# Properties

 Label 64.9.c.b.63.1 Level $64$ Weight $9$ Character 64.63 Analytic conductor $26.072$ 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] = [64,9,Mod(63,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("64.63");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 64.c (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: $$26.0722310439$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-39})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 10$$ x^2 - x + 10 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 4) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 63.1 Root $$0.500000 + 3.12250i$$ of defining polynomial Character $$\chi$$ $$=$$ 64.63 Dual form 64.9.c.b.63.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-99.9200i q^{3} -610.000 q^{5} +1398.88i q^{7} -3423.00 q^{9} +O(q^{10})$$ $$q-99.9200i q^{3} -610.000 q^{5} +1398.88i q^{7} -3423.00 q^{9} +18485.2i q^{11} +5470.00 q^{13} +60951.2i q^{15} +73090.0 q^{17} +19484.4i q^{19} +139776. q^{21} -237210. i q^{23} -18525.0 q^{25} -313549. i q^{27} +128222. q^{29} -67945.6i q^{31} +1.84704e6 q^{33} -853317. i q^{35} +3.47203e6 q^{37} -546562. i q^{39} +2.14688e6 q^{41} +5.92815e6i q^{43} +2.08803e6 q^{45} +7.62629e6i q^{47} +3.80794e6 q^{49} -7.30315e6i q^{51} -824290. q^{53} -1.12760e7i q^{55} +1.94688e6 q^{57} +3.72552e6i q^{59} +1.47461e7 q^{61} -4.78836e6i q^{63} -3.33670e6 q^{65} -1.52567e7i q^{67} -2.37020e7 q^{69} -1.19604e6i q^{71} -5.72563e6 q^{73} +1.85102e6i q^{75} -2.58586e7 q^{77} +3.59132e7i q^{79} -5.37881e7 q^{81} +5.19603e7i q^{83} -4.45849e7 q^{85} -1.28119e7i q^{87} -8.33242e7 q^{89} +7.65187e6i q^{91} -6.78912e6 q^{93} -1.18855e7i q^{95} +1.20619e8 q^{97} -6.32748e7i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 1220 q^{5} - 6846 q^{9}+O(q^{10})$$ 2 * q - 1220 * q^5 - 6846 * q^9 $$2 q - 1220 q^{5} - 6846 q^{9} + 10940 q^{13} + 146180 q^{17} + 279552 q^{21} - 37050 q^{25} + 256444 q^{29} + 3694080 q^{33} + 6944060 q^{37} + 4293764 q^{41} + 4176060 q^{45} + 7615874 q^{49} - 1648580 q^{53} + 3893760 q^{57} + 29492156 q^{61} - 6673400 q^{65} - 47404032 q^{69} - 11451260 q^{73} - 51717120 q^{77} - 107576190 q^{81} - 89169800 q^{85} - 166648444 q^{89} - 13578240 q^{93} + 241238020 q^{97}+O(q^{100})$$ 2 * q - 1220 * q^5 - 6846 * q^9 + 10940 * q^13 + 146180 * q^17 + 279552 * q^21 - 37050 * q^25 + 256444 * q^29 + 3694080 * q^33 + 6944060 * q^37 + 4293764 * q^41 + 4176060 * q^45 + 7615874 * q^49 - 1648580 * q^53 + 3893760 * q^57 + 29492156 * q^61 - 6673400 * q^65 - 47404032 * q^69 - 11451260 * q^73 - 51717120 * q^77 - 107576190 * q^81 - 89169800 * q^85 - 166648444 * q^89 - 13578240 * q^93 + 241238020 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$63$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$3$$ − 99.9200i − 1.23358i −0.787128 0.616790i $$-0.788433\pi$$
0.787128 0.616790i $$-0.211567\pi$$
$$4$$ 0 0
$$5$$ −610.000 −0.976000 −0.488000 0.872844i $$-0.662273\pi$$
−0.488000 + 0.872844i $$0.662273\pi$$
$$6$$ 0 0
$$7$$ 1398.88i 0.582624i 0.956628 + 0.291312i $$0.0940917\pi$$
−0.956628 + 0.291312i $$0.905908\pi$$
$$8$$ 0 0
$$9$$ −3423.00 −0.521719
$$10$$ 0 0
$$11$$ 18485.2i 1.26256i 0.775554 + 0.631282i $$0.217471\pi$$
−0.775554 + 0.631282i $$0.782529\pi$$
$$12$$ 0 0
$$13$$ 5470.00 0.191520 0.0957600 0.995404i $$-0.469472\pi$$
0.0957600 + 0.995404i $$0.469472\pi$$
$$14$$ 0 0
$$15$$ 60951.2i 1.20397i
$$16$$ 0 0
$$17$$ 73090.0 0.875109 0.437555 0.899192i $$-0.355845\pi$$
0.437555 + 0.899192i $$0.355845\pi$$
$$18$$ 0 0
$$19$$ 19484.4i 0.149511i 0.997202 + 0.0747554i $$0.0238176\pi$$
−0.997202 + 0.0747554i $$0.976182\pi$$
$$20$$ 0 0
$$21$$ 139776. 0.718713
$$22$$ 0 0
$$23$$ − 237210.i − 0.847660i −0.905742 0.423830i $$-0.860685\pi$$
0.905742 0.423830i $$-0.139315\pi$$
$$24$$ 0 0
$$25$$ −18525.0 −0.0474240
$$26$$ 0 0
$$27$$ − 313549.i − 0.589997i
$$28$$ 0 0
$$29$$ 128222. 0.181289 0.0906443 0.995883i $$-0.471107\pi$$
0.0906443 + 0.995883i $$0.471107\pi$$
$$30$$ 0 0
$$31$$ − 67945.6i − 0.0735723i −0.999323 0.0367862i $$-0.988288\pi$$
0.999323 0.0367862i $$-0.0117120\pi$$
$$32$$ 0 0
$$33$$ 1.84704e6 1.55747
$$34$$ 0 0
$$35$$ − 853317.i − 0.568641i
$$36$$ 0 0
$$37$$ 3.47203e6 1.85258 0.926289 0.376813i $$-0.122980\pi$$
0.926289 + 0.376813i $$0.122980\pi$$
$$38$$ 0 0
$$39$$ − 546562.i − 0.236255i
$$40$$ 0 0
$$41$$ 2.14688e6 0.759754 0.379877 0.925037i $$-0.375966\pi$$
0.379877 + 0.925037i $$0.375966\pi$$
$$42$$ 0 0
$$43$$ 5.92815e6i 1.73399i 0.498321 + 0.866993i $$0.333950\pi$$
−0.498321 + 0.866993i $$0.666050\pi$$
$$44$$ 0 0
$$45$$ 2.08803e6 0.509198
$$46$$ 0 0
$$47$$ 7.62629e6i 1.56287i 0.623989 + 0.781433i $$0.285511\pi$$
−0.623989 + 0.781433i $$0.714489\pi$$
$$48$$ 0 0
$$49$$ 3.80794e6 0.660550
$$50$$ 0 0
$$51$$ − 7.30315e6i − 1.07952i
$$52$$ 0 0
$$53$$ −824290. −0.104466 −0.0522332 0.998635i $$-0.516634\pi$$
−0.0522332 + 0.998635i $$0.516634\pi$$
$$54$$ 0 0
$$55$$ − 1.12760e7i − 1.23226i
$$56$$ 0 0
$$57$$ 1.94688e6 0.184433
$$58$$ 0 0
$$59$$ 3.72552e6i 0.307453i 0.988113 + 0.153726i $$0.0491274\pi$$
−0.988113 + 0.153726i $$0.950873\pi$$
$$60$$ 0 0
$$61$$ 1.47461e7 1.06502 0.532509 0.846424i $$-0.321249\pi$$
0.532509 + 0.846424i $$0.321249\pi$$
$$62$$ 0 0
$$63$$ − 4.78836e6i − 0.303966i
$$64$$ 0 0
$$65$$ −3.33670e6 −0.186923
$$66$$ 0 0
$$67$$ − 1.52567e7i − 0.757113i −0.925578 0.378557i $$-0.876421\pi$$
0.925578 0.378557i $$-0.123579\pi$$
$$68$$ 0 0
$$69$$ −2.37020e7 −1.04566
$$70$$ 0 0
$$71$$ − 1.19604e6i − 0.0470666i −0.999723 0.0235333i $$-0.992508\pi$$
0.999723 0.0235333i $$-0.00749158\pi$$
$$72$$ 0 0
$$73$$ −5.72563e6 −0.201619 −0.100810 0.994906i $$-0.532143\pi$$
−0.100810 + 0.994906i $$0.532143\pi$$
$$74$$ 0 0
$$75$$ 1.85102e6i 0.0585013i
$$76$$ 0 0
$$77$$ −2.58586e7 −0.735600
$$78$$ 0 0
$$79$$ 3.59132e7i 0.922032i 0.887392 + 0.461016i $$0.152515\pi$$
−0.887392 + 0.461016i $$0.847485\pi$$
$$80$$ 0 0
$$81$$ −5.37881e7 −1.24953
$$82$$ 0 0
$$83$$ 5.19603e7i 1.09486i 0.836851 + 0.547431i $$0.184394\pi$$
−0.836851 + 0.547431i $$0.815606\pi$$
$$84$$ 0 0
$$85$$ −4.45849e7 −0.854107
$$86$$ 0 0
$$87$$ − 1.28119e7i − 0.223634i
$$88$$ 0 0
$$89$$ −8.33242e7 −1.32804 −0.664020 0.747715i $$-0.731151\pi$$
−0.664020 + 0.747715i $$0.731151\pi$$
$$90$$ 0 0
$$91$$ 7.65187e6i 0.111584i
$$92$$ 0 0
$$93$$ −6.78912e6 −0.0907573
$$94$$ 0 0
$$95$$ − 1.18855e7i − 0.145923i
$$96$$ 0 0
$$97$$ 1.20619e8 1.36248 0.681238 0.732062i $$-0.261442\pi$$
0.681238 + 0.732062i $$0.261442\pi$$
$$98$$ 0 0
$$99$$ − 6.32748e7i − 0.658704i
$$100$$ 0 0
$$101$$ −2.77246e7 −0.266428 −0.133214 0.991087i $$-0.542530\pi$$
−0.133214 + 0.991087i $$0.542530\pi$$
$$102$$ 0 0
$$103$$ 1.04501e8i 0.928477i 0.885710 + 0.464238i $$0.153672\pi$$
−0.885710 + 0.464238i $$0.846328\pi$$
$$104$$ 0 0
$$105$$ −8.52634e7 −0.701464
$$106$$ 0 0
$$107$$ − 1.00328e8i − 0.765394i −0.923874 0.382697i $$-0.874995\pi$$
0.923874 0.382697i $$-0.125005\pi$$
$$108$$ 0 0
$$109$$ 5.90716e7 0.418478 0.209239 0.977865i $$-0.432901\pi$$
0.209239 + 0.977865i $$0.432901\pi$$
$$110$$ 0 0
$$111$$ − 3.46925e8i − 2.28530i
$$112$$ 0 0
$$113$$ 5.50849e7 0.337846 0.168923 0.985629i $$-0.445971\pi$$
0.168923 + 0.985629i $$0.445971\pi$$
$$114$$ 0 0
$$115$$ 1.44698e8i 0.827316i
$$116$$ 0 0
$$117$$ −1.87238e7 −0.0999196
$$118$$ 0 0
$$119$$ 1.02244e8i 0.509859i
$$120$$ 0 0
$$121$$ −1.27344e8 −0.594067
$$122$$ 0 0
$$123$$ − 2.14516e8i − 0.937217i
$$124$$ 0 0
$$125$$ 2.49581e8 1.02229
$$126$$ 0 0
$$127$$ 2.57160e8i 0.988529i 0.869312 + 0.494264i $$0.164562\pi$$
−0.869312 + 0.494264i $$0.835438\pi$$
$$128$$ 0 0
$$129$$ 5.92341e8 2.13901
$$130$$ 0 0
$$131$$ 3.12175e8i 1.06002i 0.847992 + 0.530009i $$0.177812\pi$$
−0.847992 + 0.530009i $$0.822188\pi$$
$$132$$ 0 0
$$133$$ −2.72563e7 −0.0871085
$$134$$ 0 0
$$135$$ 1.91265e8i 0.575838i
$$136$$ 0 0
$$137$$ 2.21980e8 0.630132 0.315066 0.949070i $$-0.397973\pi$$
0.315066 + 0.949070i $$0.397973\pi$$
$$138$$ 0 0
$$139$$ 2.95030e8i 0.790328i 0.918611 + 0.395164i $$0.129312\pi$$
−0.918611 + 0.395164i $$0.870688\pi$$
$$140$$ 0 0
$$141$$ 7.62019e8 1.92792
$$142$$ 0 0
$$143$$ 1.01114e8i 0.241806i
$$144$$ 0 0
$$145$$ −7.82154e7 −0.176938
$$146$$ 0 0
$$147$$ − 3.80489e8i − 0.814841i
$$148$$ 0 0
$$149$$ −4.03603e8 −0.818859 −0.409429 0.912342i $$-0.634272\pi$$
−0.409429 + 0.912342i $$0.634272\pi$$
$$150$$ 0 0
$$151$$ − 8.36985e8i − 1.60994i −0.593316 0.804970i $$-0.702181\pi$$
0.593316 0.804970i $$-0.297819\pi$$
$$152$$ 0 0
$$153$$ −2.50187e8 −0.456561
$$154$$ 0 0
$$155$$ 4.14468e7i 0.0718066i
$$156$$ 0 0
$$157$$ 2.71319e8 0.446561 0.223281 0.974754i $$-0.428323\pi$$
0.223281 + 0.974754i $$0.428323\pi$$
$$158$$ 0 0
$$159$$ 8.23630e7i 0.128868i
$$160$$ 0 0
$$161$$ 3.31828e8 0.493867
$$162$$ 0 0
$$163$$ − 5.78509e8i − 0.819520i −0.912193 0.409760i $$-0.865612\pi$$
0.912193 0.409760i $$-0.134388\pi$$
$$164$$ 0 0
$$165$$ −1.12669e9 −1.52009
$$166$$ 0 0
$$167$$ 4.68118e8i 0.601852i 0.953647 + 0.300926i $$0.0972958\pi$$
−0.953647 + 0.300926i $$0.902704\pi$$
$$168$$ 0 0
$$169$$ −7.85810e8 −0.963320
$$170$$ 0 0
$$171$$ − 6.66951e7i − 0.0780026i
$$172$$ 0 0
$$173$$ 2.06197e8 0.230196 0.115098 0.993354i $$-0.463282\pi$$
0.115098 + 0.993354i $$0.463282\pi$$
$$174$$ 0 0
$$175$$ − 2.59142e7i − 0.0276303i
$$176$$ 0 0
$$177$$ 3.72253e8 0.379268
$$178$$ 0 0
$$179$$ − 1.41911e8i − 0.138230i −0.997609 0.0691152i $$-0.977982\pi$$
0.997609 0.0691152i $$-0.0220176\pi$$
$$180$$ 0 0
$$181$$ −4.82566e8 −0.449616 −0.224808 0.974403i $$-0.572176\pi$$
−0.224808 + 0.974403i $$0.572176\pi$$
$$182$$ 0 0
$$183$$ − 1.47343e9i − 1.31379i
$$184$$ 0 0
$$185$$ −2.11794e9 −1.80812
$$186$$ 0 0
$$187$$ 1.35108e9i 1.10488i
$$188$$ 0 0
$$189$$ 4.38617e8 0.343747
$$190$$ 0 0
$$191$$ 9.92461e8i 0.745727i 0.927886 + 0.372864i $$0.121624\pi$$
−0.927886 + 0.372864i $$0.878376\pi$$
$$192$$ 0 0
$$193$$ 1.17593e9 0.847526 0.423763 0.905773i $$-0.360709\pi$$
0.423763 + 0.905773i $$0.360709\pi$$
$$194$$ 0 0
$$195$$ 3.33403e8i 0.230585i
$$196$$ 0 0
$$197$$ −1.70538e9 −1.13229 −0.566144 0.824306i $$-0.691565\pi$$
−0.566144 + 0.824306i $$0.691565\pi$$
$$198$$ 0 0
$$199$$ 2.49036e9i 1.58800i 0.607919 + 0.793999i $$0.292004\pi$$
−0.607919 + 0.793999i $$0.707996\pi$$
$$200$$ 0 0
$$201$$ −1.52445e9 −0.933960
$$202$$ 0 0
$$203$$ 1.79367e8i 0.105623i
$$204$$ 0 0
$$205$$ −1.30960e9 −0.741519
$$206$$ 0 0
$$207$$ 8.11970e8i 0.442241i
$$208$$ 0 0
$$209$$ −3.60173e8 −0.188767
$$210$$ 0 0
$$211$$ − 1.46774e9i − 0.740491i −0.928934 0.370245i $$-0.879274\pi$$
0.928934 0.370245i $$-0.120726\pi$$
$$212$$ 0 0
$$213$$ −1.19508e8 −0.0580604
$$214$$ 0 0
$$215$$ − 3.61617e9i − 1.69237i
$$216$$ 0 0
$$217$$ 9.50477e7 0.0428650
$$218$$ 0 0
$$219$$ 5.72105e8i 0.248713i
$$220$$ 0 0
$$221$$ 3.99802e8 0.167601
$$222$$ 0 0
$$223$$ − 1.47920e9i − 0.598147i −0.954230 0.299073i $$-0.903322\pi$$
0.954230 0.299073i $$-0.0966776\pi$$
$$224$$ 0 0
$$225$$ 6.34111e7 0.0247420
$$226$$ 0 0
$$227$$ 7.50054e8i 0.282481i 0.989975 + 0.141241i $$0.0451091\pi$$
−0.989975 + 0.141241i $$0.954891\pi$$
$$228$$ 0 0
$$229$$ 2.84784e9 1.03556 0.517778 0.855515i $$-0.326759\pi$$
0.517778 + 0.855515i $$0.326759\pi$$
$$230$$ 0 0
$$231$$ 2.58379e9i 0.907421i
$$232$$ 0 0
$$233$$ 2.20621e8 0.0748553 0.0374276 0.999299i $$-0.488084\pi$$
0.0374276 + 0.999299i $$0.488084\pi$$
$$234$$ 0 0
$$235$$ − 4.65204e9i − 1.52536i
$$236$$ 0 0
$$237$$ 3.58845e9 1.13740
$$238$$ 0 0
$$239$$ − 4.04493e9i − 1.23971i −0.784717 0.619855i $$-0.787192\pi$$
0.784717 0.619855i $$-0.212808\pi$$
$$240$$ 0 0
$$241$$ 6.17983e9 1.83193 0.915964 0.401260i $$-0.131427\pi$$
0.915964 + 0.401260i $$0.131427\pi$$
$$242$$ 0 0
$$243$$ 3.31731e9i 0.951395i
$$244$$ 0 0
$$245$$ −2.32284e9 −0.644696
$$246$$ 0 0
$$247$$ 1.06580e8i 0.0286343i
$$248$$ 0 0
$$249$$ 5.19187e9 1.35060
$$250$$ 0 0
$$251$$ − 5.21367e9i − 1.31356i −0.754084 0.656778i $$-0.771919\pi$$
0.754084 0.656778i $$-0.228081\pi$$
$$252$$ 0 0
$$253$$ 4.38487e9 1.07022
$$254$$ 0 0
$$255$$ 4.45492e9i 1.05361i
$$256$$ 0 0
$$257$$ −6.13693e9 −1.40676 −0.703378 0.710816i $$-0.748326\pi$$
−0.703378 + 0.710816i $$0.748326\pi$$
$$258$$ 0 0
$$259$$ 4.85695e9i 1.07936i
$$260$$ 0 0
$$261$$ −4.38904e8 −0.0945818
$$262$$ 0 0
$$263$$ 6.96916e9i 1.45666i 0.685228 + 0.728329i $$0.259703\pi$$
−0.685228 + 0.728329i $$0.740297\pi$$
$$264$$ 0 0
$$265$$ 5.02817e8 0.101959
$$266$$ 0 0
$$267$$ 8.32575e9i 1.63824i
$$268$$ 0 0
$$269$$ −2.70720e9 −0.517025 −0.258513 0.966008i $$-0.583232\pi$$
−0.258513 + 0.966008i $$0.583232\pi$$
$$270$$ 0 0
$$271$$ − 7.99032e9i − 1.48145i −0.671808 0.740725i $$-0.734482\pi$$
0.671808 0.740725i $$-0.265518\pi$$
$$272$$ 0 0
$$273$$ 7.64575e8 0.137648
$$274$$ 0 0
$$275$$ − 3.42438e8i − 0.0598758i
$$276$$ 0 0
$$277$$ 8.22965e9 1.39786 0.698928 0.715192i $$-0.253661\pi$$
0.698928 + 0.715192i $$0.253661\pi$$
$$278$$ 0 0
$$279$$ 2.32578e8i 0.0383841i
$$280$$ 0 0
$$281$$ 3.08105e9 0.494167 0.247083 0.968994i $$-0.420528\pi$$
0.247083 + 0.968994i $$0.420528\pi$$
$$282$$ 0 0
$$283$$ − 1.17112e9i − 0.182582i −0.995824 0.0912908i $$-0.970901\pi$$
0.995824 0.0912908i $$-0.0290993\pi$$
$$284$$ 0 0
$$285$$ −1.18760e9 −0.180007
$$286$$ 0 0
$$287$$ 3.00323e9i 0.442650i
$$288$$ 0 0
$$289$$ −1.63361e9 −0.234184
$$290$$ 0 0
$$291$$ − 1.20522e10i − 1.68072i
$$292$$ 0 0
$$293$$ −4.80980e9 −0.652614 −0.326307 0.945264i $$-0.605804\pi$$
−0.326307 + 0.945264i $$0.605804\pi$$
$$294$$ 0 0
$$295$$ − 2.27256e9i − 0.300074i
$$296$$ 0 0
$$297$$ 5.79601e9 0.744909
$$298$$ 0 0
$$299$$ − 1.29754e9i − 0.162344i
$$300$$ 0 0
$$301$$ −8.29277e9 −1.01026
$$302$$ 0 0
$$303$$ 2.77025e9i 0.328661i
$$304$$ 0 0
$$305$$ −8.99511e9 −1.03946
$$306$$ 0 0
$$307$$ 3.49176e9i 0.393089i 0.980495 + 0.196545i $$0.0629721\pi$$
−0.980495 + 0.196545i $$0.937028\pi$$
$$308$$ 0 0
$$309$$ 1.04417e10 1.14535
$$310$$ 0 0
$$311$$ − 1.29807e10i − 1.38757i −0.720182 0.693785i $$-0.755942\pi$$
0.720182 0.693785i $$-0.244058\pi$$
$$312$$ 0 0
$$313$$ −6.31165e9 −0.657606 −0.328803 0.944399i $$-0.606645\pi$$
−0.328803 + 0.944399i $$0.606645\pi$$
$$314$$ 0 0
$$315$$ 2.92090e9i 0.296671i
$$316$$ 0 0
$$317$$ −1.65902e10 −1.64291 −0.821455 0.570273i $$-0.806837\pi$$
−0.821455 + 0.570273i $$0.806837\pi$$
$$318$$ 0 0
$$319$$ 2.37021e9i 0.228888i
$$320$$ 0 0
$$321$$ −1.00247e10 −0.944175
$$322$$ 0 0
$$323$$ 1.42411e9i 0.130838i
$$324$$ 0 0
$$325$$ −1.01332e8 −0.00908264
$$326$$ 0 0
$$327$$ − 5.90243e9i − 0.516226i
$$328$$ 0 0
$$329$$ −1.06683e10 −0.910563
$$330$$ 0 0
$$331$$ − 5.48640e9i − 0.457062i −0.973537 0.228531i $$-0.926608\pi$$
0.973537 0.228531i $$-0.0733922\pi$$
$$332$$ 0 0
$$333$$ −1.18848e10 −0.966526
$$334$$ 0 0
$$335$$ 9.30657e9i 0.738942i
$$336$$ 0 0
$$337$$ −3.56226e8 −0.0276189 −0.0138095 0.999905i $$-0.504396\pi$$
−0.0138095 + 0.999905i $$0.504396\pi$$
$$338$$ 0 0
$$339$$ − 5.50408e9i − 0.416760i
$$340$$ 0 0
$$341$$ 1.25599e9 0.0928897
$$342$$ 0 0
$$343$$ 1.33911e10i 0.967476i
$$344$$ 0 0
$$345$$ 1.44582e10 1.02056
$$346$$ 0 0
$$347$$ 1.59731e10i 1.10172i 0.834599 + 0.550859i $$0.185700\pi$$
−0.834599 + 0.550859i $$0.814300\pi$$
$$348$$ 0 0
$$349$$ −1.03634e10 −0.698553 −0.349277 0.937020i $$-0.613573\pi$$
−0.349277 + 0.937020i $$0.613573\pi$$
$$350$$ 0 0
$$351$$ − 1.71511e9i − 0.112996i
$$352$$ 0 0
$$353$$ −1.30979e10 −0.843536 −0.421768 0.906704i $$-0.638590\pi$$
−0.421768 + 0.906704i $$0.638590\pi$$
$$354$$ 0 0
$$355$$ 7.29586e8i 0.0459370i
$$356$$ 0 0
$$357$$ 1.02162e10 0.628952
$$358$$ 0 0
$$359$$ 3.31454e9i 0.199547i 0.995010 + 0.0997737i $$0.0318119\pi$$
−0.995010 + 0.0997737i $$0.968188\pi$$
$$360$$ 0 0
$$361$$ 1.66039e10 0.977647
$$362$$ 0 0
$$363$$ 1.27242e10i 0.732829i
$$364$$ 0 0
$$365$$ 3.49263e9 0.196780
$$366$$ 0 0
$$367$$ 1.96628e10i 1.08388i 0.840418 + 0.541939i $$0.182309\pi$$
−0.840418 + 0.541939i $$0.817691\pi$$
$$368$$ 0 0
$$369$$ −7.34878e9 −0.396378
$$370$$ 0 0
$$371$$ − 1.15308e9i − 0.0608646i
$$372$$ 0 0
$$373$$ 2.10063e10 1.08521 0.542606 0.839987i $$-0.317438\pi$$
0.542606 + 0.839987i $$0.317438\pi$$
$$374$$ 0 0
$$375$$ − 2.49382e10i − 1.26107i
$$376$$ 0 0
$$377$$ 7.01374e8 0.0347204
$$378$$ 0 0
$$379$$ − 3.04816e9i − 0.147734i −0.997268 0.0738670i $$-0.976466\pi$$
0.997268 0.0738670i $$-0.0235340\pi$$
$$380$$ 0 0
$$381$$ 2.56955e10 1.21943
$$382$$ 0 0
$$383$$ − 2.23357e10i − 1.03802i −0.854770 0.519008i $$-0.826302\pi$$
0.854770 0.519008i $$-0.173698\pi$$
$$384$$ 0 0
$$385$$ 1.57737e10 0.717945
$$386$$ 0 0
$$387$$ − 2.02921e10i − 0.904654i
$$388$$ 0 0
$$389$$ −3.13680e10 −1.36990 −0.684948 0.728592i $$-0.740175\pi$$
−0.684948 + 0.728592i $$0.740175\pi$$
$$390$$ 0 0
$$391$$ − 1.73377e10i − 0.741795i
$$392$$ 0 0
$$393$$ 3.11926e10 1.30762
$$394$$ 0 0
$$395$$ − 2.19071e10i − 0.899904i
$$396$$ 0 0
$$397$$ −7.65788e9 −0.308281 −0.154140 0.988049i $$-0.549261\pi$$
−0.154140 + 0.988049i $$0.549261\pi$$
$$398$$ 0 0
$$399$$ 2.72345e9i 0.107455i
$$400$$ 0 0
$$401$$ −3.26120e10 −1.26125 −0.630623 0.776089i $$-0.717201\pi$$
−0.630623 + 0.776089i $$0.717201\pi$$
$$402$$ 0 0
$$403$$ − 3.71662e8i − 0.0140906i
$$404$$ 0 0
$$405$$ 3.28107e10 1.21954
$$406$$ 0 0
$$407$$ 6.41811e10i 2.33900i
$$408$$ 0 0
$$409$$ 2.26168e10 0.808236 0.404118 0.914707i $$-0.367578\pi$$
0.404118 + 0.914707i $$0.367578\pi$$
$$410$$ 0 0
$$411$$ − 2.21802e10i − 0.777318i
$$412$$ 0 0
$$413$$ −5.21155e9 −0.179129
$$414$$ 0 0
$$415$$ − 3.16958e10i − 1.06858i
$$416$$ 0 0
$$417$$ 2.94794e10 0.974932
$$418$$ 0 0
$$419$$ 4.94503e10i 1.60440i 0.597054 + 0.802201i $$0.296338\pi$$
−0.597054 + 0.802201i $$0.703662\pi$$
$$420$$ 0 0
$$421$$ 3.34077e10 1.06345 0.531726 0.846916i $$-0.321543\pi$$
0.531726 + 0.846916i $$0.321543\pi$$
$$422$$ 0 0
$$423$$ − 2.61048e10i − 0.815378i
$$424$$ 0 0
$$425$$ −1.35399e9 −0.0415012
$$426$$ 0 0
$$427$$ 2.06280e10i 0.620505i
$$428$$ 0 0
$$429$$ 1.01033e10 0.298287
$$430$$ 0 0
$$431$$ − 3.06956e10i − 0.889544i −0.895644 0.444772i $$-0.853285\pi$$
0.895644 0.444772i $$-0.146715\pi$$
$$432$$ 0 0
$$433$$ 2.88433e9 0.0820529 0.0410265 0.999158i $$-0.486937\pi$$
0.0410265 + 0.999158i $$0.486937\pi$$
$$434$$ 0 0
$$435$$ 7.81528e9i 0.218267i
$$436$$ 0 0
$$437$$ 4.62189e9 0.126734
$$438$$ 0 0
$$439$$ 6.92422e10i 1.86429i 0.362088 + 0.932144i $$0.382064\pi$$
−0.362088 + 0.932144i $$0.617936\pi$$
$$440$$ 0 0
$$441$$ −1.30346e10 −0.344621
$$442$$ 0 0
$$443$$ − 2.06609e10i − 0.536455i −0.963356 0.268228i $$-0.913562\pi$$
0.963356 0.268228i $$-0.0864379\pi$$
$$444$$ 0 0
$$445$$ 5.08278e10 1.29617
$$446$$ 0 0
$$447$$ 4.03280e10i 1.01013i
$$448$$ 0 0
$$449$$ 2.11092e10 0.519382 0.259691 0.965692i $$-0.416379\pi$$
0.259691 + 0.965692i $$0.416379\pi$$
$$450$$ 0 0
$$451$$ 3.96855e10i 0.959237i
$$452$$ 0 0
$$453$$ −8.36315e10 −1.98599
$$454$$ 0 0
$$455$$ − 4.66764e9i − 0.108906i
$$456$$ 0 0
$$457$$ −2.06831e10 −0.474188 −0.237094 0.971487i $$-0.576195\pi$$
−0.237094 + 0.971487i $$0.576195\pi$$
$$458$$ 0 0
$$459$$ − 2.29173e10i − 0.516312i
$$460$$ 0 0
$$461$$ −7.65072e10 −1.69394 −0.846971 0.531640i $$-0.821576\pi$$
−0.846971 + 0.531640i $$0.821576\pi$$
$$462$$ 0 0
$$463$$ 3.41303e9i 0.0742704i 0.999310 + 0.0371352i $$0.0118232\pi$$
−0.999310 + 0.0371352i $$0.988177\pi$$
$$464$$ 0 0
$$465$$ 4.14136e9 0.0885791
$$466$$ 0 0
$$467$$ − 1.92903e10i − 0.405576i −0.979223 0.202788i $$-0.935000\pi$$
0.979223 0.202788i $$-0.0650002\pi$$
$$468$$ 0 0
$$469$$ 2.13423e10 0.441112
$$470$$ 0 0
$$471$$ − 2.71102e10i − 0.550869i
$$472$$ 0 0
$$473$$ −1.09583e11 −2.18927
$$474$$ 0 0
$$475$$ − 3.60948e8i − 0.00709040i
$$476$$ 0 0
$$477$$ 2.82154e9 0.0545021
$$478$$ 0 0
$$479$$ − 2.43887e10i − 0.463282i −0.972801 0.231641i $$-0.925590\pi$$
0.972801 0.231641i $$-0.0744095\pi$$
$$480$$ 0 0
$$481$$ 1.89920e10 0.354806
$$482$$ 0 0
$$483$$ − 3.31563e10i − 0.609224i
$$484$$ 0 0
$$485$$ −7.35776e10 −1.32978
$$486$$ 0 0
$$487$$ − 9.30801e10i − 1.65478i −0.561626 0.827391i $$-0.689824\pi$$
0.561626 0.827391i $$-0.310176\pi$$
$$488$$ 0 0
$$489$$ −5.78046e10 −1.01094
$$490$$ 0 0
$$491$$ 2.12850e9i 0.0366225i 0.999832 + 0.0183113i $$0.00582898\pi$$
−0.999832 + 0.0183113i $$0.994171\pi$$
$$492$$ 0 0
$$493$$ 9.37175e9 0.158647
$$494$$ 0 0
$$495$$ 3.85976e10i 0.642895i
$$496$$ 0 0
$$497$$ 1.67312e9 0.0274221
$$498$$ 0 0
$$499$$ − 1.04101e10i − 0.167901i −0.996470 0.0839503i $$-0.973246\pi$$
0.996470 0.0839503i $$-0.0267537\pi$$
$$500$$ 0 0
$$501$$ 4.67744e10 0.742433
$$502$$ 0 0
$$503$$ 3.93019e10i 0.613962i 0.951716 + 0.306981i $$0.0993188\pi$$
−0.951716 + 0.306981i $$0.900681\pi$$
$$504$$ 0 0
$$505$$ 1.69120e10 0.260034
$$506$$ 0 0
$$507$$ 7.85181e10i 1.18833i
$$508$$ 0 0
$$509$$ 3.25113e10 0.484354 0.242177 0.970232i $$-0.422139\pi$$
0.242177 + 0.970232i $$0.422139\pi$$
$$510$$ 0 0
$$511$$ − 8.00947e9i − 0.117468i
$$512$$ 0 0
$$513$$ 6.10931e9 0.0882110
$$514$$ 0 0
$$515$$ − 6.37455e10i − 0.906194i
$$516$$ 0 0
$$517$$ −1.40973e11 −1.97322
$$518$$ 0 0
$$519$$ − 2.06032e10i − 0.283965i
$$520$$ 0 0
$$521$$ 1.84550e9 0.0250475 0.0125237 0.999922i $$-0.496013\pi$$
0.0125237 + 0.999922i $$0.496013\pi$$
$$522$$ 0 0
$$523$$ − 6.23770e10i − 0.833715i −0.908972 0.416858i $$-0.863131\pi$$
0.908972 0.416858i $$-0.136869\pi$$
$$524$$ 0 0
$$525$$ −2.58935e9 −0.0340842
$$526$$ 0 0
$$527$$ − 4.96614e9i − 0.0643838i
$$528$$ 0 0
$$529$$ 2.20424e10 0.281473
$$530$$ 0 0
$$531$$ − 1.27524e10i − 0.160404i
$$532$$ 0 0
$$533$$ 1.17434e10 0.145508
$$534$$ 0 0
$$535$$ 6.11998e10i 0.747025i
$$536$$ 0 0
$$537$$ −1.41797e10 −0.170518
$$538$$ 0 0
$$539$$ 7.03905e10i 0.833986i
$$540$$ 0 0
$$541$$ 7.45917e10 0.870766 0.435383 0.900245i $$-0.356613\pi$$
0.435383 + 0.900245i $$0.356613\pi$$
$$542$$ 0 0
$$543$$ 4.82179e10i 0.554638i
$$544$$ 0 0
$$545$$ −3.60337e10 −0.408435
$$546$$ 0 0
$$547$$ − 1.41531e9i − 0.0158089i −0.999969 0.00790445i $$-0.997484\pi$$
0.999969 0.00790445i $$-0.00251609\pi$$
$$548$$ 0 0
$$549$$ −5.04758e10 −0.555641
$$550$$ 0 0
$$551$$ 2.49833e9i 0.0271046i
$$552$$ 0 0
$$553$$ −5.02383e10 −0.537198
$$554$$ 0 0
$$555$$ 2.11624e11i 2.23046i
$$556$$ 0 0
$$557$$ 1.37543e11 1.42895 0.714475 0.699661i $$-0.246666\pi$$
0.714475 + 0.699661i $$0.246666\pi$$
$$558$$ 0 0
$$559$$ 3.24270e10i 0.332093i
$$560$$ 0 0
$$561$$ 1.35000e11 1.36296
$$562$$ 0 0
$$563$$ 1.06415e11i 1.05918i 0.848255 + 0.529589i $$0.177654\pi$$
−0.848255 + 0.529589i $$0.822346\pi$$
$$564$$ 0 0
$$565$$ −3.36018e10 −0.329738
$$566$$ 0 0
$$567$$ − 7.52431e10i − 0.728005i
$$568$$ 0 0
$$569$$ 4.02429e10 0.383919 0.191960 0.981403i $$-0.438516\pi$$
0.191960 + 0.981403i $$0.438516\pi$$
$$570$$ 0 0
$$571$$ − 1.50341e11i − 1.41427i −0.707077 0.707137i $$-0.749986\pi$$
0.707077 0.707137i $$-0.250014\pi$$
$$572$$ 0 0
$$573$$ 9.91667e10 0.919914
$$574$$ 0 0
$$575$$ 4.39432e9i 0.0401994i
$$576$$ 0 0
$$577$$ 4.96477e9 0.0447915 0.0223958 0.999749i $$-0.492871\pi$$
0.0223958 + 0.999749i $$0.492871\pi$$
$$578$$ 0 0
$$579$$ − 1.17499e11i − 1.04549i
$$580$$ 0 0
$$581$$ −7.26862e10 −0.637892
$$582$$ 0 0
$$583$$ − 1.52372e10i − 0.131895i
$$584$$ 0 0
$$585$$ 1.14215e10 0.0975216
$$586$$ 0 0
$$587$$ − 1.53440e11i − 1.29237i −0.763181 0.646185i $$-0.776363\pi$$
0.763181 0.646185i $$-0.223637\pi$$
$$588$$ 0 0
$$589$$ 1.32388e9 0.0109999
$$590$$ 0 0
$$591$$ 1.70402e11i 1.39677i
$$592$$ 0 0
$$593$$ 2.06036e11 1.66619 0.833094 0.553131i $$-0.186567\pi$$
0.833094 + 0.553131i $$0.186567\pi$$
$$594$$ 0 0
$$595$$ − 6.23689e10i − 0.497623i
$$596$$ 0 0
$$597$$ 2.48837e11 1.95892
$$598$$ 0 0
$$599$$ − 2.30634e11i − 1.79150i −0.444558 0.895750i $$-0.646639\pi$$
0.444558 0.895750i $$-0.353361\pi$$
$$600$$ 0 0
$$601$$ 1.01422e11 0.777382 0.388691 0.921368i $$-0.372927\pi$$
0.388691 + 0.921368i $$0.372927\pi$$
$$602$$ 0 0
$$603$$ 5.22236e10i 0.395001i
$$604$$ 0 0
$$605$$ 7.76795e10 0.579809
$$606$$ 0 0
$$607$$ − 1.97883e11i − 1.45765i −0.684700 0.728825i $$-0.740067\pi$$
0.684700 0.728825i $$-0.259933\pi$$
$$608$$ 0 0
$$609$$ 1.79224e10 0.130294
$$610$$ 0 0
$$611$$ 4.17158e10i 0.299320i
$$612$$ 0 0
$$613$$ −1.27158e11 −0.900538 −0.450269 0.892893i $$-0.648672\pi$$
−0.450269 + 0.892893i $$0.648672\pi$$
$$614$$ 0 0
$$615$$ 1.30855e11i 0.914723i
$$616$$ 0 0
$$617$$ −5.06702e10 −0.349632 −0.174816 0.984601i $$-0.555933\pi$$
−0.174816 + 0.984601i $$0.555933\pi$$
$$618$$ 0 0
$$619$$ 7.06748e10i 0.481395i 0.970600 + 0.240698i $$0.0773762\pi$$
−0.970600 + 0.240698i $$0.922624\pi$$
$$620$$ 0 0
$$621$$ −7.43769e10 −0.500117
$$622$$ 0 0
$$623$$ − 1.16561e11i − 0.773748i
$$624$$ 0 0
$$625$$ −1.45008e11 −0.950327
$$626$$ 0 0
$$627$$ 3.59885e10i 0.232859i
$$628$$ 0 0
$$629$$ 2.53771e11 1.62121
$$630$$ 0 0
$$631$$ 1.65273e11i 1.04252i 0.853399 + 0.521259i $$0.174537\pi$$
−0.853399 + 0.521259i $$0.825463\pi$$
$$632$$ 0 0
$$633$$ −1.46657e11 −0.913454
$$634$$ 0 0
$$635$$ − 1.56868e11i − 0.964804i
$$636$$ 0 0
$$637$$ 2.08294e10 0.126508
$$638$$ 0 0
$$639$$ 4.09405e9i 0.0245556i
$$640$$ 0 0
$$641$$ 1.12013e11 0.663490 0.331745 0.943369i $$-0.392363\pi$$
0.331745 + 0.943369i $$0.392363\pi$$
$$642$$ 0 0
$$643$$ 2.65913e11i 1.55559i 0.628518 + 0.777795i $$0.283662\pi$$
−0.628518 + 0.777795i $$0.716338\pi$$
$$644$$ 0 0
$$645$$ −3.61328e11 −2.08767
$$646$$ 0 0
$$647$$ 2.71996e11i 1.55219i 0.630614 + 0.776097i $$0.282803\pi$$
−0.630614 + 0.776097i $$0.717197\pi$$
$$648$$ 0 0
$$649$$ −6.88669e10 −0.388179
$$650$$ 0 0
$$651$$ − 9.49716e9i − 0.0528774i
$$652$$ 0 0
$$653$$ −3.03789e11 −1.67078 −0.835391 0.549656i $$-0.814759\pi$$
−0.835391 + 0.549656i $$0.814759\pi$$
$$654$$ 0 0
$$655$$ − 1.90427e11i − 1.03458i
$$656$$ 0 0
$$657$$ 1.95988e10 0.105189
$$658$$ 0 0
$$659$$ − 4.18575e10i − 0.221938i −0.993824 0.110969i $$-0.964605\pi$$
0.993824 0.110969i $$-0.0353954\pi$$
$$660$$ 0 0
$$661$$ 2.46529e11 1.29141 0.645703 0.763589i $$-0.276565\pi$$
0.645703 + 0.763589i $$0.276565\pi$$
$$662$$ 0 0
$$663$$ − 3.99482e10i − 0.206749i
$$664$$ 0 0
$$665$$ 1.66264e10 0.0850179
$$666$$ 0 0
$$667$$ − 3.04155e10i − 0.153671i
$$668$$ 0 0
$$669$$ −1.47802e11 −0.737862
$$670$$ 0 0
$$671$$ 2.72584e11i 1.34465i
$$672$$ 0 0
$$673$$ −3.15336e11 −1.53714 −0.768569 0.639767i $$-0.779031\pi$$
−0.768569 + 0.639767i $$0.779031\pi$$
$$674$$ 0 0
$$675$$ 5.80849e9i 0.0279800i
$$676$$ 0 0
$$677$$ 2.47236e10 0.117695 0.0588475 0.998267i $$-0.481257\pi$$
0.0588475 + 0.998267i $$0.481257\pi$$
$$678$$ 0 0
$$679$$ 1.68731e11i 0.793811i
$$680$$ 0 0
$$681$$ 7.49454e10 0.348463
$$682$$ 0 0
$$683$$ − 7.20843e10i − 0.331251i −0.986189 0.165626i $$-0.947036\pi$$
0.986189 0.165626i $$-0.0529644\pi$$
$$684$$ 0 0
$$685$$ −1.35408e11 −0.615009
$$686$$ 0 0
$$687$$ − 2.84556e11i − 1.27744i
$$688$$ 0 0
$$689$$ −4.50887e9 −0.0200074
$$690$$ 0 0
$$691$$ 2.95424e11i 1.29578i 0.761732 + 0.647892i $$0.224349\pi$$
−0.761732 + 0.647892i $$0.775651\pi$$
$$692$$ 0 0
$$693$$ 8.85139e10 0.383776
$$694$$ 0 0
$$695$$ − 1.79968e11i − 0.771360i
$$696$$ 0 0
$$697$$ 1.56916e11 0.664867
$$698$$ 0 0
$$699$$ − 2.20444e10i − 0.0923400i
$$700$$ 0 0
$$701$$ 2.87925e11 1.19236 0.596180 0.802851i $$-0.296685\pi$$
0.596180 + 0.802851i $$0.296685\pi$$
$$702$$ 0 0
$$703$$ 6.76504e10i 0.276980i
$$704$$ 0 0
$$705$$ −4.64831e11 −1.88165
$$706$$ 0 0
$$707$$ − 3.87834e10i − 0.155227i
$$708$$ 0 0
$$709$$ −2.51685e11 −0.996030 −0.498015 0.867168i $$-0.665938\pi$$
−0.498015 + 0.867168i $$0.665938\pi$$
$$710$$ 0 0
$$711$$ − 1.22931e11i − 0.481042i
$$712$$ 0 0
$$713$$ −1.61174e10 −0.0623643
$$714$$ 0 0
$$715$$ − 6.16795e10i − 0.236003i
$$716$$ 0 0
$$717$$ −4.04170e11 −1.52928
$$718$$ 0 0
$$719$$ − 1.38856e11i − 0.519574i −0.965666 0.259787i $$-0.916348\pi$$
0.965666 0.259787i $$-0.0836524\pi$$
$$720$$ 0 0
$$721$$ −1.46184e11 −0.540953
$$722$$ 0 0
$$723$$ − 6.17489e11i − 2.25983i
$$724$$ 0 0
$$725$$ −2.37531e9 −0.00859743
$$726$$ 0 0
$$727$$ − 1.79083e11i − 0.641088i −0.947234 0.320544i $$-0.896134\pi$$
0.947234 0.320544i $$-0.103866\pi$$
$$728$$ 0 0
$$729$$ −2.14381e10 −0.0759061
$$730$$ 0 0
$$731$$ 4.33289e11i 1.51743i
$$732$$ 0 0
$$733$$ −2.17618e11 −0.753839 −0.376920 0.926246i $$-0.623017\pi$$
−0.376920 + 0.926246i $$0.623017\pi$$
$$734$$ 0 0
$$735$$ 2.32098e11i 0.795285i
$$736$$ 0 0
$$737$$ 2.82023e11 0.955904
$$738$$ 0 0
$$739$$ − 4.84950e11i − 1.62599i −0.582268 0.812997i $$-0.697834\pi$$
0.582268 0.812997i $$-0.302166\pi$$
$$740$$ 0 0
$$741$$ 1.06494e10 0.0353227
$$742$$ 0 0
$$743$$ 2.03509e11i 0.667771i 0.942614 + 0.333886i $$0.108360\pi$$
−0.942614 + 0.333886i $$0.891640\pi$$
$$744$$ 0 0
$$745$$ 2.46198e11 0.799206
$$746$$ 0 0
$$747$$ − 1.77860e11i − 0.571210i
$$748$$ 0 0
$$749$$ 1.40346e11 0.445937
$$750$$ 0 0
$$751$$ 2.34693e11i 0.737804i 0.929468 + 0.368902i $$0.120266\pi$$
−0.929468 + 0.368902i $$0.879734\pi$$
$$752$$ 0 0
$$753$$ −5.20950e11 −1.62038
$$754$$ 0 0
$$755$$ 5.10561e11i 1.57130i
$$756$$ 0 0
$$757$$ 3.84882e11 1.17204 0.586022 0.810295i $$-0.300693\pi$$
0.586022 + 0.810295i $$0.300693\pi$$
$$758$$ 0 0
$$759$$ − 4.38136e11i − 1.32021i
$$760$$ 0 0
$$761$$ 2.39209e11 0.713244 0.356622 0.934249i $$-0.383928\pi$$
0.356622 + 0.934249i $$0.383928\pi$$
$$762$$ 0 0
$$763$$ 8.26340e10i 0.243815i
$$764$$ 0 0
$$765$$ 1.52614e11 0.445604
$$766$$ 0 0
$$767$$ 2.03786e10i 0.0588833i
$$768$$ 0 0
$$769$$ 2.08457e11 0.596089 0.298045 0.954552i $$-0.403666\pi$$
0.298045 + 0.954552i $$0.403666\pi$$
$$770$$ 0 0
$$771$$ 6.13202e11i 1.73535i
$$772$$ 0 0
$$773$$ 5.54469e10 0.155296 0.0776478 0.996981i $$-0.475259\pi$$
0.0776478 + 0.996981i $$0.475259\pi$$
$$774$$ 0 0
$$775$$ 1.25869e9i 0.00348909i
$$776$$ 0 0
$$777$$ 4.85306e11 1.33147
$$778$$ 0 0
$$779$$ 4.18307e10i 0.113591i
$$780$$ 0 0
$$781$$ 2.21091e10 0.0594246
$$782$$ 0 0
$$783$$ − 4.02039e10i − 0.106960i
$$784$$ 0 0
$$785$$ −1.65504e11 −0.435844
$$786$$ 0 0
$$787$$ − 4.05908e11i − 1.05811i −0.848589 0.529053i $$-0.822547\pi$$
0.848589 0.529053i $$-0.177453\pi$$
$$788$$ 0 0
$$789$$ 6.96358e11 1.79690
$$790$$ 0 0
$$791$$ 7.70572e10i 0.196837i
$$792$$ 0 0
$$793$$ 8.06610e10 0.203972
$$794$$ 0 0
$$795$$ − 5.02414e10i − 0.125775i
$$796$$ 0 0
$$797$$ 3.09015e11 0.765855 0.382927 0.923778i $$-0.374916\pi$$
0.382927 + 0.923778i $$0.374916\pi$$
$$798$$ 0 0
$$799$$ 5.57406e11i 1.36768i
$$800$$ 0 0
$$801$$ 2.85219e11 0.692864
$$802$$ 0 0
$$803$$ − 1.05839e11i − 0.254557i
$$804$$ 0 0
$$805$$ −2.02415e11 −0.482014
$$806$$ 0 0
$$807$$ 2.70504e11i 0.637792i
$$808$$ 0 0
$$809$$ 4.77958e11 1.11582 0.557912 0.829900i $$-0.311603\pi$$
0.557912 + 0.829900i $$0.311603\pi$$
$$810$$ 0 0
$$811$$ 6.37503e11i 1.47366i 0.676075 + 0.736832i $$0.263679\pi$$
−0.676075 + 0.736832i $$0.736321\pi$$
$$812$$ 0 0
$$813$$ −7.98393e11 −1.82749
$$814$$ 0 0
$$815$$ 3.52890e11i 0.799851i
$$816$$ 0 0
$$817$$ −1.15506e11 −0.259250
$$818$$ 0 0
$$819$$ − 2.61924e10i − 0.0582155i
$$820$$ 0 0
$$821$$ 8.43824e11 1.85729 0.928644 0.370973i $$-0.120976\pi$$
0.928644 + 0.370973i $$0.120976\pi$$
$$822$$ 0 0
$$823$$ − 2.60916e11i − 0.568723i −0.958717 0.284362i $$-0.908218\pi$$
0.958717 0.284362i $$-0.0917817\pi$$
$$824$$ 0 0
$$825$$ −3.42164e10 −0.0738616
$$826$$ 0 0
$$827$$ − 2.78675e11i − 0.595765i −0.954602 0.297883i $$-0.903720\pi$$
0.954602 0.297883i $$-0.0962804\pi$$
$$828$$ 0 0
$$829$$ −4.75156e11 −1.00605 −0.503023 0.864273i $$-0.667779\pi$$
−0.503023 + 0.864273i $$0.667779\pi$$
$$830$$ 0 0
$$831$$ − 8.22306e11i − 1.72437i
$$832$$ 0 0
$$833$$ 2.78322e11 0.578053
$$834$$ 0 0
$$835$$ − 2.85552e11i − 0.587408i
$$836$$ 0 0
$$837$$ −2.13043e10 −0.0434075
$$838$$ 0 0
$$839$$ − 3.35440e11i − 0.676966i −0.940973 0.338483i $$-0.890086\pi$$
0.940973 0.338483i $$-0.109914\pi$$
$$840$$ 0 0
$$841$$ −4.83806e11 −0.967134
$$842$$ 0 0
$$843$$ − 3.07858e11i − 0.609594i
$$844$$ 0 0
$$845$$ 4.79344e11 0.940200
$$846$$ 0 0
$$847$$ − 1.78138e11i − 0.346117i
$$848$$ 0 0
$$849$$ −1.17019e11 −0.225229
$$850$$ 0 0
$$851$$ − 8.23600e11i − 1.57036i
$$852$$ 0 0
$$853$$ 9.75408e10 0.184243 0.0921213 0.995748i $$-0.470635\pi$$
0.0921213 + 0.995748i $$0.470635\pi$$
$$854$$ 0 0
$$855$$ 4.06840e10i 0.0761306i
$$856$$ 0 0
$$857$$ −7.94769e10 −0.147339 −0.0736695 0.997283i $$-0.523471\pi$$
−0.0736695 + 0.997283i $$0.523471\pi$$
$$858$$ 0 0
$$859$$ 4.15618e11i 0.763347i 0.924297 + 0.381673i $$0.124652\pi$$
−0.924297 + 0.381673i $$0.875348\pi$$
$$860$$ 0 0
$$861$$ 3.00083e11 0.546045
$$862$$ 0 0
$$863$$ − 4.80012e11i − 0.865383i −0.901542 0.432692i $$-0.857564\pi$$
0.901542 0.432692i $$-0.142436\pi$$
$$864$$ 0 0
$$865$$ −1.25780e11 −0.224671
$$866$$ 0 0
$$867$$ 1.63230e11i 0.288884i
$$868$$ 0 0
$$869$$ −6.63863e11 −1.16412
$$870$$ 0 0
$$871$$ − 8.34540e10i − 0.145002i
$$872$$ 0 0
$$873$$ −4.12879e11 −0.710830
$$874$$ 0 0
$$875$$ 3.49134e11i 0.595608i
$$876$$ 0 0
$$877$$ −2.74155e11 −0.463444 −0.231722 0.972782i $$-0.574436\pi$$
−0.231722 + 0.972782i $$0.574436\pi$$
$$878$$ 0 0
$$879$$ 4.80595e11i 0.805051i
$$880$$ 0 0
$$881$$ 8.01838e11 1.33101 0.665507 0.746391i $$-0.268215\pi$$
0.665507 + 0.746391i $$0.268215\pi$$
$$882$$ 0 0
$$883$$ 9.95008e11i 1.63676i 0.574681 + 0.818378i $$0.305126\pi$$
−0.574681 + 0.818378i $$0.694874\pi$$
$$884$$ 0 0
$$885$$ −2.27075e11 −0.370165
$$886$$ 0 0
$$887$$ 5.46038e11i 0.882122i 0.897477 + 0.441061i $$0.145398\pi$$
−0.897477 + 0.441061i $$0.854602\pi$$
$$888$$ 0 0
$$889$$ −3.59736e11 −0.575940
$$890$$ 0 0
$$891$$ − 9.94283e11i − 1.57761i
$$892$$ 0 0
$$893$$ −1.48594e11 −0.233665
$$894$$ 0 0
$$895$$ 8.65656e10i 0.134913i
$$896$$ 0 0
$$897$$ −1.29650e11 −0.200264
$$898$$ 0 0
$$899$$ − 8.71212e9i − 0.0133378i
$$900$$ 0 0
$$901$$ −6.02474e10 −0.0914195
$$902$$ 0 0
$$903$$ 8.28613e11i 1.24624i
$$904$$ 0 0
$$905$$ 2.94365e11 0.438826
$$906$$ 0 0
$$907$$ − 6.55018e11i − 0.967886i −0.875100 0.483943i $$-0.839204\pi$$
0.875100 0.483943i $$-0.160796\pi$$
$$908$$ 0 0
$$909$$ 9.49014e10 0.139001
$$910$$ 0 0
$$911$$ − 1.19425e11i − 0.173389i −0.996235 0.0866943i $$-0.972370\pi$$
0.996235 0.0866943i $$-0.0276304\pi$$
$$912$$ 0 0
$$913$$ −9.60496e11 −1.38233
$$914$$ 0 0
$$915$$ 8.98791e11i 1.28225i
$$916$$ 0 0
$$917$$ −4.36696e11 −0.617592
$$918$$ 0 0
$$919$$ 5.33989e10i 0.0748635i 0.999299 + 0.0374318i $$0.0119177\pi$$
−0.999299 + 0.0374318i $$0.988082\pi$$
$$920$$ 0 0
$$921$$ 3.48897e11 0.484907
$$922$$ 0 0
$$923$$ − 6.54235e9i − 0.00901420i
$$924$$ 0 0
$$925$$ −6.43194e10 −0.0878567
$$926$$ 0 0
$$927$$ − 3.57707e11i − 0.484404i
$$928$$ 0 0
$$929$$ −6.30991e11 −0.847150 −0.423575 0.905861i $$-0.639225\pi$$
−0.423575 + 0.905861i $$0.639225\pi$$
$$930$$ 0 0
$$931$$ 7.41953e10i 0.0987593i
$$932$$ 0 0
$$933$$ −1.29703e12 −1.71168
$$934$$ 0 0
$$935$$ − 8.24161e11i − 1.07836i
$$936$$ 0 0
$$937$$ −8.41436e11 −1.09160 −0.545799 0.837916i $$-0.683774\pi$$
−0.545799 + 0.837916i $$0.683774\pi$$
$$938$$ 0 0
$$939$$ 6.30660e11i 0.811209i
$$940$$ 0 0
$$941$$ 4.52935e11 0.577666 0.288833 0.957379i $$-0.406733\pi$$
0.288833 + 0.957379i $$0.406733\pi$$
$$942$$ 0 0
$$943$$ − 5.09262e11i − 0.644013i
$$944$$ 0 0
$$945$$ −2.67556e11 −0.335497
$$946$$ 0 0
$$947$$ 1.00309e12i 1.24721i 0.781739 + 0.623605i $$0.214333\pi$$
−0.781739 + 0.623605i $$0.785667\pi$$
$$948$$ 0 0
$$949$$ −3.13192e10 −0.0386141
$$950$$ 0 0
$$951$$ 1.65769e12i 2.02666i
$$952$$ 0 0
$$953$$ −1.39040e12 −1.68565 −0.842826 0.538186i $$-0.819110\pi$$
−0.842826 + 0.538186i $$0.819110\pi$$
$$954$$ 0 0
$$955$$ − 6.05401e11i − 0.727830i
$$956$$ 0 0
$$957$$ 2.36831e11 0.282352
$$958$$ 0 0
$$959$$ 3.10523e11i 0.367130i
$$960$$ 0 0
$$961$$ 8.48274e11 0.994587
$$962$$ 0 0
$$963$$ 3.43421e11i 0.399321i
$$964$$ 0 0
$$965$$ −7.17318e11 −0.827185
$$966$$ 0 0
$$967$$ − 9.01843e11i − 1.03140i −0.856771 0.515698i $$-0.827533\pi$$
0.856771 0.515698i $$-0.172467\pi$$
$$968$$ 0 0
$$969$$ 1.42297e11 0.161399
$$970$$ 0 0
$$971$$ − 1.28411e12i − 1.44452i −0.691621 0.722261i $$-0.743103\pi$$
0.691621 0.722261i $$-0.256897\pi$$
$$972$$ 0 0
$$973$$ −4.12712e11 −0.460464
$$974$$ 0 0
$$975$$ 1.01251e10i 0.0112042i
$$976$$ 0 0
$$977$$ 1.20120e12 1.31837 0.659183 0.751983i $$-0.270902\pi$$
0.659183 + 0.751983i $$0.270902\pi$$
$$978$$ 0 0
$$979$$ − 1.54026e12i − 1.67674i
$$980$$ 0 0
$$981$$ −2.02202e11 −0.218328
$$982$$ 0 0
$$983$$ 3.77388e11i 0.404179i 0.979367 + 0.202090i $$0.0647733\pi$$
−0.979367 + 0.202090i $$0.935227\pi$$
$$984$$ 0 0
$$985$$ 1.04028e12 1.10511
$$986$$ 0 0
$$987$$ 1.06597e12i 1.12325i
$$988$$ 0 0
$$989$$ 1.40622e12 1.46983
$$990$$ 0 0
$$991$$ − 3.35563e11i − 0.347920i −0.984753 0.173960i $$-0.944344\pi$$
0.984753 0.173960i $$-0.0556563\pi$$
$$992$$ 0 0
$$993$$ −5.48200e11 −0.563822
$$994$$ 0 0
$$995$$ − 1.51912e12i − 1.54989i
$$996$$ 0 0
$$997$$ 1.03163e12 1.04410 0.522050 0.852915i $$-0.325167\pi$$
0.522050 + 0.852915i $$0.325167\pi$$
$$998$$ 0 0
$$999$$ − 1.08865e12i − 1.09302i
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 64.9.c.b.63.1 2
4.3 odd 2 inner 64.9.c.b.63.2 2
8.3 odd 2 4.9.b.b.3.1 2
8.5 even 2 4.9.b.b.3.2 yes 2
16.3 odd 4 256.9.d.e.127.2 4
16.5 even 4 256.9.d.e.127.1 4
16.11 odd 4 256.9.d.e.127.3 4
16.13 even 4 256.9.d.e.127.4 4
24.5 odd 2 36.9.d.b.19.1 2
24.11 even 2 36.9.d.b.19.2 2
40.3 even 4 100.9.d.b.99.2 4
40.13 odd 4 100.9.d.b.99.4 4
40.19 odd 2 100.9.b.c.51.2 2
40.27 even 4 100.9.d.b.99.3 4
40.29 even 2 100.9.b.c.51.1 2
40.37 odd 4 100.9.d.b.99.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
4.9.b.b.3.1 2 8.3 odd 2
4.9.b.b.3.2 yes 2 8.5 even 2
36.9.d.b.19.1 2 24.5 odd 2
36.9.d.b.19.2 2 24.11 even 2
64.9.c.b.63.1 2 1.1 even 1 trivial
64.9.c.b.63.2 2 4.3 odd 2 inner
100.9.b.c.51.1 2 40.29 even 2
100.9.b.c.51.2 2 40.19 odd 2
100.9.d.b.99.1 4 40.37 odd 4
100.9.d.b.99.2 4 40.3 even 4
100.9.d.b.99.3 4 40.27 even 4
100.9.d.b.99.4 4 40.13 odd 4
256.9.d.e.127.1 4 16.5 even 4
256.9.d.e.127.2 4 16.3 odd 4
256.9.d.e.127.3 4 16.11 odd 4
256.9.d.e.127.4 4 16.13 even 4