Properties

 Label 144.9.g.c.127.2 Level $144$ Weight $9$ Character 144.127 Analytic conductor $58.663$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,9,Mod(127,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

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

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

chi := DirichletCharacter("144.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 144.g (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: $$58.6625198488$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 127.2 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 144.127 Dual form 144.9.g.c.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-726.000 q^{5} +3055.34i q^{7} +O(q^{10})$$ $$q-726.000 q^{5} +3055.34i q^{7} +13281.4i q^{11} +39034.0 q^{13} +65814.0 q^{17} +130257. i q^{19} -502073. i q^{23} +136451. q^{25} -202062. q^{29} +1.19563e6i q^{31} -2.21818e6i q^{35} -1.87603e6 q^{37} -3.09105e6 q^{41} +2.26388e6i q^{43} +6.35672e6i q^{47} -3.57029e6 q^{49} +1.06648e6 q^{53} -9.64227e6i q^{55} -5.76355e6i q^{59} +1.71542e7 q^{61} -2.83387e7 q^{65} -2.74275e7i q^{67} +3.98336e7i q^{71} -5.32860e7 q^{73} -4.05791e7 q^{77} +1.82696e7i q^{79} -7.78905e6i q^{83} -4.77810e7 q^{85} -8.66672e7 q^{89} +1.19262e8i q^{91} -9.45667e7i q^{95} -7.39018e7 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 1452 q^{5}+O(q^{10})$$ 2 * q - 1452 * q^5 $$2 q - 1452 q^{5} + 78068 q^{13} + 131628 q^{17} + 272902 q^{25} - 404124 q^{29} - 3752060 q^{37} - 6182100 q^{41} - 7140574 q^{49} + 2132964 q^{53} + 34308388 q^{61} - 56677368 q^{65} - 106572028 q^{73} - 81158112 q^{77} - 95561928 q^{85} - 173334468 q^{89} - 147803644 q^{97}+O(q^{100})$$ 2 * q - 1452 * q^5 + 78068 * q^13 + 131628 * q^17 + 272902 * q^25 - 404124 * q^29 - 3752060 * q^37 - 6182100 * q^41 - 7140574 * q^49 + 2132964 * q^53 + 34308388 * q^61 - 56677368 * q^65 - 106572028 * q^73 - 81158112 * q^77 - 95561928 * q^85 - 173334468 * q^89 - 147803644 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −726.000 −1.16160 −0.580800 0.814046i $$-0.697260\pi$$
−0.580800 + 0.814046i $$0.697260\pi$$
$$6$$ 0 0
$$7$$ 3055.34i 1.27253i 0.771472 + 0.636264i $$0.219521\pi$$
−0.771472 + 0.636264i $$0.780479\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 13281.4i 0.907135i 0.891222 + 0.453568i $$0.149849\pi$$
−0.891222 + 0.453568i $$0.850151\pi$$
$$12$$ 0 0
$$13$$ 39034.0 1.36669 0.683344 0.730096i $$-0.260525\pi$$
0.683344 + 0.730096i $$0.260525\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 65814.0 0.787993 0.393997 0.919112i $$-0.371092\pi$$
0.393997 + 0.919112i $$0.371092\pi$$
$$18$$ 0 0
$$19$$ 130257.i 0.999510i 0.866167 + 0.499755i $$0.166577\pi$$
−0.866167 + 0.499755i $$0.833423\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 502073.i − 1.79414i −0.441892 0.897068i $$-0.645692\pi$$
0.441892 0.897068i $$-0.354308\pi$$
$$24$$ 0 0
$$25$$ 136451. 0.349315
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −202062. −0.285688 −0.142844 0.989745i $$-0.545625\pi$$
−0.142844 + 0.989745i $$0.545625\pi$$
$$30$$ 0 0
$$31$$ 1.19563e6i 1.29465i 0.762215 + 0.647324i $$0.224112\pi$$
−0.762215 + 0.647324i $$0.775888\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 2.21818e6i − 1.47817i
$$36$$ 0 0
$$37$$ −1.87603e6 −1.00100 −0.500499 0.865737i $$-0.666850\pi$$
−0.500499 + 0.865737i $$0.666850\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −3.09105e6 −1.09388 −0.546941 0.837171i $$-0.684208\pi$$
−0.546941 + 0.837171i $$0.684208\pi$$
$$42$$ 0 0
$$43$$ 2.26388e6i 0.662186i 0.943598 + 0.331093i $$0.107417\pi$$
−0.943598 + 0.331093i $$0.892583\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.35672e6i 1.30269i 0.758781 + 0.651346i $$0.225795\pi$$
−0.758781 + 0.651346i $$0.774205\pi$$
$$48$$ 0 0
$$49$$ −3.57029e6 −0.619325
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.06648e6 0.135161 0.0675803 0.997714i $$-0.478472\pi$$
0.0675803 + 0.997714i $$0.478472\pi$$
$$54$$ 0 0
$$55$$ − 9.64227e6i − 1.05373i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 5.76355e6i − 0.475644i −0.971309 0.237822i $$-0.923566\pi$$
0.971309 0.237822i $$-0.0764335\pi$$
$$60$$ 0 0
$$61$$ 1.71542e7 1.23894 0.619471 0.785020i $$-0.287347\pi$$
0.619471 + 0.785020i $$0.287347\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2.83387e7 −1.58755
$$66$$ 0 0
$$67$$ − 2.74275e7i − 1.36109i −0.732706 0.680546i $$-0.761743\pi$$
0.732706 0.680546i $$-0.238257\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 3.98336e7i 1.56753i 0.621056 + 0.783766i $$0.286704\pi$$
−0.621056 + 0.783766i $$0.713296\pi$$
$$72$$ 0 0
$$73$$ −5.32860e7 −1.87638 −0.938192 0.346115i $$-0.887501\pi$$
−0.938192 + 0.346115i $$0.887501\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −4.05791e7 −1.15435
$$78$$ 0 0
$$79$$ 1.82696e7i 0.469052i 0.972110 + 0.234526i $$0.0753538\pi$$
−0.972110 + 0.234526i $$0.924646\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 7.78905e6i − 0.164124i −0.996627 0.0820620i $$-0.973849\pi$$
0.996627 0.0820620i $$-0.0261506\pi$$
$$84$$ 0 0
$$85$$ −4.77810e7 −0.915333
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −8.66672e7 −1.38132 −0.690661 0.723179i $$-0.742680\pi$$
−0.690661 + 0.723179i $$0.742680\pi$$
$$90$$ 0 0
$$91$$ 1.19262e8i 1.73915i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 9.45667e7i − 1.16103i
$$96$$ 0 0
$$97$$ −7.39018e7 −0.834773 −0.417386 0.908729i $$-0.637054\pi$$
−0.417386 + 0.908729i $$0.637054\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1.91310e8 −1.83845 −0.919227 0.393727i $$-0.871185\pi$$
−0.919227 + 0.393727i $$0.871185\pi$$
$$102$$ 0 0
$$103$$ − 1.62781e8i − 1.44629i −0.690699 0.723143i $$-0.742697\pi$$
0.690699 0.723143i $$-0.257303\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 2.00810e8i − 1.53197i −0.642857 0.765986i $$-0.722251\pi$$
0.642857 0.765986i $$-0.277749\pi$$
$$108$$ 0 0
$$109$$ 6.86083e7 0.486039 0.243019 0.970021i $$-0.421862\pi$$
0.243019 + 0.970021i $$0.421862\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −3.30831e7 −0.202905 −0.101452 0.994840i $$-0.532349\pi$$
−0.101452 + 0.994840i $$0.532349\pi$$
$$114$$ 0 0
$$115$$ 3.64505e8i 2.08407i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 2.01084e8i 1.00274i
$$120$$ 0 0
$$121$$ 3.79642e7 0.177106
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.84530e8 0.755836
$$126$$ 0 0
$$127$$ − 2.70471e8i − 1.03970i −0.854259 0.519848i $$-0.825989\pi$$
0.854259 0.519848i $$-0.174011\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 3.02851e8i 1.02836i 0.857683 + 0.514178i $$0.171903\pi$$
−0.857683 + 0.514178i $$0.828097\pi$$
$$132$$ 0 0
$$133$$ −3.97980e8 −1.27190
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −6.40316e8 −1.81766 −0.908828 0.417170i $$-0.863022\pi$$
−0.908828 + 0.417170i $$0.863022\pi$$
$$138$$ 0 0
$$139$$ − 4.90714e8i − 1.31453i −0.753661 0.657263i $$-0.771714\pi$$
0.753661 0.657263i $$-0.228286\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 5.18425e8i 1.23977i
$$144$$ 0 0
$$145$$ 1.46697e8 0.331856
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 8.11121e7 0.164566 0.0822831 0.996609i $$-0.473779\pi$$
0.0822831 + 0.996609i $$0.473779\pi$$
$$150$$ 0 0
$$151$$ − 1.77325e8i − 0.341086i −0.985350 0.170543i $$-0.945448\pi$$
0.985350 0.170543i $$-0.0545521\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 8.68031e8i − 1.50386i
$$156$$ 0 0
$$157$$ −2.14784e7 −0.0353511 −0.0176755 0.999844i $$-0.505627\pi$$
−0.0176755 + 0.999844i $$0.505627\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1.53400e9 2.28309
$$162$$ 0 0
$$163$$ 2.42230e8i 0.343144i 0.985172 + 0.171572i $$0.0548847\pi$$
−0.985172 + 0.171572i $$0.945115\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3.89012e8i 0.500146i 0.968227 + 0.250073i $$0.0804547\pi$$
−0.968227 + 0.250073i $$0.919545\pi$$
$$168$$ 0 0
$$169$$ 7.07922e8 0.867838
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 6.35072e7 0.0708988 0.0354494 0.999371i $$-0.488714\pi$$
0.0354494 + 0.999371i $$0.488714\pi$$
$$174$$ 0 0
$$175$$ 4.16904e8i 0.444512i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 5.33629e8i − 0.519789i −0.965637 0.259895i $$-0.916312\pi$$
0.965637 0.259895i $$-0.0836878\pi$$
$$180$$ 0 0
$$181$$ 8.56360e8 0.797888 0.398944 0.916975i $$-0.369377\pi$$
0.398944 + 0.916975i $$0.369377\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1.36200e9 1.16276
$$186$$ 0 0
$$187$$ 8.74100e8i 0.714817i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 4.75759e8i − 0.357481i −0.983896 0.178741i $$-0.942798\pi$$
0.983896 0.178741i $$-0.0572023\pi$$
$$192$$ 0 0
$$193$$ 8.76708e8 0.631867 0.315933 0.948781i $$-0.397682\pi$$
0.315933 + 0.948781i $$0.397682\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.76762e9 1.83756 0.918780 0.394771i $$-0.129176\pi$$
0.918780 + 0.394771i $$0.129176\pi$$
$$198$$ 0 0
$$199$$ 1.42932e9i 0.911420i 0.890128 + 0.455710i $$0.150615\pi$$
−0.890128 + 0.455710i $$0.849385\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 6.17368e8i − 0.363546i
$$204$$ 0 0
$$205$$ 2.24410e9 1.27065
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1.72999e9 −0.906691
$$210$$ 0 0
$$211$$ 4.61738e8i 0.232952i 0.993194 + 0.116476i $$0.0371598\pi$$
−0.993194 + 0.116476i $$0.962840\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ − 1.64358e9i − 0.769195i
$$216$$ 0 0
$$217$$ −3.65307e9 −1.64747
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 2.56898e9 1.07694
$$222$$ 0 0
$$223$$ − 3.40037e9i − 1.37501i −0.726179 0.687506i $$-0.758705\pi$$
0.726179 0.687506i $$-0.241295\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 4.52697e9i 1.70492i 0.522792 + 0.852460i $$0.324891\pi$$
−0.522792 + 0.852460i $$0.675109\pi$$
$$228$$ 0 0
$$229$$ −9.90176e8 −0.360056 −0.180028 0.983661i $$-0.557619\pi$$
−0.180028 + 0.983661i $$0.557619\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2.23709e9 −0.759032 −0.379516 0.925185i $$-0.623909\pi$$
−0.379516 + 0.925185i $$0.623909\pi$$
$$234$$ 0 0
$$235$$ − 4.61498e9i − 1.51321i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 2.63524e9i − 0.807659i −0.914834 0.403830i $$-0.867679\pi$$
0.914834 0.403830i $$-0.132321\pi$$
$$240$$ 0 0
$$241$$ −6.19651e8 −0.183687 −0.0918436 0.995773i $$-0.529276\pi$$
−0.0918436 + 0.995773i $$0.529276\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 2.59203e9 0.719408
$$246$$ 0 0
$$247$$ 5.08446e9i 1.36602i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 6.81334e9i 1.71659i 0.513161 + 0.858293i $$0.328474\pi$$
−0.513161 + 0.858293i $$0.671526\pi$$
$$252$$ 0 0
$$253$$ 6.66822e9 1.62752
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −3.95756e9 −0.907184 −0.453592 0.891209i $$-0.649858\pi$$
−0.453592 + 0.891209i $$0.649858\pi$$
$$258$$ 0 0
$$259$$ − 5.73191e9i − 1.27380i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 1.86129e9i 0.389037i 0.980899 + 0.194518i $$0.0623144\pi$$
−0.980899 + 0.194518i $$0.937686\pi$$
$$264$$ 0 0
$$265$$ −7.74266e8 −0.157003
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −1.17367e9 −0.224149 −0.112074 0.993700i $$-0.535750\pi$$
−0.112074 + 0.993700i $$0.535750\pi$$
$$270$$ 0 0
$$271$$ 1.90505e9i 0.353207i 0.984282 + 0.176604i $$0.0565110\pi$$
−0.984282 + 0.176604i $$0.943489\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.81226e9i 0.316876i
$$276$$ 0 0
$$277$$ −5.03752e9 −0.855654 −0.427827 0.903861i $$-0.640721\pi$$
−0.427827 + 0.903861i $$0.640721\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.66317e9 −1.06870 −0.534350 0.845264i $$-0.679443\pi$$
−0.534350 + 0.845264i $$0.679443\pi$$
$$282$$ 0 0
$$283$$ 5.54295e9i 0.864162i 0.901835 + 0.432081i $$0.142221\pi$$
−0.901835 + 0.432081i $$0.857779\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 9.44420e9i − 1.39199i
$$288$$ 0 0
$$289$$ −2.64427e9 −0.379066
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −6.67390e9 −0.905543 −0.452772 0.891627i $$-0.649565\pi$$
−0.452772 + 0.891627i $$0.649565\pi$$
$$294$$ 0 0
$$295$$ 4.18434e9i 0.552508i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 1.95979e10i − 2.45203i
$$300$$ 0 0
$$301$$ −6.91692e9 −0.842649
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −1.24539e10 −1.43916
$$306$$ 0 0
$$307$$ − 6.49752e8i − 0.0731466i −0.999331 0.0365733i $$-0.988356\pi$$
0.999331 0.0365733i $$-0.0116442\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 3.97832e8i 0.0425264i 0.999774 + 0.0212632i $$0.00676879\pi$$
−0.999774 + 0.0212632i $$0.993231\pi$$
$$312$$ 0 0
$$313$$ −1.58217e10 −1.64845 −0.824223 0.566266i $$-0.808388\pi$$
−0.824223 + 0.566266i $$0.808388\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 9.60836e9 0.951507 0.475754 0.879579i $$-0.342175\pi$$
0.475754 + 0.879579i $$0.342175\pi$$
$$318$$ 0 0
$$319$$ − 2.68366e9i − 0.259158i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 8.57274e9i 0.787607i
$$324$$ 0 0
$$325$$ 5.32623e9 0.477404
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −1.94219e10 −1.65771
$$330$$ 0 0
$$331$$ 1.41991e10i 1.18290i 0.806341 + 0.591451i $$0.201445\pi$$
−0.806341 + 0.591451i $$0.798555\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 1.99124e10i 1.58104i
$$336$$ 0 0
$$337$$ 3.39383e9 0.263130 0.131565 0.991308i $$-0.458000\pi$$
0.131565 + 0.991308i $$0.458000\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −1.58797e10 −1.17442
$$342$$ 0 0
$$343$$ 6.70498e9i 0.484419i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 1.35188e8i − 0.00932439i −0.999989 0.00466219i $$-0.998516\pi$$
0.999989 0.00466219i $$-0.00148403\pi$$
$$348$$ 0 0
$$349$$ 1.13213e10 0.763122 0.381561 0.924344i $$-0.375387\pi$$
0.381561 + 0.924344i $$0.375387\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 1.42650e10 0.918697 0.459348 0.888256i $$-0.348083\pi$$
0.459348 + 0.888256i $$0.348083\pi$$
$$354$$ 0 0
$$355$$ − 2.89192e10i − 1.82085i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 8.15636e9i − 0.491042i −0.969391 0.245521i $$-0.921041\pi$$
0.969391 0.245521i $$-0.0789590\pi$$
$$360$$ 0 0
$$361$$ 1.66382e7 0.000979664 0
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3.86856e10 2.17961
$$366$$ 0 0
$$367$$ − 2.06760e10i − 1.13973i −0.821738 0.569865i $$-0.806995\pi$$
0.821738 0.569865i $$-0.193005\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 3.25846e9i 0.171996i
$$372$$ 0 0
$$373$$ −7.71358e9 −0.398493 −0.199247 0.979949i $$-0.563849\pi$$
−0.199247 + 0.979949i $$0.563849\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −7.88729e9 −0.390447
$$378$$ 0 0
$$379$$ 1.53767e9i 0.0745256i 0.999306 + 0.0372628i $$0.0118639\pi$$
−0.999306 + 0.0372628i $$0.988136\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 3.20555e10i 1.48973i 0.667216 + 0.744864i $$0.267486\pi$$
−0.667216 + 0.744864i $$0.732514\pi$$
$$384$$ 0 0
$$385$$ 2.94604e10 1.34090
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 2.99296e10 1.30708 0.653541 0.756891i $$-0.273283\pi$$
0.653541 + 0.756891i $$0.273283\pi$$
$$390$$ 0 0
$$391$$ − 3.30434e10i − 1.41377i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 1.32637e10i − 0.544851i
$$396$$ 0 0
$$397$$ 1.32156e10 0.532016 0.266008 0.963971i $$-0.414295\pi$$
0.266008 + 0.963971i $$0.414295\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.51637e10 −0.973190 −0.486595 0.873628i $$-0.661761\pi$$
−0.486595 + 0.873628i $$0.661761\pi$$
$$402$$ 0 0
$$403$$ 4.66704e10i 1.76938i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 2.49162e10i − 0.908040i
$$408$$ 0 0
$$409$$ −3.78473e10 −1.35251 −0.676257 0.736666i $$-0.736399\pi$$
−0.676257 + 0.736666i $$0.736399\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 1.76096e10 0.605270
$$414$$ 0 0
$$415$$ 5.65485e9i 0.190647i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ − 1.88088e10i − 0.610246i −0.952313 0.305123i $$-0.901302\pi$$
0.952313 0.305123i $$-0.0986976\pi$$
$$420$$ 0 0
$$421$$ 6.04555e9 0.192445 0.0962227 0.995360i $$-0.469324\pi$$
0.0962227 + 0.995360i $$0.469324\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 8.98039e9 0.275258
$$426$$ 0 0
$$427$$ 5.24119e10i 1.57659i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 4.60486e10i 1.33447i 0.744849 + 0.667233i $$0.232521\pi$$
−0.744849 + 0.667233i $$0.767479\pi$$
$$432$$ 0 0
$$433$$ 1.85654e9 0.0528145 0.0264072 0.999651i $$-0.491593\pi$$
0.0264072 + 0.999651i $$0.491593\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 6.53986e10 1.79326
$$438$$ 0 0
$$439$$ 1.24165e10i 0.334303i 0.985931 + 0.167152i $$0.0534569\pi$$
−0.985931 + 0.167152i $$0.946543\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 7.57779e9i − 0.196756i −0.995149 0.0983779i $$-0.968635\pi$$
0.995149 0.0983779i $$-0.0313654\pi$$
$$444$$ 0 0
$$445$$ 6.29204e10 1.60454
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 3.37970e10 0.831559 0.415780 0.909465i $$-0.363509\pi$$
0.415780 + 0.909465i $$0.363509\pi$$
$$450$$ 0 0
$$451$$ − 4.10534e10i − 0.992299i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ − 8.65842e10i − 2.02020i
$$456$$ 0 0
$$457$$ −2.01366e10 −0.461659 −0.230829 0.972994i $$-0.574144\pi$$
−0.230829 + 0.972994i $$0.574144\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 2.54155e10 0.562724 0.281362 0.959602i $$-0.409214\pi$$
0.281362 + 0.959602i $$0.409214\pi$$
$$462$$ 0 0
$$463$$ − 1.19712e9i − 0.0260504i −0.999915 0.0130252i $$-0.995854\pi$$
0.999915 0.0130252i $$-0.00414617\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 2.92676e10i − 0.615347i −0.951492 0.307673i $$-0.900450\pi$$
0.951492 0.307673i $$-0.0995504\pi$$
$$468$$ 0 0
$$469$$ 8.38003e10 1.73203
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −3.00674e10 −0.600692
$$474$$ 0 0
$$475$$ 1.77737e10i 0.349143i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 2.51066e10i 0.476919i 0.971152 + 0.238460i $$0.0766425\pi$$
−0.971152 + 0.238460i $$0.923357\pi$$
$$480$$ 0 0
$$481$$ −7.32290e10 −1.36805
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 5.36527e10 0.969672
$$486$$ 0 0
$$487$$ 5.82581e10i 1.03571i 0.855467 + 0.517857i $$0.173270\pi$$
−0.855467 + 0.517857i $$0.826730\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 3.61816e10i 0.622532i 0.950323 + 0.311266i $$0.100753\pi$$
−0.950323 + 0.311266i $$0.899247\pi$$
$$492$$ 0 0
$$493$$ −1.32985e10 −0.225121
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −1.21705e11 −1.99473
$$498$$ 0 0
$$499$$ − 5.58440e10i − 0.900687i −0.892855 0.450344i $$-0.851301\pi$$
0.892855 0.450344i $$-0.148699\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 1.84340e10i 0.287971i 0.989580 + 0.143985i $$0.0459918\pi$$
−0.989580 + 0.143985i $$0.954008\pi$$
$$504$$ 0 0
$$505$$ 1.38891e11 2.13555
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −1.42165e10 −0.211798 −0.105899 0.994377i $$-0.533772\pi$$
−0.105899 + 0.994377i $$0.533772\pi$$
$$510$$ 0 0
$$511$$ − 1.62807e11i − 2.38775i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 1.18179e11i 1.68001i
$$516$$ 0 0
$$517$$ −8.44260e10 −1.18172
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 6.81614e10 0.925098 0.462549 0.886594i $$-0.346935\pi$$
0.462549 + 0.886594i $$0.346935\pi$$
$$522$$ 0 0
$$523$$ − 5.63922e10i − 0.753724i −0.926269 0.376862i $$-0.877003\pi$$
0.926269 0.376862i $$-0.122997\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 7.86895e10i 1.02017i
$$528$$ 0 0
$$529$$ −1.73766e11 −2.21893
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −1.20656e11 −1.49500
$$534$$ 0 0
$$535$$ 1.45788e11i 1.77954i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 4.74183e10i − 0.561812i
$$540$$ 0 0
$$541$$ −7.61478e10 −0.888932 −0.444466 0.895796i $$-0.646607\pi$$
−0.444466 + 0.895796i $$0.646607\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −4.98096e10 −0.564583
$$546$$ 0 0
$$547$$ − 5.84939e10i − 0.653373i −0.945133 0.326686i $$-0.894068\pi$$
0.945133 0.326686i $$-0.105932\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 2.63200e10i − 0.285548i
$$552$$ 0 0
$$553$$ −5.58198e10 −0.596881
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −1.61301e11 −1.67577 −0.837887 0.545844i $$-0.816209\pi$$
−0.837887 + 0.545844i $$0.816209\pi$$
$$558$$ 0 0
$$559$$ 8.83683e10i 0.905002i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 6.88172e9i 0.0684957i 0.999413 + 0.0342479i $$0.0109036\pi$$
−0.999413 + 0.0342479i $$0.989096\pi$$
$$564$$ 0 0
$$565$$ 2.40183e10 0.235694
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −9.38382e10 −0.895221 −0.447611 0.894229i $$-0.647725\pi$$
−0.447611 + 0.894229i $$0.647725\pi$$
$$570$$ 0 0
$$571$$ 1.92744e10i 0.181316i 0.995882 + 0.0906582i $$0.0288971\pi$$
−0.995882 + 0.0906582i $$0.971103\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 6.85084e10i − 0.626718i
$$576$$ 0 0
$$577$$ 1.65488e11 1.49301 0.746507 0.665378i $$-0.231730\pi$$
0.746507 + 0.665378i $$0.231730\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 2.37982e10 0.208852
$$582$$ 0 0
$$583$$ 1.41643e10i 0.122609i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 2.09633e11i − 1.76566i −0.469695 0.882829i $$-0.655636\pi$$
0.469695 0.882829i $$-0.344364\pi$$
$$588$$ 0 0
$$589$$ −1.55740e11 −1.29401
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 7.33746e10 0.593372 0.296686 0.954975i $$-0.404119\pi$$
0.296686 + 0.954975i $$0.404119\pi$$
$$594$$ 0 0
$$595$$ − 1.45987e11i − 1.16479i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ − 1.33326e11i − 1.03563i −0.855491 0.517817i $$-0.826745\pi$$
0.855491 0.517817i $$-0.173255\pi$$
$$600$$ 0 0
$$601$$ 2.01691e11 1.54593 0.772965 0.634449i $$-0.218773\pi$$
0.772965 + 0.634449i $$0.218773\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −2.75620e10 −0.205726
$$606$$ 0 0
$$607$$ 1.55515e11i 1.14556i 0.819710 + 0.572779i $$0.194134\pi$$
−0.819710 + 0.572779i $$0.805866\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 2.48128e11i 1.78038i
$$612$$ 0 0
$$613$$ 3.19775e10 0.226466 0.113233 0.993568i $$-0.463879\pi$$
0.113233 + 0.993568i $$0.463879\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −5.63108e10 −0.388553 −0.194277 0.980947i $$-0.562236\pi$$
−0.194277 + 0.980947i $$0.562236\pi$$
$$618$$ 0 0
$$619$$ 2.66432e11i 1.81478i 0.420287 + 0.907391i $$0.361929\pi$$
−0.420287 + 0.907391i $$0.638071\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 2.64798e11i − 1.75777i
$$624$$ 0 0
$$625$$ −1.87270e11 −1.22729
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −1.23469e11 −0.788779
$$630$$ 0 0
$$631$$ 8.55727e9i 0.0539781i 0.999636 + 0.0269891i $$0.00859193\pi$$
−0.999636 + 0.0269891i $$0.991408\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 1.96362e11i 1.20771i
$$636$$ 0 0
$$637$$ −1.39363e11 −0.846425
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2.90248e10 −0.171924 −0.0859620 0.996298i $$-0.527396\pi$$
−0.0859620 + 0.996298i $$0.527396\pi$$
$$642$$ 0 0
$$643$$ − 5.13563e10i − 0.300435i −0.988653 0.150217i $$-0.952003\pi$$
0.988653 0.150217i $$-0.0479973\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 5.58175e10i − 0.318532i −0.987236 0.159266i $$-0.949087\pi$$
0.987236 0.159266i $$-0.0509128\pi$$
$$648$$ 0 0
$$649$$ 7.65478e10 0.431473
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 6.40717e10 0.352382 0.176191 0.984356i $$-0.443622\pi$$
0.176191 + 0.984356i $$0.443622\pi$$
$$654$$ 0 0
$$655$$ − 2.19870e11i − 1.19454i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 3.17581e11i 1.68389i 0.539567 + 0.841943i $$0.318588\pi$$
−0.539567 + 0.841943i $$0.681412\pi$$
$$660$$ 0 0
$$661$$ −1.33716e11 −0.700449 −0.350224 0.936666i $$-0.613895\pi$$
−0.350224 + 0.936666i $$0.613895\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2.88933e11 1.47744
$$666$$ 0 0
$$667$$ 1.01450e11i 0.512564i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 2.27831e11i 1.12389i
$$672$$ 0 0
$$673$$ −4.12429e10 −0.201043 −0.100521 0.994935i $$-0.532051\pi$$
−0.100521 + 0.994935i $$0.532051\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 5.06159e10 0.240953 0.120476 0.992716i $$-0.461558\pi$$
0.120476 + 0.992716i $$0.461558\pi$$
$$678$$ 0 0
$$679$$ − 2.25795e11i − 1.06227i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 3.59716e11i − 1.65302i −0.562924 0.826508i $$-0.690324\pi$$
0.562924 0.826508i $$-0.309676\pi$$
$$684$$ 0 0
$$685$$ 4.64869e11 2.11139
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 4.16291e10 0.184722
$$690$$ 0 0
$$691$$ − 1.38563e11i − 0.607763i −0.952710 0.303882i $$-0.901717\pi$$
0.952710 0.303882i $$-0.0982827\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 3.56258e11i 1.52695i
$$696$$ 0 0
$$697$$ −2.03434e11 −0.861972
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −2.39409e11 −0.991445 −0.495722 0.868481i $$-0.665097\pi$$
−0.495722 + 0.868481i $$0.665097\pi$$
$$702$$ 0 0
$$703$$ − 2.44366e11i − 1.00051i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 5.84518e11i − 2.33948i
$$708$$ 0 0
$$709$$ 1.08904e11 0.430981 0.215490 0.976506i $$-0.430865\pi$$
0.215490 + 0.976506i $$0.430865\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 6.00296e11 2.32278
$$714$$ 0 0
$$715$$ − 3.76376e11i − 1.44012i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 2.60327e11i 0.974101i 0.873374 + 0.487051i $$0.161927\pi$$
−0.873374 + 0.487051i $$0.838073\pi$$
$$720$$ 0 0
$$721$$ 4.97350e11 1.84044
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −2.75716e10 −0.0997951
$$726$$ 0 0
$$727$$ 4.81879e11i 1.72504i 0.506020 + 0.862522i $$0.331116\pi$$
−0.506020 + 0.862522i $$0.668884\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1.48995e11i 0.521798i
$$732$$ 0 0
$$733$$ −1.82274e11 −0.631405 −0.315702 0.948858i $$-0.602240\pi$$
−0.315702 + 0.948858i $$0.602240\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 3.64275e11 1.23469
$$738$$ 0 0
$$739$$ 5.49813e11i 1.84347i 0.387815 + 0.921737i $$0.373230\pi$$
−0.387815 + 0.921737i $$0.626770\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 2.24817e11i 0.737690i 0.929491 + 0.368845i $$0.120247\pi$$
−0.929491 + 0.368845i $$0.879753\pi$$
$$744$$ 0 0
$$745$$ −5.88874e10 −0.191160
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 6.13543e11 1.94948
$$750$$ 0 0
$$751$$ − 4.17556e11i − 1.31267i −0.754470 0.656334i $$-0.772106\pi$$
0.754470 0.656334i $$-0.227894\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 1.28738e11i 0.396205i
$$756$$ 0 0
$$757$$ 6.29371e11 1.91656 0.958282 0.285826i $$-0.0922679\pi$$
0.958282 + 0.285826i $$0.0922679\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 1.72289e11 0.513710 0.256855 0.966450i $$-0.417314\pi$$
0.256855 + 0.966450i $$0.417314\pi$$
$$762$$ 0 0
$$763$$ 2.09622e11i 0.618497i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 2.24974e11i − 0.650058i
$$768$$ 0 0
$$769$$ 9.21192e10 0.263418 0.131709 0.991288i $$-0.457954\pi$$
0.131709 + 0.991288i $$0.457954\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −4.70219e11 −1.31699 −0.658495 0.752585i $$-0.728806\pi$$
−0.658495 + 0.752585i $$0.728806\pi$$
$$774$$ 0 0
$$775$$ 1.63146e11i 0.452239i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 4.02631e11i − 1.09335i
$$780$$ 0 0
$$781$$ −5.29045e11 −1.42196
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1.55933e10 0.0410638
$$786$$ 0 0
$$787$$ 5.86015e11i 1.52760i 0.645452 + 0.763801i $$0.276669\pi$$
−0.645452 + 0.763801i $$0.723331\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 1.01080e11i − 0.258202i
$$792$$ 0 0
$$793$$ 6.69597e11 1.69325
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 2.08165e11 0.515912 0.257956 0.966157i $$-0.416951\pi$$
0.257956 + 0.966157i $$0.416951\pi$$
$$798$$ 0 0
$$799$$ 4.18361e11i 1.02651i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 7.07711e11i − 1.70213i
$$804$$ 0 0
$$805$$ −1.11369e12 −2.65203
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −3.24105e11 −0.756645 −0.378322 0.925674i $$-0.623499\pi$$
−0.378322 + 0.925674i $$0.623499\pi$$
$$810$$ 0 0
$$811$$ − 6.94202e11i − 1.60473i −0.596833 0.802366i $$-0.703574\pi$$
0.596833 0.802366i $$-0.296426\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 1.75859e11i − 0.398597i
$$816$$ 0 0
$$817$$ −2.94887e11 −0.661861
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −4.79835e11 −1.05614 −0.528068 0.849202i $$-0.677083\pi$$
−0.528068 + 0.849202i $$0.677083\pi$$
$$822$$ 0 0
$$823$$ − 3.34155e11i − 0.728365i −0.931328 0.364183i $$-0.881348\pi$$
0.931328 0.364183i $$-0.118652\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 6.22860e9i − 0.0133158i −0.999978 0.00665792i $$-0.997881\pi$$
0.999978 0.00665792i $$-0.00211930\pi$$
$$828$$ 0 0
$$829$$ 6.97808e11 1.47747 0.738733 0.673998i $$-0.235424\pi$$
0.738733 + 0.673998i $$0.235424\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −2.34975e11 −0.488024
$$834$$ 0 0
$$835$$ − 2.82423e11i − 0.580970i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 1.01762e11i 0.205370i 0.994714 + 0.102685i $$0.0327433\pi$$
−0.994714 + 0.102685i $$0.967257\pi$$
$$840$$ 0 0
$$841$$ −4.59417e11 −0.918382
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −5.13952e11 −1.00808
$$846$$ 0 0
$$847$$ 1.15993e11i 0.225372i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 9.41904e11i 1.79593i
$$852$$ 0 0
$$853$$ −8.81799e10 −0.166561 −0.0832805 0.996526i $$-0.526540\pi$$
−0.0832805 + 0.996526i $$0.526540\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 7.95365e11 1.47449 0.737247 0.675623i $$-0.236125\pi$$
0.737247 + 0.675623i $$0.236125\pi$$
$$858$$ 0 0
$$859$$ 7.01767e11i 1.28890i 0.764645 + 0.644452i $$0.222914\pi$$
−0.764645 + 0.644452i $$0.777086\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 2.80009e11i 0.504811i 0.967622 + 0.252405i $$0.0812217\pi$$
−0.967622 + 0.252405i $$0.918778\pi$$
$$864$$ 0 0
$$865$$ −4.61062e10 −0.0823560
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −2.42645e11 −0.425493
$$870$$ 0 0
$$871$$ − 1.07061e12i − 1.86019i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 5.63802e11i 0.961822i
$$876$$ 0 0
$$877$$ −6.13288e11 −1.03673 −0.518365 0.855159i $$-0.673459\pi$$
−0.518365 + 0.855159i $$0.673459\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 2.48326e11 0.412210 0.206105 0.978530i $$-0.433921\pi$$
0.206105 + 0.978530i $$0.433921\pi$$
$$882$$ 0 0
$$883$$ 3.43124e11i 0.564428i 0.959351 + 0.282214i $$0.0910689\pi$$
−0.959351 + 0.282214i $$0.908931\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 1.46020e11i − 0.235895i −0.993020 0.117947i $$-0.962369\pi$$
0.993020 0.117947i $$-0.0376314\pi$$
$$888$$ 0 0
$$889$$ 8.26381e11 1.32304
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −8.28009e11 −1.30205
$$894$$ 0 0
$$895$$ 3.87415e11i 0.603787i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 2.41592e11i − 0.369866i
$$900$$ 0 0
$$901$$ 7.01894e10 0.106506
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −6.21717e11 −0.926827
$$906$$ 0 0
$$907$$ − 9.39725e11i − 1.38858i −0.719695 0.694291i $$-0.755718\pi$$
0.719695 0.694291i $$-0.244282\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 4.25662e11i 0.618004i 0.951061 + 0.309002i $$0.0999950\pi$$
−0.951061 + 0.309002i $$0.900005\pi$$
$$912$$ 0 0
$$913$$ 1.03449e11 0.148883
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −9.25312e11 −1.30861
$$918$$ 0 0
$$919$$ − 1.21782e12i − 1.70734i −0.520815 0.853670i $$-0.674372\pi$$
0.520815 0.853670i $$-0.325628\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 1.55487e12i 2.14233i
$$924$$ 0 0
$$925$$ −2.55986e11 −0.349663
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −9.97179e11 −1.33878 −0.669392 0.742910i $$-0.733445\pi$$
−0.669392 + 0.742910i $$0.733445\pi$$
$$930$$ 0 0
$$931$$ − 4.65055e11i − 0.619022i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ − 6.34596e11i − 0.830331i
$$936$$ 0 0
$$937$$ 3.43206e11 0.445243 0.222621 0.974905i $$-0.428539\pi$$
0.222621 + 0.974905i $$0.428539\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −1.72310e11 −0.219762 −0.109881 0.993945i $$-0.535047\pi$$
−0.109881 + 0.993945i $$0.535047\pi$$
$$942$$ 0 0
$$943$$ 1.55193e12i 1.96257i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 8.75107e11i − 1.08808i −0.839059 0.544041i $$-0.816894\pi$$
0.839059 0.544041i $$-0.183106\pi$$
$$948$$ 0 0
$$949$$ −2.07997e12 −2.56443
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 1.43349e12 1.73789 0.868946 0.494907i $$-0.164798\pi$$
0.868946 + 0.494907i $$0.164798\pi$$
$$954$$ 0 0
$$955$$ 3.45401e11i 0.415250i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ − 1.95638e12i − 2.31302i
$$960$$ 0 0
$$961$$ −5.76651e11 −0.676114
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −6.36490e11 −0.733977
$$966$$ 0 0
$$967$$ 1.53627e12i 1.75696i 0.477777 + 0.878481i $$0.341443\pi$$
−0.477777 + 0.878481i $$0.658557\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 2.88911e11i 0.325003i 0.986708 + 0.162501i $$0.0519562\pi$$
−0.986708 + 0.162501i $$0.948044\pi$$
$$972$$ 0 0
$$973$$ 1.49930e12 1.67277
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −4.59815e11 −0.504667 −0.252333 0.967640i $$-0.581198\pi$$
−0.252333 + 0.967640i $$0.581198\pi$$
$$978$$ 0 0
$$979$$ − 1.15106e12i − 1.25305i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 3.07463e11i − 0.329291i −0.986353 0.164645i $$-0.947352\pi$$
0.986353 0.164645i $$-0.0526479\pi$$
$$984$$ 0 0
$$985$$ −2.00929e12 −2.13451
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.13663e12 1.18805
$$990$$ 0 0
$$991$$ 1.10256e12i 1.14316i 0.820547 + 0.571579i $$0.193669\pi$$
−0.820547 + 0.571579i $$0.806331\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 1.03769e12i − 1.05871i
$$996$$ 0 0
$$997$$ 5.02913e10 0.0508993 0.0254497 0.999676i $$-0.491898\pi$$
0.0254497 + 0.999676i $$0.491898\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 144.9.g.c.127.2 2
3.2 odd 2 48.9.g.b.31.2 yes 2
4.3 odd 2 inner 144.9.g.c.127.1 2
12.11 even 2 48.9.g.b.31.1 2
24.5 odd 2 192.9.g.a.127.1 2
24.11 even 2 192.9.g.a.127.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
48.9.g.b.31.1 2 12.11 even 2
48.9.g.b.31.2 yes 2 3.2 odd 2
144.9.g.c.127.1 2 4.3 odd 2 inner
144.9.g.c.127.2 2 1.1 even 1 trivial
192.9.g.a.127.1 2 24.5 odd 2
192.9.g.a.127.2 2 24.11 even 2