LMFDB - Embedded newform 54.10.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,10,Mod(1,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("54.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	54.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:	$27.8119351528$
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	$\chi$	$=$	54.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+16.0000 q^{2} +256.000 q^{4} -435.000 q^{5} -2527.00 q^{7} +4096.00 q^{8} +O(q^{10})$

\(q+16.0000 q^{2} +256.000 q^{4} -435.000 q^{5} -2527.00 q^{7} +4096.00 q^{8} -6960.00 q^{10} -9123.00 q^{11} -79180.0 q^{13} -40432.0 q^{14} +65536.0 q^{16} -437976. q^{17} +116966. q^{19} -111360. q^{20} -145968. q^{22} -261102. q^{23} -1.76390e6 q^{25} -1.26688e6 q^{26} -646912. q^{28} -396150. q^{29} -5.88253e6 q^{31} +1.04858e6 q^{32} -7.00762e6 q^{34} +1.09924e6 q^{35} -8.98625e6 q^{37} +1.87146e6 q^{38} -1.78176e6 q^{40} +1.74496e7 q^{41} -3.20946e7 q^{43} -2.33549e6 q^{44} -4.17763e6 q^{46} +2.09658e7 q^{47} -3.39679e7 q^{49} -2.82224e7 q^{50} -2.02701e7 q^{52} +4.06690e7 q^{53} +3.96850e6 q^{55} -1.03506e7 q^{56} -6.33840e6 q^{58} -8.43831e7 q^{59} -1.48038e8 q^{61} -9.41205e7 q^{62} +1.67772e7 q^{64} +3.44433e7 q^{65} +1.54939e8 q^{67} -1.12122e8 q^{68} +1.75879e7 q^{70} +1.68344e8 q^{71} +4.18698e8 q^{73} -1.43780e8 q^{74} +2.99433e7 q^{76} +2.30538e7 q^{77} +2.10598e8 q^{79} -2.85082e7 q^{80} +2.79193e8 q^{82} +7.76395e8 q^{83} +1.90520e8 q^{85} -5.13514e8 q^{86} -3.73678e7 q^{88} +3.70838e8 q^{89} +2.00088e8 q^{91} -6.68421e7 q^{92} +3.35453e8 q^{94} -5.08802e7 q^{95} +3.09842e8 q^{97} -5.43486e8 q^{98} +O(q^{100})\)

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 $

$2$	16.0000	0.707107
$2$	16.0000	0.707107
$3$	0	0
$3$	0	0
$4$	256.000	0.500000
$5$	−435.000	−0.311261	−0.155630	−	0.987815i	$-0.549741\pi$
$5$	−435.000	−0.311261	−0.155630	+	0.987815i	$0.549741\pi$
$6$	0	0
$7$	−2527.00	−0.397799	−0.198900	−	0.980020i	$-0.563737\pi$
$7$	−2527.00	−0.397799	−0.198900	+	0.980020i	$0.563737\pi$
$8$	4096.00	0.353553
$9$	0	0
$10$	−6960.00	−0.220095
$11$	−9123.00	−0.187876	−0.0939378	−	0.995578i	$-0.529945\pi$
$11$	−9123.00	−0.187876	−0.0939378	+	0.995578i	$0.529945\pi$
$12$	0	0
$13$	−79180.0	−0.768901	−0.384450	−	0.923146i	$-0.625609\pi$
$13$	−79180.0	−0.768901	−0.384450	+	0.923146i	$0.625609\pi$
$14$	−40432.0	−0.281287
$15$	0	0
$16$	65536.0	0.250000
$17$	−437976.	−1.27183	−0.635917	−	0.771758i	$-0.719378\pi$
$17$	−437976.	−1.27183	−0.635917	+	0.771758i	$0.719378\pi$
$18$	0	0
$19$	116966.	0.205906	0.102953	−	0.994686i	$-0.467171\pi$
$19$	116966.	0.205906	0.102953	+	0.994686i	$0.467171\pi$
$20$	−111360.	−0.155630
$21$	0	0
$22$	−145968.	−0.132848
$23$	−261102.	−0.194552	−0.0972758	−	0.995257i	$-0.531013\pi$
$23$	−261102.	−0.194552	−0.0972758	+	0.995257i	$0.531013\pi$
$24$	0	0
$25$	−1.76390e6	−0.903117
$26$	−1.26688e6	−0.543695
$27$	0	0
$28$	−646912.	−0.198900
$29$	−396150.	−0.104008	−0.0520042	−	0.998647i	$-0.516561\pi$
$29$	−396150.	−0.104008	−0.0520042	+	0.998647i	$0.516561\pi$
$30$	0	0
$31$	−5.88253e6	−1.14403	−0.572014	−	0.820244i	$-0.693838\pi$
$31$	−5.88253e6	−1.14403	−0.572014	+	0.820244i	$0.693838\pi$
$32$	1.04858e6	0.176777
$33$	0	0
$34$	−7.00762e6	−0.899322
$35$	1.09924e6	0.123819
$36$	0	0
$37$	−8.98625e6	−0.788262	−0.394131	−	0.919054i	$-0.628954\pi$
$37$	−8.98625e6	−0.788262	−0.394131	+	0.919054i	$0.628954\pi$
$38$	1.87146e6	0.145597
$39$	0	0
$40$	−1.78176e6	−0.110047
$41$	1.74496e7	0.964400	0.482200	−	0.876061i	$-0.339838\pi$
$41$	1.74496e7	0.964400	0.482200	+	0.876061i	$0.339838\pi$
$42$	0	0
$43$	−3.20946e7	−1.43161	−0.715805	−	0.698301i	$-0.753940\pi$
$43$	−3.20946e7	−1.43161	−0.715805	+	0.698301i	$0.753940\pi$
$44$	−2.33549e6	−0.0939378
$45$	0	0
$46$	−4.17763e6	−0.137569
$47$	2.09658e7	0.626716	0.313358	−	0.949635i	$-0.398546\pi$
$47$	2.09658e7	0.626716	0.313358	+	0.949635i	$0.398546\pi$
$48$	0	0
$49$	−3.39679e7	−0.841756
$50$	−2.82224e7	−0.638600
$51$	0	0
$52$	−2.02701e7	−0.384450
$53$	4.06690e7	0.707983	0.353991	−	0.935249i	$-0.384824\pi$
$53$	4.06690e7	0.707983	0.353991	+	0.935249i	$0.384824\pi$
$54$	0	0
$55$	3.96850e6	0.0584783
$56$	−1.03506e7	−0.140643
$57$	0	0
$58$	−6.33840e6	−0.0735451
$59$	−8.43831e7	−0.906612	−0.453306	−	0.891355i	$-0.649755\pi$
$59$	−8.43831e7	−0.906612	−0.453306	+	0.891355i	$0.649755\pi$
$60$	0	0
$61$	−1.48038e8	−1.36896	−0.684479	−	0.729032i	$-0.739970\pi$
$61$	−1.48038e8	−1.36896	−0.684479	+	0.729032i	$0.739970\pi$
$62$	−9.41205e7	−0.808950
$63$	0	0
$64$	1.67772e7	0.125000
$65$	3.44433e7	0.239329
$66$	0	0
$67$	1.54939e8	0.939343	0.469672	−	0.882841i	$-0.344372\pi$
$67$	1.54939e8	0.939343	0.469672	+	0.882841i	$0.344372\pi$
$68$	−1.12122e8	−0.635917
$69$	0	0
$70$	1.75879e7	0.0875535
$71$	1.68344e8	0.786202	0.393101	−	0.919495i	$-0.371402\pi$
$71$	1.68344e8	0.786202	0.393101	+	0.919495i	$0.371402\pi$
$72$	0	0
$73$	4.18698e8	1.72563	0.862816	−	0.505519i	$-0.168699\pi$
$73$	4.18698e8	1.72563	0.862816	+	0.505519i	$0.168699\pi$
$74$	−1.43780e8	−0.557385
$75$	0	0
$76$	2.99433e7	0.102953
$77$	2.30538e7	0.0747368
$78$	0	0
$79$	2.10598e8	0.608320	0.304160	−	0.952621i	$-0.401624\pi$
$79$	2.10598e8	0.608320	0.304160	+	0.952621i	$0.401624\pi$
$80$	−2.85082e7	−0.0778152
$81$	0	0
$82$	2.79193e8	0.681934
$83$	7.76395e8	1.79569	0.897844	−	0.440313i	$-0.145133\pi$
$83$	7.76395e8	1.79569	0.897844	+	0.440313i	$0.145133\pi$
$84$	0	0
$85$	1.90520e8	0.395872
$86$	−5.13514e8	−1.01230
$87$	0	0
$88$	−3.73678e7	−0.0664241
$89$	3.70838e8	0.626511	0.313256	−	0.949669i	$-0.398580\pi$
$89$	3.70838e8	0.626511	0.313256	+	0.949669i	$0.398580\pi$
$90$	0	0
$91$	2.00088e8	0.305868
$92$	−6.68421e7	−0.0972758
$93$	0	0
$94$	3.35453e8	0.443155
$95$	−5.08802e7	−0.0640904
$96$	0	0
$97$	3.09842e8	0.355359	0.177680	−	0.984088i	$-0.443141\pi$
$97$	3.09842e8	0.355359	0.177680	+	0.984088i	$0.443141\pi$
$98$	−5.43486e8	−0.595211
$99$	0	0
$100$	−4.51558e8	−0.451558
$101$	1.75966e9	1.68260	0.841301	−	0.540566i	$-0.181790\pi$
$101$	1.75966e9	1.68260	0.841301	+	0.540566i	$0.181790\pi$
$102$	0	0
$103$	−1.33403e7	−0.0116788	−0.00583939	−	0.999983i	$-0.501859\pi$
$103$	−1.33403e7	−0.0116788	−0.00583939	+	0.999983i	$0.501859\pi$
$104$	−3.24321e8	−0.271848
$105$	0	0
$106$	6.50705e8	0.500619
$107$	−1.02477e8	−0.0755787	−0.0377894	−	0.999286i	$-0.512032\pi$
$107$	−1.02477e8	−0.0755787	−0.0377894	+	0.999286i	$0.512032\pi$
$108$	0	0
$109$	−6.48714e8	−0.440184	−0.220092	−	0.975479i	$-0.570636\pi$
$109$	−6.48714e8	−0.440184	−0.220092	+	0.975479i	$0.570636\pi$
$110$	6.34961e7	0.0413504
$111$	0	0
$112$	−1.65609e8	−0.0994498
$113$	−3.07251e9	−1.77272	−0.886360	−	0.462996i	$-0.846775\pi$
$113$	−3.07251e9	−1.77272	−0.886360	+	0.462996i	$0.846775\pi$
$114$	0	0
$115$	1.13579e8	0.0605563
$116$	−1.01414e8	−0.0520042
$117$	0	0
$118$	−1.35013e9	−0.641071
$119$	1.10677e9	0.505934
$120$	0	0
$121$	−2.27472e9	−0.964703
$122$	−2.36861e9	−0.968000
$123$	0	0
$124$	−1.50593e9	−0.572014
$125$	1.61691e9	0.592365
$126$	0	0
$127$	−3.34109e9	−1.13965	−0.569825	−	0.821766i	$-0.692989\pi$
$127$	−3.34109e9	−1.13965	−0.569825	+	0.821766i	$0.692989\pi$
$128$	2.68435e8	0.0883883
$129$	0	0
$130$	5.51093e8	0.169231
$131$	−4.85065e9	−1.43906	−0.719530	−	0.694462i	$-0.755643\pi$
$131$	−4.85065e9	−1.43906	−0.719530	+	0.694462i	$0.755643\pi$
$132$	0	0
$133$	−2.95573e8	−0.0819092
$134$	2.47903e9	0.664216
$135$	0	0
$136$	−1.79395e9	−0.449661
$137$	3.90021e9	0.945900	0.472950	−	0.881089i	$-0.343189\pi$
$137$	3.90021e9	0.945900	0.472950	+	0.881089i	$0.343189\pi$
$138$	0	0
$139$	−5.34365e9	−1.21415	−0.607074	−	0.794646i	$-0.707657\pi$
$139$	−5.34365e9	−1.21415	−0.607074	+	0.794646i	$0.707657\pi$
$140$	2.81407e8	0.0619096
$141$	0	0
$142$	2.69350e9	0.555929
$143$	7.22359e8	0.144458
$144$	0	0
$145$	1.72325e8	0.0323737
$146$	6.69917e9	1.22021
$147$	0	0
$148$	−2.30048e9	−0.394131
$149$	−5.94028e9	−0.987344	−0.493672	−	0.869648i	$-0.664346\pi$
$149$	−5.94028e9	−0.987344	−0.493672	+	0.869648i	$0.664346\pi$
$150$	0	0
$151$	1.24779e9	0.195320	0.0976598	−	0.995220i	$-0.468864\pi$
$151$	1.24779e9	0.195320	0.0976598	+	0.995220i	$0.468864\pi$
$152$	4.79093e8	0.0727987
$153$	0	0
$154$	3.68861e8	0.0528469
$155$	2.55890e9	0.356091
$156$	0	0
$157$	5.12444e9	0.673129	0.336564	−	0.941660i	$-0.390735\pi$
$157$	5.12444e9	0.673129	0.336564	+	0.941660i	$0.390735\pi$
$158$	3.36957e9	0.430148
$159$	0	0
$160$	−4.56131e8	−0.0550236
$161$	6.59805e8	0.0773925
$162$	0	0
$163$	1.17910e10	1.30830	0.654151	−	0.756364i	$-0.273026\pi$
$163$	1.17910e10	1.30830	0.654151	+	0.756364i	$0.273026\pi$
$164$	4.46709e9	0.482200
$165$	0	0
$166$	1.24223e10	1.26974
$167$	−1.04075e10	−1.03544	−0.517718	−	0.855551i	$-0.673218\pi$
$167$	−1.04075e10	−1.03544	−0.517718	+	0.855551i	$0.673218\pi$
$168$	0	0
$169$	−4.33503e9	−0.408791
$170$	3.04831e9	0.279924
$171$	0	0
$172$	−8.21623e9	−0.715805
$173$	−3.50370e9	−0.297385	−0.148693	−	0.988883i	$-0.547506\pi$
$173$	−3.50370e9	−0.297385	−0.148693	+	0.988883i	$0.547506\pi$
$174$	0	0
$175$	4.45738e9	0.359259
$176$	−5.97885e8	−0.0469689
$177$	0	0
$178$	5.93340e9	0.443010
$179$	−1.89515e10	−1.37976	−0.689882	−	0.723922i	$-0.742338\pi$
$179$	−1.89515e10	−1.37976	−0.689882	+	0.723922i	$0.742338\pi$
$180$	0	0
$181$	2.02932e10	1.40539	0.702695	−	0.711492i	$-0.251980\pi$
$181$	2.02932e10	1.40539	0.702695	+	0.711492i	$0.251980\pi$
$182$	3.20141e9	0.216282
$183$	0	0
$184$	−1.06947e9	−0.0687844
$185$	3.90902e9	0.245355
$186$	0	0
$187$	3.99566e9	0.238947
$188$	5.36724e9	0.313358
$189$	0	0
$190$	−8.14083e8	−0.0453187
$191$	−1.19165e10	−0.647887	−0.323943	−	0.946076i	$-0.605009\pi$
$191$	−1.19165e10	−0.647887	−0.323943	+	0.946076i	$0.605009\pi$
$192$	0	0
$193$	−5.02968e9	−0.260935	−0.130467	−	0.991453i	$-0.541648\pi$
$193$	−5.02968e9	−0.260935	−0.130467	+	0.991453i	$0.541648\pi$
$194$	4.95747e9	0.251277
$195$	0	0
$196$	−8.69578e9	−0.420878
$197$	−5.95780e8	−0.0281831	−0.0140915	−	0.999901i	$-0.504486\pi$
$197$	−5.95780e8	−0.0281831	−0.0140915	+	0.999901i	$0.504486\pi$
$198$	0	0
$199$	1.94694e10	0.880065	0.440032	−	0.897982i	$-0.354967\pi$
$199$	1.94694e10	0.880065	0.440032	+	0.897982i	$0.354967\pi$
$200$	−7.22493e9	−0.319300
$201$	0	0
$202$	2.81545e10	1.18978
$203$	1.00107e9	0.0413745
$204$	0	0
$205$	−7.59056e9	−0.300180
$206$	−2.13445e8	−0.00825815
$207$	0	0
$208$	−5.18914e9	−0.192225
$209$	−1.06708e9	−0.0386847
$210$	0	0
$211$	−2.09449e10	−0.727457	−0.363728	−	0.931505i	$-0.618496\pi$
$211$	−2.09449e10	−0.727457	−0.363728	+	0.931505i	$0.618496\pi$
$212$	1.04113e10	0.353991
$213$	0	0
$214$	−1.63963e9	−0.0534422
$215$	1.39612e10	0.445604
$216$	0	0
$217$	1.48652e10	0.455094
$218$	−1.03794e10	−0.311257
$219$	0	0
$220$	1.01594e9	0.0292392
$221$	3.46789e10	0.977914
$222$	0	0
$223$	−3.74240e10	−1.01339	−0.506697	−	0.862124i	$-0.669134\pi$
$223$	−3.74240e10	−1.01339	−0.506697	+	0.862124i	$0.669134\pi$
$224$	−2.64975e9	−0.0703217
$225$	0	0
$226$	−4.91602e10	−1.25350
$227$	−1.53871e10	−0.384628	−0.192314	−	0.981333i	$-0.561599\pi$
$227$	−1.53871e10	−0.384628	−0.192314	+	0.981333i	$0.561599\pi$
$228$	0	0
$229$	−7.13954e10	−1.71558	−0.857789	−	0.514001i	$-0.828163\pi$
$229$	−7.13954e10	−1.71558	−0.857789	+	0.514001i	$0.828163\pi$
$230$	1.81727e9	0.0428198
$231$	0	0
$232$	−1.62263e9	−0.0367725
$233$	4.39177e10	0.976199	0.488099	−	0.872788i	$-0.337690\pi$
$233$	4.39177e10	0.976199	0.488099	+	0.872788i	$0.337690\pi$
$234$	0	0
$235$	−9.12012e9	−0.195072
$236$	−2.16021e10	−0.453306
$237$	0	0
$238$	1.77082e10	0.357750
$239$	7.69544e10	1.52561	0.762804	−	0.646630i	$-0.223822\pi$
$239$	7.69544e10	1.52561	0.762804	+	0.646630i	$0.223822\pi$
$240$	0	0
$241$	−2.27096e10	−0.433643	−0.216822	−	0.976211i	$-0.569569\pi$
$241$	−2.27096e10	−0.433643	−0.216822	+	0.976211i	$0.569569\pi$
$242$	−3.63955e10	−0.682148
$243$	0	0
$244$	−3.78978e10	−0.684479
$245$	1.47760e10	0.262005
$246$	0	0
$247$	−9.26137e9	−0.158321
$248$	−2.40949e10	−0.404475
$249$	0	0
$250$	2.58705e10	0.418866
$251$	−8.27276e10	−1.31558	−0.657792	−	0.753199i	$-0.728510\pi$
$251$	−8.27276e10	−1.31558	−0.657792	+	0.753199i	$0.728510\pi$
$252$	0	0
$253$	2.38203e9	0.0365515
$254$	−5.34574e10	−0.805854
$255$	0	0
$256$	4.29497e9	0.0625000
$257$	8.79790e10	1.25800	0.628999	−	0.777406i	$-0.283465\pi$
$257$	8.79790e10	1.25800	0.628999	+	0.777406i	$0.283465\pi$
$258$	0	0
$259$	2.27082e10	0.313570
$260$	8.81748e9	0.119664
$261$	0	0
$262$	−7.76103e10	−1.01757
$263$	−1.50737e11	−1.94276	−0.971381	−	0.237526i	$-0.923664\pi$
$263$	−1.50737e11	−1.94276	−0.971381	+	0.237526i	$0.923664\pi$
$264$	0	0
$265$	−1.76910e10	−0.220367
$266$	−4.72917e9	−0.0579185
$267$	0	0
$268$	3.96644e10	0.469672
$269$	6.86181e10	0.799012	0.399506	−	0.916731i	$-0.369182\pi$
$269$	6.86181e10	0.799012	0.399506	+	0.916731i	$0.369182\pi$
$270$	0	0
$271$	2.35686e10	0.265444	0.132722	−	0.991153i	$-0.457628\pi$
$271$	2.35686e10	0.265444	0.132722	+	0.991153i	$0.457628\pi$
$272$	−2.87032e10	−0.317958
$273$	0	0
$274$	6.24033e10	0.668852
$275$	1.60921e10	0.169674
$276$	0	0
$277$	1.22855e11	1.25381	0.626907	−	0.779094i	$-0.284321\pi$
$277$	1.22855e11	1.25381	0.626907	+	0.779094i	$0.284321\pi$
$278$	−8.54984e10	−0.858532
$279$	0	0
$280$	4.50251e9	0.0437767
$281$	3.49164e10	0.334080	0.167040	−	0.985950i	$-0.446579\pi$
$281$	3.49164e10	0.334080	0.167040	+	0.985950i	$0.446579\pi$
$282$	0	0
$283$	−7.61645e10	−0.705852	−0.352926	−	0.935651i	$-0.614813\pi$
$283$	−7.61645e10	−0.705852	−0.352926	+	0.935651i	$0.614813\pi$
$284$	4.30960e10	0.393101
$285$	0	0
$286$	1.15577e10	0.102147
$287$	−4.40951e10	−0.383638
$288$	0	0
$289$	7.32351e10	0.617560
$290$	2.75720e9	0.0228917
$291$	0	0
$292$	1.07187e11	0.862816
$293$	1.77323e11	1.40560	0.702799	−	0.711388i	$-0.251933\pi$
$293$	1.77323e11	1.40560	0.702799	+	0.711388i	$0.251933\pi$
$294$	0	0
$295$	3.67066e10	0.282193
$296$	−3.68077e10	−0.278693
$297$	0	0
$298$	−9.50444e10	−0.698158
$299$	2.06741e10	0.149591
$300$	0	0
$301$	8.11032e10	0.569493
$302$	1.99647e10	0.138112
$303$	0	0
$304$	7.66548e9	0.0514764
$305$	6.43967e10	0.426103
$306$	0	0
$307$	1.91554e11	1.23075	0.615374	−	0.788235i	$-0.289005\pi$
$307$	1.91554e11	1.23075	0.615374	+	0.788235i	$0.289005\pi$
$308$	5.90178e9	0.0373684
$309$	0	0
$310$	4.09424e10	0.251794
$311$	−1.52905e11	−0.926827	−0.463414	−	0.886142i	$-0.653376\pi$
$311$	−1.52905e11	−0.926827	−0.463414	+	0.886142i	$0.653376\pi$
$312$	0	0
$313$	−1.23800e10	−0.0729076	−0.0364538	−	0.999335i	$-0.511606\pi$
$313$	−1.23800e10	−0.0729076	−0.0364538	+	0.999335i	$0.511606\pi$
$314$	8.19911e10	0.475974
$315$	0	0
$316$	5.39131e10	0.304160
$317$	6.02948e10	0.335361	0.167681	−	0.985841i	$-0.446372\pi$
$317$	6.02948e10	0.335361	0.167681	+	0.985841i	$0.446372\pi$
$318$	0	0
$319$	3.61408e9	0.0195407
$320$	−7.29809e9	−0.0389076
$321$	0	0
$322$	1.05569e10	0.0547248
$323$	−5.12283e10	−0.261878
$324$	0	0
$325$	1.39666e11	0.694407
$326$	1.88657e11	0.925109
$327$	0	0
$328$	7.14734e10	0.340967
$329$	−5.29805e10	−0.249307
$330$	0	0
$331$	−7.45035e10	−0.341154	−0.170577	−	0.985344i	$-0.554563\pi$
$331$	−7.45035e10	−0.341154	−0.170577	+	0.985344i	$0.554563\pi$
$332$	1.98757e11	0.897844
$333$	0	0
$334$	−1.66520e11	−0.732164
$335$	−6.73985e10	−0.292381
$336$	0	0
$337$	−1.92661e11	−0.813690	−0.406845	−	0.913497i	$-0.633371\pi$
$337$	−1.92661e11	−0.813690	−0.406845	+	0.913497i	$0.633371\pi$
$338$	−6.93604e10	−0.289059
$339$	0	0
$340$	4.87730e10	0.197936
$341$	5.36663e10	0.214935
$342$	0	0
$343$	1.87810e11	0.732649
$344$	−1.31460e11	−0.506150
$345$	0	0
$346$	−5.60592e10	−0.210283
$347$	−5.28835e11	−1.95811	−0.979056	−	0.203591i	$-0.934739\pi$
$347$	−5.28835e11	−1.95811	−0.979056	+	0.203591i	$0.934739\pi$
$348$	0	0
$349$	−1.94816e11	−0.702927	−0.351464	−	0.936202i	$-0.614316\pi$
$349$	−1.94816e11	−0.702927	−0.351464	+	0.936202i	$0.614316\pi$
$350$	7.13180e10	0.254035
$351$	0	0
$352$	−9.56616e9	−0.0332120
$353$	8.43923e10	0.289279	0.144639	−	0.989484i	$-0.453798\pi$
$353$	8.43923e10	0.289279	0.144639	+	0.989484i	$0.453798\pi$
$354$	0	0
$355$	−7.32294e10	−0.244714
$356$	9.49345e10	0.313256
$357$	0	0
$358$	−3.03224e11	−0.975641
$359$	−2.49683e11	−0.793347	−0.396674	−	0.917960i	$-0.629836\pi$
$359$	−2.49683e11	−0.793347	−0.396674	+	0.917960i	$0.629836\pi$
$360$	0	0
$361$	−3.09007e11	−0.957603
$362$	3.24691e11	0.993760
$363$	0	0
$364$	5.12225e10	0.152934
$365$	−1.82134e11	−0.537121
$366$	0	0
$367$	−5.53325e11	−1.59215	−0.796073	−	0.605201i	$-0.793093\pi$
$367$	−5.53325e11	−1.59215	−0.796073	+	0.605201i	$0.793093\pi$
$368$	−1.71116e10	−0.0486379
$369$	0	0
$370$	6.25443e10	0.173492
$371$	−1.02771e11	−0.281635
$372$	0	0
$373$	−1.30552e11	−0.349216	−0.174608	−	0.984638i	$-0.555866\pi$
$373$	−1.30552e11	−0.349216	−0.174608	+	0.984638i	$0.555866\pi$
$374$	6.39305e10	0.168961
$375$	0	0
$376$	8.58758e10	0.221578
$377$	3.13672e10	0.0799722
$378$	0	0
$379$	−3.21465e11	−0.800307	−0.400154	−	0.916448i	$-0.631043\pi$
$379$	−3.21465e11	−0.800307	−0.400154	+	0.916448i	$0.631043\pi$
$380$	−1.30253e10	−0.0320452
$381$	0	0
$382$	−1.90664e11	−0.458125
$383$	−6.80223e11	−1.61531	−0.807656	−	0.589654i	$-0.799264\pi$
$383$	−6.80223e11	−1.61531	−0.807656	+	0.589654i	$0.799264\pi$
$384$	0	0
$385$	−1.00284e10	−0.0232626
$386$	−8.04748e10	−0.184509
$387$	0	0
$388$	7.93195e10	0.177680
$389$	7.94587e11	1.75941	0.879707	−	0.475515i	$-0.157738\pi$
$389$	7.94587e11	1.75941	0.879707	+	0.475515i	$0.157738\pi$
$390$	0	0
$391$	1.14356e11	0.247437
$392$	−1.39132e11	−0.297606
$393$	0	0
$394$	−9.53249e9	−0.0199284
$395$	−9.16101e10	−0.189346
$396$	0	0
$397$	1.97913e11	0.399868	0.199934	−	0.979809i	$-0.435927\pi$
$397$	1.97913e11	0.399868	0.199934	+	0.979809i	$0.435927\pi$
$398$	3.11511e11	0.622300
$399$	0	0
$400$	−1.15599e11	−0.225779
$401$	9.56475e10	0.184724	0.0923622	−	0.995725i	$-0.470558\pi$
$401$	9.56475e10	0.184724	0.0923622	+	0.995725i	$0.470558\pi$
$402$	0	0
$403$	4.65779e11	0.879644
$404$	4.50472e11	0.841301
$405$	0	0
$406$	1.60171e10	0.0292562
$407$	8.19815e10	0.148095
$408$	0	0
$409$	2.82203e11	0.498663	0.249332	−	0.968418i	$-0.419789\pi$
$409$	2.82203e11	0.498663	0.249332	+	0.968418i	$0.419789\pi$
$410$	−1.21449e11	−0.212259
$411$	0	0
$412$	−3.41511e9	−0.00583939
$413$	2.13236e11	0.360650
$414$	0	0
$415$	−3.37732e11	−0.558927
$416$	−8.30262e10	−0.135924
$417$	0	0
$418$	−1.70733e10	−0.0273542
$419$	1.52503e11	0.241722	0.120861	−	0.992669i	$-0.461435\pi$
$419$	1.52503e11	0.241722	0.120861	+	0.992669i	$0.461435\pi$
$420$	0	0
$421$	−1.07657e11	−0.167022	−0.0835111	−	0.996507i	$-0.526613\pi$
$421$	−1.07657e11	−0.167022	−0.0835111	+	0.996507i	$0.526613\pi$
$422$	−3.35118e11	−0.514390
$423$	0	0
$424$	1.66580e11	0.250310
$425$	7.72546e11	1.14861
$426$	0	0
$427$	3.74093e11	0.544571
$428$	−2.62341e10	−0.0377894
$429$	0	0
$430$	2.23379e11	0.315089
$431$	−7.77462e10	−0.108525	−0.0542627	−	0.998527i	$-0.517281\pi$
$431$	−7.77462e10	−0.108525	−0.0542627	+	0.998527i	$0.517281\pi$
$432$	0	0
$433$	1.20952e11	0.165356	0.0826778	−	0.996576i	$-0.473653\pi$
$433$	1.20952e11	0.165356	0.0826778	+	0.996576i	$0.473653\pi$
$434$	2.37843e11	0.321800
$435$	0	0
$436$	−1.66071e11	−0.220092
$437$	−3.05401e10	−0.0400593
$438$	0	0
$439$	1.03193e12	1.32605	0.663026	−	0.748597i	$-0.269272\pi$
$439$	1.03193e12	1.32605	0.663026	+	0.748597i	$0.269272\pi$
$440$	1.62550e10	0.0206752
$441$	0	0
$442$	5.54863e11	0.691489
$443$	−6.20515e11	−0.765483	−0.382742	−	0.923855i	$-0.625020\pi$
$443$	−6.20515e11	−0.765483	−0.382742	+	0.923855i	$0.625020\pi$
$444$	0	0
$445$	−1.61314e11	−0.195008
$446$	−5.98784e11	−0.716578
$447$	0	0
$448$	−4.23960e10	−0.0497249
$449$	7.51040e11	0.872076	0.436038	−	0.899928i	$-0.356381\pi$
$449$	7.51040e11	0.872076	0.436038	+	0.899928i	$0.356381\pi$
$450$	0	0
$451$	−1.59192e11	−0.181187
$452$	−7.86563e11	−0.886360
$453$	0	0
$454$	−2.46194e11	−0.271973
$455$	−8.70382e10	−0.0952048
$456$	0	0
$457$	1.33370e12	1.43033	0.715165	−	0.698955i	$-0.246351\pi$
$457$	1.33370e12	1.43033	0.715165	+	0.698955i	$0.246351\pi$
$458$	−1.14233e12	−1.21310
$459$	0	0
$460$	2.90763e10	0.0302781
$461$	−5.27899e11	−0.544373	−0.272187	−	0.962244i	$-0.587747\pi$
$461$	−5.27899e11	−0.544373	−0.272187	+	0.962244i	$0.587747\pi$
$462$	0	0
$463$	1.08746e12	1.09976	0.549879	−	0.835244i	$-0.314674\pi$
$463$	1.08746e12	1.09976	0.549879	+	0.835244i	$0.314674\pi$
$464$	−2.59621e10	−0.0260021
$465$	0	0
$466$	7.02684e11	0.690277
$467$	8.01186e11	0.779484	0.389742	−	0.920924i	$-0.372564\pi$
$467$	8.01186e11	0.779484	0.389742	+	0.920924i	$0.372564\pi$
$468$	0	0
$469$	−3.91531e11	−0.373670
$470$	−1.45922e11	−0.137937
$471$	0	0
$472$	−3.45633e11	−0.320536
$473$	2.92799e11	0.268965
$474$	0	0
$475$	−2.06316e11	−0.185957
$476$	2.83332e11	0.252967
$477$	0	0
$478$	1.23127e12	1.07877
$479$	3.89798e11	0.338322	0.169161	−	0.985588i	$-0.445894\pi$
$479$	3.89798e11	0.338322	0.169161	+	0.985588i	$0.445894\pi$
$480$	0	0
$481$	7.11531e11	0.606095
$482$	−3.63353e11	−0.306632
$483$	0	0
$484$	−5.82328e11	−0.482351
$485$	−1.34781e11	−0.110609
$486$	0	0
$487$	−1.90992e12	−1.53863	−0.769317	−	0.638868i	$-0.779403\pi$
$487$	−1.90992e12	−1.53863	−0.769317	+	0.638868i	$0.779403\pi$
$488$	−6.06365e11	−0.484000
$489$	0	0
$490$	2.36416e11	0.185266
$491$	3.88418e11	0.301601	0.150800	−	0.988564i	$-0.451815\pi$
$491$	3.88418e11	0.301601	0.150800	+	0.988564i	$0.451815\pi$
$492$	0	0
$493$	1.73504e11	0.132281
$494$	−1.48182e11	−0.111950
$495$	0	0
$496$	−3.85518e11	−0.286007
$497$	−4.25404e11	−0.312750
$498$	0	0
$499$	−5.18044e10	−0.0374037	−0.0187018	−	0.999825i	$-0.505953\pi$
$499$	−5.18044e10	−0.0374037	−0.0187018	+	0.999825i	$0.505953\pi$
$500$	4.13928e11	0.296183
$501$	0	0
$502$	−1.32364e12	−0.930259
$503$	−9.91895e11	−0.690891	−0.345446	−	0.938439i	$-0.612272\pi$
$503$	−9.91895e11	−0.690891	−0.345446	+	0.938439i	$0.612272\pi$
$504$	0	0
$505$	−7.65450e11	−0.523728
$506$	3.81125e10	0.0258458
$507$	0	0
$508$	−8.55319e11	−0.569825
$509$	2.74642e12	1.81358	0.906792	−	0.421579i	$-0.138524\pi$
$509$	2.74642e12	1.81358	0.906792	+	0.421579i	$0.138524\pi$
$510$	0	0
$511$	−1.05805e12	−0.686455
$512$	6.87195e10	0.0441942
$513$	0	0
$514$	1.40766e12	0.889539
$515$	5.80302e9	0.00363515
$516$	0	0
$517$	−1.91271e11	−0.117745
$518$	3.63332e11	0.221728
$519$	0	0
$520$	1.41080e11	0.0846154
$521$	−5.16949e11	−0.307382	−0.153691	−	0.988119i	$-0.549116\pi$
$521$	−5.16949e11	−0.307382	−0.153691	+	0.988119i	$0.549116\pi$
$522$	0	0
$523$	−3.12202e11	−0.182464	−0.0912322	−	0.995830i	$-0.529081\pi$
$523$	−3.12202e11	−0.182464	−0.0912322	+	0.995830i	$0.529081\pi$
$524$	−1.24177e12	−0.719530
$525$	0	0
$526$	−2.41180e12	−1.37374
$527$	2.57641e12	1.45501
$528$	0	0
$529$	−1.73298e12	−0.962150
$530$	−2.83057e11	−0.155823
$531$	0	0
$532$	−7.56667e10	−0.0409546
$533$	−1.38166e12	−0.741528
$534$	0	0
$535$	4.45775e10	0.0235247
$536$	6.34631e11	0.332108
$537$	0	0
$538$	1.09789e12	0.564987
$539$	3.09889e11	0.158145
$540$	0	0
$541$	−1.35533e12	−0.680234	−0.340117	−	0.940383i	$-0.610467\pi$
$541$	−1.35533e12	−0.680234	−0.340117	+	0.940383i	$0.610467\pi$
$542$	3.77098e11	0.187697
$543$	0	0
$544$	−4.59251e11	−0.224830
$545$	2.82191e11	0.137012
$546$	0	0
$547$	−2.76918e12	−1.32254	−0.661269	−	0.750149i	$-0.729982\pi$
$547$	−2.76918e12	−1.32254	−0.661269	+	0.750149i	$0.729982\pi$
$548$	9.98453e11	0.472950
$549$	0	0
$550$	2.57473e11	0.119977
$551$	−4.63361e10	−0.0214159
$552$	0	0
$553$	−5.32181e11	−0.241989
$554$	1.96568e12	0.886580
$555$	0	0
$556$	−1.36797e12	−0.607074
$557$	−8.90709e11	−0.392091	−0.196046	−	0.980595i	$-0.562810\pi$
$557$	−8.90709e11	−0.392091	−0.196046	+	0.980595i	$0.562810\pi$
$558$	0	0
$559$	2.54125e12	1.10077
$560$	7.20401e10	0.0309548
$561$	0	0
$562$	5.58662e11	0.236230
$563$	−2.27487e10	−0.00954264	−0.00477132	−	0.999989i	$-0.501519\pi$
$563$	−2.27487e10	−0.00954264	−0.00477132	+	0.999989i	$0.501519\pi$
$564$	0	0
$565$	1.33654e12	0.551778
$566$	−1.21863e12	−0.499113
$567$	0	0
$568$	6.89535e11	0.277964
$569$	−4.33772e12	−1.73483	−0.867415	−	0.497586i	$-0.834220\pi$
$569$	−4.33772e12	−1.73483	−0.867415	+	0.497586i	$0.834220\pi$
$570$	0	0
$571$	2.48077e12	0.976617	0.488308	−	0.872671i	$-0.337614\pi$
$571$	2.48077e12	0.976617	0.488308	+	0.872671i	$0.337614\pi$
$572$	1.84924e11	0.0722289
$573$	0	0
$574$	−7.05521e11	−0.271273
$575$	4.60558e11	0.175703
$576$	0	0
$577$	−1.50387e12	−0.564831	−0.282415	−	0.959292i	$-0.591136\pi$
$577$	−1.50387e12	−0.564831	−0.282415	+	0.959292i	$0.591136\pi$
$578$	1.17176e12	0.436681
$579$	0	0
$580$	4.41153e10	0.0161869
$581$	−1.96195e12	−0.714324
$582$	0	0
$583$	−3.71024e11	−0.133013
$584$	1.71499e12	0.610103
$585$	0	0
$586$	2.83717e12	0.993908
$587$	−1.80767e12	−0.628416	−0.314208	−	0.949354i	$-0.601739\pi$
$587$	−1.80767e12	−0.628416	−0.314208	+	0.949354i	$0.601739\pi$
$588$	0	0
$589$	−6.88056e11	−0.235562
$590$	5.87306e11	0.199540
$591$	0	0
$592$	−5.88923e11	−0.197065
$593$	−4.13005e12	−1.37154	−0.685771	−	0.727818i	$-0.740535\pi$
$593$	−4.13005e12	−1.37154	−0.685771	+	0.727818i	$0.740535\pi$
$594$	0	0
$595$	−4.81443e11	−0.157477
$596$	−1.52071e12	−0.493672
$597$	0	0
$598$	3.30785e11	0.105777
$599$	1.62323e12	0.515182	0.257591	−	0.966254i	$-0.417071\pi$
$599$	1.62323e12	0.515182	0.257591	+	0.966254i	$0.417071\pi$
$600$	0	0
$601$	1.87930e12	0.587573	0.293787	−	0.955871i	$-0.405084\pi$
$601$	1.87930e12	0.587573	0.293787	+	0.955871i	$0.405084\pi$
$602$	1.29765e12	0.402693
$603$	0	0
$604$	3.19435e11	0.0976598
$605$	9.89503e11	0.300274
$606$	0	0
$607$	5.43492e11	0.162497	0.0812483	−	0.996694i	$-0.474109\pi$
$607$	5.43492e11	0.162497	0.0812483	+	0.996694i	$0.474109\pi$
$608$	1.22648e11	0.0363993
$609$	0	0
$610$	1.03035e12	0.301300
$611$	−1.66007e12	−0.481882
$612$	0	0
$613$	3.90638e12	1.11738	0.558692	−	0.829375i	$-0.311303\pi$
$613$	3.90638e12	1.11738	0.558692	+	0.829375i	$0.311303\pi$
$614$	3.06487e12	0.870271
$615$	0	0
$616$	9.44285e10	0.0264235
$617$	−4.16450e12	−1.15686	−0.578428	−	0.815733i	$-0.696334\pi$
$617$	−4.16450e12	−1.15686	−0.578428	+	0.815733i	$0.696334\pi$
$618$	0	0
$619$	2.42966e12	0.665177	0.332589	−	0.943072i	$-0.392078\pi$
$619$	2.42966e12	0.665177	0.332589	+	0.943072i	$0.392078\pi$
$620$	6.55079e11	0.178045
$621$	0	0
$622$	−2.44647e12	−0.655366
$623$	−9.37107e11	−0.249226
$624$	0	0
$625$	2.74176e12	0.718737
$626$	−1.98081e11	−0.0515534
$627$	0	0
$628$	1.31186e12	0.336564
$629$	3.93576e12	1.00254
$630$	0	0
$631$	−1.52098e12	−0.381936	−0.190968	−	0.981596i	$-0.561163\pi$
$631$	−1.52098e12	−0.381936	−0.190968	+	0.981596i	$0.561163\pi$
$632$	8.62610e11	0.215074
$633$	0	0
$634$	9.64717e11	0.237136
$635$	1.45337e12	0.354728
$636$	0	0
$637$	2.68958e12	0.647227
$638$	5.78252e10	0.0138173
$639$	0	0
$640$	−1.16769e11	−0.0275118
$641$	6.09689e11	0.142642	0.0713210	−	0.997453i	$-0.477279\pi$
$641$	6.09689e11	0.142642	0.0713210	+	0.997453i	$0.477279\pi$
$642$	0	0
$643$	4.95241e12	1.14253	0.571265	−	0.820766i	$-0.306453\pi$
$643$	4.95241e12	1.14253	0.571265	+	0.820766i	$0.306453\pi$
$644$	1.68910e11	0.0386963
$645$	0	0
$646$	−8.19653e11	−0.185176
$647$	3.20285e12	0.718567	0.359284	−	0.933228i	$-0.383021\pi$
$647$	3.20285e12	0.718567	0.359284	+	0.933228i	$0.383021\pi$
$648$	0	0
$649$	7.69827e11	0.170330
$650$	2.23465e12	0.491020
$651$	0	0
$652$	3.01851e12	0.654151
$653$	−6.19750e12	−1.33385	−0.666925	−	0.745125i	$-0.732390\pi$
$653$	−6.19750e12	−1.33385	−0.666925	+	0.745125i	$0.732390\pi$
$654$	0	0
$655$	2.11003e12	0.447923
$656$	1.14357e12	0.241100
$657$	0	0
$658$	−8.47688e11	−0.176287
$659$	8.01019e12	1.65447	0.827235	−	0.561857i	$-0.189913\pi$
$659$	8.01019e12	1.65447	0.827235	+	0.561857i	$0.189913\pi$
$660$	0	0
$661$	−8.94817e12	−1.82317	−0.911586	−	0.411109i	$-0.865141\pi$
$661$	−8.94817e12	−1.82317	−0.911586	+	0.411109i	$0.865141\pi$
$662$	−1.19206e12	−0.241232
$663$	0	0
$664$	3.18011e12	0.634872
$665$	1.28574e11	0.0254951
$666$	0	0
$667$	1.03436e11	0.0202350
$668$	−2.66433e12	−0.517718
$669$	0	0
$670$	−1.07838e12	−0.206744
$671$	1.35055e12	0.257194
$672$	0	0
$673$	5.32555e12	1.00068	0.500341	−	0.865828i	$-0.333208\pi$
$673$	5.32555e12	1.00068	0.500341	+	0.865828i	$0.333208\pi$
$674$	−3.08257e12	−0.575366
$675$	0	0
$676$	−1.10977e12	−0.204396
$677$	1.10212e11	0.0201642	0.0100821	−	0.999949i	$-0.496791\pi$
$677$	1.10212e11	0.0201642	0.0100821	+	0.999949i	$0.496791\pi$
$678$	0	0
$679$	−7.82971e11	−0.141362
$680$	7.80368e11	0.139962
$681$	0	0
$682$	8.58662e11	0.151982
$683$	3.48463e12	0.612722	0.306361	−	0.951915i	$-0.400889\pi$
$683$	3.48463e12	0.612722	0.306361	+	0.951915i	$0.400889\pi$
$684$	0	0
$685$	−1.69659e12	−0.294421
$686$	3.00497e12	0.518061
$687$	0	0
$688$	−2.10335e12	−0.357902
$689$	−3.22018e12	−0.544369
$690$	0	0
$691$	−1.10169e13	−1.83827	−0.919137	−	0.393939i	$-0.871112\pi$
$691$	−1.10169e13	−1.83827	−0.919137	+	0.393939i	$0.871112\pi$
$692$	−8.96947e11	−0.148693
$693$	0	0
$694$	−8.46136e12	−1.38459
$695$	2.32449e12	0.377916
$696$	0	0
$697$	−7.64249e12	−1.22656
$698$	−3.11706e12	−0.497045
$699$	0	0
$700$	1.14109e12	0.179630
$701$	−8.81897e12	−1.37939	−0.689695	−	0.724100i	$-0.742255\pi$
$701$	−8.81897e12	−1.37939	−0.689695	+	0.724100i	$0.742255\pi$
$702$	0	0
$703$	−1.05109e12	−0.162308
$704$	−1.53059e11	−0.0234845
$705$	0	0
$706$	1.35028e12	0.204551
$707$	−4.44665e12	−0.669338
$708$	0	0
$709$	−1.84914e12	−0.274829	−0.137414	−	0.990514i	$-0.543879\pi$
$709$	−1.84914e12	−0.274829	−0.137414	+	0.990514i	$0.543879\pi$
$710$	−1.17167e12	−0.173039
$711$	0	0
$712$	1.51895e12	0.221505
$713$	1.53594e12	0.222573
$714$	0	0
$715$	−3.14226e11	−0.0449640
$716$	−4.85158e12	−0.689882
$717$	0	0
$718$	−3.99492e12	−0.560981
$719$	6.42135e12	0.896078	0.448039	−	0.894014i	$-0.352122\pi$
$719$	6.42135e12	0.896078	0.448039	+	0.894014i	$0.352122\pi$
$720$	0	0
$721$	3.37109e10	0.00464581
$722$	−4.94411e12	−0.677127
$723$	0	0
$724$	5.19505e12	0.702695
$725$	6.98769e11	0.0939318
$726$	0	0
$727$	−5.64920e12	−0.750036	−0.375018	−	0.927018i	$-0.622363\pi$
$727$	−5.64920e12	−0.750036	−0.375018	+	0.927018i	$0.622363\pi$
$728$	8.19560e11	0.108141
$729$	0	0
$730$	−2.91414e12	−0.379802
$731$	1.40567e13	1.82077
$732$	0	0
$733$	−1.24309e12	−0.159051	−0.0795254	−	0.996833i	$-0.525340\pi$
$733$	−1.24309e12	−0.159051	−0.0795254	+	0.996833i	$0.525340\pi$
$734$	−8.85319e12	−1.12582
$735$	0	0
$736$	−2.73785e11	−0.0343922
$737$	−1.41351e12	−0.176480
$738$	0	0
$739$	−2.20942e12	−0.272507	−0.136253	−	0.990674i	$-0.543506\pi$
$739$	−2.20942e12	−0.272507	−0.136253	+	0.990674i	$0.543506\pi$
$740$	1.00071e12	0.122677
$741$	0	0
$742$	−1.64433e12	−0.199146
$743$	1.66986e11	0.0201016	0.0100508	−	0.999949i	$-0.496801\pi$
$743$	1.66986e11	0.0201016	0.0100508	+	0.999949i	$0.496801\pi$
$744$	0	0
$745$	2.58402e12	0.307321
$746$	−2.08884e12	−0.246933
$747$	0	0
$748$	1.02289e12	0.119473
$749$	2.58959e11	0.0300652
$750$	0	0
$751$	−4.97060e12	−0.570202	−0.285101	−	0.958497i	$-0.592027\pi$
$751$	−4.97060e12	−0.570202	−0.285101	+	0.958497i	$0.592027\pi$
$752$	1.37401e12	0.156679
$753$	0	0
$754$	5.01875e11	0.0565489
$755$	−5.42790e11	−0.0607953
$756$	0	0
$757$	9.48178e12	1.04944	0.524721	−	0.851274i	$-0.324170\pi$
$757$	9.48178e12	1.04944	0.524721	+	0.851274i	$0.324170\pi$
$758$	−5.14343e12	−0.565903
$759$	0	0
$760$	−2.08405e11	−0.0226594
$761$	−9.47065e12	−1.02364	−0.511822	−	0.859092i	$-0.671029\pi$
$761$	−9.47065e12	−1.02364	−0.511822	+	0.859092i	$0.671029\pi$
$762$	0	0
$763$	1.63930e12	0.175105
$764$	−3.05063e12	−0.323943
$765$	0	0
$766$	−1.08836e13	−1.14220
$767$	6.68145e12	0.697095
$768$	0	0
$769$	−1.02727e13	−1.05930	−0.529649	−	0.848217i	$-0.677676\pi$
$769$	−1.02727e13	−1.05930	−0.529649	+	0.848217i	$0.677676\pi$
$770$	−1.60455e11	−0.0164492
$771$	0	0
$772$	−1.28760e12	−0.130467
$773$	1.13305e13	1.14141	0.570704	−	0.821156i	$-0.306670\pi$
$773$	1.13305e13	1.14141	0.570704	+	0.821156i	$0.306670\pi$
$774$	0	0
$775$	1.03762e13	1.03319
$776$	1.26911e12	0.125638
$777$	0	0
$778$	1.27134e13	1.24409
$779$	2.04101e12	0.198576
$780$	0	0
$781$	−1.53580e12	−0.147708
$782$	1.82970e12	0.174965
$783$	0	0
$784$	−2.22612e12	−0.210439
$785$	−2.22913e12	−0.209518
$786$	0	0
$787$	6.86916e12	0.638289	0.319144	−	0.947706i	$-0.396605\pi$
$787$	6.86916e12	0.638289	0.319144	+	0.947706i	$0.396605\pi$
$788$	−1.52520e11	−0.0140915
$789$	0	0
$790$	−1.46576e12	−0.133888
$791$	7.76423e12	0.705187
$792$	0	0
$793$	1.17217e13	1.05259
$794$	3.16660e12	0.282749
$795$	0	0
$796$	4.98418e12	0.440032
$797$	8.35867e12	0.733795	0.366898	−	0.930261i	$-0.380420\pi$
$797$	8.35867e12	0.733795	0.366898	+	0.930261i	$0.380420\pi$
$798$	0	0
$799$	−9.18251e12	−0.797078
$800$	−1.84958e12	−0.159650
$801$	0	0
$802$	1.53036e12	0.130620
$803$	−3.81978e12	−0.324204
$804$	0	0
$805$	−2.87015e11	−0.0240892
$806$	7.45246e12	0.622003
$807$	0	0
$808$	7.20755e12	0.594890
$809$	−5.37009e12	−0.440771	−0.220386	−	0.975413i	$-0.570732\pi$
$809$	−5.37009e12	−0.440771	−0.220386	+	0.975413i	$0.570732\pi$
$810$	0	0
$811$	6.15396e12	0.499529	0.249764	−	0.968307i	$-0.419647\pi$
$811$	6.15396e12	0.499529	0.249764	+	0.968307i	$0.419647\pi$
$812$	2.56274e11	0.0206872
$813$	0	0
$814$	1.31170e12	0.104719
$815$	−5.12910e12	−0.407223
$816$	0	0
$817$	−3.75398e12	−0.294777
$818$	4.51525e12	0.352608
$819$	0	0
$820$	−1.94318e12	−0.150090
$821$	7.73743e12	0.594364	0.297182	−	0.954821i	$-0.403953\pi$
$821$	7.73743e12	0.594364	0.297182	+	0.954821i	$0.403953\pi$
$822$	0	0
$823$	2.19595e13	1.66849	0.834244	−	0.551395i	$-0.185904\pi$
$823$	2.19595e13	1.66849	0.834244	+	0.551395i	$0.185904\pi$
$824$	−5.46418e10	−0.00412907
$825$	0	0
$826$	3.41178e12	0.255018
$827$	1.89761e12	0.141069	0.0705346	−	0.997509i	$-0.477529\pi$
$827$	1.89761e12	0.141069	0.0705346	+	0.997509i	$0.477529\pi$
$828$	0	0
$829$	1.14625e13	0.842912	0.421456	−	0.906849i	$-0.361519\pi$
$829$	1.14625e13	0.842912	0.421456	+	0.906849i	$0.361519\pi$
$830$	−5.40371e12	−0.395221
$831$	0	0
$832$	−1.32842e12	−0.0961126
$833$	1.48771e13	1.07057
$834$	0	0
$835$	4.52727e12	0.322290
$836$	−2.73173e11	−0.0193423
$837$	0	0
$838$	2.44005e12	0.170923
$839$	−2.40490e13	−1.67559	−0.837794	−	0.545986i	$-0.816155\pi$
$839$	−2.40490e13	−1.67559	−0.837794	+	0.545986i	$0.816155\pi$
$840$	0	0
$841$	−1.43502e13	−0.989182
$842$	−1.72252e12	−0.118103
$843$	0	0
$844$	−5.36189e12	−0.363728
$845$	1.88574e12	0.127241
$846$	0	0
$847$	5.74821e12	0.383758
$848$	2.66529e12	0.176996
$849$	0	0
$850$	1.23607e13	0.812193
$851$	2.34633e12	0.153358
$852$	0	0
$853$	2.28559e13	1.47818	0.739089	−	0.673608i	$-0.235256\pi$
$853$	2.28559e13	1.47818	0.739089	+	0.673608i	$0.235256\pi$
$854$	5.98549e12	0.385070
$855$	0	0
$856$	−4.19746e11	−0.0267211
$857$	−2.61764e13	−1.65766	−0.828832	−	0.559497i	$-0.810994\pi$
$857$	−2.61764e13	−1.65766	−0.828832	+	0.559497i	$0.810994\pi$
$858$	0	0
$859$	−3.56167e12	−0.223195	−0.111598	−	0.993753i	$-0.535597\pi$
$859$	−3.56167e12	−0.223195	−0.111598	+	0.993753i	$0.535597\pi$
$860$	3.57406e12	0.222802
$861$	0	0
$862$	−1.24394e12	−0.0767390
$863$	−2.73564e13	−1.67884	−0.839421	−	0.543481i	$-0.817106\pi$
$863$	−2.73564e13	−1.67884	−0.839421	+	0.543481i	$0.817106\pi$
$864$	0	0
$865$	1.52411e12	0.0925643
$866$	1.93524e12	0.116924
$867$	0	0
$868$	3.80548e12	0.227547
$869$	−1.92129e12	−0.114289
$870$	0	0
$871$	−1.22681e13	−0.722262
$872$	−2.65713e12	−0.155629
$873$	0	0
$874$	−4.88641e11	−0.0283262
$875$	−4.08592e12	−0.235643
$876$	0	0
$877$	−1.10519e13	−0.630867	−0.315434	−	0.948948i	$-0.602150\pi$
$877$	−1.10519e13	−0.630867	−0.315434	+	0.948948i	$0.602150\pi$
$878$	1.65109e13	0.937660
$879$	0	0
$880$	2.60080e11	0.0146196
$881$	−2.76007e13	−1.54358	−0.771789	−	0.635879i	$-0.780638\pi$
$881$	−2.76007e13	−1.54358	−0.771789	+	0.635879i	$0.780638\pi$
$882$	0	0
$883$	2.98889e11	0.0165458	0.00827288	−	0.999966i	$-0.497367\pi$
$883$	2.98889e11	0.0165458	0.00827288	+	0.999966i	$0.497367\pi$
$884$	8.87781e12	0.488957
$885$	0	0
$886$	−9.92824e12	−0.541278
$887$	3.08545e13	1.67364	0.836819	−	0.547479i	$-0.184412\pi$
$887$	3.08545e13	1.67364	0.836819	+	0.547479i	$0.184412\pi$
$888$	0	0
$889$	8.44293e12	0.453352
$890$	−2.58103e12	−0.137892
$891$	0	0
$892$	−9.58055e12	−0.506697
$893$	2.45228e12	0.129044
$894$	0	0
$895$	8.24390e12	0.429466
$896$	−6.78336e11	−0.0351608
$897$	0	0
$898$	1.20166e13	0.616651
$899$	2.33037e12	0.118989
$900$	0	0
$901$	−1.78121e13	−0.900436
$902$	−2.54708e12	−0.128119
$903$	0	0
$904$	−1.25850e13	−0.626752
$905$	−8.82753e12	−0.437442
$906$	0	0
$907$	−2.33746e13	−1.14686	−0.573431	−	0.819254i	$-0.694388\pi$
$907$	−2.33746e13	−1.14686	−0.573431	+	0.819254i	$0.694388\pi$
$908$	−3.93910e12	−0.192314
$909$	0	0
$910$	−1.39261e12	−0.0673199
$911$	−2.73111e13	−1.31373	−0.656865	−	0.754009i	$-0.728118\pi$
$911$	−2.73111e13	−1.31373	−0.656865	+	0.754009i	$0.728118\pi$
$912$	0	0
$913$	−7.08305e12	−0.337366
$914$	2.13393e13	1.01140
$915$	0	0
$916$	−1.82772e13	−0.857789
$917$	1.22576e13	0.572457
$918$	0	0
$919$	−2.88513e13	−1.33427	−0.667137	−	0.744935i	$-0.732481\pi$
$919$	−2.88513e13	−1.33427	−0.667137	+	0.744935i	$0.732481\pi$
$920$	4.65221e11	0.0214099
$921$	0	0
$922$	−8.44639e12	−0.384930
$923$	−1.33294e13	−0.604511
$924$	0	0
$925$	1.58508e13	0.711893
$926$	1.73993e13	0.777647
$927$	0	0
$928$	−4.15393e11	−0.0183863
$929$	4.94460e12	0.217801	0.108901	−	0.994053i	$-0.465267\pi$
$929$	4.94460e12	0.217801	0.108901	+	0.994053i	$0.465267\pi$
$930$	0	0
$931$	−3.97309e12	−0.173322
$932$	1.12429e13	0.488099
$933$	0	0
$934$	1.28190e13	0.551179
$935$	−1.73811e12	−0.0743747
$936$	0	0
$937$	−3.28563e13	−1.39248	−0.696242	−	0.717807i	$-0.745146\pi$
$937$	−3.28563e13	−1.39248	−0.696242	+	0.717807i	$0.745146\pi$
$938$	−6.26450e12	−0.264225
$939$	0	0
$940$	−2.33475e12	−0.0975360
$941$	4.60517e13	1.91466	0.957332	−	0.288991i	$-0.0933199\pi$
$941$	4.60517e13	1.91466	0.957332	+	0.288991i	$0.0933199\pi$
$942$	0	0
$943$	−4.55612e12	−0.187626
$944$	−5.53013e12	−0.226653
$945$	0	0
$946$	4.68479e12	0.190187
$947$	−2.89440e13	−1.16946	−0.584728	−	0.811229i	$-0.698799\pi$
$947$	−2.89440e13	−1.16946	−0.584728	+	0.811229i	$0.698799\pi$
$948$	0	0
$949$	−3.31525e13	−1.32684
$950$	−3.30106e12	−0.131491
$951$	0	0
$952$	4.53331e12	0.178875
$953$	2.49105e13	0.978284	0.489142	−	0.872204i	$-0.337310\pi$
$953$	2.49105e13	0.978284	0.489142	+	0.872204i	$0.337310\pi$
$954$	0	0
$955$	5.18369e12	0.201662
$956$	1.97003e13	0.762804
$957$	0	0
$958$	6.23677e12	0.239230
$959$	−9.85583e12	−0.376278
$960$	0	0
$961$	8.16457e12	0.308801
$962$	1.13845e13	0.428574
$963$	0	0
$964$	−5.81366e12	−0.216822
$965$	2.18791e12	0.0812188
$966$	0	0
$967$	3.98414e13	1.46526	0.732632	−	0.680625i	$-0.238292\pi$
$967$	3.98414e13	1.46526	0.732632	+	0.680625i	$0.238292\pi$
$968$	−9.31725e12	−0.341074
$969$	0	0
$970$	−2.15650e12	−0.0782126
$971$	−1.76953e13	−0.638809	−0.319405	−	0.947618i	$-0.603483\pi$
$971$	−1.76953e13	−0.638809	−0.319405	+	0.947618i	$0.603483\pi$
$972$	0	0
$973$	1.35034e13	0.482987
$974$	−3.05587e13	−1.08798
$975$	0	0
$976$	−9.70185e12	−0.342240
$977$	−3.24515e12	−0.113949	−0.0569744	−	0.998376i	$-0.518145\pi$
$977$	−3.24515e12	−0.113949	−0.0569744	+	0.998376i	$0.518145\pi$
$978$	0	0
$979$	−3.38315e12	−0.117706
$980$	3.78266e12	0.131003
$981$	0	0
$982$	6.21469e12	0.213264
$983$	4.17615e13	1.42654	0.713272	−	0.700887i	$-0.247212\pi$
$983$	4.17615e13	1.42654	0.713272	+	0.700887i	$0.247212\pi$
$984$	0	0
$985$	2.59164e11	0.00877228
$986$	2.77607e12	0.0935371
$987$	0	0
$988$	−2.37091e12	−0.0791606
$989$	8.37998e12	0.278522
$990$	0	0
$991$	−5.72174e13	−1.88450	−0.942251	−	0.334907i	$-0.891295\pi$
$991$	−5.72174e13	−1.88450	−0.942251	+	0.334907i	$0.891295\pi$
$992$	−6.16828e12	−0.202238
$993$	0	0
$994$	−6.80647e12	−0.221148
$995$	−8.46921e12	−0.273930
$996$	0	0
$997$	−4.42027e12	−0.141684	−0.0708420	−	0.997488i	$-0.522569\pi$
$997$	−4.42027e12	−0.141684	−0.0708420	+	0.997488i	$0.522569\pi$
$998$	−8.28870e11	−0.0264484
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.10.a.c.1.1	yes	1
3.2	odd	2		54.10.a.b.1.1	✓	1
9.2	odd	6		162.10.c.g.109.1		2
9.4	even	3		162.10.c.d.55.1		2
9.5	odd	6		162.10.c.g.55.1		2
9.7	even	3		162.10.c.d.109.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.10.a.b.1.1	✓	1	3.2	odd	2
54.10.a.c.1.1	yes	1	1.1	even	1	trivial
162.10.c.d.55.1		2	9.4	even	3
162.10.c.d.109.1		2	9.7	even	3
162.10.c.g.55.1		2	9.5	odd	6
162.10.c.g.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:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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