# Properties

 Label 48.9.g.b.31.2 Level $48$ Weight $9$ Character 48.31 Analytic conductor $19.554$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,9,Mod(31,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, 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("48.31");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 48.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: $$19.5541732829$$ 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: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 31.2 Root $$0.500000 - 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 48.31 Dual form 48.9.g.b.31.1

## $q$-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 $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$-0.302740\pi$$
0.580800 + 0.814046i $$0.302740\pi$$
$$6$$ 0 0
$$7$$ 3055.34i 1.27253i 0.771472 + 0.636264i $$0.219521\pi$$
−0.771472 + 0.636264i $$0.780479\pi$$
$$8$$ 0 0
$$9$$ −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 $$-0.260525\pi$$
0.683344 + 0.730096i $$0.260525\pi$$
$$14$$ 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 $$0.354308\pi$$
−0.441892 + 0.897068i $$0.645692\pi$$
$$24$$ 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 $$-0.454375\pi$$
0.142844 + 0.989745i $$0.454375\pi$$
$$30$$ 0 0
$$31$$ 1.19563e6i 1.29465i 0.762215 + 0.647324i $$0.224112\pi$$
−0.762215 + 0.647324i $$0.775888\pi$$
$$32$$ 0 0
$$33$$ 621108. 0.523735
$$34$$ 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
$$39$$ 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 $$-0.774205\pi$$
0.758781 0.651346i $$-0.225795\pi$$
$$48$$ 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 $$-0.521528\pi$$
−0.0675803 + 0.997714i $$0.521528\pi$$
$$54$$ 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 $$-0.287347\pi$$
0.619471 + 0.785020i $$0.287347\pi$$
$$62$$ 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 $$-0.713296\pi$$
0.621056 0.783766i $$-0.286704\pi$$
$$72$$ 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 $$0.0753538\pi$$
−0.972110 + 0.234526i $$0.924646\pi$$
$$80$$ 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 $$-0.128815\pi$$
0.919227 + 0.393727i $$0.128815\pi$$
$$102$$ 0 0
$$103$$ − 1.62781e8i − 1.44629i −0.690699 0.723143i $$-0.742697\pi$$
0.690699 0.723143i $$-0.257303\pi$$
$$104$$ 0 0
$$105$$ −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 $$-0.421862\pi$$
0.243019 + 0.970021i $$0.421862\pi$$
$$110$$ 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 $$-0.825989\pi$$
0.854259 0.519848i $$-0.174011\pi$$
$$128$$ 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 $$-0.526221\pi$$
−0.0822831 + 0.996609i $$0.526221\pi$$
$$150$$ 0 0
$$151$$ − 1.77325e8i − 0.341086i −0.985350 0.170543i $$-0.945448\pi$$
0.985350 0.170543i $$-0.0545521\pi$$
$$152$$ 0 0
$$153$$ 1.43935e8 0.262664
$$154$$ 0 0
$$155$$ 8.68031e8i 1.50386i
$$156$$ 0 0
$$157$$ −2.14784e7 −0.0353511 −0.0176755 0.999844i $$-0.505627\pi$$
−0.0176755 + 0.999844i $$0.505627\pi$$
$$158$$ 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 $$-0.919545\pi$$
0.968227 0.250073i $$-0.0804547\pi$$
$$168$$ 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 $$-0.511286\pi$$
−0.0354494 + 0.999371i $$0.511286\pi$$
$$174$$ 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 $$-0.369377\pi$$
0.398944 + 0.916975i $$0.369377\pi$$
$$182$$ 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 $$0.0572023\pi$$
−0.983896 + 0.178741i $$0.942798\pi$$
$$192$$ 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 $$-0.870824\pi$$
−0.918780 + 0.394771i $$0.870824\pi$$
$$198$$ 0 0
$$199$$ 1.42932e9i 0.911420i 0.890128 + 0.455710i $$0.150615\pi$$
−0.890128 + 0.455710i $$0.849385\pi$$
$$200$$ 0 0
$$201$$ 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 $$-0.758705\pi$$
0.726179 0.687506i $$-0.241295\pi$$
$$224$$ 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 $$-0.557619\pi$$
−0.180028 + 0.983661i $$0.557619\pi$$
$$230$$ 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 $$0.132321\pi$$
−0.914834 + 0.403830i $$0.867679\pi$$
$$240$$ 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 $$-0.937686\pi$$
0.980899 0.194518i $$-0.0623144\pi$$
$$264$$ 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 $$-0.464250\pi$$
0.112074 + 0.993700i $$0.464250\pi$$
$$270$$ 0 0
$$271$$ 1.90505e9i 0.353207i 0.984282 + 0.176604i $$0.0565110\pi$$
−0.984282 + 0.176604i $$0.943489\pi$$
$$272$$ 0 0
$$273$$ −5.57733e9 −1.00410
$$274$$ 0 0
$$275$$ − 1.81226e9i − 0.316876i
$$276$$ 0 0
$$277$$ −5.03752e9 −0.855654 −0.427827 0.903861i $$-0.640721\pi$$
−0.427827 + 0.903861i $$0.640721\pi$$
$$278$$ 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 $$-0.350435\pi$$
0.452772 + 0.891627i $$0.350435\pi$$
$$294$$ 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 $$-0.993231\pi$$
0.999774 0.0212632i $$-0.00676879\pi$$
$$312$$ 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 $$-0.657825\pi$$
−0.475754 + 0.879579i $$0.657825\pi$$
$$318$$ 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 $$-0.375387\pi$$
0.381561 + 0.924344i $$0.375387\pi$$
$$350$$ 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 $$0.0789590\pi$$
−0.969391 + 0.245521i $$0.921041\pi$$
$$360$$ 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 $$-0.806995\pi$$
0.821738 0.569865i $$-0.193005\pi$$
$$368$$ 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 $$-0.563849\pi$$
−0.199247 + 0.979949i $$0.563849\pi$$
$$374$$ 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 $$-0.732514\pi$$
0.667216 0.744864i $$-0.267486\pi$$
$$384$$ 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 $$-0.726717\pi$$
−0.653541 + 0.756891i $$0.726717\pi$$
$$390$$ 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 $$-0.414295\pi$$
0.266008 + 0.963971i $$0.414295\pi$$
$$398$$ 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 $$-0.469324\pi$$
0.0962227 + 0.995360i $$0.469324\pi$$
$$422$$ 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 $$-0.767479\pi$$
0.744849 0.667233i $$-0.232521\pi$$
$$432$$ 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 $$0.0534569\pi$$
−0.985931 + 0.167152i $$0.946543\pi$$
$$440$$ 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 $$-0.590786\pi$$
−0.281362 + 0.959602i $$0.590786\pi$$
$$462$$ 0 0
$$463$$ − 1.19712e9i − 0.0260504i −0.999915 0.0130252i $$-0.995854\pi$$
0.999915 0.0130252i $$-0.00414617\pi$$
$$464$$ 0 0
$$465$$ −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 $$-0.923357\pi$$
0.971152 0.238460i $$-0.0766425\pi$$
$$480$$ 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 $$0.173270\pi$$
−0.855467 + 0.517857i $$0.826730\pi$$
$$488$$ 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 $$-0.954008\pi$$
0.989580 0.143985i $$-0.0459918\pi$$
$$504$$ 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 $$-0.466228\pi$$
0.105899 + 0.994377i $$0.466228\pi$$
$$510$$ 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 $$-0.646607\pi$$
−0.444466 + 0.895796i $$0.646607\pi$$
$$542$$ 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 $$-0.183791\pi$$
0.837887 + 0.545844i $$0.183791\pi$$
$$558$$ 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 $$0.173255\pi$$
−0.855491 + 0.517817i $$0.826745\pi$$
$$600$$ 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 $$0.194134\pi$$
−0.819710 + 0.572779i $$0.805866\pi$$
$$608$$ 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 $$-0.463879\pi$$
0.113233 + 0.993568i $$0.463879\pi$$
$$614$$ 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 $$0.00859193\pi$$
−0.999636 + 0.0269891i $$0.991408\pi$$
$$632$$ 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 $$0.0509128\pi$$
−0.987236 + 0.159266i $$0.949087\pi$$
$$648$$ 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 $$-0.556378\pi$$
−0.176191 + 0.984356i $$0.556378\pi$$
$$654$$ 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 $$-0.613895\pi$$
−0.350224 + 0.936666i $$0.613895\pi$$
$$662$$ 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 $$-0.538442\pi$$
−0.120476 + 0.992716i $$0.538442\pi$$
$$678$$ 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 $$-0.334903\pi$$
0.495722 + 0.868481i $$0.334903\pi$$
$$702$$ 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 $$-0.430865\pi$$
0.215490 + 0.976506i $$0.430865\pi$$
$$710$$ 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 $$-0.838073\pi$$
0.873374 0.487051i $$-0.161927\pi$$
$$720$$ 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 $$0.331116\pi$$
−0.506020 + 0.862522i $$0.668884\pi$$
$$728$$ 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 $$-0.602240\pi$$
−0.315702 + 0.948858i $$0.602240\pi$$
$$734$$ 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 $$-0.879753\pi$$
0.929491 0.368845i $$-0.120247\pi$$
$$744$$ 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 $$-0.772106\pi$$
0.754470 0.656334i $$-0.227894\pi$$
$$752$$ 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 $$-0.0922679\pi$$
0.958282 + 0.285826i $$0.0922679\pi$$
$$758$$ 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 $$-0.271194\pi$$
0.658495 + 0.752585i $$0.271194\pi$$
$$774$$ 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 $$-0.583049\pi$$
−0.257956 + 0.966157i $$0.583049\pi$$
$$798$$ 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 $$-0.322917\pi$$
0.528068 + 0.849202i $$0.322917\pi$$
$$822$$ 0 0
$$823$$ − 3.34155e11i − 0.728365i −0.931328 0.364183i $$-0.881348\pi$$
0.931328 0.364183i $$-0.118652\pi$$
$$824$$ 0 0
$$825$$ 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 $$-0.235424\pi$$
0.738733 + 0.673998i $$0.235424\pi$$
$$830$$ 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 $$-0.967257\pi$$
0.994714 0.102685i $$-0.0327433\pi$$
$$840$$ 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 $$-0.526540\pi$$
−0.0832805 + 0.996526i $$0.526540\pi$$
$$854$$ 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 $$-0.918778\pi$$
0.967622 0.252405i $$-0.0812217\pi$$
$$864$$ 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 $$-0.673459\pi$$
−0.518365 + 0.855159i $$0.673459\pi$$
$$878$$ 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 $$0.0376314\pi$$
−0.993020 + 0.117947i $$0.962369\pi$$
$$888$$ 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 $$-0.900005\pi$$
0.951061 0.309002i $$-0.0999950\pi$$
$$912$$ 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 $$-0.674372\pi$$
0.520815 0.853670i $$-0.325628\pi$$
$$920$$ 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 $$-0.464953\pi$$
0.109881 + 0.993945i $$0.464953\pi$$
$$942$$ 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 $$0.341443\pi$$
−0.477777 + 0.878481i $$0.658557\pi$$
$$968$$ 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 $$0.0526479\pi$$
−0.986353 + 0.164645i $$0.947352\pi$$
$$984$$ 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 $$0.193669\pi$$
−0.820547 + 0.571579i $$0.806331\pi$$
$$992$$ 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 $$-0.491898\pi$$
0.0254497 + 0.999676i $$0.491898\pi$$
$$998$$ 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 48.9.g.b.31.2 yes 2
3.2 odd 2 144.9.g.c.127.2 2
4.3 odd 2 inner 48.9.g.b.31.1 2
8.3 odd 2 192.9.g.a.127.2 2
8.5 even 2 192.9.g.a.127.1 2
12.11 even 2 144.9.g.c.127.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
48.9.g.b.31.1 2 4.3 odd 2 inner
48.9.g.b.31.2 yes 2 1.1 even 1 trivial
144.9.g.c.127.1 2 12.11 even 2
144.9.g.c.127.2 2 3.2 odd 2
192.9.g.a.127.1 2 8.5 even 2
192.9.g.a.127.2 2 8.3 odd 2