# Properties

 Label 16.22.a.e.1.2 Level $16$ Weight $22$ Character 16.1 Self dual yes Analytic conductor $44.716$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,22,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 16.a (trivial)

## 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: yes Analytic conductor: $$44.7163750859$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{358549})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 89637$$ x^2 - x - 89637 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{7}\cdot 3$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-298.895$$ of defining polynomial Character $$\chi$$ $$=$$ 16.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+167684. q^{3} +3.35342e6 q^{5} -7.07779e8 q^{7} +1.76574e10 q^{9} +O(q^{10})$$ $$q+167684. q^{3} +3.35342e6 q^{5} -7.07779e8 q^{7} +1.76574e10 q^{9} -8.75119e10 q^{11} -7.41212e11 q^{13} +5.62314e11 q^{15} -6.82853e12 q^{17} -5.17730e13 q^{19} -1.18683e14 q^{21} +3.13428e14 q^{23} -4.65592e14 q^{25} +1.20683e15 q^{27} +1.46400e15 q^{29} +6.42150e15 q^{31} -1.46743e16 q^{33} -2.37348e15 q^{35} -6.93987e15 q^{37} -1.24289e17 q^{39} +3.25339e16 q^{41} -4.37536e16 q^{43} +5.92128e16 q^{45} +5.34426e16 q^{47} -5.75946e16 q^{49} -1.14503e18 q^{51} -1.19249e18 q^{53} -2.93464e17 q^{55} -8.68148e18 q^{57} +4.48047e17 q^{59} -9.05611e17 q^{61} -1.24976e19 q^{63} -2.48560e18 q^{65} +6.06820e17 q^{67} +5.25568e19 q^{69} +2.18611e19 q^{71} -6.65250e19 q^{73} -7.80721e19 q^{75} +6.19391e19 q^{77} +1.81938e18 q^{79} +1.76631e19 q^{81} +2.58951e20 q^{83} -2.28989e19 q^{85} +2.45490e20 q^{87} -1.80968e20 q^{89} +5.24614e20 q^{91} +1.07678e21 q^{93} -1.73617e20 q^{95} +4.09632e20 q^{97} -1.54524e21 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 105432 q^{3} + 2108140 q^{5} - 444771792 q^{7} + 11072347578 q^{9}+O(q^{10})$$ 2 * q + 105432 * q^3 + 2108140 * q^5 - 444771792 * q^7 + 11072347578 * q^9 $$2 q + 105432 q^{3} + 2108140 q^{5} - 444771792 q^{7} + 11072347578 q^{9} - 53806403320 q^{11} - 490366676932 q^{13} + 639834721680 q^{15} - 6593864672092 q^{17} - 19302397925320 q^{19} - 135055584824256 q^{21} + 409737865776272 q^{23} - 940878149007650 q^{25} + 22\!\cdots\!56 q^{27}+ \cdots - 17\!\cdots\!04 q^{99}+O(q^{100})$$ 2 * q + 105432 * q^3 + 2108140 * q^5 - 444771792 * q^7 + 11072347578 * q^9 - 53806403320 * q^11 - 490366676932 * q^13 + 639834721680 * q^15 - 6593864672092 * q^17 - 19302397925320 * q^19 - 135055584824256 * q^21 + 409737865776272 * q^23 - 940878149007650 * q^25 + 2267939450073456 * q^27 - 2404787522145060 * q^29 + 8689907170559168 * q^31 - 16772541164782752 * q^33 - 2701000503537120 * q^35 - 2186204096251860 * q^37 - 139904623071296688 * q^39 + 68178038573558676 * q^41 - 264529652266004024 * q^43 + 67413140092548540 * q^45 - 426494411558622432 * q^47 - 546967577640131534 * q^49 - 1159640456555021520 * q^51 - 3055980275589518132 * q^53 - 335437371694825040 * q^55 - 10702822880368260192 * q^57 - 783424997522814424 * q^59 - 7177279049078597092 * q^61 - 14229493501717343376 * q^63 - 2797969869756700760 * q^65 - 16674123174011538088 * q^67 + 46561318352256083136 * q^69 - 9448263149848716368 * q^71 - 11586140334503007532 * q^73 - 48484754640473875800 * q^75 + 70803926825553273024 * q^77 + 85280702218715897824 * q^79 + 20489906394726573618 * q^81 + 381814622040086245816 * q^83 - 23191160664932407400 * q^85 + 486328037598631978704 * q^87 - 59742932430695979660 * q^89 + 590588534777956016544 * q^91 + 935567489996727394560 * q^93 - 214051632007883873840 * q^95 + 783394660926711950788 * q^97 - 1767190682286719892504 * q^99

## 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
$$3$$ 167684. 1.63952 0.819761 0.572705i $$-0.194106\pi$$
0.819761 + 0.572705i $$0.194106\pi$$
$$4$$ 0 0
$$5$$ 3.35342e6 0.153569 0.0767844 0.997048i $$-0.475535\pi$$
0.0767844 + 0.997048i $$0.475535\pi$$
$$6$$ 0 0
$$7$$ −7.07779e8 −0.947040 −0.473520 0.880783i $$-0.657017\pi$$
−0.473520 + 0.880783i $$0.657017\pi$$
$$8$$ 0 0
$$9$$ 1.76574e10 1.68803
$$10$$ 0 0
$$11$$ −8.75119e10 −1.01729 −0.508644 0.860977i $$-0.669853\pi$$
−0.508644 + 0.860977i $$0.669853\pi$$
$$12$$ 0 0
$$13$$ −7.41212e11 −1.49120 −0.745602 0.666391i $$-0.767838\pi$$
−0.745602 + 0.666391i $$0.767838\pi$$
$$14$$ 0 0
$$15$$ 5.62314e11 0.251780
$$16$$ 0 0
$$17$$ −6.82853e12 −0.821511 −0.410756 0.911746i $$-0.634735\pi$$
−0.410756 + 0.911746i $$0.634735\pi$$
$$18$$ 0 0
$$19$$ −5.17730e13 −1.93727 −0.968635 0.248487i $$-0.920067\pi$$
−0.968635 + 0.248487i $$0.920067\pi$$
$$20$$ 0 0
$$21$$ −1.18683e14 −1.55269
$$22$$ 0 0
$$23$$ 3.13428e14 1.57760 0.788798 0.614653i $$-0.210704\pi$$
0.788798 + 0.614653i $$0.210704\pi$$
$$24$$ 0 0
$$25$$ −4.65592e14 −0.976417
$$26$$ 0 0
$$27$$ 1.20683e15 1.12805
$$28$$ 0 0
$$29$$ 1.46400e15 0.646195 0.323097 0.946366i $$-0.395276\pi$$
0.323097 + 0.946366i $$0.395276\pi$$
$$30$$ 0 0
$$31$$ 6.42150e15 1.40714 0.703572 0.710624i $$-0.251587\pi$$
0.703572 + 0.710624i $$0.251587\pi$$
$$32$$ 0 0
$$33$$ −1.46743e16 −1.66787
$$34$$ 0 0
$$35$$ −2.37348e15 −0.145436
$$36$$ 0 0
$$37$$ −6.93987e15 −0.237265 −0.118632 0.992938i $$-0.537851\pi$$
−0.118632 + 0.992938i $$0.537851\pi$$
$$38$$ 0 0
$$39$$ −1.24289e17 −2.44486
$$40$$ 0 0
$$41$$ 3.25339e16 0.378534 0.189267 0.981926i $$-0.439389\pi$$
0.189267 + 0.981926i $$0.439389\pi$$
$$42$$ 0 0
$$43$$ −4.37536e16 −0.308741 −0.154371 0.988013i $$-0.549335\pi$$
−0.154371 + 0.988013i $$0.549335\pi$$
$$44$$ 0 0
$$45$$ 5.92128e16 0.259230
$$46$$ 0 0
$$47$$ 5.34426e16 0.148204 0.0741020 0.997251i $$-0.476391\pi$$
0.0741020 + 0.997251i $$0.476391\pi$$
$$48$$ 0 0
$$49$$ −5.75946e16 −0.103115
$$50$$ 0 0
$$51$$ −1.14503e18 −1.34689
$$52$$ 0 0
$$53$$ −1.19249e18 −0.936611 −0.468306 0.883567i $$-0.655135\pi$$
−0.468306 + 0.883567i $$0.655135\pi$$
$$54$$ 0 0
$$55$$ −2.93464e17 −0.156224
$$56$$ 0 0
$$57$$ −8.68148e18 −3.17620
$$58$$ 0 0
$$59$$ 4.48047e17 0.114124 0.0570620 0.998371i $$-0.481827\pi$$
0.0570620 + 0.998371i $$0.481827\pi$$
$$60$$ 0 0
$$61$$ −9.05611e17 −0.162547 −0.0812734 0.996692i $$-0.525899\pi$$
−0.0812734 + 0.996692i $$0.525899\pi$$
$$62$$ 0 0
$$63$$ −1.24976e19 −1.59864
$$64$$ 0 0
$$65$$ −2.48560e18 −0.229003
$$66$$ 0 0
$$67$$ 6.06820e17 0.0406700 0.0203350 0.999793i $$-0.493527\pi$$
0.0203350 + 0.999793i $$0.493527\pi$$
$$68$$ 0 0
$$69$$ 5.25568e19 2.58650
$$70$$ 0 0
$$71$$ 2.18611e19 0.797003 0.398501 0.917168i $$-0.369530\pi$$
0.398501 + 0.917168i $$0.369530\pi$$
$$72$$ 0 0
$$73$$ −6.65250e19 −1.81174 −0.905868 0.423559i $$-0.860781\pi$$
−0.905868 + 0.423559i $$0.860781\pi$$
$$74$$ 0 0
$$75$$ −7.80721e19 −1.60086
$$76$$ 0 0
$$77$$ 6.19391e19 0.963413
$$78$$ 0 0
$$79$$ 1.81938e18 0.0216191 0.0108096 0.999942i $$-0.496559\pi$$
0.0108096 + 0.999942i $$0.496559\pi$$
$$80$$ 0 0
$$81$$ 1.76631e19 0.161426
$$82$$ 0 0
$$83$$ 2.58951e20 1.83188 0.915940 0.401316i $$-0.131447\pi$$
0.915940 + 0.401316i $$0.131447\pi$$
$$84$$ 0 0
$$85$$ −2.28989e19 −0.126159
$$86$$ 0 0
$$87$$ 2.45490e20 1.05945
$$88$$ 0 0
$$89$$ −1.80968e20 −0.615187 −0.307594 0.951518i $$-0.599524\pi$$
−0.307594 + 0.951518i $$0.599524\pi$$
$$90$$ 0 0
$$91$$ 5.24614e20 1.41223
$$92$$ 0 0
$$93$$ 1.07678e21 2.30704
$$94$$ 0 0
$$95$$ −1.73617e20 −0.297504
$$96$$ 0 0
$$97$$ 4.09632e20 0.564016 0.282008 0.959412i $$-0.409000\pi$$
0.282008 + 0.959412i $$0.409000\pi$$
$$98$$ 0 0
$$99$$ −1.54524e21 −1.71722
$$100$$ 0 0
$$101$$ −1.49956e21 −1.35080 −0.675398 0.737454i $$-0.736028\pi$$
−0.675398 + 0.737454i $$0.736028\pi$$
$$102$$ 0 0
$$103$$ −5.14179e20 −0.376984 −0.188492 0.982075i $$-0.560360\pi$$
−0.188492 + 0.982075i $$0.560360\pi$$
$$104$$ 0 0
$$105$$ −3.97994e20 −0.238445
$$106$$ 0 0
$$107$$ −1.11062e21 −0.545801 −0.272900 0.962042i $$-0.587983\pi$$
−0.272900 + 0.962042i $$0.587983\pi$$
$$108$$ 0 0
$$109$$ −1.75592e21 −0.710437 −0.355219 0.934783i $$-0.615594\pi$$
−0.355219 + 0.934783i $$0.615594\pi$$
$$110$$ 0 0
$$111$$ −1.16370e21 −0.389001
$$112$$ 0 0
$$113$$ 5.38798e21 1.49315 0.746573 0.665303i $$-0.231698\pi$$
0.746573 + 0.665303i $$0.231698\pi$$
$$114$$ 0 0
$$115$$ 1.05106e21 0.242270
$$116$$ 0 0
$$117$$ −1.30879e22 −2.51721
$$118$$ 0 0
$$119$$ 4.83309e21 0.778004
$$120$$ 0 0
$$121$$ 2.58090e20 0.0348758
$$122$$ 0 0
$$123$$ 5.45540e21 0.620616
$$124$$ 0 0
$$125$$ −3.16036e21 −0.303516
$$126$$ 0 0
$$127$$ −1.11104e22 −0.903212 −0.451606 0.892217i $$-0.649149\pi$$
−0.451606 + 0.892217i $$0.649149\pi$$
$$128$$ 0 0
$$129$$ −7.33676e21 −0.506189
$$130$$ 0 0
$$131$$ −2.53513e22 −1.48817 −0.744084 0.668086i $$-0.767114\pi$$
−0.744084 + 0.668086i $$0.767114\pi$$
$$132$$ 0 0
$$133$$ 3.66438e22 1.83467
$$134$$ 0 0
$$135$$ 4.04702e21 0.173233
$$136$$ 0 0
$$137$$ 1.99626e22 0.732235 0.366117 0.930569i $$-0.380687\pi$$
0.366117 + 0.930569i $$0.380687\pi$$
$$138$$ 0 0
$$139$$ 2.94117e22 0.926540 0.463270 0.886217i $$-0.346676\pi$$
0.463270 + 0.886217i $$0.346676\pi$$
$$140$$ 0 0
$$141$$ 8.96144e21 0.242984
$$142$$ 0 0
$$143$$ 6.48649e22 1.51699
$$144$$ 0 0
$$145$$ 4.90942e21 0.0992354
$$146$$ 0 0
$$147$$ −9.65767e21 −0.169060
$$148$$ 0 0
$$149$$ 1.21478e23 1.84520 0.922598 0.385762i $$-0.126061\pi$$
0.922598 + 0.385762i $$0.126061\pi$$
$$150$$ 0 0
$$151$$ 5.95169e22 0.785928 0.392964 0.919554i $$-0.371450\pi$$
0.392964 + 0.919554i $$0.371450\pi$$
$$152$$ 0 0
$$153$$ −1.20574e23 −1.38674
$$154$$ 0 0
$$155$$ 2.15340e22 0.216093
$$156$$ 0 0
$$157$$ −1.28748e23 −1.12926 −0.564630 0.825344i $$-0.690981\pi$$
−0.564630 + 0.825344i $$0.690981\pi$$
$$158$$ 0 0
$$159$$ −1.99962e23 −1.53560
$$160$$ 0 0
$$161$$ −2.21838e23 −1.49405
$$162$$ 0 0
$$163$$ 5.66875e22 0.335365 0.167682 0.985841i $$-0.446372\pi$$
0.167682 + 0.985841i $$0.446372\pi$$
$$164$$ 0 0
$$165$$ −4.92092e22 −0.256133
$$166$$ 0 0
$$167$$ 1.99516e23 0.915071 0.457535 0.889191i $$-0.348732\pi$$
0.457535 + 0.889191i $$0.348732\pi$$
$$168$$ 0 0
$$169$$ 3.02331e23 1.22369
$$170$$ 0 0
$$171$$ −9.14178e23 −3.27018
$$172$$ 0 0
$$173$$ 3.56107e23 1.12745 0.563723 0.825964i $$-0.309368\pi$$
0.563723 + 0.825964i $$0.309368\pi$$
$$174$$ 0 0
$$175$$ 3.29536e23 0.924706
$$176$$ 0 0
$$177$$ 7.51301e22 0.187109
$$178$$ 0 0
$$179$$ −1.56436e23 −0.346243 −0.173122 0.984900i $$-0.555385\pi$$
−0.173122 + 0.984900i $$0.555385\pi$$
$$180$$ 0 0
$$181$$ −7.11852e23 −1.40205 −0.701027 0.713135i $$-0.747275\pi$$
−0.701027 + 0.713135i $$0.747275\pi$$
$$182$$ 0 0
$$183$$ −1.51856e23 −0.266499
$$184$$ 0 0
$$185$$ −2.32723e22 −0.0364365
$$186$$ 0 0
$$187$$ 5.97578e23 0.835714
$$188$$ 0 0
$$189$$ −8.54171e23 −1.06831
$$190$$ 0 0
$$191$$ 1.15222e23 0.129029 0.0645143 0.997917i $$-0.479450\pi$$
0.0645143 + 0.997917i $$0.479450\pi$$
$$192$$ 0 0
$$193$$ 9.86701e23 0.990453 0.495227 0.868764i $$-0.335085\pi$$
0.495227 + 0.868764i $$0.335085\pi$$
$$194$$ 0 0
$$195$$ −4.16794e23 −0.375455
$$196$$ 0 0
$$197$$ −4.06615e23 −0.329070 −0.164535 0.986371i $$-0.552612\pi$$
−0.164535 + 0.986371i $$0.552612\pi$$
$$198$$ 0 0
$$199$$ 1.80332e24 1.31255 0.656276 0.754521i $$-0.272131\pi$$
0.656276 + 0.754521i $$0.272131\pi$$
$$200$$ 0 0
$$201$$ 1.01754e23 0.0666795
$$202$$ 0 0
$$203$$ −1.03619e24 −0.611972
$$204$$ 0 0
$$205$$ 1.09100e23 0.0581311
$$206$$ 0 0
$$207$$ 5.53434e24 2.66304
$$208$$ 0 0
$$209$$ 4.53075e24 1.97076
$$210$$ 0 0
$$211$$ −3.74550e23 −0.147416 −0.0737080 0.997280i $$-0.523483\pi$$
−0.0737080 + 0.997280i $$0.523483\pi$$
$$212$$ 0 0
$$213$$ 3.66575e24 1.30670
$$214$$ 0 0
$$215$$ −1.46724e23 −0.0474131
$$216$$ 0 0
$$217$$ −4.54500e24 −1.33262
$$218$$ 0 0
$$219$$ −1.11552e25 −2.97038
$$220$$ 0 0
$$221$$ 5.06139e24 1.22504
$$222$$ 0 0
$$223$$ −3.09034e24 −0.680465 −0.340232 0.940341i $$-0.610506\pi$$
−0.340232 + 0.940341i $$0.610506\pi$$
$$224$$ 0 0
$$225$$ −8.22116e24 −1.64823
$$226$$ 0 0
$$227$$ 9.67101e23 0.176685 0.0883426 0.996090i $$-0.471843\pi$$
0.0883426 + 0.996090i $$0.471843\pi$$
$$228$$ 0 0
$$229$$ −3.37848e24 −0.562922 −0.281461 0.959573i $$-0.590819\pi$$
−0.281461 + 0.959573i $$0.590819\pi$$
$$230$$ 0 0
$$231$$ 1.03862e25 1.57954
$$232$$ 0 0
$$233$$ −8.86566e24 −1.23161 −0.615806 0.787898i $$-0.711169\pi$$
−0.615806 + 0.787898i $$0.711169\pi$$
$$234$$ 0 0
$$235$$ 1.79215e23 0.0227595
$$236$$ 0 0
$$237$$ 3.05079e23 0.0354450
$$238$$ 0 0
$$239$$ −9.61276e24 −1.02252 −0.511258 0.859427i $$-0.670820\pi$$
−0.511258 + 0.859427i $$0.670820\pi$$
$$240$$ 0 0
$$241$$ −1.04750e25 −1.02088 −0.510442 0.859912i $$-0.670518\pi$$
−0.510442 + 0.859912i $$0.670518\pi$$
$$242$$ 0 0
$$243$$ −9.66209e24 −0.863386
$$244$$ 0 0
$$245$$ −1.93139e23 −0.0158353
$$246$$ 0 0
$$247$$ 3.83747e25 2.88887
$$248$$ 0 0
$$249$$ 4.34218e25 3.00341
$$250$$ 0 0
$$251$$ −2.60451e25 −1.65635 −0.828174 0.560471i $$-0.810620\pi$$
−0.828174 + 0.560471i $$0.810620\pi$$
$$252$$ 0 0
$$253$$ −2.74287e25 −1.60487
$$254$$ 0 0
$$255$$ −3.83978e24 −0.206840
$$256$$ 0 0
$$257$$ −1.83013e25 −0.908208 −0.454104 0.890949i $$-0.650041\pi$$
−0.454104 + 0.890949i $$0.650041\pi$$
$$258$$ 0 0
$$259$$ 4.91189e24 0.224699
$$260$$ 0 0
$$261$$ 2.58506e25 1.09080
$$262$$ 0 0
$$263$$ −7.36593e24 −0.286874 −0.143437 0.989659i $$-0.545816\pi$$
−0.143437 + 0.989659i $$0.545816\pi$$
$$264$$ 0 0
$$265$$ −3.99893e24 −0.143834
$$266$$ 0 0
$$267$$ −3.03454e25 −1.00861
$$268$$ 0 0
$$269$$ 2.31544e25 0.711599 0.355800 0.934562i $$-0.384209\pi$$
0.355800 + 0.934562i $$0.384209\pi$$
$$270$$ 0 0
$$271$$ −1.17274e25 −0.333445 −0.166722 0.986004i $$-0.553318\pi$$
−0.166722 + 0.986004i $$0.553318\pi$$
$$272$$ 0 0
$$273$$ 8.79692e25 2.31538
$$274$$ 0 0
$$275$$ 4.07448e25 0.993297
$$276$$ 0 0
$$277$$ −5.46772e25 −1.23529 −0.617644 0.786458i $$-0.711913\pi$$
−0.617644 + 0.786458i $$0.711913\pi$$
$$278$$ 0 0
$$279$$ 1.13387e26 2.37531
$$280$$ 0 0
$$281$$ 6.73811e24 0.130955 0.0654775 0.997854i $$-0.479143\pi$$
0.0654775 + 0.997854i $$0.479143\pi$$
$$282$$ 0 0
$$283$$ 3.72254e25 0.671555 0.335778 0.941941i $$-0.391001\pi$$
0.335778 + 0.941941i $$0.391001\pi$$
$$284$$ 0 0
$$285$$ −2.91127e25 −0.487765
$$286$$ 0 0
$$287$$ −2.30268e25 −0.358487
$$288$$ 0 0
$$289$$ −2.24631e25 −0.325119
$$290$$ 0 0
$$291$$ 6.86886e25 0.924716
$$292$$ 0 0
$$293$$ 6.42074e24 0.0804406 0.0402203 0.999191i $$-0.487194\pi$$
0.0402203 + 0.999191i $$0.487194\pi$$
$$294$$ 0 0
$$295$$ 1.50249e24 0.0175259
$$296$$ 0 0
$$297$$ −1.05612e26 −1.14755
$$298$$ 0 0
$$299$$ −2.32317e26 −2.35252
$$300$$ 0 0
$$301$$ 3.09679e25 0.292390
$$302$$ 0 0
$$303$$ −2.51452e26 −2.21466
$$304$$ 0 0
$$305$$ −3.03690e24 −0.0249621
$$306$$ 0 0
$$307$$ 7.52469e25 0.577478 0.288739 0.957408i $$-0.406764\pi$$
0.288739 + 0.957408i $$0.406764\pi$$
$$308$$ 0 0
$$309$$ −8.62194e25 −0.618074
$$310$$ 0 0
$$311$$ 2.48646e25 0.166570 0.0832850 0.996526i $$-0.473459\pi$$
0.0832850 + 0.996526i $$0.473459\pi$$
$$312$$ 0 0
$$313$$ −1.07554e26 −0.673615 −0.336808 0.941574i $$-0.609347\pi$$
−0.336808 + 0.941574i $$0.609347\pi$$
$$314$$ 0 0
$$315$$ −4.19096e25 −0.245501
$$316$$ 0 0
$$317$$ 2.82324e26 1.54748 0.773742 0.633501i $$-0.218383\pi$$
0.773742 + 0.633501i $$0.218383\pi$$
$$318$$ 0 0
$$319$$ −1.28118e26 −0.657366
$$320$$ 0 0
$$321$$ −1.86232e26 −0.894853
$$322$$ 0 0
$$323$$ 3.53533e26 1.59149
$$324$$ 0 0
$$325$$ 3.45102e26 1.45604
$$326$$ 0 0
$$327$$ −2.94438e26 −1.16478
$$328$$ 0 0
$$329$$ −3.78255e25 −0.140355
$$330$$ 0 0
$$331$$ 8.70208e25 0.302991 0.151495 0.988458i $$-0.451591\pi$$
0.151495 + 0.988458i $$0.451591\pi$$
$$332$$ 0 0
$$333$$ −1.22540e26 −0.400511
$$334$$ 0 0
$$335$$ 2.03492e24 0.00624565
$$336$$ 0 0
$$337$$ 1.22441e26 0.353030 0.176515 0.984298i $$-0.443518\pi$$
0.176515 + 0.984298i $$0.443518\pi$$
$$338$$ 0 0
$$339$$ 9.03476e26 2.44805
$$340$$ 0 0
$$341$$ −5.61958e26 −1.43147
$$342$$ 0 0
$$343$$ 4.36091e26 1.04469
$$344$$ 0 0
$$345$$ 1.76245e26 0.397206
$$346$$ 0 0
$$347$$ −3.10647e26 −0.658882 −0.329441 0.944176i $$-0.606860\pi$$
−0.329441 + 0.944176i $$0.606860\pi$$
$$348$$ 0 0
$$349$$ −3.62929e26 −0.724695 −0.362347 0.932043i $$-0.618025\pi$$
−0.362347 + 0.932043i $$0.618025\pi$$
$$350$$ 0 0
$$351$$ −8.94519e26 −1.68215
$$352$$ 0 0
$$353$$ −9.02974e25 −0.159971 −0.0799854 0.996796i $$-0.525487\pi$$
−0.0799854 + 0.996796i $$0.525487\pi$$
$$354$$ 0 0
$$355$$ 7.33096e25 0.122395
$$356$$ 0 0
$$357$$ 8.10430e26 1.27556
$$358$$ 0 0
$$359$$ 5.84199e26 0.867100 0.433550 0.901130i $$-0.357261\pi$$
0.433550 + 0.901130i $$0.357261\pi$$
$$360$$ 0 0
$$361$$ 1.96623e27 2.75302
$$362$$ 0 0
$$363$$ 4.32774e25 0.0571796
$$364$$ 0 0
$$365$$ −2.23087e26 −0.278226
$$366$$ 0 0
$$367$$ −7.68446e26 −0.904940 −0.452470 0.891780i $$-0.649457\pi$$
−0.452470 + 0.891780i $$0.649457\pi$$
$$368$$ 0 0
$$369$$ 5.74465e26 0.638979
$$370$$ 0 0
$$371$$ 8.44022e26 0.887008
$$372$$ 0 0
$$373$$ −6.12745e26 −0.608607 −0.304304 0.952575i $$-0.598424\pi$$
−0.304304 + 0.952575i $$0.598424\pi$$
$$374$$ 0 0
$$375$$ −5.29941e26 −0.497621
$$376$$ 0 0
$$377$$ −1.08514e27 −0.963609
$$378$$ 0 0
$$379$$ 1.02134e27 0.857945 0.428972 0.903318i $$-0.358876\pi$$
0.428972 + 0.903318i $$0.358876\pi$$
$$380$$ 0 0
$$381$$ −1.86303e27 −1.48084
$$382$$ 0 0
$$383$$ −2.10822e27 −1.58609 −0.793045 0.609163i $$-0.791506\pi$$
−0.793045 + 0.609163i $$0.791506\pi$$
$$384$$ 0 0
$$385$$ 2.07708e26 0.147950
$$386$$ 0 0
$$387$$ −7.72576e26 −0.521166
$$388$$ 0 0
$$389$$ 5.16494e26 0.330061 0.165030 0.986288i $$-0.447228\pi$$
0.165030 + 0.986288i $$0.447228\pi$$
$$390$$ 0 0
$$391$$ −2.14025e27 −1.29601
$$392$$ 0 0
$$393$$ −4.25100e27 −2.43989
$$394$$ 0 0
$$395$$ 6.10113e24 0.00332002
$$396$$ 0 0
$$397$$ 2.07736e27 1.07204 0.536019 0.844206i $$-0.319927\pi$$
0.536019 + 0.844206i $$0.319927\pi$$
$$398$$ 0 0
$$399$$ 6.14457e27 3.00799
$$400$$ 0 0
$$401$$ −3.03566e27 −1.41006 −0.705029 0.709179i $$-0.749066\pi$$
−0.705029 + 0.709179i $$0.749066\pi$$
$$402$$ 0 0
$$403$$ −4.75969e27 −2.09834
$$404$$ 0 0
$$405$$ 5.92318e25 0.0247900
$$406$$ 0 0
$$407$$ 6.07321e26 0.241367
$$408$$ 0 0
$$409$$ 1.38404e27 0.522462 0.261231 0.965276i $$-0.415872\pi$$
0.261231 + 0.965276i $$0.415872\pi$$
$$410$$ 0 0
$$411$$ 3.34739e27 1.20052
$$412$$ 0 0
$$413$$ −3.17118e26 −0.108080
$$414$$ 0 0
$$415$$ 8.68370e26 0.281320
$$416$$ 0 0
$$417$$ 4.93185e27 1.51908
$$418$$ 0 0
$$419$$ −2.05547e27 −0.602093 −0.301046 0.953610i $$-0.597336\pi$$
−0.301046 + 0.953610i $$0.597336\pi$$
$$420$$ 0 0
$$421$$ 4.61061e27 1.28468 0.642341 0.766419i $$-0.277963\pi$$
0.642341 + 0.766419i $$0.277963\pi$$
$$422$$ 0 0
$$423$$ 9.43659e26 0.250173
$$424$$ 0 0
$$425$$ 3.17931e27 0.802137
$$426$$ 0 0
$$427$$ 6.40972e26 0.153938
$$428$$ 0 0
$$429$$ 1.08768e28 2.48713
$$430$$ 0 0
$$431$$ −5.49501e27 −1.19662 −0.598312 0.801264i $$-0.704161\pi$$
−0.598312 + 0.801264i $$0.704161\pi$$
$$432$$ 0 0
$$433$$ −5.65719e27 −1.17349 −0.586743 0.809773i $$-0.699590\pi$$
−0.586743 + 0.809773i $$0.699590\pi$$
$$434$$ 0 0
$$435$$ 8.23230e26 0.162699
$$436$$ 0 0
$$437$$ −1.62271e28 −3.05623
$$438$$ 0 0
$$439$$ 2.13975e27 0.384136 0.192068 0.981382i $$-0.438481\pi$$
0.192068 + 0.981382i $$0.438481\pi$$
$$440$$ 0 0
$$441$$ −1.01697e27 −0.174062
$$442$$ 0 0
$$443$$ −5.22577e27 −0.852926 −0.426463 0.904505i $$-0.640241\pi$$
−0.426463 + 0.904505i $$0.640241\pi$$
$$444$$ 0 0
$$445$$ −6.06863e26 −0.0944736
$$446$$ 0 0
$$447$$ 2.03699e28 3.02524
$$448$$ 0 0
$$449$$ 3.58499e27 0.508044 0.254022 0.967198i $$-0.418246\pi$$
0.254022 + 0.967198i $$0.418246\pi$$
$$450$$ 0 0
$$451$$ −2.84710e27 −0.385079
$$452$$ 0 0
$$453$$ 9.98001e27 1.28855
$$454$$ 0 0
$$455$$ 1.75925e27 0.216875
$$456$$ 0 0
$$457$$ 1.08470e27 0.127700 0.0638498 0.997960i $$-0.479662\pi$$
0.0638498 + 0.997960i $$0.479662\pi$$
$$458$$ 0 0
$$459$$ −8.24089e27 −0.926704
$$460$$ 0 0
$$461$$ −1.61325e28 −1.73317 −0.866585 0.499029i $$-0.833690\pi$$
−0.866585 + 0.499029i $$0.833690\pi$$
$$462$$ 0 0
$$463$$ 1.26030e28 1.29382 0.646909 0.762567i $$-0.276061\pi$$
0.646909 + 0.762567i $$0.276061\pi$$
$$464$$ 0 0
$$465$$ 3.61090e27 0.354290
$$466$$ 0 0
$$467$$ −1.01065e28 −0.947923 −0.473961 0.880546i $$-0.657176\pi$$
−0.473961 + 0.880546i $$0.657176\pi$$
$$468$$ 0 0
$$469$$ −4.29495e26 −0.0385162
$$470$$ 0 0
$$471$$ −2.15889e28 −1.85145
$$472$$ 0 0
$$473$$ 3.82896e27 0.314079
$$474$$ 0 0
$$475$$ 2.41051e28 1.89158
$$476$$ 0 0
$$477$$ −2.10564e28 −1.58103
$$478$$ 0 0
$$479$$ 7.85064e27 0.564134 0.282067 0.959395i $$-0.408980\pi$$
0.282067 + 0.959395i $$0.408980\pi$$
$$480$$ 0 0
$$481$$ 5.14391e27 0.353810
$$482$$ 0 0
$$483$$ −3.71986e28 −2.44952
$$484$$ 0 0
$$485$$ 1.37367e27 0.0866152
$$486$$ 0 0
$$487$$ −1.35819e28 −0.820172 −0.410086 0.912047i $$-0.634501\pi$$
−0.410086 + 0.912047i $$0.634501\pi$$
$$488$$ 0 0
$$489$$ 9.50556e27 0.549838
$$490$$ 0 0
$$491$$ −3.02246e28 −1.67496 −0.837482 0.546465i $$-0.815973\pi$$
−0.837482 + 0.546465i $$0.815973\pi$$
$$492$$ 0 0
$$493$$ −9.99700e27 −0.530856
$$494$$ 0 0
$$495$$ −5.18183e27 −0.263711
$$496$$ 0 0
$$497$$ −1.54728e28 −0.754793
$$498$$ 0 0
$$499$$ 1.52172e28 0.711669 0.355835 0.934549i $$-0.384197\pi$$
0.355835 + 0.934549i $$0.384197\pi$$
$$500$$ 0 0
$$501$$ 3.34556e28 1.50028
$$502$$ 0 0
$$503$$ −9.52959e27 −0.409836 −0.204918 0.978779i $$-0.565693\pi$$
−0.204918 + 0.978779i $$0.565693\pi$$
$$504$$ 0 0
$$505$$ −5.02866e27 −0.207440
$$506$$ 0 0
$$507$$ 5.06959e28 2.00627
$$508$$ 0 0
$$509$$ 3.87204e28 1.47029 0.735146 0.677909i $$-0.237114\pi$$
0.735146 + 0.677909i $$0.237114\pi$$
$$510$$ 0 0
$$511$$ 4.70850e28 1.71579
$$512$$ 0 0
$$513$$ −6.24813e28 −2.18533
$$514$$ 0 0
$$515$$ −1.72426e27 −0.0578931
$$516$$ 0 0
$$517$$ −4.67686e27 −0.150766
$$518$$ 0 0
$$519$$ 5.97132e28 1.84847
$$520$$ 0 0
$$521$$ 8.03085e26 0.0238762 0.0119381 0.999929i $$-0.496200\pi$$
0.0119381 + 0.999929i $$0.496200\pi$$
$$522$$ 0 0
$$523$$ 1.91554e28 0.547047 0.273523 0.961865i $$-0.411811\pi$$
0.273523 + 0.961865i $$0.411811\pi$$
$$524$$ 0 0
$$525$$ 5.52578e28 1.51608
$$526$$ 0 0
$$527$$ −4.38494e28 −1.15598
$$528$$ 0 0
$$529$$ 5.87656e28 1.48881
$$530$$ 0 0
$$531$$ 7.91136e27 0.192645
$$532$$ 0 0
$$533$$ −2.41145e28 −0.564472
$$534$$ 0 0
$$535$$ −3.72436e27 −0.0838180
$$536$$ 0 0
$$537$$ −2.62318e28 −0.567673
$$538$$ 0 0
$$539$$ 5.04021e27 0.104898
$$540$$ 0 0
$$541$$ −3.68313e27 −0.0737303 −0.0368651 0.999320i $$-0.511737\pi$$
−0.0368651 + 0.999320i $$0.511737\pi$$
$$542$$ 0 0
$$543$$ −1.19366e29 −2.29870
$$544$$ 0 0
$$545$$ −5.88833e27 −0.109101
$$546$$ 0 0
$$547$$ 4.83617e28 0.862253 0.431127 0.902291i $$-0.358116\pi$$
0.431127 + 0.902291i $$0.358116\pi$$
$$548$$ 0 0
$$549$$ −1.59908e28 −0.274385
$$550$$ 0 0
$$551$$ −7.57959e28 −1.25185
$$552$$ 0 0
$$553$$ −1.28772e27 −0.0204742
$$554$$ 0 0
$$555$$ −3.90238e27 −0.0597384
$$556$$ 0 0
$$557$$ −6.17549e28 −0.910316 −0.455158 0.890411i $$-0.650417\pi$$
−0.455158 + 0.890411i $$0.650417\pi$$
$$558$$ 0 0
$$559$$ 3.24307e28 0.460397
$$560$$ 0 0
$$561$$ 1.00204e29 1.37017
$$562$$ 0 0
$$563$$ 3.51350e28 0.462809 0.231405 0.972858i $$-0.425668\pi$$
0.231405 + 0.972858i $$0.425668\pi$$
$$564$$ 0 0
$$565$$ 1.80682e28 0.229301
$$566$$ 0 0
$$567$$ −1.25016e28 −0.152877
$$568$$ 0 0
$$569$$ 1.45054e28 0.170943 0.0854716 0.996341i $$-0.472760\pi$$
0.0854716 + 0.996341i $$0.472760\pi$$
$$570$$ 0 0
$$571$$ −6.07413e28 −0.689930 −0.344965 0.938616i $$-0.612109\pi$$
−0.344965 + 0.938616i $$0.612109\pi$$
$$572$$ 0 0
$$573$$ 1.93209e28 0.211545
$$574$$ 0 0
$$575$$ −1.45930e29 −1.54039
$$576$$ 0 0
$$577$$ 2.71266e28 0.276089 0.138044 0.990426i $$-0.455918\pi$$
0.138044 + 0.990426i $$0.455918\pi$$
$$578$$ 0 0
$$579$$ 1.65454e29 1.62387
$$580$$ 0 0
$$581$$ −1.83280e29 −1.73486
$$582$$ 0 0
$$583$$ 1.04357e29 0.952804
$$584$$ 0 0
$$585$$ −4.38893e28 −0.386564
$$586$$ 0 0
$$587$$ −2.51262e28 −0.213514 −0.106757 0.994285i $$-0.534047\pi$$
−0.106757 + 0.994285i $$0.534047\pi$$
$$588$$ 0 0
$$589$$ −3.32460e29 −2.72602
$$590$$ 0 0
$$591$$ −6.81827e28 −0.539517
$$592$$ 0 0
$$593$$ −1.95530e29 −1.49328 −0.746638 0.665231i $$-0.768333\pi$$
−0.746638 + 0.665231i $$0.768333\pi$$
$$594$$ 0 0
$$595$$ 1.62074e28 0.119477
$$596$$ 0 0
$$597$$ 3.02388e29 2.15196
$$598$$ 0 0
$$599$$ 1.11818e29 0.768299 0.384149 0.923271i $$-0.374495\pi$$
0.384149 + 0.923271i $$0.374495\pi$$
$$600$$ 0 0
$$601$$ 2.32726e29 1.54405 0.772027 0.635590i $$-0.219243\pi$$
0.772027 + 0.635590i $$0.219243\pi$$
$$602$$ 0 0
$$603$$ 1.07149e28 0.0686525
$$604$$ 0 0
$$605$$ 8.65483e26 0.00535583
$$606$$ 0 0
$$607$$ −2.39690e29 −1.43274 −0.716371 0.697720i $$-0.754198\pi$$
−0.716371 + 0.697720i $$0.754198\pi$$
$$608$$ 0 0
$$609$$ −1.73752e29 −1.00334
$$610$$ 0 0
$$611$$ −3.96123e28 −0.221002
$$612$$ 0 0
$$613$$ 1.89238e29 1.02017 0.510086 0.860123i $$-0.329613\pi$$
0.510086 + 0.860123i $$0.329613\pi$$
$$614$$ 0 0
$$615$$ 1.82943e28 0.0953072
$$616$$ 0 0
$$617$$ 1.13109e29 0.569510 0.284755 0.958600i $$-0.408088\pi$$
0.284755 + 0.958600i $$0.408088\pi$$
$$618$$ 0 0
$$619$$ −1.89565e29 −0.922585 −0.461293 0.887248i $$-0.652614\pi$$
−0.461293 + 0.887248i $$0.652614\pi$$
$$620$$ 0 0
$$621$$ 3.78255e29 1.77960
$$622$$ 0 0
$$623$$ 1.28086e29 0.582607
$$624$$ 0 0
$$625$$ 2.11413e29 0.929806
$$626$$ 0 0
$$627$$ 7.59733e29 3.23111
$$628$$ 0 0
$$629$$ 4.73891e28 0.194916
$$630$$ 0 0
$$631$$ −1.39864e29 −0.556415 −0.278208 0.960521i $$-0.589740\pi$$
−0.278208 + 0.960521i $$0.589740\pi$$
$$632$$ 0 0
$$633$$ −6.28059e28 −0.241692
$$634$$ 0 0
$$635$$ −3.72578e28 −0.138705
$$636$$ 0 0
$$637$$ 4.26898e28 0.153766
$$638$$ 0 0
$$639$$ 3.86011e29 1.34537
$$640$$ 0 0
$$641$$ −7.73898e28 −0.261020 −0.130510 0.991447i $$-0.541662\pi$$
−0.130510 + 0.991447i $$0.541662\pi$$
$$642$$ 0 0
$$643$$ −2.02401e29 −0.660691 −0.330345 0.943860i $$-0.607165\pi$$
−0.330345 + 0.943860i $$0.607165\pi$$
$$644$$ 0 0
$$645$$ −2.46032e28 −0.0777348
$$646$$ 0 0
$$647$$ 6.36949e28 0.194809 0.0974046 0.995245i $$-0.468946\pi$$
0.0974046 + 0.995245i $$0.468946\pi$$
$$648$$ 0 0
$$649$$ −3.92094e28 −0.116097
$$650$$ 0 0
$$651$$ −7.62122e29 −2.18486
$$652$$ 0 0
$$653$$ −1.78655e29 −0.495938 −0.247969 0.968768i $$-0.579763\pi$$
−0.247969 + 0.968768i $$0.579763\pi$$
$$654$$ 0 0
$$655$$ −8.50137e28 −0.228536
$$656$$ 0 0
$$657$$ −1.17466e30 −3.05827
$$658$$ 0 0
$$659$$ −5.55707e29 −1.40136 −0.700678 0.713477i $$-0.747119\pi$$
−0.700678 + 0.713477i $$0.747119\pi$$
$$660$$ 0 0
$$661$$ 2.72969e29 0.666804 0.333402 0.942785i $$-0.391803\pi$$
0.333402 + 0.942785i $$0.391803\pi$$
$$662$$ 0 0
$$663$$ 8.48712e29 2.00848
$$664$$ 0 0
$$665$$ 1.22882e29 0.281749
$$666$$ 0 0
$$667$$ 4.58860e29 1.01943
$$668$$ 0 0
$$669$$ −5.18200e29 −1.11564
$$670$$ 0 0
$$671$$ 7.92518e28 0.165357
$$672$$ 0 0
$$673$$ 3.87928e29 0.784499 0.392250 0.919859i $$-0.371697\pi$$
0.392250 + 0.919859i $$0.371697\pi$$
$$674$$ 0 0
$$675$$ −5.61892e29 −1.10145
$$676$$ 0 0
$$677$$ −7.66988e29 −1.45750 −0.728748 0.684782i $$-0.759898\pi$$
−0.728748 + 0.684782i $$0.759898\pi$$
$$678$$ 0 0
$$679$$ −2.89929e29 −0.534145
$$680$$ 0 0
$$681$$ 1.62167e29 0.289680
$$682$$ 0 0
$$683$$ 5.57644e29 0.965917 0.482959 0.875643i $$-0.339562\pi$$
0.482959 + 0.875643i $$0.339562\pi$$
$$684$$ 0 0
$$685$$ 6.69429e28 0.112448
$$686$$ 0 0
$$687$$ −5.66515e29 −0.922923
$$688$$ 0 0
$$689$$ 8.83890e29 1.39668
$$690$$ 0 0
$$691$$ −6.52122e29 −0.999561 −0.499781 0.866152i $$-0.666586\pi$$
−0.499781 + 0.866152i $$0.666586\pi$$
$$692$$ 0 0
$$693$$ 1.09369e30 1.62627
$$694$$ 0 0
$$695$$ 9.86297e28 0.142288
$$696$$ 0 0
$$697$$ −2.22158e29 −0.310970
$$698$$ 0 0
$$699$$ −1.48663e30 −2.01925
$$700$$ 0 0
$$701$$ 1.49391e29 0.196917 0.0984586 0.995141i $$-0.468609\pi$$
0.0984586 + 0.995141i $$0.468609\pi$$
$$702$$ 0 0
$$703$$ 3.59297e29 0.459646
$$704$$ 0 0
$$705$$ 3.00515e28 0.0373147
$$706$$ 0 0
$$707$$ 1.06136e30 1.27926
$$708$$ 0 0
$$709$$ −1.42826e30 −1.67117 −0.835585 0.549361i $$-0.814871\pi$$
−0.835585 + 0.549361i $$0.814871\pi$$
$$710$$ 0 0
$$711$$ 3.21255e28 0.0364938
$$712$$ 0 0
$$713$$ 2.01268e30 2.21990
$$714$$ 0 0
$$715$$ 2.17519e29 0.232962
$$716$$ 0 0
$$717$$ −1.61190e30 −1.67644
$$718$$ 0 0
$$719$$ 1.67477e30 1.69162 0.845810 0.533484i $$-0.179118\pi$$
0.845810 + 0.533484i $$0.179118\pi$$
$$720$$ 0 0
$$721$$ 3.63925e29 0.357019
$$722$$ 0 0
$$723$$ −1.75649e30 −1.67376
$$724$$ 0 0
$$725$$ −6.81628e29 −0.630955
$$726$$ 0 0
$$727$$ 1.28121e30 1.15215 0.576073 0.817398i $$-0.304584\pi$$
0.576073 + 0.817398i $$0.304584\pi$$
$$728$$ 0 0
$$729$$ −1.80494e30 −1.57697
$$730$$ 0 0
$$731$$ 2.98773e29 0.253634
$$732$$ 0 0
$$733$$ −3.55290e29 −0.293083 −0.146542 0.989205i $$-0.546814\pi$$
−0.146542 + 0.989205i $$0.546814\pi$$
$$734$$ 0 0
$$735$$ −3.23862e28 −0.0259623
$$736$$ 0 0
$$737$$ −5.31040e28 −0.0413732
$$738$$ 0 0
$$739$$ −3.02616e29 −0.229153 −0.114576 0.993414i $$-0.536551\pi$$
−0.114576 + 0.993414i $$0.536551\pi$$
$$740$$ 0 0
$$741$$ 6.43482e30 4.73636
$$742$$ 0 0
$$743$$ −3.53318e29 −0.252803 −0.126402 0.991979i $$-0.540343\pi$$
−0.126402 + 0.991979i $$0.540343\pi$$
$$744$$ 0 0
$$745$$ 4.07368e29 0.283365
$$746$$ 0 0
$$747$$ 4.57240e30 3.09228
$$748$$ 0 0
$$749$$ 7.86071e29 0.516895
$$750$$ 0 0
$$751$$ 1.66286e30 1.06325 0.531625 0.846980i $$-0.321581\pi$$
0.531625 + 0.846980i $$0.321581\pi$$
$$752$$ 0 0
$$753$$ −4.36733e30 −2.71562
$$754$$ 0 0
$$755$$ 1.99585e29 0.120694
$$756$$ 0 0
$$757$$ −1.26493e30 −0.743979 −0.371989 0.928237i $$-0.621324\pi$$
−0.371989 + 0.928237i $$0.621324\pi$$
$$758$$ 0 0
$$759$$ −4.59934e30 −2.63122
$$760$$ 0 0
$$761$$ 3.19527e30 1.77815 0.889075 0.457762i $$-0.151349\pi$$
0.889075 + 0.457762i $$0.151349\pi$$
$$762$$ 0 0
$$763$$ 1.24280e30 0.672812
$$764$$ 0 0
$$765$$ −4.04337e29 −0.212960
$$766$$ 0 0
$$767$$ −3.32098e29 −0.170182
$$768$$ 0 0
$$769$$ −2.81246e30 −1.40236 −0.701182 0.712983i $$-0.747344\pi$$
−0.701182 + 0.712983i $$0.747344\pi$$
$$770$$ 0 0
$$771$$ −3.06884e30 −1.48903
$$772$$ 0 0
$$773$$ 2.43052e30 1.14766 0.573831 0.818974i $$-0.305457\pi$$
0.573831 + 0.818974i $$0.305457\pi$$
$$774$$ 0 0
$$775$$ −2.98980e30 −1.37396
$$776$$ 0 0
$$777$$ 8.23644e29 0.368399
$$778$$ 0 0
$$779$$ −1.68438e30 −0.733323
$$780$$ 0 0
$$781$$ −1.91311e30 −0.810782
$$782$$ 0 0
$$783$$ 1.76681e30 0.728939
$$784$$ 0 0
$$785$$ −4.31746e29 −0.173419
$$786$$ 0 0
$$787$$ −2.79697e30 −1.09384 −0.546919 0.837185i $$-0.684200\pi$$
−0.546919 + 0.837185i $$0.684200\pi$$
$$788$$ 0 0
$$789$$ −1.23514e30 −0.470337
$$790$$ 0 0
$$791$$ −3.81350e30 −1.41407
$$792$$ 0 0
$$793$$ 6.71250e29 0.242390
$$794$$ 0 0
$$795$$ −6.70556e29 −0.235820
$$796$$ 0 0
$$797$$ −2.37782e30 −0.814456 −0.407228 0.913327i $$-0.633504\pi$$
−0.407228 + 0.913327i $$0.633504\pi$$
$$798$$ 0 0
$$799$$ −3.64934e29 −0.121751
$$800$$ 0 0
$$801$$ −3.19544e30 −1.03846
$$802$$ 0 0
$$803$$ 5.82173e30 1.84306
$$804$$ 0 0
$$805$$ −7.43916e29 −0.229439
$$806$$ 0 0
$$807$$ 3.88262e30 1.16668
$$808$$ 0 0
$$809$$ 6.13505e29 0.179622 0.0898108 0.995959i $$-0.471374\pi$$
0.0898108 + 0.995959i $$0.471374\pi$$
$$810$$ 0 0
$$811$$ 4.24348e30 1.21061 0.605303 0.795995i $$-0.293052\pi$$
0.605303 + 0.795995i $$0.293052\pi$$
$$812$$ 0 0
$$813$$ −1.96649e30 −0.546690
$$814$$ 0 0
$$815$$ 1.90097e29 0.0515016
$$816$$ 0 0
$$817$$ 2.26525e30 0.598116
$$818$$ 0 0
$$819$$ 9.26335e30 2.38389
$$820$$ 0 0
$$821$$ 2.33090e30 0.584683 0.292342 0.956314i $$-0.405566\pi$$
0.292342 + 0.956314i $$0.405566\pi$$
$$822$$ 0 0
$$823$$ −4.92628e30 −1.20454 −0.602270 0.798292i $$-0.705737\pi$$
−0.602270 + 0.798292i $$0.705737\pi$$
$$824$$ 0 0
$$825$$ 6.83224e30 1.62853
$$826$$ 0 0
$$827$$ −3.48099e30 −0.808899 −0.404450 0.914560i $$-0.632537\pi$$
−0.404450 + 0.914560i $$0.632537\pi$$
$$828$$ 0 0
$$829$$ −7.46934e30 −1.69223 −0.846116 0.532999i $$-0.821065\pi$$
−0.846116 + 0.532999i $$0.821065\pi$$
$$830$$ 0 0
$$831$$ −9.16847e30 −2.02528
$$832$$ 0 0
$$833$$ 3.93286e29 0.0847103
$$834$$ 0 0
$$835$$ 6.69061e29 0.140526
$$836$$ 0 0
$$837$$ 7.74968e30 1.58733
$$838$$ 0 0
$$839$$ 7.84188e30 1.56646 0.783231 0.621731i $$-0.213570\pi$$
0.783231 + 0.621731i $$0.213570\pi$$
$$840$$ 0 0
$$841$$ −2.98953e30 −0.582432
$$842$$ 0 0
$$843$$ 1.12987e30 0.214704
$$844$$ 0 0
$$845$$ 1.01384e30 0.187921
$$846$$ 0 0
$$847$$ −1.82670e29 −0.0330288
$$848$$ 0 0
$$849$$ 6.24209e30 1.10103
$$850$$ 0 0
$$851$$ −2.17515e30 −0.374308
$$852$$ 0 0
$$853$$ −1.56244e30 −0.262324 −0.131162 0.991361i $$-0.541871\pi$$
−0.131162 + 0.991361i $$0.541871\pi$$
$$854$$ 0 0
$$855$$ −3.06562e30 −0.502198
$$856$$ 0 0
$$857$$ 3.39912e29 0.0543336 0.0271668 0.999631i $$-0.491351\pi$$
0.0271668 + 0.999631i $$0.491351\pi$$
$$858$$ 0 0
$$859$$ 3.06417e30 0.477953 0.238976 0.971025i $$-0.423188\pi$$
0.238976 + 0.971025i $$0.423188\pi$$
$$860$$ 0 0
$$861$$ −3.86122e30 −0.587748
$$862$$ 0 0
$$863$$ −1.16441e30 −0.172979 −0.0864896 0.996253i $$-0.527565\pi$$
−0.0864896 + 0.996253i $$0.527565\pi$$
$$864$$ 0 0
$$865$$ 1.19418e30 0.173141
$$866$$ 0 0
$$867$$ −3.76670e30 −0.533041
$$868$$ 0 0
$$869$$ −1.59217e29 −0.0219929
$$870$$ 0 0
$$871$$ −4.49783e29 −0.0606474
$$872$$ 0 0
$$873$$ 7.23306e30 0.952078
$$874$$ 0 0
$$875$$ 2.23684e30 0.287442
$$876$$ 0 0
$$877$$ 1.22709e31 1.53950 0.769752 0.638343i $$-0.220380\pi$$
0.769752 + 0.638343i $$0.220380\pi$$
$$878$$ 0 0
$$879$$ 1.07665e30 0.131884
$$880$$ 0 0
$$881$$ −7.15175e30 −0.855392 −0.427696 0.903923i $$-0.640675\pi$$
−0.427696 + 0.903923i $$0.640675\pi$$
$$882$$ 0 0
$$883$$ −4.43259e30 −0.517691 −0.258845 0.965919i $$-0.583342\pi$$
−0.258845 + 0.965919i $$0.583342\pi$$
$$884$$ 0 0
$$885$$ 2.51943e29 0.0287341
$$886$$ 0 0
$$887$$ 1.71592e31 1.91117 0.955586 0.294712i $$-0.0952238\pi$$
0.955586 + 0.294712i $$0.0952238\pi$$
$$888$$ 0 0
$$889$$ 7.86369e30 0.855378
$$890$$ 0 0
$$891$$ −1.54573e30 −0.164217
$$892$$ 0 0
$$893$$ −2.76688e30 −0.287111
$$894$$ 0 0
$$895$$ −5.24597e29 −0.0531722
$$896$$ 0 0
$$897$$ −3.89557e31 −3.85701
$$898$$ 0 0
$$899$$ 9.40110e30 0.909289
$$900$$ 0 0
$$901$$ 8.14298e30 0.769437
$$902$$ 0 0
$$903$$ 5.19280e30 0.479381
$$904$$ 0 0
$$905$$ −2.38714e30 −0.215312
$$906$$ 0 0
$$907$$ −1.99778e31 −1.76064 −0.880322 0.474377i $$-0.842673\pi$$
−0.880322 + 0.474377i $$0.842673\pi$$
$$908$$ 0 0
$$909$$ −2.64784e31 −2.28019
$$910$$ 0 0
$$911$$ −1.13275e31 −0.953219 −0.476610 0.879115i $$-0.658134\pi$$
−0.476610 + 0.879115i $$0.658134\pi$$
$$912$$ 0 0
$$913$$ −2.26613e31 −1.86355
$$914$$ 0 0
$$915$$ −5.09238e29 −0.0409260
$$916$$ 0 0
$$917$$ 1.79431e31 1.40935
$$918$$ 0 0
$$919$$ −4.22159e30 −0.324089 −0.162044 0.986783i $$-0.551809\pi$$
−0.162044 + 0.986783i $$0.551809\pi$$
$$920$$ 0 0
$$921$$ 1.26177e31 0.946789
$$922$$ 0 0
$$923$$ −1.62037e31 −1.18849
$$924$$ 0 0
$$925$$ 3.23114e30 0.231669
$$926$$ 0 0
$$927$$ −9.07908e30 −0.636363
$$928$$ 0 0
$$929$$ 7.45447e30 0.510801 0.255400 0.966835i $$-0.417793\pi$$
0.255400 + 0.966835i $$0.417793\pi$$
$$930$$ 0 0
$$931$$ 2.98184e30 0.199762
$$932$$ 0 0
$$933$$ 4.16938e30 0.273095
$$934$$ 0 0
$$935$$ 2.00393e30 0.128340
$$936$$ 0 0
$$937$$ 2.86894e31 1.79662 0.898308 0.439366i $$-0.144797\pi$$
0.898308 + 0.439366i $$0.144797\pi$$
$$938$$ 0 0
$$939$$ −1.80351e31 −1.10441
$$940$$ 0 0
$$941$$ −1.96096e31 −1.17430 −0.587148 0.809480i $$-0.699749\pi$$
−0.587148 + 0.809480i $$0.699749\pi$$
$$942$$ 0 0
$$943$$ 1.01970e31 0.597174
$$944$$ 0 0
$$945$$ −2.86440e30 −0.164059
$$946$$ 0 0
$$947$$ 6.20608e30 0.347650 0.173825 0.984777i $$-0.444387\pi$$
0.173825 + 0.984777i $$0.444387\pi$$
$$948$$ 0 0
$$949$$ 4.93092e31 2.70167
$$950$$ 0 0
$$951$$ 4.73412e31 2.53714
$$952$$ 0 0
$$953$$ −2.74365e31 −1.43831 −0.719156 0.694849i $$-0.755471\pi$$
−0.719156 + 0.694849i $$0.755471\pi$$
$$954$$ 0 0
$$955$$ 3.86389e29 0.0198148
$$956$$ 0 0
$$957$$ −2.14833e31 −1.07777
$$958$$ 0 0
$$959$$ −1.41291e31 −0.693456
$$960$$ 0 0
$$961$$ 2.04101e31 0.980054
$$962$$ 0 0
$$963$$ −1.96106e31 −0.921331
$$964$$ 0 0
$$965$$ 3.30883e30 0.152103
$$966$$ 0 0
$$967$$ 2.94115e31 1.32294 0.661469 0.749973i $$-0.269933\pi$$
0.661469 + 0.749973i $$0.269933\pi$$
$$968$$ 0 0
$$969$$ 5.92817e31 2.60928
$$970$$ 0 0
$$971$$ 3.73435e30 0.160847 0.0804236 0.996761i $$-0.474373\pi$$
0.0804236 + 0.996761i $$0.474373\pi$$
$$972$$ 0 0
$$973$$ −2.08170e31 −0.877470
$$974$$ 0 0
$$975$$ 5.78680e31 2.38721
$$976$$ 0 0
$$977$$ 4.55570e30 0.183934 0.0919670 0.995762i $$-0.470685\pi$$
0.0919670 + 0.995762i $$0.470685\pi$$
$$978$$ 0 0
$$979$$ 1.58369e31 0.625823
$$980$$ 0 0
$$981$$ −3.10050e31 −1.19924
$$982$$ 0 0
$$983$$ −4.81915e31 −1.82456 −0.912280 0.409567i $$-0.865680\pi$$
−0.912280 + 0.409567i $$0.865680\pi$$
$$984$$ 0 0
$$985$$ −1.36355e30 −0.0505349
$$986$$ 0 0
$$987$$ −6.34272e30 −0.230115
$$988$$ 0 0
$$989$$ −1.37136e31 −0.487069
$$990$$ 0 0
$$991$$ −2.54898e31 −0.886324 −0.443162 0.896441i $$-0.646143\pi$$
−0.443162 + 0.896441i $$0.646143\pi$$
$$992$$ 0 0
$$993$$ 1.45920e31 0.496760
$$994$$ 0 0
$$995$$ 6.04731e30 0.201567
$$996$$ 0 0
$$997$$ −4.55000e31 −1.48495 −0.742476 0.669873i $$-0.766349\pi$$
−0.742476 + 0.669873i $$0.766349\pi$$
$$998$$ 0 0
$$999$$ −8.37526e30 −0.267646
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 16.22.a.e.1.2 2
4.3 odd 2 8.22.a.a.1.1 2
8.3 odd 2 64.22.a.k.1.2 2
8.5 even 2 64.22.a.h.1.1 2
12.11 even 2 72.22.a.b.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
8.22.a.a.1.1 2 4.3 odd 2
16.22.a.e.1.2 2 1.1 even 1 trivial
64.22.a.h.1.1 2 8.5 even 2
64.22.a.k.1.2 2 8.3 odd 2
72.22.a.b.1.1 2 12.11 even 2