LMFDB - Embedded newform 54.10.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,10,Mod(1,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("54.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	54.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:	$27.8119351528$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	54.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+16.0000 q^{2} +256.000 q^{4} +1176.00 q^{5} -11473.0 q^{7} +4096.00 q^{8} +O(q^{10})$

\(q+16.0000 q^{2} +256.000 q^{4} +1176.00 q^{5} -11473.0 q^{7} +4096.00 q^{8} +18816.0 q^{10} -58488.0 q^{11} -106171. q^{13} -183568. q^{14} +65536.0 q^{16} +593352. q^{17} -210967. q^{19} +301056. q^{20} -935808. q^{22} -2.08783e6 q^{23} -570149. q^{25} -1.69874e6 q^{26} -2.93709e6 q^{28} -2.39942e6 q^{29} -1.18877e6 q^{31} +1.04858e6 q^{32} +9.49363e6 q^{34} -1.34922e7 q^{35} +1.15782e7 q^{37} -3.37547e6 q^{38} +4.81690e6 q^{40} -2.39416e7 q^{41} -1.06598e7 q^{43} -1.49729e7 q^{44} -3.34053e7 q^{46} -3.40540e7 q^{47} +9.12761e7 q^{49} -9.12238e6 q^{50} -2.71798e7 q^{52} +4.27411e7 q^{53} -6.87819e7 q^{55} -4.69934e7 q^{56} -3.83908e7 q^{58} -7.42079e7 q^{59} +8.10241e7 q^{61} -1.90204e7 q^{62} +1.67772e7 q^{64} -1.24857e8 q^{65} -1.96509e7 q^{67} +1.51898e8 q^{68} -2.15876e8 q^{70} +1.84185e8 q^{71} -2.57038e8 q^{73} +1.85251e8 q^{74} -5.40076e7 q^{76} +6.71033e8 q^{77} +6.51592e8 q^{79} +7.70703e7 q^{80} -3.83066e8 q^{82} +1.76386e7 q^{83} +6.97782e8 q^{85} -1.70557e8 q^{86} -2.39567e8 q^{88} +5.16255e8 q^{89} +1.21810e9 q^{91} -5.34485e8 q^{92} -5.44864e8 q^{94} -2.48097e8 q^{95} -4.34233e8 q^{97} +1.46042e9 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	16.0000	0.707107
$2$	16.0000	0.707107
$3$	0	0
$3$	0	0
$4$	256.000	0.500000
$5$	1176.00	0.841477	0.420739	−	0.907182i	$-0.361771\pi$
$5$	1176.00	0.841477	0.420739	+	0.907182i	$0.361771\pi$
$6$	0	0
$7$	−11473.0	−1.80608	−0.903038	−	0.429562i	$-0.858668\pi$
$7$	−11473.0	−1.80608	−0.903038	+	0.429562i	$0.858668\pi$
$8$	4096.00	0.353553
$9$	0	0
$10$	18816.0	0.595014
$11$	−58488.0	−1.20448	−0.602240	−	0.798315i	$-0.705725\pi$
$11$	−58488.0	−1.20448	−0.602240	+	0.798315i	$0.705725\pi$
$12$	0	0
$13$	−106171.	−1.03101	−0.515503	−	0.856888i	$-0.672395\pi$
$13$	−106171.	−1.03101	−0.515503	+	0.856888i	$0.672395\pi$
$14$	−183568.	−1.27709
$15$	0	0
$16$	65536.0	0.250000
$17$	593352.	1.72303	0.861514	−	0.507734i	$-0.169517\pi$
$17$	593352.	1.72303	0.861514	+	0.507734i	$0.169517\pi$
$18$	0	0
$19$	−210967.	−0.371384	−0.185692	−	0.982608i	$-0.559453\pi$
$19$	−210967.	−0.371384	−0.185692	+	0.982608i	$0.559453\pi$
$20$	301056.	0.420739
$21$	0	0
$22$	−935808.	−0.851696
$23$	−2.08783e6	−1.55568	−0.777840	−	0.628462i	$-0.783685\pi$
$23$	−2.08783e6	−1.55568	−0.777840	+	0.628462i	$0.783685\pi$
$24$	0	0
$25$	−570149.	−0.291916
$26$	−1.69874e6	−0.729031
$27$	0	0
$28$	−2.93709e6	−0.903038
$29$	−2.39942e6	−0.629964	−0.314982	−	0.949098i	$-0.601999\pi$
$29$	−2.39942e6	−0.629964	−0.314982	+	0.949098i	$0.601999\pi$
$30$	0	0
$31$	−1.18877e6	−0.231191	−0.115596	−	0.993296i	$-0.536878\pi$
$31$	−1.18877e6	−0.231191	−0.115596	+	0.993296i	$0.536878\pi$
$32$	1.04858e6	0.176777
$33$	0	0
$34$	9.49363e6	1.21836
$35$	−1.34922e7	−1.51977
$36$	0	0
$37$	1.15782e7	1.01562	0.507812	−	0.861468i	$-0.330455\pi$
$37$	1.15782e7	1.01562	0.507812	+	0.861468i	$0.330455\pi$
$38$	−3.37547e6	−0.262608
$39$	0	0
$40$	4.81690e6	0.297507
$41$	−2.39416e7	−1.32320	−0.661601	−	0.749856i	$-0.730123\pi$
$41$	−2.39416e7	−1.32320	−0.661601	+	0.749856i	$0.730123\pi$
$42$	0	0
$43$	−1.06598e7	−0.475491	−0.237746	−	0.971327i	$-0.576408\pi$
$43$	−1.06598e7	−0.475491	−0.237746	+	0.971327i	$0.576408\pi$
$44$	−1.49729e7	−0.602240
$45$	0	0
$46$	−3.34053e7	−1.10003
$47$	−3.40540e7	−1.01795	−0.508977	−	0.860780i	$-0.669976\pi$
$47$	−3.40540e7	−1.01795	−0.508977	+	0.860780i	$0.669976\pi$
$48$	0	0
$49$	9.12761e7	2.26191
$50$	−9.12238e6	−0.206416
$51$	0	0
$52$	−2.71798e7	−0.515503
$53$	4.27411e7	0.744053	0.372027	−	0.928222i	$-0.378663\pi$
$53$	4.27411e7	0.744053	0.372027	+	0.928222i	$0.378663\pi$
$54$	0	0
$55$	−6.87819e7	−1.01354
$56$	−4.69934e7	−0.638544
$57$	0	0
$58$	−3.83908e7	−0.445452
$59$	−7.42079e7	−0.797290	−0.398645	−	0.917105i	$-0.630519\pi$
$59$	−7.42079e7	−0.797290	−0.398645	+	0.917105i	$0.630519\pi$
$60$	0	0
$61$	8.10241e7	0.749255	0.374628	−	0.927175i	$-0.377771\pi$
$61$	8.10241e7	0.749255	0.374628	+	0.927175i	$0.377771\pi$
$62$	−1.90204e7	−0.163477
$63$	0	0
$64$	1.67772e7	0.125000
$65$	−1.24857e8	−0.867567
$66$	0	0
$67$	−1.96509e7	−0.119137	−0.0595683	−	0.998224i	$-0.518972\pi$
$67$	−1.96509e7	−0.119137	−0.0595683	+	0.998224i	$0.518972\pi$
$68$	1.51898e8	0.861514
$69$	0	0
$70$	−2.15876e8	−1.07464
$71$	1.84185e8	0.860186	0.430093	−	0.902785i	$-0.358481\pi$
$71$	1.84185e8	0.860186	0.430093	+	0.902785i	$0.358481\pi$
$72$	0	0
$73$	−2.57038e8	−1.05936	−0.529680	−	0.848197i	$-0.677688\pi$
$73$	−2.57038e8	−1.05936	−0.529680	+	0.848197i	$0.677688\pi$
$74$	1.85251e8	0.718154
$75$	0	0
$76$	−5.40076e7	−0.185692
$77$	6.71033e8	2.17538
$78$	0	0
$79$	6.51592e8	1.88215	0.941075	−	0.338199i	$-0.109818\pi$
$79$	6.51592e8	1.88215	0.941075	+	0.338199i	$0.109818\pi$
$80$	7.70703e7	0.210369
$81$	0	0
$82$	−3.83066e8	−0.935646
$83$	1.76386e7	0.0407956	0.0203978	−	0.999792i	$-0.493507\pi$
$83$	1.76386e7	0.0407956	0.0203978	+	0.999792i	$0.493507\pi$
$84$	0	0
$85$	6.97782e8	1.44989
$86$	−1.70557e8	−0.336223
$87$	0	0
$88$	−2.39567e8	−0.425848
$89$	5.16255e8	0.872186	0.436093	−	0.899902i	$-0.356362\pi$
$89$	5.16255e8	0.872186	0.436093	+	0.899902i	$0.356362\pi$
$90$	0	0
$91$	1.21810e9	1.86207
$92$	−5.34485e8	−0.777840
$93$	0	0
$94$	−5.44864e8	−0.719802
$95$	−2.48097e8	−0.312511
$96$	0	0
$97$	−4.34233e8	−0.498023	−0.249012	−	0.968500i	$-0.580106\pi$
$97$	−4.34233e8	−0.498023	−0.249012	+	0.968500i	$0.580106\pi$
$98$	1.46042e9	1.59941
$99$	0	0
$100$	−1.45958e8	−0.145958
$101$	−8.46603e7	−0.0809531	−0.0404766	−	0.999180i	$-0.512888\pi$
$101$	−8.46603e7	−0.0809531	−0.0404766	+	0.999180i	$0.512888\pi$
$102$	0	0
$103$	6.69801e7	0.0586379	0.0293189	−	0.999570i	$-0.490666\pi$
$103$	6.69801e7	0.0586379	0.0293189	+	0.999570i	$0.490666\pi$
$104$	−4.34876e8	−0.364515
$105$	0	0
$106$	6.83857e8	0.526125
$107$	2.22121e9	1.63818	0.819091	−	0.573663i	$-0.194478\pi$
$107$	2.22121e9	1.63818	0.819091	+	0.573663i	$0.194478\pi$
$108$	0	0
$109$	−4.45158e8	−0.302061	−0.151031	−	0.988529i	$-0.548259\pi$
$109$	−4.45158e8	−0.302061	−0.151031	+	0.988529i	$0.548259\pi$
$110$	−1.10051e9	−0.716683
$111$	0	0
$112$	−7.51895e8	−0.451519
$113$	−7.50383e8	−0.432942	−0.216471	−	0.976289i	$-0.569455\pi$
$113$	−7.50383e8	−0.432942	−0.216471	+	0.976289i	$0.569455\pi$
$114$	0	0
$115$	−2.45529e9	−1.30907
$116$	−6.14253e8	−0.314982
$117$	0	0
$118$	−1.18733e9	−0.563769
$119$	−6.80753e9	−3.11192
$120$	0	0
$121$	1.06290e9	0.450773
$122$	1.29639e9	0.529804
$123$	0	0
$124$	−3.04326e8	−0.115596
$125$	−2.96737e9	−1.08712
$126$	0	0
$127$	4.12423e9	1.40678	0.703391	−	0.710803i	$-0.251668\pi$
$127$	4.12423e9	1.40678	0.703391	+	0.710803i	$0.251668\pi$
$128$	2.68435e8	0.0883883
$129$	0	0
$130$	−1.99771e9	−0.613463
$131$	4.30361e9	1.27677	0.638384	−	0.769718i	$-0.279603\pi$
$131$	4.30361e9	1.27677	0.638384	+	0.769718i	$0.279603\pi$
$132$	0	0
$133$	2.42042e9	0.670748
$134$	−3.14414e8	−0.0842422
$135$	0	0
$136$	2.43037e9	0.609182
$137$	−6.60354e9	−1.60153	−0.800764	−	0.598980i	$-0.795573\pi$
$137$	−6.60354e9	−1.60153	−0.800764	+	0.598980i	$0.795573\pi$
$138$	0	0
$139$	−8.53076e9	−1.93830	−0.969150	−	0.246471i	$-0.920729\pi$
$139$	−8.53076e9	−1.93830	−0.969150	+	0.246471i	$0.920729\pi$
$140$	−3.45402e9	−0.759885
$141$	0	0
$142$	2.94697e9	0.608244
$143$	6.20973e9	1.24183
$144$	0	0
$145$	−2.82172e9	−0.530101
$146$	−4.11260e9	−0.749081
$147$	0	0
$148$	2.96402e9	0.507812
$149$	−7.84160e9	−1.30337	−0.651683	−	0.758491i	$-0.725937\pi$
$149$	−7.84160e9	−1.30337	−0.651683	+	0.758491i	$0.725937\pi$
$150$	0	0
$151$	−2.93588e9	−0.459560	−0.229780	−	0.973243i	$-0.573801\pi$
$151$	−2.93588e9	−0.459560	−0.229780	+	0.973243i	$0.573801\pi$
$152$	−8.64121e8	−0.131304
$153$	0	0
$154$	1.07365e10	1.53823
$155$	−1.39800e9	−0.194542
$156$	0	0
$157$	8.69133e9	1.14166	0.570832	−	0.821067i	$-0.306621\pi$
$157$	8.69133e9	1.14166	0.570832	+	0.821067i	$0.306621\pi$
$158$	1.04255e10	1.33088
$159$	0	0
$160$	1.23313e9	0.148754
$161$	2.39537e10	2.80968
$162$	0	0
$163$	−9.70380e9	−1.07671	−0.538353	−	0.842719i	$-0.680953\pi$
$163$	−9.70380e9	−1.07671	−0.538353	+	0.842719i	$0.680953\pi$
$164$	−6.12906e9	−0.661601
$165$	0	0
$166$	2.82218e8	0.0288468
$167$	1.95126e9	0.194129	0.0970647	−	0.995278i	$-0.469055\pi$
$167$	1.95126e9	0.194129	0.0970647	+	0.995278i	$0.469055\pi$
$168$	0	0
$169$	6.67782e8	0.0629716
$170$	1.11645e10	1.02523
$171$	0	0
$172$	−2.72892e9	−0.237746
$173$	5.23394e9	0.444244	0.222122	−	0.975019i	$-0.428702\pi$
$173$	5.23394e9	0.444244	0.222122	+	0.975019i	$0.428702\pi$
$174$	0	0
$175$	6.54132e9	0.527223
$176$	−3.83307e9	−0.301120
$177$	0	0
$178$	8.26008e9	0.616729
$179$	−1.85510e10	−1.35061	−0.675304	−	0.737539i	$-0.735988\pi$
$179$	−1.85510e10	−1.35061	−0.675304	+	0.737539i	$0.735988\pi$
$180$	0	0
$181$	−2.37080e10	−1.64188	−0.820940	−	0.571015i	$-0.806550\pi$
$181$	−2.37080e10	−1.64188	−0.820940	+	0.571015i	$0.806550\pi$
$182$	1.94896e10	1.31668
$183$	0	0
$184$	−8.55176e9	−0.550016
$185$	1.36159e10	0.854624
$186$	0	0
$187$	−3.47040e10	−2.07535
$188$	−8.71783e9	−0.508977
$189$	0	0
$190$	−3.96956e9	−0.220979
$191$	1.07995e10	0.587157	0.293578	−	0.955935i	$-0.405154\pi$
$191$	1.07995e10	0.587157	0.293578	+	0.955935i	$0.405154\pi$
$192$	0	0
$193$	1.77510e10	0.920906	0.460453	−	0.887684i	$-0.347687\pi$
$193$	1.77510e10	0.920906	0.460453	+	0.887684i	$0.347687\pi$
$194$	−6.94772e9	−0.352156
$195$	0	0
$196$	2.33667e10	1.13095
$197$	1.21392e9	0.0574239	0.0287120	−	0.999588i	$-0.490859\pi$
$197$	1.21392e9	0.0574239	0.0287120	+	0.999588i	$0.490859\pi$
$198$	0	0
$199$	1.57326e10	0.711151	0.355575	−	0.934648i	$-0.384285\pi$
$199$	1.57326e10	0.711151	0.355575	+	0.934648i	$0.384285\pi$
$200$	−2.33533e9	−0.103208
$201$	0	0
$202$	−1.35456e9	−0.0572425
$203$	2.75286e10	1.13776
$204$	0	0
$205$	−2.81554e10	−1.11344
$206$	1.07168e9	0.0414632
$207$	0	0
$208$	−6.95802e9	−0.257751
$209$	1.23390e10	0.447325
$210$	0	0
$211$	8.36693e9	0.290600	0.145300	−	0.989388i	$-0.453585\pi$
$211$	8.36693e9	0.290600	0.145300	+	0.989388i	$0.453585\pi$
$212$	1.09417e10	0.372027
$213$	0	0
$214$	3.55393e10	1.15837
$215$	−1.25360e10	−0.400115
$216$	0	0
$217$	1.36388e10	0.417548
$218$	−7.12253e9	−0.213589
$219$	0	0
$220$	−1.76082e10	−0.506771
$221$	−6.29968e10	−1.77645
$222$	0	0
$223$	4.79121e10	1.29740	0.648699	−	0.761045i	$-0.275313\pi$
$223$	4.79121e10	1.29740	0.648699	+	0.761045i	$0.275313\pi$
$224$	−1.20303e10	−0.319272
$225$	0	0
$226$	−1.20061e10	−0.306136
$227$	−2.71171e10	−0.677840	−0.338920	−	0.940815i	$-0.610062\pi$
$227$	−2.71171e10	−0.677840	−0.338920	+	0.940815i	$0.610062\pi$
$228$	0	0
$229$	5.35161e10	1.28595	0.642976	−	0.765887i	$-0.277700\pi$
$229$	5.35161e10	1.28595	0.642976	+	0.765887i	$0.277700\pi$
$230$	−3.92846e10	−0.925652
$231$	0	0
$232$	−9.82804e9	−0.222726
$233$	−7.18897e10	−1.59796	−0.798978	−	0.601360i	$-0.794626\pi$
$233$	−7.18897e10	−1.59796	−0.798978	+	0.601360i	$0.794626\pi$
$234$	0	0
$235$	−4.00475e10	−0.856584
$236$	−1.89972e10	−0.398645
$237$	0	0
$238$	−1.08920e11	−2.20046
$239$	−2.77049e10	−0.549246	−0.274623	−	0.961552i	$-0.588553\pi$
$239$	−2.77049e10	−0.549246	−0.274623	+	0.961552i	$0.588553\pi$
$240$	0	0
$241$	2.33904e10	0.446643	0.223321	−	0.974745i	$-0.428310\pi$
$241$	2.33904e10	0.446643	0.223321	+	0.974745i	$0.428310\pi$
$242$	1.70064e10	0.318744
$243$	0	0
$244$	2.07422e10	0.374628
$245$	1.07341e11	1.90334
$246$	0	0
$247$	2.23986e10	0.382899
$248$	−4.86921e9	−0.0817384
$249$	0	0
$250$	−4.74779e10	−0.768708
$251$	−9.43150e10	−1.49985	−0.749927	−	0.661521i	$-0.769911\pi$
$251$	−9.43150e10	−1.49985	−0.749927	+	0.661521i	$0.769911\pi$
$252$	0	0
$253$	1.22113e11	1.87379
$254$	6.59878e10	0.994745
$255$	0	0
$256$	4.29497e9	0.0625000
$257$	1.15053e10	0.164512	0.0822559	−	0.996611i	$-0.473788\pi$
$257$	1.15053e10	0.164512	0.0822559	+	0.996611i	$0.473788\pi$
$258$	0	0
$259$	−1.32837e11	−1.83429
$260$	−3.19634e10	−0.433784
$261$	0	0
$262$	6.88578e10	0.902812
$263$	1.70523e9	0.0219776	0.0109888	−	0.999940i	$-0.496502\pi$
$263$	1.70523e9	0.0219776	0.0109888	+	0.999940i	$0.496502\pi$
$264$	0	0
$265$	5.02635e10	0.626104
$266$	3.87268e10	0.474290
$267$	0	0
$268$	−5.03062e9	−0.0595683
$269$	−9.57785e10	−1.11528	−0.557638	−	0.830084i	$-0.688292\pi$
$269$	−9.57785e10	−1.11528	−0.557638	+	0.830084i	$0.688292\pi$
$270$	0	0
$271$	−1.62837e11	−1.83397	−0.916985	−	0.398921i	$-0.869385\pi$
$271$	−1.62837e11	−1.83397	−0.916985	+	0.398921i	$0.869385\pi$
$272$	3.88859e10	0.430757
$273$	0	0
$274$	−1.05657e11	−1.13245
$275$	3.33469e10	0.351607
$276$	0	0
$277$	5.06966e10	0.517392	0.258696	−	0.965959i	$-0.416707\pi$
$277$	5.06966e10	0.517392	0.258696	+	0.965959i	$0.416707\pi$
$278$	−1.36492e11	−1.37059
$279$	0	0
$280$	−5.52642e10	−0.537320
$281$	1.11599e11	1.06778	0.533890	−	0.845554i	$-0.320730\pi$
$281$	1.11599e11	1.06778	0.533890	+	0.845554i	$0.320730\pi$
$282$	0	0
$283$	−2.58811e9	−0.0239852	−0.0119926	−	0.999928i	$-0.503817\pi$
$283$	−2.58811e9	−0.0239852	−0.0119926	+	0.999928i	$0.503817\pi$
$284$	4.71515e10	0.430093
$285$	0	0
$286$	9.93557e10	0.878103
$287$	2.74682e11	2.38980
$288$	0	0
$289$	2.33479e11	1.96882
$290$	−4.51476e10	−0.374838
$291$	0	0
$292$	−6.58017e10	−0.529680
$293$	−6.10847e10	−0.484204	−0.242102	−	0.970251i	$-0.577837\pi$
$293$	−6.10847e10	−0.484204	−0.242102	+	0.970251i	$0.577837\pi$
$294$	0	0
$295$	−8.72685e10	−0.670901
$296$	4.74243e10	0.359077
$297$	0	0
$298$	−1.25466e11	−0.921619
$299$	2.21667e11	1.60391
$300$	0	0
$301$	1.22300e11	0.858772
$302$	−4.69741e10	−0.324958
$303$	0	0
$304$	−1.38259e10	−0.0928460
$305$	9.52843e10	0.630481
$306$	0	0
$307$	−7.67436e9	−0.0493082	−0.0246541	−	0.999696i	$-0.507848\pi$
$307$	−7.67436e9	−0.0493082	−0.0246541	+	0.999696i	$0.507848\pi$
$308$	1.71784e11	1.08769
$309$	0	0
$310$	−2.23679e10	−0.137562
$311$	−9.58718e10	−0.581124	−0.290562	−	0.956856i	$-0.593842\pi$
$311$	−9.58718e10	−0.581124	−0.290562	+	0.956856i	$0.593842\pi$
$312$	0	0
$313$	−3.95897e10	−0.233148	−0.116574	−	0.993182i	$-0.537191\pi$
$313$	−3.95897e10	−0.233148	−0.116574	+	0.993182i	$0.537191\pi$
$314$	1.39061e11	0.807278
$315$	0	0
$316$	1.66808e11	0.941075
$317$	−2.78579e11	−1.54946	−0.774732	−	0.632289i	$-0.782115\pi$
$317$	−2.78579e11	−1.54946	−0.774732	+	0.632289i	$0.782115\pi$
$318$	0	0
$319$	1.40338e11	0.758780
$320$	1.97300e10	0.105185
$321$	0	0
$322$	3.83259e11	1.98674
$323$	−1.25178e11	−0.639905
$324$	0	0
$325$	6.05333e10	0.300967
$326$	−1.55261e11	−0.761347
$327$	0	0
$328$	−9.80649e10	−0.467823
$329$	3.90702e11	1.83850
$330$	0	0
$331$	−1.00317e11	−0.459356	−0.229678	−	0.973267i	$-0.573767\pi$
$331$	−1.00317e11	−0.459356	−0.229678	+	0.973267i	$0.573767\pi$
$332$	4.51548e9	0.0203978
$333$	0	0
$334$	3.12202e10	0.137270
$335$	−2.31094e10	−0.100251
$336$	0	0
$337$	−2.88216e11	−1.21726	−0.608630	−	0.793455i	$-0.708280\pi$
$337$	−2.88216e11	−1.21726	−0.608630	+	0.793455i	$0.708280\pi$
$338$	1.06845e10	0.0445276
$339$	0	0
$340$	1.78632e11	0.724944
$341$	6.95289e10	0.278465
$342$	0	0
$343$	−5.84234e11	−2.27910
$344$	−4.36627e10	−0.168111
$345$	0	0
$346$	8.37431e10	0.314128
$347$	6.57660e10	0.243511	0.121755	−	0.992560i	$-0.461148\pi$
$347$	6.57660e10	0.243511	0.121755	+	0.992560i	$0.461148\pi$
$348$	0	0
$349$	−1.72257e11	−0.621529	−0.310765	−	0.950487i	$-0.600585\pi$
$349$	−1.72257e11	−0.621529	−0.310765	+	0.950487i	$0.600585\pi$
$350$	1.04661e11	0.372803
$351$	0	0
$352$	−6.13291e10	−0.212924
$353$	−3.51189e11	−1.20380	−0.601900	−	0.798571i	$-0.705589\pi$
$353$	−3.51189e11	−1.20380	−0.601900	+	0.798571i	$0.705589\pi$
$354$	0	0
$355$	2.16602e11	0.723827
$356$	1.32161e11	0.436093
$357$	0	0
$358$	−2.96816e11	−0.955024
$359$	−4.18672e11	−1.33030	−0.665149	−	0.746711i	$-0.731632\pi$
$359$	−4.18672e11	−1.33030	−0.665149	+	0.746711i	$0.731632\pi$
$360$	0	0
$361$	−2.78181e11	−0.862074
$362$	−3.79328e11	−1.16098
$363$	0	0
$364$	3.11834e11	0.931036
$365$	−3.02276e11	−0.891428
$366$	0	0
$367$	−4.16704e11	−1.19903	−0.599515	−	0.800364i	$-0.704640\pi$
$367$	−4.16704e11	−1.19903	−0.599515	+	0.800364i	$0.704640\pi$
$368$	−1.36828e11	−0.388920
$369$	0	0
$370$	2.17855e11	0.604310
$371$	−4.90368e11	−1.34382
$372$	0	0
$373$	6.07761e11	1.62571	0.812855	−	0.582466i	$-0.197912\pi$
$373$	6.07761e11	1.62571	0.812855	+	0.582466i	$0.197912\pi$
$374$	−5.55264e11	−1.46750
$375$	0	0
$376$	−1.39485e11	−0.359901
$377$	2.54749e11	0.649497
$378$	0	0
$379$	−7.79404e9	−0.0194038	−0.00970189	−	0.999953i	$-0.503088\pi$
$379$	−7.79404e9	−0.0194038	−0.00970189	+	0.999953i	$0.503088\pi$
$380$	−6.35129e10	−0.156256
$381$	0	0
$382$	1.72792e11	0.415182
$383$	−1.38125e11	−0.328003	−0.164002	−	0.986460i	$-0.552440\pi$
$383$	−1.38125e11	−0.328003	−0.164002	+	0.986460i	$0.552440\pi$
$384$	0	0
$385$	7.89135e11	1.83053
$386$	2.84016e11	0.651179
$387$	0	0
$388$	−1.11164e11	−0.249012
$389$	−7.34395e10	−0.162613	−0.0813067	−	0.996689i	$-0.525909\pi$
$389$	−7.34395e10	−0.162613	−0.0813067	+	0.996689i	$0.525909\pi$
$390$	0	0
$391$	−1.23882e12	−2.68048
$392$	3.73867e11	0.799705
$393$	0	0
$394$	1.94228e10	0.0406048
$395$	7.66273e11	1.58379
$396$	0	0
$397$	2.58425e10	0.0522128	0.0261064	−	0.999659i	$-0.491689\pi$
$397$	2.58425e10	0.0522128	0.0261064	+	0.999659i	$0.491689\pi$
$398$	2.51722e11	0.502859
$399$	0	0
$400$	−3.73653e10	−0.0729791
$401$	7.48013e10	0.144464	0.0722319	−	0.997388i	$-0.476988\pi$
$401$	7.48013e10	0.144464	0.0722319	+	0.997388i	$0.476988\pi$
$402$	0	0
$403$	1.26213e11	0.238359
$404$	−2.16730e10	−0.0404766
$405$	0	0
$406$	4.40457e11	0.804520
$407$	−6.77185e11	−1.22330
$408$	0	0
$409$	5.49491e11	0.970969	0.485485	−	0.874245i	$-0.338643\pi$
$409$	5.49491e11	0.970969	0.485485	+	0.874245i	$0.338643\pi$
$410$	−4.50486e11	−0.787324
$411$	0	0
$412$	1.71469e10	0.0293189
$413$	8.51388e11	1.43997
$414$	0	0
$415$	2.07430e10	0.0343285
$416$	−1.11328e11	−0.182258
$417$	0	0
$418$	1.97425e11	0.316306
$419$	−3.52117e11	−0.558116	−0.279058	−	0.960274i	$-0.590022\pi$
$419$	−3.52117e11	−0.558116	−0.279058	+	0.960274i	$0.590022\pi$
$420$	0	0
$421$	1.30073e10	0.0201798	0.0100899	−	0.999949i	$-0.496788\pi$
$421$	1.30073e10	0.0201798	0.0100899	+	0.999949i	$0.496788\pi$
$422$	1.33871e11	0.205485
$423$	0	0
$424$	1.75067e11	0.263063
$425$	−3.38299e11	−0.502980
$426$	0	0
$427$	−9.29589e11	−1.35321
$428$	5.68629e11	0.819091
$429$	0	0
$430$	−2.00575e11	−0.282924
$431$	6.43135e10	0.0897748	0.0448874	−	0.998992i	$-0.485707\pi$
$431$	6.43135e10	0.0897748	0.0448874	+	0.998992i	$0.485707\pi$
$432$	0	0
$433$	7.95481e11	1.08751	0.543756	−	0.839243i	$-0.317002\pi$
$433$	7.95481e11	1.08751	0.543756	+	0.839243i	$0.317002\pi$
$434$	2.18220e11	0.295251
$435$	0	0
$436$	−1.13960e11	−0.151031
$437$	4.40464e11	0.577755
$438$	0	0
$439$	1.66668e11	0.214172	0.107086	−	0.994250i	$-0.465848\pi$
$439$	1.66668e11	0.214172	0.107086	+	0.994250i	$0.465848\pi$
$440$	−2.81731e11	−0.358341
$441$	0	0
$442$	−1.00795e12	−1.25614
$443$	1.03327e12	1.27467	0.637334	−	0.770587i	$-0.280037\pi$
$443$	1.03327e12	1.27467	0.637334	+	0.770587i	$0.280037\pi$
$444$	0	0
$445$	6.07116e11	0.733924
$446$	7.66594e11	0.917399
$447$	0	0
$448$	−1.92485e11	−0.225759
$449$	−1.05633e12	−1.22656	−0.613281	−	0.789865i	$-0.710151\pi$
$449$	−1.05633e12	−1.22656	−0.613281	+	0.789865i	$0.710151\pi$
$450$	0	0
$451$	1.40030e12	1.59377
$452$	−1.92098e11	−0.216471
$453$	0	0
$454$	−4.33874e11	−0.479305
$455$	1.43249e12	1.56689
$456$	0	0
$457$	6.28099e11	0.673604	0.336802	−	0.941575i	$-0.390655\pi$
$457$	6.28099e11	0.673604	0.336802	+	0.941575i	$0.390655\pi$
$458$	8.56257e11	0.909305
$459$	0	0
$460$	−6.28554e11	−0.654535
$461$	8.92984e11	0.920851	0.460426	−	0.887698i	$-0.347697\pi$
$461$	8.92984e11	0.920851	0.460426	+	0.887698i	$0.347697\pi$
$462$	0	0
$463$	−1.12541e12	−1.13814	−0.569071	−	0.822289i	$-0.692697\pi$
$463$	−1.12541e12	−1.13814	−0.569071	+	0.822289i	$0.692697\pi$
$464$	−1.57249e11	−0.157491
$465$	0	0
$466$	−1.15023e12	−1.12993
$467$	1.04729e12	1.01892	0.509461	−	0.860494i	$-0.329845\pi$
$467$	1.04729e12	1.01892	0.509461	+	0.860494i	$0.329845\pi$
$468$	0	0
$469$	2.25454e11	0.215170
$470$	−6.40760e11	−0.605697
$471$	0	0
$472$	−3.03956e11	−0.281885
$473$	6.23472e11	0.572720
$474$	0	0
$475$	1.20283e11	0.108413
$476$	−1.74273e12	−1.55596
$477$	0	0
$478$	−4.43279e11	−0.388375
$479$	1.27628e12	1.10774	0.553869	−	0.832604i	$-0.313151\pi$
$479$	1.27628e12	1.10774	0.553869	+	0.832604i	$0.313151\pi$
$480$	0	0
$481$	−1.22927e12	−1.04711
$482$	3.74246e11	0.315824
$483$	0	0
$484$	2.72102e11	0.225386
$485$	−5.10658e11	−0.419075
$486$	0	0
$487$	6.62479e11	0.533693	0.266847	−	0.963739i	$-0.414018\pi$
$487$	6.62479e11	0.533693	0.266847	+	0.963739i	$0.414018\pi$
$488$	3.31875e11	0.264902
$489$	0	0
$490$	1.71745e12	1.34587
$491$	−2.32551e12	−1.80572	−0.902861	−	0.429932i	$-0.858538\pi$
$491$	−2.32551e12	−1.80572	−0.902861	+	0.429932i	$0.858538\pi$
$492$	0	0
$493$	−1.42370e12	−1.08545
$494$	3.58377e11	0.270750
$495$	0	0
$496$	−7.79074e10	−0.0577978
$497$	−2.11316e12	−1.55356
$498$	0	0
$499$	−3.11454e10	−0.0224875	−0.0112438	−	0.999937i	$-0.503579\pi$
$499$	−3.11454e10	−0.0224875	−0.0112438	+	0.999937i	$0.503579\pi$
$500$	−7.59647e11	−0.543559
$501$	0	0
$502$	−1.50904e12	−1.06056
$503$	−5.25877e11	−0.366293	−0.183146	−	0.983086i	$-0.558628\pi$
$503$	−5.25877e11	−0.366293	−0.183146	+	0.983086i	$0.558628\pi$
$504$	0	0
$505$	−9.95605e10	−0.0681202
$506$	1.95381e12	1.32497
$507$	0	0
$508$	1.05580e12	0.703391
$509$	−8.47631e11	−0.559727	−0.279864	−	0.960040i	$-0.590289\pi$
$509$	−8.47631e11	−0.559727	−0.279864	+	0.960040i	$0.590289\pi$
$510$	0	0
$511$	2.94899e12	1.91329
$512$	6.87195e10	0.0441942
$513$	0	0
$514$	1.84084e11	0.116327
$515$	7.87686e10	0.0493424
$516$	0	0
$517$	1.99175e12	1.22610
$518$	−2.12538e12	−1.29704
$519$	0	0
$520$	−5.11415e11	−0.306731
$521$	−6.24007e11	−0.371039	−0.185520	−	0.982641i	$-0.559397\pi$
$521$	−6.24007e11	−0.371039	−0.185520	+	0.982641i	$0.559397\pi$
$522$	0	0
$523$	−2.13963e12	−1.25049	−0.625247	−	0.780427i	$-0.715002\pi$
$523$	−2.13963e12	−1.25049	−0.625247	+	0.780427i	$0.715002\pi$
$524$	1.10172e12	0.638384
$525$	0	0
$526$	2.72836e10	0.0155405
$527$	−7.05360e11	−0.398349
$528$	0	0
$529$	2.55789e12	1.42014
$530$	8.04216e11	0.442722
$531$	0	0
$532$	6.19629e11	0.335374
$533$	2.54191e12	1.36423
$534$	0	0
$535$	2.61214e12	1.37849
$536$	−8.04899e10	−0.0421211
$537$	0	0
$538$	−1.53246e12	−0.788620
$539$	−5.33856e12	−2.72442
$540$	0	0
$541$	−1.28484e12	−0.644853	−0.322427	−	0.946594i	$-0.604499\pi$
$541$	−1.28484e12	−0.644853	−0.322427	+	0.946594i	$0.604499\pi$
$542$	−2.60540e12	−1.29681
$543$	0	0
$544$	6.22175e11	0.304591
$545$	−5.23506e11	−0.254177
$546$	0	0
$547$	1.66902e12	0.797112	0.398556	−	0.917144i	$-0.369511\pi$
$547$	1.66902e12	0.797112	0.398556	+	0.917144i	$0.369511\pi$
$548$	−1.69051e12	−0.800764
$549$	0	0
$550$	5.33550e11	0.248624
$551$	5.06199e11	0.233959
$552$	0	0
$553$	−7.47572e12	−3.39930
$554$	8.11146e11	0.365852
$555$	0	0
$556$	−2.18387e12	−0.969150
$557$	−3.34603e12	−1.47293	−0.736464	−	0.676477i	$-0.763506\pi$
$557$	−3.34603e12	−1.47293	−0.736464	+	0.676477i	$0.763506\pi$
$558$	0	0
$559$	1.13177e12	0.490234
$560$	−8.84228e11	−0.379943
$561$	0	0
$562$	1.78558e12	0.755034
$563$	1.34110e12	0.562564	0.281282	−	0.959625i	$-0.409240\pi$
$563$	1.34110e12	0.562564	0.281282	+	0.959625i	$0.409240\pi$
$564$	0	0
$565$	−8.82450e11	−0.364311
$566$	−4.14098e10	−0.0169601
$567$	0	0
$568$	7.54423e11	0.304122
$569$	4.49869e12	1.79921	0.899604	−	0.436707i	$-0.143855\pi$
$569$	4.49869e12	1.79921	0.899604	+	0.436707i	$0.143855\pi$
$570$	0	0
$571$	−7.36379e11	−0.289894	−0.144947	−	0.989439i	$-0.546301\pi$
$571$	−7.36379e11	−0.289894	−0.144947	+	0.989439i	$0.546301\pi$
$572$	1.58969e12	0.620913
$573$	0	0
$574$	4.39492e12	1.68985
$575$	1.19038e12	0.454128
$576$	0	0
$577$	−3.19860e11	−0.120135	−0.0600673	−	0.998194i	$-0.519132\pi$
$577$	−3.19860e11	−0.120135	−0.0600673	+	0.998194i	$0.519132\pi$
$578$	3.73566e12	1.39217
$579$	0	0
$580$	−7.22361e11	−0.265050
$581$	−2.02368e11	−0.0736798
$582$	0	0
$583$	−2.49984e12	−0.896197
$584$	−1.05283e12	−0.374541
$585$	0	0
$586$	−9.77356e11	−0.342384
$587$	2.83544e11	0.0985709	0.0492855	−	0.998785i	$-0.484306\pi$
$587$	2.83544e11	0.0985709	0.0492855	+	0.998785i	$0.484306\pi$
$588$	0	0
$589$	2.50792e11	0.0858607
$590$	−1.39630e12	−0.474399
$591$	0	0
$592$	7.58788e11	0.253906
$593$	1.27449e12	0.423245	0.211622	−	0.977352i	$-0.432125\pi$
$593$	1.27449e12	0.423245	0.211622	+	0.977352i	$0.432125\pi$
$594$	0	0
$595$	−8.00565e12	−2.61861
$596$	−2.00745e12	−0.651683
$597$	0	0
$598$	3.54668e12	1.13414
$599$	2.17615e12	0.690666	0.345333	−	0.938480i	$-0.387766\pi$
$599$	2.17615e12	0.690666	0.345333	+	0.938480i	$0.387766\pi$
$600$	0	0
$601$	−3.74900e12	−1.17214	−0.586072	−	0.810259i	$-0.699326\pi$
$601$	−3.74900e12	−1.17214	−0.586072	+	0.810259i	$0.699326\pi$
$602$	1.95680e12	0.607244
$603$	0	0
$604$	−7.51586e11	−0.229780
$605$	1.24997e12	0.379315
$606$	0	0
$607$	2.79049e12	0.834318	0.417159	−	0.908834i	$-0.363026\pi$
$607$	2.79049e12	0.834318	0.417159	+	0.908834i	$0.363026\pi$
$608$	−2.21215e11	−0.0656521
$609$	0	0
$610$	1.52455e12	0.445818
$611$	3.61555e12	1.04952
$612$	0	0
$613$	6.25456e12	1.78906	0.894530	−	0.447008i	$-0.147510\pi$
$613$	6.25456e12	1.78906	0.894530	+	0.447008i	$0.147510\pi$
$614$	−1.22790e11	−0.0348662
$615$	0	0
$616$	2.74855e12	0.769114
$617$	−3.55693e12	−0.988081	−0.494040	−	0.869439i	$-0.664481\pi$
$617$	−3.55693e12	−0.988081	−0.494040	+	0.869439i	$0.664481\pi$
$618$	0	0
$619$	2.95300e12	0.808455	0.404228	−	0.914658i	$-0.367540\pi$
$619$	2.95300e12	0.808455	0.404228	+	0.914658i	$0.367540\pi$
$620$	−3.57887e11	−0.0972710
$621$	0	0
$622$	−1.53395e12	−0.410917
$623$	−5.92299e12	−1.57523
$624$	0	0
$625$	−2.37606e12	−0.622869
$626$	−6.33435e11	−0.164861
$627$	0	0
$628$	2.22498e12	0.570832
$629$	6.86994e12	1.74995
$630$	0	0
$631$	1.18375e12	0.297254	0.148627	−	0.988893i	$-0.452515\pi$
$631$	1.18375e12	0.297254	0.148627	+	0.988893i	$0.452515\pi$
$632$	2.66892e12	0.665440
$633$	0	0
$634$	−4.45726e12	−1.09564
$635$	4.85010e12	1.18377
$636$	0	0
$637$	−9.69088e12	−2.33204
$638$	2.24540e12	0.536538
$639$	0	0
$640$	3.15680e11	0.0743768
$641$	−3.41746e12	−0.799544	−0.399772	−	0.916615i	$-0.630911\pi$
$641$	−3.41746e12	−0.799544	−0.399772	+	0.916615i	$0.630911\pi$
$642$	0	0
$643$	−1.97614e12	−0.455898	−0.227949	−	0.973673i	$-0.573202\pi$
$643$	−1.97614e12	−0.455898	−0.227949	+	0.973673i	$0.573202\pi$
$644$	6.13215e12	1.40484
$645$	0	0
$646$	−2.00284e12	−0.452481
$647$	1.33446e12	0.299388	0.149694	−	0.988732i	$-0.452171\pi$
$647$	1.33446e12	0.299388	0.149694	+	0.988732i	$0.452171\pi$
$648$	0	0
$649$	4.34027e12	0.960320
$650$	9.68533e11	0.212816
$651$	0	0
$652$	−2.48417e12	−0.538353
$653$	3.96835e12	0.854084	0.427042	−	0.904232i	$-0.359556\pi$
$653$	3.96835e12	0.854084	0.427042	+	0.904232i	$0.359556\pi$
$654$	0	0
$655$	5.06105e12	1.07437
$656$	−1.56904e12	−0.330801
$657$	0	0
$658$	6.25123e12	1.30002
$659$	4.97515e12	1.02759	0.513797	−	0.857912i	$-0.328238\pi$
$659$	4.97515e12	1.02759	0.513797	+	0.857912i	$0.328238\pi$
$660$	0	0
$661$	4.92013e12	1.00247	0.501234	−	0.865312i	$-0.332880\pi$
$661$	4.92013e12	1.00247	0.501234	+	0.865312i	$0.332880\pi$
$662$	−1.60508e12	−0.324814
$663$	0	0
$664$	7.22477e10	0.0144234
$665$	2.84642e12	0.564419
$666$	0	0
$667$	5.00959e12	0.980023
$668$	4.99523e11	0.0970647
$669$	0	0
$670$	−3.69751e11	−0.0708879
$671$	−4.73894e12	−0.902463
$672$	0	0
$673$	−9.56217e10	−0.0179675	−0.00898377	−	0.999960i	$-0.502860\pi$
$673$	−9.56217e10	−0.0179675	−0.00898377	+	0.999960i	$0.502860\pi$
$674$	−4.61145e12	−0.860732
$675$	0	0
$676$	1.70952e11	0.0314858
$677$	−3.35880e12	−0.614518	−0.307259	−	0.951626i	$-0.599412\pi$
$677$	−3.35880e12	−0.614518	−0.307259	+	0.951626i	$0.599412\pi$
$678$	0	0
$679$	4.98195e12	0.899468
$680$	2.85811e12	0.512613
$681$	0	0
$682$	1.11246e12	0.196904
$683$	−5.37711e12	−0.945487	−0.472743	−	0.881200i	$-0.656736\pi$
$683$	−5.37711e12	−0.945487	−0.472743	+	0.881200i	$0.656736\pi$
$684$	0	0
$685$	−7.76577e12	−1.34765
$686$	−9.34774e12	−1.61157
$687$	0	0
$688$	−6.98603e11	−0.118873
$689$	−4.53786e12	−0.767123
$690$	0	0
$691$	6.82194e12	1.13830	0.569150	−	0.822234i	$-0.307272\pi$
$691$	6.82194e12	1.13830	0.569150	+	0.822234i	$0.307272\pi$
$692$	1.33989e12	0.222122
$693$	0	0
$694$	1.05226e12	0.172188
$695$	−1.00322e13	−1.63104
$696$	0	0
$697$	−1.42058e13	−2.27992
$698$	−2.75611e12	−0.439487
$699$	0	0
$700$	1.67458e12	0.263611
$701$	−4.41615e12	−0.690737	−0.345369	−	0.938467i	$-0.612246\pi$
$701$	−4.41615e12	−0.690737	−0.345369	+	0.938467i	$0.612246\pi$
$702$	0	0
$703$	−2.44262e12	−0.377186
$704$	−9.81266e11	−0.150560
$705$	0	0
$706$	−5.61902e12	−0.851215
$707$	9.71307e11	0.146207
$708$	0	0
$709$	−8.20276e12	−1.21914	−0.609568	−	0.792734i	$-0.708657\pi$
$709$	−8.20276e12	−1.21914	−0.609568	+	0.792734i	$0.708657\pi$
$710$	3.46563e12	0.511823
$711$	0	0
$712$	2.11458e12	0.308364
$713$	2.48196e12	0.359659
$714$	0	0
$715$	7.30264e12	1.04497
$716$	−4.74906e12	−0.675304
$717$	0	0
$718$	−6.69875e12	−0.940663
$719$	9.34677e12	1.30431	0.652156	−	0.758084i	$-0.273865\pi$
$719$	9.34677e12	1.30431	0.652156	+	0.758084i	$0.273865\pi$
$720$	0	0
$721$	−7.68463e11	−0.105904
$722$	−4.45089e12	−0.609578
$723$	0	0
$724$	−6.06925e12	−0.820940
$725$	1.36803e12	0.183897
$726$	0	0
$727$	1.15218e13	1.52973	0.764866	−	0.644189i	$-0.222805\pi$
$727$	1.15218e13	1.52973	0.764866	+	0.644189i	$0.222805\pi$
$728$	4.98934e12	0.658342
$729$	0	0
$730$	−4.83642e12	−0.630335
$731$	−6.32503e12	−0.819284
$732$	0	0
$733$	7.43504e12	0.951296	0.475648	−	0.879636i	$-0.342214\pi$
$733$	7.43504e12	0.951296	0.475648	+	0.879636i	$0.342214\pi$
$734$	−6.66726e12	−0.847842
$735$	0	0
$736$	−2.18925e12	−0.275008
$737$	1.14934e12	0.143498
$738$	0	0
$739$	5.94698e12	0.733494	0.366747	−	0.930321i	$-0.380471\pi$
$739$	5.94698e12	0.733494	0.366747	+	0.930321i	$0.380471\pi$
$740$	3.48568e12	0.427312
$741$	0	0
$742$	−7.84589e12	−0.950221
$743$	4.38079e12	0.527354	0.263677	−	0.964611i	$-0.415065\pi$
$743$	4.38079e12	0.527354	0.263677	+	0.964611i	$0.415065\pi$
$744$	0	0
$745$	−9.22172e12	−1.09675
$746$	9.72418e12	1.14955
$747$	0	0
$748$	−8.88422e12	−1.03768
$749$	−2.54839e13	−2.95868
$750$	0	0
$751$	−5.24654e11	−0.0601856	−0.0300928	−	0.999547i	$-0.509580\pi$
$751$	−5.24654e11	−0.0601856	−0.0300928	+	0.999547i	$0.509580\pi$
$752$	−2.23176e12	−0.254488
$753$	0	0
$754$	4.07599e12	0.459263
$755$	−3.45260e12	−0.386710
$756$	0	0
$757$	3.95445e12	0.437678	0.218839	−	0.975761i	$-0.429773\pi$
$757$	3.95445e12	0.437678	0.218839	+	0.975761i	$0.429773\pi$
$758$	−1.24705e11	−0.0137205
$759$	0	0
$760$	−1.01621e12	−0.110489
$761$	−1.44239e13	−1.55902	−0.779508	−	0.626392i	$-0.784531\pi$
$761$	−1.44239e13	−1.55902	−0.779508	+	0.626392i	$0.784531\pi$
$762$	0	0
$763$	5.10730e12	0.545545
$764$	2.76468e12	0.293578
$765$	0	0
$766$	−2.21000e12	−0.231933
$767$	7.87873e12	0.822010
$768$	0	0
$769$	1.83772e13	1.89501	0.947505	−	0.319740i	$-0.103596\pi$
$769$	1.83772e13	1.89501	0.947505	+	0.319740i	$0.103596\pi$
$770$	1.26262e13	1.29438
$771$	0	0
$772$	4.54426e12	0.460453
$773$	5.50573e12	0.554635	0.277318	−	0.960778i	$-0.410555\pi$
$773$	5.50573e12	0.554635	0.277318	+	0.960778i	$0.410555\pi$
$774$	0	0
$775$	6.77777e11	0.0674884
$776$	−1.77862e12	−0.176078
$777$	0	0
$778$	−1.17503e12	−0.114985
$779$	5.05089e12	0.491416
$780$	0	0
$781$	−1.07726e13	−1.03608
$782$	−1.98211e13	−1.89539
$783$	0	0
$784$	5.98187e12	0.565477
$785$	1.02210e13	0.960683
$786$	0	0
$787$	1.19140e13	1.10706	0.553531	−	0.832828i	$-0.313280\pi$
$787$	1.19140e13	1.10706	0.553531	+	0.832828i	$0.313280\pi$
$788$	3.10764e11	0.0287120
$789$	0	0
$790$	1.22604e13	1.11991
$791$	8.60914e12	0.781926
$792$	0	0
$793$	−8.60241e12	−0.772486
$794$	4.13480e11	0.0369200
$795$	0	0
$796$	4.02754e12	0.355575
$797$	1.72903e13	1.51789	0.758946	−	0.651153i	$-0.225714\pi$
$797$	1.72903e13	1.51789	0.758946	+	0.651153i	$0.225714\pi$
$798$	0	0
$799$	−2.02060e13	−1.75396
$800$	−5.97845e11	−0.0516040
$801$	0	0
$802$	1.19682e12	0.102151
$803$	1.50336e13	1.27598
$804$	0	0
$805$	2.81695e13	2.36428
$806$	2.01941e12	0.168545
$807$	0	0
$808$	−3.46769e11	−0.0286212
$809$	−5.01450e12	−0.411584	−0.205792	−	0.978596i	$-0.565977\pi$
$809$	−5.01450e12	−0.411584	−0.205792	+	0.978596i	$0.565977\pi$
$810$	0	0
$811$	−7.01801e12	−0.569666	−0.284833	−	0.958577i	$-0.591938\pi$
$811$	−7.01801e12	−0.569666	−0.284833	+	0.958577i	$0.591938\pi$
$812$	7.04732e12	0.568882
$813$	0	0
$814$	−1.08350e13	−0.865003
$815$	−1.14117e13	−0.906024
$816$	0	0
$817$	2.24887e12	0.176590
$818$	8.79185e12	0.686579
$819$	0	0
$820$	−7.20777e12	−0.556722
$821$	4.20014e12	0.322641	0.161320	−	0.986902i	$-0.448425\pi$
$821$	4.20014e12	0.322641	0.161320	+	0.986902i	$0.448425\pi$
$822$	0	0
$823$	−1.57266e13	−1.19491	−0.597455	−	0.801903i	$-0.703821\pi$
$823$	−1.57266e13	−1.19491	−0.597455	+	0.801903i	$0.703821\pi$
$824$	2.74350e11	0.0207316
$825$	0	0
$826$	1.36222e13	1.01821
$827$	−1.63254e13	−1.21364	−0.606820	−	0.794840i	$-0.707555\pi$
$827$	−1.63254e13	−1.21364	−0.606820	+	0.794840i	$0.707555\pi$
$828$	0	0
$829$	1.72909e12	0.127152	0.0635759	−	0.997977i	$-0.479749\pi$
$829$	1.72909e12	0.127152	0.0635759	+	0.997977i	$0.479749\pi$
$830$	3.31888e11	0.0242739
$831$	0	0
$832$	−1.78125e12	−0.128876
$833$	5.41589e13	3.89733
$834$	0	0
$835$	2.29468e12	0.163355
$836$	3.15879e12	0.223662
$837$	0	0
$838$	−5.63388e12	−0.394647
$839$	−1.46998e13	−1.02420	−0.512098	−	0.858927i	$-0.671132\pi$
$839$	−1.46998e13	−1.02420	−0.512098	+	0.858927i	$0.671132\pi$
$840$	0	0
$841$	−8.74991e12	−0.603145
$842$	2.08116e11	0.0142693
$843$	0	0
$844$	2.14193e12	0.145300
$845$	7.85311e11	0.0529891
$846$	0	0
$847$	−1.21946e13	−0.814129
$848$	2.80108e12	0.186013
$849$	0	0
$850$	−5.41278e12	−0.355660
$851$	−2.41733e13	−1.57999
$852$	0	0
$853$	1.49168e13	0.964730	0.482365	−	0.875970i	$-0.339778\pi$
$853$	1.49168e13	0.964730	0.482365	+	0.875970i	$0.339778\pi$
$854$	−1.48734e13	−0.956865
$855$	0	0
$856$	9.09807e12	0.579185
$857$	2.71254e13	1.71776	0.858880	−	0.512176i	$-0.171161\pi$
$857$	2.71254e13	1.71776	0.858880	+	0.512176i	$0.171161\pi$
$858$	0	0
$859$	3.63710e11	0.0227922	0.0113961	−	0.999935i	$-0.496372\pi$
$859$	3.63710e11	0.0227922	0.0113961	+	0.999935i	$0.496372\pi$
$860$	−3.20921e12	−0.200057
$861$	0	0
$862$	1.02902e12	0.0634804
$863$	2.24882e13	1.38009	0.690044	−	0.723768i	$-0.257591\pi$
$863$	2.24882e13	1.38009	0.690044	+	0.723768i	$0.257591\pi$
$864$	0	0
$865$	6.15512e12	0.373821
$866$	1.27277e13	0.768988
$867$	0	0
$868$	3.49153e12	0.208774
$869$	−3.81103e13	−2.26701
$870$	0	0
$871$	2.08635e12	0.122830
$872$	−1.82337e12	−0.106795
$873$	0	0
$874$	7.04742e12	0.408534
$875$	3.40446e13	1.96342
$876$	0	0
$877$	−2.15332e13	−1.22917	−0.614584	−	0.788851i	$-0.710676\pi$
$877$	−2.15332e13	−1.22917	−0.614584	+	0.788851i	$0.710676\pi$
$878$	2.66669e12	0.151443
$879$	0	0
$880$	−4.50769e12	−0.253386
$881$	−1.34923e13	−0.754558	−0.377279	−	0.926100i	$-0.623140\pi$
$881$	−1.34923e13	−0.754558	−0.377279	+	0.926100i	$0.623140\pi$
$882$	0	0
$883$	−7.07524e12	−0.391668	−0.195834	−	0.980637i	$-0.562741\pi$
$883$	−7.07524e12	−0.391668	−0.195834	+	0.980637i	$0.562741\pi$
$884$	−1.61272e13	−0.888225
$885$	0	0
$886$	1.65323e13	0.901327
$887$	−1.18463e13	−0.642581	−0.321291	−	0.946981i	$-0.604117\pi$
$887$	−1.18463e13	−0.642581	−0.321291	+	0.946981i	$0.604117\pi$
$888$	0	0
$889$	−4.73173e13	−2.54075
$890$	9.71385e12	0.518963
$891$	0	0
$892$	1.22655e13	0.648699
$893$	7.18427e12	0.378052
$894$	0	0
$895$	−2.18160e13	−1.13651
$896$	−3.07976e12	−0.159636
$897$	0	0
$898$	−1.69012e13	−0.867310
$899$	2.85237e12	0.145642
$900$	0	0
$901$	2.53605e13	1.28202
$902$	2.24048e13	1.12697
$903$	0	0
$904$	−3.07357e12	−0.153068
$905$	−2.78806e13	−1.38160
$906$	0	0
$907$	2.20971e13	1.08418	0.542092	−	0.840319i	$-0.317632\pi$
$907$	2.20971e13	1.08418	0.542092	+	0.840319i	$0.317632\pi$
$908$	−6.94198e12	−0.338920
$909$	0	0
$910$	2.29198e13	1.10796
$911$	−7.98152e12	−0.383931	−0.191965	−	0.981402i	$-0.561486\pi$
$911$	−7.98152e12	−0.383931	−0.191965	+	0.981402i	$0.561486\pi$
$912$	0	0
$913$	−1.03165e12	−0.0491374
$914$	1.00496e13	0.476310
$915$	0	0
$916$	1.37001e13	0.642976
$917$	−4.93753e13	−2.30594
$918$	0	0
$919$	−3.54669e13	−1.64023	−0.820113	−	0.572202i	$-0.806090\pi$
$919$	−3.54669e13	−1.64023	−0.820113	+	0.572202i	$0.806090\pi$
$920$	−1.00569e13	−0.462826
$921$	0	0
$922$	1.42877e13	0.651140
$923$	−1.95551e13	−0.886857
$924$	0	0
$925$	−6.60129e12	−0.296477
$926$	−1.80066e13	−0.804787
$927$	0	0
$928$	−2.51598e12	−0.111363
$929$	−1.84302e13	−0.811821	−0.405910	−	0.913913i	$-0.633046\pi$
$929$	−1.84302e13	−0.811821	−0.405910	+	0.913913i	$0.633046\pi$
$930$	0	0
$931$	−1.92562e13	−0.840036
$932$	−1.84038e13	−0.798978
$933$	0	0
$934$	1.67566e13	0.720486
$935$	−4.08119e13	−1.74636
$936$	0	0
$937$	−2.09388e13	−0.887410	−0.443705	−	0.896173i	$-0.646336\pi$
$937$	−2.09388e13	−0.887410	−0.443705	+	0.896173i	$0.646336\pi$
$938$	3.60727e12	0.152148
$939$	0	0
$940$	−1.02522e13	−0.428292
$941$	2.48893e13	1.03481	0.517403	−	0.855742i	$-0.326899\pi$
$941$	2.48893e13	1.03481	0.517403	+	0.855742i	$0.326899\pi$
$942$	0	0
$943$	4.99861e13	2.05848
$944$	−4.86329e12	−0.199322
$945$	0	0
$946$	9.97556e12	0.404974
$947$	−1.24148e13	−0.501608	−0.250804	−	0.968038i	$-0.580695\pi$
$947$	−1.24148e13	−0.501608	−0.250804	+	0.968038i	$0.580695\pi$
$948$	0	0
$949$	2.72899e13	1.09221
$950$	1.92452e12	0.0766596
$951$	0	0
$952$	−2.78836e13	−1.10023
$953$	3.02285e13	1.18713	0.593566	−	0.804785i	$-0.297719\pi$
$953$	3.02285e13	1.18713	0.593566	+	0.804785i	$0.297719\pi$
$954$	0	0
$955$	1.27002e13	0.494079
$956$	−7.09247e12	−0.274623
$957$	0	0
$958$	2.04205e13	0.783289
$959$	7.57624e13	2.89248
$960$	0	0
$961$	−2.50264e13	−0.946551
$962$	−1.96683e13	−0.740421
$963$	0	0
$964$	5.98793e12	0.223321
$965$	2.08752e13	0.774921
$966$	0	0
$967$	2.11452e13	0.777664	0.388832	−	0.921309i	$-0.372879\pi$
$967$	2.11452e13	0.777664	0.388832	+	0.921309i	$0.372879\pi$
$968$	4.35363e12	0.159372
$969$	0	0
$970$	−8.17052e12	−0.296331
$971$	−4.31181e13	−1.55659	−0.778293	−	0.627901i	$-0.783914\pi$
$971$	−4.31181e13	−1.55659	−0.778293	+	0.627901i	$0.783914\pi$
$972$	0	0
$973$	9.78734e13	3.50072
$974$	1.05997e13	0.377378
$975$	0	0
$976$	5.31000e12	0.187314
$977$	−2.83858e13	−0.996725	−0.498362	−	0.866969i	$-0.666065\pi$
$977$	−2.83858e13	−0.996725	−0.498362	+	0.866969i	$0.666065\pi$
$978$	0	0
$979$	−3.01947e13	−1.05053
$980$	2.74792e13	0.951672
$981$	0	0
$982$	−3.72081e13	−1.27684
$983$	3.87980e13	1.32531	0.662657	−	0.748923i	$-0.269429\pi$
$983$	3.87980e13	1.32531	0.662657	+	0.748923i	$0.269429\pi$
$984$	0	0
$985$	1.42757e12	0.0483209
$986$	−2.27792e13	−0.767526
$987$	0	0
$988$	5.73404e12	0.191449
$989$	2.22559e13	0.739712
$990$	0	0
$991$	−3.91150e13	−1.28829	−0.644143	−	0.764905i	$-0.722786\pi$
$991$	−3.91150e13	−1.28829	−0.644143	+	0.764905i	$0.722786\pi$
$992$	−1.24652e12	−0.0408692
$993$	0	0
$994$	−3.38105e13	−1.09853
$995$	1.85015e13	0.598417
$996$	0	0
$997$	1.02904e13	0.329842	0.164921	−	0.986307i	$-0.447263\pi$
$997$	1.02904e13	0.329842	0.164921	+	0.986307i	$0.447263\pi$
$998$	−4.98326e11	−0.0159011
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.10.a.d.1.1	yes	1
3.2	odd	2		54.10.a.a.1.1	✓	1
9.2	odd	6		162.10.c.j.109.1		2
9.4	even	3		162.10.c.a.55.1		2
9.5	odd	6		162.10.c.j.55.1		2
9.7	even	3		162.10.c.a.109.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.10.a.a.1.1	✓	1	3.2	odd	2
54.10.a.d.1.1	yes	1	1.1	even	1	trivial
162.10.c.a.55.1		2	9.4	even	3
162.10.c.a.109.1		2	9.7	even	3
162.10.c.j.55.1		2	9.5	odd	6
162.10.c.j.109.1		2	9.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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