# Properties

 Label 68.9.d.a.67.1 Level $68$ Weight $9$ Character 68.67 Self dual yes Analytic conductor $27.702$ Analytic rank $0$ Dimension $1$ CM discriminant -68 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,9,Mod(67,68)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("68.67");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$68 = 2^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 68.d (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: yes Analytic conductor: $$27.7017454842$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 67.1 Character $$\chi$$ $$=$$ 68.67

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+16.0000 q^{2} -94.0000 q^{3} +256.000 q^{4} -1504.00 q^{6} +706.000 q^{7} +4096.00 q^{8} +2275.00 q^{9} +O(q^{10})$$ $$q+16.0000 q^{2} -94.0000 q^{3} +256.000 q^{4} -1504.00 q^{6} +706.000 q^{7} +4096.00 q^{8} +2275.00 q^{9} -13982.0 q^{11} -24064.0 q^{12} +17954.0 q^{13} +11296.0 q^{14} +65536.0 q^{16} +83521.0 q^{17} +36400.0 q^{18} -66364.0 q^{21} -223712. q^{22} +190018. q^{23} -385024. q^{24} +390625. q^{25} +287264. q^{26} +402884. q^{27} +180736. q^{28} +1.78151e6 q^{31} +1.04858e6 q^{32} +1.31431e6 q^{33} +1.33634e6 q^{34} +582400. q^{36} -1.68768e6 q^{39} -1.06182e6 q^{42} -3.57939e6 q^{44} +3.04029e6 q^{46} -6.16038e6 q^{48} -5.26636e6 q^{49} +6.25000e6 q^{50} -7.85097e6 q^{51} +4.59622e6 q^{52} +1.64131e6 q^{53} +6.44614e6 q^{54} +2.89178e6 q^{56} +2.85041e7 q^{62} +1.60615e6 q^{63} +1.67772e7 q^{64} +2.10289e7 q^{66} +2.13814e7 q^{68} -1.78617e7 q^{69} -1.16499e7 q^{71} +9.31840e6 q^{72} -3.67188e7 q^{75} -9.87129e6 q^{77} -2.70028e7 q^{78} +7.08458e7 q^{79} -5.27974e7 q^{81} -1.69892e7 q^{84} -5.72703e7 q^{88} -1.25191e8 q^{89} +1.26755e7 q^{91} +4.86446e7 q^{92} -1.67462e8 q^{93} -9.85661e7 q^{96} -8.42618e7 q^{98} -3.18091e7 q^{99} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$$.

 $$n$$ $$35$$ $$37$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 16.0000 1.00000
$$3$$ −94.0000 −1.16049 −0.580247 0.814441i $$-0.697044\pi$$
−0.580247 + 0.814441i $$0.697044\pi$$
$$4$$ 256.000 1.00000
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ −1504.00 −1.16049
$$7$$ 706.000 0.294044 0.147022 0.989133i $$-0.453031\pi$$
0.147022 + 0.989133i $$0.453031\pi$$
$$8$$ 4096.00 1.00000
$$9$$ 2275.00 0.346746
$$10$$ 0 0
$$11$$ −13982.0 −0.954989 −0.477495 0.878635i $$-0.658455\pi$$
−0.477495 + 0.878635i $$0.658455\pi$$
$$12$$ −24064.0 −1.16049
$$13$$ 17954.0 0.628619 0.314310 0.949320i $$-0.398227\pi$$
0.314310 + 0.949320i $$0.398227\pi$$
$$14$$ 11296.0 0.294044
$$15$$ 0 0
$$16$$ 65536.0 1.00000
$$17$$ 83521.0 1.00000
$$18$$ 36400.0 0.346746
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ −66364.0 −0.341236
$$22$$ −223712. −0.954989
$$23$$ 190018. 0.679021 0.339511 0.940602i $$-0.389739\pi$$
0.339511 + 0.940602i $$0.389739\pi$$
$$24$$ −385024. −1.16049
$$25$$ 390625. 1.00000
$$26$$ 287264. 0.628619
$$27$$ 402884. 0.758097
$$28$$ 180736. 0.294044
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 1.78151e6 1.92904 0.964518 0.264016i $$-0.0850471\pi$$
0.964518 + 0.264016i $$0.0850471\pi$$
$$32$$ 1.04858e6 1.00000
$$33$$ 1.31431e6 1.10826
$$34$$ 1.33634e6 1.00000
$$35$$ 0 0
$$36$$ 582400. 0.346746
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ −1.68768e6 −0.729509
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ −1.06182e6 −0.341236
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ −3.57939e6 −0.954989
$$45$$ 0 0
$$46$$ 3.04029e6 0.679021
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ −6.16038e6 −1.16049
$$49$$ −5.26636e6 −0.913538
$$50$$ 6.25000e6 1.00000
$$51$$ −7.85097e6 −1.16049
$$52$$ 4.59622e6 0.628619
$$53$$ 1.64131e6 0.208012 0.104006 0.994577i $$-0.466834\pi$$
0.104006 + 0.994577i $$0.466834\pi$$
$$54$$ 6.44614e6 0.758097
$$55$$ 0 0
$$56$$ 2.89178e6 0.294044
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 2.85041e7 1.92904
$$63$$ 1.60615e6 0.101959
$$64$$ 1.67772e7 1.00000
$$65$$ 0 0
$$66$$ 2.10289e7 1.10826
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 2.13814e7 1.00000
$$69$$ −1.78617e7 −0.788000
$$70$$ 0 0
$$71$$ −1.16499e7 −0.458445 −0.229222 0.973374i $$-0.573618\pi$$
−0.229222 + 0.973374i $$0.573618\pi$$
$$72$$ 9.31840e6 0.346746
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ −3.67188e7 −1.16049
$$76$$ 0 0
$$77$$ −9.87129e6 −0.280809
$$78$$ −2.70028e7 −0.729509
$$79$$ 7.08458e7 1.81889 0.909444 0.415827i $$-0.136508\pi$$
0.909444 + 0.415827i $$0.136508\pi$$
$$80$$ 0 0
$$81$$ −5.27974e7 −1.22651
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ −1.69892e7 −0.341236
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ −5.72703e7 −0.954989
$$89$$ −1.25191e8 −1.99532 −0.997659 0.0683856i $$-0.978215\pi$$
−0.997659 + 0.0683856i $$0.978215\pi$$
$$90$$ 0 0
$$91$$ 1.26755e7 0.184842
$$92$$ 4.86446e7 0.679021
$$93$$ −1.67462e8 −2.23864
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −9.85661e7 −1.16049
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ −8.42618e7 −0.913538
$$99$$ −3.18091e7 −0.331139
$$100$$ 1.00000e8 1.00000
$$101$$ −1.16229e8 −1.11694 −0.558471 0.829524i $$-0.688612\pi$$
−0.558471 + 0.829524i $$0.688612\pi$$
$$102$$ −1.25616e8 −1.16049
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 7.35396e7 0.628619
$$105$$ 0 0
$$106$$ 2.62610e7 0.208012
$$107$$ 1.66589e8 1.27090 0.635450 0.772142i $$-0.280815\pi$$
0.635450 + 0.772142i $$0.280815\pi$$
$$108$$ 1.03138e8 0.758097
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.62684e7 0.294044
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 4.08454e7 0.217971
$$118$$ 0 0
$$119$$ 5.89658e7 0.294044
$$120$$ 0 0
$$121$$ −1.88626e7 −0.0879952
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 4.56066e8 1.92904
$$125$$ 0 0
$$126$$ 2.56984e7 0.101959
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 2.68435e8 1.00000
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −4.77414e8 −1.62110 −0.810551 0.585668i $$-0.800832\pi$$
−0.810551 + 0.585668i $$0.800832\pi$$
$$132$$ 3.36463e8 1.10826
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 3.42102e8 1.00000
$$137$$ 6.28721e8 1.78474 0.892372 0.451300i $$-0.149040\pi$$
0.892372 + 0.451300i $$0.149040\pi$$
$$138$$ −2.85787e8 −0.788000
$$139$$ −5.83021e8 −1.56180 −0.780900 0.624657i $$-0.785239\pi$$
−0.780900 + 0.624657i $$0.785239\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1.86398e8 −0.458445
$$143$$ −2.51033e8 −0.600325
$$144$$ 1.49094e8 0.346746
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 4.95038e8 1.06016
$$148$$ 0 0
$$149$$ 9.26194e8 1.87913 0.939565 0.342369i $$-0.111229\pi$$
0.939565 + 0.342369i $$0.111229\pi$$
$$150$$ −5.87500e8 −1.16049
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 1.90010e8 0.346746
$$154$$ −1.57941e8 −0.280809
$$155$$ 0 0
$$156$$ −4.32045e8 −0.729509
$$157$$ 3.45578e8 0.568784 0.284392 0.958708i $$-0.408208\pi$$
0.284392 + 0.958708i $$0.408208\pi$$
$$158$$ 1.13353e9 1.81889
$$159$$ −1.54284e8 −0.241397
$$160$$ 0 0
$$161$$ 1.34153e8 0.199662
$$162$$ −8.44758e8 −1.22651
$$163$$ 5.99802e8 0.849683 0.424842 0.905268i $$-0.360330\pi$$
0.424842 + 0.905268i $$0.360330\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.18202e9 1.51971 0.759853 0.650095i $$-0.225271\pi$$
0.759853 + 0.650095i $$0.225271\pi$$
$$168$$ −2.71827e8 −0.341236
$$169$$ −4.93385e8 −0.604838
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 2.75781e8 0.294044
$$176$$ −9.16324e8 −0.954989
$$177$$ 0 0
$$178$$ −2.00305e9 −1.99532
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 2.02808e8 0.184842
$$183$$ 0 0
$$184$$ 7.78314e8 0.679021
$$185$$ 0 0
$$186$$ −2.67939e9 −2.23864
$$187$$ −1.16779e9 −0.954989
$$188$$ 0 0
$$189$$ 2.84436e8 0.222914
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ −1.57706e9 −1.16049
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −1.34819e9 −0.913538
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ −5.08945e8 −0.331139
$$199$$ 2.65036e9 1.69003 0.845013 0.534746i $$-0.179593\pi$$
0.845013 + 0.534746i $$0.179593\pi$$
$$200$$ 1.60000e9 1.00000
$$201$$ 0 0
$$202$$ −1.85967e9 −1.11694
$$203$$ 0 0
$$204$$ −2.00985e9 −1.16049
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.32291e8 0.235448
$$208$$ 1.17663e9 0.628619
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −3.67033e9 −1.85172 −0.925859 0.377869i $$-0.876657\pi$$
−0.925859 + 0.377869i $$0.876657\pi$$
$$212$$ 4.20176e8 0.208012
$$213$$ 1.09509e9 0.532022
$$214$$ 2.66542e9 1.27090
$$215$$ 0 0
$$216$$ 1.65021e9 0.758097
$$217$$ 1.25774e9 0.567222
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1.49954e9 0.628619
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 7.40295e8 0.294044
$$225$$ 8.88672e8 0.346746
$$226$$ 0 0
$$227$$ −5.30347e9 −1.99736 −0.998681 0.0513528i $$-0.983647\pi$$
−0.998681 + 0.0513528i $$0.983647\pi$$
$$228$$ 0 0
$$229$$ 1.08994e8 0.0396334 0.0198167 0.999804i $$-0.493692\pi$$
0.0198167 + 0.999804i $$0.493692\pi$$
$$230$$ 0 0
$$231$$ 9.27901e8 0.325877
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 6.53526e8 0.217971
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −6.65951e9 −2.11081
$$238$$ 9.43453e8 0.294044
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ −3.01801e8 −0.0879952
$$243$$ 2.31963e9 0.665264
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 7.29705e9 1.92904
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 4.11174e8 0.101959
$$253$$ −2.65683e9 −0.648458
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 4.29497e9 1.00000
$$257$$ −8.33461e8 −0.191053 −0.0955263 0.995427i $$-0.530453\pi$$
−0.0955263 + 0.995427i $$0.530453\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −7.63863e9 −1.62110
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 5.38341e9 1.10826
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 1.17679e10 2.31555
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 5.47363e9 1.00000
$$273$$ −1.19150e9 −0.214508
$$274$$ 1.00595e10 1.78474
$$275$$ −5.46172e9 −0.954989
$$276$$ −4.57259e9 −0.788000
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ −9.32834e9 −1.56180
$$279$$ 4.05293e9 0.668886
$$280$$ 0 0
$$281$$ 1.63283e9 0.261889 0.130944 0.991390i $$-0.458199\pi$$
0.130944 + 0.991390i $$0.458199\pi$$
$$282$$ 0 0
$$283$$ −1.26845e10 −1.97756 −0.988778 0.149390i $$-0.952269\pi$$
−0.988778 + 0.149390i $$0.952269\pi$$
$$284$$ −2.98236e9 −0.458445
$$285$$ 0 0
$$286$$ −4.01653e9 −0.600325
$$287$$ 0 0
$$288$$ 2.38551e9 0.346746
$$289$$ 6.97576e9 1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −9.39620e9 −1.27492 −0.637458 0.770485i $$-0.720014\pi$$
−0.637458 + 0.770485i $$0.720014\pi$$
$$294$$ 7.92061e9 1.06016
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −5.63312e9 −0.723975
$$298$$ 1.48191e10 1.87913
$$299$$ 3.41158e9 0.426846
$$300$$ −9.40000e9 −1.16049
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 1.09256e10 1.29620
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 3.04016e9 0.346746
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ −2.52705e9 −0.280809
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2.71676e9 0.290409 0.145204 0.989402i $$-0.453616\pi$$
0.145204 + 0.989402i $$0.453616\pi$$
$$312$$ −6.91272e9 −0.729509
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 5.52924e9 0.568784
$$315$$ 0 0
$$316$$ 1.81365e10 1.81889
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ −2.46854e9 −0.241397
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −1.56594e10 −1.47487
$$322$$ 2.14644e9 0.199662
$$323$$ 0 0
$$324$$ −1.35161e10 −1.22651
$$325$$ 7.01328e9 0.628619
$$326$$ 9.59682e9 0.849683
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 1.89123e10 1.51971
$$335$$ 0 0
$$336$$ −4.34923e9 −0.341236
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ −7.89415e9 −0.604838
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2.49090e10 −1.84221
$$342$$ 0 0
$$343$$ −7.78800e9 −0.562665
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −2.89187e10 −1.99462 −0.997310 0.0732981i $$-0.976648\pi$$
−0.997310 + 0.0732981i $$0.976648\pi$$
$$348$$ 0 0
$$349$$ 2.61223e10 1.76080 0.880400 0.474233i $$-0.157274\pi$$
0.880400 + 0.474233i $$0.157274\pi$$
$$350$$ 4.41250e9 0.294044
$$351$$ 7.23338e9 0.476555
$$352$$ −1.46612e10 −0.954989
$$353$$ −2.03852e10 −1.31285 −0.656425 0.754391i $$-0.727932\pi$$
−0.656425 + 0.754391i $$0.727932\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −3.20488e10 −1.99532
$$357$$ −5.54279e9 −0.341236
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 1.69836e10 1.00000
$$362$$ 0 0
$$363$$ 1.77308e9 0.102118
$$364$$ 3.24493e9 0.184842
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −3.62639e10 −1.99899 −0.999495 0.0317713i $$-0.989885\pi$$
−0.999495 + 0.0317713i $$0.989885\pi$$
$$368$$ 1.24530e10 0.679021
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 1.15877e9 0.0611647
$$372$$ −4.28702e10 −2.23864
$$373$$ 1.35835e10 0.701742 0.350871 0.936424i $$-0.385886\pi$$
0.350871 + 0.936424i $$0.385886\pi$$
$$374$$ −1.86846e10 −0.954989
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 4.55098e9 0.222914
$$379$$ 3.58744e10 1.73871 0.869356 0.494186i $$-0.164534\pi$$
0.869356 + 0.494186i $$0.164534\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ −2.52329e10 −1.16049
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −3.94188e10 −1.72149 −0.860746 0.509035i $$-0.830002\pi$$
−0.860746 + 0.509035i $$0.830002\pi$$
$$390$$ 0 0
$$391$$ 1.58705e10 0.679021
$$392$$ −2.15710e10 −0.913538
$$393$$ 4.48770e10 1.88128
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −8.14312e9 −0.331139
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 4.24058e10 1.69003
$$399$$ 0 0
$$400$$ 2.56000e10 1.00000
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 3.19852e10 1.21263
$$404$$ −2.97547e10 −1.11694
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ −3.21576e10 −1.16049
$$409$$ 2.93717e10 1.04963 0.524815 0.851216i $$-0.324134\pi$$
0.524815 + 0.851216i $$0.324134\pi$$
$$410$$ 0 0
$$411$$ −5.90998e10 −2.07118
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 6.91666e9 0.235448
$$415$$ 0 0
$$416$$ 1.88261e10 0.628619
$$417$$ 5.48040e10 1.81246
$$418$$ 0 0
$$419$$ 4.23423e10 1.37378 0.686892 0.726759i $$-0.258974\pi$$
0.686892 + 0.726759i $$0.258974\pi$$
$$420$$ 0 0
$$421$$ 4.24676e10 1.35185 0.675926 0.736969i $$-0.263744\pi$$
0.675926 + 0.736969i $$0.263744\pi$$
$$422$$ −5.87252e10 −1.85172
$$423$$ 0 0
$$424$$ 6.72282e9 0.208012
$$425$$ 3.26254e10 1.00000
$$426$$ 1.75214e10 0.532022
$$427$$ 0 0
$$428$$ 4.26468e10 1.27090
$$429$$ 2.35971e10 0.696673
$$430$$ 0 0
$$431$$ 6.50925e10 1.88635 0.943175 0.332298i $$-0.107824\pi$$
0.943175 + 0.332298i $$0.107824\pi$$
$$432$$ 2.64034e10 0.758097
$$433$$ 7.01123e10 1.99454 0.997270 0.0738399i $$-0.0235254\pi$$
0.997270 + 0.0738399i $$0.0235254\pi$$
$$434$$ 2.01239e10 0.567222
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 3.25623e10 0.876711 0.438355 0.898802i $$-0.355561\pi$$
0.438355 + 0.898802i $$0.355561\pi$$
$$440$$ 0 0
$$441$$ −1.19810e10 −0.316766
$$442$$ 2.39926e10 0.628619
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −8.70623e10 −2.18072
$$448$$ 1.18447e10 0.294044
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 1.42188e10 0.346746
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ −8.48555e10 −1.99736
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.53892e10 −1.49914 −0.749568 0.661927i $$-0.769739\pi$$
−0.749568 + 0.661927i $$0.769739\pi$$
$$458$$ 1.74391e9 0.0396334
$$459$$ 3.36493e10 0.758097
$$460$$ 0 0
$$461$$ −5.78432e10 −1.28070 −0.640351 0.768082i $$-0.721211\pi$$
−0.640351 + 0.768082i $$0.721211\pi$$
$$462$$ 1.48464e10 0.325877
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 1.04564e10 0.217971
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −3.24843e10 −0.660070
$$472$$ 0 0
$$473$$ 0 0
$$474$$ −1.06552e11 −2.11081
$$475$$ 0 0
$$476$$ 1.50953e10 0.294044
$$477$$ 3.73399e9 0.0721273
$$478$$ 0 0
$$479$$ −7.31624e10 −1.38978 −0.694890 0.719116i $$-0.744547\pi$$
−0.694890 + 0.719116i $$0.744547\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −1.26104e10 −0.231707
$$484$$ −4.82881e9 −0.0879952
$$485$$ 0 0
$$486$$ 3.71141e10 0.665264
$$487$$ −1.12072e11 −1.99241 −0.996207 0.0870146i $$-0.972267\pi$$
−0.996207 + 0.0870146i $$0.972267\pi$$
$$488$$ 0 0
$$489$$ −5.63813e10 −0.986052
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 1.16753e11 1.92904
$$497$$ −8.22480e9 −0.134803
$$498$$ 0 0
$$499$$ 1.04363e11 1.68323 0.841614 0.540080i $$-0.181606\pi$$
0.841614 + 0.540080i $$0.181606\pi$$
$$500$$ 0 0
$$501$$ −1.11110e11 −1.76361
$$502$$ 0 0
$$503$$ −1.10148e11 −1.72070 −0.860350 0.509704i $$-0.829755\pi$$
−0.860350 + 0.509704i $$0.829755\pi$$
$$504$$ 6.57879e9 0.101959
$$505$$ 0 0
$$506$$ −4.25093e10 −0.648458
$$507$$ 4.63782e10 0.701910
$$508$$ 0 0
$$509$$ −1.32775e11 −1.97808 −0.989042 0.147633i $$-0.952835\pi$$
−0.989042 + 0.147633i $$0.952835\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 6.87195e10 1.00000
$$513$$ 0 0
$$514$$ −1.33354e10 −0.191053
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ −1.22218e11 −1.62110
$$525$$ −2.59234e10 −0.341236
$$526$$ 0 0
$$527$$ 1.48793e11 1.92904
$$528$$ 8.61345e10 1.10826
$$529$$ −4.22041e10 −0.538930
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 1.88287e11 2.31555
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 7.36343e10 0.872419
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 8.75781e10 1.00000
$$545$$ 0 0
$$546$$ −1.90640e10 −0.214508
$$547$$ 7.07426e10 0.790190 0.395095 0.918640i $$-0.370712\pi$$
0.395095 + 0.918640i $$0.370712\pi$$
$$548$$ 1.60953e11 1.78474
$$549$$ 0 0
$$550$$ −8.73875e10 −0.954989
$$551$$ 0 0
$$552$$ −7.31615e10 −0.788000
$$553$$ 5.00172e10 0.534833
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −1.49253e11 −1.56180
$$557$$ −1.91461e11 −1.98912 −0.994558 0.104181i $$-0.966778\pi$$
−0.994558 + 0.104181i $$0.966778\pi$$
$$558$$ 6.48468e10 0.668886
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 1.09772e11 1.10826
$$562$$ 2.61253e10 0.261889
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −2.02953e11 −1.97756
$$567$$ −3.72749e10 −0.360649
$$568$$ −4.77178e10 −0.458445
$$569$$ −9.96321e10 −0.950496 −0.475248 0.879852i $$-0.657642\pi$$
−0.475248 + 0.879852i $$0.657642\pi$$
$$570$$ 0 0
$$571$$ −2.10436e11 −1.97959 −0.989793 0.142509i $$-0.954483\pi$$
−0.989793 + 0.142509i $$0.954483\pi$$
$$572$$ −6.42644e10 −0.600325
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 7.42258e10 0.679021
$$576$$ 3.81682e10 0.346746
$$577$$ 1.07152e11 0.966714 0.483357 0.875423i $$-0.339417\pi$$
0.483357 + 0.875423i $$0.339417\pi$$
$$578$$ 1.11612e11 1.00000
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −2.29489e10 −0.198649
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −1.50339e11 −1.27492
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ 1.26730e11 1.06016
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 5.14859e10 0.416361 0.208180 0.978090i $$-0.433246\pi$$
0.208180 + 0.978090i $$0.433246\pi$$
$$594$$ −9.01300e10 −0.723975
$$595$$ 0 0
$$596$$ 2.37106e11 1.87913
$$597$$ −2.49134e11 −1.96126
$$598$$ 5.45853e10 0.426846
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ −1.50400e11 −1.16049
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 1.74809e11 1.29620
$$607$$ 2.70040e11 1.98917 0.994587 0.103906i $$-0.0331341\pi$$
0.994587 + 0.103906i $$0.0331341\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 4.86426e10 0.346746
$$613$$ 2.67782e11 1.89644 0.948222 0.317607i $$-0.102879\pi$$
0.948222 + 0.317607i $$0.102879\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −4.04328e10 −0.280809
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ −6.03146e10 −0.410828 −0.205414 0.978675i $$-0.565854\pi$$
−0.205414 + 0.978675i $$0.565854\pi$$
$$620$$ 0 0
$$621$$ 7.65552e10 0.514764
$$622$$ 4.34682e10 0.290409
$$623$$ −8.83846e10 −0.586712
$$624$$ −1.10604e11 −0.729509
$$625$$ 1.52588e11 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 8.84679e10 0.568784
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 2.90185e11 1.81889
$$633$$ 3.45011e11 2.14891
$$634$$ 0 0
$$635$$ 0 0
$$636$$ −3.94966e10 −0.241397
$$637$$ −9.45523e10 −0.574268
$$638$$ 0 0
$$639$$ −2.65034e10 −0.158964
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ −2.50550e11 −1.47487
$$643$$ −3.59809e10 −0.210488 −0.105244 0.994446i $$-0.533562\pi$$
−0.105244 + 0.994446i $$0.533562\pi$$
$$644$$ 3.43431e10 0.199662
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ −2.16258e11 −1.22651
$$649$$ 0 0
$$650$$ 1.12212e11 0.628619
$$651$$ −1.18228e11 −0.658258
$$652$$ 1.53549e11 0.849683
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ −2.26537e11 −1.18668 −0.593340 0.804952i $$-0.702191\pi$$
−0.593340 + 0.804952i $$0.702191\pi$$
$$662$$ 0 0
$$663$$ −1.40956e11 −0.729509
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 3.02597e11 1.51971
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −6.95877e10 −0.341236
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 1.57377e11 0.758097
$$676$$ −1.26306e11 −0.604838
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 4.98526e11 2.31793
$$682$$ −3.98544e11 −1.84221
$$683$$ −3.31844e11 −1.52493 −0.762467 0.647028i $$-0.776012\pi$$
−0.762467 + 0.647028i $$0.776012\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.24608e11 −0.562665
$$687$$ −1.02455e10 −0.0459944
$$688$$ 0 0
$$689$$ 2.94682e10 0.130760
$$690$$ 0 0
$$691$$ 3.85775e10 0.169208 0.0846042 0.996415i $$-0.473037\pi$$
0.0846042 + 0.996415i $$0.473037\pi$$
$$692$$ 0 0
$$693$$ −2.24572e10 −0.0973694
$$694$$ −4.62698e11 −1.99462
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 4.17957e11 1.76080
$$699$$ 0 0
$$700$$ 7.06000e10 0.294044
$$701$$ 2.31248e11 0.957646 0.478823 0.877911i $$-0.341064\pi$$
0.478823 + 0.877911i $$0.341064\pi$$
$$702$$ 1.15734e11 0.476555
$$703$$ 0 0
$$704$$ −2.34579e11 −0.954989
$$705$$ 0 0
$$706$$ −3.26162e11 −1.31285
$$707$$ −8.20580e10 −0.328430
$$708$$ 0 0
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 0 0
$$711$$ 1.61174e11 0.630692
$$712$$ −5.12781e11 −1.99532
$$713$$ 3.38518e11 1.30986
$$714$$ −8.86846e10 −0.341236
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 4.80421e11 1.79765 0.898827 0.438304i $$-0.144421\pi$$
0.898827 + 0.438304i $$0.144421\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 2.71737e11 1.00000
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 2.83693e10 0.102118
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 5.19189e10 0.184842
$$729$$ 1.28358e11 0.454479
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −5.65203e11 −1.95789 −0.978946 0.204121i $$-0.934566\pi$$
−0.978946 + 0.204121i $$0.934566\pi$$
$$734$$ −5.80223e11 −1.99899
$$735$$ 0 0
$$736$$ 1.99248e11 0.679021
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 1.85403e10 0.0611647
$$743$$ −1.85977e11 −0.610243 −0.305122 0.952313i $$-0.598697\pi$$
−0.305122 + 0.952313i $$0.598697\pi$$
$$744$$ −6.85923e11 −2.23864
$$745$$ 0 0
$$746$$ 2.17336e11 0.701742
$$747$$ 0 0
$$748$$ −2.98954e11 −0.954989
$$749$$ 1.17612e11 0.373701
$$750$$ 0 0
$$751$$ −8.07947e10 −0.253994 −0.126997 0.991903i $$-0.540534\pi$$
−0.126997 + 0.991903i $$0.540534\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 7.28156e10 0.222914
$$757$$ −1.59460e11 −0.485590 −0.242795 0.970078i $$-0.578064\pi$$
−0.242795 + 0.970078i $$0.578064\pi$$
$$758$$ 5.73990e11 1.73871
$$759$$ 2.49742e11 0.752532
$$760$$ 0 0
$$761$$ −5.68097e11 −1.69389 −0.846943 0.531684i $$-0.821559\pi$$
−0.846943 + 0.531684i $$0.821559\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −4.03727e11 −1.16049
$$769$$ 4.65460e11 1.33100 0.665498 0.746399i $$-0.268219\pi$$
0.665498 + 0.746399i $$0.268219\pi$$
$$770$$ 0 0
$$771$$ 7.83454e10 0.221715
$$772$$ 0 0
$$773$$ −6.44277e11 −1.80449 −0.902245 0.431223i $$-0.858082\pi$$
−0.902245 + 0.431223i $$0.858082\pi$$
$$774$$ 0 0
$$775$$ 6.95901e11 1.92904
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −6.30701e11 −1.72149
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 1.62888e11 0.437810
$$782$$ 2.53928e11 0.679021
$$783$$ 0 0
$$784$$ −3.45136e11 −0.913538
$$785$$ 0 0
$$786$$ 7.18031e11 1.88128
$$787$$ 1.93787e11 0.505157 0.252578 0.967576i $$-0.418721\pi$$
0.252578 + 0.967576i $$0.418721\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −1.30290e11 −0.331139
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 6.78493e11 1.69003
$$797$$ −5.79198e11 −1.43547 −0.717735 0.696317i $$-0.754821\pi$$
−0.717735 + 0.696317i $$0.754821\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 4.09600e11 1.00000
$$801$$ −2.84809e11 −0.691868
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 5.11763e11 1.21263
$$807$$ 0 0
$$808$$ −4.76076e11 −1.11694
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ −8.18198e11 −1.89136 −0.945682 0.325093i $$-0.894604\pi$$
−0.945682 + 0.325093i $$0.894604\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −5.14521e11 −1.16049
$$817$$ 0 0
$$818$$ 4.69948e11 1.04963
$$819$$ 2.88368e10 0.0640932
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ −9.45597e11 −2.07118
$$823$$ −8.51393e11 −1.85580 −0.927900 0.372830i $$-0.878387\pi$$
−0.927900 + 0.372830i $$0.878387\pi$$
$$824$$ 0 0
$$825$$ 5.13402e11 1.10826
$$826$$ 0 0
$$827$$ 1.64971e11 0.352684 0.176342 0.984329i $$-0.443574\pi$$
0.176342 + 0.984329i $$0.443574\pi$$
$$828$$ 1.10666e11 0.235448
$$829$$ −8.65995e11 −1.83357 −0.916785 0.399382i $$-0.869225\pi$$
−0.916785 + 0.399382i $$0.869225\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 3.01218e11 0.628619
$$833$$ −4.39852e11 −0.913538
$$834$$ 8.76864e11 1.81246
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 7.17740e11 1.46240
$$838$$ 6.77477e11 1.37378
$$839$$ −9.48615e11 −1.91444 −0.957221 0.289359i $$-0.906558\pi$$
−0.957221 + 0.289359i $$0.906558\pi$$
$$840$$ 0 0
$$841$$ 5.00246e11 1.00000
$$842$$ 6.79482e11 1.35185
$$843$$ −1.53486e11 −0.303920
$$844$$ −9.39604e11 −1.85172
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −1.33170e10 −0.0258745
$$848$$ 1.07565e11 0.208012
$$849$$ 1.19235e12 2.29494
$$850$$ 5.22006e11 1.00000
$$851$$ 0 0
$$852$$ 2.80342e11 0.532022
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 6.82349e11 1.27090
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 3.77553e11 0.696673
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 1.04148e12 1.88635
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 4.22454e11 0.758097
$$865$$ 0 0
$$866$$ 1.12180e12 1.99454
$$867$$ −6.55721e11 −1.16049
$$868$$ 3.21982e11 0.567222
$$869$$ −9.90566e11 −1.73702
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 5.20996e11 0.876711
$$879$$ 8.83243e11 1.47953
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ −1.91696e11 −0.316766
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ 3.83881e11 0.628619
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 6.00722e11 0.970463 0.485231 0.874386i $$-0.338735\pi$$
0.485231 + 0.874386i $$0.338735\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 7.38213e11 1.17131
$$892$$ 0 0
$$893$$ 0 0
$$894$$ −1.39300e12 −2.18072
$$895$$ 0 0
$$896$$ 1.89515e11 0.294044
$$897$$ −3.20689e11 −0.495352
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 2.27500e11 0.346746
$$901$$ 1.37084e11 0.208012
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 1.89048e10 0.0279347 0.0139673 0.999902i $$-0.495554\pi$$
0.0139673 + 0.999902i $$0.495554\pi$$
$$908$$ −1.35769e12 −1.99736
$$909$$ −2.64422e11 −0.387295
$$910$$ 0 0
$$911$$ 1.37408e12 1.99498 0.997489 0.0708249i $$-0.0225632\pi$$
0.997489 + 0.0708249i $$0.0225632\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −1.04623e12 −1.49914
$$915$$ 0 0
$$916$$ 2.79025e10 0.0396334
$$917$$ −3.37055e11 −0.476676
$$918$$ 5.38388e11 0.758097
$$919$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −9.25491e11 −1.28070
$$923$$ −2.09161e11 −0.288187
$$924$$ 2.37543e11 0.325877
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −2.55375e11 −0.337018
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 1.67303e11 0.217971
$$937$$ −1.51499e12 −1.96541 −0.982704 0.185185i $$-0.940712\pi$$
−0.982704 + 0.185185i $$0.940712\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ −5.19749e11 −0.660070
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −9.91404e11 −1.23268 −0.616341 0.787480i $$-0.711386\pi$$
−0.616341 + 0.787480i $$0.711386\pi$$
$$948$$ −1.70483e12 −2.11081
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 2.41524e11 0.294044
$$953$$ −4.45589e11 −0.540210 −0.270105 0.962831i $$-0.587058\pi$$
−0.270105 + 0.962831i $$0.587058\pi$$
$$954$$ 5.97438e10 0.0721273
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −1.17060e12 −1.38978
$$959$$ 4.43877e11 0.524794
$$960$$ 0 0
$$961$$ 2.32087e12 2.72118
$$962$$ 0 0
$$963$$ 3.78990e11 0.440679
$$964$$ 0 0
$$965$$ 0 0
$$966$$ −2.01766e11 −0.231707
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ −7.72610e10 −0.0879952
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ 5.93826e11 0.665264
$$973$$ −4.11613e11 −0.459238
$$974$$ −1.79315e12 −1.99241
$$975$$ −6.59248e11 −0.729509
$$976$$ 0 0
$$977$$ −1.82215e12 −1.99989 −0.999943 0.0106585i $$-0.996607\pi$$
−0.999943 + 0.0106585i $$0.996607\pi$$
$$978$$ −9.02101e11 −0.986052
$$979$$ 1.75042e12 1.90551
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −1.23434e12 −1.32197 −0.660984 0.750400i $$-0.729861\pi$$
−0.660984 + 0.750400i $$0.729861\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −1.67034e12 −1.73185 −0.865925 0.500173i $$-0.833270\pi$$
−0.865925 + 0.500173i $$0.833270\pi$$
$$992$$ 1.86804e12 1.92904
$$993$$ 0 0
$$994$$ −1.31597e11 −0.134803
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ 1.66980e12 1.68323
$$999$$ 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 68.9.d.a.67.1 1
4.3 odd 2 68.9.d.b.67.1 yes 1
17.16 even 2 68.9.d.b.67.1 yes 1
68.67 odd 2 CM 68.9.d.a.67.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
68.9.d.a.67.1 1 1.1 even 1 trivial
68.9.d.a.67.1 1 68.67 odd 2 CM
68.9.d.b.67.1 yes 1 4.3 odd 2
68.9.d.b.67.1 yes 1 17.16 even 2