LMFDB - Embedded newform 18.14.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,14,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, 14, names="a")

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 14 $
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:	$19.3015672113$
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	$\chi$	$=$	18.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-64.0000 q^{2} +4096.00 q^{4} -15936.0 q^{5} +98252.0 q^{7} -262144. q^{8} +O(q^{10})$

\(q-64.0000 q^{2} +4096.00 q^{4} -15936.0 q^{5} +98252.0 q^{7} -262144. q^{8} +1.01990e6 q^{10} +1.63046e6 q^{11} +2.23163e7 q^{13} -6.28813e6 q^{14} +1.67772e7 q^{16} -1.22938e8 q^{17} -7.42730e7 q^{19} -6.52739e7 q^{20} -1.04350e8 q^{22} -1.06951e9 q^{23} -9.66747e8 q^{25} -1.42824e9 q^{26} +4.02440e8 q^{28} -5.60709e9 q^{29} +2.16203e9 q^{31} -1.07374e9 q^{32} +7.86803e9 q^{34} -1.56574e9 q^{35} -5.95945e9 q^{37} +4.75347e9 q^{38} +4.17753e9 q^{40} +2.16769e10 q^{41} -6.11010e10 q^{43} +6.67838e9 q^{44} +6.84486e10 q^{46} +1.36472e11 q^{47} -8.72356e10 q^{49} +6.18718e10 q^{50} +9.14076e10 q^{52} +5.57016e8 q^{53} -2.59831e10 q^{55} -2.57562e10 q^{56} +3.58854e11 q^{58} -3.02212e11 q^{59} -1.90535e11 q^{61} -1.38370e11 q^{62} +6.87195e10 q^{64} -3.55633e11 q^{65} -9.18343e11 q^{67} -5.03554e11 q^{68} +1.00208e11 q^{70} -1.08659e12 q^{71} -7.72759e10 q^{73} +3.81405e11 q^{74} -3.04222e11 q^{76} +1.60196e11 q^{77} +2.62436e12 q^{79} -2.67362e11 q^{80} -1.38732e12 q^{82} +3.26961e12 q^{83} +1.95914e12 q^{85} +3.91047e12 q^{86} -4.27416e11 q^{88} +5.92223e12 q^{89} +2.19262e12 q^{91} -4.38071e12 q^{92} -8.73420e12 q^{94} +1.18361e12 q^{95} +5.34013e12 q^{97} +5.58308e12 q^{98} +O(q^{100})\)

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 $

$2$	−64.0000	−0.707107
$2$	−64.0000	−0.707107
$3$	0	0
$3$	0	0
$4$	4096.00	0.500000
$5$	−15936.0	−0.456115	−0.228057	−	0.973648i	$-0.573237\pi$
$5$	−15936.0	−0.456115	−0.228057	+	0.973648i	$0.573237\pi$
$6$	0	0
$7$	98252.0	0.315649	0.157824	−	0.987467i	$-0.449552\pi$
$7$	98252.0	0.315649	0.157824	+	0.987467i	$0.449552\pi$
$8$	−262144.	−0.353553
$9$	0	0
$10$	1.01990e6	0.322522
$11$	1.63046e6	0.277497	0.138749	−	0.990328i	$-0.455692\pi$
$11$	1.63046e6	0.277497	0.138749	+	0.990328i	$0.455692\pi$
$12$	0	0
$13$	2.23163e7	1.28230	0.641151	−	0.767415i	$-0.278457\pi$
$13$	2.23163e7	1.28230	0.641151	+	0.767415i	$0.278457\pi$
$14$	−6.28813e6	−0.223197
$15$	0	0
$16$	1.67772e7	0.250000
$17$	−1.22938e8	−1.23529	−0.617644	−	0.786458i	$-0.711913\pi$
$17$	−1.22938e8	−1.23529	−0.617644	+	0.786458i	$0.711913\pi$
$18$	0	0
$19$	−7.42730e7	−0.362187	−0.181093	−	0.983466i	$-0.557964\pi$
$19$	−7.42730e7	−0.362187	−0.181093	+	0.983466i	$0.557964\pi$
$20$	−6.52739e7	−0.228057
$21$	0	0
$22$	−1.04350e8	−0.196220
$23$	−1.06951e9	−1.50645	−0.753223	−	0.657765i	$-0.771502\pi$
$23$	−1.06951e9	−1.50645	−0.753223	+	0.657765i	$0.771502\pi$
$24$	0	0
$25$	−9.66747e8	−0.791959
$26$	−1.42824e9	−0.906725
$27$	0	0
$28$	4.02440e8	0.157824
$29$	−5.60709e9	−1.75045	−0.875227	−	0.483713i	$-0.839288\pi$
$29$	−5.60709e9	−1.75045	−0.875227	+	0.483713i	$0.839288\pi$
$30$	0	0
$31$	2.16203e9	0.437533	0.218767	−	0.975777i	$-0.429797\pi$
$31$	2.16203e9	0.437533	0.218767	+	0.975777i	$0.429797\pi$
$32$	−1.07374e9	−0.176777
$33$	0	0
$34$	7.86803e9	0.873480
$35$	−1.56574e9	−0.143972
$36$	0	0
$37$	−5.95945e9	−0.381852	−0.190926	−	0.981604i	$-0.561149\pi$
$37$	−5.95945e9	−0.381852	−0.190926	+	0.981604i	$0.561149\pi$
$38$	4.75347e9	0.256105
$39$	0	0
$40$	4.17753e9	0.161261
$41$	2.16769e10	0.712691	0.356345	−	0.934354i	$-0.384023\pi$
$41$	2.16769e10	0.712691	0.356345	+	0.934354i	$0.384023\pi$
$42$	0	0
$43$	−6.11010e10	−1.47402	−0.737010	−	0.675881i	$-0.763763\pi$
$43$	−6.11010e10	−1.47402	−0.737010	+	0.675881i	$0.763763\pi$
$44$	6.67838e9	0.138749
$45$	0	0
$46$	6.84486e10	1.06522
$47$	1.36472e11	1.84675	0.923373	−	0.383903i	$-0.125420\pi$
$47$	1.36472e11	1.84675	0.923373	+	0.383903i	$0.125420\pi$
$48$	0	0
$49$	−8.72356e10	−0.900366
$50$	6.18718e10	0.560000
$51$	0	0
$52$	9.14076e10	0.641151
$53$	5.57016e8	0.00345203	0.00172601	−	0.999999i	$-0.499451\pi$
$53$	5.57016e8	0.00345203	0.00172601	+	0.999999i	$0.499451\pi$
$54$	0	0
$55$	−2.59831e10	−0.126571
$56$	−2.57562e10	−0.111599
$57$	0	0
$58$	3.58854e11	1.23776
$59$	−3.02212e11	−0.932768	−0.466384	−	0.884582i	$-0.654443\pi$
$59$	−3.02212e11	−0.932768	−0.466384	+	0.884582i	$0.654443\pi$
$60$	0	0
$61$	−1.90535e11	−0.473513	−0.236756	−	0.971569i	$-0.576084\pi$
$61$	−1.90535e11	−0.473513	−0.236756	+	0.971569i	$0.576084\pi$
$62$	−1.38370e11	−0.309383
$63$	0	0
$64$	6.87195e10	0.125000
$65$	−3.55633e11	−0.584877
$66$	0	0
$67$	−9.18343e11	−1.24028	−0.620139	−	0.784492i	$-0.712924\pi$
$67$	−9.18343e11	−1.24028	−0.620139	+	0.784492i	$0.712924\pi$
$68$	−5.03554e11	−0.617644
$69$	0	0
$70$	1.00208e11	0.101804
$71$	−1.08659e12	−1.00667	−0.503336	−	0.864091i	$-0.667894\pi$
$71$	−1.08659e12	−1.00667	−0.503336	+	0.864091i	$0.667894\pi$
$72$	0	0
$73$	−7.72759e10	−0.0597648	−0.0298824	−	0.999553i	$-0.509513\pi$
$73$	−7.72759e10	−0.0597648	−0.0298824	+	0.999553i	$0.509513\pi$
$74$	3.81405e11	0.270010
$75$	0	0
$76$	−3.04222e11	−0.181093
$77$	1.60196e11	0.0875917
$78$	0	0
$79$	2.62436e12	1.21464	0.607320	−	0.794457i	$-0.292244\pi$
$79$	2.62436e12	1.21464	0.607320	+	0.794457i	$0.292244\pi$
$80$	−2.67362e11	−0.114029
$81$	0	0
$82$	−1.38732e12	−0.503948
$83$	3.26961e12	1.09771	0.548856	−	0.835917i	$-0.315064\pi$
$83$	3.26961e12	1.09771	0.548856	+	0.835917i	$0.315064\pi$
$84$	0	0
$85$	1.95914e12	0.563433
$86$	3.91047e12	1.04229
$87$	0	0
$88$	−4.27416e11	−0.0981101
$89$	5.92223e12	1.26314	0.631568	−	0.775320i	$-0.282412\pi$
$89$	5.92223e12	1.26314	0.631568	+	0.775320i	$0.282412\pi$
$90$	0	0
$91$	2.19262e12	0.404757
$92$	−4.38071e12	−0.753223
$93$	0	0
$94$	−8.73420e12	−1.30585
$95$	1.18361e12	0.165199
$96$	0	0
$97$	5.34013e12	0.650932	0.325466	−	0.945554i	$-0.394479\pi$
$97$	5.34013e12	0.650932	0.325466	+	0.945554i	$0.394479\pi$
$98$	5.58308e12	0.636655
$99$	0	0
$100$	−3.95980e12	−0.395980
$101$	−7.40002e12	−0.693656	−0.346828	−	0.937929i	$-0.612741\pi$
$101$	−7.40002e12	−0.693656	−0.346828	+	0.937929i	$0.612741\pi$
$102$	0	0
$103$	−9.13810e12	−0.754074	−0.377037	−	0.926198i	$-0.623057\pi$
$103$	−9.13810e12	−0.754074	−0.377037	+	0.926198i	$0.623057\pi$
$104$	−5.85009e12	−0.453362
$105$	0	0
$106$	−3.56490e10	−0.00244095
$107$	−6.61381e12	−0.426047	−0.213023	−	0.977047i	$-0.568331\pi$
$107$	−6.61381e12	−0.426047	−0.213023	+	0.977047i	$0.568331\pi$
$108$	0	0
$109$	1.46236e13	0.835185	0.417592	−	0.908634i	$-0.362874\pi$
$109$	1.46236e13	0.835185	0.417592	+	0.908634i	$0.362874\pi$
$110$	1.66292e12	0.0894990
$111$	0	0
$112$	1.64840e12	0.0789122
$113$	−2.97612e13	−1.34475	−0.672374	−	0.740212i	$-0.734725\pi$
$113$	−2.97612e13	−1.34475	−0.672374	+	0.740212i	$0.734725\pi$
$114$	0	0
$115$	1.70437e13	0.687113
$116$	−2.29666e13	−0.875227
$117$	0	0
$118$	1.93416e13	0.659566
$119$	−1.20789e13	−0.389917
$120$	0	0
$121$	−3.18643e13	−0.922995
$122$	1.21943e13	0.334824
$123$	0	0
$124$	8.85568e12	0.218767
$125$	3.48592e13	0.817339
$126$	0	0
$127$	−5.79603e13	−1.22576	−0.612881	−	0.790175i	$-0.709990\pi$
$127$	−5.79603e13	−1.22576	−0.612881	+	0.790175i	$0.709990\pi$
$128$	−4.39805e12	−0.0883883
$129$	0	0
$130$	2.27605e13	0.413571
$131$	3.17740e13	0.549298	0.274649	−	0.961545i	$-0.411438\pi$
$131$	3.17740e13	0.549298	0.274649	+	0.961545i	$0.411438\pi$
$132$	0	0
$133$	−7.29747e12	−0.114324
$134$	5.87740e13	0.877008
$135$	0	0
$136$	3.22275e13	0.436740
$137$	6.89176e13	0.890526	0.445263	−	0.895400i	$-0.353110\pi$
$137$	6.89176e13	0.890526	0.445263	+	0.895400i	$0.353110\pi$
$138$	0	0
$139$	−1.18252e14	−1.39064	−0.695318	−	0.718702i	$-0.744737\pi$
$139$	−1.18252e14	−1.39064	−0.695318	+	0.718702i	$0.744737\pi$
$140$	−6.41329e12	−0.0719861
$141$	0	0
$142$	6.95420e13	0.711824
$143$	3.63859e13	0.355836
$144$	0	0
$145$	8.93546e13	0.798408
$146$	4.94566e12	0.0422601
$147$	0	0
$148$	−2.44099e13	−0.190926
$149$	1.55654e12	0.0116533	0.00582666	−	0.999983i	$-0.498145\pi$
$149$	1.55654e12	0.0116533	0.00582666	+	0.999983i	$0.498145\pi$
$150$	0	0
$151$	3.55424e13	0.244004	0.122002	−	0.992530i	$-0.461069\pi$
$151$	3.55424e13	0.244004	0.122002	+	0.992530i	$0.461069\pi$
$152$	1.94702e13	0.128052
$153$	0	0
$154$	−1.02526e13	−0.0619367
$155$	−3.44541e13	−0.199565
$156$	0	0
$157$	1.53865e14	0.819958	0.409979	−	0.912095i	$-0.365536\pi$
$157$	1.53865e14	0.819958	0.409979	+	0.912095i	$0.365536\pi$
$158$	−1.67959e14	−0.858881
$159$	0	0
$160$	1.71111e13	0.0806305
$161$	−1.05081e14	−0.475508
$162$	0	0
$163$	−1.99640e14	−0.833736	−0.416868	−	0.908967i	$-0.636872\pi$
$163$	−1.99640e14	−0.833736	−0.416868	+	0.908967i	$0.636872\pi$
$164$	8.87884e13	0.356345
$165$	0	0
$166$	−2.09255e14	−0.776199
$167$	−4.64940e14	−1.65859	−0.829297	−	0.558809i	$-0.811259\pi$
$167$	−4.64940e14	−1.65859	−0.829297	+	0.558809i	$0.811259\pi$
$168$	0	0
$169$	1.95142e14	0.644300
$170$	−1.25385e14	−0.398407
$171$	0	0
$172$	−2.50270e14	−0.737010
$173$	−3.76731e14	−1.06840	−0.534198	−	0.845360i	$-0.679386\pi$
$173$	−3.76731e14	−1.06840	−0.534198	+	0.845360i	$0.679386\pi$
$174$	0	0
$175$	−9.49848e13	−0.249981
$176$	2.73546e13	0.0693743
$177$	0	0
$178$	−3.79023e14	−0.893172
$179$	7.32554e14	1.66454	0.832271	−	0.554369i	$-0.187040\pi$
$179$	7.32554e14	1.66454	0.832271	+	0.554369i	$0.187040\pi$
$180$	0	0
$181$	7.37497e14	1.55901	0.779506	−	0.626395i	$-0.215470\pi$
$181$	7.37497e14	1.55901	0.779506	+	0.626395i	$0.215470\pi$
$182$	−1.40328e14	−0.286207
$183$	0	0
$184$	2.80365e14	0.532609
$185$	9.49698e13	0.174168
$186$	0	0
$187$	−2.00446e14	−0.342789
$188$	5.58989e14	0.923373
$189$	0	0
$190$	−7.57513e13	−0.116813
$191$	2.20757e14	0.329001	0.164501	−	0.986377i	$-0.447399\pi$
$191$	2.20757e14	0.329001	0.164501	+	0.986377i	$0.447399\pi$
$192$	0	0
$193$	1.11051e15	1.54667	0.773337	−	0.633995i	$-0.218586\pi$
$193$	1.11051e15	1.54667	0.773337	+	0.633995i	$0.218586\pi$
$194$	−3.41769e14	−0.460279
$195$	0	0
$196$	−3.57317e14	−0.450183
$197$	1.37055e15	1.67057	0.835287	−	0.549815i	$-0.185302\pi$
$197$	1.37055e15	1.67057	0.835287	+	0.549815i	$0.185302\pi$
$198$	0	0
$199$	−9.76425e14	−1.11454	−0.557268	−	0.830333i	$-0.688150\pi$
$199$	−9.76425e14	−1.11454	−0.557268	+	0.830333i	$0.688150\pi$
$200$	2.53427e14	0.280000
$201$	0	0
$202$	4.73602e14	0.490489
$203$	−5.50908e14	−0.552529
$204$	0	0
$205$	−3.45442e14	−0.325069
$206$	5.84838e14	0.533211
$207$	0	0
$208$	3.74405e14	0.320576
$209$	−1.21099e14	−0.100506
$210$	0	0
$211$	−1.13103e15	−0.882343	−0.441171	−	0.897423i	$-0.645437\pi$
$211$	−1.13103e15	−0.882343	−0.441171	+	0.897423i	$0.645437\pi$
$212$	2.28154e12	0.00172601
$213$	0	0
$214$	4.23284e14	0.301261
$215$	9.73706e14	0.672323
$216$	0	0
$217$	2.12424e14	0.138107
$218$	−9.35912e14	−0.590565
$219$	0	0
$220$	−1.06427e14	−0.0632853
$221$	−2.74352e15	−1.58401
$222$	0	0
$223$	−3.49575e15	−1.90352	−0.951762	−	0.306837i	$-0.900729\pi$
$223$	−3.49575e15	−1.90352	−0.951762	+	0.306837i	$0.900729\pi$
$224$	−1.05497e14	−0.0557994
$225$	0	0
$226$	1.90472e15	0.950880
$227$	−2.16551e15	−1.05049	−0.525245	−	0.850951i	$-0.676027\pi$
$227$	−2.16551e15	−1.05049	−0.525245	+	0.850951i	$0.676027\pi$
$228$	0	0
$229$	2.65691e15	1.21744	0.608719	−	0.793386i	$-0.291684\pi$
$229$	2.65691e15	1.21744	0.608719	+	0.793386i	$0.291684\pi$
$230$	−1.09080e15	−0.485862
$231$	0	0
$232$	1.46987e15	0.618879
$233$	1.28597e15	0.526525	0.263262	−	0.964724i	$-0.415202\pi$
$233$	1.28597e15	0.526525	0.263262	+	0.964724i	$0.415202\pi$
$234$	0	0
$235$	−2.17482e15	−0.842329
$236$	−1.23786e15	−0.466384
$237$	0	0
$238$	7.73050e14	0.275713
$239$	−3.90092e15	−1.35388	−0.676940	−	0.736038i	$-0.736694\pi$
$239$	−3.90092e15	−1.35388	−0.676940	+	0.736038i	$0.736694\pi$
$240$	0	0
$241$	3.93072e15	1.29229	0.646147	−	0.763213i	$-0.276379\pi$
$241$	3.93072e15	1.29229	0.646147	+	0.763213i	$0.276379\pi$
$242$	2.03932e15	0.652656
$243$	0	0
$244$	−7.80433e14	−0.236756
$245$	1.39019e15	0.410670
$246$	0	0
$247$	−1.65750e15	−0.464433
$248$	−5.66763e14	−0.154691
$249$	0	0
$250$	−2.23099e15	−0.577946
$251$	6.50267e15	1.64139	0.820696	−	0.571365i	$-0.193586\pi$
$251$	6.50267e15	1.64139	0.820696	+	0.571365i	$0.193586\pi$
$252$	0	0
$253$	−1.74380e15	−0.418035
$254$	3.70946e15	0.866745
$255$	0	0
$256$	2.81475e14	0.0625000
$257$	−2.69802e15	−0.584090	−0.292045	−	0.956405i	$-0.594336\pi$
$257$	−2.69802e15	−0.584090	−0.292045	+	0.956405i	$0.594336\pi$
$258$	0	0
$259$	−5.85528e14	−0.120531
$260$	−1.45667e15	−0.292439
$261$	0	0
$262$	−2.03353e15	−0.388412
$263$	5.08958e15	0.948352	0.474176	−	0.880430i	$-0.342746\pi$
$263$	5.08958e15	0.948352	0.474176	+	0.880430i	$0.342746\pi$
$264$	0	0
$265$	−8.87660e12	−0.00157452
$266$	4.67038e14	0.0808391
$267$	0	0
$268$	−3.76153e15	−0.620139
$269$	7.06228e15	1.13646	0.568231	−	0.822869i	$-0.307628\pi$
$269$	7.06228e15	1.13646	0.568231	+	0.822869i	$0.307628\pi$
$270$	0	0
$271$	5.25952e15	0.806576	0.403288	−	0.915073i	$-0.367867\pi$
$271$	5.25952e15	0.806576	0.403288	+	0.915073i	$0.367867\pi$
$272$	−2.06256e15	−0.308822
$273$	0	0
$274$	−4.41073e15	−0.629697
$275$	−1.57625e15	−0.219767
$276$	0	0
$277$	1.34084e16	1.78344	0.891722	−	0.452584i	$-0.149498\pi$
$277$	1.34084e16	1.78344	0.891722	+	0.452584i	$0.149498\pi$
$278$	7.56815e15	0.983329
$279$	0	0
$280$	4.10450e14	0.0509018
$281$	−1.82029e15	−0.220572	−0.110286	−	0.993900i	$-0.535177\pi$
$281$	−1.82029e15	−0.220572	−0.110286	+	0.993900i	$0.535177\pi$
$282$	0	0
$283$	−4.56765e15	−0.528544	−0.264272	−	0.964448i	$-0.585132\pi$
$283$	−4.56765e15	−0.528544	−0.264272	+	0.964448i	$0.585132\pi$
$284$	−4.45069e15	−0.503336
$285$	0	0
$286$	−2.32870e15	−0.251614
$287$	2.12979e15	0.224960
$288$	0	0
$289$	5.20917e15	0.525936
$290$	−5.71869e15	−0.564560
$291$	0	0
$292$	−3.16522e14	−0.0298824
$293$	−1.34769e16	−1.24437	−0.622186	−	0.782869i	$-0.713755\pi$
$293$	−1.34769e16	−1.24437	−0.622186	+	0.782869i	$0.713755\pi$
$294$	0	0
$295$	4.81605e15	0.425449
$296$	1.56223e15	0.135005
$297$	0	0
$298$	−9.96186e13	−0.00824014
$299$	−2.38675e16	−1.93172
$300$	0	0
$301$	−6.00330e15	−0.465273
$302$	−2.27472e15	−0.172537
$303$	0	0
$304$	−1.24609e15	−0.0905467
$305$	3.03637e15	0.215976
$306$	0	0
$307$	−3.20152e15	−0.218252	−0.109126	−	0.994028i	$-0.534805\pi$
$307$	−3.20152e15	−0.218252	−0.109126	+	0.994028i	$0.534805\pi$
$308$	6.56164e14	0.0437958
$309$	0	0
$310$	2.20506e15	0.141114
$311$	2.27156e15	0.142358	0.0711789	−	0.997464i	$-0.477324\pi$
$311$	2.27156e15	0.142358	0.0711789	+	0.997464i	$0.477324\pi$
$312$	0	0
$313$	−2.03562e16	−1.22365	−0.611827	−	0.790991i	$-0.709565\pi$
$313$	−2.03562e16	−1.22365	−0.611827	+	0.790991i	$0.709565\pi$
$314$	−9.84735e15	−0.579798
$315$	0	0
$316$	1.07494e16	0.607320
$317$	4.16681e15	0.230631	0.115316	−	0.993329i	$-0.463212\pi$
$317$	4.16681e15	0.230631	0.115316	+	0.993329i	$0.463212\pi$
$318$	0	0
$319$	−9.14216e15	−0.485746
$320$	−1.09511e15	−0.0570144
$321$	0	0
$322$	6.72521e15	0.336235
$323$	9.13097e15	0.447405
$324$	0	0
$325$	−2.15742e16	−1.01553
$326$	1.27770e16	0.589541
$327$	0	0
$328$	−5.68246e15	−0.251974
$329$	1.34086e16	0.582923
$330$	0	0
$331$	4.19092e16	1.75157	0.875785	−	0.482702i	$-0.160344\pi$
$331$	4.19092e16	1.75157	0.875785	+	0.482702i	$0.160344\pi$
$332$	1.33923e16	0.548856
$333$	0	0
$334$	2.97562e16	1.17280
$335$	1.46347e16	0.565709
$336$	0	0
$337$	−1.06740e16	−0.396946	−0.198473	−	0.980106i	$-0.563598\pi$
$337$	−1.06740e16	−0.396946	−0.198473	+	0.980106i	$0.563598\pi$
$338$	−1.24891e16	−0.455589
$339$	0	0
$340$	8.02464e15	0.281717
$341$	3.52511e15	0.121414
$342$	0	0
$343$	−1.80906e16	−0.599848
$344$	1.60173e16	0.521145
$345$	0	0
$346$	2.41108e16	0.755470
$347$	2.04361e16	0.628429	0.314214	−	0.949352i	$-0.398259\pi$
$347$	2.04361e16	0.628429	0.314214	+	0.949352i	$0.398259\pi$
$348$	0	0
$349$	−1.64023e16	−0.485893	−0.242947	−	0.970040i	$-0.578114\pi$
$349$	−1.64023e16	−0.485893	−0.242947	+	0.970040i	$0.578114\pi$
$350$	6.07903e15	0.176763
$351$	0	0
$352$	−1.75070e15	−0.0490551
$353$	1.55775e16	0.428510	0.214255	−	0.976778i	$-0.431268\pi$
$353$	1.55775e16	0.428510	0.214255	+	0.976778i	$0.431268\pi$
$354$	0	0
$355$	1.73160e16	0.459158
$356$	2.42575e16	0.631568
$357$	0	0
$358$	−4.68834e16	−1.17701
$359$	−1.79339e16	−0.442140	−0.221070	−	0.975258i	$-0.570955\pi$
$359$	−1.79339e16	−0.442140	−0.221070	+	0.975258i	$0.570955\pi$
$360$	0	0
$361$	−3.65365e16	−0.868821
$362$	−4.71998e16	−1.10239
$363$	0	0
$364$	8.98098e15	0.202379
$365$	1.23147e15	0.0272596
$366$	0	0
$367$	4.84657e16	1.03539	0.517697	−	0.855564i	$-0.326790\pi$
$367$	4.84657e16	1.03539	0.517697	+	0.855564i	$0.326790\pi$
$368$	−1.79434e16	−0.376611
$369$	0	0
$370$	−6.07807e15	−0.123156
$371$	5.47279e13	0.00108963
$372$	0	0
$373$	−3.64940e16	−0.701640	−0.350820	−	0.936443i	$-0.614097\pi$
$373$	−3.64940e16	−0.701640	−0.350820	+	0.936443i	$0.614097\pi$
$374$	1.28285e16	0.242388
$375$	0	0
$376$	−3.57753e16	−0.652924
$377$	−1.25130e17	−2.24461
$378$	0	0
$379$	3.25386e16	0.563955	0.281977	−	0.959421i	$-0.409010\pi$
$379$	3.25386e16	0.563955	0.281977	+	0.959421i	$0.409010\pi$
$380$	4.84808e15	0.0825994
$381$	0	0
$382$	−1.41284e16	−0.232639
$383$	−3.03495e16	−0.491314	−0.245657	−	0.969357i	$-0.579004\pi$
$383$	−3.03495e16	−0.491314	−0.245657	+	0.969357i	$0.579004\pi$
$384$	0	0
$385$	−2.55289e15	−0.0399519
$386$	−7.10724e16	−1.09366
$387$	0	0
$388$	2.18732e16	0.325466
$389$	4.62220e15	0.0676358	0.0338179	−	0.999428i	$-0.489233\pi$
$389$	4.62220e15	0.0676358	0.0338179	+	0.999428i	$0.489233\pi$
$390$	0	0
$391$	1.31483e17	1.86089
$392$	2.28683e16	0.318327
$393$	0	0
$394$	−8.77154e16	−1.18127
$395$	−4.18219e16	−0.554016
$396$	0	0
$397$	4.28843e15	0.0549743	0.0274872	−	0.999622i	$-0.491249\pi$
$397$	4.28843e15	0.0549743	0.0274872	+	0.999622i	$0.491249\pi$
$398$	6.24912e16	0.788096
$399$	0	0
$400$	−1.62193e16	−0.197990
$401$	1.12464e17	1.35075	0.675376	−	0.737473i	$-0.263981\pi$
$401$	1.12464e17	1.35075	0.675376	+	0.737473i	$0.263981\pi$
$402$	0	0
$403$	4.82485e16	0.561050
$404$	−3.03105e16	−0.346828
$405$	0	0
$406$	3.52581e16	0.390697
$407$	−9.71667e15	−0.105963
$408$	0	0
$409$	4.47607e16	0.472820	0.236410	−	0.971653i	$-0.424029\pi$
$409$	4.47607e16	0.472820	0.236410	+	0.971653i	$0.424029\pi$
$410$	2.21083e16	0.229858
$411$	0	0
$412$	−3.74296e16	−0.377037
$413$	−2.96929e16	−0.294427
$414$	0	0
$415$	−5.21045e16	−0.500682
$416$	−2.39620e16	−0.226681
$417$	0	0
$418$	7.75036e15	0.0710683
$419$	1.39060e17	1.25549	0.627743	−	0.778421i	$-0.283979\pi$
$419$	1.39060e17	1.25549	0.627743	+	0.778421i	$0.283979\pi$
$420$	0	0
$421$	−4.69201e16	−0.410700	−0.205350	−	0.978689i	$-0.565833\pi$
$421$	−4.69201e16	−0.410700	−0.205350	+	0.978689i	$0.565833\pi$
$422$	7.23858e16	0.623911
$423$	0	0
$424$	−1.46018e14	−0.00122048
$425$	1.18850e17	0.978297
$426$	0	0
$427$	−1.87205e16	−0.149464
$428$	−2.70902e16	−0.213023
$429$	0	0
$430$	−6.23172e16	−0.475404
$431$	3.57649e16	0.268754	0.134377	−	0.990930i	$-0.457097\pi$
$431$	3.57649e16	0.268754	0.134377	+	0.990930i	$0.457097\pi$
$432$	0	0
$433$	−1.90804e17	−1.39129	−0.695644	−	0.718386i	$-0.744881\pi$
$433$	−1.90804e17	−1.39129	−0.695644	+	0.718386i	$0.744881\pi$
$434$	−1.35951e16	−0.0976563
$435$	0	0
$436$	5.98984e16	0.417592
$437$	7.94356e16	0.545615
$438$	0	0
$439$	1.80923e16	0.120635	0.0603176	−	0.998179i	$-0.480789\pi$
$439$	1.80923e16	0.120635	0.0603176	+	0.998179i	$0.480789\pi$
$440$	6.81131e15	0.0447495
$441$	0	0
$442$	1.75585e17	1.12007
$443$	−2.24850e16	−0.141341	−0.0706705	−	0.997500i	$-0.522514\pi$
$443$	−2.24850e16	−0.141341	−0.0706705	+	0.997500i	$0.522514\pi$
$444$	0	0
$445$	−9.43767e16	−0.576135
$446$	2.23728e17	1.34599
$447$	0	0
$448$	6.75183e15	0.0394561
$449$	−1.74031e17	−1.00236	−0.501181	−	0.865343i	$-0.667101\pi$
$449$	−1.74031e17	−1.00236	−0.501181	+	0.865343i	$0.667101\pi$
$450$	0	0
$451$	3.53433e16	0.197770
$452$	−1.21902e17	−0.672374
$453$	0	0
$454$	1.38593e17	0.742809
$455$	−3.49416e16	−0.184616
$456$	0	0
$457$	2.39682e17	1.23078	0.615389	−	0.788224i	$-0.288999\pi$
$457$	2.39682e17	1.23078	0.615389	+	0.788224i	$0.288999\pi$
$458$	−1.70042e17	−0.860858
$459$	0	0
$460$	6.98110e16	0.343556
$461$	−4.80424e16	−0.233114	−0.116557	−	0.993184i	$-0.537186\pi$
$461$	−4.80424e16	−0.233114	−0.116557	+	0.993184i	$0.537186\pi$
$462$	0	0
$463$	−2.34366e17	−1.10565	−0.552825	−	0.833298i	$-0.686450\pi$
$463$	−2.34366e17	−1.10565	−0.552825	+	0.833298i	$0.686450\pi$
$464$	−9.40714e16	−0.437613
$465$	0	0
$466$	−8.23023e16	−0.372309
$467$	−3.01241e17	−1.34386	−0.671931	−	0.740614i	$-0.734535\pi$
$467$	−3.01241e17	−1.34386	−0.671931	+	0.740614i	$0.734535\pi$
$468$	0	0
$469$	−9.02290e16	−0.391492
$470$	1.39188e17	0.595616
$471$	0	0
$472$	7.92230e16	0.329783
$473$	−9.96230e16	−0.409037
$474$	0	0
$475$	7.18032e16	0.286837
$476$	−4.94752e16	−0.194959
$477$	0	0
$478$	2.49659e17	0.957338
$479$	−1.06328e17	−0.402224	−0.201112	−	0.979568i	$-0.564456\pi$
$479$	−1.06328e17	−0.402224	−0.201112	+	0.979568i	$0.564456\pi$
$480$	0	0
$481$	−1.32993e17	−0.489650
$482$	−2.51566e17	−0.913790
$483$	0	0
$484$	−1.30516e17	−0.461498
$485$	−8.51004e16	−0.296900
$486$	0	0
$487$	−5.55030e17	−1.88529	−0.942646	−	0.333795i	$-0.891671\pi$
$487$	−5.55030e17	−1.88529	−0.942646	+	0.333795i	$0.891671\pi$
$488$	4.99477e16	0.167412
$489$	0	0
$490$	−8.89719e16	−0.290388
$491$	4.75553e17	1.53168	0.765841	−	0.643029i	$-0.222323\pi$
$491$	4.75553e17	1.53168	0.765841	+	0.643029i	$0.222323\pi$
$492$	0	0
$493$	6.89324e17	2.16231
$494$	1.06080e17	0.328404
$495$	0	0
$496$	3.62729e16	0.109383
$497$	−1.06760e17	−0.317755
$498$	0	0
$499$	−1.85876e17	−0.538977	−0.269489	−	0.963004i	$-0.586855\pi$
$499$	−1.85876e17	−0.538977	−0.269489	+	0.963004i	$0.586855\pi$
$500$	1.42783e17	0.408670
$501$	0	0
$502$	−4.16171e17	−1.16064
$503$	1.23322e17	0.339508	0.169754	−	0.985486i	$-0.445703\pi$
$503$	1.23322e17	0.339508	0.169754	+	0.985486i	$0.445703\pi$
$504$	0	0
$505$	1.17927e17	0.316387
$506$	1.11603e17	0.295595
$507$	0	0
$508$	−2.37405e17	−0.612881
$509$	−7.00587e16	−0.178565	−0.0892826	−	0.996006i	$-0.528457\pi$
$509$	−7.00587e16	−0.178565	−0.0892826	+	0.996006i	$0.528457\pi$
$510$	0	0
$511$	−7.59251e15	−0.0188647
$512$	−1.80144e16	−0.0441942
$513$	0	0
$514$	1.72673e17	0.413014
$515$	1.45625e17	0.343944
$516$	0	0
$517$	2.22513e17	0.512467
$518$	3.74738e16	0.0852284
$519$	0	0
$520$	9.32270e16	0.206785
$521$	8.63330e17	1.89118	0.945588	−	0.325368i	$-0.105488\pi$
$521$	8.63330e17	1.89118	0.945588	+	0.325368i	$0.105488\pi$
$522$	0	0
$523$	4.05623e17	0.866685	0.433343	−	0.901229i	$-0.357334\pi$
$523$	4.05623e17	0.866685	0.433343	+	0.901229i	$0.357334\pi$
$524$	1.30146e17	0.274649
$525$	0	0
$526$	−3.25733e17	−0.670586
$527$	−2.65796e17	−0.540479
$528$	0	0
$529$	6.39813e17	1.26938
$530$	5.68102e14	0.00111335
$531$	0	0
$532$	−2.98904e16	−0.0571619
$533$	4.83747e17	0.913885
$534$	0	0
$535$	1.05398e17	0.194326
$536$	2.40738e17	0.438504
$537$	0	0
$538$	−4.51986e17	−0.803601
$539$	−1.42234e17	−0.249849
$540$	0	0
$541$	9.11068e16	0.156232	0.0781158	−	0.996944i	$-0.475110\pi$
$541$	9.11068e16	0.156232	0.0781158	+	0.996944i	$0.475110\pi$
$542$	−3.36609e17	−0.570335
$543$	0	0
$544$	1.32004e17	0.218370
$545$	−2.33042e17	−0.380940
$546$	0	0
$547$	−2.00896e17	−0.320667	−0.160334	−	0.987063i	$-0.551257\pi$
$547$	−2.00896e17	−0.320667	−0.160334	+	0.987063i	$0.551257\pi$
$548$	2.82286e17	0.445263
$549$	0	0
$550$	1.00880e17	0.155398
$551$	4.16455e17	0.633991
$552$	0	0
$553$	2.57849e17	0.383400
$554$	−8.58139e17	−1.26109
$555$	0	0
$556$	−4.84362e17	−0.695318
$557$	−6.99866e17	−0.993016	−0.496508	−	0.868032i	$-0.665385\pi$
$557$	−6.99866e17	−0.993016	−0.496508	+	0.868032i	$0.665385\pi$
$558$	0	0
$559$	−1.36355e18	−1.89014
$560$	−2.62688e16	−0.0359930
$561$	0	0
$562$	1.16499e17	0.155968
$563$	−1.36211e18	−1.80263	−0.901317	−	0.433161i	$-0.857398\pi$
$563$	−1.36211e18	−1.80263	−0.901317	+	0.433161i	$0.857398\pi$
$564$	0	0
$565$	4.74275e17	0.613359
$566$	2.92330e17	0.373737
$567$	0	0
$568$	2.84844e17	0.355912
$569$	−1.39312e18	−1.72091	−0.860455	−	0.509526i	$-0.829821\pi$
$569$	−1.39312e18	−1.72091	−0.860455	+	0.509526i	$0.829821\pi$
$570$	0	0
$571$	−7.08264e17	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$571$	−7.08264e17	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$572$	1.49037e17	0.177918
$573$	0	0
$574$	−1.36307e17	−0.159071
$575$	1.03394e18	1.19304
$576$	0	0
$577$	2.33945e17	0.263920	0.131960	−	0.991255i	$-0.457873\pi$
$577$	2.33945e17	0.263920	0.131960	+	0.991255i	$0.457873\pi$
$578$	−3.33387e17	−0.371893
$579$	0	0
$580$	3.65996e17	0.399204
$581$	3.21246e17	0.346491
$582$	0	0
$583$	9.08194e14	0.000957928 0
$584$	2.02574e16	0.0211300
$585$	0	0
$586$	8.62521e17	0.879904
$587$	−1.11658e18	−1.12653	−0.563264	−	0.826277i	$-0.690455\pi$
$587$	−1.11658e18	−1.12653	−0.563264	+	0.826277i	$0.690455\pi$
$588$	0	0
$589$	−1.60581e17	−0.158469
$590$	−3.08227e17	−0.300838
$591$	0	0
$592$	−9.99830e16	−0.0954630
$593$	1.28180e18	1.21050	0.605251	−	0.796034i	$-0.293073\pi$
$593$	1.28180e18	1.21050	0.605251	+	0.796034i	$0.293073\pi$
$594$	0	0
$595$	1.92489e17	0.177847
$596$	6.37559e15	0.00582666
$597$	0	0
$598$	1.52752e18	1.36593
$599$	−6.88211e17	−0.608761	−0.304381	−	0.952550i	$-0.598449\pi$
$599$	−6.88211e17	−0.608761	−0.304381	+	0.952550i	$0.598449\pi$
$600$	0	0
$601$	−9.33658e17	−0.808172	−0.404086	−	0.914721i	$-0.632410\pi$
$601$	−9.33658e17	−0.808172	−0.404086	+	0.914721i	$0.632410\pi$
$602$	3.84211e17	0.328998
$603$	0	0
$604$	1.45582e17	0.122002
$605$	5.07789e17	0.420992
$606$	0	0
$607$	−4.16897e17	−0.338300	−0.169150	−	0.985590i	$-0.554102\pi$
$607$	−4.16897e17	−0.338300	−0.169150	+	0.985590i	$0.554102\pi$
$608$	7.97500e16	0.0640262
$609$	0	0
$610$	−1.94328e17	−0.152718
$611$	3.04555e18	2.36809
$612$	0	0
$613$	−4.83422e17	−0.367988	−0.183994	−	0.982927i	$-0.558903\pi$
$613$	−4.83422e17	−0.367988	−0.183994	+	0.982927i	$0.558903\pi$
$614$	2.04898e17	0.154327
$615$	0	0
$616$	−4.19945e16	−0.0309683
$617$	−1.45200e18	−1.05953	−0.529764	−	0.848145i	$-0.677719\pi$
$617$	−1.45200e18	−1.05953	−0.529764	+	0.848145i	$0.677719\pi$
$618$	0	0
$619$	1.42177e18	1.01588	0.507938	−	0.861394i	$-0.330408\pi$
$619$	1.42177e18	1.01588	0.507938	+	0.861394i	$0.330408\pi$
$620$	−1.41124e17	−0.0997827
$621$	0	0
$622$	−1.45380e17	−0.100662
$623$	5.81871e17	0.398708
$624$	0	0
$625$	6.24595e17	0.419158
$626$	1.30280e18	0.865254
$627$	0	0
$628$	6.30230e17	0.409979
$629$	7.32643e17	0.471697
$630$	0	0
$631$	−2.75897e18	−1.74003	−0.870013	−	0.493029i	$-0.835890\pi$
$631$	−2.75897e18	−1.74003	−0.870013	+	0.493029i	$0.835890\pi$
$632$	−6.87961e17	−0.429440
$633$	0	0
$634$	−2.66676e17	−0.163081
$635$	9.23655e17	0.559089
$636$	0	0
$637$	−1.94678e18	−1.15454
$638$	5.85098e17	0.343474
$639$	0	0
$640$	7.00873e16	0.0403152
$641$	2.14378e18	1.22068	0.610341	−	0.792139i	$-0.291033\pi$
$641$	2.14378e18	1.22068	0.610341	+	0.792139i	$0.291033\pi$
$642$	0	0
$643$	2.18189e18	1.21748	0.608739	−	0.793371i	$-0.291676\pi$
$643$	2.18189e18	1.21748	0.608739	+	0.793371i	$0.291676\pi$
$644$	−4.30413e17	−0.237754
$645$	0	0
$646$	−5.84382e17	−0.316363
$647$	−2.20644e18	−1.18254	−0.591269	−	0.806475i	$-0.701373\pi$
$647$	−2.20644e18	−1.18254	−0.591269	+	0.806475i	$0.701373\pi$
$648$	0	0
$649$	−4.92746e17	−0.258841
$650$	1.38075e18	0.718089
$651$	0	0
$652$	−8.17727e17	−0.416868
$653$	−1.68563e18	−0.850797	−0.425398	−	0.905006i	$-0.639866\pi$
$653$	−1.68563e18	−0.850797	−0.425398	+	0.905006i	$0.639866\pi$
$654$	0	0
$655$	−5.06350e17	−0.250543
$656$	3.63677e17	0.178173
$657$	0	0
$658$	−8.58153e17	−0.412189
$659$	−1.13582e18	−0.540198	−0.270099	−	0.962833i	$-0.587056\pi$
$659$	−1.13582e18	−0.540198	−0.270099	+	0.962833i	$0.587056\pi$
$660$	0	0
$661$	1.50291e18	0.700850	0.350425	−	0.936591i	$-0.386037\pi$
$661$	1.50291e18	0.700850	0.350425	+	0.936591i	$0.386037\pi$
$662$	−2.68219e18	−1.23855
$663$	0	0
$664$	−8.57108e17	−0.388100
$665$	1.16292e17	0.0521448
$666$	0	0
$667$	5.99683e18	2.63696
$668$	−1.90439e18	−0.829297
$669$	0	0
$670$	−9.36622e17	−0.400017
$671$	−3.10661e17	−0.131399
$672$	0	0
$673$	−2.18721e17	−0.0907387	−0.0453693	−	0.998970i	$-0.514446\pi$
$673$	−2.18721e17	−0.0907387	−0.0453693	+	0.998970i	$0.514446\pi$
$674$	6.83134e17	0.280683
$675$	0	0
$676$	7.99303e17	0.322150
$677$	−3.22640e18	−1.28793	−0.643965	−	0.765055i	$-0.722712\pi$
$677$	−3.22640e18	−1.28793	−0.643965	+	0.765055i	$0.722712\pi$
$678$	0	0
$679$	5.24679e17	0.205466
$680$	−5.13577e17	−0.199204
$681$	0	0
$682$	−2.25607e17	−0.0858528
$683$	−2.67590e18	−1.00864	−0.504319	−	0.863517i	$-0.668257\pi$
$683$	−2.67590e18	−1.00864	−0.504319	+	0.863517i	$0.668257\pi$
$684$	0	0
$685$	−1.09827e18	−0.406182
$686$	1.15780e18	0.424157
$687$	0	0
$688$	−1.02511e18	−0.368505
$689$	1.24305e16	0.00442654
$690$	0	0
$691$	3.91753e18	1.36900	0.684502	−	0.729011i	$-0.260020\pi$
$691$	3.91753e18	1.36900	0.684502	+	0.729011i	$0.260020\pi$
$692$	−1.54309e18	−0.534198
$693$	0	0
$694$	−1.30791e18	−0.444366
$695$	1.88447e18	0.634290
$696$	0	0
$697$	−2.66491e18	−0.880378
$698$	1.04975e18	0.343578
$699$	0	0
$700$	−3.89058e17	−0.124990
$701$	3.13316e18	0.997277	0.498638	−	0.866810i	$-0.333834\pi$
$701$	3.13316e18	0.997277	0.498638	+	0.866810i	$0.333834\pi$
$702$	0	0
$703$	4.42626e17	0.138302
$704$	1.12045e17	0.0346872
$705$	0	0
$706$	−9.96957e17	−0.303002
$707$	−7.27067e17	−0.218952
$708$	0	0
$709$	−1.61211e18	−0.476643	−0.238322	−	0.971186i	$-0.576597\pi$
$709$	−1.61211e18	−0.476643	−0.238322	+	0.971186i	$0.576597\pi$
$710$	−1.10822e18	−0.324674
$711$	0	0
$712$	−1.55248e18	−0.446586
$713$	−2.31231e18	−0.659120
$714$	0	0
$715$	−5.79846e17	−0.162302
$716$	3.00054e18	0.832271
$717$	0	0
$718$	1.14777e18	0.312640
$719$	−2.04122e18	−0.551000	−0.275500	−	0.961301i	$-0.588843\pi$
$719$	−2.04122e18	−0.551000	−0.275500	+	0.961301i	$0.588843\pi$
$720$	0	0
$721$	−8.97836e17	−0.238022
$722$	2.33834e18	0.614349
$723$	0	0
$724$	3.02079e18	0.779506
$725$	5.42064e18	1.38629
$726$	0	0
$727$	−1.72741e18	−0.433932	−0.216966	−	0.976179i	$-0.569616\pi$
$727$	−1.72741e18	−0.433932	−0.216966	+	0.976179i	$0.569616\pi$
$728$	−5.74783e17	−0.143103
$729$	0	0
$730$	−7.88140e16	−0.0192755
$731$	7.51164e18	1.82084
$732$	0	0
$733$	−3.12183e18	−0.743419	−0.371709	−	0.928349i	$-0.621228\pi$
$733$	−3.12183e18	−0.743419	−0.371709	+	0.928349i	$0.621228\pi$
$734$	−3.10181e18	−0.732133
$735$	0	0
$736$	1.14838e18	0.266305
$737$	−1.49733e18	−0.344174
$738$	0	0
$739$	4.01232e18	0.906164	0.453082	−	0.891469i	$-0.350325\pi$
$739$	4.01232e18	0.906164	0.453082	+	0.891469i	$0.350325\pi$
$740$	3.88996e17	0.0870842
$741$	0	0
$742$	−3.50259e15	−0.000770484 0
$743$	4.94213e18	1.07767	0.538837	−	0.842410i	$-0.318864\pi$
$743$	4.94213e18	1.07767	0.538837	+	0.842410i	$0.318864\pi$
$744$	0	0
$745$	−2.48050e16	−0.00531525
$746$	2.33562e18	0.496134
$747$	0	0
$748$	−8.21027e17	−0.171394
$749$	−6.49820e17	−0.134481
$750$	0	0
$751$	−1.93899e18	−0.394381	−0.197191	−	0.980365i	$-0.563182\pi$
$751$	−1.93899e18	−0.394381	−0.197191	+	0.980365i	$0.563182\pi$
$752$	2.28962e18	0.461687
$753$	0	0
$754$	8.00829e18	1.58718
$755$	−5.66404e17	−0.111294
$756$	0	0
$757$	−4.50167e17	−0.0869461	−0.0434730	−	0.999055i	$-0.513842\pi$
$757$	−4.50167e17	−0.0869461	−0.0434730	+	0.999055i	$0.513842\pi$
$758$	−2.08247e18	−0.398776
$759$	0	0
$760$	−3.10277e17	−0.0584066
$761$	5.48520e18	1.02375	0.511873	−	0.859061i	$-0.328952\pi$
$761$	5.48520e18	1.02375	0.511873	+	0.859061i	$0.328952\pi$
$762$	0	0
$763$	1.43680e18	0.263625
$764$	9.04220e17	0.164501
$765$	0	0
$766$	1.94237e18	0.347411
$767$	−6.74425e18	−1.19609
$768$	0	0
$769$	−6.62553e18	−1.15531	−0.577656	−	0.816280i	$-0.696033\pi$
$769$	−6.62553e18	−1.15531	−0.577656	+	0.816280i	$0.696033\pi$
$770$	1.63385e17	0.0282502
$771$	0	0
$772$	4.54864e18	0.773337
$773$	−7.66142e18	−1.29164	−0.645822	−	0.763488i	$-0.723485\pi$
$773$	−7.66142e18	−1.29164	−0.645822	+	0.763488i	$0.723485\pi$
$774$	0	0
$775$	−2.09014e18	−0.346508
$776$	−1.39988e18	−0.230139
$777$	0	0
$778$	−2.95821e17	−0.0478257
$779$	−1.61000e18	−0.258127
$780$	0	0
$781$	−1.77165e18	−0.279349
$782$	−8.41493e18	−1.31585
$783$	0	0
$784$	−1.46357e18	−0.225091
$785$	−2.45199e18	−0.373995
$786$	0	0
$787$	9.71301e18	1.45720	0.728598	−	0.684941i	$-0.240172\pi$
$787$	9.71301e18	1.45720	0.728598	+	0.684941i	$0.240172\pi$
$788$	5.61379e18	0.835287
$789$	0	0
$790$	2.67660e18	0.391748
$791$	−2.92410e18	−0.424468
$792$	0	0
$793$	−4.25205e18	−0.607187
$794$	−2.74459e17	−0.0388727
$795$	0	0
$796$	−3.99944e18	−0.557268
$797$	−7.05558e18	−0.975110	−0.487555	−	0.873092i	$-0.662111\pi$
$797$	−7.05558e18	−0.975110	−0.487555	+	0.873092i	$0.662111\pi$
$798$	0	0
$799$	−1.67776e19	−2.28126
$800$	1.03804e18	0.140000
$801$	0	0
$802$	−7.19771e18	−0.955126
$803$	−1.25996e17	−0.0165846
$804$	0	0
$805$	1.67458e18	0.216886
$806$	−3.08791e18	−0.396722
$807$	0	0
$808$	1.93987e18	0.245244
$809$	−7.32496e18	−0.918628	−0.459314	−	0.888274i	$-0.651905\pi$
$809$	−7.32496e18	−0.918628	−0.459314	+	0.888274i	$0.651905\pi$
$810$	0	0
$811$	1.06716e19	1.31703	0.658513	−	0.752570i	$-0.271186\pi$
$811$	1.06716e19	1.31703	0.658513	+	0.752570i	$0.271186\pi$
$812$	−2.25652e18	−0.276264
$813$	0	0
$814$	6.21867e17	0.0749271
$815$	3.18147e18	0.380280
$816$	0	0
$817$	4.53816e18	0.533871
$818$	−2.86469e18	−0.334334
$819$	0	0
$820$	−1.41493e18	−0.162534
$821$	1.68217e19	1.91707	0.958537	−	0.284969i	$-0.0919832\pi$
$821$	1.68217e19	1.91707	0.958537	+	0.284969i	$0.0919832\pi$
$822$	0	0
$823$	−1.47808e19	−1.65806	−0.829028	−	0.559207i	$-0.811106\pi$
$823$	−1.47808e19	−1.65806	−0.829028	+	0.559207i	$0.811106\pi$
$824$	2.39550e18	0.266605
$825$	0	0
$826$	1.90035e18	0.208191
$827$	−7.19689e18	−0.782274	−0.391137	−	0.920332i	$-0.627918\pi$
$827$	−7.19689e18	−0.782274	−0.391137	+	0.920332i	$0.627918\pi$
$828$	0	0
$829$	1.66095e19	1.77727	0.888634	−	0.458617i	$-0.151655\pi$
$829$	1.66095e19	1.77727	0.888634	+	0.458617i	$0.151655\pi$
$830$	3.33469e18	0.354036
$831$	0	0
$832$	1.53356e18	0.160288
$833$	1.07246e19	1.11221
$834$	0	0
$835$	7.40928e18	0.756509
$836$	−4.96023e17	−0.0502529
$837$	0	0
$838$	−8.89986e18	−0.887763
$839$	−2.37199e18	−0.234779	−0.117390	−	0.993086i	$-0.537453\pi$
$839$	−2.37199e18	−0.234779	−0.117390	+	0.993086i	$0.537453\pi$
$840$	0	0
$841$	2.11788e19	2.06409
$842$	3.00288e18	0.290409
$843$	0	0
$844$	−4.63269e18	−0.441171
$845$	−3.10979e18	−0.293875
$846$	0	0
$847$	−3.13073e18	−0.291342
$848$	9.34517e15	0.000863007 0
$849$	0	0
$850$	−7.60640e18	−0.691761
$851$	6.37369e18	0.575240
$852$	0	0
$853$	−9.23784e18	−0.821111	−0.410556	−	0.911836i	$-0.634665\pi$
$853$	−9.23784e18	−0.821111	−0.410556	+	0.911836i	$0.634665\pi$
$854$	1.19811e18	0.105687
$855$	0	0
$856$	1.73377e18	0.150630
$857$	4.40680e18	0.379969	0.189984	−	0.981787i	$-0.439156\pi$
$857$	4.40680e18	0.379969	0.189984	+	0.981787i	$0.439156\pi$
$858$	0	0
$859$	9.82119e18	0.834082	0.417041	−	0.908888i	$-0.363067\pi$
$859$	9.82119e18	0.834082	0.417041	+	0.908888i	$0.363067\pi$
$860$	3.98830e18	0.336161
$861$	0	0
$862$	−2.28895e18	−0.190038
$863$	1.47168e19	1.21267	0.606334	−	0.795210i	$-0.292639\pi$
$863$	1.47168e19	1.21267	0.606334	+	0.795210i	$0.292639\pi$
$864$	0	0
$865$	6.00359e18	0.487311
$866$	1.22115e19	0.983790
$867$	0	0
$868$	8.70088e17	0.0690534
$869$	4.27893e18	0.337060
$870$	0	0
$871$	−2.04940e19	−1.59041
$872$	−3.83349e18	−0.295282
$873$	0	0
$874$	−5.08388e18	−0.385808
$875$	3.42499e18	0.257992
$876$	0	0
$877$	1.14468e19	0.849548	0.424774	−	0.905300i	$-0.360354\pi$
$877$	1.14468e19	0.849548	0.424774	+	0.905300i	$0.360354\pi$
$878$	−1.15791e18	−0.0853019
$879$	0	0
$880$	−4.35924e17	−0.0316427
$881$	−6.44660e18	−0.464502	−0.232251	−	0.972656i	$-0.574609\pi$
$881$	−6.44660e18	−0.464502	−0.232251	+	0.972656i	$0.574609\pi$
$882$	0	0
$883$	1.62047e18	0.115053	0.0575263	−	0.998344i	$-0.481679\pi$
$883$	1.62047e18	0.115053	0.0575263	+	0.998344i	$0.481679\pi$
$884$	−1.12375e19	−0.792006
$885$	0	0
$886$	1.43904e18	0.0999432
$887$	5.24274e18	0.361455	0.180728	−	0.983533i	$-0.442155\pi$
$887$	5.24274e18	0.361455	0.180728	+	0.983533i	$0.442155\pi$
$888$	0	0
$889$	−5.69471e18	−0.386910
$890$	6.04011e18	0.407389
$891$	0	0
$892$	−1.43186e19	−0.951762
$893$	−1.01362e19	−0.668867
$894$	0	0
$895$	−1.16740e19	−0.759223
$896$	−4.32117e17	−0.0278997
$897$	0	0
$898$	1.11380e19	0.708777
$899$	−1.21227e19	−0.765881
$900$	0	0
$901$	−6.84784e16	−0.00426425
$902$	−2.26197e18	−0.139844
$903$	0	0
$904$	7.80172e18	0.475440
$905$	−1.17527e19	−0.711088
$906$	0	0
$907$	−2.17095e18	−0.129480	−0.0647399	−	0.997902i	$-0.520622\pi$
$907$	−2.17095e18	−0.129480	−0.0647399	+	0.997902i	$0.520622\pi$
$908$	−8.86993e18	−0.525245
$909$	0	0
$910$	2.23626e18	0.130543
$911$	−8.79267e18	−0.509626	−0.254813	−	0.966990i	$-0.582014\pi$
$911$	−8.79267e18	−0.509626	−0.254813	+	0.966990i	$0.582014\pi$
$912$	0	0
$913$	5.33098e18	0.304612
$914$	−1.53396e19	−0.870291
$915$	0	0
$916$	1.08827e19	0.608719
$917$	3.12185e18	0.173385
$918$	0	0
$919$	−2.51608e19	−1.37776	−0.688880	−	0.724876i	$-0.741897\pi$
$919$	−2.51608e19	−1.37776	−0.688880	+	0.724876i	$0.741897\pi$
$920$	−4.46790e18	−0.242931
$921$	0	0
$922$	3.07471e18	0.164837
$923$	−2.42487e19	−1.29086
$924$	0	0
$925$	5.76128e18	0.302411
$926$	1.49994e19	0.781812
$927$	0	0
$928$	6.02057e18	0.309439
$929$	−1.29149e18	−0.0659157	−0.0329579	−	0.999457i	$-0.510493\pi$
$929$	−1.29149e18	−0.0659157	−0.0329579	+	0.999457i	$0.510493\pi$
$930$	0	0
$931$	6.47924e18	0.326101
$932$	5.26735e18	0.263262
$933$	0	0
$934$	1.92794e19	0.950254
$935$	3.19431e18	0.156351
$936$	0	0
$937$	2.58468e19	1.24767	0.623835	−	0.781556i	$-0.285574\pi$
$937$	2.58468e19	1.24767	0.623835	+	0.781556i	$0.285574\pi$
$938$	5.77466e18	0.276827
$939$	0	0
$940$	−8.90805e18	−0.421164
$941$	−1.64109e19	−0.770550	−0.385275	−	0.922802i	$-0.625893\pi$
$941$	−1.64109e19	−0.770550	−0.385275	+	0.922802i	$0.625893\pi$
$942$	0	0
$943$	−2.31836e19	−1.07363
$944$	−5.07028e18	−0.233192
$945$	0	0
$946$	6.37587e18	0.289233
$947$	6.99992e18	0.315369	0.157684	−	0.987490i	$-0.449597\pi$
$947$	6.99992e18	0.315369	0.157684	+	0.987490i	$0.449597\pi$
$948$	0	0
$949$	−1.72451e18	−0.0766366
$950$	−4.59540e18	−0.202824
$951$	0	0
$952$	3.16641e18	0.137856
$953$	2.33363e19	1.00909	0.504543	−	0.863387i	$-0.331661\pi$
$953$	2.33363e19	1.00909	0.504543	+	0.863387i	$0.331661\pi$
$954$	0	0
$955$	−3.51798e18	−0.150062
$956$	−1.59782e19	−0.676940
$957$	0	0
$958$	6.80501e18	0.284415
$959$	6.77129e18	0.281093
$960$	0	0
$961$	−1.97432e19	−0.808565
$962$	8.51155e18	0.346235
$963$	0	0
$964$	1.61002e19	0.646147
$965$	−1.76970e19	−0.705461
$966$	0	0
$967$	−2.71778e19	−1.06891	−0.534456	−	0.845197i	$-0.679483\pi$
$967$	−2.71778e19	−1.06891	−0.534456	+	0.845197i	$0.679483\pi$
$968$	8.35303e18	0.326328
$969$	0	0
$970$	5.44642e18	0.209940
$971$	3.52698e19	1.35045	0.675225	−	0.737612i	$-0.264047\pi$
$971$	3.52698e19	1.35045	0.675225	+	0.737612i	$0.264047\pi$
$972$	0	0
$973$	−1.16185e19	−0.438953
$974$	3.55219e19	1.33310
$975$	0	0
$976$	−3.19665e18	−0.118378
$977$	1.34520e18	0.0494849	0.0247424	−	0.999694i	$-0.492123\pi$
$977$	1.34520e18	0.0494849	0.0247424	+	0.999694i	$0.492123\pi$
$978$	0	0
$979$	9.65598e18	0.350517
$980$	5.69420e18	0.205335
$981$	0	0
$982$	−3.04354e19	−1.08306
$983$	−1.07665e19	−0.380607	−0.190304	−	0.981725i	$-0.560947\pi$
$983$	−1.07665e19	−0.380607	−0.190304	+	0.981725i	$0.560947\pi$
$984$	0	0
$985$	−2.18411e19	−0.761973
$986$	−4.41168e19	−1.52899
$987$	0	0
$988$	−6.78911e18	−0.232216
$989$	6.53481e19	2.22053
$990$	0	0
$991$	6.52510e18	0.218831	0.109415	−	0.993996i	$-0.465102\pi$
$991$	6.52510e18	0.218831	0.109415	+	0.993996i	$0.465102\pi$
$992$	−2.32146e18	−0.0773457
$993$	0	0
$994$	6.83264e18	0.224686
$995$	1.55603e19	0.508356
$996$	0	0
$997$	2.97797e19	0.960290	0.480145	−	0.877189i	$-0.340584\pi$
$997$	2.97797e19	0.960290	0.480145	+	0.877189i	$0.340584\pi$
$998$	1.18961e19	0.381115
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.14.a.b.1.1	✓	1
3.2	odd	2		18.14.a.e.1.1	yes	1
4.3	odd	2		144.14.a.c.1.1		1
12.11	even	2		144.14.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.14.a.b.1.1	✓	1	1.1	even	1	trivial
18.14.a.e.1.1	yes	1	3.2	odd	2
144.14.a.c.1.1		1	4.3	odd	2
144.14.a.h.1.1		1	12.11	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:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more