - 5280 * q^10 - 83648 * q^13 + 32768 * q^16 - 72608 * q^19 + 456576 * q^22 + 672350 * q^25 + 904192 * q^28 - 942392 * q^31 - 2144736 * q^34 - 6014804 * q^37 + 675840 * q^40 + 7247440 * q^43 - 9359232 * q^46 + 13420446 * q^49 + 10706944 * q^52 + 9416880 * q^55 - 6091104 * q^58 - 10881260 * q^61 - 4194304 * q^64 - 12243152 * q^67 + 18648960 * q^70 - 98062304 * q^73 + 9293824 * q^76 + 16715512 * q^79 + 38813664 * q^82 - 44235180 * q^85 - 58441728 * q^88 + 295444736 * q^91 + 136095360 * q^94 + 40862656 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 11.3137i 0.707107i
$$3$$ 0 0
$$4$$ −128.000 −0.500000
$$5$$ 233.345i 0.373352i 0.982421 + 0.186676i $$0.0597715\pi$$
−0.982421 + 0.186676i $$0.940228\pi$$
$$6$$ 0 0
$$7$$ −3532.00 −1.47105 −0.735527 0.677496i $$-0.763065\pi$$
−0.735527 + 0.677496i $$0.763065\pi$$
$$8$$ − 1448.15i − 0.353553i
$$9$$ 0 0
$$10$$ −2640.00 −0.264000
$$11$$ − 20178.0i − 1.37818i −0.724674 0.689092i $$-0.758009\pi$$
0.724674 0.689092i $$-0.241991\pi$$
$$12$$ 0 0
$$13$$ −41824.0 −1.46437 −0.732187 0.681103i $$-0.761500\pi$$
−0.732187 + 0.681103i $$0.761500\pi$$
$$14$$ − 39960.0i − 1.04019i
$$15$$ 0 0
$$16$$ 16384.0 0.250000
$$17$$ 94784.8i 1.13486i 0.823421 + 0.567431i $$0.192063\pi$$
−0.823421 + 0.567431i $$0.807937\pi$$
$$18$$ 0 0
$$19$$ −36304.0 −0.278574 −0.139287 0.990252i $$-0.544481\pi$$
−0.139287 + 0.990252i $$0.544481\pi$$
$$20$$ − 29868.2i − 0.186676i
$$21$$ 0 0
$$22$$ 228288. 0.974524
$$23$$ 413624.i 1.47807i 0.673669 + 0.739033i $$0.264717\pi$$
−0.673669 + 0.739033i $$0.735283\pi$$
$$24$$ 0 0
$$25$$ 336175. 0.860608
$$26$$ − 473185.i − 1.03547i
$$27$$ 0 0
$$28$$ 452096. 0.735527
$$29$$ 269191.i 0.380600i 0.981726 + 0.190300i $$0.0609461\pi$$
−0.981726 + 0.190300i $$0.939054\pi$$
$$30$$ 0 0
$$31$$ −471196. −0.510217 −0.255108 0.966912i $$-0.582111\pi$$
−0.255108 + 0.966912i $$0.582111\pi$$
$$32$$ 185364.i 0.176777i
$$33$$ 0 0
$$34$$ −1.07237e6 −0.802469
$$35$$ − 824175.i − 0.549221i
$$36$$ 0 0
$$37$$ −3.00740e6 −1.60467 −0.802333 0.596877i $$-0.796408\pi$$
−0.802333 + 0.596877i $$0.796408\pi$$
$$38$$ − 410733.i − 0.196981i
$$39$$ 0 0
$$40$$ 337920. 0.132000
$$41$$ − 1.71534e6i − 0.607036i −0.952826 0.303518i $$-0.901839\pi$$
0.952826 0.303518i $$-0.0981612\pi$$
$$42$$ 0 0
$$43$$ 3.62372e6 1.05994 0.529969 0.848017i $$-0.322203\pi$$
0.529969 + 0.848017i $$0.322203\pi$$
$$44$$ 2.58278e6i 0.689092i
$$45$$ 0 0
$$46$$ −4.67962e6 −1.04515
$$47$$ − 6.01462e6i − 1.23259i −0.787517 0.616293i $$-0.788634\pi$$
0.787517 0.616293i $$-0.211366\pi$$
$$48$$ 0 0
$$49$$ 6.71022e6 1.16400
$$50$$ 3.80339e6i 0.608542i
$$51$$ 0 0
$$52$$ 5.35347e6 0.732187
$$53$$ 1.02767e7i 1.30241i 0.758901 + 0.651206i $$0.225737\pi$$
−0.758901 + 0.651206i $$0.774263\pi$$
$$54$$ 0 0
$$55$$ 4.70844e6 0.514548
$$56$$ 5.11488e6i 0.520096i
$$57$$ 0 0
$$58$$ −3.04555e6 −0.269125
$$59$$ 2.68810e6i 0.221839i 0.993829 + 0.110919i $$0.0353796\pi$$
−0.993829 + 0.110919i $$0.964620\pi$$
$$60$$ 0 0
$$61$$ −5.44063e6 −0.392943 −0.196472 0.980510i $$-0.562948\pi$$
−0.196472 + 0.980510i $$0.562948\pi$$
$$62$$ − 5.33097e6i − 0.360778i
$$63$$ 0 0
$$64$$ −2.09715e6 −0.125000
$$65$$ − 9.75943e6i − 0.546728i
$$66$$ 0 0
$$67$$ −6.12158e6 −0.303783 −0.151892 0.988397i $$-0.548536\pi$$
−0.151892 + 0.988397i $$0.548536\pi$$
$$68$$ − 1.21325e7i − 0.567431i
$$69$$ 0 0
$$70$$ 9.32448e6 0.388358
$$71$$ − 2.11941e7i − 0.834029i −0.908900 0.417014i $$-0.863076\pi$$
0.908900 0.417014i $$-0.136924\pi$$
$$72$$ 0 0
$$73$$ −4.90312e7 −1.72656 −0.863278 0.504729i $$-0.831593\pi$$
−0.863278 + 0.504729i $$0.831593\pi$$
$$74$$ − 3.40249e7i − 1.13467i
$$75$$ 0 0
$$76$$ 4.64691e6 0.139287
$$77$$ 7.12687e7i 2.02738i
$$78$$ 0 0
$$79$$ 8.35776e6 0.214576 0.107288 0.994228i $$-0.465783\pi$$
0.107288 + 0.994228i $$0.465783\pi$$
$$80$$ 3.82313e6i 0.0933381i
$$81$$ 0 0
$$82$$ 1.94068e7 0.429239
$$83$$ − 5.13918e7i − 1.08288i −0.840738 0.541441i $$-0.817879\pi$$
0.840738 0.541441i $$-0.182121\pi$$
$$84$$ 0 0
$$85$$ −2.21176e7 −0.423704
$$86$$ 4.09977e7i 0.749490i
$$87$$ 0 0
$$88$$ −2.92209e7 −0.487262
$$89$$ 1.07337e8i 1.71076i 0.517997 + 0.855382i $$0.326678\pi$$
−0.517997 + 0.855382i $$0.673322\pi$$
$$90$$ 0 0
$$91$$ 1.47722e8 2.15417
$$92$$ − 5.29438e7i − 0.739033i
$$93$$ 0 0
$$94$$ 6.80477e7 0.871569
$$95$$ − 8.47137e6i − 0.104006i
$$96$$ 0 0
$$97$$ 2.04313e7 0.230786 0.115393 0.993320i $$-0.463187\pi$$
0.115393 + 0.993320i $$0.463187\pi$$
$$98$$ 7.59175e7i 0.823072i
$$99$$ 0 0
$$100$$ −4.30304e7 −0.430304
$$101$$ − 1.69583e8i − 1.62966i −0.579700 0.814830i $$-0.696830\pi$$
0.579700 0.814830i $$-0.303170\pi$$
$$102$$ 0 0
$$103$$ −2.98252e7 −0.264993 −0.132497 0.991183i $$-0.542299\pi$$
−0.132497 + 0.991183i $$0.542299\pi$$
$$104$$ 6.05676e7i 0.517735i
$$105$$ 0 0
$$106$$ −1.16267e8 −0.920945
$$107$$ − 1.22823e8i − 0.937014i −0.883460 0.468507i $$-0.844792\pi$$
0.883460 0.468507i $$-0.155208\pi$$
$$108$$ 0 0
$$109$$ −4.88844e7 −0.346310 −0.173155 0.984895i $$-0.555396\pi$$
−0.173155 + 0.984895i $$0.555396\pi$$
$$110$$ 5.32699e7i 0.363841i
$$111$$ 0 0
$$112$$ −5.78683e7 −0.367763
$$113$$ 1.89245e8i 1.16068i 0.814376 + 0.580338i $$0.197080\pi$$
−0.814376 + 0.580338i $$0.802920\pi$$
$$114$$ 0 0
$$115$$ −9.65171e7 −0.551840
$$116$$ − 3.44565e7i − 0.190300i
$$117$$ 0 0
$$118$$ −3.04124e7 −0.156864
$$119$$ − 3.34780e8i − 1.66944i
$$120$$ 0 0
$$121$$ −1.92793e8 −0.899392
$$122$$ − 6.15537e7i − 0.277853i
$$123$$ 0 0
$$124$$ 6.03131e7 0.255108
$$125$$ 1.69595e8i 0.694662i
$$126$$ 0 0
$$127$$ 3.39908e8 1.30661 0.653306 0.757094i $$-0.273381\pi$$
0.653306 + 0.757094i $$0.273381\pi$$
$$128$$ − 2.37266e7i − 0.0883883i
$$129$$ 0 0
$$130$$ 1.10415e8 0.386595
$$131$$ 7.69237e7i 0.261201i 0.991435 + 0.130600i $$0.0416905\pi$$
−0.991435 + 0.130600i $$0.958309\pi$$
$$132$$ 0 0
$$133$$ 1.28226e8 0.409797
$$134$$ − 6.92577e7i − 0.214807i
$$135$$ 0 0
$$136$$ 1.37263e8 0.401234
$$137$$ 1.27859e8i 0.362952i 0.983395 + 0.181476i $$0.0580875\pi$$
−0.983395 + 0.181476i $$0.941913\pi$$
$$138$$ 0 0
$$139$$ 9.08929e6 0.0243484 0.0121742 0.999926i $$-0.496125\pi$$
0.0121742 + 0.999926i $$0.496125\pi$$
$$140$$ 1.05494e8i 0.274611i
$$141$$ 0 0
$$142$$ 2.39784e8 0.589748
$$143$$ 8.43925e8i 2.01818i
$$144$$ 0 0
$$145$$ −6.28145e7 −0.142098
$$146$$ − 5.54724e8i − 1.22086i
$$147$$ 0 0
$$148$$ 3.84947e8 0.802333
$$149$$ 1.85204e8i 0.375755i 0.982193 + 0.187877i $$0.0601607\pi$$
−0.982193 + 0.187877i $$0.939839\pi$$
$$150$$ 0 0
$$151$$ −3.35601e8 −0.645529 −0.322764 0.946479i $$-0.604612\pi$$
−0.322764 + 0.946479i $$0.604612\pi$$
$$152$$ 5.25738e7i 0.0984907i
$$153$$ 0 0
$$154$$ −8.06313e8 −1.43358
$$155$$ − 1.09951e8i − 0.190491i
$$156$$ 0 0
$$157$$ −2.01631e8 −0.331863 −0.165931 0.986137i $$-0.553063\pi$$
−0.165931 + 0.986137i $$0.553063\pi$$
$$158$$ 9.45572e7i 0.151728i
$$159$$ 0 0
$$160$$ −4.32538e7 −0.0660000
$$161$$ − 1.46092e9i − 2.17431i
$$162$$ 0 0
$$163$$ −5.27661e8 −0.747488 −0.373744 0.927532i $$-0.621926\pi$$
−0.373744 + 0.927532i $$0.621926\pi$$
$$164$$ 2.19563e8i 0.303518i
$$165$$ 0 0
$$166$$ 5.81432e8 0.765714
$$167$$ − 1.81615e8i − 0.233500i −0.993161 0.116750i $$-0.962752\pi$$
0.993161 0.116750i $$-0.0372476\pi$$
$$168$$ 0 0
$$169$$ 9.33516e8 1.14439
$$170$$ − 2.50232e8i − 0.299604i
$$171$$ 0 0
$$172$$ −4.63836e8 −0.529969
$$173$$ − 7.06460e7i − 0.0788684i −0.999222 0.0394342i $$-0.987444\pi$$
0.999222 0.0394342i $$-0.0125556\pi$$
$$174$$ 0 0
$$175$$ −1.18737e9 −1.26600
$$176$$ − 3.30596e8i − 0.344546i
$$177$$ 0 0
$$178$$ −1.21438e9 −1.20969
$$179$$ − 7.90371e7i − 0.0769873i −0.999259 0.0384936i $$-0.987744\pi$$
0.999259 0.0384936i $$-0.0122559\pi$$
$$180$$ 0 0
$$181$$ 5.48168e8 0.510739 0.255370 0.966844i $$-0.417803\pi$$
0.255370 + 0.966844i $$0.417803\pi$$
$$182$$ 1.67129e9i 1.52323i
$$183$$ 0 0
$$184$$ 5.98991e8 0.522575
$$185$$ − 7.01763e8i − 0.599106i
$$186$$ 0 0
$$187$$ 1.91257e9 1.56405
$$188$$ 7.69872e8i 0.616293i
$$189$$ 0 0
$$190$$ 9.58426e7 0.0735435
$$191$$ 2.24597e9i 1.68761i 0.536653 + 0.843803i $$0.319688\pi$$
−0.536653 + 0.843803i $$0.680312\pi$$
$$192$$ 0 0
$$193$$ 6.55575e8 0.472491 0.236245 0.971693i $$-0.424083\pi$$
0.236245 + 0.971693i $$0.424083\pi$$
$$194$$ 2.31154e8i 0.163190i
$$195$$ 0 0
$$196$$ −8.58909e8 −0.582000
$$197$$ − 4.48231e8i − 0.297603i −0.988867 0.148801i $$-0.952458\pi$$
0.988867 0.148801i $$-0.0475415\pi$$
$$198$$ 0 0
$$199$$ 7.34930e8 0.468634 0.234317 0.972160i $$-0.424715\pi$$
0.234317 + 0.972160i $$0.424715\pi$$
$$200$$ − 4.86833e8i − 0.304271i
$$201$$ 0 0
$$202$$ 1.91861e9 1.15234
$$203$$ − 9.50784e8i − 0.559883i
$$204$$ 0 0
$$205$$ 4.00266e8 0.226638
$$206$$ − 3.37434e8i − 0.187379i
$$207$$ 0 0
$$208$$ −6.85244e8 −0.366094
$$209$$ 7.32542e8i 0.383926i
$$210$$ 0 0
$$211$$ −3.26800e9 −1.64874 −0.824370 0.566051i $$-0.808471\pi$$
−0.824370 + 0.566051i $$0.808471\pi$$
$$212$$ − 1.31541e9i − 0.651206i
$$213$$ 0 0
$$214$$ 1.38959e9 0.662569
$$215$$ 8.45578e8i 0.395731i
$$216$$ 0 0
$$217$$ 1.66426e9 0.750556
$$218$$ − 5.53064e8i − 0.244878i
$$219$$ 0 0
$$220$$ −6.02680e8 −0.257274
$$221$$ − 3.96428e9i − 1.66186i
$$222$$ 0 0
$$223$$ −3.80841e9 −1.54001 −0.770005 0.638037i $$-0.779746\pi$$
−0.770005 + 0.638037i $$0.779746\pi$$
$$224$$ − 6.54705e8i − 0.260048i
$$225$$ 0 0
$$226$$ −2.14107e9 −0.820722
$$227$$ 4.34651e9i 1.63696i 0.574537 + 0.818478i $$0.305182\pi$$
−0.574537 + 0.818478i $$0.694818\pi$$
$$228$$ 0 0
$$229$$ −2.86397e9 −1.04142 −0.520711 0.853733i $$-0.674333\pi$$
−0.520711 + 0.853733i $$0.674333\pi$$
$$230$$ − 1.09197e9i − 0.390209i
$$231$$ 0 0
$$232$$ 3.89831e8 0.134563
$$233$$ 5.57102e8i 0.189022i 0.995524 + 0.0945108i $$0.0301287\pi$$
−0.995524 + 0.0945108i $$0.969871\pi$$
$$234$$ 0 0
$$235$$ 1.40348e9 0.460189
$$236$$ − 3.44077e8i − 0.110919i
$$237$$ 0 0
$$238$$ 3.78760e9 1.18047
$$239$$ 1.30831e9i 0.400977i 0.979696 + 0.200488i $$0.0642529\pi$$
−0.979696 + 0.200488i $$0.935747\pi$$
$$240$$ 0 0
$$241$$ −4.84182e9 −1.43529 −0.717647 0.696407i $$-0.754781\pi$$
−0.717647 + 0.696407i $$0.754781\pi$$
$$242$$ − 2.18120e9i − 0.635967i
$$243$$ 0 0
$$244$$ 6.96401e8 0.196472
$$245$$ 1.56580e9i 0.434582i
$$246$$ 0 0
$$247$$ 1.51838e9 0.407936
$$248$$ 6.82365e8i 0.180389i
$$249$$ 0 0
$$250$$ −1.91875e9 −0.491201
$$251$$ − 7.77711e9i − 1.95940i −0.200466 0.979701i $$-0.564246\pi$$
0.200466 0.979701i $$-0.435754\pi$$
$$252$$ 0 0
$$253$$ 8.34610e9 2.03705
$$254$$ 3.84562e9i 0.923915i
$$255$$ 0 0
$$256$$ 2.68435e8 0.0625000
$$257$$ 5.68198e9i 1.30247i 0.758877 + 0.651234i $$0.225749\pi$$
−0.758877 + 0.651234i $$0.774251\pi$$
$$258$$ 0 0
$$259$$ 1.06221e10 2.36055
$$260$$ 1.24921e9i 0.273364i
$$261$$ 0 0
$$262$$ −8.70292e8 −0.184697
$$263$$ − 1.04532e9i − 0.218486i −0.994015 0.109243i $$-0.965157\pi$$
0.994015 0.109243i $$-0.0348427\pi$$
$$264$$ 0 0
$$265$$ −2.39801e9 −0.486259
$$266$$ 1.45071e9i 0.289770i
$$267$$ 0 0
$$268$$ 7.83562e8 0.151892
$$269$$ 4.92684e9i 0.940935i 0.882417 + 0.470467i $$0.155915\pi$$
−0.882417 + 0.470467i $$0.844085\pi$$
$$270$$ 0 0
$$271$$ −6.59224e9 −1.22224 −0.611119 0.791539i $$-0.709280\pi$$
−0.611119 + 0.791539i $$0.709280\pi$$
$$272$$ 1.55295e9i 0.283716i
$$273$$ 0 0
$$274$$ −1.44656e9 −0.256646
$$275$$ − 6.78334e9i − 1.18608i
$$276$$ 0 0
$$277$$ −1.21590e8 −0.0206528 −0.0103264 0.999947i $$-0.503287\pi$$
−0.0103264 + 0.999947i $$0.503287\pi$$
$$278$$ 1.02834e8i 0.0172169i
$$279$$ 0 0
$$280$$ −1.19353e9 −0.194179
$$281$$ − 5.60638e9i − 0.899203i −0.893229 0.449601i $$-0.851566\pi$$
0.893229 0.449601i $$-0.148434\pi$$
$$282$$ 0 0
$$283$$ 1.62560e9 0.253435 0.126718 0.991939i $$-0.459556\pi$$
0.126718 + 0.991939i $$0.459556\pi$$
$$284$$ 2.71284e9i 0.417014i
$$285$$ 0 0
$$286$$ −9.54792e9 −1.42707
$$287$$ 6.05857e9i 0.892982i
$$288$$ 0 0
$$289$$ −2.00841e9 −0.287912
$$290$$ − 7.10665e8i − 0.100478i
$$291$$ 0 0
$$292$$ 6.27599e9 0.863278
$$293$$ − 4.21249e9i − 0.571569i −0.958294 0.285784i $$-0.907746\pi$$
0.958294 0.285784i $$-0.0922542\pi$$
$$294$$ 0 0
$$295$$ −6.27256e8 −0.0828241
$$296$$ 4.35518e9i 0.567335i
$$297$$ 0 0
$$298$$ −2.09534e9 −0.265699
$$299$$ − 1.72994e10i − 2.16444i
$$300$$ 0 0
$$301$$ −1.27990e10 −1.55923
$$302$$ − 3.79689e9i − 0.456458i
$$303$$ 0 0
$$304$$ −5.94805e8 −0.0696434
$$305$$ − 1.26955e9i − 0.146706i
$$306$$ 0 0
$$307$$ 5.88475e9 0.662483 0.331241 0.943546i $$-0.392533\pi$$
0.331241 + 0.943546i $$0.392533\pi$$
$$308$$ − 9.12239e9i − 1.01369i
$$309$$ 0 0
$$310$$ 1.24396e9 0.134697
$$311$$ 9.80006e9i 1.04758i 0.851847 + 0.523790i $$0.175482\pi$$
−0.851847 + 0.523790i $$0.824518\pi$$
$$312$$ 0 0
$$313$$ 3.76162e9 0.391920 0.195960 0.980612i $$-0.437218\pi$$
0.195960 + 0.980612i $$0.437218\pi$$
$$314$$ − 2.28119e9i − 0.234662i
$$315$$ 0 0
$$316$$ −1.06979e9 −0.107288
$$317$$ 1.69349e10i 1.67704i 0.544868 + 0.838522i $$0.316580\pi$$
−0.544868 + 0.838522i $$0.683420\pi$$
$$318$$ 0 0
$$319$$ 5.43174e9 0.524537
$$320$$ − 4.89360e8i − 0.0466690i
$$321$$ 0 0
$$322$$ 1.65284e10 1.53747
$$323$$ − 3.44107e9i − 0.316143i
$$324$$ 0 0
$$325$$ −1.40602e10 −1.26025
$$326$$ − 5.96980e9i − 0.528554i
$$327$$ 0 0
$$328$$ −2.48407e9 −0.214620
$$329$$ 2.12436e10i 1.81320i
$$330$$ 0 0
$$331$$ 3.18998e9 0.265752 0.132876 0.991133i $$-0.457579\pi$$
0.132876 + 0.991133i $$0.457579\pi$$
$$332$$ 6.57815e9i 0.541441i
$$333$$ 0 0
$$334$$ 2.05474e9 0.165109
$$335$$ − 1.42844e9i − 0.113418i
$$336$$ 0 0
$$337$$ 1.66378e10 1.28996 0.644978 0.764201i $$-0.276866\pi$$
0.644978 + 0.764201i $$0.276866\pi$$
$$338$$ 1.05615e10i 0.809208i
$$339$$ 0 0
$$340$$ 2.83105e9 0.211852
$$341$$ 9.50779e9i 0.703173i
$$342$$ 0 0
$$343$$ −3.33923e9 −0.241251
$$344$$ − 5.24771e9i − 0.374745i
$$345$$ 0 0
$$346$$ 7.99268e8 0.0557684
$$347$$ 1.07257e9i 0.0739792i 0.999316 + 0.0369896i $$0.0117768\pi$$
−0.999316 + 0.0369896i $$0.988223\pi$$
$$348$$ 0 0
$$349$$ −2.32302e9 −0.156586 −0.0782928 0.996930i $$-0.524947\pi$$
−0.0782928 + 0.996930i $$0.524947\pi$$
$$350$$ − 1.34336e10i − 0.895198i
$$351$$ 0 0
$$352$$ 3.74027e9 0.243631
$$353$$ − 1.98506e10i − 1.27843i −0.769030 0.639213i $$-0.779260\pi$$
0.769030 0.639213i $$-0.220740\pi$$
$$354$$ 0 0
$$355$$ 4.94554e9 0.311387
$$356$$ − 1.37392e10i − 0.855382i
$$357$$ 0 0
$$358$$ 8.94203e8 0.0544382
$$359$$ 1.03679e10i 0.624187i 0.950051 + 0.312094i $$0.101030\pi$$
−0.950051 + 0.312094i $$0.898970\pi$$
$$360$$ 0 0
$$361$$ −1.56656e10 −0.922397
$$362$$ 6.20181e9i 0.361147i
$$363$$ 0 0
$$364$$ −1.89085e10 −1.07709
$$365$$ − 1.14412e10i − 0.644614i
$$366$$ 0 0
$$367$$ −2.02880e10 −1.11834 −0.559171 0.829052i $$-0.688880\pi$$
−0.559171 + 0.829052i $$0.688880\pi$$
$$368$$ 6.77681e9i 0.369517i
$$369$$ 0 0
$$370$$ 7.93954e9 0.423632
$$371$$ − 3.62972e10i − 1.91592i
$$372$$ 0 0
$$373$$ 2.22111e10 1.14745 0.573726 0.819047i $$-0.305497\pi$$
0.573726 + 0.819047i $$0.305497\pi$$
$$374$$ 2.16382e10i 1.10595i
$$375$$ 0 0
$$376$$ −8.71010e9 −0.435785
$$377$$ − 1.12587e10i − 0.557341i
$$378$$ 0 0
$$379$$ 1.23790e10 0.599967 0.299983 0.953944i $$-0.403019\pi$$
0.299983 + 0.953944i $$0.403019\pi$$
$$380$$ 1.08433e9i 0.0520031i
$$381$$ 0 0
$$382$$ −2.54103e10 −1.19332
$$383$$ 2.90368e10i 1.34944i 0.738074 + 0.674720i $$0.235735\pi$$
−0.738074 + 0.674720i $$0.764265\pi$$
$$384$$ 0 0
$$385$$ −1.66302e10 −0.756928
$$386$$ 7.41699e9i 0.334101i
$$387$$ 0 0
$$388$$ −2.61521e9 −0.115393
$$389$$ − 2.93738e10i − 1.28281i −0.767203 0.641404i $$-0.778352\pi$$
0.767203 0.641404i $$-0.221648\pi$$
$$390$$ 0 0
$$391$$ −3.92052e10 −1.67740
$$392$$ − 9.71744e9i − 0.411536i
$$393$$ 0 0
$$394$$ 5.07116e9 0.210437
$$395$$ 1.95024e9i 0.0801125i
$$396$$ 0 0
$$397$$ 2.58158e10 1.03926 0.519630 0.854391i $$-0.326070\pi$$
0.519630 + 0.854391i $$0.326070\pi$$
$$398$$ 8.31479e9i 0.331374i
$$399$$ 0 0
$$400$$ 5.50789e9 0.215152
$$401$$ − 6.54944e9i − 0.253295i −0.991948 0.126647i $$-0.959578\pi$$
0.991948 0.126647i $$-0.0404217\pi$$
$$402$$ 0 0
$$403$$ 1.97073e10 0.747149
$$404$$ 2.17066e10i 0.814830i
$$405$$ 0 0
$$406$$ 1.07569e10 0.395897
$$407$$ 6.06834e10i 2.21153i
$$408$$ 0 0
$$409$$ 4.41813e10 1.57887 0.789433 0.613836i $$-0.210374\pi$$
0.789433 + 0.613836i $$0.210374\pi$$
$$410$$ 4.52849e9i 0.160257i
$$411$$ 0 0
$$412$$ 3.81763e9 0.132497
$$413$$ − 9.49438e9i − 0.326337i
$$414$$ 0 0
$$415$$ 1.19920e10 0.404297
$$416$$ − 7.75266e9i − 0.258867i
$$417$$ 0 0
$$418$$ −8.28777e9 −0.271477
$$419$$ − 4.52526e10i − 1.46821i −0.679038 0.734103i $$-0.737603\pi$$
0.679038 0.734103i $$-0.262397\pi$$
$$420$$ 0 0
$$421$$ 2.19193e10 0.697747 0.348873 0.937170i $$-0.386564\pi$$
0.348873 + 0.937170i $$0.386564\pi$$
$$422$$ − 3.69732e10i − 1.16584i
$$423$$ 0 0
$$424$$ 1.48822e10 0.460472
$$425$$ 3.18643e10i 0.976672i
$$426$$ 0 0
$$427$$ 1.92163e10 0.578041
$$428$$ 1.57214e10i 0.468507i
$$429$$ 0 0
$$430$$ −9.56662e9 −0.279824
$$431$$ 9.62022e9i 0.278789i 0.990237 + 0.139395i $$0.0445157\pi$$
−0.990237 + 0.139395i $$0.955484\pi$$
$$432$$ 0 0
$$433$$ −4.64805e10 −1.32227 −0.661134 0.750268i $$-0.729924\pi$$
−0.661134 + 0.750268i $$0.729924\pi$$
$$434$$ 1.88290e10i 0.530724i
$$435$$ 0 0
$$436$$ 6.25721e9 0.173155
$$437$$ − 1.50162e10i − 0.411750i
$$438$$ 0 0
$$439$$ −3.72274e10 −1.00232 −0.501158 0.865356i $$-0.667092\pi$$
−0.501158 + 0.865356i $$0.667092\pi$$
$$440$$ − 6.81855e9i − 0.181920i
$$441$$ 0 0
$$442$$ 4.48507e10 1.17511
$$443$$ 5.33802e10i 1.38601i 0.720935 + 0.693003i $$0.243713\pi$$
−0.720935 + 0.693003i $$0.756287\pi$$
$$444$$ 0 0
$$445$$ −2.50466e10 −0.638718
$$446$$ − 4.30872e10i − 1.08895i
$$447$$ 0 0
$$448$$ 7.40714e9 0.183882
$$449$$ − 4.16287e9i − 0.102425i −0.998688 0.0512126i $$-0.983691\pi$$
0.998688 0.0512126i $$-0.0163086\pi$$
$$450$$ 0 0
$$451$$ −3.46121e10 −0.836607
$$452$$ − 2.42234e10i − 0.580338i
$$453$$ 0 0
$$454$$ −4.91751e10 −1.15750
$$455$$ 3.44703e10i 0.804266i
$$456$$ 0 0
$$457$$ −2.91975e10 −0.669393 −0.334697 0.942326i $$-0.608634\pi$$
−0.334697 + 0.942326i $$0.608634\pi$$
$$458$$ − 3.24021e10i − 0.736396i
$$459$$ 0 0
$$460$$ 1.23542e10 0.275920
$$461$$ − 1.65767e10i − 0.367025i −0.983017 0.183512i $$-0.941253\pi$$
0.983017 0.183512i $$-0.0587467\pi$$
$$462$$ 0 0
$$463$$ −3.66847e10 −0.798291 −0.399146 0.916888i $$-0.630693\pi$$
−0.399146 + 0.916888i $$0.630693\pi$$
$$464$$ 4.41043e9i 0.0951501i
$$465$$ 0 0
$$466$$ −6.30289e9 −0.133658
$$467$$ − 1.80368e10i − 0.379220i −0.981860 0.189610i $$-0.939278\pi$$
0.981860 0.189610i $$-0.0607223\pi$$
$$468$$ 0 0
$$469$$ 2.16214e10 0.446882
$$470$$ 1.58786e10i 0.325402i
$$471$$ 0 0
$$472$$ 3.89279e9 0.0784319
$$473$$ − 7.31194e10i − 1.46079i
$$474$$ 0 0
$$475$$ −1.22045e10 −0.239743
$$476$$ 4.28518e10i 0.834722i
$$477$$ 0 0
$$478$$ −1.48018e10 −0.283533
$$479$$ 2.85119e9i 0.0541606i 0.999633 + 0.0270803i $$0.00862099\pi$$
−0.999633 + 0.0270803i $$0.991379\pi$$
$$480$$ 0 0
$$481$$ 1.25782e11 2.34983
$$482$$ − 5.47790e10i − 1.01491i
$$483$$ 0 0
$$484$$ 2.46775e10 0.449696
$$485$$ 4.76755e9i 0.0861645i
$$486$$ 0 0
$$487$$ −1.01307e11 −1.80104 −0.900522 0.434810i $$-0.856816\pi$$
−0.900522 + 0.434810i $$0.856816\pi$$
$$488$$ 7.87887e9i 0.138926i
$$489$$ 0 0
$$490$$ −1.77150e10 −0.307296
$$491$$ − 1.17469e10i − 0.202114i −0.994881 0.101057i $$-0.967778\pi$$
0.994881 0.101057i $$-0.0322225\pi$$
$$492$$ 0 0
$$493$$ −2.55153e10 −0.431929
$$494$$ 1.71785e10i 0.288454i
$$495$$ 0 0
$$496$$ −7.72008e9 −0.127554
$$497$$ 7.48575e10i 1.22690i
$$498$$ 0 0
$$499$$ 1.14667e11 1.84942 0.924708 0.380676i $$-0.124309\pi$$
0.924708 + 0.380676i $$0.124309\pi$$
$$500$$ − 2.17082e10i − 0.347331i
$$501$$ 0 0
$$502$$ 8.79880e10 1.38551
$$503$$ − 1.63604e10i − 0.255577i −0.991801 0.127789i $$-0.959212\pi$$
0.991801 0.127789i $$-0.0407879\pi$$
$$504$$ 0 0
$$505$$ 3.95714e10 0.608437
$$506$$ 9.44253e10i 1.44041i
$$507$$ 0 0
$$508$$ −4.35083e10 −0.653306
$$509$$ 3.92710e10i 0.585061i 0.956256 + 0.292531i $$0.0944973\pi$$
−0.956256 + 0.292531i $$0.905503\pi$$
$$510$$ 0 0
$$511$$ 1.73178e11 2.53986
$$512$$ 3.03700e9i 0.0441942i
$$513$$ 0 0
$$514$$ −6.42843e10 −0.920984
$$515$$ − 6.95958e9i − 0.0989359i
$$516$$ 0 0
$$517$$ −1.21363e11 −1.69873
$$518$$ 1.20176e11i 1.66916i
$$519$$ 0 0
$$520$$ −1.41332e10 −0.193297
$$521$$ − 3.71996e10i − 0.504879i −0.967613 0.252439i $$-0.918767\pi$$
0.967613 0.252439i $$-0.0812328\pi$$
$$522$$ 0 0
$$523$$ 3.03301e10 0.405385 0.202692 0.979242i $$-0.435031\pi$$
0.202692 + 0.979242i $$0.435031\pi$$
$$524$$ − 9.84623e9i − 0.130600i
$$525$$ 0 0
$$526$$ 1.18264e10 0.154493
$$527$$ − 4.46622e10i − 0.579026i
$$528$$ 0 0
$$529$$ −9.27734e10 −1.18468
$$530$$ − 2.71304e10i − 0.343837i
$$531$$ 0 0
$$532$$ −1.64129e10 −0.204898
$$533$$ 7.17423e10i 0.888928i
$$534$$ 0 0
$$535$$ 2.86603e10 0.349836
$$536$$ 8.86499e9i 0.107404i
$$537$$ 0 0
$$538$$ −5.57409e10 −0.665341
$$539$$ − 1.35399e11i − 1.60421i
$$540$$ 0 0
$$541$$ 7.54262e10 0.880508 0.440254 0.897873i $$-0.354888\pi$$
0.440254 + 0.897873i $$0.354888\pi$$
$$542$$ − 7.45826e10i − 0.864252i
$$543$$ 0 0
$$544$$ −1.75697e10 −0.200617
$$545$$ − 1.14069e10i − 0.129295i
$$546$$ 0 0
$$547$$ −2.04221e10 −0.228113 −0.114057 0.993474i $$-0.536385\pi$$
−0.114057 + 0.993474i $$0.536385\pi$$
$$548$$ − 1.63660e10i − 0.181476i
$$549$$ 0 0
$$550$$ 7.67447e10 0.838683
$$551$$ − 9.77272e9i − 0.106025i
$$552$$ 0 0
$$553$$ −2.95196e10 −0.315653
$$554$$ − 1.37563e9i − 0.0146037i
$$555$$ 0 0
$$556$$ −1.16343e9 −0.0121742
$$557$$ 2.53750e10i 0.263625i 0.991275 + 0.131812i $$0.0420796\pi$$
−0.991275 + 0.131812i $$0.957920\pi$$
$$558$$ 0 0
$$559$$ −1.51558e11 −1.55215
$$560$$ − 1.35033e10i − 0.137305i
$$561$$ 0 0
$$562$$ 6.34290e10 0.635832
$$563$$ − 1.03444e11i − 1.02960i −0.857309 0.514802i $$-0.827866\pi$$
0.857309 0.514802i $$-0.172134\pi$$
$$564$$ 0 0
$$565$$ −4.41595e10 −0.433341
$$566$$ 1.83915e10i 0.179206i
$$567$$ 0 0
$$568$$ −3.06923e10 −0.294874
$$569$$ − 9.50812e10i − 0.907080i −0.891236 0.453540i $$-0.850161\pi$$
0.891236 0.453540i $$-0.149839\pi$$
$$570$$ 0 0
$$571$$ −1.03046e11 −0.969368 −0.484684 0.874689i $$-0.661065\pi$$
−0.484684 + 0.874689i $$0.661065\pi$$
$$572$$ − 1.08022e11i − 1.00909i
$$573$$ 0 0
$$574$$ −6.85449e10 −0.631434
$$575$$ 1.39050e11i 1.27204i
$$576$$ 0 0
$$577$$ −8.28869e10 −0.747795 −0.373898 0.927470i $$-0.621979\pi$$
−0.373898 + 0.927470i $$0.621979\pi$$
$$578$$ − 2.27225e10i − 0.203585i
$$579$$ 0 0
$$580$$ 8.04026e9 0.0710490
$$581$$ 1.81516e11i 1.59298i
$$582$$ 0 0
$$583$$ 2.07363e11 1.79497
$$584$$ 7.10047e10i 0.610430i
$$585$$ 0 0
$$586$$ 4.76589e10 0.404160
$$587$$ − 1.12503e10i − 0.0947573i −0.998877 0.0473787i $$-0.984913\pi$$
0.998877 0.0473787i $$-0.0150867\pi$$
$$588$$ 0 0
$$589$$ 1.71063e10 0.142133
$$590$$ − 7.09659e9i − 0.0585655i
$$591$$ 0 0
$$592$$ −4.92733e10 −0.401166
$$593$$ 1.94961e11i 1.57663i 0.615272 + 0.788315i $$0.289046\pi$$
−0.615272 + 0.788315i $$0.710954\pi$$
$$594$$ 0 0
$$595$$ 7.81193e10 0.623291
$$596$$ − 2.37061e10i − 0.187877i
$$597$$ 0 0
$$598$$ 1.95720e11 1.53049
$$599$$ − 3.03920e10i − 0.236076i −0.993009 0.118038i $$-0.962340\pi$$
0.993009 0.118038i $$-0.0376605\pi$$
$$600$$ 0 0
$$601$$ −1.19015e11 −0.912227 −0.456114 0.889921i $$-0.650759\pi$$
−0.456114 + 0.889921i $$0.650759\pi$$
$$602$$ − 1.44804e11i − 1.10254i
$$603$$ 0 0
$$604$$ 4.29569e10 0.322764
$$605$$ − 4.49873e10i − 0.335790i
$$606$$ 0 0
$$607$$ −1.13080e11 −0.832973 −0.416486 0.909142i $$-0.636739\pi$$
−0.416486 + 0.909142i $$0.636739\pi$$
$$608$$ − 6.72945e9i − 0.0492453i
$$609$$ 0 0
$$610$$ 1.43633e10 0.103737
$$611$$ 2.51556e11i 1.80497i
$$612$$ 0 0
$$613$$ 1.93018e11 1.36696 0.683481 0.729968i $$-0.260465\pi$$
0.683481 + 0.729968i $$0.260465\pi$$
$$614$$ 6.65783e10i 0.468446i
$$615$$ 0 0
$$616$$ 1.03208e11 0.716788
$$617$$ 2.28160e11i 1.57434i 0.616734 + 0.787172i $$0.288455\pi$$
−0.616734 + 0.787172i $$0.711545\pi$$
$$618$$ 0 0
$$619$$ 1.21474e11 0.827413 0.413706 0.910410i $$-0.364234\pi$$
0.413706 + 0.910410i $$0.364234\pi$$
$$620$$ 1.40738e10i 0.0952453i
$$621$$ 0 0
$$622$$ −1.10875e11 −0.740751
$$623$$ − 3.79115e11i − 2.51663i
$$624$$ 0 0
$$625$$ 9.17441e10 0.601254
$$626$$ 4.25579e10i 0.277129i
$$627$$ 0 0
$$628$$ 2.58087e10 0.165931
$$629$$ − 2.85056e11i − 1.82107i
$$630$$ 0 0
$$631$$ 3.03109e11 1.91197 0.955986 0.293412i $$-0.0947908\pi$$
0.955986 + 0.293412i $$0.0947908\pi$$
$$632$$ − 1.21033e10i − 0.0758641i
$$633$$ 0 0
$$634$$ −1.91596e11 −1.18585
$$635$$ 7.93160e10i 0.487827i
$$636$$ 0 0
$$637$$ −2.80648e11 −1.70453
$$638$$ 6.14531e10i 0.370904i
$$639$$ 0 0
$$640$$ 5.53648e9 0.0330000
$$641$$ 2.84037e10i 0.168246i 0.996455 + 0.0841228i $$0.0268088\pi$$
−0.996455 + 0.0841228i $$0.973191\pi$$
$$642$$ 0 0
$$643$$ −3.38883e11 −1.98247 −0.991234 0.132118i $$-0.957822\pi$$
−0.991234 + 0.132118i $$0.957822\pi$$
$$644$$ 1.86998e11i 1.08716i
$$645$$ 0 0
$$646$$ 3.89312e10 0.223547
$$647$$ − 1.91011e11i − 1.09004i −0.838424 0.545019i $$-0.816523\pi$$
0.838424 0.545019i $$-0.183477\pi$$
$$648$$ 0 0
$$649$$ 5.42405e10 0.305735
$$650$$ − 1.59073e11i − 0.891133i
$$651$$ 0 0
$$652$$ 6.75406e10 0.373744
$$653$$ − 4.76514e10i − 0.262073i −0.991378 0.131037i $$-0.958169\pi$$
0.991378 0.131037i $$-0.0418306\pi$$
$$654$$ 0 0
$$655$$ −1.79498e10 −0.0975200
$$656$$ − 2.81041e10i − 0.151759i
$$657$$ 0 0
$$658$$ −2.40344e11 −1.28213
$$659$$ 8.16161e10i 0.432747i 0.976311 + 0.216374i $$0.0694229\pi$$
−0.976311 + 0.216374i $$0.930577\pi$$
$$660$$ 0 0
$$661$$ 5.88857e10 0.308463 0.154232 0.988035i $$-0.450710\pi$$
0.154232 + 0.988035i $$0.450710\pi$$
$$662$$ 3.60905e10i 0.187915i
$$663$$ 0 0
$$664$$ −7.44233e10 −0.382857
$$665$$ 2.99209e10i 0.152999i
$$666$$ 0 0
$$667$$ −1.11344e11 −0.562552
$$668$$ 2.32468e10i 0.116750i
$$669$$ 0 0
$$670$$ 1.61610e10 0.0801988
$$671$$ 1.09781e11i 0.541548i
$$672$$ 0 0
$$673$$ 1.65634e8 0.000807400 0 0.000403700 1.00000i $$-0.499871\pi$$
0.000403700 1.00000i $$0.499871\pi$$
$$674$$ 1.88235e11i 0.912137i
$$675$$ 0 0
$$676$$ −1.19490e11 −0.572196
$$677$$ − 2.58779e11i − 1.23190i −0.787786 0.615949i $$-0.788773\pi$$
0.787786 0.615949i $$-0.211227\pi$$
$$678$$ 0 0
$$679$$ −7.21635e10 −0.339499
$$680$$ 3.20297e10i 0.149802i
$$681$$ 0 0
$$682$$ −1.07568e11 −0.497218
$$683$$ − 1.24132e11i − 0.570428i −0.958464 0.285214i $$-0.907935\pi$$
0.958464 0.285214i $$-0.0920647\pi$$
$$684$$ 0 0
$$685$$ −2.98353e10 −0.135509
$$686$$ − 3.77791e10i − 0.170591i
$$687$$ 0 0
$$688$$ 5.93710e10 0.264985
$$689$$ − 4.29811e11i − 1.90722i
$$690$$ 0 0
$$691$$ 1.19734e11 0.525175 0.262588 0.964908i $$-0.415424\pi$$
0.262588 + 0.964908i $$0.415424\pi$$
$$692$$ 9.04269e9i 0.0394342i
$$693$$ 0 0
$$694$$ −1.21348e10 −0.0523112
$$695$$ 2.12094e9i 0.00909054i
$$696$$ 0 0
$$697$$ 1.62588e11 0.688902
$$698$$ − 2.62820e10i − 0.110723i
$$699$$ 0 0
$$700$$ 1.51983e11 0.633000
$$701$$ 1.43654e11i 0.594902i 0.954737 + 0.297451i $$0.0961365\pi$$
−0.954737 + 0.297451i $$0.903864\pi$$
$$702$$ 0 0
$$703$$ 1.09181e11 0.447018
$$704$$ 4.23163e10i 0.172273i
$$705$$ 0 0
$$706$$ 2.24584e11 0.903984
$$707$$ 5.98967e11i 2.39732i
$$708$$ 0 0
$$709$$ 4.14241e11 1.63934 0.819669 0.572838i $$-0.194158\pi$$
0.819669 + 0.572838i $$0.194158\pi$$
$$710$$ 5.59524e10i 0.220184i
$$711$$ 0 0
$$712$$ 1.55441e11 0.604847
$$713$$ − 1.94898e11i − 0.754134i
$$714$$ 0 0
$$715$$ −1.96926e11 −0.753492
$$716$$ 1.01168e10i 0.0384936i
$$717$$ 0 0
$$718$$ −1.17300e11 −0.441367
$$719$$ 3.07755e11i 1.15157i 0.817602 + 0.575784i $$0.195303\pi$$
−0.817602 + 0.575784i $$0.804697\pi$$
$$720$$ 0 0
$$721$$ 1.05343e11 0.389819
$$722$$ − 1.77236e11i − 0.652233i
$$723$$ 0 0
$$724$$ −7.01655e10 −0.255370
$$725$$ 9.04954e10i 0.327548i
$$726$$ 0 0
$$727$$ −1.59396e10 −0.0570610 −0.0285305 0.999593i $$-0.509083\pi$$
−0.0285305 + 0.999593i $$0.509083\pi$$
$$728$$ − 2.13925e11i − 0.761615i
$$729$$ 0 0
$$730$$ 1.29442e11 0.455811
$$731$$ 3.43474e11i 1.20288i
$$732$$ 0 0
$$733$$ 1.67831e10 0.0581374 0.0290687 0.999577i $$-0.490746\pi$$
0.0290687 + 0.999577i $$0.490746\pi$$
$$734$$ − 2.29532e11i − 0.790787i
$$735$$ 0 0
$$736$$ −7.66708e10 −0.261288
$$737$$ 1.23521e11i 0.418670i
$$738$$ 0 0
$$739$$ −3.67552e11 −1.23237 −0.616185 0.787601i $$-0.711323\pi$$
−0.616185 + 0.787601i $$0.711323\pi$$
$$740$$ 8.98257e10i 0.299553i
$$741$$ 0 0
$$742$$ 4.10656e11 1.35476
$$743$$ − 3.34339e11i − 1.09706i −0.836130 0.548531i $$-0.815187\pi$$
0.836130 0.548531i $$-0.184813\pi$$
$$744$$ 0 0
$$745$$ −4.32164e10 −0.140289
$$746$$ 2.51290e11i 0.811371i
$$747$$ 0 0
$$748$$ −2.44809e11 −0.782025
$$749$$ 4.33812e11i 1.37840i
$$750$$ 0 0
$$751$$ −1.79372e11 −0.563892 −0.281946 0.959430i $$-0.590980\pi$$
−0.281946 + 0.959430i $$0.590980\pi$$
$$752$$ − 9.85436e10i − 0.308146i
$$753$$ 0 0
$$754$$ 1.27377e11 0.394100
$$755$$ − 7.83109e10i − 0.241010i
$$756$$ 0 0
$$757$$ −1.95683e11 −0.595894 −0.297947 0.954582i $$-0.596302\pi$$
−0.297947 + 0.954582i $$0.596302\pi$$
$$758$$ 1.40052e11i 0.424241i
$$759$$ 0 0
$$760$$ −1.22678e10 −0.0367717
$$761$$ − 3.29292e11i − 0.981843i −0.871204 0.490922i $$-0.836660\pi$$
0.871204 0.490922i $$-0.163340\pi$$
$$762$$ 0 0
$$763$$ 1.72660e11 0.509440
$$764$$ − 2.87485e11i − 0.843803i
$$765$$ 0 0
$$766$$ −3.28514e11 −0.954198
$$767$$ − 1.12427e11i − 0.324855i
$$768$$ 0 0
$$769$$ −8.67201e10 −0.247979 −0.123989 0.992284i $$-0.539569\pi$$
−0.123989 + 0.992284i $$0.539569\pi$$
$$770$$ − 1.88149e11i − 0.535229i
$$771$$ 0 0
$$772$$ −8.39136e10 −0.236245
$$773$$ − 7.04193e8i − 0.00197230i −1.00000 0.000986152i $$-0.999686\pi$$
1.00000 0.000986152i $$-0.000313902\pi$$
$$774$$ 0 0
$$775$$ −1.58404e11 −0.439097
$$776$$ − 2.95877e10i − 0.0815952i
$$777$$ 0 0
$$778$$ 3.32326e11 0.907082
$$779$$ 6.22736e10i 0.169104i
$$780$$ 0 0
$$781$$ −4.27654e11 −1.14945
$$782$$ − 4.43557e11i − 1.18610i
$$783$$ 0 0
$$784$$ 1.09940e11 0.291000
$$785$$ − 4.70496e10i − 0.123902i
$$786$$ 0 0
$$787$$ −4.04090e11 −1.05337 −0.526683 0.850062i $$-0.676565\pi$$
−0.526683 + 0.850062i $$0.676565\pi$$
$$788$$ 5.73736e10i 0.148801i
$$789$$ 0 0
$$790$$ −2.20645e10 −0.0566481
$$791$$ − 6.68414e11i − 1.70742i
$$792$$ 0 0
$$793$$ 2.27549e11 0.575416
$$794$$ 2.92073e11i 0.734868i
$$795$$ 0 0
$$796$$ −9.40711e10 −0.234317
$$797$$ − 3.97163e11i − 0.984318i −0.870505 0.492159i $$-0.836208\pi$$
0.870505 0.492159i $$-0.163792\pi$$
$$798$$ 0 0
$$799$$ 5.70095e11 1.39881
$$800$$ 6.23147e10i 0.152135i
$$801$$ 0 0
$$802$$ 7.40984e10 0.179107
$$803$$ 9.89351e11i 2.37951i
$$804$$ 0 0
$$805$$ 3.40898e11 0.811786
$$806$$ 2.22963e11i 0.528314i
$$807$$ 0 0
$$808$$ −2.45583e11 −0.576172
$$809$$ − 5.98958e11i − 1.39831i −0.714972 0.699153i $$-0.753561\pi$$
0.714972 0.699153i $$-0.246439\pi$$
$$810$$ 0 0
$$811$$ −1.62271e11 −0.375109 −0.187554 0.982254i $$-0.560056\pi$$
−0.187554 + 0.982254i $$0.560056\pi$$
$$812$$ 1.21700e11i 0.279942i
$$813$$ 0 0
$$814$$ −6.86554e11 −1.56378
$$815$$ − 1.23127e11i − 0.279077i
$$816$$ 0 0
$$817$$ −1.31556e11 −0.295271
$$818$$ 4.99855e11i 1.11643i
$$819$$ 0 0
$$820$$ −5.12340e10 −0.113319
$$821$$ − 2.35494e11i − 0.518332i −0.965833 0.259166i $$-0.916552\pi$$
0.965833 0.259166i $$-0.0834476\pi$$
$$822$$ 0 0
$$823$$ 3.64579e11 0.794681 0.397340 0.917671i $$-0.369933\pi$$
0.397340 + 0.917671i $$0.369933\pi$$
$$824$$ 4.31916e10i 0.0936893i
$$825$$ 0 0
$$826$$ 1.07417e11 0.230755
$$827$$ 7.48202e11i 1.59955i 0.600302 + 0.799774i $$0.295047\pi$$
−0.600302 + 0.799774i $$0.704953\pi$$
$$828$$ 0 0
$$829$$ 4.91506e11 1.04066 0.520332 0.853964i $$-0.325808\pi$$
0.520332 + 0.853964i $$0.325808\pi$$
$$830$$ 1.35674e11i 0.285881i
$$831$$ 0 0
$$832$$ 8.77113e10 0.183047
$$833$$ 6.36027e11i 1.32098i
$$834$$ 0 0
$$835$$ 4.23791e10 0.0871778
$$836$$ − 9.37654e10i − 0.191963i
$$837$$ 0 0
$$838$$ 5.11974e11 1.03818
$$839$$ − 6.85496e10i − 0.138343i −0.997605 0.0691715i $$-0.977964\pi$$
0.997605 0.0691715i $$-0.0220356\pi$$
$$840$$ 0 0
$$841$$ 4.27782e11 0.855143
$$842$$ 2.47988e11i 0.493381i
$$843$$ 0 0
$$844$$ 4.18304e11 0.824370
$$845$$ 2.17832e11i 0.427262i
$$846$$ 0 0
$$847$$ 6.80944e11 1.32305
$$848$$ 1.68373e11i 0.325603i
$$849$$ 0 0
$$850$$ −3.60503e11 −0.690611
$$851$$ − 1.24393e12i − 2.37180i
$$852$$ 0 0
$$853$$ −7.31087e11 −1.38093 −0.690467 0.723364i $$-0.742595\pi$$
−0.690467 + 0.723364i $$0.742595\pi$$
$$854$$ 2.17408e11i 0.408736i
$$855$$ 0 0
$$856$$ −1.77867e11 −0.331284
$$857$$ 8.31857e11i 1.54215i 0.636747 + 0.771073i $$0.280280\pi$$
−0.636747 + 0.771073i $$0.719720\pi$$
$$858$$ 0 0
$$859$$ −2.64366e11 −0.485549 −0.242775 0.970083i $$-0.578058\pi$$
−0.242775 + 0.970083i $$0.578058\pi$$
$$860$$ − 1.08234e11i − 0.197865i
$$861$$ 0 0
$$862$$ −1.08840e11 −0.197134
$$863$$ − 7.30956e11i − 1.31780i −0.752232 0.658898i $$-0.771023\pi$$
0.752232 0.658898i $$-0.228977\pi$$
$$864$$ 0 0
$$865$$ 1.64849e10 0.0294457
$$866$$ − 5.25867e11i − 0.934984i
$$867$$ 0 0
$$868$$ −2.13026e11 −0.375278
$$869$$ − 1.68643e11i − 0.295725i
$$870$$ 0 0
$$871$$ 2.56029e11 0.444853
$$872$$ 7.07922e10i 0.122439i
$$873$$ 0 0
$$874$$ 1.69889e11 0.291151
$$875$$ − 5.99011e11i − 1.02189i
$$876$$ 0 0
$$877$$ 6.63180e11 1.12107 0.560536 0.828130i $$-0.310595\pi$$
0.560536 + 0.828130i $$0.310595\pi$$
$$878$$ − 4.21180e11i − 0.708744i
$$879$$ 0 0
$$880$$ 7.71431e10 0.128637
$$881$$ 4.47616e11i 0.743023i 0.928428 + 0.371511i $$0.121160\pi$$
−0.928428 + 0.371511i $$0.878840\pi$$
$$882$$ 0 0
$$883$$ −2.27693e11 −0.374548 −0.187274 0.982308i $$-0.559965\pi$$
−0.187274 + 0.982308i $$0.559965\pi$$
$$884$$ 5.07428e11i 0.830932i
$$885$$ 0 0
$$886$$ −6.03928e11 −0.980054
$$887$$ 1.92088e11i 0.310316i 0.987890 + 0.155158i $$0.0495887\pi$$
−0.987890 + 0.155158i $$0.950411\pi$$
$$888$$ 0 0
$$889$$ −1.20056e12 −1.92210
$$890$$ − 2.83370e11i − 0.451642i
$$891$$ 0 0
$$892$$ 4.87476e11 0.770005
$$893$$ 2.18355e11i 0.343366i
$$894$$ 0 0
$$895$$ 1.84429e10 0.0287434
$$896$$ 8.38022e10i 0.130024i
$$897$$ 0 0
$$898$$ 4.70975e10 0.0724256
$$899$$ − 1.26842e11i − 0.194189i
$$900$$ 0 0
$$901$$ −9.74072e11 −1.47806
$$902$$ − 3.91591e11i − 0.591571i
$$903$$ 0 0
$$904$$ 2.74056e11 0.410361
$$905$$ 1.27912e11i 0.190686i
$$906$$ 0 0
$$907$$ 6.50953e11 0.961879 0.480939 0.876754i $$-0.340296\pi$$
0.480939 + 0.876754i $$0.340296\pi$$
$$908$$ − 5.56353e11i − 0.818478i
$$909$$ 0 0
$$910$$ −3.89987e11 −0.568702
$$911$$ − 2.94797e11i − 0.428005i −0.976833 0.214003i $$-0.931350\pi$$
0.976833 0.214003i $$-0.0686501\pi$$
$$912$$ 0 0
$$913$$ −1.03698e12 −1.49241
$$914$$ − 3.30332e11i − 0.473333i
$$915$$ 0 0
$$916$$ 3.66588e11 0.520711
$$917$$ − 2.71694e11i − 0.384241i
$$918$$ 0 0
$$919$$ −8.25357e11 −1.15712 −0.578562 0.815639i $$-0.696386\pi$$
−0.578562 + 0.815639i $$0.696386\pi$$
$$920$$ 1.39772e11i 0.195105i
$$921$$ 0 0
$$922$$ 1.87544e11 0.259526
$$923$$ 8.86421e11i 1.22133i
$$924$$ 0 0
$$925$$ −1.01101e12 −1.38099
$$926$$ − 4.15040e11i − 0.564477i
$$927$$ 0 0
$$928$$ −4.98983e10 −0.0672813
$$929$$ 1.07779e12i 1.44701i 0.690320 + 0.723504i $$0.257470\pi$$
−0.690320 + 0.723504i $$0.742530\pi$$
$$930$$ 0 0
$$931$$ −2.43608e11 −0.324259
$$932$$ − 7.13091e10i − 0.0945108i
$$933$$ 0 0
$$934$$ 2.04063e11 0.268149
$$935$$ 4.46289e11i 0.583942i
$$936$$ 0 0
$$937$$ 1.44214e12 1.87090 0.935449 0.353463i $$-0.114996\pi$$
0.935449 + 0.353463i $$0.114996\pi$$
$$938$$ 2.44618e11i 0.315993i
$$939$$ 0 0
$$940$$ −1.79646e11 −0.230094
$$941$$ 4.67480e11i 0.596217i 0.954532 + 0.298109i $$0.0963558\pi$$
−0.954532 + 0.298109i $$0.903644\pi$$
$$942$$ 0 0
$$943$$ 7.09504e11 0.897239
$$944$$ 4.40419e10i 0.0554597i
$$945$$ 0 0
$$946$$ 8.27252e11 1.03294
$$947$$ − 1.07029e12i − 1.33077i −0.746502 0.665384i $$-0.768268\pi$$
0.746502 0.665384i $$-0.231732\pi$$
$$948$$ 0 0
$$949$$ 2.05068e12 2.52832
$$950$$ − 1.38078e11i − 0.169524i
$$951$$ 0 0
$$952$$ −4.84813e11 −0.590237
$$953$$ − 5.68036e11i − 0.688659i −0.938849 0.344330i $$-0.888106\pi$$
0.938849 0.344330i $$-0.111894\pi$$
$$954$$ 0 0
$$955$$ −5.24087e11 −0.630072
$$956$$ − 1.67464e11i − 0.200488i
$$957$$ 0 0
$$958$$ −3.22575e10 −0.0382973
$$959$$ − 4.51598e11i − 0.533922i
$$960$$ 0 0
$$961$$ −6.30865e11 −0.739679
$$962$$ 1.42306e12i 1.66158i
$$963$$ 0 0
$$964$$ 6.19753e11 0.717647
$$965$$ 1.52975e11i 0.176406i
$$966$$ 0 0
$$967$$ 5.58775e11 0.639044 0.319522 0.947579i $$-0.396478\pi$$
0.319522 + 0.947579i $$0.396478\pi$$
$$968$$ 2.79194e11i 0.317983i
$$969$$ 0 0
$$970$$ −5.39387e10 −0.0609275
$$971$$ 7.11343e11i 0.800207i 0.916470 + 0.400103i $$0.131026\pi$$
−0.916470 + 0.400103i $$0.868974\pi$$
$$972$$ 0 0
$$973$$ −3.21034e10 −0.0358178
$$974$$ − 1.14616e12i − 1.27353i
$$975$$ 0 0
$$976$$ −8.91393e10 −0.0982358
$$977$$ 2.73431e11i 0.300103i 0.988678 + 0.150051i $$0.0479439\pi$$
−0.988678 + 0.150051i $$0.952056\pi$$
$$978$$ 0 0
$$979$$ 2.16585e12 2.35775
$$980$$ − 2.00422e11i − 0.217291i
$$981$$ 0 0
$$982$$ 1.32901e11 0.142916
$$983$$ 2.95740e11i 0.316735i 0.987380 + 0.158368i $$0.0506231\pi$$
−0.987380 + 0.158368i $$0.949377\pi$$
$$984$$ 0 0
$$985$$ 1.04593e11 0.111111
$$986$$ − 2.88672e11i − 0.305420i
$$987$$ 0 0
$$988$$ −1.94352e11 −0.203968
$$989$$ 1.49886e12i 1.56666i
$$990$$ 0 0
$$991$$ −1.29048e12 −1.33800 −0.669000 0.743263i $$-0.733277\pi$$
−0.669000 + 0.743263i $$0.733277\pi$$
$$992$$ − 8.73427e10i − 0.0901945i
$$993$$ 0 0
$$994$$ −8.46916e11 −0.867550
$$995$$ 1.71492e11i 0.174966i
$$996$$ 0 0
$$997$$ −2.21939e11 −0.224623 −0.112311 0.993673i $$-0.535825\pi$$
−0.112311 + 0.993673i $$0.535825\pi$$
$$998$$ 1.29730e12i 1.30774i
$$999$$ 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 18.9.b.a.17.2 yes 2
3.2 odd 2 inner 18.9.b.a.17.1 2
4.3 odd 2 144.9.e.d.17.2 2
5.2 odd 4 450.9.b.a.449.1 4
5.3 odd 4 450.9.b.a.449.4 4
5.4 even 2 450.9.d.b.251.1 2
9.2 odd 6 162.9.d.d.53.1 4
9.4 even 3 162.9.d.d.107.1 4
9.5 odd 6 162.9.d.d.107.2 4
9.7 even 3 162.9.d.d.53.2 4
12.11 even 2 144.9.e.d.17.1 2
15.2 even 4 450.9.b.a.449.3 4
15.8 even 4 450.9.b.a.449.2 4
15.14 odd 2 450.9.d.b.251.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
18.9.b.a.17.1 2 3.2 odd 2 inner
18.9.b.a.17.2 yes 2 1.1 even 1 trivial
144.9.e.d.17.1 2 12.11 even 2
144.9.e.d.17.2 2 4.3 odd 2
162.9.d.d.53.1 4 9.2 odd 6
162.9.d.d.53.2 4 9.7 even 3
162.9.d.d.107.1 4 9.4 even 3
162.9.d.d.107.2 4 9.5 odd 6
450.9.b.a.449.1 4 5.2 odd 4
450.9.b.a.449.2 4 15.8 even 4
450.9.b.a.449.3 4 15.2 even 4
450.9.b.a.449.4 4 5.3 odd 4
450.9.d.b.251.1 2 5.4 even 2
450.9.d.b.251.2 2 15.14 odd 2