LMFDB - Embedded newform 144.14.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,14,Mod(1,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 14, names="a")

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

chi := DirichletCharacter("144.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	144.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:	$154.412537691$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	144.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+15936.0 q^{5} -98252.0 q^{7} +O(q^{10})$

$q+15936.0 q^{5} -98252.0 q^{7} +1.63046e6 q^{11} +2.23163e7 q^{13} +1.22938e8 q^{17} +7.42730e7 q^{19} -1.06951e9 q^{23} -9.66747e8 q^{25} +5.60709e9 q^{29} -2.16203e9 q^{31} -1.56574e9 q^{35} -5.95945e9 q^{37} -2.16769e10 q^{41} +6.11010e10 q^{43} +1.36472e11 q^{47} -8.72356e10 q^{49} -5.57016e8 q^{53} +2.59831e10 q^{55} -3.02212e11 q^{59} -1.90535e11 q^{61} +3.55633e11 q^{65} +9.18343e11 q^{67} -1.08659e12 q^{71} -7.72759e10 q^{73} -1.60196e11 q^{77} -2.62436e12 q^{79} +3.26961e12 q^{83} +1.95914e12 q^{85} -5.92223e12 q^{89} -2.19262e12 q^{91} +1.18361e12 q^{95} +5.34013e12 q^{97} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	15936.0	0.456115	0.228057	−	0.973648i	$-0.426763\pi$
$5$	15936.0	0.456115	0.228057	+	0.973648i	$0.426763\pi$
$6$	0	0
$7$	−98252.0	−0.315649	−0.157824	−	0.987467i	$-0.550448\pi$
$7$	−98252.0	−0.315649	−0.157824	+	0.987467i	$0.550448\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	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	$0.278457\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.22938e8	1.23529	0.617644	−	0.786458i	$-0.288087\pi$
$17$	1.22938e8	1.23529	0.617644	+	0.786458i	$0.288087\pi$
$18$	0	0
$19$	7.42730e7	0.362187	0.181093	−	0.983466i	$-0.442036\pi$
$19$	7.42730e7	0.362187	0.181093	+	0.983466i	$0.442036\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−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
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.60709e9	1.75045	0.875227	−	0.483713i	$-0.160712\pi$
$29$	5.60709e9	1.75045	0.875227	+	0.483713i	$0.160712\pi$
$30$	0	0
$31$	−2.16203e9	−0.437533	−0.218767	−	0.975777i	$-0.570203\pi$
$31$	−2.16203e9	−0.437533	−0.218767	+	0.975777i	$0.570203\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−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	$0.561149\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.16769e10	−0.712691	−0.356345	−	0.934354i	$-0.615977\pi$
$41$	−2.16769e10	−0.712691	−0.356345	+	0.934354i	$0.615977\pi$
$42$	0	0
$43$	6.11010e10	1.47402	0.737010	−	0.675881i	$-0.236237\pi$
$43$	6.11010e10	1.47402	0.737010	+	0.675881i	$0.236237\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	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
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.57016e8	−0.00345203	−0.00172601	−	0.999999i	$-0.500549\pi$
$53$	−5.57016e8	−0.00345203	−0.00172601	+	0.999999i	$0.500549\pi$
$54$	0	0
$55$	2.59831e10	0.126571
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−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	$0.576084\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.55633e11	0.584877
$66$	0	0
$67$	9.18343e11	1.24028	0.620139	−	0.784492i	$-0.287076\pi$
$67$	9.18343e11	1.24028	0.620139	+	0.784492i	$0.287076\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−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	$0.509513\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.60196e11	−0.0875917
$78$	0	0
$79$	−2.62436e12	−1.21464	−0.607320	−	0.794457i	$-0.707756\pi$
$79$	−2.62436e12	−1.21464	−0.607320	+	0.794457i	$0.707756\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	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
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.92223e12	−1.26314	−0.631568	−	0.775320i	$-0.717588\pi$
$89$	−5.92223e12	−1.26314	−0.631568	+	0.775320i	$0.717588\pi$
$90$	0	0
$91$	−2.19262e12	−0.404757
$92$	0	0
$93$	0	0
$94$	0	0
$95$	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	$0.394479\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.40002e12	0.693656	0.346828	−	0.937929i	$-0.387259\pi$
$101$	7.40002e12	0.693656	0.346828	+	0.937929i	$0.387259\pi$
$102$	0	0
$103$	9.13810e12	0.754074	0.377037	−	0.926198i	$-0.376943\pi$
$103$	9.13810e12	0.754074	0.377037	+	0.926198i	$0.376943\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−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	$0.362874\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.97612e13	1.34475	0.672374	−	0.740212i	$-0.265275\pi$
$113$	2.97612e13	1.34475	0.672374	+	0.740212i	$0.265275\pi$
$114$	0	0
$115$	−1.70437e13	−0.687113
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.20789e13	−0.389917
$120$	0	0
$121$	−3.18643e13	−0.922995
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.48592e13	−0.817339
$126$	0	0
$127$	5.79603e13	1.22576	0.612881	−	0.790175i	$-0.290010\pi$
$127$	5.79603e13	1.22576	0.612881	+	0.790175i	$0.290010\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	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
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.89176e13	−0.890526	−0.445263	−	0.895400i	$-0.646890\pi$
$137$	−6.89176e13	−0.890526	−0.445263	+	0.895400i	$0.646890\pi$
$138$	0	0
$139$	1.18252e14	1.39064	0.695318	−	0.718702i	$-0.255263\pi$
$139$	1.18252e14	1.39064	0.695318	+	0.718702i	$0.255263\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.63859e13	0.355836
$144$	0	0
$145$	8.93546e13	0.798408
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.55654e12	−0.0116533	−0.00582666	−	0.999983i	$-0.501855\pi$
$149$	−1.55654e12	−0.0116533	−0.00582666	+	0.999983i	$0.501855\pi$
$150$	0	0
$151$	−3.55424e13	−0.244004	−0.122002	−	0.992530i	$-0.538931\pi$
$151$	−3.55424e13	−0.244004	−0.122002	+	0.992530i	$0.538931\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−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	$0.365536\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.05081e14	0.475508
$162$	0	0
$163$	1.99640e14	0.833736	0.416868	−	0.908967i	$-0.363128\pi$
$163$	1.99640e14	0.833736	0.416868	+	0.908967i	$0.363128\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−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
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.76731e14	1.06840	0.534198	−	0.845360i	$-0.320614\pi$
$173$	3.76731e14	1.06840	0.534198	+	0.845360i	$0.320614\pi$
$174$	0	0
$175$	9.49848e13	0.249981
$176$	0	0
$177$	0	0
$178$	0	0
$179$	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	$0.215470\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.49698e13	−0.174168
$186$	0	0
$187$	2.00446e14	0.342789
$188$	0	0
$189$	0	0
$190$	0	0
$191$	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	$0.218586\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.37055e15	−1.67057	−0.835287	−	0.549815i	$-0.814698\pi$
$197$	−1.37055e15	−1.67057	−0.835287	+	0.549815i	$0.814698\pi$
$198$	0	0
$199$	9.76425e14	1.11454	0.557268	−	0.830333i	$-0.311850\pi$
$199$	9.76425e14	1.11454	0.557268	+	0.830333i	$0.311850\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.50908e14	−0.552529
$204$	0	0
$205$	−3.45442e14	−0.325069
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.21099e14	0.100506
$210$	0	0
$211$	1.13103e15	0.882343	0.441171	−	0.897423i	$-0.354563\pi$
$211$	1.13103e15	0.882343	0.441171	+	0.897423i	$0.354563\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.73706e14	0.672323
$216$	0	0
$217$	2.12424e14	0.138107
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.74352e15	1.58401
$222$	0	0
$223$	3.49575e15	1.90352	0.951762	−	0.306837i	$-0.0992708\pi$
$223$	3.49575e15	1.90352	0.951762	+	0.306837i	$0.0992708\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−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	$0.291684\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.28597e15	−0.526525	−0.263262	−	0.964724i	$-0.584798\pi$
$233$	−1.28597e15	−0.526525	−0.263262	+	0.964724i	$0.584798\pi$
$234$	0	0
$235$	2.17482e15	0.842329
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−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	$0.276379\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.39019e15	−0.410670
$246$	0	0
$247$	1.65750e15	0.464433
$248$	0	0
$249$	0	0
$250$	0	0
$251$	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
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.69802e15	0.584090	0.292045	−	0.956405i	$-0.405664\pi$
$257$	2.69802e15	0.584090	0.292045	+	0.956405i	$0.405664\pi$
$258$	0	0
$259$	5.85528e14	0.120531
$260$	0	0
$261$	0	0
$262$	0	0
$263$	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
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.06228e15	−1.13646	−0.568231	−	0.822869i	$-0.692372\pi$
$269$	−7.06228e15	−1.13646	−0.568231	+	0.822869i	$0.692372\pi$
$270$	0	0
$271$	−5.25952e15	−0.806576	−0.403288	−	0.915073i	$-0.632133\pi$
$271$	−5.25952e15	−0.806576	−0.403288	+	0.915073i	$0.632133\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−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	$0.149498\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.82029e15	0.220572	0.110286	−	0.993900i	$-0.464823\pi$
$281$	1.82029e15	0.220572	0.110286	+	0.993900i	$0.464823\pi$
$282$	0	0
$283$	4.56765e15	0.528544	0.264272	−	0.964448i	$-0.414868\pi$
$283$	4.56765e15	0.528544	0.264272	+	0.964448i	$0.414868\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.12979e15	0.224960
$288$	0	0
$289$	5.20917e15	0.525936
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.34769e16	1.24437	0.622186	−	0.782869i	$-0.286245\pi$
$293$	1.34769e16	1.24437	0.622186	+	0.782869i	$0.286245\pi$
$294$	0	0
$295$	−4.81605e15	−0.425449
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.38675e16	−1.93172
$300$	0	0
$301$	−6.00330e15	−0.465273
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.03637e15	−0.215976
$306$	0	0
$307$	3.20152e15	0.218252	0.109126	−	0.994028i	$-0.465195\pi$
$307$	3.20152e15	0.218252	0.109126	+	0.994028i	$0.465195\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	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	$0.709565\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.16681e15	−0.230631	−0.115316	−	0.993329i	$-0.536788\pi$
$317$	−4.16681e15	−0.230631	−0.115316	+	0.993329i	$0.536788\pi$
$318$	0	0
$319$	9.14216e15	0.485746
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.13097e15	0.447405
$324$	0	0
$325$	−2.15742e16	−1.01553
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.34086e16	−0.582923
$330$	0	0
$331$	−4.19092e16	−1.75157	−0.875785	−	0.482702i	$-0.839656\pi$
$331$	−4.19092e16	−1.75157	−0.875785	+	0.482702i	$0.839656\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	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	$0.563598\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.52511e15	−0.121414
$342$	0	0
$343$	1.80906e16	0.599848
$344$	0	0
$345$	0	0
$346$	0	0
$347$	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	$0.578114\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.55775e16	−0.428510	−0.214255	−	0.976778i	$-0.568732\pi$
$353$	−1.55775e16	−0.428510	−0.214255	+	0.976778i	$0.568732\pi$
$354$	0	0
$355$	−1.73160e16	−0.459158
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−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
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.23147e15	−0.0272596
$366$	0	0
$367$	−4.84657e16	−1.03539	−0.517697	−	0.855564i	$-0.673210\pi$
$367$	−4.84657e16	−1.03539	−0.517697	+	0.855564i	$0.673210\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	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	$0.614097\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.25130e17	2.24461
$378$	0	0
$379$	−3.25386e16	−0.563955	−0.281977	−	0.959421i	$-0.590990\pi$
$379$	−3.25386e16	−0.563955	−0.281977	+	0.959421i	$0.590990\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−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
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.62220e15	−0.0676358	−0.0338179	−	0.999428i	$-0.510767\pi$
$389$	−4.62220e15	−0.0676358	−0.0338179	+	0.999428i	$0.510767\pi$
$390$	0	0
$391$	−1.31483e17	−1.86089
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−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	$0.491249\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.12464e17	−1.35075	−0.675376	−	0.737473i	$-0.736019\pi$
$401$	−1.12464e17	−1.35075	−0.675376	+	0.737473i	$0.736019\pi$
$402$	0	0
$403$	−4.82485e16	−0.561050
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−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	$0.424029\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.96929e16	0.294427
$414$	0	0
$415$	5.21045e16	0.500682
$416$	0	0
$417$	0	0
$418$	0	0
$419$	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	$0.565833\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.18850e17	−0.978297
$426$	0	0
$427$	1.87205e16	0.149464
$428$	0	0
$429$	0	0
$430$	0	0
$431$	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	$0.744881\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.94356e16	−0.545615
$438$	0	0
$439$	−1.80923e16	−0.120635	−0.0603176	−	0.998179i	$-0.519211\pi$
$439$	−1.80923e16	−0.120635	−0.0603176	+	0.998179i	$0.519211\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−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
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.74031e17	1.00236	0.501181	−	0.865343i	$-0.332899\pi$
$449$	1.74031e17	1.00236	0.501181	+	0.865343i	$0.332899\pi$
$450$	0	0
$451$	−3.53433e16	−0.197770
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−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	$0.288999\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.80424e16	0.233114	0.116557	−	0.993184i	$-0.462814\pi$
$461$	4.80424e16	0.233114	0.116557	+	0.993184i	$0.462814\pi$
$462$	0	0
$463$	2.34366e17	1.10565	0.552825	−	0.833298i	$-0.313550\pi$
$463$	2.34366e17	1.10565	0.552825	+	0.833298i	$0.313550\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−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
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.96230e16	0.409037
$474$	0	0
$475$	−7.18032e16	−0.286837
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−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
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.51004e16	0.296900
$486$	0	0
$487$	5.55030e17	1.88529	0.942646	−	0.333795i	$-0.108329\pi$
$487$	5.55030e17	1.88529	0.942646	+	0.333795i	$0.108329\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	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
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.06760e17	0.317755
$498$	0	0
$499$	1.85876e17	0.538977	0.269489	−	0.963004i	$-0.413145\pi$
$499$	1.85876e17	0.538977	0.269489	+	0.963004i	$0.413145\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	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
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.00587e16	0.178565	0.0892826	−	0.996006i	$-0.471543\pi$
$509$	7.00587e16	0.178565	0.0892826	+	0.996006i	$0.471543\pi$
$510$	0	0
$511$	7.59251e15	0.0188647
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.45625e17	0.343944
$516$	0	0
$517$	2.22513e17	0.512467
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8.63330e17	−1.89118	−0.945588	−	0.325368i	$-0.894512\pi$
$521$	−8.63330e17	−1.89118	−0.945588	+	0.325368i	$0.894512\pi$
$522$	0	0
$523$	−4.05623e17	−0.866685	−0.433343	−	0.901229i	$-0.642666\pi$
$523$	−4.05623e17	−0.866685	−0.433343	+	0.901229i	$0.642666\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.65796e17	−0.540479
$528$	0	0
$529$	6.39813e17	1.26938
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.83747e17	−0.913885
$534$	0	0
$535$	−1.05398e17	−0.194326
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−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	$0.475110\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.33042e17	0.380940
$546$	0	0
$547$	2.00896e17	0.320667	0.160334	−	0.987063i	$-0.448743\pi$
$547$	2.00896e17	0.320667	0.160334	+	0.987063i	$0.448743\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.16455e17	0.633991
$552$	0	0
$553$	2.57849e17	0.383400
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.99866e17	0.993016	0.496508	−	0.868032i	$-0.334615\pi$
$557$	6.99866e17	0.993016	0.496508	+	0.868032i	$0.334615\pi$
$558$	0	0
$559$	1.36355e18	1.89014
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−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
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.39312e18	1.72091	0.860455	−	0.509526i	$-0.170179\pi$
$569$	1.39312e18	1.72091	0.860455	+	0.509526i	$0.170179\pi$
$570$	0	0
$571$	7.08264e17	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$571$	7.08264e17	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	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	$0.457873\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.21246e17	−0.346491
$582$	0	0
$583$	−9.08194e14	−0.000957928 0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−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
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.28180e18	−1.21050	−0.605251	−	0.796034i	$-0.706927\pi$
$593$	−1.28180e18	−1.21050	−0.605251	+	0.796034i	$0.706927\pi$
$594$	0	0
$595$	−1.92489e17	−0.177847
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−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	$0.632410\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5.07789e17	−0.420992
$606$	0	0
$607$	4.16897e17	0.338300	0.169150	−	0.985590i	$-0.445898\pi$
$607$	4.16897e17	0.338300	0.169150	+	0.985590i	$0.445898\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	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	$0.558903\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.45200e18	1.05953	0.529764	−	0.848145i	$-0.322281\pi$
$617$	1.45200e18	1.05953	0.529764	+	0.848145i	$0.322281\pi$
$618$	0	0
$619$	−1.42177e18	−1.01588	−0.507938	−	0.861394i	$-0.669592\pi$
$619$	−1.42177e18	−1.01588	−0.507938	+	0.861394i	$0.669592\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.81871e17	0.398708
$624$	0	0
$625$	6.24595e17	0.419158
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.32643e17	−0.471697
$630$	0	0
$631$	2.75897e18	1.74003	0.870013	−	0.493029i	$-0.164110\pi$
$631$	2.75897e18	1.74003	0.870013	+	0.493029i	$0.164110\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.23655e17	0.559089
$636$	0	0
$637$	−1.94678e18	−1.15454
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.14378e18	−1.22068	−0.610341	−	0.792139i	$-0.708967\pi$
$641$	−2.14378e18	−1.22068	−0.610341	+	0.792139i	$0.708967\pi$
$642$	0	0
$643$	−2.18189e18	−1.21748	−0.608739	−	0.793371i	$-0.708324\pi$
$643$	−2.18189e18	−1.21748	−0.608739	+	0.793371i	$0.708324\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−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
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.68563e18	0.850797	0.425398	−	0.905006i	$-0.360134\pi$
$653$	1.68563e18	0.850797	0.425398	+	0.905006i	$0.360134\pi$
$654$	0	0
$655$	5.06350e17	0.250543
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−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	$0.386037\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.16292e17	−0.0521448
$666$	0	0
$667$	−5.99683e18	−2.63696
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−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	$0.514446\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.22640e18	1.28793	0.643965	−	0.765055i	$-0.277288\pi$
$677$	3.22640e18	1.28793	0.643965	+	0.765055i	$0.277288\pi$
$678$	0	0
$679$	−5.24679e17	−0.205466
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−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
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.24305e16	−0.00442654
$690$	0	0
$691$	−3.91753e18	−1.36900	−0.684502	−	0.729011i	$-0.739980\pi$
$691$	−3.91753e18	−1.36900	−0.684502	+	0.729011i	$0.739980\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.88447e18	0.634290
$696$	0	0
$697$	−2.66491e18	−0.880378
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.13316e18	−0.997277	−0.498638	−	0.866810i	$-0.666166\pi$
$701$	−3.13316e18	−0.997277	−0.498638	+	0.866810i	$0.666166\pi$
$702$	0	0
$703$	−4.42626e17	−0.138302
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−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	$0.576597\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.31231e18	0.659120
$714$	0	0
$715$	5.79846e17	0.162302
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−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
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.42064e18	−1.38629
$726$	0	0
$727$	1.72741e18	0.433932	0.216966	−	0.976179i	$-0.430384\pi$
$727$	1.72741e18	0.433932	0.216966	+	0.976179i	$0.430384\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	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	$0.621228\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.49733e18	0.344174
$738$	0	0
$739$	−4.01232e18	−0.906164	−0.453082	−	0.891469i	$-0.649675\pi$
$739$	−4.01232e18	−0.906164	−0.453082	+	0.891469i	$0.649675\pi$
$740$	0	0
$741$	0	0
$742$	0	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
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.49820e17	0.134481
$750$	0	0
$751$	1.93899e18	0.394381	0.197191	−	0.980365i	$-0.436818\pi$
$751$	1.93899e18	0.394381	0.197191	+	0.980365i	$0.436818\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−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	$0.513842\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.48520e18	−1.02375	−0.511873	−	0.859061i	$-0.671048\pi$
$761$	−5.48520e18	−1.02375	−0.511873	+	0.859061i	$0.671048\pi$
$762$	0	0
$763$	−1.43680e18	−0.263625
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−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	$0.696033\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.66142e18	1.29164	0.645822	−	0.763488i	$-0.276515\pi$
$773$	7.66142e18	1.29164	0.645822	+	0.763488i	$0.276515\pi$
$774$	0	0
$775$	2.09014e18	0.346508
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.61000e18	−0.258127
$780$	0	0
$781$	−1.77165e18	−0.279349
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.45199e18	0.373995
$786$	0	0
$787$	−9.71301e18	−1.45720	−0.728598	−	0.684941i	$-0.759828\pi$
$787$	−9.71301e18	−1.45720	−0.728598	+	0.684941i	$0.759828\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.92410e18	−0.424468
$792$	0	0
$793$	−4.25205e18	−0.607187
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.05558e18	0.975110	0.487555	−	0.873092i	$-0.337889\pi$
$797$	7.05558e18	0.975110	0.487555	+	0.873092i	$0.337889\pi$
$798$	0	0
$799$	1.67776e19	2.28126
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.25996e17	−0.0165846
$804$	0	0
$805$	1.67458e18	0.216886
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.32496e18	0.918628	0.459314	−	0.888274i	$-0.348095\pi$
$809$	7.32496e18	0.918628	0.459314	+	0.888274i	$0.348095\pi$
$810$	0	0
$811$	−1.06716e19	−1.31703	−0.658513	−	0.752570i	$-0.728814\pi$
$811$	−1.06716e19	−1.31703	−0.658513	+	0.752570i	$0.728814\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.18147e18	0.380280
$816$	0	0
$817$	4.53816e18	0.533871
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.68217e19	−1.91707	−0.958537	−	0.284969i	$-0.908017\pi$
$821$	−1.68217e19	−1.91707	−0.958537	+	0.284969i	$0.908017\pi$
$822$	0	0
$823$	1.47808e19	1.65806	0.829028	−	0.559207i	$-0.188894\pi$
$823$	1.47808e19	1.65806	0.829028	+	0.559207i	$0.188894\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−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	$0.151655\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.07246e19	−1.11221
$834$	0	0
$835$	−7.40928e18	−0.756509
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−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
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.10979e18	0.293875
$846$	0	0
$847$	3.13073e18	0.291342
$848$	0	0
$849$	0	0
$850$	0	0
$851$	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	$0.634665\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.40680e18	−0.379969	−0.189984	−	0.981787i	$-0.560844\pi$
$857$	−4.40680e18	−0.379969	−0.189984	+	0.981787i	$0.560844\pi$
$858$	0	0
$859$	−9.82119e18	−0.834082	−0.417041	−	0.908888i	$-0.636933\pi$
$859$	−9.82119e18	−0.834082	−0.417041	+	0.908888i	$0.636933\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	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
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.27893e18	−0.337060
$870$	0	0
$871$	2.04940e19	1.59041
$872$	0	0
$873$	0	0
$874$	0	0
$875$	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	$0.360354\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.44660e18	0.464502	0.232251	−	0.972656i	$-0.425391\pi$
$881$	6.44660e18	0.464502	0.232251	+	0.972656i	$0.425391\pi$
$882$	0	0
$883$	−1.62047e18	−0.115053	−0.0575263	−	0.998344i	$-0.518321\pi$
$883$	−1.62047e18	−0.115053	−0.0575263	+	0.998344i	$0.518321\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	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
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.01362e19	0.668867
$894$	0	0
$895$	1.16740e19	0.759223
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.21227e19	−0.765881
$900$	0	0
$901$	−6.84784e16	−0.00426425
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.17527e19	0.711088
$906$	0	0
$907$	2.17095e18	0.129480	0.0647399	−	0.997902i	$-0.479378\pi$
$907$	2.17095e18	0.129480	0.0647399	+	0.997902i	$0.479378\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−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
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.12185e18	−0.173385
$918$	0	0
$919$	2.51608e19	1.37776	0.688880	−	0.724876i	$-0.258103\pi$
$919$	2.51608e19	1.37776	0.688880	+	0.724876i	$0.258103\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.42487e19	−1.29086
$924$	0	0
$925$	5.76128e18	0.302411
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.29149e18	0.0659157	0.0329579	−	0.999457i	$-0.489507\pi$
$929$	1.29149e18	0.0659157	0.0329579	+	0.999457i	$0.489507\pi$
$930$	0	0
$931$	−6.47924e18	−0.326101
$932$	0	0
$933$	0	0
$934$	0	0
$935$	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	$0.285574\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.64109e19	0.770550	0.385275	−	0.922802i	$-0.374107\pi$
$941$	1.64109e19	0.770550	0.385275	+	0.922802i	$0.374107\pi$
$942$	0	0
$943$	2.31836e19	1.07363
$944$	0	0
$945$	0	0
$946$	0	0
$947$	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
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.33363e19	−1.00909	−0.504543	−	0.863387i	$-0.668339\pi$
$953$	−2.33363e19	−1.00909	−0.504543	+	0.863387i	$0.668339\pi$
$954$	0	0
$955$	3.51798e18	0.150062
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.77129e18	0.281093
$960$	0	0
$961$	−1.97432e19	−0.808565
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.76970e19	0.705461
$966$	0	0
$967$	2.71778e19	1.06891	0.534456	−	0.845197i	$-0.320517\pi$
$967$	2.71778e19	1.06891	0.534456	+	0.845197i	$0.320517\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	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
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.34520e18	−0.0494849	−0.0247424	−	0.999694i	$-0.507877\pi$
$977$	−1.34520e18	−0.0494849	−0.0247424	+	0.999694i	$0.507877\pi$
$978$	0	0
$979$	−9.65598e18	−0.350517
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−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
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.53481e19	−2.22053
$990$	0	0
$991$	−6.52510e18	−0.218831	−0.109415	−	0.993996i	$-0.534898\pi$
$991$	−6.52510e18	−0.218831	−0.109415	+	0.993996i	$0.534898\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	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	$0.340584\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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