LMFDB - Embedded newform 18.12.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,12,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, 12, names="a")

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 12 $
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:	$13.8301772501$
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)

$f(q)$

$=$

$q+32.0000 q^{2} +1024.00 q^{4} -5280.00 q^{5} -49036.0 q^{7} +32768.0 q^{8} +O(q^{10})$

\(q+32.0000 q^{2} +1024.00 q^{4} -5280.00 q^{5} -49036.0 q^{7} +32768.0 q^{8} -168960. q^{10} -414336. q^{11} -522982. q^{13} -1.56915e6 q^{14} +1.04858e6 q^{16} -9.49997e6 q^{17} +1.30539e7 q^{19} -5.40672e6 q^{20} -1.32588e7 q^{22} -5.87558e7 q^{23} -2.09497e7 q^{25} -1.67354e7 q^{26} -5.02129e7 q^{28} +1.17143e8 q^{29} +1.42907e8 q^{31} +3.35544e7 q^{32} -3.03999e8 q^{34} +2.58910e8 q^{35} +7.18522e8 q^{37} +4.17726e8 q^{38} -1.73015e8 q^{40} +6.68055e8 q^{41} +1.41576e8 q^{43} -4.24280e8 q^{44} -1.88019e9 q^{46} -7.29235e8 q^{47} +4.27203e8 q^{49} -6.70391e8 q^{50} -5.35534e8 q^{52} -4.91723e9 q^{53} +2.18769e9 q^{55} -1.60681e9 q^{56} +3.74857e9 q^{58} -1.40802e9 q^{59} -3.22333e9 q^{61} +4.57303e9 q^{62} +1.07374e9 q^{64} +2.76134e9 q^{65} -2.35868e9 q^{67} -9.72797e9 q^{68} +8.28512e9 q^{70} +2.22451e10 q^{71} -2.80366e10 q^{73} +2.29927e10 q^{74} +1.33672e10 q^{76} +2.03174e10 q^{77} -2.06850e10 q^{79} -5.53648e9 q^{80} +2.13778e10 q^{82} -3.78186e10 q^{83} +5.01598e10 q^{85} +4.53043e9 q^{86} -1.35770e10 q^{88} -1.12887e10 q^{89} +2.56449e10 q^{91} -6.01660e10 q^{92} -2.33355e10 q^{94} -6.89248e10 q^{95} -1.15724e11 q^{97} +1.36705e10 q^{98} +O(q^{100})\)

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 $

$2$	32.0000	0.707107
$2$	32.0000	0.707107
$3$	0	0
$3$	0	0
$4$	1024.00	0.500000
$5$	−5280.00	−0.755612	−0.377806	−	0.925885i	$-0.623321\pi$
$5$	−5280.00	−0.755612	−0.377806	+	0.925885i	$0.623321\pi$
$6$	0	0
$7$	−49036.0	−1.10275	−0.551373	−	0.834259i	$-0.685896\pi$
$7$	−49036.0	−1.10275	−0.551373	+	0.834259i	$0.685896\pi$
$8$	32768.0	0.353553
$9$	0	0
$10$	−168960.	−0.534298
$11$	−414336.	−0.775698	−0.387849	−	0.921723i	$-0.626782\pi$
$11$	−414336.	−0.775698	−0.387849	+	0.921723i	$0.626782\pi$
$12$	0	0
$13$	−522982.	−0.390659	−0.195330	−	0.980738i	$-0.562578\pi$
$13$	−522982.	−0.390659	−0.195330	+	0.980738i	$0.562578\pi$
$14$	−1.56915e6	−0.779760
$15$	0	0
$16$	1.04858e6	0.250000
$17$	−9.49997e6	−1.62276	−0.811378	−	0.584522i	$-0.801282\pi$
$17$	−9.49997e6	−1.62276	−0.811378	+	0.584522i	$0.801282\pi$
$18$	0	0
$19$	1.30539e7	1.20948	0.604738	−	0.796425i	$-0.293278\pi$
$19$	1.30539e7	1.20948	0.604738	+	0.796425i	$0.293278\pi$
$20$	−5.40672e6	−0.377806
$21$	0	0
$22$	−1.32588e7	−0.548502
$23$	−5.87558e7	−1.90348	−0.951739	−	0.306908i	$-0.900706\pi$
$23$	−5.87558e7	−1.90348	−0.951739	+	0.306908i	$0.900706\pi$
$24$	0	0
$25$	−2.09497e7	−0.429050
$26$	−1.67354e7	−0.276238
$27$	0	0
$28$	−5.02129e7	−0.551373
$29$	1.17143e8	1.06054	0.530270	−	0.847829i	$-0.322091\pi$
$29$	1.17143e8	1.06054	0.530270	+	0.847829i	$0.322091\pi$
$30$	0	0
$31$	1.42907e8	0.896530	0.448265	−	0.893901i	$-0.352042\pi$
$31$	1.42907e8	0.896530	0.448265	+	0.893901i	$0.352042\pi$
$32$	3.35544e7	0.176777
$33$	0	0
$34$	−3.03999e8	−1.14746
$35$	2.58910e8	0.833249
$36$	0	0
$37$	7.18522e8	1.70345	0.851727	−	0.523986i	$-0.175555\pi$
$37$	7.18522e8	1.70345	0.851727	+	0.523986i	$0.175555\pi$
$38$	4.17726e8	0.855228
$39$	0	0
$40$	−1.73015e8	−0.267149
$41$	6.68055e8	0.900536	0.450268	−	0.892893i	$-0.351328\pi$
$41$	6.68055e8	0.900536	0.450268	+	0.892893i	$0.351328\pi$
$42$	0	0
$43$	1.41576e8	0.146863	0.0734316	−	0.997300i	$-0.476605\pi$
$43$	1.41576e8	0.146863	0.0734316	+	0.997300i	$0.476605\pi$
$44$	−4.24280e8	−0.387849
$45$	0	0
$46$	−1.88019e9	−1.34596
$47$	−7.29235e8	−0.463799	−0.231899	−	0.972740i	$-0.574494\pi$
$47$	−7.29235e8	−0.463799	−0.231899	+	0.972740i	$0.574494\pi$
$48$	0	0
$49$	4.27203e8	0.216051
$50$	−6.70391e8	−0.303384
$51$	0	0
$52$	−5.35534e8	−0.195330
$53$	−4.91723e9	−1.61511	−0.807556	−	0.589790i	$-0.799210\pi$
$53$	−4.91723e9	−1.61511	−0.807556	+	0.589790i	$0.799210\pi$
$54$	0	0
$55$	2.18769e9	0.586127
$56$	−1.60681e9	−0.389880
$57$	0	0
$58$	3.74857e9	0.749915
$59$	−1.40802e9	−0.256402	−0.128201	−	0.991748i	$-0.540920\pi$
$59$	−1.40802e9	−0.256402	−0.128201	+	0.991748i	$0.540920\pi$
$60$	0	0
$61$	−3.22333e9	−0.488641	−0.244321	−	0.969694i	$-0.578565\pi$
$61$	−3.22333e9	−0.488641	−0.244321	+	0.969694i	$0.578565\pi$
$62$	4.57303e9	0.633942
$63$	0	0
$64$	1.07374e9	0.125000
$65$	2.76134e9	0.295187
$66$	0	0
$67$	−2.35868e9	−0.213431	−0.106716	−	0.994290i	$-0.534033\pi$
$67$	−2.35868e9	−0.213431	−0.106716	+	0.994290i	$0.534033\pi$
$68$	−9.72797e9	−0.811378
$69$	0	0
$70$	8.28512e9	0.589196
$71$	2.22451e10	1.46323	0.731616	−	0.681717i	$-0.238766\pi$
$71$	2.22451e10	1.46323	0.731616	+	0.681717i	$0.238766\pi$
$72$	0	0
$73$	−2.80366e10	−1.58289	−0.791443	−	0.611243i	$-0.790670\pi$
$73$	−2.80366e10	−1.58289	−0.791443	+	0.611243i	$0.790670\pi$
$74$	2.29927e10	1.20452
$75$	0	0
$76$	1.33672e10	0.604738
$77$	2.03174e10	0.855399
$78$	0	0
$79$	−2.06850e10	−0.756323	−0.378162	−	0.925740i	$-0.623444\pi$
$79$	−2.06850e10	−0.756323	−0.378162	+	0.925740i	$0.623444\pi$
$80$	−5.53648e9	−0.188903
$81$	0	0
$82$	2.13778e10	0.636775
$83$	−3.78186e10	−1.05384	−0.526922	−	0.849914i	$-0.676654\pi$
$83$	−3.78186e10	−1.05384	−0.526922	+	0.849914i	$0.676654\pi$
$84$	0	0
$85$	5.01598e10	1.22617
$86$	4.53043e9	0.103848
$87$	0	0
$88$	−1.35770e10	−0.274251
$89$	−1.12887e10	−0.214289	−0.107144	−	0.994243i	$-0.534171\pi$
$89$	−1.12887e10	−0.214289	−0.107144	+	0.994243i	$0.534171\pi$
$90$	0	0
$91$	2.56449e10	0.430798
$92$	−6.01660e10	−0.951739
$93$	0	0
$94$	−2.33355e10	−0.327955
$95$	−6.89248e10	−0.913894
$96$	0	0
$97$	−1.15724e11	−1.36830	−0.684148	−	0.729343i	$-0.739826\pi$
$97$	−1.15724e11	−1.36830	−0.684148	+	0.729343i	$0.739826\pi$
$98$	1.36705e10	0.152771
$99$	0	0
$100$	−2.14525e10	−0.214525
$101$	−6.11783e10	−0.579202	−0.289601	−	0.957147i	$-0.593523\pi$
$101$	−6.11783e10	−0.579202	−0.289601	+	0.957147i	$0.593523\pi$
$102$	0	0
$103$	1.76207e11	1.49768	0.748839	−	0.662752i	$-0.230612\pi$
$103$	1.76207e11	1.49768	0.748839	+	0.662752i	$0.230612\pi$
$104$	−1.71371e10	−0.138119
$105$	0	0
$106$	−1.57351e11	−1.14206
$107$	8.77786e10	0.605031	0.302516	−	0.953144i	$-0.402174\pi$
$107$	8.77786e10	0.605031	0.302516	+	0.953144i	$0.402174\pi$
$108$	0	0
$109$	9.07853e9	0.0565158	0.0282579	−	0.999601i	$-0.491004\pi$
$109$	9.07853e9	0.0565158	0.0282579	+	0.999601i	$0.491004\pi$
$110$	7.00062e10	0.414454
$111$	0	0
$112$	−5.14180e10	−0.275687
$113$	−1.71667e9	−0.00876505	−0.00438253	−	0.999990i	$-0.501395\pi$
$113$	−1.71667e9	−0.00876505	−0.00438253	+	0.999990i	$0.501395\pi$
$114$	0	0
$115$	3.10231e11	1.43829
$116$	1.19954e11	0.530270
$117$	0	0
$118$	−4.50565e10	−0.181304
$119$	4.65840e11	1.78949
$120$	0	0
$121$	−1.13637e11	−0.398292
$122$	−1.03146e11	−0.345522
$123$	0	0
$124$	1.46337e11	0.448265
$125$	3.68427e11	1.07981
$126$	0	0
$127$	9.96723e10	0.267703	0.133852	−	0.991001i	$-0.457265\pi$
$127$	9.96723e10	0.267703	0.133852	+	0.991001i	$0.457265\pi$
$128$	3.43597e10	0.0883883
$129$	0	0
$130$	8.83630e10	0.208729
$131$	−7.55892e11	−1.71186	−0.855928	−	0.517094i	$-0.827014\pi$
$131$	−7.55892e11	−1.71186	−0.855928	+	0.517094i	$0.827014\pi$
$132$	0	0
$133$	−6.40113e11	−1.33374
$134$	−7.54778e10	−0.150919
$135$	0	0
$136$	−3.11295e11	−0.573731
$137$	−5.14820e11	−0.911365	−0.455683	−	0.890142i	$-0.650605\pi$
$137$	−5.14820e11	−0.911365	−0.455683	+	0.890142i	$0.650605\pi$
$138$	0	0
$139$	−2.17974e11	−0.356306	−0.178153	−	0.984003i	$-0.557012\pi$
$139$	−2.17974e11	−0.356306	−0.178153	+	0.984003i	$0.557012\pi$
$140$	2.65124e11	0.416624
$141$	0	0
$142$	7.11843e11	1.03466
$143$	2.16690e11	0.303034
$144$	0	0
$145$	−6.18515e11	−0.801357
$146$	−8.97171e11	−1.11927
$147$	0	0
$148$	7.35766e11	0.851727
$149$	−1.00196e12	−1.11770	−0.558852	−	0.829267i	$-0.688758\pi$
$149$	−1.00196e12	−1.11770	−0.558852	+	0.829267i	$0.688758\pi$
$150$	0	0
$151$	1.15528e12	1.19760	0.598801	−	0.800898i	$-0.295644\pi$
$151$	1.15528e12	1.19760	0.598801	+	0.800898i	$0.295644\pi$
$152$	4.27752e11	0.427614
$153$	0	0
$154$	6.50156e11	0.604858
$155$	−7.54550e11	−0.677429
$156$	0	0
$157$	−2.14998e12	−1.79882	−0.899409	−	0.437108i	$-0.856003\pi$
$157$	−2.14998e12	−1.79882	−0.899409	+	0.437108i	$0.856003\pi$
$158$	−6.61921e11	−0.534801
$159$	0	0
$160$	−1.77167e11	−0.133575
$161$	2.88115e12	2.09905
$162$	0	0
$163$	2.08743e12	1.42096	0.710478	−	0.703720i	$-0.248479\pi$
$163$	2.08743e12	1.42096	0.710478	+	0.703720i	$0.248479\pi$
$164$	6.84089e11	0.450268
$165$	0	0
$166$	−1.21020e12	−0.745180
$167$	−4.97824e10	−0.0296575	−0.0148288	−	0.999890i	$-0.504720\pi$
$167$	−4.97824e10	−0.0296575	−0.0148288	+	0.999890i	$0.504720\pi$
$168$	0	0
$169$	−1.51865e12	−0.847385
$170$	1.60511e12	0.867036
$171$	0	0
$172$	1.44974e11	0.0734316
$173$	3.63256e11	0.178221	0.0891107	−	0.996022i	$-0.471598\pi$
$173$	3.63256e11	0.178221	0.0891107	+	0.996022i	$0.471598\pi$
$174$	0	0
$175$	1.02729e12	0.473134
$176$	−4.34463e11	−0.193925
$177$	0	0
$178$	−3.61239e11	−0.151525
$179$	4.11586e12	1.67405	0.837027	−	0.547162i	$-0.184292\pi$
$179$	4.11586e12	1.67405	0.837027	+	0.547162i	$0.184292\pi$
$180$	0	0
$181$	−1.47606e12	−0.564772	−0.282386	−	0.959301i	$-0.591126\pi$
$181$	−1.47606e12	−0.564772	−0.282386	+	0.959301i	$0.591126\pi$
$182$	8.20638e11	0.304620
$183$	0	0
$184$	−1.92531e12	−0.672981
$185$	−3.79380e12	−1.28715
$186$	0	0
$187$	3.93618e12	1.25877
$188$	−7.46737e11	−0.231899
$189$	0	0
$190$	−2.20559e12	−0.646221
$191$	−1.14505e12	−0.325942	−0.162971	−	0.986631i	$-0.552108\pi$
$191$	−1.14505e12	−0.325942	−0.162971	+	0.986631i	$0.552108\pi$
$192$	0	0
$193$	4.61179e12	1.23967	0.619833	−	0.784734i	$-0.287200\pi$
$193$	4.61179e12	1.23967	0.619833	+	0.784734i	$0.287200\pi$
$194$	−3.70318e12	−0.967532
$195$	0	0
$196$	4.37455e11	0.108025
$197$	3.31733e11	0.0796570	0.0398285	−	0.999207i	$-0.487319\pi$
$197$	3.31733e11	0.0796570	0.0398285	+	0.999207i	$0.487319\pi$
$198$	0	0
$199$	3.84736e12	0.873919	0.436959	−	0.899481i	$-0.356055\pi$
$199$	3.84736e12	0.873919	0.436959	+	0.899481i	$0.356055\pi$
$200$	−6.86481e11	−0.151692
$201$	0	0
$202$	−1.95771e12	−0.409558
$203$	−5.74422e12	−1.16951
$204$	0	0
$205$	−3.52733e12	−0.680456
$206$	5.63862e12	1.05902
$207$	0	0
$208$	−5.48386e11	−0.0976649
$209$	−5.40872e12	−0.938188
$210$	0	0
$211$	−4.76529e12	−0.784398	−0.392199	−	0.919880i	$-0.628285\pi$
$211$	−4.76529e12	−0.784398	−0.392199	+	0.919880i	$0.628285\pi$
$212$	−5.03524e12	−0.807556
$213$	0	0
$214$	2.80892e12	0.427822
$215$	−7.47521e11	−0.110972
$216$	0	0
$217$	−7.00760e12	−0.988645
$218$	2.90513e11	0.0399627
$219$	0	0
$220$	2.24020e12	0.293064
$221$	4.96831e12	0.633945
$222$	0	0
$223$	−6.74459e11	−0.0818990	−0.0409495	−	0.999161i	$-0.513038\pi$
$223$	−6.74459e11	−0.0818990	−0.0409495	+	0.999161i	$0.513038\pi$
$224$	−1.64538e12	−0.194940
$225$	0	0
$226$	−5.49333e10	−0.00619783
$227$	9.04626e12	0.996154	0.498077	−	0.867133i	$-0.334040\pi$
$227$	9.04626e12	0.996154	0.498077	+	0.867133i	$0.334040\pi$
$228$	0	0
$229$	5.21223e12	0.546925	0.273463	−	0.961883i	$-0.411831\pi$
$229$	5.21223e12	0.546925	0.273463	+	0.961883i	$0.411831\pi$
$230$	9.92739e12	1.01703
$231$	0	0
$232$	3.83854e12	0.374958
$233$	1.87172e13	1.78559	0.892797	−	0.450460i	$-0.148740\pi$
$233$	1.87172e13	1.78559	0.892797	+	0.450460i	$0.148740\pi$
$234$	0	0
$235$	3.85036e12	0.350452
$236$	−1.44181e12	−0.128201
$237$	0	0
$238$	1.49069e13	1.26536
$239$	−5.07132e12	−0.420661	−0.210331	−	0.977630i	$-0.567454\pi$
$239$	−5.07132e12	−0.420661	−0.210331	+	0.977630i	$0.567454\pi$
$240$	0	0
$241$	−2.02971e13	−1.60820	−0.804099	−	0.594495i	$-0.797352\pi$
$241$	−2.02971e13	−1.60820	−0.804099	+	0.594495i	$0.797352\pi$
$242$	−3.63640e12	−0.281635
$243$	0	0
$244$	−3.30069e12	−0.244321
$245$	−2.25563e12	−0.163250
$246$	0	0
$247$	−6.82698e12	−0.472493
$248$	4.68278e12	0.316971
$249$	0	0
$250$	1.17897e13	0.763539
$251$	2.67098e12	0.169225	0.0846125	−	0.996414i	$-0.473035\pi$
$251$	2.67098e12	0.169225	0.0846125	+	0.996414i	$0.473035\pi$
$252$	0	0
$253$	2.43447e13	1.47653
$254$	3.18951e12	0.189295
$255$	0	0
$256$	1.09951e12	0.0625000
$257$	−1.79537e13	−0.998900	−0.499450	−	0.866343i	$-0.666465\pi$
$257$	−1.79537e13	−0.998900	−0.499450	+	0.866343i	$0.666465\pi$
$258$	0	0
$259$	−3.52334e13	−1.87848
$260$	2.82762e12	0.147593
$261$	0	0
$262$	−2.41885e13	−1.21047
$263$	−6.14166e12	−0.300974	−0.150487	−	0.988612i	$-0.548084\pi$
$263$	−6.14166e12	−0.300974	−0.150487	+	0.988612i	$0.548084\pi$
$264$	0	0
$265$	2.59629e13	1.22040
$266$	−2.04836e13	−0.943100
$267$	0	0
$268$	−2.41529e12	−0.106716
$269$	−2.56644e13	−1.11095	−0.555474	−	0.831534i	$-0.687463\pi$
$269$	−2.56644e13	−1.11095	−0.555474	+	0.831534i	$0.687463\pi$
$270$	0	0
$271$	−1.53150e13	−0.636480	−0.318240	−	0.948010i	$-0.603092\pi$
$271$	−1.53150e13	−0.636480	−0.318240	+	0.948010i	$0.603092\pi$
$272$	−9.96144e12	−0.405689
$273$	0	0
$274$	−1.64742e13	−0.644433
$275$	8.68023e12	0.332814
$276$	0	0
$277$	2.09481e13	0.771804	0.385902	−	0.922540i	$-0.373890\pi$
$277$	2.09481e13	0.771804	0.385902	+	0.922540i	$0.373890\pi$
$278$	−6.97516e12	−0.251946
$279$	0	0
$280$	8.48397e12	0.294598
$281$	1.02475e12	0.0348925	0.0174463	−	0.999848i	$-0.494446\pi$
$281$	1.02475e12	0.0348925	0.0174463	+	0.999848i	$0.494446\pi$
$282$	0	0
$283$	1.41891e13	0.464652	0.232326	−	0.972638i	$-0.425366\pi$
$283$	1.41891e13	0.464652	0.232326	+	0.972638i	$0.425366\pi$
$284$	2.27790e13	0.731616
$285$	0	0
$286$	6.93409e12	0.214277
$287$	−3.27588e13	−0.993064
$288$	0	0
$289$	5.59775e13	1.63334
$290$	−1.97925e13	−0.566645
$291$	0	0
$292$	−2.87095e13	−0.791443
$293$	3.37240e13	0.912362	0.456181	−	0.889887i	$-0.349217\pi$
$293$	3.37240e13	0.912362	0.456181	+	0.889887i	$0.349217\pi$
$294$	0	0
$295$	7.43432e12	0.193740
$296$	2.35445e13	0.602262
$297$	0	0
$298$	−3.20628e13	−0.790336
$299$	3.07282e13	0.743612
$300$	0	0
$301$	−6.94231e12	−0.161953
$302$	3.69689e13	0.846833
$303$	0	0
$304$	1.36881e13	0.302369
$305$	1.70192e13	0.369223
$306$	0	0
$307$	−3.05293e13	−0.638933	−0.319467	−	0.947598i	$-0.603504\pi$
$307$	−3.05293e13	−0.638933	−0.319467	+	0.947598i	$0.603504\pi$
$308$	2.08050e13	0.427699
$309$	0	0
$310$	−2.41456e13	−0.479014
$311$	9.85338e13	1.92045	0.960225	−	0.279228i	$-0.0900784\pi$
$311$	9.85338e13	1.92045	0.960225	+	0.279228i	$0.0900784\pi$
$312$	0	0
$313$	1.03973e13	0.195627	0.0978133	−	0.995205i	$-0.468815\pi$
$313$	1.03973e13	0.195627	0.0978133	+	0.995205i	$0.468815\pi$
$314$	−6.87995e13	−1.27196
$315$	0	0
$316$	−2.11815e13	−0.378162
$317$	1.63419e12	0.0286733	0.0143367	−	0.999897i	$-0.495436\pi$
$317$	1.63419e12	0.0286733	0.0143367	+	0.999897i	$0.495436\pi$
$318$	0	0
$319$	−4.85365e13	−0.822659
$320$	−5.66936e12	−0.0944515
$321$	0	0
$322$	9.21968e13	1.48426
$323$	−1.24012e14	−1.96268
$324$	0	0
$325$	1.09563e13	0.167613
$326$	6.67978e13	1.00477
$327$	0	0
$328$	2.18908e13	0.318388
$329$	3.57588e13	0.511452
$330$	0	0
$331$	−8.86557e13	−1.22646	−0.613229	−	0.789905i	$-0.710130\pi$
$331$	−8.86557e13	−1.22646	−0.613229	+	0.789905i	$0.710130\pi$
$332$	−3.87263e13	−0.526922
$333$	0	0
$334$	−1.59304e12	−0.0209710
$335$	1.24538e13	0.161271
$336$	0	0
$337$	2.25857e13	0.283053	0.141527	−	0.989934i	$-0.454799\pi$
$337$	2.25857e13	0.283053	0.141527	+	0.989934i	$0.454799\pi$
$338$	−4.85968e13	−0.599192
$339$	0	0
$340$	5.13637e13	0.613087
$341$	−5.92116e13	−0.695437
$342$	0	0
$343$	7.60119e13	0.864498
$344$	4.63916e12	0.0519239
$345$	0	0
$346$	1.16242e13	0.126022
$347$	−1.08734e14	−1.16025	−0.580126	−	0.814526i	$-0.696997\pi$
$347$	−1.08734e14	−1.16025	−0.580126	+	0.814526i	$0.696997\pi$
$348$	0	0
$349$	−2.52635e13	−0.261188	−0.130594	−	0.991436i	$-0.541688\pi$
$349$	−2.52635e13	−0.261188	−0.130594	+	0.991436i	$0.541688\pi$
$350$	3.28733e13	0.334556
$351$	0	0
$352$	−1.39028e13	−0.137125
$353$	−1.16788e14	−1.13407	−0.567033	−	0.823695i	$-0.691909\pi$
$353$	−1.16788e14	−1.13407	−0.567033	+	0.823695i	$0.691909\pi$
$354$	0	0
$355$	−1.17454e14	−1.10564
$356$	−1.15596e13	−0.107144
$357$	0	0
$358$	1.31708e14	1.18373
$359$	−6.20478e13	−0.549170	−0.274585	−	0.961563i	$-0.588540\pi$
$359$	−6.20478e13	−0.549170	−0.274585	+	0.961563i	$0.588540\pi$
$360$	0	0
$361$	5.39152e13	0.462830
$362$	−4.72341e13	−0.399354
$363$	0	0
$364$	2.62604e13	0.215399
$365$	1.48033e14	1.19605
$366$	0	0
$367$	−1.36092e14	−1.06701	−0.533507	−	0.845796i	$-0.679126\pi$
$367$	−1.36092e14	−1.06701	−0.533507	+	0.845796i	$0.679126\pi$
$368$	−6.16100e13	−0.475870
$369$	0	0
$370$	−1.21401e14	−0.910153
$371$	2.41121e14	1.78106
$372$	0	0
$373$	−1.41264e14	−1.01305	−0.506526	−	0.862225i	$-0.669071\pi$
$373$	−1.41264e14	−1.01305	−0.506526	+	0.862225i	$0.669071\pi$
$374$	1.25958e14	0.890084
$375$	0	0
$376$	−2.38956e13	−0.163978
$377$	−6.12637e13	−0.414310
$378$	0	0
$379$	1.73436e14	1.13926	0.569631	−	0.821901i	$-0.307086\pi$
$379$	1.73436e14	1.13926	0.569631	+	0.821901i	$0.307086\pi$
$380$	−7.05790e13	−0.456947
$381$	0	0
$382$	−3.66415e13	−0.230476
$383$	1.84328e14	1.14287	0.571437	−	0.820646i	$-0.306386\pi$
$383$	1.84328e14	1.14287	0.571437	+	0.820646i	$0.306386\pi$
$384$	0	0
$385$	−1.07276e14	−0.646350
$386$	1.47577e14	0.876576
$387$	0	0
$388$	−1.18502e14	−0.684148
$389$	−7.94999e13	−0.452526	−0.226263	−	0.974066i	$-0.572651\pi$
$389$	−7.94999e13	−0.452526	−0.226263	+	0.974066i	$0.572651\pi$
$390$	0	0
$391$	5.58179e14	3.08888
$392$	1.39986e13	0.0763854
$393$	0	0
$394$	1.06154e13	0.0563260
$395$	1.09217e14	0.571487
$396$	0	0
$397$	8.73187e13	0.444385	0.222193	−	0.975003i	$-0.428679\pi$
$397$	8.73187e13	0.444385	0.222193	+	0.975003i	$0.428679\pi$
$398$	1.23116e14	0.617954
$399$	0	0
$400$	−2.19674e13	−0.107263
$401$	1.77759e14	0.856128	0.428064	−	0.903748i	$-0.359196\pi$
$401$	1.77759e14	0.856128	0.428064	+	0.903748i	$0.359196\pi$
$402$	0	0
$403$	−7.47379e13	−0.350238
$404$	−6.26466e13	−0.289601
$405$	0	0
$406$	−1.83815e14	−0.826967
$407$	−2.97709e14	−1.32137
$408$	0	0
$409$	−1.26724e14	−0.547495	−0.273748	−	0.961802i	$-0.588263\pi$
$409$	−1.26724e14	−0.547495	−0.273748	+	0.961802i	$0.588263\pi$
$410$	−1.12875e14	−0.481155
$411$	0	0
$412$	1.80436e14	0.748839
$413$	6.90434e13	0.282746
$414$	0	0
$415$	1.99682e14	0.796297
$416$	−1.75484e13	−0.0690595
$417$	0	0
$418$	−1.73079e14	−0.663399
$419$	−4.17470e14	−1.57924	−0.789619	−	0.613597i	$-0.789722\pi$
$419$	−4.17470e14	−1.57924	−0.789619	+	0.613597i	$0.789722\pi$
$420$	0	0
$421$	−6.66324e13	−0.245547	−0.122773	−	0.992435i	$-0.539179\pi$
$421$	−6.66324e13	−0.245547	−0.122773	+	0.992435i	$0.539179\pi$
$422$	−1.52489e14	−0.554653
$423$	0	0
$424$	−1.61128e14	−0.571029
$425$	1.99022e14	0.696244
$426$	0	0
$427$	1.58059e14	0.538848
$428$	8.98853e13	0.302516
$429$	0	0
$430$	−2.39207e13	−0.0784687
$431$	1.85689e14	0.601396	0.300698	−	0.953719i	$-0.402780\pi$
$431$	1.85689e14	0.601396	0.300698	+	0.953719i	$0.402780\pi$
$432$	0	0
$433$	−3.31842e14	−1.04773	−0.523864	−	0.851802i	$-0.675510\pi$
$433$	−3.31842e14	−1.04773	−0.523864	+	0.851802i	$0.675510\pi$
$434$	−2.24243e14	−0.699078
$435$	0	0
$436$	9.29642e12	0.0282579
$437$	−7.66995e14	−2.30221
$438$	0	0
$439$	5.11462e13	0.149713	0.0748563	−	0.997194i	$-0.476150\pi$
$439$	5.11462e13	0.149713	0.0748563	+	0.997194i	$0.476150\pi$
$440$	7.16864e13	0.207227
$441$	0	0
$442$	1.58986e14	0.448267
$443$	−9.39135e13	−0.261522	−0.130761	−	0.991414i	$-0.541742\pi$
$443$	−9.39135e13	−0.261522	−0.130761	+	0.991414i	$0.541742\pi$
$444$	0	0
$445$	5.96044e13	0.161919
$446$	−2.15827e13	−0.0579113
$447$	0	0
$448$	−5.26520e13	−0.137843
$449$	−8.14882e13	−0.210737	−0.105368	−	0.994433i	$-0.533602\pi$
$449$	−8.14882e13	−0.210737	−0.105368	+	0.994433i	$0.533602\pi$
$450$	0	0
$451$	−2.76799e14	−0.698545
$452$	−1.75787e12	−0.00438253
$453$	0	0
$454$	2.89480e14	0.704388
$455$	−1.35405e14	−0.325517
$456$	0	0
$457$	−4.89054e14	−1.14767	−0.573835	−	0.818971i	$-0.694545\pi$
$457$	−4.89054e14	−1.14767	−0.573835	+	0.818971i	$0.694545\pi$
$458$	1.66791e14	0.386735
$459$	0	0
$460$	3.17676e14	0.719146
$461$	8.53552e14	1.90930	0.954651	−	0.297727i	$-0.0962284\pi$
$461$	8.53552e14	1.90930	0.954651	+	0.297727i	$0.0962284\pi$
$462$	0	0
$463$	4.23617e14	0.925291	0.462645	−	0.886543i	$-0.346900\pi$
$463$	4.23617e14	0.925291	0.462645	+	0.886543i	$0.346900\pi$
$464$	1.22833e14	0.265135
$465$	0	0
$466$	5.98949e14	1.26260
$467$	6.43575e14	1.34078	0.670388	−	0.742011i	$-0.266128\pi$
$467$	6.43575e14	1.34078	0.670388	+	0.742011i	$0.266128\pi$
$468$	0	0
$469$	1.15660e14	0.235361
$470$	1.23212e14	0.247807
$471$	0	0
$472$	−4.61378e13	−0.0906518
$473$	−5.86600e13	−0.113921
$474$	0	0
$475$	−2.73477e14	−0.518926
$476$	4.77021e14	0.894744
$477$	0	0
$478$	−1.62282e14	−0.297452
$479$	−2.16256e14	−0.391852	−0.195926	−	0.980619i	$-0.562771\pi$
$479$	−2.16256e14	−0.391852	−0.195926	+	0.980619i	$0.562771\pi$
$480$	0	0
$481$	−3.75774e14	−0.665470
$482$	−6.49506e14	−1.13717
$483$	0	0
$484$	−1.16365e14	−0.199146
$485$	6.11025e14	1.03390
$486$	0	0
$487$	7.09676e14	1.17395	0.586976	−	0.809604i	$-0.300318\pi$
$487$	7.09676e14	1.17395	0.586976	+	0.809604i	$0.300318\pi$
$488$	−1.05622e14	−0.172761
$489$	0	0
$490$	−7.21801e13	−0.115435
$491$	9.93022e14	1.57040	0.785200	−	0.619242i	$-0.212560\pi$
$491$	9.93022e14	1.57040	0.785200	+	0.619242i	$0.212560\pi$
$492$	0	0
$493$	−1.11285e15	−1.72100
$494$	−2.18463e14	−0.334103
$495$	0	0
$496$	1.49849e14	0.224132
$497$	−1.09081e15	−1.61357
$498$	0	0
$499$	−1.04065e15	−1.50575	−0.752875	−	0.658163i	$-0.771334\pi$
$499$	−1.04065e15	−1.50575	−0.752875	+	0.658163i	$0.771334\pi$
$500$	3.77269e14	0.539904
$501$	0	0
$502$	8.54712e13	0.119660
$503$	−2.63018e13	−0.0364219	−0.0182109	−	0.999834i	$-0.505797\pi$
$503$	−2.63018e13	−0.0364219	−0.0182109	+	0.999834i	$0.505797\pi$
$504$	0	0
$505$	3.23022e14	0.437652
$506$	7.79029e14	1.04406
$507$	0	0
$508$	1.02064e14	0.133852
$509$	−8.22718e14	−1.06734	−0.533670	−	0.845693i	$-0.679188\pi$
$509$	−8.22718e14	−1.06734	−0.533670	+	0.845693i	$0.679188\pi$
$510$	0	0
$511$	1.37480e15	1.74552
$512$	3.51844e13	0.0441942
$513$	0	0
$514$	−5.74519e14	−0.706329
$515$	−9.30373e14	−1.13166
$516$	0	0
$517$	3.02148e14	0.359768
$518$	−1.12747e15	−1.32829
$519$	0	0
$520$	9.04838e13	0.104364
$521$	9.18768e14	1.04857	0.524286	−	0.851542i	$-0.324332\pi$
$521$	9.18768e14	1.04857	0.524286	+	0.851542i	$0.324332\pi$
$522$	0	0
$523$	−5.78310e14	−0.646252	−0.323126	−	0.946356i	$-0.604734\pi$
$523$	−5.78310e14	−0.646252	−0.323126	+	0.946356i	$0.604734\pi$
$524$	−7.74033e14	−0.855928
$525$	0	0
$526$	−1.96533e14	−0.212821
$527$	−1.35761e15	−1.45485
$528$	0	0
$529$	2.49944e15	2.62323
$530$	8.30814e14	0.862952
$531$	0	0
$532$	−6.55476e14	−0.666872
$533$	−3.49381e14	−0.351803
$534$	0	0
$535$	−4.63471e14	−0.457169
$536$	−7.72893e13	−0.0754593
$537$	0	0
$538$	−8.21260e14	−0.785558
$539$	−1.77005e14	−0.167590
$540$	0	0
$541$	5.73476e14	0.532023	0.266011	−	0.963970i	$-0.414294\pi$
$541$	5.73476e14	0.532023	0.266011	+	0.963970i	$0.414294\pi$
$542$	−4.90079e14	−0.450059
$543$	0	0
$544$	−3.18766e14	−0.286865
$545$	−4.79346e13	−0.0427040
$546$	0	0
$547$	−7.21000e14	−0.629513	−0.314756	−	0.949172i	$-0.601923\pi$
$547$	−7.21000e14	−0.629513	−0.314756	+	0.949172i	$0.601923\pi$
$548$	−5.27176e14	−0.455683
$549$	0	0
$550$	2.77767e14	0.235335
$551$	1.52918e15	1.28270
$552$	0	0
$553$	1.01431e15	0.834033
$554$	6.70341e14	0.545748
$555$	0	0
$556$	−2.23205e14	−0.178153
$557$	−3.02540e14	−0.239100	−0.119550	−	0.992828i	$-0.538145\pi$
$557$	−3.02540e14	−0.239100	−0.119550	+	0.992828i	$0.538145\pi$
$558$	0	0
$559$	−7.40416e13	−0.0573735
$560$	2.71487e14	0.208312
$561$	0	0
$562$	3.27919e13	0.0246727
$563$	2.25846e13	0.0168273	0.00841367	−	0.999965i	$-0.497322\pi$
$563$	2.25846e13	0.0168273	0.00841367	+	0.999965i	$0.497322\pi$
$564$	0	0
$565$	9.06400e12	0.00662298
$566$	4.54050e14	0.328559
$567$	0	0
$568$	7.28927e14	0.517331
$569$	1.79161e15	1.25929	0.629646	−	0.776883i	$-0.283200\pi$
$569$	1.79161e15	1.25929	0.629646	+	0.776883i	$0.283200\pi$
$570$	0	0
$571$	−2.68809e15	−1.85330	−0.926649	−	0.375928i	$-0.877324\pi$
$571$	−2.68809e15	−1.85330	−0.926649	+	0.375928i	$0.877324\pi$
$572$	2.21891e14	0.151517
$573$	0	0
$574$	−1.04828e15	−0.702202
$575$	1.23092e15	0.816688
$576$	0	0
$577$	−1.09334e15	−0.711683	−0.355841	−	0.934546i	$-0.615806\pi$
$577$	−1.09334e15	−0.711683	−0.355841	+	0.934546i	$0.615806\pi$
$578$	1.79128e15	1.15494
$579$	0	0
$580$	−6.33359e14	−0.400679
$581$	1.85447e15	1.16212
$582$	0	0
$583$	2.03738e15	1.25284
$584$	−9.18703e14	−0.559635
$585$	0	0
$586$	1.07917e15	0.645137
$587$	−2.03438e15	−1.20482	−0.602409	−	0.798187i	$-0.705793\pi$
$587$	−2.03438e15	−1.20482	−0.602409	+	0.798187i	$0.705793\pi$
$588$	0	0
$589$	1.86550e15	1.08433
$590$	2.37898e14	0.136995
$591$	0	0
$592$	7.53425e14	0.425864
$593$	−1.58660e15	−0.888518	−0.444259	−	0.895898i	$-0.646533\pi$
$593$	−1.58660e15	−0.888518	−0.444259	+	0.895898i	$0.646533\pi$
$594$	0	0
$595$	−2.45964e15	−1.35216
$596$	−1.02601e15	−0.558852
$597$	0	0
$598$	9.83304e14	0.525813
$599$	8.83347e14	0.468041	0.234020	−	0.972232i	$-0.424812\pi$
$599$	8.83347e14	0.468041	0.234020	+	0.972232i	$0.424812\pi$
$600$	0	0
$601$	2.76844e14	0.144021	0.0720105	−	0.997404i	$-0.477058\pi$
$601$	2.76844e14	0.144021	0.0720105	+	0.997404i	$0.477058\pi$
$602$	−2.22154e14	−0.114518
$603$	0	0
$604$	1.18300e15	0.598801
$605$	6.00005e14	0.300954
$606$	0	0
$607$	1.00453e15	0.494796	0.247398	−	0.968914i	$-0.420424\pi$
$607$	1.00453e15	0.494796	0.247398	+	0.968914i	$0.420424\pi$
$608$	4.38018e14	0.213807
$609$	0	0
$610$	5.44613e14	0.261080
$611$	3.81377e14	0.181187
$612$	0	0
$613$	1.75795e14	0.0820303	0.0410152	−	0.999159i	$-0.486941\pi$
$613$	1.75795e14	0.0820303	0.0410152	+	0.999159i	$0.486941\pi$
$614$	−9.76937e14	−0.451794
$615$	0	0
$616$	6.65760e14	0.302429
$617$	−8.66387e14	−0.390071	−0.195035	−	0.980796i	$-0.562482\pi$
$617$	−8.66387e14	−0.390071	−0.195035	+	0.980796i	$0.562482\pi$
$618$	0	0
$619$	−4.48165e15	−1.98216	−0.991082	−	0.133252i	$-0.957458\pi$
$619$	−4.48165e15	−1.98216	−0.991082	+	0.133252i	$0.957458\pi$
$620$	−7.72659e14	−0.338714
$621$	0	0
$622$	3.15308e15	1.35796
$623$	5.53553e14	0.236306
$624$	0	0
$625$	−9.22359e14	−0.386865
$626$	3.32715e14	0.138329
$627$	0	0
$628$	−2.20158e15	−0.899409
$629$	−6.82593e15	−2.76429
$630$	0	0
$631$	−1.06749e15	−0.424816	−0.212408	−	0.977181i	$-0.568131\pi$
$631$	−1.06749e15	−0.424816	−0.212408	+	0.977181i	$0.568131\pi$
$632$	−6.77808e14	−0.267401
$633$	0	0
$634$	5.22942e13	0.0202751
$635$	−5.26270e14	−0.202280
$636$	0	0
$637$	−2.23419e14	−0.0844022
$638$	−1.55317e15	−0.581708
$639$	0	0
$640$	−1.81419e14	−0.0667873
$641$	−9.84911e14	−0.359483	−0.179741	−	0.983714i	$-0.557526\pi$
$641$	−9.84911e14	−0.359483	−0.179741	+	0.983714i	$0.557526\pi$
$642$	0	0
$643$	−7.00399e13	−0.0251296	−0.0125648	−	0.999921i	$-0.504000\pi$
$643$	−7.00399e13	−0.0251296	−0.0125648	+	0.999921i	$0.504000\pi$
$644$	2.95030e15	1.04953
$645$	0	0
$646$	−3.96839e15	−1.38783
$647$	−3.09515e15	−1.07327	−0.536633	−	0.843816i	$-0.680304\pi$
$647$	−3.09515e15	−1.07327	−0.536633	+	0.843816i	$0.680304\pi$
$648$	0	0
$649$	5.83391e14	0.198891
$650$	3.50603e14	0.118520
$651$	0	0
$652$	2.13753e15	0.710478
$653$	4.27370e15	1.40858	0.704290	−	0.709913i	$-0.251266\pi$
$653$	4.27370e15	1.40858	0.704290	+	0.709913i	$0.251266\pi$
$654$	0	0
$655$	3.99111e15	1.29350
$656$	7.00507e14	0.225134
$657$	0	0
$658$	1.14428e15	0.361651
$659$	2.32939e15	0.730083	0.365041	−	0.930991i	$-0.381055\pi$
$659$	2.32939e15	0.730083	0.365041	+	0.930991i	$0.381055\pi$
$660$	0	0
$661$	1.82318e15	0.561979	0.280990	−	0.959711i	$-0.409337\pi$
$661$	1.82318e15	0.561979	0.280990	+	0.959711i	$0.409337\pi$
$662$	−2.83698e15	−0.867237
$663$	0	0
$664$	−1.23924e15	−0.372590
$665$	3.37980e15	1.00779
$666$	0	0
$667$	−6.88283e15	−2.01872
$668$	−5.09771e13	−0.0148288
$669$	0	0
$670$	3.98523e14	0.114036
$671$	1.33554e15	0.379038
$672$	0	0
$673$	3.37196e15	0.941454	0.470727	−	0.882279i	$-0.343992\pi$
$673$	3.37196e15	0.941454	0.470727	+	0.882279i	$0.343992\pi$
$674$	7.22741e14	0.200149
$675$	0	0
$676$	−1.55510e15	−0.423693
$677$	3.26034e15	0.881100	0.440550	−	0.897728i	$-0.354783\pi$
$677$	3.26034e15	0.881100	0.440550	+	0.897728i	$0.354783\pi$
$678$	0	0
$679$	5.67466e15	1.50889
$680$	1.64364e15	0.433518
$681$	0	0
$682$	−1.89477e15	−0.491748
$683$	−1.78782e15	−0.460268	−0.230134	−	0.973159i	$-0.573916\pi$
$683$	−1.78782e15	−0.460268	−0.230134	+	0.973159i	$0.573916\pi$
$684$	0	0
$685$	2.71825e15	0.688639
$686$	2.43238e15	0.611292
$687$	0	0
$688$	1.48453e14	0.0367158
$689$	2.57162e15	0.630959
$690$	0	0
$691$	2.48246e15	0.599450	0.299725	−	0.954026i	$-0.403105\pi$
$691$	2.48246e15	0.599450	0.299725	+	0.954026i	$0.403105\pi$
$692$	3.71975e14	0.0891107
$693$	0	0
$694$	−3.47948e15	−0.820423
$695$	1.15090e15	0.269229
$696$	0	0
$697$	−6.34650e15	−1.46135
$698$	−8.08431e14	−0.184688
$699$	0	0
$700$	1.05195e15	0.236567
$701$	1.80932e15	0.403707	0.201853	−	0.979416i	$-0.435304\pi$
$701$	1.80932e15	0.403707	0.201853	+	0.979416i	$0.435304\pi$
$702$	0	0
$703$	9.37954e15	2.06029
$704$	−4.44890e14	−0.0969623
$705$	0	0
$706$	−3.73722e15	−0.801905
$707$	2.99994e15	0.638713
$708$	0	0
$709$	1.01905e14	0.0213621	0.0106810	−	0.999943i	$-0.496600\pi$
$709$	1.01905e14	0.0213621	0.0106810	+	0.999943i	$0.496600\pi$
$710$	−3.75853e15	−0.781803
$711$	0	0
$712$	−3.69909e14	−0.0757625
$713$	−8.39663e15	−1.70652
$714$	0	0
$715$	−1.14412e15	−0.228976
$716$	4.21465e15	0.837027
$717$	0	0
$718$	−1.98553e15	−0.388322
$719$	−4.32247e15	−0.838925	−0.419463	−	0.907773i	$-0.637782\pi$
$719$	−4.32247e15	−0.838925	−0.419463	+	0.907773i	$0.637782\pi$
$720$	0	0
$721$	−8.64049e15	−1.65156
$722$	1.72529e15	0.327270
$723$	0	0
$724$	−1.51149e15	−0.282386
$725$	−2.45411e15	−0.455025
$726$	0	0
$727$	2.02959e15	0.370654	0.185327	−	0.982677i	$-0.440666\pi$
$727$	2.02959e15	0.370654	0.185327	+	0.982677i	$0.440666\pi$
$728$	8.40334e14	0.152310
$729$	0	0
$730$	4.73706e15	0.845733
$731$	−1.34497e15	−0.238323
$732$	0	0
$733$	−7.80111e15	−1.36171	−0.680855	−	0.732419i	$-0.738391\pi$
$733$	−7.80111e15	−1.36171	−0.680855	+	0.732419i	$0.738391\pi$
$734$	−4.35495e15	−0.754493
$735$	0	0
$736$	−1.97152e15	−0.336491
$737$	9.77287e14	0.165558
$738$	0	0
$739$	−9.59860e14	−0.160200	−0.0801001	−	0.996787i	$-0.525524\pi$
$739$	−9.59860e14	−0.160200	−0.0801001	+	0.996787i	$0.525524\pi$
$740$	−3.88485e15	−0.643575
$741$	0	0
$742$	7.71587e15	1.25940
$743$	1.43941e15	0.233209	0.116605	−	0.993178i	$-0.462799\pi$
$743$	1.43941e15	0.233209	0.116605	+	0.993178i	$0.462799\pi$
$744$	0	0
$745$	5.29036e15	0.844551
$746$	−4.52044e15	−0.716336
$747$	0	0
$748$	4.03065e15	0.629384
$749$	−4.30431e15	−0.667197
$750$	0	0
$751$	8.64808e15	1.32099	0.660496	−	0.750829i	$-0.270346\pi$
$751$	8.64808e15	1.32099	0.660496	+	0.750829i	$0.270346\pi$
$752$	−7.64659e14	−0.115950
$753$	0	0
$754$	−1.96044e15	−0.292961
$755$	−6.09986e15	−0.904923
$756$	0	0
$757$	−4.64600e15	−0.679285	−0.339643	−	0.940555i	$-0.610306\pi$
$757$	−4.64600e15	−0.679285	−0.339643	+	0.940555i	$0.610306\pi$
$758$	5.54995e15	0.805579
$759$	0	0
$760$	−2.25853e15	−0.323110
$761$	−4.02578e15	−0.571788	−0.285894	−	0.958261i	$-0.592290\pi$
$761$	−4.02578e15	−0.571788	−0.285894	+	0.958261i	$0.592290\pi$
$762$	0	0
$763$	−4.45175e14	−0.0623226
$764$	−1.17253e15	−0.162971
$765$	0	0
$766$	5.89850e15	0.808134
$767$	7.36367e14	0.100166
$768$	0	0
$769$	−2.43691e14	−0.0326772	−0.0163386	−	0.999867i	$-0.505201\pi$
$769$	−2.43691e14	−0.0326772	−0.0163386	+	0.999867i	$0.505201\pi$
$770$	−3.43282e15	−0.457038
$771$	0	0
$772$	4.72247e15	0.619833
$773$	3.77235e15	0.491614	0.245807	−	0.969319i	$-0.420947\pi$
$773$	3.77235e15	0.491614	0.245807	+	0.969319i	$0.420947\pi$
$774$	0	0
$775$	−2.99387e15	−0.384656
$776$	−3.79206e15	−0.483766
$777$	0	0
$778$	−2.54400e15	−0.319984
$779$	8.72076e15	1.08918
$780$	0	0
$781$	−9.21694e15	−1.13503
$782$	1.78617e16	2.18417
$783$	0	0
$784$	4.47954e14	0.0540126
$785$	1.13519e16	1.35921
$786$	0	0
$787$	−8.96483e15	−1.05848	−0.529238	−	0.848474i	$-0.677522\pi$
$787$	−8.96483e15	−1.05848	−0.529238	+	0.848474i	$0.677522\pi$
$788$	3.39694e14	0.0398285
$789$	0	0
$790$	3.49495e15	0.404102
$791$	8.41785e13	0.00966564
$792$	0	0
$793$	1.68574e15	0.190892
$794$	2.79420e15	0.314228
$795$	0	0
$796$	3.93970e15	0.436959
$797$	−7.85740e15	−0.865482	−0.432741	−	0.901518i	$-0.642453\pi$
$797$	−7.85740e15	−0.865482	−0.432741	+	0.901518i	$0.642453\pi$
$798$	0	0
$799$	6.92771e15	0.752632
$800$	−7.02956e14	−0.0758461
$801$	0	0
$802$	5.68830e15	0.605374
$803$	1.16166e16	1.22784
$804$	0	0
$805$	−1.52125e16	−1.58607
$806$	−2.39161e15	−0.247655
$807$	0	0
$808$	−2.00469e15	−0.204779
$809$	9.92038e15	1.00650	0.503248	−	0.864142i	$-0.332138\pi$
$809$	9.92038e15	1.00650	0.503248	+	0.864142i	$0.332138\pi$
$810$	0	0
$811$	−3.16056e15	−0.316336	−0.158168	−	0.987412i	$-0.550559\pi$
$811$	−3.16056e15	−0.316336	−0.158168	+	0.987412i	$0.550559\pi$
$812$	−5.88208e15	−0.584754
$813$	0	0
$814$	−9.52670e15	−0.934347
$815$	−1.10216e16	−1.07369
$816$	0	0
$817$	1.84812e15	0.177627
$818$	−4.05516e15	−0.387138
$819$	0	0
$820$	−3.61199e15	−0.340228
$821$	−1.95351e15	−0.182780	−0.0913900	−	0.995815i	$-0.529131\pi$
$821$	−1.95351e15	−0.182780	−0.0913900	+	0.995815i	$0.529131\pi$
$822$	0	0
$823$	1.32212e16	1.22059	0.610296	−	0.792174i	$-0.291051\pi$
$823$	1.32212e16	1.22059	0.610296	+	0.792174i	$0.291051\pi$
$824$	5.77395e15	0.529509
$825$	0	0
$826$	2.20939e15	0.199932
$827$	2.05440e16	1.84674	0.923368	−	0.383916i	$-0.125425\pi$
$827$	2.05440e16	1.84674	0.923368	+	0.383916i	$0.125425\pi$
$828$	0	0
$829$	1.33785e16	1.18675	0.593375	−	0.804926i	$-0.297795\pi$
$829$	1.33785e16	1.18675	0.593375	+	0.804926i	$0.297795\pi$
$830$	6.38983e15	0.563067
$831$	0	0
$832$	−5.61548e14	−0.0488324
$833$	−4.05841e15	−0.350597
$834$	0	0
$835$	2.62851e14	0.0224096
$836$	−5.53853e15	−0.469094
$837$	0	0
$838$	−1.33590e16	−1.11669
$839$	−1.29496e16	−1.07539	−0.537695	−	0.843140i	$-0.680705\pi$
$839$	−1.29496e16	−1.07539	−0.537695	+	0.843140i	$0.680705\pi$
$840$	0	0
$841$	1.52196e15	0.124746
$842$	−2.13224e15	−0.173628
$843$	0	0
$844$	−4.87966e15	−0.392199
$845$	8.01847e15	0.640295
$846$	0	0
$847$	5.57232e15	0.439215
$848$	−5.15608e15	−0.403778
$849$	0	0
$850$	6.36869e15	0.492319
$851$	−4.22174e16	−3.24249
$852$	0	0
$853$	4.94214e14	0.0374710	0.0187355	−	0.999824i	$-0.494036\pi$
$853$	4.94214e14	0.0374710	0.0187355	+	0.999824i	$0.494036\pi$
$854$	5.05789e15	0.381023
$855$	0	0
$856$	2.87633e15	0.213911
$857$	−8.07998e15	−0.597057	−0.298528	−	0.954401i	$-0.596496\pi$
$857$	−8.07998e15	−0.597057	−0.298528	+	0.954401i	$0.596496\pi$
$858$	0	0
$859$	1.88003e16	1.37152	0.685761	−	0.727827i	$-0.259469\pi$
$859$	1.88003e16	1.37152	0.685761	+	0.727827i	$0.259469\pi$
$860$	−7.65461e14	−0.0554858
$861$	0	0
$862$	5.94204e15	0.425251
$863$	1.30928e16	0.931054	0.465527	−	0.885034i	$-0.345865\pi$
$863$	1.30928e16	0.931054	0.465527	+	0.885034i	$0.345865\pi$
$864$	0	0
$865$	−1.91799e15	−0.134666
$866$	−1.06190e16	−0.740855
$867$	0	0
$868$	−7.17578e15	−0.494323
$869$	8.57056e15	0.586679
$870$	0	0
$871$	1.23355e15	0.0833789
$872$	2.97485e14	0.0199814
$873$	0	0
$874$	−2.45439e16	−1.62791
$875$	−1.80662e16	−1.19075
$876$	0	0
$877$	8.26079e15	0.537680	0.268840	−	0.963185i	$-0.413360\pi$
$877$	8.26079e15	0.537680	0.268840	+	0.963185i	$0.413360\pi$
$878$	1.63668e15	0.105863
$879$	0	0
$880$	2.29396e15	0.146532
$881$	−2.60109e16	−1.65116	−0.825578	−	0.564287i	$-0.809151\pi$
$881$	−2.60109e16	−1.65116	−0.825578	+	0.564287i	$0.809151\pi$
$882$	0	0
$883$	−2.29371e16	−1.43799	−0.718994	−	0.695016i	$-0.755397\pi$
$883$	−2.29371e16	−1.43799	−0.718994	+	0.695016i	$0.755397\pi$
$884$	5.08755e15	0.316972
$885$	0	0
$886$	−3.00523e15	−0.184924
$887$	2.41118e16	1.47452	0.737259	−	0.675610i	$-0.236120\pi$
$887$	2.41118e16	1.47452	0.737259	+	0.675610i	$0.236120\pi$
$888$	0	0
$889$	−4.88753e15	−0.295209
$890$	1.90734e15	0.114494
$891$	0	0
$892$	−6.90646e14	−0.0409495
$893$	−9.51940e15	−0.560953
$894$	0	0
$895$	−2.17318e16	−1.26494
$896$	−1.68486e15	−0.0974700
$897$	0	0
$898$	−2.60762e15	−0.149013
$899$	1.67406e16	0.950806
$900$	0	0
$901$	4.67135e16	2.62093
$902$	−8.85758e15	−0.493946
$903$	0	0
$904$	−5.62517e13	−0.00309891
$905$	7.79362e15	0.426749
$906$	0	0
$907$	−2.65410e16	−1.43574	−0.717872	−	0.696175i	$-0.754884\pi$
$907$	−2.65410e16	−1.43574	−0.717872	+	0.696175i	$0.754884\pi$
$908$	9.26337e15	0.498077
$909$	0	0
$910$	−4.33297e15	−0.230175
$911$	2.34817e16	1.23988	0.619939	−	0.784650i	$-0.287157\pi$
$911$	2.34817e16	1.23988	0.619939	+	0.784650i	$0.287157\pi$
$912$	0	0
$913$	1.56696e16	0.817464
$914$	−1.56497e16	−0.811526
$915$	0	0
$916$	5.33732e15	0.273463
$917$	3.70659e16	1.88774
$918$	0	0
$919$	4.23763e15	0.213249	0.106625	−	0.994299i	$-0.465996\pi$
$919$	4.23763e15	0.213249	0.106625	+	0.994299i	$0.465996\pi$
$920$	1.01656e16	0.508513
$921$	0	0
$922$	2.73137e16	1.35008
$923$	−1.16338e16	−0.571626
$924$	0	0
$925$	−1.50528e16	−0.730868
$926$	1.35557e16	0.654279
$927$	0	0
$928$	3.93066e15	0.187479
$929$	2.07347e16	0.983133	0.491566	−	0.870840i	$-0.336425\pi$
$929$	2.07347e16	0.983133	0.491566	+	0.870840i	$0.336425\pi$
$930$	0	0
$931$	5.57668e15	0.261308
$932$	1.91664e16	0.892797
$933$	0	0
$934$	2.05944e16	0.948071
$935$	−2.07830e16	−0.951141
$936$	0	0
$937$	−7.69850e15	−0.348208	−0.174104	−	0.984727i	$-0.555703\pi$
$937$	−7.69850e15	−0.348208	−0.174104	+	0.984727i	$0.555703\pi$
$938$	3.70113e15	0.166425
$939$	0	0
$940$	3.94277e15	0.175226
$941$	−3.71019e16	−1.63928	−0.819639	−	0.572880i	$-0.805826\pi$
$941$	−3.71019e16	−1.63928	−0.819639	+	0.572880i	$0.805826\pi$
$942$	0	0
$943$	−3.92522e16	−1.71415
$944$	−1.47641e15	−0.0641005
$945$	0	0
$946$	−1.87712e15	−0.0805547
$947$	2.86020e16	1.22031	0.610156	−	0.792281i	$-0.291107\pi$
$947$	2.86020e16	1.22031	0.610156	+	0.792281i	$0.291107\pi$
$948$	0	0
$949$	1.46626e16	0.618369
$950$	−8.75125e15	−0.366936
$951$	0	0
$952$	1.52647e16	0.632680
$953$	−7.65574e15	−0.315483	−0.157742	−	0.987480i	$-0.550421\pi$
$953$	−7.65574e15	−0.315483	−0.157742	+	0.987480i	$0.550421\pi$
$954$	0	0
$955$	6.04585e15	0.246285
$956$	−5.19303e15	−0.210331
$957$	0	0
$958$	−6.92018e15	−0.277081
$959$	2.52447e16	1.00501
$960$	0	0
$961$	−4.98602e15	−0.196235
$962$	−1.20248e16	−0.470559
$963$	0	0
$964$	−2.07842e16	−0.804099
$965$	−2.43503e16	−0.936706
$966$	0	0
$967$	−2.60484e16	−0.990684	−0.495342	−	0.868698i	$-0.664957\pi$
$967$	−2.60484e16	−0.990684	−0.495342	+	0.868698i	$0.664957\pi$
$968$	−3.72367e15	−0.140817
$969$	0	0
$970$	1.95528e16	0.731079
$971$	−1.56465e16	−0.581717	−0.290859	−	0.956766i	$-0.593941\pi$
$971$	−1.56465e16	−0.581717	−0.290859	+	0.956766i	$0.593941\pi$
$972$	0	0
$973$	1.06886e16	0.392915
$974$	2.27096e16	0.830110
$975$	0	0
$976$	−3.37990e15	−0.122160
$977$	2.62552e16	0.943615	0.471808	−	0.881702i	$-0.343602\pi$
$977$	2.62552e16	0.943615	0.471808	+	0.881702i	$0.343602\pi$
$978$	0	0
$979$	4.67732e15	0.166223
$980$	−2.30976e15	−0.0816252
$981$	0	0
$982$	3.17767e16	1.11044
$983$	−3.52027e16	−1.22330	−0.611648	−	0.791130i	$-0.709493\pi$
$983$	−3.52027e16	−1.22330	−0.611648	+	0.791130i	$0.709493\pi$
$984$	0	0
$985$	−1.75155e15	−0.0601898
$986$	−3.56113e16	−1.21693
$987$	0	0
$988$	−6.99083e15	−0.236246
$989$	−8.31841e15	−0.279551
$990$	0	0
$991$	−4.08271e16	−1.35689	−0.678443	−	0.734653i	$-0.737345\pi$
$991$	−4.08271e16	−1.35689	−0.678443	+	0.734653i	$0.737345\pi$
$992$	4.79517e15	0.158486
$993$	0	0
$994$	−3.49059e16	−1.14097
$995$	−2.03141e16	−0.660343
$996$	0	0
$997$	−3.74291e16	−1.20333	−0.601666	−	0.798747i	$-0.705496\pi$
$997$	−3.74291e16	−1.20333	−0.601666	+	0.798747i	$0.705496\pi$
$998$	−3.33009e16	−1.06473
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.12.a.d.1.1	yes	1
3.2	odd	2		18.12.a.b.1.1	✓	1
4.3	odd	2		144.12.a.c.1.1		1
9.2	odd	6		162.12.c.h.109.1		2
9.4	even	3		162.12.c.c.55.1		2
9.5	odd	6		162.12.c.h.55.1		2
9.7	even	3		162.12.c.c.109.1		2
12.11	even	2		144.12.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.12.a.b.1.1	✓	1	3.2	odd	2
18.12.a.d.1.1	yes	1	1.1	even	1	trivial
144.12.a.c.1.1		1	4.3	odd	2
144.12.a.k.1.1		1	12.11	even	2
162.12.c.c.55.1		2	9.4	even	3
162.12.c.c.109.1		2	9.7	even	3
162.12.c.h.55.1		2	9.5	odd	6
162.12.c.h.109.1		2	9.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more