# Properties

 Label 192.9.g.a.127.2 Level $192$ Weight $9$ Character 192.127 Analytic conductor $78.217$ 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] = [192,9,Mod(127,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, 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("192.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 192.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$78.2166931317$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ 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 $$\chi$$ $$=$$ 192.127 Dual form 192.9.g.a.127.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+46.7654i q^{3} -726.000 q^{5} -3055.34i q^{7} -2187.00 q^{9} +O(q^{10})$$ $$q+46.7654i q^{3} -726.000 q^{5} -3055.34i q^{7} -2187.00 q^{9} -13281.4i q^{11} -39034.0 q^{13} -33951.7i q^{15} -65814.0 q^{17} +130257. i q^{19} +142884. q^{21} -502073. i q^{23} +136451. q^{25} -102276. i q^{27} -202062. q^{29} -1.19563e6i q^{31} +621108. q^{33} +2.21818e6i q^{35} +1.87603e6 q^{37} -1.82544e6i q^{39} +3.09105e6 q^{41} +2.26388e6i q^{43} +1.58776e6 q^{45} +6.35672e6i q^{47} -3.57029e6 q^{49} -3.07782e6i q^{51} +1.06648e6 q^{53} +9.64227e6i q^{55} -6.09152e6 q^{57} +5.76355e6i q^{59} -1.71542e7 q^{61} +6.68202e6i q^{63} +2.83387e7 q^{65} -2.74275e7i q^{67} +2.34796e7 q^{69} +3.98336e7i q^{71} -5.32860e7 q^{73} +6.38118e6i q^{75} -4.05791e7 q^{77} -1.82696e7i q^{79} +4.78297e6 q^{81} +7.78905e6i q^{83} +4.77810e7 q^{85} -9.44950e6i q^{87} +8.66672e7 q^{89} +1.19262e8i q^{91} +5.59143e7 q^{93} -9.45667e7i q^{95} -7.39018e7 q^{97} +2.90463e7i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 1452 q^{5} - 4374 q^{9}+O(q^{10})$$ 2 * q - 1452 * q^5 - 4374 * q^9 $$2 q - 1452 q^{5} - 4374 q^{9} - 78068 q^{13} - 131628 q^{17} + 285768 q^{21} + 272902 q^{25} - 404124 q^{29} + 1242216 q^{33} + 3752060 q^{37} + 6182100 q^{41} + 3175524 q^{45} - 7140574 q^{49} + 2132964 q^{53} - 12183048 q^{57} - 34308388 q^{61} + 56677368 q^{65} + 46959264 q^{69} - 106572028 q^{73} - 81158112 q^{77} + 9565938 q^{81} + 95561928 q^{85} + 173334468 q^{89} + 111828600 q^{93} - 147803644 q^{97}+O(q^{100})$$ 2 * q - 1452 * q^5 - 4374 * q^9 - 78068 * q^13 - 131628 * q^17 + 285768 * q^21 + 272902 * q^25 - 404124 * q^29 + 1242216 * q^33 + 3752060 * q^37 + 6182100 * q^41 + 3175524 * q^45 - 7140574 * q^49 + 2132964 * q^53 - 12183048 * q^57 - 34308388 * q^61 + 56677368 * q^65 + 46959264 * q^69 - 106572028 * q^73 - 81158112 * q^77 + 9565938 * q^81 + 95561928 * q^85 + 173334468 * q^89 + 111828600 * q^93 - 147803644 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 46.7654i 0.577350i
$$4$$ 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 $$-0.780479\pi$$
0.771472 0.636264i $$-0.219521\pi$$
$$8$$ 0 0
$$9$$ −2187.00 −0.333333
$$10$$ 0 0
$$11$$ − 13281.4i − 0.907135i −0.891222 0.453568i $$-0.850151\pi$$
0.891222 0.453568i $$-0.149849\pi$$
$$12$$ 0 0
$$13$$ −39034.0 −1.36669 −0.683344 0.730096i $$-0.739475\pi$$
−0.683344 + 0.730096i $$0.739475\pi$$
$$14$$ 0 0
$$15$$ − 33951.7i − 0.670650i
$$16$$ 0 0
$$17$$ −65814.0 −0.787993 −0.393997 0.919112i $$-0.628908\pi$$
−0.393997 + 0.919112i $$0.628908\pi$$
$$18$$ 0 0
$$19$$ 130257.i 0.999510i 0.866167 + 0.499755i $$0.166577\pi$$
−0.866167 + 0.499755i $$0.833423\pi$$
$$20$$ 0 0
$$21$$ 142884. 0.734694
$$22$$ 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
$$27$$ − 102276.i − 0.192450i
$$28$$ 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 $$-0.775888\pi$$
0.762215 0.647324i $$-0.224112\pi$$
$$32$$ 0 0
$$33$$ 621108. 0.523735
$$34$$ 0 0
$$35$$ 2.21818e6i 1.47817i
$$36$$ 0 0
$$37$$ 1.87603e6 1.00100 0.500499 0.865737i $$-0.333150\pi$$
0.500499 + 0.865737i $$0.333150\pi$$
$$38$$ 0 0
$$39$$ − 1.82544e6i − 0.789058i
$$40$$ 0 0
$$41$$ 3.09105e6 1.09388 0.546941 0.837171i $$-0.315792\pi$$
0.546941 + 0.837171i $$0.315792\pi$$
$$42$$ 0 0
$$43$$ 2.26388e6i 0.662186i 0.943598 + 0.331093i $$0.107417\pi$$
−0.943598 + 0.331093i $$0.892583\pi$$
$$44$$ 0 0
$$45$$ 1.58776e6 0.387200
$$46$$ 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
$$51$$ − 3.07782e6i − 0.454948i
$$52$$ 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
$$57$$ −6.09152e6 −0.577067
$$58$$ 0 0
$$59$$ 5.76355e6i 0.475644i 0.971309 + 0.237822i $$0.0764335\pi$$
−0.971309 + 0.237822i $$0.923566\pi$$
$$60$$ 0 0
$$61$$ −1.71542e7 −1.23894 −0.619471 0.785020i $$-0.712653\pi$$
−0.619471 + 0.785020i $$0.712653\pi$$
$$62$$ 0 0
$$63$$ 6.68202e6i 0.424176i
$$64$$ 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
$$69$$ 2.34796e7 1.03585
$$70$$ 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
$$75$$ 6.38118e6i 0.201677i
$$76$$ 0 0
$$77$$ −4.05791e7 −1.15435
$$78$$ 0 0
$$79$$ − 1.82696e7i − 0.469052i −0.972110 0.234526i $$-0.924646\pi$$
0.972110 0.234526i $$-0.0753538\pi$$
$$80$$ 0 0
$$81$$ 4.78297e6 0.111111
$$82$$ 0 0
$$83$$ 7.78905e6i 0.164124i 0.996627 + 0.0820620i $$0.0261506\pi$$
−0.996627 + 0.0820620i $$0.973849\pi$$
$$84$$ 0 0
$$85$$ 4.77810e7 0.915333
$$86$$ 0 0
$$87$$ − 9.44950e6i − 0.164942i
$$88$$ 0 0
$$89$$ 8.66672e7 1.38132 0.690661 0.723179i $$-0.257320\pi$$
0.690661 + 0.723179i $$0.257320\pi$$
$$90$$ 0 0
$$91$$ 1.19262e8i 1.73915i
$$92$$ 0 0
$$93$$ 5.59143e7 0.747465
$$94$$ 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
$$99$$ 2.90463e7i 0.302378i
$$100$$ 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 $$0.257303\pi$$
−0.690699 + 0.723143i $$0.742697\pi$$
$$104$$ 0 0
$$105$$ −1.03734e8 −0.853420
$$106$$ 0 0
$$107$$ 2.00810e8i 1.53197i 0.642857 + 0.765986i $$0.277749\pi$$
−0.642857 + 0.765986i $$0.722251\pi$$
$$108$$ 0 0
$$109$$ −6.86083e7 −0.486039 −0.243019 0.970021i $$-0.578138\pi$$
−0.243019 + 0.970021i $$0.578138\pi$$
$$110$$ 0 0
$$111$$ 8.77332e7i 0.577926i
$$112$$ 0 0
$$113$$ 3.30831e7 0.202905 0.101452 0.994840i $$-0.467651\pi$$
0.101452 + 0.994840i $$0.467651\pi$$
$$114$$ 0 0
$$115$$ 3.64505e8i 2.08407i
$$116$$ 0 0
$$117$$ 8.53674e7 0.455563
$$118$$ 0 0
$$119$$ 2.01084e8i 1.00274i
$$120$$ 0 0
$$121$$ 3.79642e7 0.177106
$$122$$ 0 0
$$123$$ 1.44554e8i 0.631553i
$$124$$ 0 0
$$125$$ 1.84530e8 0.755836
$$126$$ 0 0
$$127$$ 2.70471e8i 1.03970i 0.854259 + 0.519848i $$0.174011\pi$$
−0.854259 + 0.519848i $$0.825989\pi$$
$$128$$ 0 0
$$129$$ −1.05871e8 −0.382313
$$130$$ 0 0
$$131$$ − 3.02851e8i − 1.02836i −0.857683 0.514178i $$-0.828097\pi$$
0.857683 0.514178i $$-0.171903\pi$$
$$132$$ 0 0
$$133$$ 3.97980e8 1.27190
$$134$$ 0 0
$$135$$ 7.42523e7i 0.223550i
$$136$$ 0 0
$$137$$ 6.40316e8 1.81766 0.908828 0.417170i $$-0.136978\pi$$
0.908828 + 0.417170i $$0.136978\pi$$
$$138$$ 0 0
$$139$$ − 4.90714e8i − 1.31453i −0.753661 0.657263i $$-0.771714\pi$$
0.753661 0.657263i $$-0.228286\pi$$
$$140$$ 0 0
$$141$$ −2.97275e8 −0.752110
$$142$$ 0 0
$$143$$ 5.18425e8i 1.23977i
$$144$$ 0 0
$$145$$ 1.46697e8 0.331856
$$146$$ 0 0
$$147$$ − 1.66966e8i − 0.357568i
$$148$$ 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 $$0.0545521\pi$$
−0.985350 + 0.170543i $$0.945448\pi$$
$$152$$ 0 0
$$153$$ 1.43935e8 0.262664
$$154$$ 0 0
$$155$$ 8.68031e8i 1.50386i
$$156$$ 0 0
$$157$$ 2.14784e7 0.0353511 0.0176755 0.999844i $$-0.494373\pi$$
0.0176755 + 0.999844i $$0.494373\pi$$
$$158$$ 0 0
$$159$$ 4.98744e7i 0.0780350i
$$160$$ 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
$$165$$ −4.50924e8 −0.608370
$$166$$ 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
$$171$$ − 2.84872e8i − 0.333170i
$$172$$ 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
$$177$$ −2.69535e8 −0.274613
$$178$$ 0 0
$$179$$ 5.33629e8i 0.519789i 0.965637 + 0.259895i $$0.0836878\pi$$
−0.965637 + 0.259895i $$0.916312\pi$$
$$180$$ 0 0
$$181$$ −8.56360e8 −0.797888 −0.398944 0.916975i $$-0.630623\pi$$
−0.398944 + 0.916975i $$0.630623\pi$$
$$182$$ 0 0
$$183$$ − 8.02222e8i − 0.715303i
$$184$$ 0 0
$$185$$ −1.36200e9 −1.16276
$$186$$ 0 0
$$187$$ 8.74100e8i 0.714817i
$$188$$ 0 0
$$189$$ −3.12487e8 −0.244898
$$190$$ 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
$$195$$ 1.32527e9i 0.916570i
$$196$$ 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 $$-0.849385\pi$$
0.890128 0.455710i $$-0.150615\pi$$
$$200$$ 0 0
$$201$$ 1.28266e9 0.785826
$$202$$ 0 0
$$203$$ 6.17368e8i 0.363546i
$$204$$ 0 0
$$205$$ −2.24410e9 −1.27065
$$206$$ 0 0
$$207$$ 1.09803e9i 0.598046i
$$208$$ 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
$$213$$ −1.86283e9 −0.905015
$$214$$ 0 0
$$215$$ − 1.64358e9i − 0.769195i
$$216$$ 0 0
$$217$$ −3.65307e9 −1.64747
$$218$$ 0 0
$$219$$ − 2.49194e9i − 1.08333i
$$220$$ 0 0
$$221$$ 2.56898e9 1.07694
$$222$$ 0 0
$$223$$ 3.40037e9i 1.37501i 0.726179 + 0.687506i $$0.241295\pi$$
−0.726179 + 0.687506i $$0.758705\pi$$
$$224$$ 0 0
$$225$$ −2.98418e8 −0.116438
$$226$$ 0 0
$$227$$ − 4.52697e9i − 1.70492i −0.522792 0.852460i $$-0.675109\pi$$
0.522792 0.852460i $$-0.324891\pi$$
$$228$$ 0 0
$$229$$ 9.90176e8 0.360056 0.180028 0.983661i $$-0.442381\pi$$
0.180028 + 0.983661i $$0.442381\pi$$
$$230$$ 0 0
$$231$$ − 1.89769e9i − 0.666467i
$$232$$ 0 0
$$233$$ 2.23709e9 0.759032 0.379516 0.925185i $$-0.376091\pi$$
0.379516 + 0.925185i $$0.376091\pi$$
$$234$$ 0 0
$$235$$ − 4.61498e9i − 1.51321i
$$236$$ 0 0
$$237$$ 8.54385e8 0.270807
$$238$$ 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
$$243$$ 2.23677e8i 0.0641500i
$$244$$ 0 0
$$245$$ 2.59203e9 0.719408
$$246$$ 0 0
$$247$$ − 5.08446e9i − 1.36602i
$$248$$ 0 0
$$249$$ −3.64258e8 −0.0947571
$$250$$ 0 0
$$251$$ − 6.81334e9i − 1.71659i −0.513161 0.858293i $$-0.671526\pi$$
0.513161 0.858293i $$-0.328474\pi$$
$$252$$ 0 0
$$253$$ −6.66822e9 −1.62752
$$254$$ 0 0
$$255$$ 2.23449e9i 0.528468i
$$256$$ 0 0
$$257$$ 3.95756e9 0.907184 0.453592 0.891209i $$-0.350142\pi$$
0.453592 + 0.891209i $$0.350142\pi$$
$$258$$ 0 0
$$259$$ − 5.73191e9i − 1.27380i
$$260$$ 0 0
$$261$$ 4.41910e8 0.0952295
$$262$$ 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
$$267$$ 4.05303e9i 0.797507i
$$268$$ 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 $$-0.943489\pi$$
0.984282 0.176604i $$-0.0565110\pi$$
$$272$$ 0 0
$$273$$ −5.57733e9 −1.00410
$$274$$ 0 0
$$275$$ − 1.81226e9i − 0.316876i
$$276$$ 0 0
$$277$$ 5.03752e9 0.855654 0.427827 0.903861i $$-0.359279\pi$$
0.427827 + 0.903861i $$0.359279\pi$$
$$278$$ 0 0
$$279$$ 2.61485e9i 0.431549i
$$280$$ 0 0
$$281$$ 6.66317e9 1.06870 0.534350 0.845264i $$-0.320557\pi$$
0.534350 + 0.845264i $$0.320557\pi$$
$$282$$ 0 0
$$283$$ 5.54295e9i 0.864162i 0.901835 + 0.432081i $$0.142221\pi$$
−0.901835 + 0.432081i $$0.857779\pi$$
$$284$$ 0 0
$$285$$ 4.42245e9 0.670321
$$286$$ 0 0
$$287$$ − 9.44420e9i − 1.39199i
$$288$$ 0 0
$$289$$ −2.64427e9 −0.379066
$$290$$ 0 0
$$291$$ − 3.45605e9i − 0.481956i
$$292$$ 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
$$297$$ −1.35836e9 −0.174578
$$298$$ 0 0
$$299$$ 1.95979e10i 2.45203i
$$300$$ 0 0
$$301$$ 6.91692e9 0.842649
$$302$$ 0 0
$$303$$ − 8.94670e9i − 1.06143i
$$304$$ 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
$$309$$ −7.61250e9 −0.835013
$$310$$ 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
$$315$$ − 4.85115e9i − 0.492723i
$$316$$ 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
$$321$$ −9.39097e9 −0.884485
$$322$$ 0 0
$$323$$ − 8.57274e9i − 0.787607i
$$324$$ 0 0
$$325$$ −5.32623e9 −0.477404
$$326$$ 0 0
$$327$$ − 3.20849e9i − 0.280615i
$$328$$ 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
$$333$$ −4.10288e9 −0.333666
$$334$$ 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
$$339$$ 1.54714e9i 0.117147i
$$340$$ 0 0
$$341$$ −1.58797e10 −1.17442
$$342$$ 0 0
$$343$$ − 6.70498e9i − 0.484419i
$$344$$ 0 0
$$345$$ −1.70462e10 −1.20324
$$346$$ 0 0
$$347$$ 1.35188e8i 0.00932439i 0.999989 + 0.00466219i $$0.00148403\pi$$
−0.999989 + 0.00466219i $$0.998516\pi$$
$$348$$ 0 0
$$349$$ −1.13213e10 −0.763122 −0.381561 0.924344i $$-0.624613\pi$$
−0.381561 + 0.924344i $$0.624613\pi$$
$$350$$ 0 0
$$351$$ 3.99224e9i 0.263019i
$$352$$ 0 0
$$353$$ −1.42650e10 −0.918697 −0.459348 0.888256i $$-0.651917\pi$$
−0.459348 + 0.888256i $$0.651917\pi$$
$$354$$ 0 0
$$355$$ − 2.89192e10i − 1.82085i
$$356$$ 0 0
$$357$$ −9.40377e9 −0.578934
$$358$$ 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
$$363$$ 1.77541e9i 0.102252i
$$364$$ 0 0
$$365$$ 3.86856e10 2.17961
$$366$$ 0 0
$$367$$ 2.06760e10i 1.13973i 0.821738 + 0.569865i $$0.193005\pi$$
−0.821738 + 0.569865i $$0.806995\pi$$
$$368$$ 0 0
$$369$$ −6.76013e9 −0.364627
$$370$$ 0 0
$$371$$ − 3.25846e9i − 0.171996i
$$372$$ 0 0
$$373$$ 7.71358e9 0.398493 0.199247 0.979949i $$-0.436151\pi$$
0.199247 + 0.979949i $$0.436151\pi$$
$$374$$ 0 0
$$375$$ 8.62963e9i 0.436382i
$$376$$ 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
$$381$$ −1.26487e10 −0.600268
$$382$$ 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
$$387$$ − 4.95111e9i − 0.220729i
$$388$$ 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
$$393$$ 1.41629e10 0.593722
$$394$$ 0 0
$$395$$ 1.32637e10i 0.544851i
$$396$$ 0 0
$$397$$ −1.32156e10 −0.532016 −0.266008 0.963971i $$-0.585705\pi$$
−0.266008 + 0.963971i $$0.585705\pi$$
$$398$$ 0 0
$$399$$ 1.86117e10i 0.734334i
$$400$$ 0 0
$$401$$ 2.51637e10 0.973190 0.486595 0.873628i $$-0.338239\pi$$
0.486595 + 0.873628i $$0.338239\pi$$
$$402$$ 0 0
$$403$$ 4.66704e10i 1.76938i
$$404$$ 0 0
$$405$$ −3.47244e9 −0.129067
$$406$$ 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
$$411$$ 2.99446e10i 1.04942i
$$412$$ 0 0
$$413$$ 1.76096e10 0.605270
$$414$$ 0 0
$$415$$ − 5.65485e9i − 0.190647i
$$416$$ 0 0
$$417$$ 2.29484e10 0.758942
$$418$$ 0 0
$$419$$ 1.88088e10i 0.610246i 0.952313 + 0.305123i $$0.0986976\pi$$
−0.952313 + 0.305123i $$0.901302\pi$$
$$420$$ 0 0
$$421$$ −6.04555e9 −0.192445 −0.0962227 0.995360i $$-0.530676\pi$$
−0.0962227 + 0.995360i $$0.530676\pi$$
$$422$$ 0 0
$$423$$ − 1.39022e10i − 0.434231i
$$424$$ 0 0
$$425$$ −8.98039e9 −0.275258
$$426$$ 0 0
$$427$$ 5.24119e10i 1.57659i
$$428$$ 0 0
$$429$$ −2.42443e10 −0.715782
$$430$$ 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
$$435$$ 6.86034e9i 0.191597i
$$436$$ 0 0
$$437$$ 6.53986e10 1.79326
$$438$$ 0 0
$$439$$ − 1.24165e10i − 0.334303i −0.985931 0.167152i $$-0.946543\pi$$
0.985931 0.167152i $$-0.0534569\pi$$
$$440$$ 0 0
$$441$$ 7.80822e9 0.206442
$$442$$ 0 0
$$443$$ 7.57779e9i 0.196756i 0.995149 + 0.0983779i $$0.0313654\pi$$
−0.995149 + 0.0983779i $$0.968635\pi$$
$$444$$ 0 0
$$445$$ −6.29204e10 −1.60454
$$446$$ 0 0
$$447$$ 3.79324e9i 0.0950123i
$$448$$ 0 0
$$449$$ −3.37970e10 −0.831559 −0.415780 0.909465i $$-0.636491\pi$$
−0.415780 + 0.909465i $$0.636491\pi$$
$$450$$ 0 0
$$451$$ − 4.10534e10i − 0.992299i
$$452$$ 0 0
$$453$$ −8.29269e9 −0.196926
$$454$$ 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
$$459$$ 6.73118e9i 0.151649i
$$460$$ 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 $$0.00414617\pi$$
−0.999915 + 0.0130252i $$0.995854\pi$$
$$464$$ 0 0
$$465$$ −4.05938e10 −0.868256
$$466$$ 0 0
$$467$$ 2.92676e10i 0.615347i 0.951492 + 0.307673i $$0.0995504\pi$$
−0.951492 + 0.307673i $$0.900450\pi$$
$$468$$ 0 0
$$469$$ −8.38003e10 −1.73203
$$470$$ 0 0
$$471$$ 1.00444e9i 0.0204100i
$$472$$ 0 0
$$473$$ 3.00674e10 0.600692
$$474$$ 0 0
$$475$$ 1.77737e10i 0.349143i
$$476$$ 0 0
$$477$$ −2.33240e9 −0.0450535
$$478$$ 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
$$483$$ − 7.17382e10i − 1.31814i
$$484$$ 0 0
$$485$$ 5.36527e10 0.969672
$$486$$ 0 0
$$487$$ − 5.82581e10i − 1.03571i −0.855467 0.517857i $$-0.826730\pi$$
0.855467 0.517857i $$-0.173270\pi$$
$$488$$ 0 0
$$489$$ −1.13280e10 −0.198115
$$490$$ 0 0
$$491$$ − 3.61816e10i − 0.622532i −0.950323 0.311266i $$-0.899247\pi$$
0.950323 0.311266i $$-0.100753\pi$$
$$492$$ 0 0
$$493$$ 1.32985e10 0.225121
$$494$$ 0 0
$$495$$ − 2.10876e10i − 0.351243i
$$496$$ 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
$$501$$ −1.81923e10 −0.288760
$$502$$ 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
$$507$$ 3.31063e10i 0.501047i
$$508$$ 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
$$513$$ 1.33222e10 0.192356
$$514$$ 0 0
$$515$$ − 1.18179e11i − 1.68001i
$$516$$ 0 0
$$517$$ 8.44260e10 1.18172
$$518$$ 0 0
$$519$$ 2.96994e9i 0.0409334i
$$520$$ 0 0
$$521$$ −6.81614e10 −0.925098 −0.462549 0.886594i $$-0.653065\pi$$
−0.462549 + 0.886594i $$0.653065\pi$$
$$522$$ 0 0
$$523$$ − 5.63922e10i − 0.753724i −0.926269 0.376862i $$-0.877003\pi$$
0.926269 0.376862i $$-0.122997\pi$$
$$524$$ 0 0
$$525$$ 1.94967e10 0.256639
$$526$$ 0 0
$$527$$ 7.86895e10i 1.02017i
$$528$$ 0 0
$$529$$ −1.73766e11 −2.21893
$$530$$ 0 0
$$531$$ − 1.26049e10i − 0.158548i
$$532$$ 0 0
$$533$$ −1.20656e11 −1.49500
$$534$$ 0 0
$$535$$ − 1.45788e11i − 1.77954i
$$536$$ 0 0
$$537$$ −2.49554e10 −0.300100
$$538$$ 0 0
$$539$$ 4.74183e10i 0.561812i
$$540$$ 0 0
$$541$$ 7.61478e10 0.888932 0.444466 0.895796i $$-0.353393\pi$$
0.444466 + 0.895796i $$0.353393\pi$$
$$542$$ 0 0
$$543$$ − 4.00480e10i − 0.460661i
$$544$$ 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
$$549$$ 3.75162e10 0.412981
$$550$$ 0 0
$$551$$ − 2.63200e10i − 0.285548i
$$552$$ 0 0
$$553$$ −5.58198e10 −0.596881
$$554$$ 0 0
$$555$$ − 6.36943e10i − 0.671319i
$$556$$ 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
$$561$$ −4.08776e10 −0.412700
$$562$$ 0 0
$$563$$ − 6.88172e9i − 0.0684957i −0.999413 0.0342479i $$-0.989096\pi$$
0.999413 0.0342479i $$-0.0109036\pi$$
$$564$$ 0 0
$$565$$ −2.40183e10 −0.235694
$$566$$ 0 0
$$567$$ − 1.46136e10i − 0.141392i
$$568$$ 0 0
$$569$$ 9.38382e10 0.895221 0.447611 0.894229i $$-0.352275\pi$$
0.447611 + 0.894229i $$0.352275\pi$$
$$570$$ 0 0
$$571$$ 1.92744e10i 0.181316i 0.995882 + 0.0906582i $$0.0288971\pi$$
−0.995882 + 0.0906582i $$0.971103\pi$$
$$572$$ 0 0
$$573$$ 2.22490e10 0.206392
$$574$$ 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
$$579$$ 4.09996e10i 0.364809i
$$580$$ 0 0
$$581$$ 2.37982e10 0.208852
$$582$$ 0 0
$$583$$ − 1.41643e10i − 0.122609i
$$584$$ 0 0
$$585$$ −6.19767e10 −0.529182
$$586$$ 0 0
$$587$$ 2.09633e11i 1.76566i 0.469695 + 0.882829i $$0.344364\pi$$
−0.469695 + 0.882829i $$0.655636\pi$$
$$588$$ 0 0
$$589$$ 1.55740e11 1.29401
$$590$$ 0 0
$$591$$ 1.29429e11i 1.06092i
$$592$$ 0 0
$$593$$ −7.33746e10 −0.593372 −0.296686 0.954975i $$-0.595881\pi$$
−0.296686 + 0.954975i $$0.595881\pi$$
$$594$$ 0 0
$$595$$ − 1.45987e11i − 1.16479i
$$596$$ 0 0
$$597$$ 6.68429e10 0.526209
$$598$$ 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
$$603$$ 5.99840e10i 0.453697i
$$604$$ 0 0
$$605$$ −2.75620e10 −0.205726
$$606$$ 0 0
$$607$$ − 1.55515e11i − 1.14556i −0.819710 0.572779i $$-0.805866\pi$$
0.819710 0.572779i $$-0.194134\pi$$
$$608$$ 0 0
$$609$$ −2.88714e10 −0.209894
$$610$$ 0 0
$$611$$ − 2.48128e11i − 1.78038i
$$612$$ 0 0
$$613$$ −3.19775e10 −0.226466 −0.113233 0.993568i $$-0.536121\pi$$
−0.113233 + 0.993568i $$0.536121\pi$$
$$614$$ 0 0
$$615$$ − 1.04946e11i − 0.733612i
$$616$$ 0 0
$$617$$ 5.63108e10 0.388553 0.194277 0.980947i $$-0.437764\pi$$
0.194277 + 0.980947i $$0.437764\pi$$
$$618$$ 0 0
$$619$$ 2.66432e11i 1.81478i 0.420287 + 0.907391i $$0.361929\pi$$
−0.420287 + 0.907391i $$0.638071\pi$$
$$620$$ 0 0
$$621$$ −5.13500e10 −0.345282
$$622$$ 0 0
$$623$$ − 2.64798e11i − 1.75777i
$$624$$ 0 0
$$625$$ −1.87270e11 −1.22729
$$626$$ 0 0
$$627$$ 8.09038e10i 0.523478i
$$628$$ 0 0
$$629$$ −1.23469e11 −0.788779
$$630$$ 0 0
$$631$$ − 8.55727e9i − 0.0539781i −0.999636 0.0269891i $$-0.991408\pi$$
0.999636 0.0269891i $$-0.00859193\pi$$
$$632$$ 0 0
$$633$$ −2.15934e10 −0.134495
$$634$$ 0 0
$$635$$ − 1.96362e11i − 1.20771i
$$636$$ 0 0
$$637$$ 1.39363e11 0.846425
$$638$$ 0 0
$$639$$ − 8.71161e10i − 0.522511i
$$640$$ 0 0
$$641$$ 2.90248e10 0.171924 0.0859620 0.996298i $$-0.472604\pi$$
0.0859620 + 0.996298i $$0.472604\pi$$
$$642$$ 0 0
$$643$$ − 5.13563e10i − 0.300435i −0.988653 0.150217i $$-0.952003\pi$$
0.988653 0.150217i $$-0.0479973\pi$$
$$644$$ 0 0
$$645$$ 7.68625e10 0.444095
$$646$$ 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
$$651$$ − 1.70837e11i − 0.951170i
$$652$$ 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
$$657$$ 1.16537e11 0.625461
$$658$$ 0 0
$$659$$ − 3.17581e11i − 1.68389i −0.539567 0.841943i $$-0.681412\pi$$
0.539567 0.841943i $$-0.318588\pi$$
$$660$$ 0 0
$$661$$ 1.33716e11 0.700449 0.350224 0.936666i $$-0.386105\pi$$
0.350224 + 0.936666i $$0.386105\pi$$
$$662$$ 0 0
$$663$$ 1.20139e11i 0.621773i
$$664$$ 0 0
$$665$$ −2.88933e11 −1.47744
$$666$$ 0 0
$$667$$ 1.01450e11i 0.512564i
$$668$$ 0 0
$$669$$ −1.59019e11 −0.793864
$$670$$ 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
$$675$$ − 1.39556e10i − 0.0672256i
$$676$$ 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
$$681$$ 2.11705e11 0.984337
$$682$$ 0 0
$$683$$ 3.59716e11i 1.65302i 0.562924 + 0.826508i $$0.309676\pi$$
−0.562924 + 0.826508i $$0.690324\pi$$
$$684$$ 0 0
$$685$$ −4.64869e11 −2.11139
$$686$$ 0 0
$$687$$ 4.63059e10i 0.207879i
$$688$$ 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
$$693$$ 8.87464e10 0.384785
$$694$$ 0 0
$$695$$ 3.56258e11i 1.52695i
$$696$$ 0 0
$$697$$ −2.03434e11 −0.861972
$$698$$ 0 0
$$699$$ 1.04618e11i 0.438227i
$$700$$ 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
$$705$$ 2.15821e11 0.873651
$$706$$ 0 0
$$707$$ 5.84518e11i 2.33948i
$$708$$ 0 0
$$709$$ −1.08904e11 −0.430981 −0.215490 0.976506i $$-0.569135\pi$$
−0.215490 + 0.976506i $$0.569135\pi$$
$$710$$ 0 0
$$711$$ 3.99556e10i 0.156351i
$$712$$ 0 0
$$713$$ −6.00296e11 −2.32278
$$714$$ 0 0
$$715$$ − 3.76376e11i − 1.44012i
$$716$$ 0 0
$$717$$ 1.23238e11 0.466302
$$718$$ 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
$$723$$ − 2.89782e10i − 0.106052i
$$724$$ 0 0
$$725$$ −2.75716e10 −0.0997951
$$726$$ 0 0
$$727$$ − 4.81879e11i − 1.72504i −0.506020 0.862522i $$-0.668884\pi$$
0.506020 0.862522i $$-0.331116\pi$$
$$728$$ 0 0
$$729$$ −1.04604e10 −0.0370370
$$730$$ 0 0
$$731$$ − 1.48995e11i − 0.521798i
$$732$$ 0 0
$$733$$ 1.82274e11 0.631405 0.315702 0.948858i $$-0.397760\pi$$
0.315702 + 0.948858i $$0.397760\pi$$
$$734$$ 0 0
$$735$$ 1.21217e11i 0.415351i
$$736$$ 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
$$741$$ 2.37777e11 0.788672
$$742$$ 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
$$747$$ − 1.70347e10i − 0.0547080i
$$748$$ 0 0
$$749$$ 6.13543e11 1.94948
$$750$$ 0 0
$$751$$ 4.17556e11i 1.31267i 0.754470 + 0.656334i $$0.227894\pi$$
−0.754470 + 0.656334i $$0.772106\pi$$
$$752$$ 0 0
$$753$$ 3.18629e11 0.991071
$$754$$ 0 0
$$755$$ − 1.28738e11i − 0.396205i
$$756$$ 0 0
$$757$$ −6.29371e11 −1.91656 −0.958282 0.285826i $$-0.907732\pi$$
−0.958282 + 0.285826i $$0.907732\pi$$
$$758$$ 0 0
$$759$$ − 3.11842e11i − 0.939652i
$$760$$ 0 0
$$761$$ −1.72289e11 −0.513710 −0.256855 0.966450i $$-0.582686\pi$$
−0.256855 + 0.966450i $$0.582686\pi$$
$$762$$ 0 0
$$763$$ 2.09622e11i 0.618497i
$$764$$ 0 0
$$765$$ −1.04497e11 −0.305111
$$766$$ 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
$$771$$ 1.85077e11i 0.523763i
$$772$$ 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
$$777$$ 2.68055e11 0.735427
$$778$$ 0 0
$$779$$ 4.02631e11i 1.09335i
$$780$$ 0 0
$$781$$ 5.29045e11 1.42196
$$782$$ 0 0
$$783$$ 2.06661e10i 0.0549808i
$$784$$ 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
$$789$$ −8.70439e10 −0.224611
$$790$$ 0 0
$$791$$ − 1.01080e11i − 0.258202i
$$792$$ 0 0
$$793$$ 6.69597e11 1.69325
$$794$$ 0 0
$$795$$ − 3.62088e10i − 0.0906455i
$$796$$ 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
$$801$$ −1.89541e11 −0.460441
$$802$$ 0 0
$$803$$ 7.07711e11i 1.70213i
$$804$$ 0 0
$$805$$ 1.11369e12 2.65203
$$806$$ 0 0
$$807$$ − 5.48871e10i − 0.129412i
$$808$$ 0 0
$$809$$ 3.24105e11 0.756645 0.378322 0.925674i $$-0.376501\pi$$
0.378322 + 0.925674i $$0.376501\pi$$
$$810$$ 0 0
$$811$$ − 6.94202e11i − 1.60473i −0.596833 0.802366i $$-0.703574\pi$$
0.596833 0.802366i $$-0.296426\pi$$
$$812$$ 0 0
$$813$$ 8.90904e10 0.203924
$$814$$ 0 0
$$815$$ − 1.75859e11i − 0.398597i
$$816$$ 0 0
$$817$$ −2.94887e11 −0.661861
$$818$$ 0 0
$$819$$ − 2.60826e11i − 0.579716i
$$820$$ 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 $$0.118652\pi$$
−0.931328 + 0.364183i $$0.881348\pi$$
$$824$$ 0 0
$$825$$ 8.47508e10 0.182948
$$826$$ 0 0
$$827$$ 6.22860e9i 0.0133158i 0.999978 + 0.00665792i $$0.00211930\pi$$
−0.999978 + 0.00665792i $$0.997881\pi$$
$$828$$ 0 0
$$829$$ −6.97808e11 −1.47747 −0.738733 0.673998i $$-0.764576\pi$$
−0.738733 + 0.673998i $$0.764576\pi$$
$$830$$ 0 0
$$831$$ 2.35582e11i 0.494012i
$$832$$ 0 0
$$833$$ 2.34975e11 0.488024
$$834$$ 0 0
$$835$$ − 2.82423e11i − 0.580970i
$$836$$ 0 0
$$837$$ −1.22285e11 −0.249155
$$838$$ 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
$$843$$ 3.11606e11i 0.617014i
$$844$$ 0 0
$$845$$ −5.13952e11 −1.00808
$$846$$ 0 0
$$847$$ − 1.15993e11i − 0.225372i
$$848$$ 0 0
$$849$$ −2.59218e11 −0.498924
$$850$$ 0 0
$$851$$ − 9.41904e11i − 1.79593i
$$852$$ 0 0
$$853$$ 8.81799e10 0.166561 0.0832805 0.996526i $$-0.473460\pi$$
0.0832805 + 0.996526i $$0.473460\pi$$
$$854$$ 0 0
$$855$$ 2.06817e11i 0.387010i
$$856$$ 0 0
$$857$$ −7.95365e11 −1.47449 −0.737247 0.675623i $$-0.763875\pi$$
−0.737247 + 0.675623i $$0.763875\pi$$
$$858$$ 0 0
$$859$$ 7.01767e11i 1.28890i 0.764645 + 0.644452i $$0.222914\pi$$
−0.764645 + 0.644452i $$0.777086\pi$$
$$860$$ 0 0
$$861$$ 4.41662e11 0.803669
$$862$$ 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
$$867$$ − 1.23660e11i − 0.218854i
$$868$$ 0 0
$$869$$ −2.42645e11 −0.425493
$$870$$ 0 0
$$871$$ 1.07061e12i 1.86019i
$$872$$ 0 0
$$873$$ 1.61623e11 0.278258
$$874$$ 0 0
$$875$$ − 5.63802e11i − 0.961822i
$$876$$ 0 0
$$877$$ 6.13288e11 1.03673 0.518365 0.855159i $$-0.326541\pi$$
0.518365 + 0.855159i $$0.326541\pi$$
$$878$$ 0 0
$$879$$ − 3.12107e11i − 0.522816i
$$880$$ 0 0
$$881$$ −2.48326e11 −0.412210 −0.206105 0.978530i $$-0.566079\pi$$
−0.206105 + 0.978530i $$0.566079\pi$$
$$882$$ 0 0
$$883$$ 3.43124e11i 0.564428i 0.959351 + 0.282214i $$0.0910689\pi$$
−0.959351 + 0.282214i $$0.908931\pi$$
$$884$$ 0 0
$$885$$ 1.95682e11 0.318991
$$886$$ 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
$$891$$ − 6.35244e10i − 0.100793i
$$892$$ 0 0
$$893$$ −8.28009e11 −1.30205
$$894$$ 0 0
$$895$$ − 3.87415e11i − 0.603787i
$$896$$ 0 0
$$897$$ −9.16504e11 −1.41568
$$898$$ 0 0
$$899$$ 2.41592e11i 0.369866i
$$900$$ 0 0
$$901$$ −7.01894e10 −0.106506
$$902$$ 0 0
$$903$$ 3.23472e11i 0.486504i
$$904$$ 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
$$909$$ 4.18396e11 0.612818
$$910$$ 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
$$915$$ 5.82413e11i 0.830897i
$$916$$ 0 0
$$917$$ −9.25312e11 −1.30861
$$918$$ 0 0
$$919$$ 1.21782e12i 1.70734i 0.520815 + 0.853670i $$0.325628\pi$$
−0.520815 + 0.853670i $$0.674372\pi$$
$$920$$ 0 0
$$921$$ 3.03859e10 0.0422312
$$922$$ 0 0
$$923$$ − 1.55487e12i − 2.14233i
$$924$$ 0 0
$$925$$ 2.55986e11 0.349663
$$926$$ 0 0
$$927$$ − 3.56001e11i − 0.482095i
$$928$$ 0 0
$$929$$ 9.97179e11 1.33878 0.669392 0.742910i $$-0.266555\pi$$
0.669392 + 0.742910i $$0.266555\pi$$
$$930$$ 0 0
$$931$$ − 4.65055e11i − 0.619022i
$$932$$ 0 0
$$933$$ −1.86048e10 −0.0245526
$$934$$ 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
$$939$$ − 7.39906e11i − 0.951730i
$$940$$ 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
$$945$$ 2.26866e11 0.284473
$$946$$ 0 0
$$947$$ 8.75107e11i 1.08808i 0.839059 + 0.544041i $$0.183106\pi$$
−0.839059 + 0.544041i $$0.816894\pi$$
$$948$$ 0 0
$$949$$ 2.07997e12 2.56443
$$950$$ 0 0
$$951$$ 4.49339e11i 0.549353i
$$952$$ 0 0
$$953$$ −1.43349e12 −1.73789 −0.868946 0.494907i $$-0.835202\pi$$
−0.868946 + 0.494907i $$0.835202\pi$$
$$954$$ 0 0
$$955$$ 3.45401e11i 0.415250i
$$956$$ 0 0
$$957$$ −1.25502e11 −0.149625
$$958$$ 0 0
$$959$$ − 1.95638e12i − 2.31302i
$$960$$ 0 0
$$961$$ −5.76651e11 −0.676114
$$962$$ 0 0
$$963$$ − 4.39172e11i − 0.510657i
$$964$$ 0 0
$$965$$ −6.36490e11 −0.733977
$$966$$ 0 0
$$967$$ − 1.53627e12i − 1.75696i −0.477777 0.878481i $$-0.658557\pi$$
0.477777 0.878481i $$-0.341443\pi$$
$$968$$ 0 0
$$969$$ 4.00908e11 0.454725
$$970$$ 0 0
$$971$$ − 2.88911e11i − 0.325003i −0.986708 0.162501i $$-0.948044\pi$$
0.986708 0.162501i $$-0.0519562\pi$$
$$972$$ 0 0
$$973$$ −1.49930e12 −1.67277
$$974$$ 0 0
$$975$$ − 2.49083e11i − 0.275630i
$$976$$ 0 0
$$977$$ 4.59815e11 0.504667 0.252333 0.967640i $$-0.418802\pi$$
0.252333 + 0.967640i $$0.418802\pi$$
$$978$$ 0 0
$$979$$ − 1.15106e12i − 1.25305i
$$980$$ 0 0
$$981$$ 1.50046e11 0.162013
$$982$$ 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
$$987$$ 9.08274e11i 0.957080i
$$988$$ 0 0
$$989$$ 1.13663e12 1.18805
$$990$$ 0 0
$$991$$ − 1.10256e12i − 1.14316i −0.820547 0.571579i $$-0.806331\pi$$
0.820547 0.571579i $$-0.193669\pi$$
$$992$$ 0 0
$$993$$ −6.64026e11 −0.682949
$$994$$ 0 0
$$995$$ 1.03769e12i 1.05871i
$$996$$ 0 0
$$997$$ −5.02913e10 −0.0508993 −0.0254497 0.999676i $$-0.508102\pi$$
−0.0254497 + 0.999676i $$0.508102\pi$$
$$998$$ 0 0
$$999$$ − 1.91873e11i − 0.192642i
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 192.9.g.a.127.2 2
4.3 odd 2 inner 192.9.g.a.127.1 2
8.3 odd 2 48.9.g.b.31.2 yes 2
8.5 even 2 48.9.g.b.31.1 2
24.5 odd 2 144.9.g.c.127.1 2
24.11 even 2 144.9.g.c.127.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
48.9.g.b.31.1 2 8.5 even 2
48.9.g.b.31.2 yes 2 8.3 odd 2
144.9.g.c.127.1 2 24.5 odd 2
144.9.g.c.127.2 2 24.11 even 2
192.9.g.a.127.1 2 4.3 odd 2 inner
192.9.g.a.127.2 2 1.1 even 1 trivial