LMFDB - Embedded newform 342.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,8,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("342.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	342.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:	$106.835678716$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} +64.0000 q^{4} -450.000 q^{5} -568.000 q^{7} +512.000 q^{8} +O(q^{10})$

\(q+8.00000 q^{2} +64.0000 q^{4} -450.000 q^{5} -568.000 q^{7} +512.000 q^{8} -3600.00 q^{10} +5880.00 q^{11} +2858.00 q^{13} -4544.00 q^{14} +4096.00 q^{16} +8958.00 q^{17} +6859.00 q^{19} -28800.0 q^{20} +47040.0 q^{22} -47832.0 q^{23} +124375. q^{25} +22864.0 q^{26} -36352.0 q^{28} +94806.0 q^{29} -26428.0 q^{31} +32768.0 q^{32} +71664.0 q^{34} +255600. q^{35} +93242.0 q^{37} +54872.0 q^{38} -230400. q^{40} +44514.0 q^{41} -944452. q^{43} +376320. q^{44} -382656. q^{46} +713448. q^{47} -500919. q^{49} +995000. q^{50} +182912. q^{52} -649218. q^{53} -2.64600e6 q^{55} -290816. q^{56} +758448. q^{58} -2.05945e6 q^{59} +955574. q^{61} -211424. q^{62} +262144. q^{64} -1.28610e6 q^{65} -2.92644e6 q^{67} +573312. q^{68} +2.04480e6 q^{70} +2.61984e6 q^{71} -6.30828e6 q^{73} +745936. q^{74} +438976. q^{76} -3.33984e6 q^{77} -7.67710e6 q^{79} -1.84320e6 q^{80} +356112. q^{82} +413616. q^{83} -4.03110e6 q^{85} -7.55562e6 q^{86} +3.01056e6 q^{88} +6.21515e6 q^{89} -1.62334e6 q^{91} -3.06125e6 q^{92} +5.70758e6 q^{94} -3.08655e6 q^{95} +6.96365e6 q^{97} -4.00735e6 q^{98} +O(q^{100})\)

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	−450.000	−1.60997	−0.804984	−	0.593296i	$-0.797826\pi$
$5$	−450.000	−1.60997	−0.804984	+	0.593296i	$0.797826\pi$
$6$	0	0
$7$	−568.000	−0.625900	−0.312950	−	0.949770i	$-0.601317\pi$
$7$	−568.000	−0.625900	−0.312950	+	0.949770i	$0.601317\pi$
$8$	512.000	0.353553
$9$	0	0
$10$	−3600.00	−1.13842
$11$	5880.00	1.33200	0.665998	−	0.745954i	$-0.268006\pi$
$11$	5880.00	1.33200	0.665998	+	0.745954i	$0.268006\pi$
$12$	0	0
$13$	2858.00	0.360795	0.180397	−	0.983594i	$-0.442262\pi$
$13$	2858.00	0.360795	0.180397	+	0.983594i	$0.442262\pi$
$14$	−4544.00	−0.442578
$15$	0	0
$16$	4096.00	0.250000
$17$	8958.00	0.442221	0.221111	−	0.975249i	$-0.429032\pi$
$17$	8958.00	0.442221	0.221111	+	0.975249i	$0.429032\pi$
$18$	0	0
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	−28800.0	−0.804984
$21$	0	0
$22$	47040.0	0.941863
$23$	−47832.0	−0.819731	−0.409865	−	0.912146i	$-0.634424\pi$
$23$	−47832.0	−0.819731	−0.409865	+	0.912146i	$0.634424\pi$
$24$	0	0
$25$	124375.	1.59200
$26$	22864.0	0.255121
$27$	0	0
$28$	−36352.0	−0.312950
$29$	94806.0	0.721843	0.360922	−	0.932596i	$-0.382462\pi$
$29$	94806.0	0.721843	0.360922	+	0.932596i	$0.382462\pi$
$30$	0	0
$31$	−26428.0	−0.159330	−0.0796651	−	0.996822i	$-0.525385\pi$
$31$	−26428.0	−0.159330	−0.0796651	+	0.996822i	$0.525385\pi$
$32$	32768.0	0.176777
$33$	0	0
$34$	71664.0	0.312698
$35$	255600.	1.00768
$36$	0	0
$37$	93242.0	0.302626	0.151313	−	0.988486i	$-0.451650\pi$
$37$	93242.0	0.302626	0.151313	+	0.988486i	$0.451650\pi$
$38$	54872.0	0.162221
$39$	0	0
$40$	−230400.	−0.569210
$41$	44514.0	0.100868	0.0504340	−	0.998727i	$-0.483940\pi$
$41$	44514.0	0.100868	0.0504340	+	0.998727i	$0.483940\pi$
$42$	0	0
$43$	−944452.	−1.81151	−0.905754	−	0.423804i	$-0.860695\pi$
$43$	−944452.	−1.81151	−0.905754	+	0.423804i	$0.860695\pi$
$44$	376320.	0.665998
$45$	0	0
$46$	−382656.	−0.579637
$47$	713448.	1.00235	0.501175	−	0.865346i	$-0.332901\pi$
$47$	713448.	1.00235	0.501175	+	0.865346i	$0.332901\pi$
$48$	0	0
$49$	−500919.	−0.608249
$50$	995000.	1.12571
$51$	0	0
$52$	182912.	0.180397
$53$	−649218.	−0.598997	−0.299499	−	0.954097i	$-0.596819\pi$
$53$	−649218.	−0.598997	−0.299499	+	0.954097i	$0.596819\pi$
$54$	0	0
$55$	−2.64600e6	−2.14447
$56$	−290816.	−0.221289
$57$	0	0
$58$	758448.	0.510420
$59$	−2.05945e6	−1.30548	−0.652739	−	0.757583i	$-0.726380\pi$
$59$	−2.05945e6	−1.30548	−0.652739	+	0.757583i	$0.726380\pi$
$60$	0	0
$61$	955574.	0.539026	0.269513	−	0.962997i	$-0.413137\pi$
$61$	955574.	0.539026	0.269513	+	0.962997i	$0.413137\pi$
$62$	−211424.	−0.112664
$63$	0	0
$64$	262144.	0.125000
$65$	−1.28610e6	−0.580869
$66$	0	0
$67$	−2.92644e6	−1.18872	−0.594358	−	0.804200i	$-0.702594\pi$
$67$	−2.92644e6	−1.18872	−0.594358	+	0.804200i	$0.702594\pi$
$68$	573312.	0.221111
$69$	0	0
$70$	2.04480e6	0.712537
$71$	2.61984e6	0.868701	0.434351	−	0.900744i	$-0.356978\pi$
$71$	2.61984e6	0.868701	0.434351	+	0.900744i	$0.356978\pi$
$72$	0	0
$73$	−6.30828e6	−1.89793	−0.948966	−	0.315377i	$-0.897869\pi$
$73$	−6.30828e6	−1.89793	−0.948966	+	0.315377i	$0.897869\pi$
$74$	745936.	0.213989
$75$	0	0
$76$	438976.	0.114708
$77$	−3.33984e6	−0.833697
$78$	0	0
$79$	−7.67710e6	−1.75187	−0.875936	−	0.482427i	$-0.839755\pi$
$79$	−7.67710e6	−1.75187	−0.875936	+	0.482427i	$0.839755\pi$
$80$	−1.84320e6	−0.402492
$81$	0	0
$82$	356112.	0.0713244
$83$	413616.	0.0794006	0.0397003	−	0.999212i	$-0.487360\pi$
$83$	413616.	0.0794006	0.0397003	+	0.999212i	$0.487360\pi$
$84$	0	0
$85$	−4.03110e6	−0.711963
$86$	−7.55562e6	−1.28093
$87$	0	0
$88$	3.01056e6	0.470932
$89$	6.21515e6	0.934516	0.467258	−	0.884121i	$-0.345242\pi$
$89$	6.21515e6	0.934516	0.467258	+	0.884121i	$0.345242\pi$
$90$	0	0
$91$	−1.62334e6	−0.225822
$92$	−3.06125e6	−0.409865
$93$	0	0
$94$	5.70758e6	0.708769
$95$	−3.08655e6	−0.369352
$96$	0	0
$97$	6.96365e6	0.774704	0.387352	−	0.921932i	$-0.373390\pi$
$97$	6.96365e6	0.774704	0.387352	+	0.921932i	$0.373390\pi$
$98$	−4.00735e6	−0.430097
$99$	0	0
$100$	7.96000e6	0.796000
$101$	−4.83561e6	−0.467010	−0.233505	−	0.972356i	$-0.575020\pi$
$101$	−4.83561e6	−0.467010	−0.233505	+	0.972356i	$0.575020\pi$
$102$	0	0
$103$	−9.62338e6	−0.867755	−0.433878	−	0.900972i	$-0.642855\pi$
$103$	−9.62338e6	−0.867755	−0.433878	+	0.900972i	$0.642855\pi$
$104$	1.46330e6	0.127560
$105$	0	0
$106$	−5.19374e6	−0.423555
$107$	8.21452e6	0.648244	0.324122	−	0.946015i	$-0.394931\pi$
$107$	8.21452e6	0.648244	0.324122	+	0.946015i	$0.394931\pi$
$108$	0	0
$109$	−1.09532e7	−0.810119	−0.405060	−	0.914290i	$-0.632749\pi$
$109$	−1.09532e7	−0.810119	−0.405060	+	0.914290i	$0.632749\pi$
$110$	−2.11680e7	−1.51637
$111$	0	0
$112$	−2.32653e6	−0.156475
$113$	−2.07110e7	−1.35029	−0.675146	−	0.737684i	$-0.735919\pi$
$113$	−2.07110e7	−1.35029	−0.675146	+	0.737684i	$0.735919\pi$
$114$	0	0
$115$	2.15244e7	1.31974
$116$	6.06758e6	0.360922
$117$	0	0
$118$	−1.64756e7	−0.923113
$119$	−5.08814e6	−0.276787
$120$	0	0
$121$	1.50872e7	0.774213
$122$	7.64459e6	0.381149
$123$	0	0
$124$	−1.69139e6	−0.0796651
$125$	−2.08125e7	−0.953102
$126$	0	0
$127$	3.59771e7	1.55852	0.779262	−	0.626698i	$-0.215594\pi$
$127$	3.59771e7	1.55852	0.779262	+	0.626698i	$0.215594\pi$
$128$	2.09715e6	0.0883883
$129$	0	0
$130$	−1.02888e7	−0.410736
$131$	4.36542e7	1.69659	0.848295	−	0.529524i	$-0.177629\pi$
$131$	4.36542e7	1.69659	0.848295	+	0.529524i	$0.177629\pi$
$132$	0	0
$133$	−3.89591e6	−0.143591
$134$	−2.34116e7	−0.840550
$135$	0	0
$136$	4.58650e6	0.156349
$137$	−1.35694e7	−0.450855	−0.225428	−	0.974260i	$-0.572378\pi$
$137$	−1.35694e7	−0.450855	−0.225428	+	0.974260i	$0.572378\pi$
$138$	0	0
$139$	−5.21660e7	−1.64754	−0.823770	−	0.566924i	$-0.808133\pi$
$139$	−5.21660e7	−1.64754	−0.823770	+	0.566924i	$0.808133\pi$
$140$	1.63584e7	0.503840
$141$	0	0
$142$	2.09587e7	0.614265
$143$	1.68050e7	0.480577
$144$	0	0
$145$	−4.26627e7	−1.16215
$146$	−5.04662e7	−1.34204
$147$	0	0
$148$	5.96749e6	0.151313
$149$	1.12219e7	0.277917	0.138958	−	0.990298i	$-0.455625\pi$
$149$	1.12219e7	0.277917	0.138958	+	0.990298i	$0.455625\pi$
$150$	0	0
$151$	3.10269e7	0.733362	0.366681	−	0.930347i	$-0.380494\pi$
$151$	3.10269e7	0.733362	0.366681	+	0.930347i	$0.380494\pi$
$152$	3.51181e6	0.0811107
$153$	0	0
$154$	−2.67187e7	−0.589513
$155$	1.18926e7	0.256517
$156$	0	0
$157$	−3.52901e7	−0.727786	−0.363893	−	0.931441i	$-0.618553\pi$
$157$	−3.52901e7	−0.727786	−0.363893	+	0.931441i	$0.618553\pi$
$158$	−6.14168e7	−1.23876
$159$	0	0
$160$	−1.47456e7	−0.284605
$161$	2.71686e7	0.513070
$162$	0	0
$163$	−7.62232e7	−1.37858	−0.689288	−	0.724488i	$-0.742076\pi$
$163$	−7.62232e7	−1.37858	−0.689288	+	0.724488i	$0.742076\pi$
$164$	2.84890e6	0.0504340
$165$	0	0
$166$	3.30893e6	0.0561447
$167$	−8.94954e7	−1.48694	−0.743469	−	0.668770i	$-0.766821\pi$
$167$	−8.94954e7	−1.48694	−0.743469	+	0.668770i	$0.766821\pi$
$168$	0	0
$169$	−5.45804e7	−0.869827
$170$	−3.22488e7	−0.503434
$171$	0	0
$172$	−6.04449e7	−0.905754
$173$	1.45296e7	0.213350	0.106675	−	0.994294i	$-0.465980\pi$
$173$	1.45296e7	0.213350	0.106675	+	0.994294i	$0.465980\pi$
$174$	0	0
$175$	−7.06450e7	−0.996433
$176$	2.40845e7	0.332999
$177$	0	0
$178$	4.97212e7	0.660803
$179$	−8.80139e6	−0.114701	−0.0573503	−	0.998354i	$-0.518265\pi$
$179$	−8.80139e6	−0.114701	−0.0573503	+	0.998354i	$0.518265\pi$
$180$	0	0
$181$	5.93503e6	0.0743958	0.0371979	−	0.999308i	$-0.488157\pi$
$181$	5.93503e6	0.0743958	0.0371979	+	0.999308i	$0.488157\pi$
$182$	−1.29868e7	−0.159680
$183$	0	0
$184$	−2.44900e7	−0.289819
$185$	−4.19589e7	−0.487218
$186$	0	0
$187$	5.26730e7	0.589037
$188$	4.56607e7	0.501175
$189$	0	0
$190$	−2.46924e7	−0.261171
$191$	−1.41230e7	−0.146659	−0.0733296	−	0.997308i	$-0.523362\pi$
$191$	−1.41230e7	−0.146659	−0.0733296	+	0.997308i	$0.523362\pi$
$192$	0	0
$193$	−1.11299e8	−1.11440	−0.557198	−	0.830380i	$-0.688123\pi$
$193$	−1.11299e8	−1.11440	−0.557198	+	0.830380i	$0.688123\pi$
$194$	5.57092e7	0.547799
$195$	0	0
$196$	−3.20588e7	−0.304124
$197$	9.03497e7	0.841967	0.420983	−	0.907068i	$-0.361685\pi$
$197$	9.03497e7	0.841967	0.420983	+	0.907068i	$0.361685\pi$
$198$	0	0
$199$	6.83998e7	0.615275	0.307638	−	0.951504i	$-0.400462\pi$
$199$	6.83998e7	0.615275	0.307638	+	0.951504i	$0.400462\pi$
$200$	6.36800e7	0.562857
$201$	0	0
$202$	−3.86849e7	−0.330226
$203$	−5.38498e7	−0.451802
$204$	0	0
$205$	−2.00313e7	−0.162394
$206$	−7.69870e7	−0.613596
$207$	0	0
$208$	1.17064e7	0.0901987
$209$	4.03309e7	0.305581
$210$	0	0
$211$	−5.84777e7	−0.428550	−0.214275	−	0.976773i	$-0.568739\pi$
$211$	−5.84777e7	−0.428550	−0.214275	+	0.976773i	$0.568739\pi$
$212$	−4.15500e7	−0.299499
$213$	0	0
$214$	6.57161e7	0.458378
$215$	4.25003e8	2.91647
$216$	0	0
$217$	1.50111e7	0.0997249
$218$	−8.76257e7	−0.572841
$219$	0	0
$220$	−1.69344e8	−1.07224
$221$	2.56020e7	0.159551
$222$	0	0
$223$	3.23189e8	1.95160	0.975798	−	0.218673i	$-0.0701729\pi$
$223$	3.23189e8	1.95160	0.975798	+	0.218673i	$0.0701729\pi$
$224$	−1.86122e7	−0.110645
$225$	0	0
$226$	−1.65688e8	−0.954800
$227$	−2.33317e8	−1.32390	−0.661952	−	0.749546i	$-0.730272\pi$
$227$	−2.33317e8	−1.32390	−0.661952	+	0.749546i	$0.730272\pi$
$228$	0	0
$229$	1.51133e8	0.831639	0.415820	−	0.909447i	$-0.363495\pi$
$229$	1.51133e8	0.831639	0.415820	+	0.909447i	$0.363495\pi$
$230$	1.72195e8	0.933198
$231$	0	0
$232$	4.85407e7	0.255210
$233$	−1.45570e8	−0.753920	−0.376960	−	0.926230i	$-0.623031\pi$
$233$	−1.45570e8	−0.753920	−0.376960	+	0.926230i	$0.623031\pi$
$234$	0	0
$235$	−3.21052e8	−1.61375
$236$	−1.31805e8	−0.652739
$237$	0	0
$238$	−4.07052e7	−0.195718
$239$	3.17260e7	0.150322	0.0751611	−	0.997171i	$-0.476053\pi$
$239$	3.17260e7	0.150322	0.0751611	+	0.997171i	$0.476053\pi$
$240$	0	0
$241$	−1.50338e8	−0.691846	−0.345923	−	0.938263i	$-0.612434\pi$
$241$	−1.50338e8	−0.691846	−0.345923	+	0.938263i	$0.612434\pi$
$242$	1.20698e8	0.547452
$243$	0	0
$244$	6.11567e7	0.269513
$245$	2.25414e8	0.979262
$246$	0	0
$247$	1.96030e7	0.0827720
$248$	−1.35311e7	−0.0563318
$249$	0	0
$250$	−1.66500e8	−0.673945
$251$	−7.71175e6	−0.0307819	−0.0153909	−	0.999882i	$-0.504899\pi$
$251$	−7.71175e6	−0.0307819	−0.0153909	+	0.999882i	$0.504899\pi$
$252$	0	0
$253$	−2.81252e8	−1.09188
$254$	2.87817e8	1.10204
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	3.93750e7	0.144695	0.0723477	−	0.997379i	$-0.476951\pi$
$257$	3.93750e7	0.144695	0.0723477	+	0.997379i	$0.476951\pi$
$258$	0	0
$259$	−5.29615e7	−0.189413
$260$	−8.23104e7	−0.290434
$261$	0	0
$262$	3.49234e8	1.19967
$263$	−3.27740e8	−1.11092	−0.555461	−	0.831542i	$-0.687458\pi$
$263$	−3.27740e8	−1.11092	−0.555461	+	0.831542i	$0.687458\pi$
$264$	0	0
$265$	2.92148e8	0.964367
$266$	−3.11673e7	−0.101534
$267$	0	0
$268$	−1.87292e8	−0.594358
$269$	−2.12975e8	−0.667109	−0.333554	−	0.942731i	$-0.608248\pi$
$269$	−2.12975e8	−0.667109	−0.333554	+	0.942731i	$0.608248\pi$
$270$	0	0
$271$	−5.86684e8	−1.79065	−0.895327	−	0.445410i	$-0.853058\pi$
$271$	−5.86684e8	−1.79065	−0.895327	+	0.445410i	$0.853058\pi$
$272$	3.66920e7	0.110555
$273$	0	0
$274$	−1.08555e8	−0.318803
$275$	7.31325e8	2.12054
$276$	0	0
$277$	1.40654e8	0.397624	0.198812	−	0.980038i	$-0.436292\pi$
$277$	1.40654e8	0.397624	0.198812	+	0.980038i	$0.436292\pi$
$278$	−4.17328e8	−1.16499
$279$	0	0
$280$	1.30867e8	0.356269
$281$	−8.85611e7	−0.238106	−0.119053	−	0.992888i	$-0.537986\pi$
$281$	−8.85611e7	−0.238106	−0.119053	+	0.992888i	$0.537986\pi$
$282$	0	0
$283$	6.32439e8	1.65869	0.829346	−	0.558735i	$-0.188713\pi$
$283$	6.32439e8	1.65869	0.829346	+	0.558735i	$0.188713\pi$
$284$	1.67670e8	0.434351
$285$	0	0
$286$	1.34440e8	0.339820
$287$	−2.52840e7	−0.0631333
$288$	0	0
$289$	−3.30093e8	−0.804440
$290$	−3.41302e8	−0.821761
$291$	0	0
$292$	−4.03730e8	−0.948966
$293$	−5.00163e8	−1.16165	−0.580825	−	0.814029i	$-0.697270\pi$
$293$	−5.00163e8	−1.16165	−0.580825	+	0.814029i	$0.697270\pi$
$294$	0	0
$295$	9.26753e8	2.10178
$296$	4.77399e7	0.106994
$297$	0	0
$298$	8.97752e7	0.196517
$299$	−1.36704e8	−0.295755
$300$	0	0
$301$	5.36449e8	1.13382
$302$	2.48215e8	0.518565
$303$	0	0
$304$	2.80945e7	0.0573539
$305$	−4.30008e8	−0.867815
$306$	0	0
$307$	−2.96364e8	−0.584577	−0.292288	−	0.956330i	$-0.594417\pi$
$307$	−2.96364e8	−0.584577	−0.292288	+	0.956330i	$0.594417\pi$
$308$	−2.13750e8	−0.416848
$309$	0	0
$310$	9.51408e7	0.181385
$311$	−2.67972e8	−0.505158	−0.252579	−	0.967576i	$-0.581279\pi$
$311$	−2.67972e8	−0.505158	−0.252579	+	0.967576i	$0.581279\pi$
$312$	0	0
$313$	4.55489e8	0.839601	0.419801	−	0.907616i	$-0.362100\pi$
$313$	4.55489e8	0.839601	0.419801	+	0.907616i	$0.362100\pi$
$314$	−2.82321e8	−0.514622
$315$	0	0
$316$	−4.91334e8	−0.875936
$317$	−1.06186e9	−1.87223	−0.936117	−	0.351688i	$-0.885608\pi$
$317$	−1.06186e9	−1.87223	−0.936117	+	0.351688i	$0.885608\pi$
$318$	0	0
$319$	5.57459e8	0.961492
$320$	−1.17965e8	−0.201246
$321$	0	0
$322$	2.17349e8	0.362795
$323$	6.14429e7	0.101453
$324$	0	0
$325$	3.55464e8	0.574386
$326$	−6.09786e8	−0.974800
$327$	0	0
$328$	2.27912e7	0.0356622
$329$	−4.05238e8	−0.627372
$330$	0	0
$331$	−5.14980e8	−0.780535	−0.390268	−	0.920701i	$-0.627618\pi$
$331$	−5.14980e8	−0.780535	−0.390268	+	0.920701i	$0.627618\pi$
$332$	2.64714e7	0.0397003
$333$	0	0
$334$	−7.15963e8	−1.05142
$335$	1.31690e9	1.91380
$336$	0	0
$337$	1.54638e8	0.220095	0.110048	−	0.993926i	$-0.464900\pi$
$337$	1.54638e8	0.220095	0.110048	+	0.993926i	$0.464900\pi$
$338$	−4.36643e8	−0.615061
$339$	0	0
$340$	−2.57990e8	−0.355981
$341$	−1.55397e8	−0.212227
$342$	0	0
$343$	7.52294e8	1.00660
$344$	−4.83559e8	−0.640465
$345$	0	0
$346$	1.16237e8	0.150861
$347$	−3.16102e8	−0.406139	−0.203069	−	0.979164i	$-0.565092\pi$
$347$	−3.16102e8	−0.406139	−0.203069	+	0.979164i	$0.565092\pi$
$348$	0	0
$349$	−8.77724e7	−0.110527	−0.0552636	−	0.998472i	$-0.517600\pi$
$349$	−8.77724e7	−0.110527	−0.0552636	+	0.998472i	$0.517600\pi$
$350$	−5.65160e8	−0.704585
$351$	0	0
$352$	1.92676e8	0.235466
$353$	3.66289e8	0.443213	0.221607	−	0.975136i	$-0.428870\pi$
$353$	3.66289e8	0.443213	0.221607	+	0.975136i	$0.428870\pi$
$354$	0	0
$355$	−1.17893e9	−1.39858
$356$	3.97770e8	0.467258
$357$	0	0
$358$	−7.04111e7	−0.0811055
$359$	6.64292e8	0.757754	0.378877	−	0.925447i	$-0.376310\pi$
$359$	6.64292e8	0.757754	0.378877	+	0.925447i	$0.376310\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	4.74803e7	0.0526057
$363$	0	0
$364$	−1.03894e8	−0.112911
$365$	2.83873e9	3.05561
$366$	0	0
$367$	1.25098e9	1.32105	0.660526	−	0.750803i	$-0.270333\pi$
$367$	1.25098e9	1.32105	0.660526	+	0.750803i	$0.270333\pi$
$368$	−1.95920e8	−0.204933
$369$	0	0
$370$	−3.35671e8	−0.344515
$371$	3.68756e8	0.374913
$372$	0	0
$373$	−6.50061e8	−0.648595	−0.324297	−	0.945955i	$-0.605128\pi$
$373$	−6.50061e8	−0.648595	−0.324297	+	0.945955i	$0.605128\pi$
$374$	4.21384e8	0.416512
$375$	0	0
$376$	3.65285e8	0.354385
$377$	2.70956e8	0.260437
$378$	0	0
$379$	2.37564e8	0.224153	0.112076	−	0.993700i	$-0.464250\pi$
$379$	2.37564e8	0.224153	0.112076	+	0.993700i	$0.464250\pi$
$380$	−1.97539e8	−0.184676
$381$	0	0
$382$	−1.12984e8	−0.103704
$383$	−4.44321e8	−0.404111	−0.202056	−	0.979374i	$-0.564762\pi$
$383$	−4.44321e8	−0.404111	−0.202056	+	0.979374i	$0.564762\pi$
$384$	0	0
$385$	1.50293e9	1.34223
$386$	−8.90389e8	−0.787996
$387$	0	0
$388$	4.45674e8	0.387352
$389$	1.97677e9	1.70268	0.851338	−	0.524618i	$-0.175792\pi$
$389$	1.97677e9	1.70268	0.851338	+	0.524618i	$0.175792\pi$
$390$	0	0
$391$	−4.28479e8	−0.362502
$392$	−2.56471e8	−0.215048
$393$	0	0
$394$	7.22797e8	0.595360
$395$	3.45470e9	2.82046
$396$	0	0
$397$	6.10958e8	0.490055	0.245027	−	0.969516i	$-0.421203\pi$
$397$	6.10958e8	0.490055	0.245027	+	0.969516i	$0.421203\pi$
$398$	5.47199e8	0.435065
$399$	0	0
$400$	5.09440e8	0.398000
$401$	−4.15454e8	−0.321749	−0.160875	−	0.986975i	$-0.551431\pi$
$401$	−4.15454e8	−0.321749	−0.160875	+	0.986975i	$0.551431\pi$
$402$	0	0
$403$	−7.55312e7	−0.0574856
$404$	−3.09479e8	−0.233505
$405$	0	0
$406$	−4.30798e8	−0.319472
$407$	5.48263e8	0.403096
$408$	0	0
$409$	1.79791e9	1.29938	0.649689	−	0.760200i	$-0.274899\pi$
$409$	1.79791e9	1.29938	0.649689	+	0.760200i	$0.274899\pi$
$410$	−1.60250e8	−0.114830
$411$	0	0
$412$	−6.15896e8	−0.433878
$413$	1.16977e9	0.817099
$414$	0	0
$415$	−1.86127e8	−0.127833
$416$	9.36509e7	0.0637801
$417$	0	0
$418$	3.22647e8	0.216078
$419$	2.30123e9	1.52831	0.764154	−	0.645034i	$-0.223157\pi$
$419$	2.30123e9	1.52831	0.764154	+	0.645034i	$0.223157\pi$
$420$	0	0
$421$	2.17533e9	1.42081	0.710406	−	0.703792i	$-0.248511\pi$
$421$	2.17533e9	1.42081	0.710406	+	0.703792i	$0.248511\pi$
$422$	−4.67821e8	−0.303031
$423$	0	0
$424$	−3.32400e8	−0.211778
$425$	1.11415e9	0.704017
$426$	0	0
$427$	−5.42766e8	−0.337377
$428$	5.25729e8	0.324122
$429$	0	0
$430$	3.40003e9	2.06226
$431$	1.00024e9	0.601776	0.300888	−	0.953660i	$-0.402717\pi$
$431$	1.00024e9	0.601776	0.300888	+	0.953660i	$0.402717\pi$
$432$	0	0
$433$	−2.66223e9	−1.57593	−0.787966	−	0.615719i	$-0.788866\pi$
$433$	−2.66223e9	−1.57593	−0.787966	+	0.615719i	$0.788866\pi$
$434$	1.20089e8	0.0705161
$435$	0	0
$436$	−7.01006e8	−0.405060
$437$	−3.28080e8	−0.188059
$438$	0	0
$439$	3.44597e9	1.94395	0.971976	−	0.235080i	$-0.0755354\pi$
$439$	3.44597e9	1.94395	0.971976	+	0.235080i	$0.0755354\pi$
$440$	−1.35475e9	−0.758185
$441$	0	0
$442$	2.04816e8	0.112820
$443$	2.54646e9	1.39163	0.695815	−	0.718221i	$-0.255044\pi$
$443$	2.54646e9	1.39163	0.695815	+	0.718221i	$0.255044\pi$
$444$	0	0
$445$	−2.79682e9	−1.50454
$446$	2.58551e9	1.37999
$447$	0	0
$448$	−1.48898e8	−0.0782375
$449$	3.17403e9	1.65481	0.827407	−	0.561602i	$-0.189815\pi$
$449$	3.17403e9	1.65481	0.827407	+	0.561602i	$0.189815\pi$
$450$	0	0
$451$	2.61742e8	0.134356
$452$	−1.32551e9	−0.675146
$453$	0	0
$454$	−1.86654e9	−0.936141
$455$	7.30505e8	0.363566
$456$	0	0
$457$	−3.33161e9	−1.63285	−0.816427	−	0.577448i	$-0.804049\pi$
$457$	−3.33161e9	−1.63285	−0.816427	+	0.577448i	$0.804049\pi$
$458$	1.20906e9	0.588058
$459$	0	0
$460$	1.37756e9	0.659870
$461$	−3.75510e9	−1.78513	−0.892563	−	0.450923i	$-0.851095\pi$
$461$	−3.75510e9	−1.78513	−0.892563	+	0.450923i	$0.851095\pi$
$462$	0	0
$463$	−7.01070e8	−0.328267	−0.164134	−	0.986438i	$-0.552483\pi$
$463$	−7.01070e8	−0.328267	−0.164134	+	0.986438i	$0.552483\pi$
$464$	3.88325e8	0.180461
$465$	0	0
$466$	−1.16456e9	−0.533102
$467$	−1.42941e9	−0.649455	−0.324727	−	0.945808i	$-0.605273\pi$
$467$	−1.42941e9	−0.649455	−0.324727	+	0.945808i	$0.605273\pi$
$468$	0	0
$469$	1.66222e9	0.744018
$470$	−2.56841e9	−1.14110
$471$	0	0
$472$	−1.05444e9	−0.461556
$473$	−5.55338e9	−2.41292
$474$	0	0
$475$	8.53088e8	0.365230
$476$	−3.25641e8	−0.138393
$477$	0	0
$478$	2.53808e8	0.106294
$479$	−2.49544e9	−1.03746	−0.518732	−	0.854937i	$-0.673596\pi$
$479$	−2.49544e9	−1.03746	−0.518732	+	0.854937i	$0.673596\pi$
$480$	0	0
$481$	2.66486e8	0.109186
$482$	−1.20271e9	−0.489209
$483$	0	0
$484$	9.65583e8	0.387107
$485$	−3.13364e9	−1.24725
$486$	0	0
$487$	−8.42235e8	−0.330432	−0.165216	−	0.986257i	$-0.552832\pi$
$487$	−8.42235e8	−0.330432	−0.165216	+	0.986257i	$0.552832\pi$
$488$	4.89254e8	0.190575
$489$	0	0
$490$	1.80331e9	0.692443
$491$	−3.94590e9	−1.50439	−0.752195	−	0.658941i	$-0.771005\pi$
$491$	−3.94590e9	−1.50439	−0.752195	+	0.658941i	$0.771005\pi$
$492$	0	0
$493$	8.49272e8	0.319215
$494$	1.56824e8	0.0585287
$495$	0	0
$496$	−1.08249e8	−0.0398326
$497$	−1.48807e9	−0.543721
$498$	0	0
$499$	−4.39302e9	−1.58275	−0.791374	−	0.611333i	$-0.790634\pi$
$499$	−4.39302e9	−1.58275	−0.791374	+	0.611333i	$0.790634\pi$
$500$	−1.33200e9	−0.476551
$501$	0	0
$502$	−6.16940e7	−0.0217661
$503$	3.23757e9	1.13431	0.567154	−	0.823612i	$-0.308045\pi$
$503$	3.23757e9	1.13431	0.567154	+	0.823612i	$0.308045\pi$
$504$	0	0
$505$	2.17602e9	0.751872
$506$	−2.25002e9	−0.772074
$507$	0	0
$508$	2.30254e9	0.779262
$509$	−4.33463e9	−1.45693	−0.728466	−	0.685082i	$-0.759766\pi$
$509$	−4.33463e9	−1.45693	−0.728466	+	0.685082i	$0.759766\pi$
$510$	0	0
$511$	3.58310e9	1.18792
$512$	1.34218e8	0.0441942
$513$	0	0
$514$	3.15000e8	0.102315
$515$	4.33052e9	1.39706
$516$	0	0
$517$	4.19507e9	1.33513
$518$	−4.23692e8	−0.133936
$519$	0	0
$520$	−6.58483e8	−0.205368
$521$	2.38455e9	0.738710	0.369355	−	0.929288i	$-0.379579\pi$
$521$	2.38455e9	0.738710	0.369355	+	0.929288i	$0.379579\pi$
$522$	0	0
$523$	8.47445e8	0.259033	0.129517	−	0.991577i	$-0.458657\pi$
$523$	8.47445e8	0.259033	0.129517	+	0.991577i	$0.458657\pi$
$524$	2.79387e9	0.848295
$525$	0	0
$526$	−2.62192e9	−0.785541
$527$	−2.36742e8	−0.0704593
$528$	0	0
$529$	−1.11693e9	−0.328042
$530$	2.33718e9	0.681911
$531$	0	0
$532$	−2.49338e8	−0.0717957
$533$	1.27221e8	0.0363926
$534$	0	0
$535$	−3.69653e9	−1.04365
$536$	−1.49834e9	−0.420275
$537$	0	0
$538$	−1.70380e9	−0.471717
$539$	−2.94540e9	−0.810185
$540$	0	0
$541$	−2.61429e9	−0.709845	−0.354922	−	0.934896i	$-0.615493\pi$
$541$	−2.61429e9	−0.709845	−0.354922	+	0.934896i	$0.615493\pi$
$542$	−4.69347e9	−1.26618
$543$	0	0
$544$	2.93536e8	0.0781745
$545$	4.92895e9	1.30427
$546$	0	0
$547$	−6.72109e9	−1.75584	−0.877918	−	0.478811i	$-0.841068\pi$
$547$	−6.72109e9	−1.75584	−0.877918	+	0.478811i	$0.841068\pi$
$548$	−8.68439e8	−0.225428
$549$	0	0
$550$	5.85060e9	1.49945
$551$	6.50274e8	0.165602
$552$	0	0
$553$	4.36059e9	1.09650
$554$	1.12523e9	0.281162
$555$	0	0
$556$	−3.33863e9	−0.823770
$557$	−3.08256e9	−0.755821	−0.377910	−	0.925842i	$-0.623357\pi$
$557$	−3.08256e9	−0.755821	−0.377910	+	0.925842i	$0.623357\pi$
$558$	0	0
$559$	−2.69924e9	−0.653583
$560$	1.04694e9	0.251920
$561$	0	0
$562$	−7.08488e8	−0.168367
$563$	2.41284e9	0.569836	0.284918	−	0.958552i	$-0.408034\pi$
$563$	2.41284e9	0.569836	0.284918	+	0.958552i	$0.408034\pi$
$564$	0	0
$565$	9.31997e9	2.17393
$566$	5.05951e9	1.17287
$567$	0	0
$568$	1.34136e9	0.307132
$569$	8.20220e9	1.86654	0.933270	−	0.359176i	$-0.116942\pi$
$569$	8.20220e9	1.86654	0.933270	+	0.359176i	$0.116942\pi$
$570$	0	0
$571$	−7.16780e9	−1.61124	−0.805619	−	0.592434i	$-0.798167\pi$
$571$	−7.16780e9	−1.61124	−0.805619	+	0.592434i	$0.798167\pi$
$572$	1.07552e9	0.240289
$573$	0	0
$574$	−2.02272e8	−0.0446420
$575$	−5.94910e9	−1.30501
$576$	0	0
$577$	2.08578e9	0.452016	0.226008	−	0.974125i	$-0.427432\pi$
$577$	2.08578e9	0.452016	0.226008	+	0.974125i	$0.427432\pi$
$578$	−2.64074e9	−0.568825
$579$	0	0
$580$	−2.73041e9	−0.581073
$581$	−2.34934e8	−0.0496969
$582$	0	0
$583$	−3.81740e9	−0.797862
$584$	−3.22984e9	−0.671021
$585$	0	0
$586$	−4.00130e9	−0.821410
$587$	−4.76488e9	−0.972341	−0.486171	−	0.873864i	$-0.661607\pi$
$587$	−4.76488e9	−0.972341	−0.486171	+	0.873864i	$0.661607\pi$
$588$	0	0
$589$	−1.81270e8	−0.0365529
$590$	7.41403e9	1.48618
$591$	0	0
$592$	3.81919e8	0.0756564
$593$	−6.62056e9	−1.30378	−0.651888	−	0.758315i	$-0.726023\pi$
$593$	−6.62056e9	−1.30378	−0.651888	+	0.758315i	$0.726023\pi$
$594$	0	0
$595$	2.28966e9	0.445618
$596$	7.18202e8	0.138958
$597$	0	0
$598$	−1.09363e9	−0.209130
$599$	1.57460e9	0.299349	0.149674	−	0.988735i	$-0.452178\pi$
$599$	1.57460e9	0.299349	0.149674	+	0.988735i	$0.452178\pi$
$600$	0	0
$601$	−1.37744e9	−0.258829	−0.129415	−	0.991591i	$-0.541310\pi$
$601$	−1.37744e9	−0.258829	−0.129415	+	0.991591i	$0.541310\pi$
$602$	4.29159e9	0.801734
$603$	0	0
$604$	1.98572e9	0.366681
$605$	−6.78925e9	−1.24646
$606$	0	0
$607$	3.29541e9	0.598067	0.299033	−	0.954243i	$-0.403336\pi$
$607$	3.29541e9	0.598067	0.299033	+	0.954243i	$0.403336\pi$
$608$	2.24756e8	0.0405554
$609$	0	0
$610$	−3.44007e9	−0.613638
$611$	2.03903e9	0.361643
$612$	0	0
$613$	6.54256e9	1.14719	0.573596	−	0.819139i	$-0.305548\pi$
$613$	6.54256e9	1.14719	0.573596	+	0.819139i	$0.305548\pi$
$614$	−2.37091e9	−0.413358
$615$	0	0
$616$	−1.71000e9	−0.294756
$617$	3.04639e8	0.0522140	0.0261070	−	0.999659i	$-0.491689\pi$
$617$	3.04639e8	0.0522140	0.0261070	+	0.999659i	$0.491689\pi$
$618$	0	0
$619$	1.00589e9	0.170464	0.0852322	−	0.996361i	$-0.472837\pi$
$619$	1.00589e9	0.170464	0.0852322	+	0.996361i	$0.472837\pi$
$620$	7.61126e8	0.128258
$621$	0	0
$622$	−2.14377e9	−0.357201
$623$	−3.53021e9	−0.584914
$624$	0	0
$625$	−3.51172e8	−0.0575360
$626$	3.64392e9	0.593688
$627$	0	0
$628$	−2.25856e9	−0.363893
$629$	8.35262e8	0.133828
$630$	0	0
$631$	3.45164e9	0.546918	0.273459	−	0.961884i	$-0.411832\pi$
$631$	3.45164e9	0.546918	0.273459	+	0.961884i	$0.411832\pi$
$632$	−3.93068e9	−0.619380
$633$	0	0
$634$	−8.49489e9	−1.32387
$635$	−1.61897e10	−2.50917
$636$	0	0
$637$	−1.43163e9	−0.219453
$638$	4.45967e9	0.679878
$639$	0	0
$640$	−9.43718e8	−0.142302
$641$	3.30072e9	0.495001	0.247500	−	0.968888i	$-0.420391\pi$
$641$	3.30072e9	0.495001	0.247500	+	0.968888i	$0.420391\pi$
$642$	0	0
$643$	−1.12578e10	−1.67000	−0.834998	−	0.550253i	$-0.814531\pi$
$643$	−1.12578e10	−1.67000	−0.834998	+	0.550253i	$0.814531\pi$
$644$	1.73879e9	0.256535
$645$	0	0
$646$	4.91543e8	0.0717378
$647$	1.33317e8	0.0193518	0.00967591	−	0.999953i	$-0.496920\pi$
$647$	1.33317e8	0.0193518	0.00967591	+	0.999953i	$0.496920\pi$
$648$	0	0
$649$	−1.21096e10	−1.73889
$650$	2.84371e9	0.406152
$651$	0	0
$652$	−4.87829e9	−0.689288
$653$	−1.35927e10	−1.91034	−0.955168	−	0.296064i	$-0.904326\pi$
$653$	−1.35927e10	−1.91034	−0.955168	+	0.296064i	$0.904326\pi$
$654$	0	0
$655$	−1.96444e10	−2.73146
$656$	1.82329e8	0.0252170
$657$	0	0
$658$	−3.24191e9	−0.443619
$659$	−1.48685e9	−0.202381	−0.101190	−	0.994867i	$-0.532265\pi$
$659$	−1.48685e9	−0.202381	−0.101190	+	0.994867i	$0.532265\pi$
$660$	0	0
$661$	1.12564e10	1.51599	0.757995	−	0.652260i	$-0.226179\pi$
$661$	1.12564e10	1.51599	0.757995	+	0.652260i	$0.226179\pi$
$662$	−4.11984e9	−0.551922
$663$	0	0
$664$	2.11771e8	0.0280724
$665$	1.75316e9	0.231178
$666$	0	0
$667$	−4.53476e9	−0.591717
$668$	−5.72770e9	−0.743469
$669$	0	0
$670$	1.05352e10	1.35326
$671$	5.61878e9	0.717981
$672$	0	0
$673$	3.60459e9	0.455830	0.227915	−	0.973681i	$-0.426809\pi$
$673$	3.60459e9	0.455830	0.227915	+	0.973681i	$0.426809\pi$
$674$	1.23710e9	0.155631
$675$	0	0
$676$	−3.49314e9	−0.434913
$677$	1.21252e10	1.50186	0.750930	−	0.660382i	$-0.229606\pi$
$677$	1.21252e10	1.50186	0.750930	+	0.660382i	$0.229606\pi$
$678$	0	0
$679$	−3.95535e9	−0.484888
$680$	−2.06392e9	−0.251717
$681$	0	0
$682$	−1.24317e9	−0.150067
$683$	−1.06657e10	−1.28091	−0.640455	−	0.767996i	$-0.721254\pi$
$683$	−1.06657e10	−1.28091	−0.640455	+	0.767996i	$0.721254\pi$
$684$	0	0
$685$	6.10621e9	0.725863
$686$	6.01836e9	0.711776
$687$	0	0
$688$	−3.86848e9	−0.452877
$689$	−1.85547e9	−0.216115
$690$	0	0
$691$	−9.86821e9	−1.13780	−0.568899	−	0.822407i	$-0.692630\pi$
$691$	−9.86821e9	−1.13780	−0.568899	+	0.822407i	$0.692630\pi$
$692$	9.29895e8	0.106675
$693$	0	0
$694$	−2.52882e9	−0.287183
$695$	2.34747e10	2.65249
$696$	0	0
$697$	3.98756e8	0.0446060
$698$	−7.02179e8	−0.0781545
$699$	0	0
$700$	−4.52128e9	−0.498217
$701$	−1.81276e9	−0.198759	−0.0993795	−	0.995050i	$-0.531686\pi$
$701$	−1.81276e9	−0.198759	−0.0993795	+	0.995050i	$0.531686\pi$
$702$	0	0
$703$	6.39547e8	0.0694271
$704$	1.54141e9	0.166500
$705$	0	0
$706$	2.93031e9	0.313399
$707$	2.74663e9	0.292302
$708$	0	0
$709$	8.65901e8	0.0912445	0.0456222	−	0.998959i	$-0.485473\pi$
$709$	8.65901e8	0.0912445	0.0456222	+	0.998959i	$0.485473\pi$
$710$	−9.43142e9	−0.988947
$711$	0	0
$712$	3.18216e9	0.330401
$713$	1.26410e9	0.130608
$714$	0	0
$715$	−7.56227e9	−0.773715
$716$	−5.63289e8	−0.0573503
$717$	0	0
$718$	5.31434e9	0.535813
$719$	−7.73636e9	−0.776221	−0.388111	−	0.921613i	$-0.626872\pi$
$719$	−7.73636e9	−0.776221	−0.388111	+	0.921613i	$0.626872\pi$
$720$	0	0
$721$	5.46608e9	0.543128
$722$	3.76367e8	0.0372161
$723$	0	0
$724$	3.79842e8	0.0371979
$725$	1.17915e10	1.14917
$726$	0	0
$727$	−1.07002e10	−1.03281	−0.516405	−	0.856344i	$-0.672730\pi$
$727$	−1.07002e10	−1.03281	−0.516405	+	0.856344i	$0.672730\pi$
$728$	−8.31152e8	−0.0798400
$729$	0	0
$730$	2.27098e10	2.16064
$731$	−8.46040e9	−0.801088
$732$	0	0
$733$	3.75616e9	0.352273	0.176137	−	0.984366i	$-0.443640\pi$
$733$	3.75616e9	0.352273	0.176137	+	0.984366i	$0.443640\pi$
$734$	1.00079e10	0.934125
$735$	0	0
$736$	−1.56736e9	−0.144909
$737$	−1.72075e10	−1.58337
$738$	0	0
$739$	6.25265e8	0.0569913	0.0284956	−	0.999594i	$-0.490928\pi$
$739$	6.25265e8	0.0569913	0.0284956	+	0.999594i	$0.490928\pi$
$740$	−2.68537e9	−0.243609
$741$	0	0
$742$	2.95005e9	0.265103
$743$	−7.79983e9	−0.697628	−0.348814	−	0.937192i	$-0.613416\pi$
$743$	−7.79983e9	−0.697628	−0.348814	+	0.937192i	$0.613416\pi$
$744$	0	0
$745$	−5.04986e9	−0.447437
$746$	−5.20049e9	−0.458626
$747$	0	0
$748$	3.37107e9	0.294519
$749$	−4.66585e9	−0.405736
$750$	0	0
$751$	3.23476e9	0.278678	0.139339	−	0.990245i	$-0.455502\pi$
$751$	3.23476e9	0.278678	0.139339	+	0.990245i	$0.455502\pi$
$752$	2.92228e9	0.250588
$753$	0	0
$754$	2.16764e9	0.184157
$755$	−1.39621e10	−1.18069
$756$	0	0
$757$	−9.99507e9	−0.837434	−0.418717	−	0.908117i	$-0.637520\pi$
$757$	−9.99507e9	−0.837434	−0.418717	+	0.908117i	$0.637520\pi$
$758$	1.90052e9	0.158500
$759$	0	0
$760$	−1.58031e9	−0.130586
$761$	−6.82542e9	−0.561414	−0.280707	−	0.959794i	$-0.590569\pi$
$761$	−6.82542e9	−0.561414	−0.280707	+	0.959794i	$0.590569\pi$
$762$	0	0
$763$	6.22143e9	0.507054
$764$	−9.03870e8	−0.0733296
$765$	0	0
$766$	−3.55457e9	−0.285750
$767$	−5.88591e9	−0.471010
$768$	0	0
$769$	−1.98960e10	−1.57769	−0.788847	−	0.614589i	$-0.789322\pi$
$769$	−1.98960e10	−1.57769	−0.788847	+	0.614589i	$0.789322\pi$
$770$	1.20234e10	0.949097
$771$	0	0
$772$	−7.12311e9	−0.557198
$773$	1.64030e10	1.27731	0.638655	−	0.769493i	$-0.279491\pi$
$773$	1.64030e10	1.27731	0.638655	+	0.769493i	$0.279491\pi$
$774$	0	0
$775$	−3.28698e9	−0.253654
$776$	3.56539e9	0.273899
$777$	0	0
$778$	1.58141e10	1.20397
$779$	3.05322e8	0.0231407
$780$	0	0
$781$	1.54047e10	1.15711
$782$	−3.42783e9	−0.256328
$783$	0	0
$784$	−2.05176e9	−0.152062
$785$	1.58805e10	1.17171
$786$	0	0
$787$	1.23265e10	0.901419	0.450709	−	0.892671i	$-0.351171\pi$
$787$	1.23265e10	0.901419	0.450709	+	0.892671i	$0.351171\pi$
$788$	5.78238e9	0.420983
$789$	0	0
$790$	2.76376e10	1.99437
$791$	1.17639e10	0.845148
$792$	0	0
$793$	2.73103e9	0.194478
$794$	4.88766e9	0.346521
$795$	0	0
$796$	4.37759e9	0.307638
$797$	2.66011e10	1.86121	0.930605	−	0.366025i	$-0.119281\pi$
$797$	2.66011e10	1.86121	0.930605	+	0.366025i	$0.119281\pi$
$798$	0	0
$799$	6.39107e9	0.443261
$800$	4.07552e9	0.281428
$801$	0	0
$802$	−3.32363e9	−0.227511
$803$	−3.70927e10	−2.52804
$804$	0	0
$805$	−1.22259e10	−0.826026
$806$	−6.04250e8	−0.0406484
$807$	0	0
$808$	−2.47583e9	−0.165113
$809$	6.91604e9	0.459238	0.229619	−	0.973281i	$-0.426252\pi$
$809$	6.91604e9	0.459238	0.229619	+	0.973281i	$0.426252\pi$
$810$	0	0
$811$	5.93036e9	0.390399	0.195199	−	0.980764i	$-0.437465\pi$
$811$	5.93036e9	0.390399	0.195199	+	0.980764i	$0.437465\pi$
$812$	−3.44639e9	−0.225901
$813$	0	0
$814$	4.38610e9	0.285032
$815$	3.43004e10	2.21946
$816$	0	0
$817$	−6.47800e9	−0.415588
$818$	1.43833e10	0.918799
$819$	0	0
$820$	−1.28200e9	−0.0811971
$821$	2.10526e10	1.32771	0.663857	−	0.747859i	$-0.268918\pi$
$821$	2.10526e10	1.32771	0.663857	+	0.747859i	$0.268918\pi$
$822$	0	0
$823$	1.97535e10	1.23522	0.617610	−	0.786485i	$-0.288101\pi$
$823$	1.97535e10	1.23522	0.617610	+	0.786485i	$0.288101\pi$
$824$	−4.92717e9	−0.306798
$825$	0	0
$826$	9.35815e9	0.577776
$827$	−1.07101e10	−0.658453	−0.329226	−	0.944251i	$-0.606788\pi$
$827$	−1.07101e10	−0.658453	−0.329226	+	0.944251i	$0.606788\pi$
$828$	0	0
$829$	2.15680e10	1.31483	0.657413	−	0.753531i	$-0.271651\pi$
$829$	2.15680e10	1.31483	0.657413	+	0.753531i	$0.271651\pi$
$830$	−1.48902e9	−0.0903913
$831$	0	0
$832$	7.49208e8	0.0450994
$833$	−4.48723e9	−0.268981
$834$	0	0
$835$	4.02729e10	2.39392
$836$	2.58118e9	0.152790
$837$	0	0
$838$	1.84098e10	1.08068
$839$	−3.65417e9	−0.213610	−0.106805	−	0.994280i	$-0.534062\pi$
$839$	−3.65417e9	−0.213610	−0.106805	+	0.994280i	$0.534062\pi$
$840$	0	0
$841$	−8.26170e9	−0.478942
$842$	1.74026e10	1.00467
$843$	0	0
$844$	−3.74257e9	−0.214275
$845$	2.45612e10	1.40039
$846$	0	0
$847$	−8.56955e9	−0.484580
$848$	−2.65920e9	−0.149749
$849$	0	0
$850$	8.91321e9	0.497815
$851$	−4.45995e9	−0.248071
$852$	0	0
$853$	−1.37485e10	−0.758463	−0.379232	−	0.925302i	$-0.623812\pi$
$853$	−1.37485e10	−0.758463	−0.379232	+	0.925302i	$0.623812\pi$
$854$	−4.34213e9	−0.238561
$855$	0	0
$856$	4.20583e9	0.229189
$857$	−4.70344e9	−0.255260	−0.127630	−	0.991822i	$-0.540737\pi$
$857$	−4.70344e9	−0.255260	−0.127630	+	0.991822i	$0.540737\pi$
$858$	0	0
$859$	1.02041e10	0.549284	0.274642	−	0.961547i	$-0.411441\pi$
$859$	1.02041e10	0.549284	0.274642	+	0.961547i	$0.411441\pi$
$860$	2.72002e10	1.45824
$861$	0	0
$862$	8.00194e9	0.425520
$863$	−3.02998e10	−1.60473	−0.802364	−	0.596834i	$-0.796425\pi$
$863$	−3.02998e10	−1.60473	−0.802364	+	0.596834i	$0.796425\pi$
$864$	0	0
$865$	−6.53832e9	−0.343487
$866$	−2.12978e10	−1.11435
$867$	0	0
$868$	9.60711e8	0.0498624
$869$	−4.51413e10	−2.33349
$870$	0	0
$871$	−8.36378e9	−0.428883
$872$	−5.60805e9	−0.286420
$873$	0	0
$874$	−2.62464e9	−0.132978
$875$	1.18215e10	0.596547
$876$	0	0
$877$	−3.75994e10	−1.88227	−0.941136	−	0.338029i	$-0.890240\pi$
$877$	−3.75994e10	−1.88227	−0.941136	+	0.338029i	$0.890240\pi$
$878$	2.75677e10	1.37458
$879$	0	0
$880$	−1.08380e10	−0.536118
$881$	−1.99561e10	−0.983243	−0.491621	−	0.870809i	$-0.663596\pi$
$881$	−1.99561e10	−0.983243	−0.491621	+	0.870809i	$0.663596\pi$
$882$	0	0
$883$	3.58736e10	1.75353	0.876764	−	0.480921i	$-0.159697\pi$
$883$	3.58736e10	1.75353	0.876764	+	0.480921i	$0.159697\pi$
$884$	1.63853e9	0.0797756
$885$	0	0
$886$	2.03717e10	0.984031
$887$	2.80006e10	1.34721	0.673604	−	0.739092i	$-0.264745\pi$
$887$	2.80006e10	1.34721	0.673604	+	0.739092i	$0.264745\pi$
$888$	0	0
$889$	−2.04350e10	−0.975481
$890$	−2.23746e10	−1.06387
$891$	0	0
$892$	2.06841e10	0.975798
$893$	4.89354e9	0.229955
$894$	0	0
$895$	3.96062e9	0.184664
$896$	−1.19118e9	−0.0553223
$897$	0	0
$898$	2.53923e10	1.17013
$899$	−2.50553e9	−0.115011
$900$	0	0
$901$	−5.81569e9	−0.264890
$902$	2.09394e9	0.0950038
$903$	0	0
$904$	−1.06041e10	−0.477400
$905$	−2.67077e9	−0.119775
$906$	0	0
$907$	1.92380e10	0.856120	0.428060	−	0.903750i	$-0.359197\pi$
$907$	1.92380e10	0.856120	0.428060	+	0.903750i	$0.359197\pi$
$908$	−1.49323e10	−0.661952
$909$	0	0
$910$	5.84404e9	0.257080
$911$	−3.57401e10	−1.56618	−0.783089	−	0.621910i	$-0.786357\pi$
$911$	−3.57401e10	−1.56618	−0.783089	+	0.621910i	$0.786357\pi$
$912$	0	0
$913$	2.43206e9	0.105761
$914$	−2.66529e10	−1.15460
$915$	0	0
$916$	9.67251e9	0.415820
$917$	−2.47956e10	−1.06190
$918$	0	0
$919$	−1.82574e9	−0.0775951	−0.0387975	−	0.999247i	$-0.512353\pi$
$919$	−1.82574e9	−0.0775951	−0.0387975	+	0.999247i	$0.512353\pi$
$920$	1.10205e10	0.466599
$921$	0	0
$922$	−3.00408e10	−1.26227
$923$	7.48750e9	0.313423
$924$	0	0
$925$	1.15970e10	0.481780
$926$	−5.60856e9	−0.232120
$927$	0	0
$928$	3.10660e9	0.127605
$929$	2.45593e10	1.00499	0.502495	−	0.864580i	$-0.332416\pi$
$929$	2.45593e10	1.00499	0.502495	+	0.864580i	$0.332416\pi$
$930$	0	0
$931$	−3.43580e9	−0.139542
$932$	−9.31646e9	−0.376960
$933$	0	0
$934$	−1.14353e10	−0.459234
$935$	−2.37029e10	−0.948332
$936$	0	0
$937$	3.92359e10	1.55810	0.779050	−	0.626962i	$-0.215702\pi$
$937$	3.92359e10	1.55810	0.779050	+	0.626962i	$0.215702\pi$
$938$	1.32978e10	0.526100
$939$	0	0
$940$	−2.05473e10	−0.806877
$941$	3.31361e10	1.29640	0.648198	−	0.761472i	$-0.275523\pi$
$941$	3.31361e10	1.29640	0.648198	+	0.761472i	$0.275523\pi$
$942$	0	0
$943$	−2.12919e9	−0.0826845
$944$	−8.43552e9	−0.326370
$945$	0	0
$946$	−4.44270e10	−1.70619
$947$	1.24574e10	0.476653	0.238327	−	0.971185i	$-0.423401\pi$
$947$	1.24574e10	0.476653	0.238327	+	0.971185i	$0.423401\pi$
$948$	0	0
$949$	−1.80291e10	−0.684765
$950$	6.82470e9	0.258257
$951$	0	0
$952$	−2.60513e9	−0.0978588
$953$	−7.50532e9	−0.280895	−0.140448	−	0.990088i	$-0.544854\pi$
$953$	−7.50532e9	−0.280895	−0.140448	+	0.990088i	$0.544854\pi$
$954$	0	0
$955$	6.35534e9	0.236117
$956$	2.03047e9	0.0751611
$957$	0	0
$958$	−1.99635e10	−0.733598
$959$	7.70739e9	0.282190
$960$	0	0
$961$	−2.68142e10	−0.974614
$962$	2.13189e9	0.0772060
$963$	0	0
$964$	−9.62165e9	−0.345923
$965$	5.00844e10	1.79414
$966$	0	0
$967$	−1.74747e10	−0.621466	−0.310733	−	0.950497i	$-0.600574\pi$
$967$	−1.74747e10	−0.621466	−0.310733	+	0.950497i	$0.600574\pi$
$968$	7.72466e9	0.273726
$969$	0	0
$970$	−2.50691e10	−0.881939
$971$	−3.44244e9	−0.120670	−0.0603349	−	0.998178i	$-0.519217\pi$
$971$	−3.44244e9	−0.120670	−0.0603349	+	0.998178i	$0.519217\pi$
$972$	0	0
$973$	2.96303e10	1.03120
$974$	−6.73788e9	−0.233651
$975$	0	0
$976$	3.91403e9	0.134757
$977$	1.28752e10	0.441695	0.220847	−	0.975308i	$-0.429118\pi$
$977$	1.28752e10	0.441695	0.220847	+	0.975308i	$0.429118\pi$
$978$	0	0
$979$	3.65451e10	1.24477
$980$	1.44265e10	0.489631
$981$	0	0
$982$	−3.15672e10	−1.06376
$983$	2.90910e8	0.00976834	0.00488417	−	0.999988i	$-0.498445\pi$
$983$	2.90910e8	0.00976834	0.00488417	+	0.999988i	$0.498445\pi$
$984$	0	0
$985$	−4.06574e10	−1.35554
$986$	6.79418e9	0.225719
$987$	0	0
$988$	1.25459e9	0.0413860
$989$	4.51750e10	1.48495
$990$	0	0
$991$	1.79076e10	0.584492	0.292246	−	0.956343i	$-0.405597\pi$
$991$	1.79076e10	0.584492	0.292246	+	0.956343i	$0.405597\pi$
$992$	−8.65993e8	−0.0281659
$993$	0	0
$994$	−1.19046e10	−0.384468
$995$	−3.07799e10	−0.990574
$996$	0	0
$997$	3.33090e10	1.06446	0.532229	−	0.846600i	$-0.321354\pi$
$997$	3.33090e10	1.06446	0.532229	+	0.846600i	$0.321354\pi$
$998$	−3.51442e10	−1.11917
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.8.a.d.1.1		1
3.2	odd	2		114.8.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.8.a.b.1.1	✓	1	3.2	odd	2
342.8.a.d.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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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