LMFDB - Embedded newform 18.20.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,20,Mod(1,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	18.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:	$41.1870053801$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	18.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-512.000 q^{2} +262144. q^{4} +5.55693e6 q^{5} -4.44964e7 q^{7} -1.34218e8 q^{8} +O(q^{10})$

\(q-512.000 q^{2} +262144. q^{4} +5.55693e6 q^{5} -4.44964e7 q^{7} -1.34218e8 q^{8} -2.84515e9 q^{10} -6.32067e9 q^{11} -3.31250e10 q^{13} +2.27822e10 q^{14} +6.87195e10 q^{16} +7.22355e11 q^{17} -1.31262e12 q^{19} +1.45672e12 q^{20} +3.23619e12 q^{22} -3.37975e12 q^{23} +1.18060e13 q^{25} +1.69600e13 q^{26} -1.16645e13 q^{28} +2.93781e13 q^{29} +1.31976e14 q^{31} -3.51844e13 q^{32} -3.69846e14 q^{34} -2.47264e14 q^{35} -4.66464e14 q^{37} +6.72062e14 q^{38} -7.45839e14 q^{40} -1.88945e15 q^{41} -4.32351e15 q^{43} -1.65693e15 q^{44} +1.73043e15 q^{46} -1.21034e16 q^{47} -9.41896e15 q^{49} -6.04466e15 q^{50} -8.68351e15 q^{52} +3.05939e16 q^{53} -3.51235e16 q^{55} +5.97221e15 q^{56} -1.50416e16 q^{58} -9.90874e15 q^{59} -9.16381e16 q^{61} -6.75720e16 q^{62} +1.80144e16 q^{64} -1.84073e17 q^{65} -1.03349e17 q^{67} +1.89361e17 q^{68} +1.26599e17 q^{70} -2.85448e17 q^{71} +8.75008e17 q^{73} +2.38830e17 q^{74} -3.44096e17 q^{76} +2.81247e17 q^{77} -1.08139e18 q^{79} +3.81869e17 q^{80} +9.67397e17 q^{82} +6.65085e17 q^{83} +4.01408e18 q^{85} +2.21364e18 q^{86} +8.48347e17 q^{88} +2.02099e18 q^{89} +1.47394e18 q^{91} -8.85982e17 q^{92} +6.19693e18 q^{94} -7.29414e18 q^{95} -1.28256e19 q^{97} +4.82251e18 q^{98} +O(q^{100})\)

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 $

$2$	−512.000	−0.707107
$2$	−512.000	−0.707107
$3$	0	0
$3$	0	0
$4$	262144.	0.500000
$5$	5.55693e6	1.27239	0.636194	−	0.771529i	$-0.280508\pi$
$5$	5.55693e6	1.27239	0.636194	+	0.771529i	$0.280508\pi$
$6$	0	0
$7$	−4.44964e7	−0.416767	−0.208384	−	0.978047i	$-0.566820\pi$
$7$	−4.44964e7	−0.416767	−0.208384	+	0.978047i	$0.566820\pi$
$8$	−1.34218e8	−0.353553
$9$	0	0
$10$	−2.84515e9	−0.899715
$11$	−6.32067e9	−0.808226	−0.404113	−	0.914709i	$-0.632420\pi$
$11$	−6.32067e9	−0.808226	−0.404113	+	0.914709i	$0.632420\pi$
$12$	0	0
$13$	−3.31250e10	−0.866351	−0.433175	−	0.901310i	$-0.642607\pi$
$13$	−3.31250e10	−0.866351	−0.433175	+	0.901310i	$0.642607\pi$
$14$	2.27822e10	0.294699
$15$	0	0
$16$	6.87195e10	0.250000
$17$	7.22355e11	1.47736	0.738680	−	0.674057i	$-0.235450\pi$
$17$	7.22355e11	1.47736	0.738680	+	0.674057i	$0.235450\pi$
$18$	0	0
$19$	−1.31262e12	−0.933211	−0.466606	−	0.884465i	$-0.654523\pi$
$19$	−1.31262e12	−0.933211	−0.466606	+	0.884465i	$0.654523\pi$
$20$	1.45672e12	0.636194
$21$	0	0
$22$	3.23619e12	0.571502
$23$	−3.37975e12	−0.391265	−0.195632	−	0.980677i	$-0.562676\pi$
$23$	−3.37975e12	−0.391265	−0.195632	+	0.980677i	$0.562676\pi$
$24$	0	0
$25$	1.18060e13	0.618974
$26$	1.69600e13	0.612602
$27$	0	0
$28$	−1.16645e13	−0.208384
$29$	2.93781e13	0.376047	0.188024	−	0.982164i	$-0.439792\pi$
$29$	2.93781e13	0.376047	0.188024	+	0.982164i	$0.439792\pi$
$30$	0	0
$31$	1.31976e14	0.896521	0.448260	−	0.893903i	$-0.352044\pi$
$31$	1.31976e14	0.896521	0.448260	+	0.893903i	$0.352044\pi$
$32$	−3.51844e13	−0.176777
$33$	0	0
$34$	−3.69846e14	−1.04465
$35$	−2.47264e14	−0.530290
$36$	0	0
$37$	−4.66464e14	−0.590068	−0.295034	−	0.955487i	$-0.595331\pi$
$37$	−4.66464e14	−0.590068	−0.295034	+	0.955487i	$0.595331\pi$
$38$	6.72062e14	0.659880
$39$	0	0
$40$	−7.45839e14	−0.449857
$41$	−1.88945e15	−0.901339	−0.450670	−	0.892691i	$-0.648815\pi$
$41$	−1.88945e15	−0.901339	−0.450670	+	0.892691i	$0.648815\pi$
$42$	0	0
$43$	−4.32351e15	−1.31186	−0.655928	−	0.754824i	$-0.727722\pi$
$43$	−4.32351e15	−1.31186	−0.655928	+	0.754824i	$0.727722\pi$
$44$	−1.65693e15	−0.404113
$45$	0	0
$46$	1.73043e15	0.276666
$47$	−1.21034e16	−1.57753	−0.788764	−	0.614696i	$-0.789279\pi$
$47$	−1.21034e16	−1.57753	−0.788764	+	0.614696i	$0.789279\pi$
$48$	0	0
$49$	−9.41896e15	−0.826305
$50$	−6.04466e15	−0.437680
$51$	0	0
$52$	−8.68351e15	−0.433175
$53$	3.05939e16	1.27355	0.636773	−	0.771051i	$-0.280269\pi$
$53$	3.05939e16	1.27355	0.636773	+	0.771051i	$0.280269\pi$
$54$	0	0
$55$	−3.51235e16	−1.02838
$56$	5.97221e15	0.147350
$57$	0	0
$58$	−1.50416e16	−0.265906
$59$	−9.90874e15	−0.148910	−0.0744551	−	0.997224i	$-0.523722\pi$
$59$	−9.90874e15	−0.148910	−0.0744551	+	0.997224i	$0.523722\pi$
$60$	0	0
$61$	−9.16381e16	−1.00333	−0.501664	−	0.865063i	$-0.667279\pi$
$61$	−9.16381e16	−1.00333	−0.501664	+	0.865063i	$0.667279\pi$
$62$	−6.75720e16	−0.633936
$63$	0	0
$64$	1.80144e16	0.125000
$65$	−1.84073e17	−1.10233
$66$	0	0
$67$	−1.03349e17	−0.464085	−0.232042	−	0.972706i	$-0.574541\pi$
$67$	−1.03349e17	−0.464085	−0.232042	+	0.972706i	$0.574541\pi$
$68$	1.89361e17	0.738680
$69$	0	0
$70$	1.26599e17	0.374972
$71$	−2.85448e17	−0.738879	−0.369439	−	0.929255i	$-0.620450\pi$
$71$	−2.85448e17	−0.738879	−0.369439	+	0.929255i	$0.620450\pi$
$72$	0	0
$73$	8.75008e17	1.73958	0.869791	−	0.493420i	$-0.164253\pi$
$73$	8.75008e17	1.73958	0.869791	+	0.493420i	$0.164253\pi$
$74$	2.38830e17	0.417241
$75$	0	0
$76$	−3.44096e17	−0.466606
$77$	2.81247e17	0.336842
$78$	0	0
$79$	−1.08139e18	−1.01514	−0.507571	−	0.861610i	$-0.669457\pi$
$79$	−1.08139e18	−1.01514	−0.507571	+	0.861610i	$0.669457\pi$
$80$	3.81869e17	0.318097
$81$	0	0
$82$	9.67397e17	0.637343
$83$	6.65085e17	0.390513	0.195256	−	0.980752i	$-0.437446\pi$
$83$	6.65085e17	0.390513	0.195256	+	0.980752i	$0.437446\pi$
$84$	0	0
$85$	4.01408e18	1.87978
$86$	2.21364e18	0.927622
$87$	0	0
$88$	8.48347e17	0.285751
$89$	2.02099e18	0.611446	0.305723	−	0.952121i	$-0.401102\pi$
$89$	2.02099e18	0.611446	0.305723	+	0.952121i	$0.401102\pi$
$90$	0	0
$91$	1.47394e18	0.361067
$92$	−8.85982e17	−0.195632
$93$	0	0
$94$	6.19693e18	1.11548
$95$	−7.29414e18	−1.18741
$96$	0	0
$97$	−1.28256e19	−1.71295	−0.856477	−	0.516186i	$-0.827351\pi$
$97$	−1.28256e19	−1.71295	−0.856477	+	0.516186i	$0.827351\pi$
$98$	4.82251e18	0.584286
$99$	0	0
$100$	3.09487e18	0.309487
$101$	−1.56528e19	−1.42410	−0.712049	−	0.702130i	$-0.752232\pi$
$101$	−1.56528e19	−1.42410	−0.712049	+	0.702130i	$0.752232\pi$
$102$	0	0
$103$	−2.05978e19	−1.55549	−0.777744	−	0.628581i	$-0.783636\pi$
$103$	−2.05978e19	−1.55549	−0.777744	+	0.628581i	$0.783636\pi$
$104$	4.44596e18	0.306301
$105$	0	0
$106$	−1.56641e19	−0.900533
$107$	1.55282e18	0.0816537	0.0408268	−	0.999166i	$-0.487001\pi$
$107$	1.55282e18	0.0816537	0.0408268	+	0.999166i	$0.487001\pi$
$108$	0	0
$109$	1.18884e19	0.524292	0.262146	−	0.965028i	$-0.415570\pi$
$109$	1.18884e19	0.524292	0.262146	+	0.965028i	$0.415570\pi$
$110$	1.79833e19	0.727173
$111$	0	0
$112$	−3.05777e18	−0.104192
$113$	−3.63303e19	−1.13769	−0.568846	−	0.822444i	$-0.692610\pi$
$113$	−3.63303e19	−1.13769	−0.568846	+	0.822444i	$0.692610\pi$
$114$	0	0
$115$	−1.87810e19	−0.497841
$116$	7.70129e18	0.188024
$117$	0	0
$118$	5.07328e18	0.105295
$119$	−3.21422e19	−0.615715
$120$	0	0
$121$	−2.12082e19	−0.346770
$122$	4.69187e19	0.709460
$123$	0	0
$124$	3.45968e19	0.448260
$125$	−4.03850e19	−0.484814
$126$	0	0
$127$	−2.91372e19	−0.300824	−0.150412	−	0.988623i	$-0.548060\pi$
$127$	−2.91372e19	−0.300824	−0.150412	+	0.988623i	$0.548060\pi$
$128$	−9.22337e18	−0.0883883
$129$	0	0
$130$	9.42455e19	0.779469
$131$	1.43034e20	1.09992	0.549961	−	0.835190i	$-0.314643\pi$
$131$	1.43034e20	1.09992	0.549961	+	0.835190i	$0.314643\pi$
$132$	0	0
$133$	5.84069e19	0.388932
$134$	5.29149e19	0.328158
$135$	0	0
$136$	−9.69529e19	−0.522325
$137$	−1.22706e19	−0.0616626	−0.0308313	−	0.999525i	$-0.509815\pi$
$137$	−1.22706e19	−0.0616626	−0.0308313	+	0.999525i	$0.509815\pi$
$138$	0	0
$139$	1.00994e20	0.442238	0.221119	−	0.975247i	$-0.429029\pi$
$139$	1.00994e20	0.442238	0.221119	+	0.975247i	$0.429029\pi$
$140$	−6.48186e19	−0.265145
$141$	0	0
$142$	1.46150e20	0.522466
$143$	2.09372e20	0.700207
$144$	0	0
$145$	1.63252e20	0.478479
$146$	−4.48004e20	−1.23007
$147$	0	0
$148$	−1.22281e20	−0.295034
$149$	−2.28276e20	−0.516644	−0.258322	−	0.966059i	$-0.583170\pi$
$149$	−2.28276e20	−0.516644	−0.258322	+	0.966059i	$0.583170\pi$
$150$	0	0
$151$	5.06196e20	1.00934	0.504670	−	0.863312i	$-0.331614\pi$
$151$	5.06196e20	1.00934	0.504670	+	0.863312i	$0.331614\pi$
$152$	1.76177e20	0.329940
$153$	0	0
$154$	−1.43999e20	−0.238184
$155$	7.33384e20	1.14072
$156$	0	0
$157$	4.13309e20	0.569152	0.284576	−	0.958653i	$-0.408147\pi$
$157$	4.13309e20	0.569152	0.284576	+	0.958653i	$0.408147\pi$
$158$	5.53674e20	0.717814
$159$	0	0
$160$	−1.95517e20	−0.224929
$161$	1.50387e20	0.163066
$162$	0	0
$163$	−1.64552e21	−1.58680	−0.793398	−	0.608703i	$-0.791690\pi$
$163$	−1.64552e21	−1.58680	−0.793398	+	0.608703i	$0.791690\pi$
$164$	−4.95307e20	−0.450670
$165$	0	0
$166$	−3.40524e20	−0.276134
$167$	−7.99547e20	−0.612403	−0.306202	−	0.951967i	$-0.599058\pi$
$167$	−7.99547e20	−0.612403	−0.306202	+	0.951967i	$0.599058\pi$
$168$	0	0
$169$	−3.64656e20	−0.249437
$170$	−2.05521e21	−1.32920
$171$	0	0
$172$	−1.13338e21	−0.655928
$173$	1.45994e21	0.799644	0.399822	−	0.916593i	$-0.369072\pi$
$173$	1.45994e21	0.799644	0.399822	+	0.916593i	$0.369072\pi$
$174$	0	0
$175$	−5.25324e20	−0.257968
$176$	−4.34353e20	−0.202057
$177$	0	0
$178$	−1.03474e21	−0.432358
$179$	−3.29643e20	−0.130599	−0.0652996	−	0.997866i	$-0.520800\pi$
$179$	−3.29643e20	−0.130599	−0.0652996	+	0.997866i	$0.520800\pi$
$180$	0	0
$181$	2.22517e21	0.793262	0.396631	−	0.917978i	$-0.370179\pi$
$181$	2.22517e21	0.793262	0.396631	+	0.917978i	$0.370179\pi$
$182$	−7.54659e20	−0.255313
$183$	0	0
$184$	4.53623e20	0.138333
$185$	−2.59211e21	−0.750796
$186$	0	0
$187$	−4.56577e21	−1.19404
$188$	−3.17283e21	−0.788764
$189$	0	0
$190$	3.73460e21	0.839624
$191$	2.86313e21	0.612384	0.306192	−	0.951970i	$-0.400945\pi$
$191$	2.86313e21	0.612384	0.306192	+	0.951970i	$0.400945\pi$
$192$	0	0
$193$	3.67465e21	0.711905	0.355952	−	0.934504i	$-0.384156\pi$
$193$	3.67465e21	0.711905	0.355952	+	0.934504i	$0.384156\pi$
$194$	6.56670e21	1.21124
$195$	0	0
$196$	−2.46912e21	−0.413152
$197$	5.40192e21	0.861229	0.430614	−	0.902536i	$-0.358297\pi$
$197$	5.40192e21	0.861229	0.430614	+	0.902536i	$0.358297\pi$
$198$	0	0
$199$	7.53120e21	1.09084	0.545419	−	0.838164i	$-0.316371\pi$
$199$	7.53120e21	1.09084	0.545419	+	0.838164i	$0.316371\pi$
$200$	−1.58457e21	−0.218840
$201$	0	0
$202$	8.01425e21	1.00699
$203$	−1.30722e21	−0.156724
$204$	0	0
$205$	−1.04995e22	−1.14685
$206$	1.05461e22	1.09990
$207$	0	0
$208$	−2.27633e21	−0.216588
$209$	8.29665e21	0.754246
$210$	0	0
$211$	2.36878e22	1.96717	0.983583	−	0.180458i	$-0.0577581\pi$
$211$	2.36878e22	1.96717	0.983583	+	0.180458i	$0.0577581\pi$
$212$	8.02002e21	0.636773
$213$	0	0
$214$	−7.95045e20	−0.0577379
$215$	−2.40254e22	−1.66919
$216$	0	0
$217$	−5.87248e21	−0.373641
$218$	−6.08688e21	−0.370730
$219$	0	0
$220$	−9.20743e21	−0.514189
$221$	−2.39280e22	−1.27991
$222$	0	0
$223$	9.25315e20	0.0454353	0.0227176	−	0.999742i	$-0.492768\pi$
$223$	9.25315e20	0.0454353	0.0227176	+	0.999742i	$0.492768\pi$
$224$	1.56558e21	0.0736748
$225$	0	0
$226$	1.86011e22	0.804469
$227$	−8.09087e21	−0.335544	−0.167772	−	0.985826i	$-0.553657\pi$
$227$	−8.09087e21	−0.335544	−0.167772	+	0.985826i	$0.553657\pi$
$228$	0	0
$229$	−1.94194e22	−0.740965	−0.370483	−	0.928839i	$-0.620808\pi$
$229$	−1.94194e22	−0.740965	−0.370483	+	0.928839i	$0.620808\pi$
$230$	9.61590e21	0.352027
$231$	0	0
$232$	−3.94306e21	−0.132953
$233$	2.02616e22	0.655833	0.327917	−	0.944707i	$-0.393654\pi$
$233$	2.02616e22	0.655833	0.327917	+	0.944707i	$0.393654\pi$
$234$	0	0
$235$	−6.72577e22	−2.00723
$236$	−2.59752e21	−0.0744551
$237$	0	0
$238$	1.64568e22	0.435376
$239$	2.16056e22	0.549270	0.274635	−	0.961549i	$-0.411443\pi$
$239$	2.16056e22	0.549270	0.274635	+	0.961549i	$0.411443\pi$
$240$	0	0
$241$	−4.62435e22	−1.08615	−0.543074	−	0.839685i	$-0.682740\pi$
$241$	−4.62435e22	−1.08615	−0.543074	+	0.839685i	$0.682740\pi$
$242$	1.08586e22	0.245204
$243$	0	0
$244$	−2.40224e22	−0.501664
$245$	−5.23405e22	−1.05138
$246$	0	0
$247$	4.34805e22	0.808488
$248$	−1.77136e22	−0.316968
$249$	0	0
$250$	2.06771e22	0.342815
$251$	−2.94186e22	−0.469592	−0.234796	−	0.972045i	$-0.575442\pi$
$251$	−2.94186e22	−0.469592	−0.234796	+	0.972045i	$0.575442\pi$
$252$	0	0
$253$	2.13623e22	0.316230
$254$	1.49183e22	0.212715
$255$	0	0
$256$	4.72237e21	0.0625000
$257$	7.23838e22	0.923159	0.461580	−	0.887099i	$-0.347283\pi$
$257$	7.23838e22	0.923159	0.461580	+	0.887099i	$0.347283\pi$
$258$	0	0
$259$	2.07560e22	0.245921
$260$	−4.82537e22	−0.551167
$261$	0	0
$262$	−7.32334e22	−0.777762
$263$	2.69226e22	0.275764	0.137882	−	0.990449i	$-0.455971\pi$
$263$	2.69226e22	0.275764	0.137882	+	0.990449i	$0.455971\pi$
$264$	0	0
$265$	1.70008e23	1.62045
$266$	−2.99043e22	−0.275017
$267$	0	0
$268$	−2.70924e22	−0.232042
$269$	9.12692e22	0.754532	0.377266	−	0.926105i	$-0.376864\pi$
$269$	9.12692e22	0.754532	0.377266	+	0.926105i	$0.376864\pi$
$270$	0	0
$271$	1.19911e22	0.0923953	0.0461976	−	0.998932i	$-0.485290\pi$
$271$	1.19911e22	0.0923953	0.0461976	+	0.998932i	$0.485290\pi$
$272$	4.96399e22	0.369340
$273$	0	0
$274$	6.28257e21	0.0436021
$275$	−7.46218e22	−0.500271
$276$	0	0
$277$	6.34729e22	0.397219	0.198610	−	0.980079i	$-0.436357\pi$
$277$	6.34729e22	0.397219	0.198610	+	0.980079i	$0.436357\pi$
$278$	−5.17091e22	−0.312710
$279$	0	0
$280$	3.31871e22	0.187486
$281$	4.94651e22	0.270140	0.135070	−	0.990836i	$-0.456874\pi$
$281$	4.94651e22	0.270140	0.135070	+	0.990836i	$0.456874\pi$
$282$	0	0
$283$	−1.84739e23	−0.943166	−0.471583	−	0.881822i	$-0.656317\pi$
$283$	−1.84739e23	−0.943166	−0.471583	+	0.881822i	$0.656317\pi$
$284$	−7.48286e22	−0.369439
$285$	0	0
$286$	−1.07199e23	−0.495121
$287$	8.40737e22	0.375649
$288$	0	0
$289$	2.82725e23	1.18259
$290$	−8.35850e22	−0.338336
$291$	0	0
$292$	2.29378e23	0.869791
$293$	3.93566e23	1.44469	0.722346	−	0.691531i	$-0.243064\pi$
$293$	3.93566e23	1.44469	0.722346	+	0.691531i	$0.243064\pi$
$294$	0	0
$295$	−5.50622e22	−0.189472
$296$	6.26078e22	0.208620
$297$	0	0
$298$	1.16878e23	0.365323
$299$	1.11954e23	0.338972
$300$	0	0
$301$	1.92381e23	0.546738
$302$	−2.59172e23	−0.713711
$303$	0	0
$304$	−9.02026e22	−0.233303
$305$	−5.09227e23	−1.27662
$306$	0	0
$307$	−1.26610e23	−0.298300	−0.149150	−	0.988815i	$-0.547654\pi$
$307$	−1.26610e23	−0.298300	−0.149150	+	0.988815i	$0.547654\pi$
$308$	7.37273e22	0.168421
$309$	0	0
$310$	−3.75493e23	−0.806613
$311$	3.64792e23	0.760014	0.380007	−	0.924984i	$-0.375922\pi$
$311$	3.64792e23	0.760014	0.380007	+	0.924984i	$0.375922\pi$
$312$	0	0
$313$	−7.65963e23	−1.50154	−0.750769	−	0.660564i	$-0.770317\pi$
$313$	−7.65963e23	−1.50154	−0.750769	+	0.660564i	$0.770317\pi$
$314$	−2.11614e23	−0.402451
$315$	0	0
$316$	−2.83481e23	−0.507571
$317$	−5.27819e23	−0.917111	−0.458556	−	0.888666i	$-0.651633\pi$
$317$	−5.27819e23	−0.917111	−0.458556	+	0.888666i	$0.651633\pi$
$318$	0	0
$319$	−1.85689e23	−0.303931
$320$	1.00105e23	0.159049
$321$	0	0
$322$	−7.69981e22	−0.115305
$323$	−9.48178e23	−1.37869
$324$	0	0
$325$	−3.91073e23	−0.536248
$326$	8.42507e23	1.12203
$327$	0	0
$328$	2.53597e23	0.318672
$329$	5.38557e23	0.657462
$330$	0	0
$331$	−3.35783e23	−0.386984	−0.193492	−	0.981102i	$-0.561981\pi$
$331$	−3.35783e23	−0.386984	−0.193492	+	0.981102i	$0.561981\pi$
$332$	1.74348e23	0.195256
$333$	0	0
$334$	4.09368e23	0.433034
$335$	−5.74306e23	−0.590497
$336$	0	0
$337$	3.39212e22	0.0329600	0.0164800	−	0.999864i	$-0.494754\pi$
$337$	3.39212e22	0.0329600	0.0164800	+	0.999864i	$0.494754\pi$
$338$	1.86704e23	0.176378
$339$	0	0
$340$	1.05227e24	0.939888
$341$	−8.34180e23	−0.724592
$342$	0	0
$343$	9.26320e23	0.761144
$344$	5.80291e23	0.463811
$345$	0	0
$346$	−7.47489e23	−0.565434
$347$	−1.68737e24	−1.24188	−0.620941	−	0.783857i	$-0.713250\pi$
$347$	−1.68737e24	−1.24188	−0.620941	+	0.783857i	$0.713250\pi$
$348$	0	0
$349$	2.07379e24	1.44519	0.722593	−	0.691274i	$-0.242950\pi$
$349$	2.07379e24	1.44519	0.722593	+	0.691274i	$0.242950\pi$
$350$	2.68966e23	0.182411
$351$	0	0
$352$	2.22389e23	0.142876
$353$	2.53500e24	1.58533	0.792663	−	0.609660i	$-0.208694\pi$
$353$	2.53500e24	1.58533	0.792663	+	0.609660i	$0.208694\pi$
$354$	0	0
$355$	−1.58622e24	−0.940141
$356$	5.29789e23	0.305723
$357$	0	0
$358$	1.68777e23	0.0923476
$359$	4.19609e23	0.223588	0.111794	−	0.993731i	$-0.464340\pi$
$359$	4.19609e23	0.223588	0.111794	+	0.993731i	$0.464340\pi$
$360$	0	0
$361$	−2.55447e23	−0.129117
$362$	−1.13929e24	−0.560921
$363$	0	0
$364$	3.86385e23	0.180533
$365$	4.86236e24	2.21343
$366$	0	0
$367$	4.30311e24	1.85975	0.929875	−	0.367877i	$-0.119915\pi$
$367$	4.30311e24	1.85975	0.929875	+	0.367877i	$0.119915\pi$
$368$	−2.32255e23	−0.0978161
$369$	0	0
$370$	1.32716e24	0.530893
$371$	−1.36132e24	−0.530772
$372$	0	0
$373$	−1.43004e24	−0.529802	−0.264901	−	0.964276i	$-0.585339\pi$
$373$	−1.43004e24	−0.529802	−0.264901	+	0.964276i	$0.585339\pi$
$374$	2.33768e24	0.844314
$375$	0	0
$376$	1.62449e24	0.557741
$377$	−9.73149e23	−0.325789
$378$	0	0
$379$	−5.42530e24	−1.72723	−0.863617	−	0.504149i	$-0.831806\pi$
$379$	−5.42530e24	−1.72723	−0.863617	+	0.504149i	$0.831806\pi$
$380$	−1.91212e24	−0.593704
$381$	0	0
$382$	−1.46592e24	−0.433021
$383$	−5.40027e24	−1.55606	−0.778032	−	0.628225i	$-0.783782\pi$
$383$	−5.40027e24	−1.55606	−0.778032	+	0.628225i	$0.783782\pi$
$384$	0	0
$385$	1.56287e24	0.428594
$386$	−1.88142e24	−0.503393
$387$	0	0
$388$	−3.36215e24	−0.856477
$389$	7.85304e23	0.195217	0.0976083	−	0.995225i	$-0.468881\pi$
$389$	7.85304e23	0.195217	0.0976083	+	0.995225i	$0.468881\pi$
$390$	0	0
$391$	−2.44138e24	−0.578038
$392$	1.26419e24	0.292143
$393$	0	0
$394$	−2.76578e24	−0.608981
$395$	−6.00923e24	−1.29166
$396$	0	0
$397$	5.59905e24	1.14711	0.573555	−	0.819167i	$-0.305564\pi$
$397$	5.59905e24	1.14711	0.573555	+	0.819167i	$0.305564\pi$
$398$	−3.85598e24	−0.771339
$399$	0	0
$400$	8.11301e23	0.154743
$401$	7.17909e24	1.33720	0.668602	−	0.743620i	$-0.266893\pi$
$401$	7.17909e24	1.33720	0.668602	+	0.743620i	$0.266893\pi$
$402$	0	0
$403$	−4.37172e24	−0.776701
$404$	−4.10329e24	−0.712049
$405$	0	0
$406$	6.69297e23	0.110821
$407$	2.94837e24	0.476908
$408$	0	0
$409$	−1.23002e24	−0.189907	−0.0949537	−	0.995482i	$-0.530270\pi$
$409$	−1.23002e24	−0.189907	−0.0949537	+	0.995482i	$0.530270\pi$
$410$	5.37576e24	0.810948
$411$	0	0
$412$	−5.39958e24	−0.777744
$413$	4.40904e23	0.0620609
$414$	0	0
$415$	3.69583e24	0.496884
$416$	1.16548e24	0.153151
$417$	0	0
$418$	−4.24788e24	−0.533332
$419$	1.05615e25	1.29626	0.648129	−	0.761531i	$-0.275552\pi$
$419$	1.05615e25	1.29626	0.648129	+	0.761531i	$0.275552\pi$
$420$	0	0
$421$	−1.64386e24	−0.192835	−0.0964174	−	0.995341i	$-0.530738\pi$
$421$	−1.64386e24	−0.192835	−0.0964174	+	0.995341i	$0.530738\pi$
$422$	−1.21281e25	−1.39100
$423$	0	0
$424$	−4.10625e24	−0.450266
$425$	8.52812e24	0.914446
$426$	0	0
$427$	4.07757e24	0.418154
$428$	4.07063e23	0.0408268
$429$	0	0
$430$	1.23010e25	1.18030
$431$	−1.68459e25	−1.58110	−0.790551	−	0.612397i	$-0.790206\pi$
$431$	−1.68459e25	−1.58110	−0.790551	+	0.612397i	$0.790206\pi$
$432$	0	0
$433$	2.88832e24	0.259424	0.129712	−	0.991552i	$-0.458595\pi$
$433$	2.88832e24	0.259424	0.129712	+	0.991552i	$0.458595\pi$
$434$	3.00671e24	0.264204
$435$	0	0
$436$	3.11648e24	0.262146
$437$	4.43633e24	0.365133
$438$	0	0
$439$	−1.10102e25	−0.867728	−0.433864	−	0.900978i	$-0.642850\pi$
$439$	−1.10102e25	−0.867728	−0.433864	+	0.900978i	$0.642850\pi$
$440$	4.71420e24	0.363587
$441$	0	0
$442$	1.22511e25	0.905034
$443$	−1.39420e24	−0.100807	−0.0504034	−	0.998729i	$-0.516051\pi$
$443$	−1.39420e24	−0.100807	−0.0504034	+	0.998729i	$0.516051\pi$
$444$	0	0
$445$	1.12305e25	0.777997
$446$	−4.73761e23	−0.0321276
$447$	0	0
$448$	−8.01576e23	−0.0520959
$449$	−4.24061e24	−0.269829	−0.134914	−	0.990857i	$-0.543076\pi$
$449$	−4.24061e24	−0.269829	−0.134914	+	0.990857i	$0.543076\pi$
$450$	0	0
$451$	1.19426e25	0.728486
$452$	−9.52378e24	−0.568846
$453$	0	0
$454$	4.14253e24	0.237265
$455$	8.19060e24	0.459417
$456$	0	0
$457$	5.97553e24	0.321494	0.160747	−	0.986996i	$-0.448610\pi$
$457$	5.97553e24	0.321494	0.160747	+	0.986996i	$0.448610\pi$
$458$	9.94271e24	0.523942
$459$	0	0
$460$	−4.92334e24	−0.248920
$461$	1.10872e25	0.549115	0.274558	−	0.961571i	$-0.411469\pi$
$461$	1.10872e25	0.549115	0.274558	+	0.961571i	$0.411469\pi$
$462$	0	0
$463$	−9.79106e24	−0.465383	−0.232691	−	0.972551i	$-0.574753\pi$
$463$	−9.79106e24	−0.465383	−0.232691	+	0.972551i	$0.574753\pi$
$464$	2.01885e24	0.0940119
$465$	0	0
$466$	−1.03740e25	−0.463744
$467$	−4.42989e25	−1.94036	−0.970181	−	0.242380i	$-0.922072\pi$
$467$	−4.42989e25	−1.94036	−0.970181	+	0.242380i	$0.922072\pi$
$468$	0	0
$469$	4.59868e24	0.193415
$470$	3.44359e25	1.41933
$471$	0	0
$472$	1.32993e24	0.0526477
$473$	2.73275e25	1.06028
$474$	0	0
$475$	−1.54968e25	−0.577633
$476$	−8.42589e24	−0.307858
$477$	0	0
$478$	−1.10621e25	−0.388393
$479$	−3.33527e24	−0.114801	−0.0574003	−	0.998351i	$-0.518281\pi$
$479$	−3.33527e24	−0.114801	−0.0574003	+	0.998351i	$0.518281\pi$
$480$	0	0
$481$	1.54516e25	0.511206
$482$	2.36767e25	0.768022
$483$	0	0
$484$	−5.55959e24	−0.173385
$485$	−7.12708e25	−2.17954
$486$	0	0
$487$	−4.70148e25	−1.38264	−0.691321	−	0.722548i	$-0.742971\pi$
$487$	−4.70148e25	−1.38264	−0.691321	+	0.722548i	$0.742971\pi$
$488$	1.22995e25	0.354730
$489$	0	0
$490$	2.67983e25	0.743439
$491$	4.37495e25	1.19042	0.595209	−	0.803571i	$-0.297069\pi$
$491$	4.37495e25	1.19042	0.595209	+	0.803571i	$0.297069\pi$
$492$	0	0
$493$	2.12214e25	0.555557
$494$	−2.22620e25	−0.571688
$495$	0	0
$496$	9.06935e24	0.224130
$497$	1.27014e25	0.307941
$498$	0	0
$499$	−1.73307e25	−0.404447	−0.202224	−	0.979339i	$-0.564817\pi$
$499$	−1.73307e25	−0.404447	−0.202224	+	0.979339i	$0.564817\pi$
$500$	−1.05867e25	−0.242407
$501$	0	0
$502$	1.50623e25	0.332052
$503$	−7.90450e25	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$503$	−7.90450e25	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$504$	0	0
$505$	−8.69816e25	−1.81201
$506$	−1.09375e25	−0.223609
$507$	0	0
$508$	−7.63815e24	−0.150412
$509$	3.75610e25	0.725969	0.362985	−	0.931795i	$-0.381758\pi$
$509$	3.75610e25	0.725969	0.362985	+	0.931795i	$0.381758\pi$
$510$	0	0
$511$	−3.89347e25	−0.725001
$512$	−2.41785e24	−0.0441942
$513$	0	0
$514$	−3.70605e25	−0.652772
$515$	−1.14460e26	−1.97919
$516$	0	0
$517$	7.65016e25	1.27500
$518$	−1.06271e25	−0.173892
$519$	0	0
$520$	2.47059e25	0.389734
$521$	7.83655e25	1.21385	0.606927	−	0.794757i	$-0.292402\pi$
$521$	7.83655e25	1.21385	0.606927	+	0.794757i	$0.292402\pi$
$522$	0	0
$523$	4.48736e25	0.670232	0.335116	−	0.942177i	$-0.391224\pi$
$523$	4.48736e25	0.670232	0.335116	+	0.942177i	$0.391224\pi$
$524$	3.74955e25	0.549961
$525$	0	0
$526$	−1.37844e25	−0.194994
$527$	9.53339e25	1.32448
$528$	0	0
$529$	−6.31927e25	−0.846912
$530$	−8.70443e25	−1.14583
$531$	0	0
$532$	1.53110e25	0.194466
$533$	6.25879e25	0.780876
$534$	0	0
$535$	8.62892e24	0.103895
$536$	1.38713e25	0.164079
$537$	0	0
$538$	−4.67298e25	−0.533535
$539$	5.95342e25	0.667841
$540$	0	0
$541$	1.23370e26	1.33609	0.668046	−	0.744120i	$-0.267131\pi$
$541$	1.23370e26	1.33609	0.668046	+	0.744120i	$0.267131\pi$
$542$	−6.13943e24	−0.0653333
$543$	0	0
$544$	−2.54156e25	−0.261163
$545$	6.60632e25	0.667103
$546$	0	0
$547$	1.76813e26	1.72439	0.862193	−	0.506580i	$-0.169091\pi$
$547$	1.76813e26	1.72439	0.862193	+	0.506580i	$0.169091\pi$
$548$	−3.21668e24	−0.0308313
$549$	0	0
$550$	3.82064e25	0.353745
$551$	−3.85623e25	−0.350932
$552$	0	0
$553$	4.81182e25	0.423078
$554$	−3.24981e25	−0.280876
$555$	0	0
$556$	2.64751e25	0.221119
$557$	−1.85225e26	−1.52081	−0.760406	−	0.649448i	$-0.775000\pi$
$557$	−1.85225e26	−1.52081	−0.760406	+	0.649448i	$0.775000\pi$
$558$	0	0
$559$	1.43216e26	1.13653
$560$	−1.69918e25	−0.132573
$561$	0	0
$562$	−2.53261e25	−0.191018
$563$	9.59717e25	0.711727	0.355863	−	0.934538i	$-0.384187\pi$
$563$	9.59717e25	0.711727	0.355863	+	0.934538i	$0.384187\pi$
$564$	0	0
$565$	−2.01885e26	−1.44759
$566$	9.45865e25	0.666919
$567$	0	0
$568$	3.83122e25	0.261233
$569$	2.08932e26	1.40100	0.700501	−	0.713651i	$-0.252960\pi$
$569$	2.08932e26	1.40100	0.700501	+	0.713651i	$0.252960\pi$
$570$	0	0
$571$	6.61413e25	0.428973	0.214487	−	0.976727i	$-0.431192\pi$
$571$	6.61413e25	0.428973	0.214487	+	0.976727i	$0.431192\pi$
$572$	5.48857e25	0.350104
$573$	0	0
$574$	−4.30457e25	−0.265624
$575$	−3.99013e25	−0.242182
$576$	0	0
$577$	−6.06938e25	−0.356430	−0.178215	−	0.983992i	$-0.557032\pi$
$577$	−6.06938e25	−0.356430	−0.178215	+	0.983992i	$0.557032\pi$
$578$	−1.44755e26	−0.836217
$579$	0	0
$580$	4.27955e25	0.239239
$581$	−2.95939e25	−0.162753
$582$	0	0
$583$	−1.93374e26	−1.02931
$584$	−1.17442e26	−0.615035
$585$	0	0
$586$	−2.01506e26	−1.02155
$587$	1.66078e26	0.828419	0.414210	−	0.910182i	$-0.364058\pi$
$587$	1.66078e26	0.828419	0.414210	+	0.910182i	$0.364058\pi$
$588$	0	0
$589$	−1.73235e26	−0.836644
$590$	2.81918e25	0.133977
$591$	0	0
$592$	−3.20552e25	−0.147517
$593$	−4.13595e26	−1.87308	−0.936539	−	0.350564i	$-0.885990\pi$
$593$	−4.13595e26	−1.87308	−0.936539	+	0.350564i	$0.885990\pi$
$594$	0	0
$595$	−1.78612e26	−0.783429
$596$	−5.98413e25	−0.258322
$597$	0	0
$598$	−5.73206e25	−0.239690
$599$	−1.71335e26	−0.705165	−0.352583	−	0.935781i	$-0.614696\pi$
$599$	−1.71335e26	−0.705165	−0.352583	+	0.935781i	$0.614696\pi$
$600$	0	0
$601$	4.29252e25	0.171161	0.0855805	−	0.996331i	$-0.472726\pi$
$601$	4.29252e25	0.171161	0.0855805	+	0.996331i	$0.472726\pi$
$602$	−9.84989e25	−0.386602
$603$	0	0
$604$	1.32696e26	0.504670
$605$	−1.17852e26	−0.441227
$606$	0	0
$607$	−1.37498e26	−0.498887	−0.249444	−	0.968389i	$-0.580248\pi$
$607$	−1.37498e26	−0.498887	−0.249444	+	0.968389i	$0.580248\pi$
$608$	4.61837e25	0.164970
$609$	0	0
$610$	2.60724e26	0.902709
$611$	4.00924e26	1.36669
$612$	0	0
$613$	1.96376e26	0.648955	0.324478	−	0.945893i	$-0.394811\pi$
$613$	1.96376e26	0.648955	0.324478	+	0.945893i	$0.394811\pi$
$614$	6.48244e25	0.210930
$615$	0	0
$616$	−3.77484e25	−0.119092
$617$	−8.56236e25	−0.266002	−0.133001	−	0.991116i	$-0.542461\pi$
$617$	−8.56236e25	−0.266002	−0.133001	+	0.991116i	$0.542461\pi$
$618$	0	0
$619$	2.67407e26	0.805584	0.402792	−	0.915291i	$-0.368040\pi$
$619$	2.67407e26	0.805584	0.402792	+	0.915291i	$0.368040\pi$
$620$	1.92252e26	0.570362
$621$	0	0
$622$	−1.86774e26	−0.537411
$623$	−8.99266e25	−0.254831
$624$	0	0
$625$	−4.49598e26	−1.23585
$626$	3.92173e26	1.06175
$627$	0	0
$628$	1.08346e26	0.284576
$629$	−3.36953e26	−0.871742
$630$	0	0
$631$	5.76941e25	0.144828	0.0724140	−	0.997375i	$-0.476930\pi$
$631$	5.76941e25	0.144828	0.0724140	+	0.997375i	$0.476930\pi$
$632$	1.45142e26	0.358907
$633$	0	0
$634$	2.70243e26	0.648496
$635$	−1.61914e26	−0.382766
$636$	0	0
$637$	3.12003e26	0.715870
$638$	9.50730e25	0.214912
$639$	0	0
$640$	−5.12536e25	−0.112464
$641$	1.10772e26	0.239486	0.119743	−	0.992805i	$-0.461793\pi$
$641$	1.10772e26	0.239486	0.119743	+	0.992805i	$0.461793\pi$
$642$	0	0
$643$	−5.89228e26	−1.23674	−0.618371	−	0.785886i	$-0.712207\pi$
$643$	−5.89228e26	−1.23674	−0.618371	+	0.785886i	$0.712207\pi$
$644$	3.94230e25	0.0815331
$645$	0	0
$646$	4.85467e26	0.974880
$647$	7.94369e26	1.57192	0.785962	−	0.618275i	$-0.212168\pi$
$647$	7.94369e26	1.57192	0.785962	+	0.618275i	$0.212168\pi$
$648$	0	0
$649$	6.26299e25	0.120353
$650$	2.00229e26	0.379185
$651$	0	0
$652$	−4.31363e26	−0.793398
$653$	−9.14897e26	−1.65843	−0.829215	−	0.558930i	$-0.811212\pi$
$653$	−9.14897e26	−1.65843	−0.829215	+	0.558930i	$0.811212\pi$
$654$	0	0
$655$	7.94830e26	1.39953
$656$	−1.29842e26	−0.225335
$657$	0	0
$658$	−2.75741e26	−0.464896
$659$	−2.55538e26	−0.424662	−0.212331	−	0.977198i	$-0.568106\pi$
$659$	−2.55538e26	−0.424662	−0.212331	+	0.977198i	$0.568106\pi$
$660$	0	0
$661$	9.40351e26	1.51837	0.759183	−	0.650878i	$-0.225599\pi$
$661$	9.40351e26	1.51837	0.759183	+	0.650878i	$0.225599\pi$
$662$	1.71921e26	0.273639
$663$	0	0
$664$	−8.92662e25	−0.138067
$665$	3.24563e26	0.494873
$666$	0	0
$667$	−9.92907e25	−0.147134
$668$	−2.09596e26	−0.306202
$669$	0	0
$670$	2.94044e26	0.417544
$671$	5.79215e26	0.810916
$672$	0	0
$673$	−4.53766e26	−0.617574	−0.308787	−	0.951131i	$-0.599923\pi$
$673$	−4.53766e26	−0.617574	−0.308787	+	0.951131i	$0.599923\pi$
$674$	−1.73677e25	−0.0233063
$675$	0	0
$676$	−9.55925e25	−0.124718
$677$	6.16629e25	0.0793290	0.0396645	−	0.999213i	$-0.487371\pi$
$677$	6.16629e25	0.0793290	0.0396645	+	0.999213i	$0.487371\pi$
$678$	0	0
$679$	5.70692e26	0.713903
$680$	−5.38760e26	−0.664601
$681$	0	0
$682$	4.27100e26	0.512364
$683$	1.77510e26	0.210003	0.105001	−	0.994472i	$-0.466515\pi$
$683$	1.77510e26	0.210003	0.105001	+	0.994472i	$0.466515\pi$
$684$	0	0
$685$	−6.81871e25	−0.0784588
$686$	−4.74276e26	−0.538210
$687$	0	0
$688$	−2.97109e26	−0.327964
$689$	−1.01342e27	−1.10334
$690$	0	0
$691$	−1.07067e27	−1.13401	−0.567003	−	0.823716i	$-0.691897\pi$
$691$	−1.07067e27	−1.13401	−0.567003	+	0.823716i	$0.691897\pi$
$692$	3.82714e26	0.399822
$693$	0	0
$694$	8.63934e26	0.878144
$695$	5.61219e26	0.562699
$696$	0	0
$697$	−1.36485e27	−1.33160
$698$	−1.06178e27	−1.02190
$699$	0	0
$700$	−1.37711e26	−0.128984
$701$	3.23143e26	0.298589	0.149295	−	0.988793i	$-0.452300\pi$
$701$	3.23143e26	0.298589	0.149295	+	0.988793i	$0.452300\pi$
$702$	0	0
$703$	6.12290e26	0.550658
$704$	−1.13863e26	−0.101028
$705$	0	0
$706$	−1.29792e27	−1.12100
$707$	6.96495e26	0.593517
$708$	0	0
$709$	−1.87451e27	−1.55506	−0.777531	−	0.628845i	$-0.783528\pi$
$709$	−1.87451e27	−1.55506	−0.777531	+	0.628845i	$0.783528\pi$
$710$	8.12143e26	0.664780
$711$	0	0
$712$	−2.71252e26	−0.216179
$713$	−4.46048e26	−0.350777
$714$	0	0
$715$	1.16347e27	0.890936
$716$	−8.64140e25	−0.0652996
$717$	0	0
$718$	−2.14840e26	−0.158100
$719$	2.64509e27	1.92095	0.960473	−	0.278372i	$-0.0897947\pi$
$719$	2.64509e27	1.92095	0.960473	+	0.278372i	$0.0897947\pi$
$720$	0	0
$721$	9.16527e26	0.648277
$722$	1.30789e26	0.0912992
$723$	0	0
$724$	5.83315e26	0.396631
$725$	3.46837e26	0.232763
$726$	0	0
$727$	−4.60373e26	−0.300977	−0.150488	−	0.988612i	$-0.548085\pi$
$727$	−4.60373e26	−0.300977	−0.150488	+	0.988612i	$0.548085\pi$
$728$	−1.97829e26	−0.127656
$729$	0	0
$730$	−2.48953e27	−1.56513
$731$	−3.12311e27	−1.93808
$732$	0	0
$733$	7.20125e26	0.435432	0.217716	−	0.976012i	$-0.430139\pi$
$733$	7.20125e26	0.435432	0.217716	+	0.976012i	$0.430139\pi$
$734$	−2.20319e27	−1.31504
$735$	0	0
$736$	1.18914e26	0.0691665
$737$	6.53238e26	0.375086
$738$	0	0
$739$	2.33531e27	1.30684	0.653420	−	0.756996i	$-0.273334\pi$
$739$	2.33531e27	1.30684	0.653420	+	0.756996i	$0.273334\pi$
$740$	−6.79506e26	−0.375398
$741$	0	0
$742$	6.96996e26	0.375313
$743$	−3.31315e27	−1.76136	−0.880679	−	0.473714i	$-0.842913\pi$
$743$	−3.31315e27	−1.76136	−0.880679	+	0.473714i	$0.842913\pi$
$744$	0	0
$745$	−1.26852e27	−0.657372
$746$	7.32180e26	0.374627
$747$	0	0
$748$	−1.19689e27	−0.597020
$749$	−6.90950e25	−0.0340306
$750$	0	0
$751$	1.59410e27	0.765482	0.382741	−	0.923856i	$-0.374980\pi$
$751$	1.59410e27	0.765482	0.382741	+	0.923856i	$0.374980\pi$
$752$	−8.31738e26	−0.394382
$753$	0	0
$754$	4.98252e26	0.230368
$755$	2.81289e27	1.28427
$756$	0	0
$757$	−2.68445e27	−1.19521	−0.597604	−	0.801791i	$-0.703881\pi$
$757$	−2.68445e27	−1.19521	−0.597604	+	0.801791i	$0.703881\pi$
$758$	2.77775e27	1.22134
$759$	0	0
$760$	9.79003e26	0.419812
$761$	1.91199e26	0.0809713	0.0404857	−	0.999180i	$-0.487109\pi$
$761$	1.91199e26	0.0809713	0.0404857	+	0.999180i	$0.487109\pi$
$762$	0	0
$763$	−5.28993e26	−0.218508
$764$	7.50552e26	0.306192
$765$	0	0
$766$	2.76494e27	1.10030
$767$	3.28227e26	0.129008
$768$	0	0
$769$	2.08520e27	0.799555	0.399777	−	0.916612i	$-0.369087\pi$
$769$	2.08520e27	0.799555	0.399777	+	0.916612i	$0.369087\pi$
$770$	−8.00191e26	−0.303062
$771$	0	0
$772$	9.63288e26	0.355952
$773$	3.98794e25	0.0145560	0.00727801	−	0.999974i	$-0.497683\pi$
$773$	3.98794e25	0.0145560	0.00727801	+	0.999974i	$0.497683\pi$
$774$	0	0
$775$	1.55811e27	0.554923
$776$	1.72142e27	0.605620
$777$	0	0
$778$	−4.02076e26	−0.138039
$779$	2.48013e27	0.841140
$780$	0	0
$781$	1.80423e27	0.597181
$782$	1.24999e27	0.408735
$783$	0	0
$784$	−6.47266e26	−0.206576
$785$	2.29673e27	0.724183
$786$	0	0
$787$	−1.49258e27	−0.459387	−0.229693	−	0.973263i	$-0.573772\pi$
$787$	−1.49258e27	−0.459387	−0.229693	+	0.973263i	$0.573772\pi$
$788$	1.41608e27	0.430614
$789$	0	0
$790$	3.07673e27	0.913338
$791$	1.61657e27	0.474153
$792$	0	0
$793$	3.03551e27	0.869234
$794$	−2.86672e27	−0.811129
$795$	0	0
$796$	1.97426e27	0.545419
$797$	−2.93619e27	−0.801549	−0.400774	−	0.916177i	$-0.631259\pi$
$797$	−2.93619e27	−0.801549	−0.400774	+	0.916177i	$0.631259\pi$
$798$	0	0
$799$	−8.74294e27	−2.33058
$800$	−4.15386e26	−0.109420
$801$	0	0
$802$	−3.67569e27	−0.945546
$803$	−5.53064e27	−1.40598
$804$	0	0
$805$	8.35690e26	0.207484
$806$	2.23832e27	0.549211
$807$	0	0
$808$	2.10089e27	0.503494
$809$	−8.71815e26	−0.206497	−0.103248	−	0.994656i	$-0.532924\pi$
$809$	−8.71815e26	−0.206497	−0.103248	+	0.994656i	$0.532924\pi$
$810$	0	0
$811$	−3.25283e27	−0.752598	−0.376299	−	0.926498i	$-0.622803\pi$
$811$	−3.25283e27	−0.752598	−0.376299	+	0.926498i	$0.622803\pi$
$812$	−3.42680e26	−0.0783622
$813$	0	0
$814$	−1.50956e27	−0.337225
$815$	−9.14404e27	−2.01902
$816$	0	0
$817$	5.67513e27	1.22424
$818$	6.29772e26	0.134285
$819$	0	0
$820$	−2.75239e27	−0.573427
$821$	7.44187e27	1.53258	0.766289	−	0.642496i	$-0.222101\pi$
$821$	7.44187e27	1.53258	0.766289	+	0.642496i	$0.222101\pi$
$822$	0	0
$823$	1.28115e27	0.257811	0.128905	−	0.991657i	$-0.458854\pi$
$823$	1.28115e27	0.257811	0.128905	+	0.991657i	$0.458854\pi$
$824$	2.76459e27	0.549948
$825$	0	0
$826$	−2.25743e26	−0.0438837
$827$	−6.37977e26	−0.122603	−0.0613017	−	0.998119i	$-0.519525\pi$
$827$	−6.37977e26	−0.122603	−0.0613017	+	0.998119i	$0.519525\pi$
$828$	0	0
$829$	−1.57977e27	−0.296706	−0.148353	−	0.988934i	$-0.547397\pi$
$829$	−1.57977e27	−0.296706	−0.148353	+	0.988934i	$0.547397\pi$
$830$	−1.89227e27	−0.351350
$831$	0	0
$832$	−5.96726e26	−0.108294
$833$	−6.80384e27	−1.22075
$834$	0	0
$835$	−4.44302e27	−0.779215
$836$	2.17492e27	0.377123
$837$	0	0
$838$	−5.40747e27	−0.916592
$839$	−2.75112e27	−0.461074	−0.230537	−	0.973064i	$-0.574048\pi$
$839$	−2.75112e27	−0.461074	−0.230537	+	0.973064i	$0.574048\pi$
$840$	0	0
$841$	−5.24019e27	−0.858588
$842$	8.41657e26	0.136355
$843$	0	0
$844$	6.20961e27	0.983583
$845$	−2.02637e27	−0.317380
$846$	0	0
$847$	9.43687e26	0.144523
$848$	2.10240e27	0.318386
$849$	0	0
$850$	−4.36639e27	−0.646611
$851$	1.57653e27	0.230873
$852$	0	0
$853$	−1.24537e28	−1.78354	−0.891769	−	0.452490i	$-0.850536\pi$
$853$	−1.24537e28	−1.78354	−0.891769	+	0.452490i	$0.850536\pi$
$854$	−2.08772e27	−0.295680
$855$	0	0
$856$	−2.08416e26	−0.0288689
$857$	7.69343e26	0.105391	0.0526953	−	0.998611i	$-0.483219\pi$
$857$	7.69343e26	0.105391	0.0526953	+	0.998611i	$0.483219\pi$
$858$	0	0
$859$	−7.99570e26	−0.107133	−0.0535663	−	0.998564i	$-0.517059\pi$
$859$	−7.99570e26	−0.107133	−0.0535663	+	0.998564i	$0.517059\pi$
$860$	−6.29812e27	−0.834595
$861$	0	0
$862$	8.62509e27	1.11801
$863$	9.51188e27	1.21945	0.609725	−	0.792613i	$-0.291280\pi$
$863$	9.51188e27	1.21945	0.609725	+	0.792613i	$0.291280\pi$
$864$	0	0
$865$	8.11278e27	1.01746
$866$	−1.47882e27	−0.183441
$867$	0	0
$868$	−1.53944e27	−0.186820
$869$	6.83514e27	0.820464
$870$	0	0
$871$	3.42345e27	0.402060
$872$	−1.59564e27	−0.185365
$873$	0	0
$874$	−2.27140e27	−0.258188
$875$	1.79699e27	0.202055
$876$	0	0
$877$	8.50499e27	0.935788	0.467894	−	0.883784i	$-0.345013\pi$
$877$	8.50499e27	0.935788	0.467894	+	0.883784i	$0.345013\pi$
$878$	5.63724e27	0.613577
$879$	0	0
$880$	−2.41367e27	−0.257095
$881$	7.43881e26	0.0783849	0.0391925	−	0.999232i	$-0.487521\pi$
$881$	7.43881e26	0.0783849	0.0391925	+	0.999232i	$0.487521\pi$
$882$	0	0
$883$	1.77154e28	1.82694	0.913469	−	0.406909i	$-0.133393\pi$
$883$	1.77154e28	1.82694	0.913469	+	0.406909i	$0.133393\pi$
$884$	−6.27258e27	−0.639955
$885$	0	0
$886$	7.13830e26	0.0712811
$887$	−7.00983e26	−0.0692521	−0.0346260	−	0.999400i	$-0.511024\pi$
$887$	−7.00983e26	−0.0692521	−0.0346260	+	0.999400i	$0.511024\pi$
$888$	0	0
$889$	1.29650e27	0.125374
$890$	−5.75000e27	−0.550127
$891$	0	0
$892$	2.42566e26	0.0227176
$893$	1.58872e28	1.47217
$894$	0	0
$895$	−1.83181e27	−0.166173
$896$	4.10407e26	0.0368374
$897$	0	0
$898$	2.17119e27	0.190798
$899$	3.87722e27	0.337134
$900$	0	0
$901$	2.20997e28	1.88148
$902$	−6.11460e27	−0.515117
$903$	0	0
$904$	4.87618e27	0.402235
$905$	1.23651e28	1.00934
$906$	0	0
$907$	−8.55845e27	−0.684110	−0.342055	−	0.939680i	$-0.611123\pi$
$907$	−8.55845e27	−0.684110	−0.342055	+	0.939680i	$0.611123\pi$
$908$	−2.12097e27	−0.167772
$909$	0	0
$910$	−4.19359e27	−0.324857
$911$	−1.70402e28	−1.30632	−0.653162	−	0.757218i	$-0.726558\pi$
$911$	−1.70402e28	−1.30632	−0.653162	+	0.757218i	$0.726558\pi$
$912$	0	0
$913$	−4.20379e27	−0.315623
$914$	−3.05947e27	−0.227331
$915$	0	0
$916$	−5.09067e27	−0.370483
$917$	−6.36450e27	−0.458412
$918$	0	0
$919$	−2.63990e28	−1.86247	−0.931237	−	0.364414i	$-0.881269\pi$
$919$	−2.63990e28	−1.86247	−0.931237	+	0.364414i	$0.881269\pi$
$920$	2.52075e27	0.176013
$921$	0	0
$922$	−5.67665e27	−0.388283
$923$	9.45547e27	0.640128
$924$	0	0
$925$	−5.50707e27	−0.365236
$926$	5.01302e27	0.329075
$927$	0	0
$928$	−1.03365e27	−0.0664764
$929$	4.71555e27	0.300181	0.150090	−	0.988672i	$-0.452043\pi$
$929$	4.71555e27	0.300181	0.150090	+	0.988672i	$0.452043\pi$
$930$	0	0
$931$	1.23635e28	0.771117
$932$	5.31147e27	0.327917
$933$	0	0
$934$	2.26811e28	1.37204
$935$	−2.53717e28	−1.51928
$936$	0	0
$937$	1.32505e28	0.777508	0.388754	−	0.921342i	$-0.372906\pi$
$937$	1.32505e28	0.777508	0.388754	+	0.921342i	$0.372906\pi$
$938$	−2.35452e27	−0.136765
$939$	0	0
$940$	−1.76312e28	−1.00361
$941$	1.96808e28	1.10903	0.554513	−	0.832175i	$-0.312904\pi$
$941$	1.96808e28	1.10903	0.554513	+	0.832175i	$0.312904\pi$
$942$	0	0
$943$	6.38587e27	0.352662
$944$	−6.80924e26	−0.0372275
$945$	0	0
$946$	−1.39917e28	−0.749728
$947$	−1.96022e28	−1.03987	−0.519935	−	0.854206i	$-0.674044\pi$
$947$	−1.96022e28	−1.03987	−0.519935	+	0.854206i	$0.674044\pi$
$948$	0	0
$949$	−2.89846e28	−1.50709
$950$	7.93435e27	0.408448
$951$	0	0
$952$	4.31406e27	0.217688
$953$	−1.88423e28	−0.941352	−0.470676	−	0.882306i	$-0.655990\pi$
$953$	−1.88423e28	−0.941352	−0.470676	+	0.882306i	$0.655990\pi$
$954$	0	0
$955$	1.59102e28	0.779191
$956$	5.66377e27	0.274635
$957$	0	0
$958$	1.70766e27	0.0811763
$959$	5.46000e26	0.0256990
$960$	0	0
$961$	−4.25287e27	−0.196250
$962$	−7.91122e27	−0.361477
$963$	0	0
$964$	−1.21225e28	−0.543074
$965$	2.04198e28	0.905820
$966$	0	0
$967$	−2.90124e28	−1.26192	−0.630961	−	0.775815i	$-0.717339\pi$
$967$	−2.90124e28	−1.26192	−0.630961	+	0.775815i	$0.717339\pi$
$968$	2.84651e27	0.122602
$969$	0	0
$970$	3.64907e28	1.54117
$971$	−3.48521e28	−1.45763	−0.728814	−	0.684712i	$-0.759928\pi$
$971$	−3.48521e28	−1.45763	−0.728814	+	0.684712i	$0.759928\pi$
$972$	0	0
$973$	−4.49389e27	−0.184311
$974$	2.40716e28	0.977676
$975$	0	0
$976$	−6.29733e27	−0.250832
$977$	−1.42249e28	−0.561113	−0.280556	−	0.959838i	$-0.590519\pi$
$977$	−1.42249e28	−0.561113	−0.280556	+	0.959838i	$0.590519\pi$
$978$	0	0
$979$	−1.27740e28	−0.494187
$980$	−1.37208e28	−0.525691
$981$	0	0
$982$	−2.23998e28	−0.841752
$983$	1.50245e28	0.559167	0.279584	−	0.960121i	$-0.409804\pi$
$983$	1.50245e28	0.559167	0.279584	+	0.960121i	$0.409804\pi$
$984$	0	0
$985$	3.00181e28	1.09582
$986$	−1.08654e28	−0.392838
$987$	0	0
$988$	1.13982e28	0.404244
$989$	1.46124e28	0.513282
$990$	0	0
$991$	−2.10392e27	−0.0724985	−0.0362492	−	0.999343i	$-0.511541\pi$
$991$	−2.10392e27	−0.0724985	−0.0362492	+	0.999343i	$0.511541\pi$
$992$	−4.64351e27	−0.158484
$993$	0	0
$994$	−6.50313e27	−0.217747
$995$	4.18504e28	1.38797
$996$	0	0
$997$	−3.57257e28	−1.16246	−0.581228	−	0.813741i	$-0.697427\pi$
$997$	−3.57257e28	−1.16246	−0.581228	+	0.813741i	$0.697427\pi$
$998$	8.87334e27	0.285988
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.20.a.c.1.1		1
3.2	odd	2		2.20.a.b.1.1	✓	1
12.11	even	2		16.20.a.d.1.1		1
15.2	even	4		50.20.b.a.49.2		2
15.8	even	4		50.20.b.a.49.1		2
15.14	odd	2		50.20.a.b.1.1		1
21.20	even	2		98.20.a.b.1.1		1
24.5	odd	2		64.20.a.i.1.1		1
24.11	even	2		64.20.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.20.a.b.1.1	✓	1	3.2	odd	2
16.20.a.d.1.1		1	12.11	even	2
18.20.a.c.1.1		1	1.1	even	1	trivial
50.20.a.b.1.1		1	15.14	odd	2
50.20.b.a.49.1		2	15.8	even	4
50.20.b.a.49.2		2	15.2	even	4
64.20.a.a.1.1		1	24.11	even	2
64.20.a.i.1.1		1	24.5	odd	2
98.20.a.b.1.1		1	21.20	even	2

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:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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