# Properties

 Label 162.9.b.a.161.5 Level $162$ Weight $9$ Character 162.161 Analytic conductor $65.995$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,9,Mod(161,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("162.161");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 162.b (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: $$65.9953348299$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 3364x^{6} + 4188433x^{4} + 2287495488x^{2} + 462682923264$$ x^8 + 3364*x^6 + 4188433*x^4 + 2287495488*x^2 + 462682923264 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{16}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 161.5 Root $$-26.3494i$$ of defining polynomial Character $$\chi$$ $$=$$ 162.161 Dual form 162.9.b.a.161.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+11.3137i q^{2} -128.000 q^{4} -1061.95i q^{5} -3233.60 q^{7} -1448.15i q^{8} +O(q^{10})$$ $$q+11.3137i q^{2} -128.000 q^{4} -1061.95i q^{5} -3233.60 q^{7} -1448.15i q^{8} +12014.6 q^{10} +7892.01i q^{11} -54494.5 q^{13} -36584.0i q^{14} +16384.0 q^{16} -85389.0i q^{17} -153936. q^{19} +135930. i q^{20} -89287.9 q^{22} -352061. i q^{23} -737118. q^{25} -616535. i q^{26} +413901. q^{28} +880286. i q^{29} +1.43554e6 q^{31} +185364. i q^{32} +966067. q^{34} +3.43393e6i q^{35} +1.22625e6 q^{37} -1.74158e6i q^{38} -1.53787e6 q^{40} +291589. i q^{41} +2.30050e6 q^{43} -1.01018e6i q^{44} +3.98311e6 q^{46} -1.84797e6i q^{47} +4.69138e6 q^{49} -8.33954e6i q^{50} +6.97530e6 q^{52} +3.72595e6i q^{53} +8.38094e6 q^{55} +4.68276e6i q^{56} -9.95929e6 q^{58} +3.50160e6i q^{59} -47311.8 q^{61} +1.62413e7i q^{62} -2.09715e6 q^{64} +5.78706e7i q^{65} -8.18647e6 q^{67} +1.09298e7i q^{68} -3.88505e7 q^{70} -6.23977e6i q^{71} +3.16653e7 q^{73} +1.38734e7i q^{74} +1.97038e7 q^{76} -2.55196e7i q^{77} -2.77912e7 q^{79} -1.73990e7i q^{80} -3.29895e6 q^{82} -8.77450e7i q^{83} -9.06791e7 q^{85} +2.60272e7i q^{86} +1.14289e7 q^{88} +8.43269e7i q^{89} +1.76214e8 q^{91} +4.50637e7i q^{92} +2.09074e7 q^{94} +1.63472e8i q^{95} +2.33853e7 q^{97} +5.30769e7i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 1024 q^{4} - 8876 q^{7}+O(q^{10})$$ 8 * q - 1024 * q^4 - 8876 * q^7 $$8 q - 1024 q^{4} - 8876 q^{7} + 8448 q^{10} - 117380 q^{13} + 131072 q^{16} + 270220 q^{19} - 210816 q^{22} - 1801672 q^{25} + 1136128 q^{28} - 393344 q^{31} - 691968 q^{34} + 1830988 q^{37} - 1081344 q^{40} + 11135236 q^{43} + 5296320 q^{46} - 13586328 q^{49} + 15024640 q^{52} - 1579716 q^{55} - 32988672 q^{58} + 12184204 q^{61} - 16777216 q^{64} - 80355716 q^{67} - 18723264 q^{70} + 197085760 q^{73} - 34588160 q^{76} + 84451852 q^{79} + 144639168 q^{82} - 582634548 q^{85} + 26984448 q^{88} + 373079588 q^{91} - 210121536 q^{94} + 341136928 q^{97}+O(q^{100})$$ 8 * q - 1024 * q^4 - 8876 * q^7 + 8448 * q^10 - 117380 * q^13 + 131072 * q^16 + 270220 * q^19 - 210816 * q^22 - 1801672 * q^25 + 1136128 * q^28 - 393344 * q^31 - 691968 * q^34 + 1830988 * q^37 - 1081344 * q^40 + 11135236 * q^43 + 5296320 * q^46 - 13586328 * q^49 + 15024640 * q^52 - 1579716 * q^55 - 32988672 * q^58 + 12184204 * q^61 - 16777216 * q^64 - 80355716 * q^67 - 18723264 * q^70 + 197085760 * q^73 - 34588160 * q^76 + 84451852 * q^79 + 144639168 * q^82 - 582634548 * q^85 + 26984448 * q^88 + 373079588 * q^91 - 210121536 * q^94 + 341136928 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

 $$n$$ $$83$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 11.3137i 0.707107i
$$3$$ 0 0
$$4$$ −128.000 −0.500000
$$5$$ − 1061.95i − 1.69912i −0.527489 0.849562i $$-0.676866\pi$$
0.527489 0.849562i $$-0.323134\pi$$
$$6$$ 0 0
$$7$$ −3233.60 −1.34677 −0.673386 0.739291i $$-0.735161\pi$$
−0.673386 + 0.739291i $$0.735161\pi$$
$$8$$ − 1448.15i − 0.353553i
$$9$$ 0 0
$$10$$ 12014.6 1.20146
$$11$$ 7892.01i 0.539035i 0.962995 + 0.269517i $$0.0868642\pi$$
−0.962995 + 0.269517i $$0.913136\pi$$
$$12$$ 0 0
$$13$$ −54494.5 −1.90800 −0.954002 0.299800i $$-0.903080\pi$$
−0.954002 + 0.299800i $$0.903080\pi$$
$$14$$ − 36584.0i − 0.952312i
$$15$$ 0 0
$$16$$ 16384.0 0.250000
$$17$$ − 85389.0i − 1.02237i −0.859472 0.511183i $$-0.829207\pi$$
0.859472 0.511183i $$-0.170793\pi$$
$$18$$ 0 0
$$19$$ −153936. −1.18120 −0.590602 0.806963i $$-0.701110\pi$$
−0.590602 + 0.806963i $$0.701110\pi$$
$$20$$ 135930.i 0.849562i
$$21$$ 0 0
$$22$$ −89287.9 −0.381155
$$23$$ − 352061.i − 1.25807i −0.777376 0.629037i $$-0.783450\pi$$
0.777376 0.629037i $$-0.216550\pi$$
$$24$$ 0 0
$$25$$ −737118. −1.88702
$$26$$ − 616535.i − 1.34916i
$$27$$ 0 0
$$28$$ 413901. 0.673386
$$29$$ 880286.i 1.24461i 0.782777 + 0.622303i $$0.213803\pi$$
−0.782777 + 0.622303i $$0.786197\pi$$
$$30$$ 0 0
$$31$$ 1.43554e6 1.55442 0.777212 0.629239i $$-0.216633\pi$$
0.777212 + 0.629239i $$0.216633\pi$$
$$32$$ 185364.i 0.176777i
$$33$$ 0 0
$$34$$ 966067. 0.722922
$$35$$ 3.43393e6i 2.28833i
$$36$$ 0 0
$$37$$ 1.22625e6 0.654293 0.327146 0.944974i $$-0.393913\pi$$
0.327146 + 0.944974i $$0.393913\pi$$
$$38$$ − 1.74158e6i − 0.835237i
$$39$$ 0 0
$$40$$ −1.53787e6 −0.600731
$$41$$ 291589.i 0.103189i 0.998668 + 0.0515947i $$0.0164304\pi$$
−0.998668 + 0.0515947i $$0.983570\pi$$
$$42$$ 0 0
$$43$$ 2.30050e6 0.672897 0.336448 0.941702i $$-0.390774\pi$$
0.336448 + 0.941702i $$0.390774\pi$$
$$44$$ − 1.01018e6i − 0.269517i
$$45$$ 0 0
$$46$$ 3.98311e6 0.889592
$$47$$ − 1.84797e6i − 0.378706i −0.981909 0.189353i $$-0.939361\pi$$
0.981909 0.189353i $$-0.0606391\pi$$
$$48$$ 0 0
$$49$$ 4.69138e6 0.813797
$$50$$ − 8.33954e6i − 1.33433i
$$51$$ 0 0
$$52$$ 6.97530e6 0.954002
$$53$$ 3.72595e6i 0.472208i 0.971728 + 0.236104i $$0.0758706\pi$$
−0.971728 + 0.236104i $$0.924129\pi$$
$$54$$ 0 0
$$55$$ 8.38094e6 0.915887
$$56$$ 4.68276e6i 0.476156i
$$57$$ 0 0
$$58$$ −9.95929e6 −0.880069
$$59$$ 3.50160e6i 0.288974i 0.989507 + 0.144487i $$0.0461532\pi$$
−0.989507 + 0.144487i $$0.953847\pi$$
$$60$$ 0 0
$$61$$ −47311.8 −0.00341704 −0.00170852 0.999999i $$-0.500544\pi$$
−0.00170852 + 0.999999i $$0.500544\pi$$
$$62$$ 1.62413e7i 1.09914i
$$63$$ 0 0
$$64$$ −2.09715e6 −0.125000
$$65$$ 5.78706e7i 3.24194i
$$66$$ 0 0
$$67$$ −8.18647e6 −0.406254 −0.203127 0.979152i $$-0.565110\pi$$
−0.203127 + 0.979152i $$0.565110\pi$$
$$68$$ 1.09298e7i 0.511183i
$$69$$ 0 0
$$70$$ −3.88505e7 −1.61810
$$71$$ − 6.23977e6i − 0.245547i −0.992435 0.122774i $$-0.960821\pi$$
0.992435 0.122774i $$-0.0391789\pi$$
$$72$$ 0 0
$$73$$ 3.16653e7 1.11505 0.557523 0.830162i $$-0.311752\pi$$
0.557523 + 0.830162i $$0.311752\pi$$
$$74$$ 1.38734e7i 0.462655i
$$75$$ 0 0
$$76$$ 1.97038e7 0.590602
$$77$$ − 2.55196e7i − 0.725958i
$$78$$ 0 0
$$79$$ −2.77912e7 −0.713509 −0.356755 0.934198i $$-0.616117\pi$$
−0.356755 + 0.934198i $$0.616117\pi$$
$$80$$ − 1.73990e7i − 0.424781i
$$81$$ 0 0
$$82$$ −3.29895e6 −0.0729660
$$83$$ − 8.77450e7i − 1.84889i −0.381321 0.924443i $$-0.624531\pi$$
0.381321 0.924443i $$-0.375469\pi$$
$$84$$ 0 0
$$85$$ −9.06791e7 −1.73713
$$86$$ 2.60272e7i 0.475810i
$$87$$ 0 0
$$88$$ 1.14289e7 0.190578
$$89$$ 8.43269e7i 1.34402i 0.740541 + 0.672011i $$0.234569\pi$$
−0.740541 + 0.672011i $$0.765431\pi$$
$$90$$ 0 0
$$91$$ 1.76214e8 2.56965
$$92$$ 4.50637e7i 0.629037i
$$93$$ 0 0
$$94$$ 2.09074e7 0.267786
$$95$$ 1.63472e8i 2.00701i
$$96$$ 0 0
$$97$$ 2.33853e7 0.264153 0.132077 0.991239i $$-0.457835\pi$$
0.132077 + 0.991239i $$0.457835\pi$$
$$98$$ 5.30769e7i 0.575441i
$$99$$ 0 0
$$100$$ 9.43511e7 0.943511
$$101$$ − 7.12639e7i − 0.684833i −0.939548 0.342416i $$-0.888755\pi$$
0.939548 0.342416i $$-0.111245\pi$$
$$102$$ 0 0
$$103$$ −1.75219e8 −1.55679 −0.778397 0.627772i $$-0.783967\pi$$
−0.778397 + 0.627772i $$0.783967\pi$$
$$104$$ 7.89165e7i 0.674581i
$$105$$ 0 0
$$106$$ −4.21543e7 −0.333901
$$107$$ 3.54632e7i 0.270547i 0.990808 + 0.135273i $$0.0431913\pi$$
−0.990808 + 0.135273i $$0.956809\pi$$
$$108$$ 0 0
$$109$$ 5.93417e7 0.420392 0.210196 0.977659i $$-0.432590\pi$$
0.210196 + 0.977659i $$0.432590\pi$$
$$110$$ 9.48195e7i 0.647630i
$$111$$ 0 0
$$112$$ −5.29793e7 −0.336693
$$113$$ 2.32353e8i 1.42506i 0.701640 + 0.712532i $$0.252452\pi$$
−0.701640 + 0.712532i $$0.747548\pi$$
$$114$$ 0 0
$$115$$ −3.73872e8 −2.13762
$$116$$ − 1.12677e8i − 0.622303i
$$117$$ 0 0
$$118$$ −3.96161e7 −0.204335
$$119$$ 2.76114e8i 1.37689i
$$120$$ 0 0
$$121$$ 1.52075e8 0.709441
$$122$$ − 535272.i − 0.00241621i
$$123$$ 0 0
$$124$$ −1.83750e8 −0.777212
$$125$$ 3.67959e8i 1.50716i
$$126$$ 0 0
$$127$$ −4.40554e7 −0.169350 −0.0846749 0.996409i $$-0.526985\pi$$
−0.0846749 + 0.996409i $$0.526985\pi$$
$$128$$ − 2.37266e7i − 0.0883883i
$$129$$ 0 0
$$130$$ −6.54731e8 −2.29239
$$131$$ 1.32203e8i 0.448906i 0.974485 + 0.224453i $$0.0720596\pi$$
−0.974485 + 0.224453i $$0.927940\pi$$
$$132$$ 0 0
$$133$$ 4.97766e8 1.59081
$$134$$ − 9.26194e7i − 0.287265i
$$135$$ 0 0
$$136$$ −1.23657e8 −0.361461
$$137$$ 5.07826e8i 1.44156i 0.693164 + 0.720780i $$0.256216\pi$$
−0.693164 + 0.720780i $$0.743784\pi$$
$$138$$ 0 0
$$139$$ 1.39585e7 0.0373920 0.0186960 0.999825i $$-0.494049\pi$$
0.0186960 + 0.999825i $$0.494049\pi$$
$$140$$ − 4.39543e8i − 1.14417i
$$141$$ 0 0
$$142$$ 7.05950e7 0.173628
$$143$$ − 4.30071e8i − 1.02848i
$$144$$ 0 0
$$145$$ 9.34821e8 2.11474
$$146$$ 3.58252e8i 0.788456i
$$147$$ 0 0
$$148$$ −1.56960e8 −0.327146
$$149$$ 2.72813e8i 0.553504i 0.960941 + 0.276752i $$0.0892580\pi$$
−0.960941 + 0.276752i $$0.910742\pi$$
$$150$$ 0 0
$$151$$ −9.29048e8 −1.78702 −0.893512 0.449039i $$-0.851767\pi$$
−0.893512 + 0.449039i $$0.851767\pi$$
$$152$$ 2.22923e8i 0.417618i
$$153$$ 0 0
$$154$$ 2.88722e8 0.513330
$$155$$ − 1.52448e9i − 2.64116i
$$156$$ 0 0
$$157$$ 1.59766e8 0.262958 0.131479 0.991319i $$-0.458027\pi$$
0.131479 + 0.991319i $$0.458027\pi$$
$$158$$ − 3.14422e8i − 0.504527i
$$159$$ 0 0
$$160$$ 1.96848e8 0.300366
$$161$$ 1.13842e9i 1.69434i
$$162$$ 0 0
$$163$$ 4.67512e8 0.662281 0.331140 0.943581i $$-0.392567\pi$$
0.331140 + 0.943581i $$0.392567\pi$$
$$164$$ − 3.73234e7i − 0.0515947i
$$165$$ 0 0
$$166$$ 9.92721e8 1.30736
$$167$$ − 3.25225e8i − 0.418136i −0.977901 0.209068i $$-0.932957\pi$$
0.977901 0.209068i $$-0.0670431\pi$$
$$168$$ 0 0
$$169$$ 2.15392e9 2.64048
$$170$$ − 1.02592e9i − 1.22833i
$$171$$ 0 0
$$172$$ −2.94464e8 −0.336448
$$173$$ − 5.39504e8i − 0.602297i −0.953577 0.301148i $$-0.902630\pi$$
0.953577 0.301148i $$-0.0973699\pi$$
$$174$$ 0 0
$$175$$ 2.38355e9 2.54139
$$176$$ 1.29303e8i 0.134759i
$$177$$ 0 0
$$178$$ −9.54050e8 −0.950367
$$179$$ 7.15170e8i 0.696622i 0.937379 + 0.348311i $$0.113245\pi$$
−0.937379 + 0.348311i $$0.886755\pi$$
$$180$$ 0 0
$$181$$ −2.01588e9 −1.87824 −0.939118 0.343594i $$-0.888356\pi$$
−0.939118 + 0.343594i $$0.888356\pi$$
$$182$$ 1.99363e9i 1.81702i
$$183$$ 0 0
$$184$$ −5.09838e8 −0.444796
$$185$$ − 1.30222e9i − 1.11172i
$$186$$ 0 0
$$187$$ 6.73891e8 0.551091
$$188$$ 2.36540e8i 0.189353i
$$189$$ 0 0
$$190$$ −1.84948e9 −1.41917
$$191$$ − 1.02979e9i − 0.773775i −0.922127 0.386888i $$-0.873550\pi$$
0.922127 0.386888i $$-0.126450\pi$$
$$192$$ 0 0
$$193$$ −1.82400e9 −1.31460 −0.657301 0.753628i $$-0.728302\pi$$
−0.657301 + 0.753628i $$0.728302\pi$$
$$194$$ 2.64575e8i 0.186785i
$$195$$ 0 0
$$196$$ −6.00496e8 −0.406899
$$197$$ − 1.21693e9i − 0.807981i −0.914763 0.403991i $$-0.867623\pi$$
0.914763 0.403991i $$-0.132377\pi$$
$$198$$ 0 0
$$199$$ 1.71825e9 1.09566 0.547829 0.836591i $$-0.315455\pi$$
0.547829 + 0.836591i $$0.315455\pi$$
$$200$$ 1.06746e9i 0.667163i
$$201$$ 0 0
$$202$$ 8.06260e8 0.484250
$$203$$ − 2.84649e9i − 1.67620i
$$204$$ 0 0
$$205$$ 3.09653e8 0.175332
$$206$$ − 1.98237e9i − 1.10082i
$$207$$ 0 0
$$208$$ −8.92838e8 −0.477001
$$209$$ − 1.21486e9i − 0.636710i
$$210$$ 0 0
$$211$$ −6.18692e8 −0.312136 −0.156068 0.987746i $$-0.549882\pi$$
−0.156068 + 0.987746i $$0.549882\pi$$
$$212$$ − 4.76921e8i − 0.236104i
$$213$$ 0 0
$$214$$ −4.01220e8 −0.191305
$$215$$ − 2.44302e9i − 1.14333i
$$216$$ 0 0
$$217$$ −4.64198e9 −2.09346
$$218$$ 6.71375e8i 0.297262i
$$219$$ 0 0
$$220$$ −1.07276e9 −0.457944
$$221$$ 4.65323e9i 1.95068i
$$222$$ 0 0
$$223$$ 3.75384e9 1.51795 0.758973 0.651122i $$-0.225701\pi$$
0.758973 + 0.651122i $$0.225701\pi$$
$$224$$ − 5.99393e8i − 0.238078i
$$225$$ 0 0
$$226$$ −2.62877e9 −1.00767
$$227$$ − 4.30166e9i − 1.62007i −0.586384 0.810033i $$-0.699449\pi$$
0.586384 0.810033i $$-0.300551\pi$$
$$228$$ 0 0
$$229$$ 4.03628e9 1.46771 0.733854 0.679308i $$-0.237720\pi$$
0.733854 + 0.679308i $$0.237720\pi$$
$$230$$ − 4.22987e9i − 1.51153i
$$231$$ 0 0
$$232$$ 1.27479e9 0.440034
$$233$$ − 1.15028e9i − 0.390282i −0.980775 0.195141i $$-0.937484\pi$$
0.980775 0.195141i $$-0.0625164\pi$$
$$234$$ 0 0
$$235$$ −1.96245e9 −0.643469
$$236$$ − 4.48205e8i − 0.144487i
$$237$$ 0 0
$$238$$ −3.12387e9 −0.973612
$$239$$ 3.60021e9i 1.10341i 0.834040 + 0.551705i $$0.186022\pi$$
−0.834040 + 0.551705i $$0.813978\pi$$
$$240$$ 0 0
$$241$$ −3.55755e8 −0.105459 −0.0527294 0.998609i $$-0.516792\pi$$
−0.0527294 + 0.998609i $$0.516792\pi$$
$$242$$ 1.72053e9i 0.501651i
$$243$$ 0 0
$$244$$ 6.05591e6 0.00170852
$$245$$ − 4.98202e9i − 1.38274i
$$246$$ 0 0
$$247$$ 8.38864e9 2.25374
$$248$$ − 2.07889e9i − 0.549572i
$$249$$ 0 0
$$250$$ −4.16299e9 −1.06572
$$251$$ 5.98696e9i 1.50838i 0.656654 + 0.754192i $$0.271971\pi$$
−0.656654 + 0.754192i $$0.728029\pi$$
$$252$$ 0 0
$$253$$ 2.77847e9 0.678146
$$254$$ − 4.98430e8i − 0.119748i
$$255$$ 0 0
$$256$$ 2.68435e8 0.0625000
$$257$$ 2.89949e9i 0.664643i 0.943166 + 0.332322i $$0.107832\pi$$
−0.943166 + 0.332322i $$0.892168\pi$$
$$258$$ 0 0
$$259$$ −3.96520e9 −0.881184
$$260$$ − 7.40743e9i − 1.62097i
$$261$$ 0 0
$$262$$ −1.49570e9 −0.317425
$$263$$ − 2.66911e9i − 0.557882i −0.960308 0.278941i $$-0.910017\pi$$
0.960308 0.278941i $$-0.0899835\pi$$
$$264$$ 0 0
$$265$$ 3.95678e9 0.802339
$$266$$ 5.63158e9i 1.12487i
$$267$$ 0 0
$$268$$ 1.04787e9 0.203127
$$269$$ 1.26012e9i 0.240660i 0.992734 + 0.120330i $$0.0383953\pi$$
−0.992734 + 0.120330i $$0.961605\pi$$
$$270$$ 0 0
$$271$$ 1.48125e9 0.274632 0.137316 0.990527i $$-0.456152\pi$$
0.137316 + 0.990527i $$0.456152\pi$$
$$272$$ − 1.39901e9i − 0.255591i
$$273$$ 0 0
$$274$$ −5.74540e9 −1.01934
$$275$$ − 5.81735e9i − 1.01717i
$$276$$ 0 0
$$277$$ 6.41212e8 0.108914 0.0544568 0.998516i $$-0.482657\pi$$
0.0544568 + 0.998516i $$0.482657\pi$$
$$278$$ 1.57922e8i 0.0264401i
$$279$$ 0 0
$$280$$ 4.97286e9 0.809048
$$281$$ 1.27667e9i 0.204763i 0.994745 + 0.102382i $$0.0326463\pi$$
−0.994745 + 0.102382i $$0.967354\pi$$
$$282$$ 0 0
$$283$$ −1.12894e10 −1.76006 −0.880029 0.474921i $$-0.842477\pi$$
−0.880029 + 0.474921i $$0.842477\pi$$
$$284$$ 7.98691e8i 0.122774i
$$285$$ 0 0
$$286$$ 4.86570e9 0.727246
$$287$$ − 9.42882e8i − 0.138973i
$$288$$ 0 0
$$289$$ −3.15528e8 −0.0452321
$$290$$ 1.05763e10i 1.49535i
$$291$$ 0 0
$$292$$ −4.05316e9 −0.557523
$$293$$ − 1.64745e9i − 0.223533i −0.993735 0.111766i $$-0.964349\pi$$
0.993735 0.111766i $$-0.0356508\pi$$
$$294$$ 0 0
$$295$$ 3.71854e9 0.491003
$$296$$ − 1.77580e9i − 0.231327i
$$297$$ 0 0
$$298$$ −3.08653e9 −0.391386
$$299$$ 1.91854e10i 2.40041i
$$300$$ 0 0
$$301$$ −7.43890e9 −0.906239
$$302$$ − 1.05110e10i − 1.26362i
$$303$$ 0 0
$$304$$ −2.52208e9 −0.295301
$$305$$ 5.02429e7i 0.00580598i
$$306$$ 0 0
$$307$$ −4.02739e9 −0.453388 −0.226694 0.973966i $$-0.572792\pi$$
−0.226694 + 0.973966i $$0.572792\pi$$
$$308$$ 3.26651e9i 0.362979i
$$309$$ 0 0
$$310$$ 1.72475e10 1.86758
$$311$$ − 2.03690e9i − 0.217735i −0.994056 0.108868i $$-0.965278\pi$$
0.994056 0.108868i $$-0.0347224\pi$$
$$312$$ 0 0
$$313$$ −5.27110e9 −0.549192 −0.274596 0.961560i $$-0.588544\pi$$
−0.274596 + 0.961560i $$0.588544\pi$$
$$314$$ 1.80755e9i 0.185939i
$$315$$ 0 0
$$316$$ 3.55728e9 0.356755
$$317$$ − 1.11628e10i − 1.10545i −0.833365 0.552723i $$-0.813589\pi$$
0.833365 0.552723i $$-0.186411\pi$$
$$318$$ 0 0
$$319$$ −6.94722e9 −0.670886
$$320$$ 2.22708e9i 0.212391i
$$321$$ 0 0
$$322$$ −1.28798e10 −1.19808
$$323$$ 1.31444e10i 1.20762i
$$324$$ 0 0
$$325$$ 4.01689e10 3.60045
$$326$$ 5.28929e9i 0.468303i
$$327$$ 0 0
$$328$$ 4.22266e8 0.0364830
$$329$$ 5.97559e9i 0.510031i
$$330$$ 0 0
$$331$$ 7.64734e9 0.637086 0.318543 0.947908i $$-0.396806\pi$$
0.318543 + 0.947908i $$0.396806\pi$$
$$332$$ 1.12314e10i 0.924443i
$$333$$ 0 0
$$334$$ 3.67950e9 0.295667
$$335$$ 8.69365e9i 0.690276i
$$336$$ 0 0
$$337$$ −1.17075e8 −0.00907703 −0.00453852 0.999990i $$-0.501445\pi$$
−0.00453852 + 0.999990i $$0.501445\pi$$
$$338$$ 2.43688e10i 1.86710i
$$339$$ 0 0
$$340$$ 1.16069e10 0.868563
$$341$$ 1.13293e10i 0.837889i
$$342$$ 0 0
$$343$$ 3.47102e9 0.250773
$$344$$ − 3.33148e9i − 0.237905i
$$345$$ 0 0
$$346$$ 6.10379e9 0.425888
$$347$$ 1.54670e10i 1.06681i 0.845859 + 0.533407i $$0.179088\pi$$
−0.845859 + 0.533407i $$0.820912\pi$$
$$348$$ 0 0
$$349$$ −1.67148e10 −1.12667 −0.563337 0.826227i $$-0.690483\pi$$
−0.563337 + 0.826227i $$0.690483\pi$$
$$350$$ 2.69668e10i 1.79703i
$$351$$ 0 0
$$352$$ −1.46289e9 −0.0952888
$$353$$ 1.76650e10i 1.13767i 0.822452 + 0.568834i $$0.192605\pi$$
−0.822452 + 0.568834i $$0.807395\pi$$
$$354$$ 0 0
$$355$$ −6.62634e9 −0.417215
$$356$$ − 1.07938e10i − 0.672011i
$$357$$ 0 0
$$358$$ −8.09123e9 −0.492586
$$359$$ − 4.69625e9i − 0.282731i −0.989957 0.141365i $$-0.954851\pi$$
0.989957 0.141365i $$-0.0451493\pi$$
$$360$$ 0 0
$$361$$ 6.71260e9 0.395241
$$362$$ − 2.28071e10i − 1.32811i
$$363$$ 0 0
$$364$$ −2.25553e10 −1.28482
$$365$$ − 3.36271e10i − 1.89460i
$$366$$ 0 0
$$367$$ 2.64435e10 1.45766 0.728829 0.684696i $$-0.240065\pi$$
0.728829 + 0.684696i $$0.240065\pi$$
$$368$$ − 5.76816e9i − 0.314518i
$$369$$ 0 0
$$370$$ 1.47329e10 0.786108
$$371$$ − 1.20482e10i − 0.635957i
$$372$$ 0 0
$$373$$ 1.53246e10 0.791685 0.395843 0.918318i $$-0.370453\pi$$
0.395843 + 0.918318i $$0.370453\pi$$
$$374$$ 7.62421e9i 0.389680i
$$375$$ 0 0
$$376$$ −2.67614e9 −0.133893
$$377$$ − 4.79707e10i − 2.37471i
$$378$$ 0 0
$$379$$ 1.88826e10 0.915175 0.457588 0.889165i $$-0.348714\pi$$
0.457588 + 0.889165i $$0.348714\pi$$
$$380$$ − 2.09245e10i − 1.00351i
$$381$$ 0 0
$$382$$ 1.16507e10 0.547142
$$383$$ 4.19990e10i 1.95184i 0.218128 + 0.975920i $$0.430005\pi$$
−0.218128 + 0.975920i $$0.569995\pi$$
$$384$$ 0 0
$$385$$ −2.71006e10 −1.23349
$$386$$ − 2.06362e10i − 0.929564i
$$387$$ 0 0
$$388$$ −2.99332e9 −0.132077
$$389$$ 2.28908e10i 0.999684i 0.866117 + 0.499842i $$0.166609\pi$$
−0.866117 + 0.499842i $$0.833391\pi$$
$$390$$ 0 0
$$391$$ −3.00621e10 −1.28621
$$392$$ − 6.79384e9i − 0.287721i
$$393$$ 0 0
$$394$$ 1.37680e10 0.571329
$$395$$ 2.95130e10i 1.21234i
$$396$$ 0 0
$$397$$ −3.22176e10 −1.29697 −0.648487 0.761225i $$-0.724598\pi$$
−0.648487 + 0.761225i $$0.724598\pi$$
$$398$$ 1.94398e10i 0.774747i
$$399$$ 0 0
$$400$$ −1.20769e10 −0.471756
$$401$$ − 3.38757e10i − 1.31012i −0.755577 0.655059i $$-0.772644\pi$$
0.755577 0.655059i $$-0.227356\pi$$
$$402$$ 0 0
$$403$$ −7.82292e10 −2.96585
$$404$$ 9.12179e9i 0.342416i
$$405$$ 0 0
$$406$$ 3.22044e10 1.18525
$$407$$ 9.67758e9i 0.352687i
$$408$$ 0 0
$$409$$ −3.28412e10 −1.17362 −0.586808 0.809726i $$-0.699616\pi$$
−0.586808 + 0.809726i $$0.699616\pi$$
$$410$$ 3.50333e9i 0.123978i
$$411$$ 0 0
$$412$$ 2.24280e10 0.778397
$$413$$ − 1.13228e10i − 0.389182i
$$414$$ 0 0
$$415$$ −9.31810e10 −3.14149
$$416$$ − 1.01013e10i − 0.337291i
$$417$$ 0 0
$$418$$ 1.37446e10 0.450222
$$419$$ 4.20810e10i 1.36531i 0.730743 + 0.682653i $$0.239174\pi$$
−0.730743 + 0.682653i $$0.760826\pi$$
$$420$$ 0 0
$$421$$ 2.16574e10 0.689410 0.344705 0.938711i $$-0.387979\pi$$
0.344705 + 0.938711i $$0.387979\pi$$
$$422$$ − 6.99970e9i − 0.220714i
$$423$$ 0 0
$$424$$ 5.39575e9 0.166951
$$425$$ 6.29418e10i 1.92923i
$$426$$ 0 0
$$427$$ 1.52988e8 0.00460198
$$428$$ − 4.53928e9i − 0.135273i
$$429$$ 0 0
$$430$$ 2.76396e10 0.808460
$$431$$ − 5.79188e9i − 0.167846i −0.996472 0.0839229i $$-0.973255\pi$$
0.996472 0.0839229i $$-0.0267450\pi$$
$$432$$ 0 0
$$433$$ −2.54361e10 −0.723601 −0.361801 0.932255i $$-0.617838\pi$$
−0.361801 + 0.932255i $$0.617838\pi$$
$$434$$ − 5.25180e10i − 1.48030i
$$435$$ 0 0
$$436$$ −7.59574e9 −0.210196
$$437$$ 5.41946e10i 1.48604i
$$438$$ 0 0
$$439$$ −1.90617e10 −0.513219 −0.256610 0.966515i $$-0.582605\pi$$
−0.256610 + 0.966515i $$0.582605\pi$$
$$440$$ − 1.21369e10i − 0.323815i
$$441$$ 0 0
$$442$$ −5.26453e10 −1.37934
$$443$$ − 1.92985e10i − 0.501083i −0.968106 0.250541i $$-0.919391\pi$$
0.968106 0.250541i $$-0.0806086\pi$$
$$444$$ 0 0
$$445$$ 8.95512e10 2.28366
$$446$$ 4.24699e10i 1.07335i
$$447$$ 0 0
$$448$$ 6.78135e9 0.168347
$$449$$ − 2.56335e9i − 0.0630700i −0.999503 0.0315350i $$-0.989960\pi$$
0.999503 0.0315350i $$-0.0100396\pi$$
$$450$$ 0 0
$$451$$ −2.30122e9 −0.0556227
$$452$$ − 2.97412e10i − 0.712532i
$$453$$ 0 0
$$454$$ 4.86677e10 1.14556
$$455$$ − 1.87130e11i − 4.36615i
$$456$$ 0 0
$$457$$ 3.22717e10 0.739873 0.369937 0.929057i $$-0.379379\pi$$
0.369937 + 0.929057i $$0.379379\pi$$
$$458$$ 4.56653e10i 1.03783i
$$459$$ 0 0
$$460$$ 4.78556e10 1.06881
$$461$$ − 6.87557e10i − 1.52232i −0.648567 0.761158i $$-0.724631\pi$$
0.648567 0.761158i $$-0.275369\pi$$
$$462$$ 0 0
$$463$$ 9.26696e9 0.201657 0.100829 0.994904i $$-0.467851\pi$$
0.100829 + 0.994904i $$0.467851\pi$$
$$464$$ 1.44226e10i 0.311151i
$$465$$ 0 0
$$466$$ 1.30139e10 0.275971
$$467$$ − 5.14063e10i − 1.08081i −0.841406 0.540404i $$-0.818271\pi$$
0.841406 0.540404i $$-0.181729\pi$$
$$468$$ 0 0
$$469$$ 2.64718e10 0.547132
$$470$$ − 2.22026e10i − 0.455001i
$$471$$ 0 0
$$472$$ 5.07086e9 0.102168
$$473$$ 1.81556e10i 0.362715i
$$474$$ 0 0
$$475$$ 1.13469e11 2.22896
$$476$$ − 3.53426e10i − 0.688447i
$$477$$ 0 0
$$478$$ −4.07318e10 −0.780228
$$479$$ − 7.06306e10i − 1.34169i −0.741600 0.670843i $$-0.765933\pi$$
0.741600 0.670843i $$-0.234067\pi$$
$$480$$ 0 0
$$481$$ −6.68239e10 −1.24839
$$482$$ − 4.02491e9i − 0.0745706i
$$483$$ 0 0
$$484$$ −1.94656e10 −0.354721
$$485$$ − 2.48341e10i − 0.448829i
$$486$$ 0 0
$$487$$ −4.61712e10 −0.820834 −0.410417 0.911898i $$-0.634617\pi$$
−0.410417 + 0.911898i $$0.634617\pi$$
$$488$$ 6.85148e7i 0.00120811i
$$489$$ 0 0
$$490$$ 5.63651e10 0.977747
$$491$$ − 3.86564e10i − 0.665112i −0.943083 0.332556i $$-0.892089\pi$$
0.943083 0.332556i $$-0.107911\pi$$
$$492$$ 0 0
$$493$$ 7.51667e10 1.27244
$$494$$ 9.49067e10i 1.59364i
$$495$$ 0 0
$$496$$ 2.35199e10 0.388606
$$497$$ 2.01769e10i 0.330697i
$$498$$ 0 0
$$499$$ 1.18884e10 0.191744 0.0958721 0.995394i $$-0.469436\pi$$
0.0958721 + 0.995394i $$0.469436\pi$$
$$500$$ − 4.70988e10i − 0.753581i
$$501$$ 0 0
$$502$$ −6.77348e10 −1.06659
$$503$$ − 7.35514e10i − 1.14900i −0.818506 0.574499i $$-0.805197\pi$$
0.818506 0.574499i $$-0.194803\pi$$
$$504$$ 0 0
$$505$$ −7.56789e10 −1.16362
$$506$$ 3.14347e10i 0.479521i
$$507$$ 0 0
$$508$$ 5.63909e9 0.0846749
$$509$$ 2.05011e10i 0.305427i 0.988271 + 0.152713i $$0.0488011\pi$$
−0.988271 + 0.152713i $$0.951199\pi$$
$$510$$ 0 0
$$511$$ −1.02393e11 −1.50171
$$512$$ 3.03700e9i 0.0441942i
$$513$$ 0 0
$$514$$ −3.28039e10 −0.469974
$$515$$ 1.86074e11i 2.64519i
$$516$$ 0 0
$$517$$ 1.45842e10 0.204136
$$518$$ − 4.48612e10i − 0.623091i
$$519$$ 0 0
$$520$$ 8.38055e10 1.14620
$$521$$ − 7.15997e10i − 0.971763i −0.874025 0.485881i $$-0.838499\pi$$
0.874025 0.485881i $$-0.161501\pi$$
$$522$$ 0 0
$$523$$ 1.01958e11 1.36275 0.681376 0.731934i $$-0.261382\pi$$
0.681376 + 0.731934i $$0.261382\pi$$
$$524$$ − 1.69220e10i − 0.224453i
$$525$$ 0 0
$$526$$ 3.01975e10 0.394482
$$527$$ − 1.22580e11i − 1.58919i
$$528$$ 0 0
$$529$$ −4.56356e10 −0.582749
$$530$$ 4.47658e10i 0.567340i
$$531$$ 0 0
$$532$$ −6.37141e10 −0.795406
$$533$$ − 1.58900e10i − 0.196886i
$$534$$ 0 0
$$535$$ 3.76602e10 0.459692
$$536$$ 1.18553e10i 0.143632i
$$537$$ 0 0
$$538$$ −1.42567e10 −0.170172
$$539$$ 3.70244e10i 0.438665i
$$540$$ 0 0
$$541$$ −7.17904e10 −0.838064 −0.419032 0.907972i $$-0.637630\pi$$
−0.419032 + 0.907972i $$0.637630\pi$$
$$542$$ 1.67584e10i 0.194194i
$$543$$ 0 0
$$544$$ 1.58280e10 0.180730
$$545$$ − 6.30181e10i − 0.714298i
$$546$$ 0 0
$$547$$ −1.28661e11 −1.43714 −0.718568 0.695456i $$-0.755202\pi$$
−0.718568 + 0.695456i $$0.755202\pi$$
$$548$$ − 6.50017e10i − 0.720780i
$$549$$ 0 0
$$550$$ 6.58157e10 0.719249
$$551$$ − 1.35507e11i − 1.47013i
$$552$$ 0 0
$$553$$ 8.98658e10 0.960935
$$554$$ 7.25448e9i 0.0770136i
$$555$$ 0 0
$$556$$ −1.78669e9 −0.0186960
$$557$$ 1.41085e10i 0.146575i 0.997311 + 0.0732874i $$0.0233490\pi$$
−0.997311 + 0.0732874i $$0.976651\pi$$
$$558$$ 0 0
$$559$$ −1.25365e11 −1.28389
$$560$$ 5.62615e10i 0.572084i
$$561$$ 0 0
$$562$$ −1.44438e10 −0.144790
$$563$$ 2.80691e10i 0.279380i 0.990195 + 0.139690i $$0.0446105\pi$$
−0.990195 + 0.139690i $$0.955389\pi$$
$$564$$ 0 0
$$565$$ 2.46748e11 2.42136
$$566$$ − 1.27725e11i − 1.24455i
$$567$$ 0 0
$$568$$ −9.03616e9 −0.0868141
$$569$$ 1.43549e11i 1.36946i 0.728796 + 0.684731i $$0.240080\pi$$
−0.728796 + 0.684731i $$0.759920\pi$$
$$570$$ 0 0
$$571$$ −6.83821e10 −0.643277 −0.321639 0.946863i $$-0.604234\pi$$
−0.321639 + 0.946863i $$0.604234\pi$$
$$572$$ 5.50491e10i 0.514240i
$$573$$ 0 0
$$574$$ 1.06675e10 0.0982686
$$575$$ 2.59510e11i 2.37401i
$$576$$ 0 0
$$577$$ 9.83443e10 0.887250 0.443625 0.896213i $$-0.353692\pi$$
0.443625 + 0.896213i $$0.353692\pi$$
$$578$$ − 3.56979e9i − 0.0319839i
$$579$$ 0 0
$$580$$ −1.19657e11 −1.05737
$$581$$ 2.83732e11i 2.49003i
$$582$$ 0 0
$$583$$ −2.94052e10 −0.254536
$$584$$ − 4.58563e10i − 0.394228i
$$585$$ 0 0
$$586$$ 1.86387e10 0.158061
$$587$$ − 3.62220e10i − 0.305085i −0.988297 0.152542i $$-0.951254\pi$$
0.988297 0.152542i $$-0.0487460\pi$$
$$588$$ 0 0
$$589$$ −2.20981e11 −1.83609
$$590$$ 4.20704e10i 0.347191i
$$591$$ 0 0
$$592$$ 2.00909e10 0.163573
$$593$$ 8.30807e10i 0.671864i 0.941886 + 0.335932i $$0.109051\pi$$
−0.941886 + 0.335932i $$0.890949\pi$$
$$594$$ 0 0
$$595$$ 2.93220e11 2.33951
$$596$$ − 3.49201e10i − 0.276752i
$$597$$ 0 0
$$598$$ −2.17058e11 −1.69735
$$599$$ 3.03534e10i 0.235776i 0.993027 + 0.117888i $$0.0376124\pi$$
−0.993027 + 0.117888i $$0.962388\pi$$
$$600$$ 0 0
$$601$$ 4.92836e10 0.377750 0.188875 0.982001i $$-0.439516\pi$$
0.188875 + 0.982001i $$0.439516\pi$$
$$602$$ − 8.41615e10i − 0.640808i
$$603$$ 0 0
$$604$$ 1.18918e11 0.893512
$$605$$ − 1.61496e11i − 1.20543i
$$606$$ 0 0
$$607$$ 2.37886e11 1.75233 0.876163 0.482015i $$-0.160095\pi$$
0.876163 + 0.482015i $$0.160095\pi$$
$$608$$ − 2.85341e10i − 0.208809i
$$609$$ 0 0
$$610$$ −5.68433e8 −0.00410545
$$611$$ 1.00704e11i 0.722573i
$$612$$ 0 0
$$613$$ 1.03907e11 0.735875 0.367938 0.929850i $$-0.380064\pi$$
0.367938 + 0.929850i $$0.380064\pi$$
$$614$$ − 4.55647e10i − 0.320594i
$$615$$ 0 0
$$616$$ −3.69564e10 −0.256665
$$617$$ 2.25154e11i 1.55360i 0.629748 + 0.776800i $$0.283158\pi$$
−0.629748 + 0.776800i $$0.716842\pi$$
$$618$$ 0 0
$$619$$ −2.27847e11 −1.55196 −0.775981 0.630756i $$-0.782745\pi$$
−0.775981 + 0.630756i $$0.782745\pi$$
$$620$$ 1.95133e11i 1.32058i
$$621$$ 0 0
$$622$$ 2.30449e10 0.153962
$$623$$ − 2.72680e11i − 1.81009i
$$624$$ 0 0
$$625$$ 1.02819e11 0.673832
$$626$$ − 5.96357e10i − 0.388337i
$$627$$ 0 0
$$628$$ −2.04501e10 −0.131479
$$629$$ − 1.04708e11i − 0.668927i
$$630$$ 0 0
$$631$$ −1.35217e11 −0.852931 −0.426466 0.904504i $$-0.640242\pi$$
−0.426466 + 0.904504i $$0.640242\pi$$
$$632$$ 4.02460e10i 0.252264i
$$633$$ 0 0
$$634$$ 1.26293e11 0.781668
$$635$$ 4.67848e10i 0.287746i
$$636$$ 0 0
$$637$$ −2.55654e11 −1.55273
$$638$$ − 7.85989e10i − 0.474388i
$$639$$ 0 0
$$640$$ −2.51965e10 −0.150183
$$641$$ 2.40867e11i 1.42674i 0.700788 + 0.713370i $$0.252832\pi$$
−0.700788 + 0.713370i $$0.747168\pi$$
$$642$$ 0 0
$$643$$ −3.23996e11 −1.89538 −0.947690 0.319193i $$-0.896588\pi$$
−0.947690 + 0.319193i $$0.896588\pi$$
$$644$$ − 1.45718e11i − 0.847170i
$$645$$ 0 0
$$646$$ −1.48712e11 −0.853918
$$647$$ 2.57799e11i 1.47118i 0.677429 + 0.735588i $$0.263094\pi$$
−0.677429 + 0.735588i $$0.736906\pi$$
$$648$$ 0 0
$$649$$ −2.76347e10 −0.155767
$$650$$ 4.54459e11i 2.54590i
$$651$$ 0 0
$$652$$ −5.98415e10 −0.331140
$$653$$ 2.12821e11i 1.17048i 0.810861 + 0.585238i $$0.198999\pi$$
−0.810861 + 0.585238i $$0.801001\pi$$
$$654$$ 0 0
$$655$$ 1.40393e11 0.762747
$$656$$ 4.77739e9i 0.0257974i
$$657$$ 0 0
$$658$$ −6.76060e10 −0.360647
$$659$$ − 7.65162e9i − 0.0405707i −0.999794 0.0202853i $$-0.993543\pi$$
0.999794 0.0202853i $$-0.00645746\pi$$
$$660$$ 0 0
$$661$$ 5.80465e10 0.304068 0.152034 0.988375i $$-0.451418\pi$$
0.152034 + 0.988375i $$0.451418\pi$$
$$662$$ 8.65197e10i 0.450488i
$$663$$ 0 0
$$664$$ −1.27068e11 −0.653680
$$665$$ − 5.28604e11i − 2.70299i
$$666$$ 0 0
$$667$$ 3.09914e11 1.56580
$$668$$ 4.16288e10i 0.209068i
$$669$$ 0 0
$$670$$ −9.83574e10 −0.488099
$$671$$ − 3.73385e8i − 0.00184190i
$$672$$ 0 0
$$673$$ 2.03908e10 0.0993973 0.0496986 0.998764i $$-0.484174\pi$$
0.0496986 + 0.998764i $$0.484174\pi$$
$$674$$ − 1.32455e9i − 0.00641843i
$$675$$ 0 0
$$676$$ −2.75702e11 −1.32024
$$677$$ 3.03449e11i 1.44454i 0.691609 + 0.722272i $$0.256902\pi$$
−0.691609 + 0.722272i $$0.743098\pi$$
$$678$$ 0 0
$$679$$ −7.56188e10 −0.355755
$$680$$ 1.31317e11i 0.614167i
$$681$$ 0 0
$$682$$ −1.28177e11 −0.592477
$$683$$ 2.08506e11i 0.958156i 0.877772 + 0.479078i $$0.159029\pi$$
−0.877772 + 0.479078i $$0.840971\pi$$
$$684$$ 0 0
$$685$$ 5.39287e11 2.44939
$$686$$ 3.92701e10i 0.177323i
$$687$$ 0 0
$$688$$ 3.76914e10 0.168224
$$689$$ − 2.03044e11i − 0.900974i
$$690$$ 0 0
$$691$$ 1.91880e10 0.0841624 0.0420812 0.999114i $$-0.486601\pi$$
0.0420812 + 0.999114i $$0.486601\pi$$
$$692$$ 6.90565e10i 0.301148i
$$693$$ 0 0
$$694$$ −1.74989e11 −0.754351
$$695$$ − 1.48232e10i − 0.0635337i
$$696$$ 0 0
$$697$$ 2.48985e10 0.105497
$$698$$ − 1.89106e11i − 0.796679i
$$699$$ 0 0
$$700$$ −3.05094e11 −1.27070
$$701$$ 3.44220e11i 1.42549i 0.701423 + 0.712745i $$0.252548\pi$$
−0.701423 + 0.712745i $$0.747452\pi$$
$$702$$ 0 0
$$703$$ −1.88764e11 −0.772853
$$704$$ − 1.65507e10i − 0.0673794i
$$705$$ 0 0
$$706$$ −1.99857e11 −0.804453
$$707$$ 2.30439e11i 0.922314i
$$708$$ 0 0
$$709$$ 2.20130e11 0.871153 0.435576 0.900152i $$-0.356545\pi$$
0.435576 + 0.900152i $$0.356545\pi$$
$$710$$ − 7.49685e10i − 0.295016i
$$711$$ 0 0
$$712$$ 1.22118e11 0.475183
$$713$$ − 5.05398e11i − 1.95558i
$$714$$ 0 0
$$715$$ −4.56715e11 −1.74752
$$716$$ − 9.15418e10i − 0.348311i
$$717$$ 0 0
$$718$$ 5.31320e10 0.199921
$$719$$ − 2.06177e11i − 0.771479i −0.922608 0.385740i $$-0.873946\pi$$
0.922608 0.385740i $$-0.126054\pi$$
$$720$$ 0 0
$$721$$ 5.66587e11 2.09665
$$722$$ 7.59444e10i 0.279478i
$$723$$ 0 0
$$724$$ 2.58033e11 0.939118
$$725$$ − 6.48875e11i − 2.34860i
$$726$$ 0 0
$$727$$ 4.32232e11 1.54732 0.773659 0.633602i $$-0.218424\pi$$
0.773659 + 0.633602i $$0.218424\pi$$
$$728$$ − 2.55184e11i − 0.908508i
$$729$$ 0 0
$$730$$ 3.80447e11 1.33969
$$731$$ − 1.96437e11i − 0.687947i
$$732$$ 0 0
$$733$$ 4.82047e11 1.66984 0.834918 0.550374i $$-0.185515\pi$$
0.834918 + 0.550374i $$0.185515\pi$$
$$734$$ 2.99175e11i 1.03072i
$$735$$ 0 0
$$736$$ 6.52593e10 0.222398
$$737$$ − 6.46077e10i − 0.218985i
$$738$$ 0 0
$$739$$ −5.20187e11 −1.74414 −0.872070 0.489380i $$-0.837223\pi$$
−0.872070 + 0.489380i $$0.837223\pi$$
$$740$$ 1.66684e11i 0.555862i
$$741$$ 0 0
$$742$$ 1.36310e11 0.449689
$$743$$ − 1.34453e11i − 0.441178i −0.975367 0.220589i $$-0.929202\pi$$
0.975367 0.220589i $$-0.0707980\pi$$
$$744$$ 0 0
$$745$$ 2.89715e11 0.940472
$$746$$ 1.73378e11i 0.559806i
$$747$$ 0 0
$$748$$ −8.62581e10 −0.275545
$$749$$ − 1.14674e11i − 0.364365i
$$750$$ 0 0
$$751$$ 1.80147e11 0.566328 0.283164 0.959071i $$-0.408616\pi$$
0.283164 + 0.959071i $$0.408616\pi$$
$$752$$ − 3.02771e10i − 0.0946766i
$$753$$ 0 0
$$754$$ 5.42727e11 1.67917
$$755$$ 9.86605e11i 3.03638i
$$756$$ 0 0
$$757$$ 5.77282e10 0.175794 0.0878971 0.996130i $$-0.471985\pi$$
0.0878971 + 0.996130i $$0.471985\pi$$
$$758$$ 2.13632e11i 0.647127i
$$759$$ 0 0
$$760$$ 2.36733e11 0.709586
$$761$$ 2.25519e11i 0.672425i 0.941786 + 0.336212i $$0.109146\pi$$
−0.941786 + 0.336212i $$0.890854\pi$$
$$762$$ 0 0
$$763$$ −1.91888e11 −0.566172
$$764$$ 1.31813e11i 0.386888i
$$765$$ 0 0
$$766$$ −4.75165e11 −1.38016
$$767$$ − 1.90818e11i − 0.551364i
$$768$$ 0 0
$$769$$ 4.70600e11 1.34569 0.672847 0.739782i $$-0.265071\pi$$
0.672847 + 0.739782i $$0.265071\pi$$
$$770$$ − 3.06609e11i − 0.872211i
$$771$$ 0 0
$$772$$ 2.33471e11 0.657301
$$773$$ − 5.56707e10i − 0.155923i −0.996956 0.0779613i $$-0.975159\pi$$
0.996956 0.0779613i $$-0.0248411\pi$$
$$774$$ 0 0
$$775$$ −1.05817e12 −2.93323
$$776$$ − 3.38655e10i − 0.0933923i
$$777$$ 0 0
$$778$$ −2.58980e11 −0.706883
$$779$$ − 4.48859e10i − 0.121888i
$$780$$ 0 0
$$781$$ 4.92443e10 0.132359
$$782$$ − 3.40114e11i − 0.909489i
$$783$$ 0 0
$$784$$ 7.68635e10 0.203449
$$785$$ − 1.69664e11i − 0.446798i
$$786$$ 0 0
$$787$$ −1.83736e11 −0.478956 −0.239478 0.970902i $$-0.576976\pi$$
−0.239478 + 0.970902i $$0.576976\pi$$
$$788$$ 1.55767e11i 0.403991i
$$789$$ 0 0
$$790$$ −3.33901e11 −0.857254
$$791$$ − 7.51337e11i − 1.91924i
$$792$$ 0 0
$$793$$ 2.57823e9 0.00651973
$$794$$ − 3.64501e11i − 0.917100i
$$795$$ 0 0
$$796$$ −2.19936e11 −0.547829
$$797$$ 5.78800e11i 1.43448i 0.696826 + 0.717241i $$0.254595\pi$$
−0.696826 + 0.717241i $$0.745405\pi$$
$$798$$ 0 0
$$799$$ −1.57796e11 −0.387176
$$800$$ − 1.36635e11i − 0.333582i
$$801$$ 0 0
$$802$$ 3.83260e11 0.926394
$$803$$ 2.49903e11i 0.601049i
$$804$$ 0 0
$$805$$ 1.20895e12 2.87889
$$806$$ − 8.85063e11i − 2.09717i
$$807$$ 0 0
$$808$$ −1.03201e11 −0.242125
$$809$$ − 2.72068e10i − 0.0635161i −0.999496 0.0317581i $$-0.989889\pi$$
0.999496 0.0317581i $$-0.0101106\pi$$
$$810$$ 0 0
$$811$$ 2.76712e11 0.639653 0.319827 0.947476i $$-0.396375\pi$$
0.319827 + 0.947476i $$0.396375\pi$$
$$812$$ 3.64351e11i 0.838100i
$$813$$ 0 0
$$814$$ −1.09489e11 −0.249387
$$815$$ − 4.96476e11i − 1.12530i
$$816$$ 0 0
$$817$$ −3.54129e11 −0.794828
$$818$$ − 3.71556e11i − 0.829872i
$$819$$ 0 0
$$820$$ −3.96356e10 −0.0876658
$$821$$ 3.61935e11i 0.796632i 0.917248 + 0.398316i $$0.130405\pi$$
−0.917248 + 0.398316i $$0.869595\pi$$
$$822$$ 0 0
$$823$$ −3.97144e11 −0.865663 −0.432831 0.901475i $$-0.642485\pi$$
−0.432831 + 0.901475i $$0.642485\pi$$
$$824$$ 2.53744e11i 0.550410i
$$825$$ 0 0
$$826$$ 1.28103e11 0.275193
$$827$$ 4.91891e11i 1.05159i 0.850611 + 0.525796i $$0.176232\pi$$
−0.850611 + 0.525796i $$0.823768\pi$$
$$828$$ 0 0
$$829$$ −6.55696e11 −1.38830 −0.694152 0.719828i $$-0.744220\pi$$
−0.694152 + 0.719828i $$0.744220\pi$$
$$830$$ − 1.05422e12i − 2.22137i
$$831$$ 0 0
$$832$$ 1.14283e11 0.238500
$$833$$ − 4.00592e11i − 0.831998i
$$834$$ 0 0
$$835$$ −3.45373e11 −0.710466
$$836$$ 1.55502e11i 0.318355i
$$837$$ 0 0
$$838$$ −4.76092e11 −0.965417
$$839$$ 6.10795e11i 1.23267i 0.787483 + 0.616336i $$0.211384\pi$$
−0.787483 + 0.616336i $$0.788616\pi$$
$$840$$ 0 0
$$841$$ −2.74656e11 −0.549042
$$842$$ 2.45025e11i 0.487487i
$$843$$ 0 0
$$844$$ 7.91925e10 0.156068
$$845$$ − 2.28736e12i − 4.48650i
$$846$$ 0 0
$$847$$ −4.91750e11 −0.955456
$$848$$ 6.10459e10i 0.118052i
$$849$$ 0 0
$$850$$ −7.12105e11 −1.36417
$$851$$ − 4.31714e11i − 0.823148i
$$852$$ 0 0
$$853$$ 4.05108e11 0.765199 0.382599 0.923914i $$-0.375029\pi$$
0.382599 + 0.923914i $$0.375029\pi$$
$$854$$ 1.73086e9i 0.00325409i
$$855$$ 0 0
$$856$$ 5.13561e10 0.0956527
$$857$$ 2.10370e10i 0.0389996i 0.999810 + 0.0194998i $$0.00620737\pi$$
−0.999810 + 0.0194998i $$0.993793\pi$$
$$858$$ 0 0
$$859$$ 2.43542e10 0.0447302 0.0223651 0.999750i $$-0.492880\pi$$
0.0223651 + 0.999750i $$0.492880\pi$$
$$860$$ 3.12707e11i 0.571667i
$$861$$ 0 0
$$862$$ 6.55277e10 0.118685
$$863$$ 6.16048e11i 1.11064i 0.831638 + 0.555318i $$0.187403\pi$$
−0.831638 + 0.555318i $$0.812597\pi$$
$$864$$ 0 0
$$865$$ −5.72928e11 −1.02338
$$866$$ − 2.87777e11i − 0.511663i
$$867$$ 0 0
$$868$$ 5.94173e11 1.04673
$$869$$ − 2.19329e11i − 0.384606i
$$870$$ 0 0
$$871$$ 4.46118e11 0.775134
$$872$$ − 8.59360e10i − 0.148631i
$$873$$ 0 0
$$874$$ −6.13142e11 −1.05079
$$875$$ − 1.18983e12i − 2.02980i
$$876$$ 0 0
$$877$$ −1.01983e12 −1.72397 −0.861987 0.506930i $$-0.830780\pi$$
−0.861987 + 0.506930i $$0.830780\pi$$
$$878$$ − 2.15658e11i − 0.362901i
$$879$$ 0 0
$$880$$ 1.37313e11 0.228972
$$881$$ − 1.04617e12i − 1.73659i −0.496050 0.868294i $$-0.665217\pi$$
0.496050 0.868294i $$-0.334783\pi$$
$$882$$ 0 0
$$883$$ −1.11194e11 −0.182911 −0.0914554 0.995809i $$-0.529152\pi$$
−0.0914554 + 0.995809i $$0.529152\pi$$
$$884$$ − 5.95614e11i − 0.975339i
$$885$$ 0 0
$$886$$ 2.18338e11 0.354319
$$887$$ − 7.81949e10i − 0.126324i −0.998003 0.0631618i $$-0.979882\pi$$
0.998003 0.0631618i $$-0.0201184\pi$$
$$888$$ 0 0
$$889$$ 1.42458e11 0.228076
$$890$$ 1.01316e12i 1.61479i
$$891$$ 0 0
$$892$$ −4.80492e11 −0.758973
$$893$$ 2.84468e11i 0.447329i
$$894$$ 0 0
$$895$$ 7.59477e11 1.18365
$$896$$ 7.67223e10i 0.119039i
$$897$$ 0 0
$$898$$ 2.90010e10 0.0445973
$$899$$ 1.26369e12i 1.93464i
$$900$$ 0 0
$$901$$ 3.18155e11 0.482769
$$902$$ − 2.60353e10i − 0.0393312i
$$903$$ 0 0
$$904$$ 3.36483e11 0.503836
$$905$$ 2.14077e12i 3.19136i
$$906$$ 0 0
$$907$$ −4.43060e11 −0.654686 −0.327343 0.944906i $$-0.606153\pi$$
−0.327343 + 0.944906i $$0.606153\pi$$
$$908$$ 5.50613e11i 0.810033i
$$909$$ 0 0
$$910$$ 2.11714e12 3.08733
$$911$$ − 1.14326e12i − 1.65986i −0.557866 0.829931i $$-0.688380\pi$$
0.557866 0.829931i $$-0.311620\pi$$
$$912$$ 0 0
$$913$$ 6.92484e11 0.996614
$$914$$ 3.65113e11i 0.523170i
$$915$$ 0 0
$$916$$ −5.16644e11 −0.733854
$$917$$ − 4.27491e11i − 0.604575i
$$918$$ 0 0
$$919$$ 1.85438e11 0.259978 0.129989 0.991515i $$-0.458506\pi$$
0.129989 + 0.991515i $$0.458506\pi$$
$$920$$ 5.41424e11i 0.755764i
$$921$$ 0 0
$$922$$ 7.77882e11 1.07644
$$923$$ 3.40033e11i 0.468505i
$$924$$ 0 0
$$925$$ −9.03891e11 −1.23467
$$926$$ 1.04844e11i 0.142593i
$$927$$ 0 0
$$928$$ −1.63173e11 −0.220017
$$929$$ − 6.47121e10i − 0.0868806i −0.999056 0.0434403i $$-0.986168\pi$$
0.999056 0.0434403i $$-0.0138318\pi$$
$$930$$ 0 0
$$931$$ −7.22170e11 −0.961260
$$932$$ 1.47235e11i 0.195141i
$$933$$ 0 0
$$934$$ 5.81595e11 0.764247
$$935$$ − 7.15640e11i − 0.936372i
$$936$$ 0 0
$$937$$ 8.18788e11 1.06222 0.531108 0.847304i $$-0.321776\pi$$
0.531108 + 0.847304i $$0.321776\pi$$
$$938$$ 2.99494e11i 0.386881i
$$939$$ 0 0
$$940$$ 2.51194e11 0.321735
$$941$$ − 8.70441e11i − 1.11015i −0.831801 0.555074i $$-0.812690\pi$$
0.831801 0.555074i $$-0.187310\pi$$
$$942$$ 0 0
$$943$$ 1.02657e11 0.129820
$$944$$ 5.73703e10i 0.0722435i
$$945$$ 0 0
$$946$$ −2.05407e11 −0.256478
$$947$$ 1.48793e12i 1.85005i 0.379906 + 0.925025i $$0.375956\pi$$
−0.379906 + 0.925025i $$0.624044\pi$$
$$948$$ 0 0
$$949$$ −1.72559e12 −2.12751
$$950$$ 1.28375e12i 1.57611i
$$951$$ 0 0
$$952$$ 3.99856e11 0.486806
$$953$$ 1.64141e12i 1.98997i 0.100019 + 0.994986i $$0.468110\pi$$
−0.100019 + 0.994986i $$0.531890\pi$$
$$954$$ 0 0
$$955$$ −1.09359e12 −1.31474
$$956$$ − 4.60827e11i − 0.551705i
$$957$$ 0 0
$$958$$ 7.99094e11 0.948715
$$959$$ − 1.64211e12i − 1.94145i
$$960$$ 0 0
$$961$$ 1.20789e12 1.41624
$$962$$ − 7.56026e11i − 0.882747i
$$963$$ 0 0
$$964$$ 4.55366e10 0.0527294
$$965$$ 1.93700e12i 2.23367i
$$966$$ 0 0
$$967$$ −7.74230e11 −0.885450 −0.442725 0.896657i $$-0.645988\pi$$
−0.442725 + 0.896657i $$0.645988\pi$$
$$968$$ − 2.20228e11i − 0.250825i
$$969$$ 0 0
$$970$$ 2.80966e11 0.317370
$$971$$ 6.61550e11i 0.744194i 0.928194 + 0.372097i $$0.121361\pi$$
−0.928194 + 0.372097i $$0.878639\pi$$
$$972$$ 0 0
$$973$$ −4.51362e10 −0.0503585
$$974$$ − 5.22368e11i − 0.580418i
$$975$$ 0 0
$$976$$ −7.75157e8 −0.000854260 0
$$977$$ 3.68827e11i 0.404803i 0.979303 + 0.202402i $$0.0648746\pi$$
−0.979303 + 0.202402i $$0.935125\pi$$
$$978$$ 0 0
$$979$$ −6.65509e11 −0.724475
$$980$$ 6.37699e11i 0.691371i
$$981$$ 0 0
$$982$$ 4.37347e11 0.470305
$$983$$ 6.01419e9i 0.00644114i 0.999995 + 0.00322057i $$0.00102514\pi$$
−0.999995 + 0.00322057i $$0.998975\pi$$
$$984$$ 0 0
$$985$$ −1.29232e12 −1.37286
$$986$$ 8.50414e11i 0.899752i
$$987$$ 0 0
$$988$$ −1.07375e12 −1.12687
$$989$$ − 8.09915e11i − 0.846553i
$$990$$ 0 0
$$991$$ 1.32877e12 1.37770 0.688851 0.724903i $$-0.258115\pi$$
0.688851 + 0.724903i $$0.258115\pi$$
$$992$$ 2.66098e11i 0.274786i
$$993$$ 0 0
$$994$$ −2.28276e11 −0.233838
$$995$$ − 1.82470e12i − 1.86166i
$$996$$ 0 0
$$997$$ 5.76153e11 0.583119 0.291559 0.956553i $$-0.405826\pi$$
0.291559 + 0.956553i $$0.405826\pi$$
$$998$$ 1.34502e11i 0.135584i
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 162.9.b.a.161.5 yes 8
3.2 odd 2 inner 162.9.b.a.161.4 8
9.2 odd 6 162.9.d.h.53.1 16
9.4 even 3 162.9.d.h.107.1 16
9.5 odd 6 162.9.d.h.107.8 16
9.7 even 3 162.9.d.h.53.8 16

By twisted newform
Twist Min Dim Char Parity Ord Type
162.9.b.a.161.4 8 3.2 odd 2 inner
162.9.b.a.161.5 yes 8 1.1 even 1 trivial
162.9.d.h.53.1 16 9.2 odd 6
162.9.d.h.53.8 16 9.7 even 3
162.9.d.h.107.1 16 9.4 even 3
162.9.d.h.107.8 16 9.5 odd 6