LMFDB - Embedded newform 18.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,8,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, 8, names="a")

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
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:	$5.62293045871$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
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)

$f(q)$

$=$

$q-8.00000 q^{2} +64.0000 q^{4} +114.000 q^{5} -1576.00 q^{7} -512.000 q^{8} +O(q^{10})$

\(q-8.00000 q^{2} +64.0000 q^{4} +114.000 q^{5} -1576.00 q^{7} -512.000 q^{8} -912.000 q^{10} -7332.00 q^{11} -3802.00 q^{13} +12608.0 q^{14} +4096.00 q^{16} +6606.00 q^{17} +24860.0 q^{19} +7296.00 q^{20} +58656.0 q^{22} -41448.0 q^{23} -65129.0 q^{25} +30416.0 q^{26} -100864. q^{28} +41610.0 q^{29} +33152.0 q^{31} -32768.0 q^{32} -52848.0 q^{34} -179664. q^{35} -36466.0 q^{37} -198880. q^{38} -58368.0 q^{40} +639078. q^{41} -156412. q^{43} -469248. q^{44} +331584. q^{46} +433776. q^{47} +1.66023e6 q^{49} +521032. q^{50} -243328. q^{52} -786078. q^{53} -835848. q^{55} +806912. q^{56} -332880. q^{58} -745140. q^{59} -1.66062e6 q^{61} -265216. q^{62} +262144. q^{64} -433428. q^{65} -3.29084e6 q^{67} +422784. q^{68} +1.43731e6 q^{70} -5.71615e6 q^{71} +2.65990e6 q^{73} +291728. q^{74} +1.59104e6 q^{76} +1.15552e7 q^{77} +3.80744e6 q^{79} +466944. q^{80} -5.11262e6 q^{82} -2.22947e6 q^{83} +753084. q^{85} +1.25130e6 q^{86} +3.75398e6 q^{88} -5.99121e6 q^{89} +5.99195e6 q^{91} -2.65267e6 q^{92} -3.47021e6 q^{94} +2.83404e6 q^{95} -4.06013e6 q^{97} -1.32819e7 q^{98} +O(q^{100})\)

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
$5$	114.000	0.407859	0.203929	−	0.978986i	$-0.434629\pi$
$5$	114.000	0.407859	0.203929	+	0.978986i	$0.434629\pi$
$6$	0	0
$7$	−1576.00	−1.73665	−0.868327	−	0.495993i	$-0.834804\pi$
$7$	−1576.00	−1.73665	−0.868327	+	0.495993i	$0.834804\pi$
$8$	−512.000	−0.353553
$9$	0	0
$10$	−912.000	−0.288400
$11$	−7332.00	−1.66092	−0.830459	−	0.557080i	$-0.811922\pi$
$11$	−7332.00	−1.66092	−0.830459	+	0.557080i	$0.811922\pi$
$12$	0	0
$13$	−3802.00	−0.479966	−0.239983	−	0.970777i	$-0.577142\pi$
$13$	−3802.00	−0.479966	−0.239983	+	0.970777i	$0.577142\pi$
$14$	12608.0	1.22800
$15$	0	0
$16$	4096.00	0.250000
$17$	6606.00	0.326112	0.163056	−	0.986617i	$-0.447865\pi$
$17$	6606.00	0.326112	0.163056	+	0.986617i	$0.447865\pi$
$18$	0	0
$19$	24860.0	0.831502	0.415751	−	0.909478i	$-0.363519\pi$
$19$	24860.0	0.831502	0.415751	+	0.909478i	$0.363519\pi$
$20$	7296.00	0.203929
$21$	0	0
$22$	58656.0	1.17445
$23$	−41448.0	−0.710323	−0.355162	−	0.934805i	$-0.615574\pi$
$23$	−41448.0	−0.710323	−0.355162	+	0.934805i	$0.615574\pi$
$24$	0	0
$25$	−65129.0	−0.833651
$26$	30416.0	0.339387
$27$	0	0
$28$	−100864.	−0.868327
$29$	41610.0	0.316814	0.158407	−	0.987374i	$-0.449364\pi$
$29$	41610.0	0.316814	0.158407	+	0.987374i	$0.449364\pi$
$30$	0	0
$31$	33152.0	0.199868	0.0999341	−	0.994994i	$-0.468137\pi$
$31$	33152.0	0.199868	0.0999341	+	0.994994i	$0.468137\pi$
$32$	−32768.0	−0.176777
$33$	0	0
$34$	−52848.0	−0.230596
$35$	−179664.	−0.708309
$36$	0	0
$37$	−36466.0	−0.118354	−0.0591769	−	0.998248i	$-0.518848\pi$
$37$	−36466.0	−0.118354	−0.0591769	+	0.998248i	$0.518848\pi$
$38$	−198880.	−0.587961
$39$	0	0
$40$	−58368.0	−0.144200
$41$	639078.	1.44814	0.724070	−	0.689727i	$-0.242269\pi$
$41$	639078.	1.44814	0.724070	+	0.689727i	$0.242269\pi$
$42$	0	0
$43$	−156412.	−0.300006	−0.150003	−	0.988686i	$-0.547928\pi$
$43$	−156412.	−0.300006	−0.150003	+	0.988686i	$0.547928\pi$
$44$	−469248.	−0.830459
$45$	0	0
$46$	331584.	0.502275
$47$	433776.	0.609429	0.304714	−	0.952444i	$-0.401439\pi$
$47$	433776.	0.609429	0.304714	+	0.952444i	$0.401439\pi$
$48$	0	0
$49$	1.66023e6	2.01596
$50$	521032.	0.589480
$51$	0	0
$52$	−243328.	−0.239983
$53$	−786078.	−0.725271	−0.362635	−	0.931931i	$-0.618123\pi$
$53$	−786078.	−0.725271	−0.362635	+	0.931931i	$0.618123\pi$
$54$	0	0
$55$	−835848.	−0.677420
$56$	806912.	0.614000
$57$	0	0
$58$	−332880.	−0.224022
$59$	−745140.	−0.472341	−0.236171	−	0.971712i	$-0.575892\pi$
$59$	−745140.	−0.472341	−0.236171	+	0.971712i	$0.575892\pi$
$60$	0	0
$61$	−1.66062e6	−0.936732	−0.468366	−	0.883535i	$-0.655157\pi$
$61$	−1.66062e6	−0.936732	−0.468366	+	0.883535i	$0.655157\pi$
$62$	−265216.	−0.141328
$63$	0	0
$64$	262144.	0.125000
$65$	−433428.	−0.195758
$66$	0	0
$67$	−3.29084e6	−1.33673	−0.668366	−	0.743832i	$-0.733006\pi$
$67$	−3.29084e6	−1.33673	−0.668366	+	0.743832i	$0.733006\pi$
$68$	422784.	0.163056
$69$	0	0
$70$	1.43731e6	0.500850
$71$	−5.71615e6	−1.89539	−0.947697	−	0.319171i	$-0.896596\pi$
$71$	−5.71615e6	−1.89539	−0.947697	+	0.319171i	$0.896596\pi$
$72$	0	0
$73$	2.65990e6	0.800267	0.400134	−	0.916457i	$-0.368964\pi$
$73$	2.65990e6	0.800267	0.400134	+	0.916457i	$0.368964\pi$
$74$	291728.	0.0836888
$75$	0	0
$76$	1.59104e6	0.415751
$77$	1.15552e7	2.88444
$78$	0	0
$79$	3.80744e6	0.868837	0.434418	−	0.900711i	$-0.356954\pi$
$79$	3.80744e6	0.868837	0.434418	+	0.900711i	$0.356954\pi$
$80$	466944.	0.101965
$81$	0	0
$82$	−5.11262e6	−1.02399
$83$	−2.22947e6	−0.427984	−0.213992	−	0.976835i	$-0.568647\pi$
$83$	−2.22947e6	−0.427984	−0.213992	+	0.976835i	$0.568647\pi$
$84$	0	0
$85$	753084.	0.133008
$86$	1.25130e6	0.212137
$87$	0	0
$88$	3.75398e6	0.587223
$89$	−5.99121e6	−0.900844	−0.450422	−	0.892816i	$-0.648726\pi$
$89$	−5.99121e6	−0.900844	−0.450422	+	0.892816i	$0.648726\pi$
$90$	0	0
$91$	5.99195e6	0.833534
$92$	−2.65267e6	−0.355162
$93$	0	0
$94$	−3.47021e6	−0.430931
$95$	2.83404e6	0.339136
$96$	0	0
$97$	−4.06013e6	−0.451688	−0.225844	−	0.974163i	$-0.572514\pi$
$97$	−4.06013e6	−0.451688	−0.225844	+	0.974163i	$0.572514\pi$
$98$	−1.32819e7	−1.42550
$99$	0	0
$100$	−4.16826e6	−0.416826
$101$	1.72819e7	1.66904	0.834522	−	0.550975i	$-0.185744\pi$
$101$	1.72819e7	1.66904	0.834522	+	0.550975i	$0.185744\pi$
$102$	0	0
$103$	−1.43623e7	−1.29507	−0.647536	−	0.762035i	$-0.724201\pi$
$103$	−1.43623e7	−1.29507	−0.647536	+	0.762035i	$0.724201\pi$
$104$	1.94662e6	0.169694
$105$	0	0
$106$	6.28862e6	0.512844
$107$	−6.45440e6	−0.509346	−0.254673	−	0.967027i	$-0.581968\pi$
$107$	−6.45440e6	−0.509346	−0.254673	+	0.967027i	$0.581968\pi$
$108$	0	0
$109$	−884410.	−0.0654125	−0.0327063	−	0.999465i	$-0.510413\pi$
$109$	−884410.	−0.0654125	−0.0327063	+	0.999465i	$0.510413\pi$
$110$	6.68678e6	0.479008
$111$	0	0
$112$	−6.45530e6	−0.434163
$113$	−1.21325e7	−0.790999	−0.395499	−	0.918466i	$-0.629428\pi$
$113$	−1.21325e7	−0.790999	−0.395499	+	0.918466i	$0.629428\pi$
$114$	0	0
$115$	−4.72507e6	−0.289712
$116$	2.66304e6	0.158407
$117$	0	0
$118$	5.96112e6	0.333996
$119$	−1.04111e7	−0.566344
$120$	0	0
$121$	3.42711e7	1.75865
$122$	1.32849e7	0.662369
$123$	0	0
$124$	2.12173e6	0.0999341
$125$	−1.63310e7	−0.747871
$126$	0	0
$127$	6.86806e6	0.297524	0.148762	−	0.988873i	$-0.452471\pi$
$127$	6.86806e6	0.297524	0.148762	+	0.988873i	$0.452471\pi$
$128$	−2.09715e6	−0.0883883
$129$	0	0
$130$	3.46742e6	0.138422
$131$	3.95208e7	1.53595	0.767973	−	0.640482i	$-0.221265\pi$
$131$	3.95208e7	1.53595	0.767973	+	0.640482i	$0.221265\pi$
$132$	0	0
$133$	−3.91794e7	−1.44403
$134$	2.63267e7	0.945212
$135$	0	0
$136$	−3.38227e6	−0.115298
$137$	−1.91741e7	−0.637078	−0.318539	−	0.947910i	$-0.603192\pi$
$137$	−1.91741e7	−0.637078	−0.318539	+	0.947910i	$0.603192\pi$
$138$	0	0
$139$	1.32449e7	0.418309	0.209154	−	0.977883i	$-0.432929\pi$
$139$	1.32449e7	0.418309	0.209154	+	0.977883i	$0.432929\pi$
$140$	−1.14985e7	−0.354155
$141$	0	0
$142$	4.57292e7	1.34025
$143$	2.78763e7	0.797184
$144$	0	0
$145$	4.74354e6	0.129215
$146$	−2.12792e7	−0.565874
$147$	0	0
$148$	−2.33382e6	−0.0591769
$149$	−5.73624e7	−1.42061	−0.710306	−	0.703893i	$-0.751444\pi$
$149$	−5.73624e7	−1.42061	−0.710306	+	0.703893i	$0.751444\pi$
$150$	0	0
$151$	−3.10873e7	−0.734790	−0.367395	−	0.930065i	$-0.619750\pi$
$151$	−3.10873e7	−0.734790	−0.367395	+	0.930065i	$0.619750\pi$
$152$	−1.27283e7	−0.293981
$153$	0	0
$154$	−9.24419e7	−2.03961
$155$	3.77933e6	0.0815180
$156$	0	0
$157$	−3.37835e7	−0.696715	−0.348358	−	0.937362i	$-0.613261\pi$
$157$	−3.37835e7	−0.696715	−0.348358	+	0.937362i	$0.613261\pi$
$158$	−3.04595e7	−0.614360
$159$	0	0
$160$	−3.73555e6	−0.0720999
$161$	6.53220e7	1.23359
$162$	0	0
$163$	6.26659e7	1.13338	0.566689	−	0.823932i	$-0.308224\pi$
$163$	6.26659e7	1.13338	0.566689	+	0.823932i	$0.308224\pi$
$164$	4.09010e7	0.724070
$165$	0	0
$166$	1.78357e7	0.302631
$167$	−6.27072e7	−1.04186	−0.520931	−	0.853599i	$-0.674415\pi$
$167$	−6.27072e7	−1.04186	−0.520931	+	0.853599i	$0.674415\pi$
$168$	0	0
$169$	−4.82933e7	−0.769633
$170$	−6.02467e6	−0.0940507
$171$	0	0
$172$	−1.00104e7	−0.150003
$173$	2.70521e7	0.397228	0.198614	−	0.980078i	$-0.436356\pi$
$173$	2.70521e7	0.397228	0.198614	+	0.980078i	$0.436356\pi$
$174$	0	0
$175$	1.02643e8	1.44776
$176$	−3.00319e7	−0.415229
$177$	0	0
$178$	4.79297e7	0.636993
$179$	1.34281e8	1.74996	0.874981	−	0.484157i	$-0.160874\pi$
$179$	1.34281e8	1.74996	0.874981	+	0.484157i	$0.160874\pi$
$180$	0	0
$181$	1.14661e8	1.43727	0.718636	−	0.695386i	$-0.244767\pi$
$181$	1.14661e8	1.43727	0.718636	+	0.695386i	$0.244767\pi$
$182$	−4.79356e7	−0.589398
$183$	0	0
$184$	2.12214e7	0.251137
$185$	−4.15712e6	−0.0482716
$186$	0	0
$187$	−4.84352e7	−0.541646
$188$	2.77617e7	0.304714
$189$	0	0
$190$	−2.26723e7	−0.239805
$191$	−1.63605e7	−0.169895	−0.0849474	−	0.996385i	$-0.527072\pi$
$191$	−1.63605e7	−0.169895	−0.0849474	+	0.996385i	$0.527072\pi$
$192$	0	0
$193$	−1.54198e8	−1.54394	−0.771968	−	0.635661i	$-0.780728\pi$
$193$	−1.54198e8	−1.54394	−0.771968	+	0.635661i	$0.780728\pi$
$194$	3.24810e7	0.319392
$195$	0	0
$196$	1.06255e8	1.00798
$197$	−8.32288e7	−0.775607	−0.387804	−	0.921742i	$-0.626766\pi$
$197$	−8.32288e7	−0.775607	−0.387804	+	0.921742i	$0.626766\pi$
$198$	0	0
$199$	−7.61722e7	−0.685190	−0.342595	−	0.939483i	$-0.611306\pi$
$199$	−7.61722e7	−0.685190	−0.342595	+	0.939483i	$0.611306\pi$
$200$	3.33460e7	0.294740
$201$	0	0
$202$	−1.38256e8	−1.18019
$203$	−6.55774e7	−0.550196
$204$	0	0
$205$	7.28549e7	0.590636
$206$	1.14898e8	0.915755
$207$	0	0
$208$	−1.55730e7	−0.119991
$209$	−1.82274e8	−1.38106
$210$	0	0
$211$	3.52446e7	0.258288	0.129144	−	0.991626i	$-0.458777\pi$
$211$	3.52446e7	0.258288	0.129144	+	0.991626i	$0.458777\pi$
$212$	−5.03090e7	−0.362635
$213$	0	0
$214$	5.16352e7	0.360162
$215$	−1.78310e7	−0.122360
$216$	0	0
$217$	−5.22476e7	−0.347102
$218$	7.07528e6	0.0462536
$219$	0	0
$220$	−5.34943e7	−0.338710
$221$	−2.51160e7	−0.156523
$222$	0	0
$223$	−1.89131e8	−1.14208	−0.571040	−	0.820922i	$-0.693460\pi$
$223$	−1.89131e8	−1.14208	−0.571040	+	0.820922i	$0.693460\pi$
$224$	5.16424e7	0.307000
$225$	0	0
$226$	9.70600e7	0.559320
$227$	1.76100e8	0.999239	0.499620	−	0.866245i	$-0.333473\pi$
$227$	1.76100e8	0.999239	0.499620	+	0.866245i	$0.333473\pi$
$228$	0	0
$229$	6.50396e7	0.357894	0.178947	−	0.983859i	$-0.442731\pi$
$229$	6.50396e7	0.357894	0.178947	+	0.983859i	$0.442731\pi$
$230$	3.78006e7	0.204857
$231$	0	0
$232$	−2.13043e7	−0.112011
$233$	2.51319e8	1.30160	0.650802	−	0.759248i	$-0.274433\pi$
$233$	2.51319e8	1.30160	0.650802	+	0.759248i	$0.274433\pi$
$234$	0	0
$235$	4.94505e7	0.248561
$236$	−4.76890e7	−0.236171
$237$	0	0
$238$	8.32884e7	0.400466
$239$	−2.13079e8	−1.00960	−0.504799	−	0.863237i	$-0.668434\pi$
$239$	−2.13079e8	−1.00960	−0.504799	+	0.863237i	$0.668434\pi$
$240$	0	0
$241$	2.57284e8	1.18400	0.592001	−	0.805937i	$-0.298338\pi$
$241$	2.57284e8	1.18400	0.592001	+	0.805937i	$0.298338\pi$
$242$	−2.74168e8	−1.24355
$243$	0	0
$244$	−1.06280e8	−0.468366
$245$	1.89267e8	0.822229
$246$	0	0
$247$	−9.45177e7	−0.399093
$248$	−1.69738e7	−0.0706641
$249$	0	0
$250$	1.30648e8	0.528824
$251$	−1.23058e8	−0.491193	−0.245596	−	0.969372i	$-0.578984\pi$
$251$	−1.23058e8	−0.491193	−0.245596	+	0.969372i	$0.578984\pi$
$252$	0	0
$253$	3.03897e8	1.17979
$254$	−5.49445e7	−0.210381
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	4.43334e8	1.62916	0.814582	−	0.580048i	$-0.196966\pi$
$257$	4.43334e8	1.62916	0.814582	+	0.580048i	$0.196966\pi$
$258$	0	0
$259$	5.74704e7	0.205539
$260$	−2.77394e7	−0.0978792
$261$	0	0
$262$	−3.16166e8	−1.08608
$263$	−2.98925e8	−1.01325	−0.506625	−	0.862166i	$-0.669107\pi$
$263$	−2.98925e8	−1.01325	−0.506625	+	0.862166i	$0.669107\pi$
$264$	0	0
$265$	−8.96129e7	−0.295808
$266$	3.13435e8	1.02108
$267$	0	0
$268$	−2.10614e8	−0.668366
$269$	−2.08908e8	−0.654368	−0.327184	−	0.944961i	$-0.606100\pi$
$269$	−2.08908e8	−0.654368	−0.327184	+	0.944961i	$0.606100\pi$
$270$	0	0
$271$	−1.12749e7	−0.0344129	−0.0172064	−	0.999852i	$-0.505477\pi$
$271$	−1.12749e7	−0.0344129	−0.0172064	+	0.999852i	$0.505477\pi$
$272$	2.70582e7	0.0815281
$273$	0	0
$274$	1.53393e8	0.450482
$275$	4.77526e8	1.38463
$276$	0	0
$277$	−6.58964e8	−1.86287	−0.931435	−	0.363907i	$-0.881443\pi$
$277$	−6.58964e8	−1.86287	−0.931435	+	0.363907i	$0.881443\pi$
$278$	−1.05959e8	−0.295789
$279$	0	0
$280$	9.19880e7	0.250425
$281$	1.05123e8	0.282634	0.141317	−	0.989964i	$-0.454866\pi$
$281$	1.05123e8	0.282634	0.141317	+	0.989964i	$0.454866\pi$
$282$	0	0
$283$	3.30161e8	0.865911	0.432956	−	0.901415i	$-0.357471\pi$
$283$	3.30161e8	0.865911	0.432956	+	0.901415i	$0.357471\pi$
$284$	−3.65834e8	−0.947697
$285$	0	0
$286$	−2.23010e8	−0.563694
$287$	−1.00719e9	−2.51492
$288$	0	0
$289$	−3.66699e8	−0.893651
$290$	−3.79483e7	−0.0913691
$291$	0	0
$292$	1.70233e8	0.400134
$293$	8.71002e7	0.202294	0.101147	−	0.994871i	$-0.467749\pi$
$293$	8.71002e7	0.202294	0.101147	+	0.994871i	$0.467749\pi$
$294$	0	0
$295$	−8.49460e7	−0.192649
$296$	1.86706e7	0.0418444
$297$	0	0
$298$	4.58899e8	1.00452
$299$	1.57585e8	0.340931
$300$	0	0
$301$	2.46505e8	0.521007
$302$	2.48698e8	0.519575
$303$	0	0
$304$	1.01827e8	0.207876
$305$	−1.89310e8	−0.382054
$306$	0	0
$307$	−3.91709e8	−0.772644	−0.386322	−	0.922364i	$-0.626255\pi$
$307$	−3.91709e8	−0.772644	−0.386322	+	0.922364i	$0.626255\pi$
$308$	7.39535e8	1.44222
$309$	0	0
$310$	−3.02346e7	−0.0576419
$311$	2.04936e8	0.386328	0.193164	−	0.981166i	$-0.438125\pi$
$311$	2.04936e8	0.386328	0.193164	+	0.981166i	$0.438125\pi$
$312$	0	0
$313$	8.77202e8	1.61694	0.808471	−	0.588536i	$-0.200295\pi$
$313$	8.77202e8	1.61694	0.808471	+	0.588536i	$0.200295\pi$
$314$	2.70268e8	0.492652
$315$	0	0
$316$	2.43676e8	0.434418
$317$	4.40831e8	0.777256	0.388628	−	0.921395i	$-0.372949\pi$
$317$	4.40831e8	0.777256	0.388628	+	0.921395i	$0.372949\pi$
$318$	0	0
$319$	−3.05085e8	−0.526202
$320$	2.98844e7	0.0509823
$321$	0	0
$322$	−5.22576e8	−0.872277
$323$	1.64225e8	0.271163
$324$	0	0
$325$	2.47620e8	0.400124
$326$	−5.01327e8	−0.801419
$327$	0	0
$328$	−3.27208e8	−0.511995
$329$	−6.83631e8	−1.05837
$330$	0	0
$331$	1.11223e9	1.68576	0.842882	−	0.538099i	$-0.180857\pi$
$331$	1.11223e9	1.68576	0.842882	+	0.538099i	$0.180857\pi$
$332$	−1.42686e8	−0.213992
$333$	0	0
$334$	5.01658e8	0.736707
$335$	−3.75155e8	−0.545198
$336$	0	0
$337$	2.88198e8	0.410191	0.205096	−	0.978742i	$-0.434249\pi$
$337$	2.88198e8	0.410191	0.205096	+	0.978742i	$0.434249\pi$
$338$	3.86347e8	0.544213
$339$	0	0
$340$	4.81974e7	0.0665039
$341$	−2.43070e8	−0.331965
$342$	0	0
$343$	−1.31862e9	−1.76438
$344$	8.00829e7	0.106068
$345$	0	0
$346$	−2.16417e8	−0.280883
$347$	1.10601e9	1.42103	0.710517	−	0.703680i	$-0.248461\pi$
$347$	1.10601e9	1.42103	0.710517	+	0.703680i	$0.248461\pi$
$348$	0	0
$349$	−1.32184e9	−1.66453	−0.832264	−	0.554379i	$-0.812956\pi$
$349$	−1.32184e9	−1.66453	−0.832264	+	0.554379i	$0.812956\pi$
$350$	−8.21146e8	−1.02372
$351$	0	0
$352$	2.40255e8	0.293612
$353$	−1.20395e9	−1.45679	−0.728396	−	0.685157i	$-0.759734\pi$
$353$	−1.20395e9	−1.45679	−0.728396	+	0.685157i	$0.759734\pi$
$354$	0	0
$355$	−6.51641e8	−0.773053
$356$	−3.83437e8	−0.450422
$357$	0	0
$358$	−1.07425e9	−1.23741
$359$	1.32057e9	1.50637	0.753185	−	0.657809i	$-0.228516\pi$
$359$	1.32057e9	1.50637	0.753185	+	0.657809i	$0.228516\pi$
$360$	0	0
$361$	−2.75852e8	−0.308604
$362$	−9.17284e8	−1.01630
$363$	0	0
$364$	3.83485e8	0.416767
$365$	3.03228e8	0.326396
$366$	0	0
$367$	1.75107e9	1.84915	0.924575	−	0.381000i	$-0.124420\pi$
$367$	1.75107e9	1.84915	0.924575	+	0.381000i	$0.124420\pi$
$368$	−1.69771e8	−0.177581
$369$	0	0
$370$	3.32570e7	0.0341332
$371$	1.23886e9	1.25954
$372$	0	0
$373$	−4.87945e8	−0.486844	−0.243422	−	0.969920i	$-0.578270\pi$
$373$	−4.87945e8	−0.486844	−0.243422	+	0.969920i	$0.578270\pi$
$374$	3.87482e8	0.383001
$375$	0	0
$376$	−2.22093e8	−0.215466
$377$	−1.58201e8	−0.152060
$378$	0	0
$379$	1.11007e9	1.04740	0.523700	−	0.851903i	$-0.324551\pi$
$379$	1.11007e9	1.04740	0.523700	+	0.851903i	$0.324551\pi$
$380$	1.81379e8	0.169568
$381$	0	0
$382$	1.30884e8	0.120134
$383$	−1.86912e9	−1.69997	−0.849983	−	0.526810i	$-0.823388\pi$
$383$	−1.86912e9	−1.69997	−0.849983	+	0.526810i	$0.823388\pi$
$384$	0	0
$385$	1.31730e9	1.17644
$386$	1.23359e9	1.09173
$387$	0	0
$388$	−2.59848e8	−0.225844
$389$	2.73895e8	0.235918	0.117959	−	0.993018i	$-0.462365\pi$
$389$	2.73895e8	0.235918	0.117959	+	0.993018i	$0.462365\pi$
$390$	0	0
$391$	−2.73805e8	−0.231645
$392$	−8.50039e8	−0.712751
$393$	0	0
$394$	6.65831e8	0.548437
$395$	4.34048e8	0.354363
$396$	0	0
$397$	6.24552e8	0.500958	0.250479	−	0.968122i	$-0.419412\pi$
$397$	6.24552e8	0.500958	0.250479	+	0.968122i	$0.419412\pi$
$398$	6.09378e8	0.484502
$399$	0	0
$400$	−2.66768e8	−0.208413
$401$	−5.55500e8	−0.430208	−0.215104	−	0.976591i	$-0.569009\pi$
$401$	−5.55500e8	−0.430208	−0.215104	+	0.976591i	$0.569009\pi$
$402$	0	0
$403$	−1.26044e8	−0.0959299
$404$	1.10604e9	0.834522
$405$	0	0
$406$	5.24619e8	0.389048
$407$	2.67369e8	0.196576
$408$	0	0
$409$	−2.15770e9	−1.55941	−0.779704	−	0.626149i	$-0.784630\pi$
$409$	−2.15770e9	−1.55941	−0.779704	+	0.626149i	$0.784630\pi$
$410$	−5.82839e8	−0.417643
$411$	0	0
$412$	−9.19188e8	−0.647536
$413$	1.17434e9	0.820293
$414$	0	0
$415$	−2.54159e8	−0.174557
$416$	1.24584e8	0.0848468
$417$	0	0
$418$	1.45819e9	0.976555
$419$	−1.67797e9	−1.11438	−0.557191	−	0.830384i	$-0.688121\pi$
$419$	−1.67797e9	−1.11438	−0.557191	+	0.830384i	$0.688121\pi$
$420$	0	0
$421$	−5.25233e8	−0.343056	−0.171528	−	0.985179i	$-0.554870\pi$
$421$	−5.25233e8	−0.343056	−0.171528	+	0.985179i	$0.554870\pi$
$422$	−2.81957e8	−0.182637
$423$	0	0
$424$	4.02472e8	0.256422
$425$	−4.30242e8	−0.271864
$426$	0	0
$427$	2.61713e9	1.62678
$428$	−4.13082e8	−0.254673
$429$	0	0
$430$	1.42648e8	0.0865218
$431$	−1.70593e8	−0.102634	−0.0513169	−	0.998682i	$-0.516342\pi$
$431$	−1.70593e8	−0.102634	−0.0513169	+	0.998682i	$0.516342\pi$
$432$	0	0
$433$	−1.68797e9	−0.999210	−0.499605	−	0.866253i	$-0.666522\pi$
$433$	−1.68797e9	−0.999210	−0.499605	+	0.866253i	$0.666522\pi$
$434$	4.17980e8	0.245438
$435$	0	0
$436$	−5.66022e7	−0.0327063
$437$	−1.03040e9	−0.590636
$438$	0	0
$439$	−1.17850e9	−0.664817	−0.332409	−	0.943135i	$-0.607861\pi$
$439$	−1.17850e9	−0.664817	−0.332409	+	0.943135i	$0.607861\pi$
$440$	4.27954e8	0.239504
$441$	0	0
$442$	2.00928e8	0.110678
$443$	−7.15755e8	−0.391157	−0.195579	−	0.980688i	$-0.562658\pi$
$443$	−7.15755e8	−0.391157	−0.195579	+	0.980688i	$0.562658\pi$
$444$	0	0
$445$	−6.82998e8	−0.367417
$446$	1.51305e9	0.807573
$447$	0	0
$448$	−4.13139e8	−0.217082
$449$	1.37358e9	0.716132	0.358066	−	0.933696i	$-0.383436\pi$
$449$	1.37358e9	0.716132	0.358066	+	0.933696i	$0.383436\pi$
$450$	0	0
$451$	−4.68572e9	−2.40524
$452$	−7.76480e8	−0.395499
$453$	0	0
$454$	−1.40880e9	−0.706569
$455$	6.83083e8	0.339964
$456$	0	0
$457$	1.84752e9	0.905488	0.452744	−	0.891641i	$-0.350445\pi$
$457$	1.84752e9	0.905488	0.452744	+	0.891641i	$0.350445\pi$
$458$	−5.20317e8	−0.253069
$459$	0	0
$460$	−3.02405e8	−0.144856
$461$	−3.09414e9	−1.47091	−0.735455	−	0.677573i	$-0.763032\pi$
$461$	−3.09414e9	−1.47091	−0.735455	+	0.677573i	$0.763032\pi$
$462$	0	0
$463$	3.00451e9	1.40682	0.703412	−	0.710782i	$-0.251659\pi$
$463$	3.00451e9	1.40682	0.703412	+	0.710782i	$0.251659\pi$
$464$	1.70435e8	0.0792036
$465$	0	0
$466$	−2.01055e9	−0.920373
$467$	2.99252e9	1.35965	0.679825	−	0.733374i	$-0.262056\pi$
$467$	2.99252e9	1.35965	0.679825	+	0.733374i	$0.262056\pi$
$468$	0	0
$469$	5.18636e9	2.32144
$470$	−3.95604e8	−0.175759
$471$	0	0
$472$	3.81512e8	0.166998
$473$	1.14681e9	0.498286
$474$	0	0
$475$	−1.61911e9	−0.693183
$476$	−6.66308e8	−0.283172
$477$	0	0
$478$	1.70464e9	0.713894
$479$	−1.84041e9	−0.765141	−0.382570	−	0.923926i	$-0.624961\pi$
$479$	−1.84041e9	−0.765141	−0.382570	+	0.923926i	$0.624961\pi$
$480$	0	0
$481$	1.38644e8	0.0568058
$482$	−2.05827e9	−0.837216
$483$	0	0
$484$	2.19335e9	0.879323
$485$	−4.62854e8	−0.184225
$486$	0	0
$487$	−4.26676e8	−0.167397	−0.0836983	−	0.996491i	$-0.526673\pi$
$487$	−4.26676e8	−0.167397	−0.0836983	+	0.996491i	$0.526673\pi$
$488$	8.50236e8	0.331185
$489$	0	0
$490$	−1.51413e9	−0.581403
$491$	−6.07547e7	−0.0231630	−0.0115815	−	0.999933i	$-0.503687\pi$
$491$	−6.07547e7	−0.0231630	−0.0115815	+	0.999933i	$0.503687\pi$
$492$	0	0
$493$	2.74876e8	0.103317
$494$	7.56142e8	0.282201
$495$	0	0
$496$	1.35791e8	0.0499671
$497$	9.00866e9	3.29164
$498$	0	0
$499$	3.24588e9	1.16945	0.584723	−	0.811233i	$-0.301203\pi$
$499$	3.24588e9	1.16945	0.584723	+	0.811233i	$0.301203\pi$
$500$	−1.04518e9	−0.373935
$501$	0	0
$502$	9.84464e8	0.347326
$503$	7.44381e8	0.260800	0.130400	−	0.991461i	$-0.458374\pi$
$503$	7.44381e8	0.260800	0.130400	+	0.991461i	$0.458374\pi$
$504$	0	0
$505$	1.97014e9	0.680734
$506$	−2.43117e9	−0.834237
$507$	0	0
$508$	4.39556e8	0.148762
$509$	4.44155e8	0.149287	0.0746436	−	0.997210i	$-0.476218\pi$
$509$	4.44155e8	0.149287	0.0746436	+	0.997210i	$0.476218\pi$
$510$	0	0
$511$	−4.19200e9	−1.38979
$512$	−1.34218e8	−0.0441942
$513$	0	0
$514$	−3.54667e9	−1.15199
$515$	−1.63730e9	−0.528207
$516$	0	0
$517$	−3.18045e9	−1.01221
$518$	−4.59763e8	−0.145338
$519$	0	0
$520$	2.21915e8	0.0692110
$521$	−3.04963e9	−0.944745	−0.472372	−	0.881399i	$-0.656602\pi$
$521$	−3.04963e9	−0.944745	−0.472372	+	0.881399i	$0.656602\pi$
$522$	0	0
$523$	−1.40306e9	−0.428866	−0.214433	−	0.976739i	$-0.568790\pi$
$523$	−1.40306e9	−0.428866	−0.214433	+	0.976739i	$0.568790\pi$
$524$	2.52933e9	0.767973
$525$	0	0
$526$	2.39140e9	0.716476
$527$	2.19002e8	0.0651795
$528$	0	0
$529$	−1.68689e9	−0.495441
$530$	7.16903e8	0.209168
$531$	0	0
$532$	−2.50748e9	−0.722016
$533$	−2.42977e9	−0.695058
$534$	0	0
$535$	−7.35802e8	−0.207741
$536$	1.68491e9	0.472606
$537$	0	0
$538$	1.67126e9	0.462708
$539$	−1.21728e10	−3.34835
$540$	0	0
$541$	4.21106e9	1.14341	0.571704	−	0.820460i	$-0.306283\pi$
$541$	4.21106e9	1.14341	0.571704	+	0.820460i	$0.306283\pi$
$542$	9.01994e7	0.0243336
$543$	0	0
$544$	−2.16465e8	−0.0576491
$545$	−1.00823e8	−0.0266791
$546$	0	0
$547$	1.99956e9	0.522371	0.261185	−	0.965289i	$-0.415887\pi$
$547$	1.99956e9	0.522371	0.261185	+	0.965289i	$0.415887\pi$
$548$	−1.22714e9	−0.318539
$549$	0	0
$550$	−3.82021e9	−0.979078
$551$	1.03442e9	0.263432
$552$	0	0
$553$	−6.00053e9	−1.50887
$554$	5.27172e9	1.31725
$555$	0	0
$556$	8.47674e8	0.209154
$557$	3.37403e9	0.827287	0.413643	−	0.910439i	$-0.364256\pi$
$557$	3.37403e9	0.827287	0.413643	+	0.910439i	$0.364256\pi$
$558$	0	0
$559$	5.94678e8	0.143993
$560$	−7.35904e8	−0.177077
$561$	0	0
$562$	−8.40983e8	−0.199853
$563$	5.58021e9	1.31787	0.658933	−	0.752201i	$-0.271008\pi$
$563$	5.58021e9	1.31787	0.658933	+	0.752201i	$0.271008\pi$
$564$	0	0
$565$	−1.38310e9	−0.322616
$566$	−2.64129e9	−0.612292
$567$	0	0
$568$	2.92667e9	0.670123
$569$	8.88310e8	0.202149	0.101074	−	0.994879i	$-0.467772\pi$
$569$	8.88310e8	0.202149	0.101074	+	0.994879i	$0.467772\pi$
$570$	0	0
$571$	−1.79171e9	−0.402755	−0.201377	−	0.979514i	$-0.564542\pi$
$571$	−1.79171e9	−0.402755	−0.201377	+	0.979514i	$0.564542\pi$
$572$	1.78408e9	0.398592
$573$	0	0
$574$	8.05750e9	1.77831
$575$	2.69947e9	0.592162
$576$	0	0
$577$	3.82103e9	0.828066	0.414033	−	0.910262i	$-0.364120\pi$
$577$	3.82103e9	0.828066	0.414033	+	0.910262i	$0.364120\pi$
$578$	2.93360e9	0.631906
$579$	0	0
$580$	3.03587e8	0.0646077
$581$	3.51364e9	0.743260
$582$	0	0
$583$	5.76352e9	1.20461
$584$	−1.36187e9	−0.282937
$585$	0	0
$586$	−6.96802e8	−0.143043
$587$	−4.36219e9	−0.890166	−0.445083	−	0.895489i	$-0.646826\pi$
$587$	−4.36219e9	−0.890166	−0.445083	+	0.895489i	$0.646826\pi$
$588$	0	0
$589$	8.24159e8	0.166191
$590$	6.79568e8	0.136223
$591$	0	0
$592$	−1.49365e8	−0.0295884
$593$	−6.38531e9	−1.25745	−0.628724	−	0.777628i	$-0.716423\pi$
$593$	−6.38531e9	−1.25745	−0.628724	+	0.777628i	$0.716423\pi$
$594$	0	0
$595$	−1.18686e9	−0.230988
$596$	−3.67120e9	−0.710306
$597$	0	0
$598$	−1.26068e9	−0.241075
$599$	−8.04297e8	−0.152905	−0.0764527	−	0.997073i	$-0.524359\pi$
$599$	−8.04297e8	−0.152905	−0.0764527	+	0.997073i	$0.524359\pi$
$600$	0	0
$601$	−4.87162e9	−0.915403	−0.457702	−	0.889106i	$-0.651327\pi$
$601$	−4.87162e9	−0.915403	−0.457702	+	0.889106i	$0.651327\pi$
$602$	−1.97204e9	−0.368408
$603$	0	0
$604$	−1.98959e9	−0.367395
$605$	3.90690e9	0.717280
$606$	0	0
$607$	7.17517e9	1.30218	0.651091	−	0.759000i	$-0.274312\pi$
$607$	7.17517e9	1.30218	0.651091	+	0.759000i	$0.274312\pi$
$608$	−8.14612e8	−0.146990
$609$	0	0
$610$	1.51448e9	0.270153
$611$	−1.64922e9	−0.292505
$612$	0	0
$613$	3.47891e9	0.610002	0.305001	−	0.952352i	$-0.401343\pi$
$613$	3.47891e9	0.610002	0.305001	+	0.952352i	$0.401343\pi$
$614$	3.13367e9	0.546342
$615$	0	0
$616$	−5.91628e9	−1.01980
$617$	2.39378e8	0.0410286	0.0205143	−	0.999790i	$-0.493470\pi$
$617$	2.39378e8	0.0410286	0.0205143	+	0.999790i	$0.493470\pi$
$618$	0	0
$619$	−5.52959e9	−0.937078	−0.468539	−	0.883443i	$-0.655219\pi$
$619$	−5.52959e9	−0.937078	−0.468539	+	0.883443i	$0.655219\pi$
$620$	2.41877e8	0.0407590
$621$	0	0
$622$	−1.63949e9	−0.273175
$623$	9.44215e9	1.56445
$624$	0	0
$625$	3.22647e9	0.528626
$626$	−7.01762e9	−1.14335
$627$	0	0
$628$	−2.16214e9	−0.348358
$629$	−2.40894e8	−0.0385966
$630$	0	0
$631$	−6.13683e9	−0.972392	−0.486196	−	0.873850i	$-0.661616\pi$
$631$	−6.13683e9	−0.972392	−0.486196	+	0.873850i	$0.661616\pi$
$632$	−1.94941e9	−0.307180
$633$	0	0
$634$	−3.52664e9	−0.549603
$635$	7.82959e8	0.121348
$636$	0	0
$637$	−6.31221e9	−0.967594
$638$	2.44068e9	0.372081
$639$	0	0
$640$	−2.39075e8	−0.0360500
$641$	−1.07038e10	−1.60522	−0.802611	−	0.596503i	$-0.796556\pi$
$641$	−1.07038e10	−1.60522	−0.802611	+	0.596503i	$0.796556\pi$
$642$	0	0
$643$	−1.39803e9	−0.207385	−0.103692	−	0.994609i	$-0.533066\pi$
$643$	−1.39803e9	−0.207385	−0.103692	+	0.994609i	$0.533066\pi$
$644$	4.18061e9	0.616793
$645$	0	0
$646$	−1.31380e9	−0.191741
$647$	5.31605e9	0.771656	0.385828	−	0.922571i	$-0.373916\pi$
$647$	5.31605e9	0.771656	0.385828	+	0.922571i	$0.373916\pi$
$648$	0	0
$649$	5.46337e9	0.784520
$650$	−1.98096e9	−0.282930
$651$	0	0
$652$	4.01062e9	0.566689
$653$	−3.24403e9	−0.455921	−0.227960	−	0.973670i	$-0.573206\pi$
$653$	−3.24403e9	−0.455921	−0.227960	+	0.973670i	$0.573206\pi$
$654$	0	0
$655$	4.50537e9	0.626449
$656$	2.61766e9	0.362035
$657$	0	0
$658$	5.46905e9	0.748378
$659$	5.16506e9	0.703034	0.351517	−	0.936181i	$-0.385666\pi$
$659$	5.16506e9	0.703034	0.351517	+	0.936181i	$0.385666\pi$
$660$	0	0
$661$	−3.22515e9	−0.434356	−0.217178	−	0.976132i	$-0.569685\pi$
$661$	−3.22515e9	−0.434356	−0.217178	+	0.976132i	$0.569685\pi$
$662$	−8.89784e9	−1.19201
$663$	0	0
$664$	1.14149e9	0.151315
$665$	−4.46645e9	−0.588961
$666$	0	0
$667$	−1.72465e9	−0.225041
$668$	−4.01326e9	−0.520931
$669$	0	0
$670$	3.00124e9	0.385513
$671$	1.21757e10	1.55583
$672$	0	0
$673$	−2.00633e9	−0.253718	−0.126859	−	0.991921i	$-0.540490\pi$
$673$	−2.00633e9	−0.253718	−0.126859	+	0.991921i	$0.540490\pi$
$674$	−2.30559e9	−0.290049
$675$	0	0
$676$	−3.09077e9	−0.384816
$677$	−1.00211e10	−1.24124	−0.620619	−	0.784112i	$-0.713119\pi$
$677$	−1.00211e10	−1.24124	−0.620619	+	0.784112i	$0.713119\pi$
$678$	0	0
$679$	6.39876e9	0.784425
$680$	−3.85579e8	−0.0470254
$681$	0	0
$682$	1.94456e9	0.234734
$683$	5.84861e9	0.702393	0.351196	−	0.936302i	$-0.385775\pi$
$683$	5.84861e9	0.702393	0.351196	+	0.936302i	$0.385775\pi$
$684$	0	0
$685$	−2.18584e9	−0.259838
$686$	1.05490e10	1.24760
$687$	0	0
$688$	−6.40664e8	−0.0750016
$689$	2.98867e9	0.348105
$690$	0	0
$691$	2.58686e9	0.298263	0.149131	−	0.988817i	$-0.452352\pi$
$691$	2.58686e9	0.298263	0.149131	+	0.988817i	$0.452352\pi$
$692$	1.73134e9	0.198614
$693$	0	0
$694$	−8.84805e9	−1.00482
$695$	1.50992e9	0.170611
$696$	0	0
$697$	4.22175e9	0.472256
$698$	1.05747e10	1.17700
$699$	0	0
$700$	6.56917e9	0.723881
$701$	1.74460e9	0.191286	0.0956429	−	0.995416i	$-0.469509\pi$
$701$	1.74460e9	0.191286	0.0956429	+	0.995416i	$0.469509\pi$
$702$	0	0
$703$	−9.06545e8	−0.0984114
$704$	−1.92204e9	−0.207615
$705$	0	0
$706$	9.63161e9	1.03011
$707$	−2.72363e10	−2.89855
$708$	0	0
$709$	−1.12051e10	−1.18074	−0.590368	−	0.807134i	$-0.701017\pi$
$709$	−1.12051e10	−1.18074	−0.590368	+	0.807134i	$0.701017\pi$
$710$	5.21313e9	0.546631
$711$	0	0
$712$	3.06750e9	0.318496
$713$	−1.37408e9	−0.141971
$714$	0	0
$715$	3.17789e9	0.325138
$716$	8.59398e9	0.874981
$717$	0	0
$718$	−1.05646e10	−1.06516
$719$	9.36568e8	0.0939698	0.0469849	−	0.998896i	$-0.485039\pi$
$719$	9.36568e8	0.0939698	0.0469849	+	0.998896i	$0.485039\pi$
$720$	0	0
$721$	2.26350e10	2.24909
$722$	2.20682e9	0.218216
$723$	0	0
$724$	7.33827e9	0.718636
$725$	−2.71002e9	−0.264113
$726$	0	0
$727$	4.20445e9	0.405825	0.202913	−	0.979197i	$-0.434959\pi$
$727$	4.20445e9	0.405825	0.202913	+	0.979197i	$0.434959\pi$
$728$	−3.06788e9	−0.294699
$729$	0	0
$730$	−2.42583e9	−0.230797
$731$	−1.03326e9	−0.0978358
$732$	0	0
$733$	−1.15491e10	−1.08314	−0.541571	−	0.840655i	$-0.682170\pi$
$733$	−1.15491e10	−1.08314	−0.541571	+	0.840655i	$0.682170\pi$
$734$	−1.40086e10	−1.30755
$735$	0	0
$736$	1.35817e9	0.125569
$737$	2.41284e10	2.22020
$738$	0	0
$739$	−1.39655e10	−1.27292	−0.636460	−	0.771310i	$-0.719602\pi$
$739$	−1.39655e10	−1.27292	−0.636460	+	0.771310i	$0.719602\pi$
$740$	−2.66056e8	−0.0241358
$741$	0	0
$742$	−9.91087e9	−0.890632
$743$	−1.43832e10	−1.28646	−0.643230	−	0.765673i	$-0.722406\pi$
$743$	−1.43832e10	−1.28646	−0.643230	+	0.765673i	$0.722406\pi$
$744$	0	0
$745$	−6.53932e9	−0.579409
$746$	3.90356e9	0.344251
$747$	0	0
$748$	−3.09985e9	−0.270823
$749$	1.01721e10	0.884557
$750$	0	0
$751$	−6.70841e8	−0.0577936	−0.0288968	−	0.999582i	$-0.509199\pi$
$751$	−6.70841e8	−0.0577936	−0.0288968	+	0.999582i	$0.509199\pi$
$752$	1.77675e9	0.152357
$753$	0	0
$754$	1.26561e9	0.107523
$755$	−3.54395e9	−0.299691
$756$	0	0
$757$	−1.91569e10	−1.60506	−0.802529	−	0.596613i	$-0.796513\pi$
$757$	−1.91569e10	−1.60506	−0.802529	+	0.596613i	$0.796513\pi$
$758$	−8.88055e9	−0.740624
$759$	0	0
$760$	−1.45103e9	−0.119903
$761$	−1.79120e10	−1.47332	−0.736660	−	0.676263i	$-0.763598\pi$
$761$	−1.79120e10	−1.47332	−0.736660	+	0.676263i	$0.763598\pi$
$762$	0	0
$763$	1.39383e9	0.113599
$764$	−1.04707e9	−0.0849474
$765$	0	0
$766$	1.49529e10	1.20206
$767$	2.83302e9	0.226708
$768$	0	0
$769$	−2.14072e10	−1.69753	−0.848765	−	0.528771i	$-0.822653\pi$
$769$	−2.14072e10	−1.69753	−0.848765	+	0.528771i	$0.822653\pi$
$770$	−1.05384e10	−0.831871
$771$	0	0
$772$	−9.86870e9	−0.771968
$773$	7.55163e8	0.0588047	0.0294024	−	0.999568i	$-0.490640\pi$
$773$	7.55163e8	0.0588047	0.0294024	+	0.999568i	$0.490640\pi$
$774$	0	0
$775$	−2.15916e9	−0.166620
$776$	2.07878e9	0.159696
$777$	0	0
$778$	−2.19116e9	−0.166819
$779$	1.58875e10	1.20413
$780$	0	0
$781$	4.19108e10	3.14809
$782$	2.19044e9	0.163798
$783$	0	0
$784$	6.80031e9	0.503991
$785$	−3.85132e9	−0.284162
$786$	0	0
$787$	2.04665e10	1.49669	0.748347	−	0.663308i	$-0.230848\pi$
$787$	2.04665e10	1.49669	0.748347	+	0.663308i	$0.230848\pi$
$788$	−5.32664e9	−0.387804
$789$	0	0
$790$	−3.47239e9	−0.250572
$791$	1.91208e10	1.37369
$792$	0	0
$793$	6.31367e9	0.449599
$794$	−4.99641e9	−0.354231
$795$	0	0
$796$	−4.87502e9	−0.342595
$797$	1.03098e10	0.721348	0.360674	−	0.932692i	$-0.382547\pi$
$797$	1.03098e10	0.721348	0.360674	+	0.932692i	$0.382547\pi$
$798$	0	0
$799$	2.86552e9	0.198742
$800$	2.13415e9	0.147370
$801$	0	0
$802$	4.44400e9	0.304203
$803$	−1.95024e10	−1.32918
$804$	0	0
$805$	7.44671e9	0.503129
$806$	1.00835e9	0.0678327
$807$	0	0
$808$	−8.84835e9	−0.590096
$809$	4.16428e8	0.0276516	0.0138258	−	0.999904i	$-0.495599\pi$
$809$	4.16428e8	0.0276516	0.0138258	+	0.999904i	$0.495599\pi$
$810$	0	0
$811$	−5.82687e9	−0.383586	−0.191793	−	0.981435i	$-0.561430\pi$
$811$	−5.82687e9	−0.383586	−0.191793	+	0.981435i	$0.561430\pi$
$812$	−4.19695e9	−0.275098
$813$	0	0
$814$	−2.13895e9	−0.139000
$815$	7.14391e9	0.462258
$816$	0	0
$817$	−3.88840e9	−0.249456
$818$	1.72616e10	1.10267
$819$	0	0
$820$	4.66271e9	0.295318
$821$	2.08333e10	1.31388	0.656941	−	0.753942i	$-0.271850\pi$
$821$	2.08333e10	1.31388	0.656941	+	0.753942i	$0.271850\pi$
$822$	0	0
$823$	4.23403e9	0.264761	0.132381	−	0.991199i	$-0.457738\pi$
$823$	4.23403e9	0.264761	0.132381	+	0.991199i	$0.457738\pi$
$824$	7.35350e9	0.457877
$825$	0	0
$826$	−9.39473e9	−0.580035
$827$	−5.70597e9	−0.350800	−0.175400	−	0.984497i	$-0.556122\pi$
$827$	−5.70597e9	−0.350800	−0.175400	+	0.984497i	$0.556122\pi$
$828$	0	0
$829$	2.51612e10	1.53388	0.766940	−	0.641719i	$-0.221779\pi$
$829$	2.51612e10	1.53388	0.766940	+	0.641719i	$0.221779\pi$
$830$	2.03327e9	0.123431
$831$	0	0
$832$	−9.96671e8	−0.0599957
$833$	1.09675e10	0.657431
$834$	0	0
$835$	−7.14862e9	−0.424932
$836$	−1.16655e10	−0.690528
$837$	0	0
$838$	1.34237e10	0.787988
$839$	−2.27048e10	−1.32724	−0.663622	−	0.748068i	$-0.730982\pi$
$839$	−2.27048e10	−1.32724	−0.663622	+	0.748068i	$0.730982\pi$
$840$	0	0
$841$	−1.55185e10	−0.899629
$842$	4.20187e9	0.242577
$843$	0	0
$844$	2.25565e9	0.129144
$845$	−5.50544e9	−0.313901
$846$	0	0
$847$	−5.40112e10	−3.05416
$848$	−3.21978e9	−0.181318
$849$	0	0
$850$	3.44194e9	0.192237
$851$	1.51144e9	0.0840695
$852$	0	0
$853$	2.86872e10	1.58258	0.791292	−	0.611439i	$-0.209409\pi$
$853$	2.86872e10	1.58258	0.791292	+	0.611439i	$0.209409\pi$
$854$	−2.09371e10	−1.15031
$855$	0	0
$856$	3.30465e9	0.180081
$857$	−5.34950e9	−0.290322	−0.145161	−	0.989408i	$-0.546370\pi$
$857$	−5.34950e9	−0.290322	−0.145161	+	0.989408i	$0.546370\pi$
$858$	0	0
$859$	−1.40330e10	−0.755394	−0.377697	−	0.925929i	$-0.623284\pi$
$859$	−1.40330e10	−0.755394	−0.377697	+	0.925929i	$0.623284\pi$
$860$	−1.14118e9	−0.0611801
$861$	0	0
$862$	1.36474e9	0.0725731
$863$	−3.24994e10	−1.72122	−0.860612	−	0.509261i	$-0.829919\pi$
$863$	−3.24994e10	−1.72122	−0.860612	+	0.509261i	$0.829919\pi$
$864$	0	0
$865$	3.08394e9	0.162013
$866$	1.35038e10	0.706549
$867$	0	0
$868$	−3.34384e9	−0.173551
$869$	−2.79162e10	−1.44307
$870$	0	0
$871$	1.25118e10	0.641586
$872$	4.52818e8	0.0231268
$873$	0	0
$874$	8.24318e9	0.417642
$875$	2.57376e10	1.29879
$876$	0	0
$877$	−3.38694e10	−1.69554	−0.847772	−	0.530361i	$-0.822056\pi$
$877$	−3.38694e10	−1.69554	−0.847772	+	0.530361i	$0.822056\pi$
$878$	9.42797e9	0.470097
$879$	0	0
$880$	−3.42363e9	−0.169355
$881$	1.52708e10	0.752397	0.376198	−	0.926539i	$-0.377231\pi$
$881$	1.52708e10	0.752397	0.376198	+	0.926539i	$0.377231\pi$
$882$	0	0
$883$	−1.12045e10	−0.547685	−0.273842	−	0.961775i	$-0.588295\pi$
$883$	−1.12045e10	−0.547685	−0.273842	+	0.961775i	$0.588295\pi$
$884$	−1.60742e9	−0.0782614
$885$	0	0
$886$	5.72604e9	0.276590
$887$	−6.97232e9	−0.335463	−0.167732	−	0.985833i	$-0.553644\pi$
$887$	−6.97232e9	−0.335463	−0.167732	+	0.985833i	$0.553644\pi$
$888$	0	0
$889$	−1.08241e10	−0.516695
$890$	5.46398e9	0.259803
$891$	0	0
$892$	−1.21044e10	−0.571040
$893$	1.07837e10	0.506742
$894$	0	0
$895$	1.53080e10	0.713737
$896$	3.30511e9	0.153500
$897$	0	0
$898$	−1.09887e10	−0.506382
$899$	1.37945e9	0.0633211
$900$	0	0
$901$	−5.19283e9	−0.236520
$902$	3.74858e10	1.70076
$903$	0	0
$904$	6.21184e9	0.279660
$905$	1.30713e10	0.586204
$906$	0	0
$907$	1.18095e9	0.0525539	0.0262770	−	0.999655i	$-0.491635\pi$
$907$	1.18095e9	0.0525539	0.0262770	+	0.999655i	$0.491635\pi$
$908$	1.12704e10	0.499620
$909$	0	0
$910$	−5.46466e9	−0.240391
$911$	−1.27915e10	−0.560541	−0.280271	−	0.959921i	$-0.590424\pi$
$911$	−1.27915e10	−0.560541	−0.280271	+	0.959921i	$0.590424\pi$
$912$	0	0
$913$	1.63465e10	0.710847
$914$	−1.47802e10	−0.640276
$915$	0	0
$916$	4.16254e9	0.178947
$917$	−6.22848e10	−2.66741
$918$	0	0
$919$	3.52353e10	1.49752	0.748761	−	0.662840i	$-0.230649\pi$
$919$	3.52353e10	1.49752	0.748761	+	0.662840i	$0.230649\pi$
$920$	2.41924e9	0.102429
$921$	0	0
$922$	2.47531e10	1.04009
$923$	2.17328e10	0.909725
$924$	0	0
$925$	2.37499e9	0.0986658
$926$	−2.40361e10	−0.994775
$927$	0	0
$928$	−1.36348e9	−0.0560054
$929$	2.17764e10	0.891111	0.445556	−	0.895254i	$-0.353006\pi$
$929$	2.17764e10	0.891111	0.445556	+	0.895254i	$0.353006\pi$
$930$	0	0
$931$	4.12734e10	1.67628
$932$	1.60844e10	0.650802
$933$	0	0
$934$	−2.39401e10	−0.961418
$935$	−5.52161e9	−0.220915
$936$	0	0
$937$	1.15795e10	0.459833	0.229916	−	0.973210i	$-0.426155\pi$
$937$	1.15795e10	0.459833	0.229916	+	0.973210i	$0.426155\pi$
$938$	−4.14909e10	−1.64151
$939$	0	0
$940$	3.16483e9	0.124280
$941$	3.83930e10	1.50207	0.751033	−	0.660265i	$-0.229556\pi$
$941$	3.83930e10	1.50207	0.751033	+	0.660265i	$0.229556\pi$
$942$	0	0
$943$	−2.64885e10	−1.02865
$944$	−3.05209e9	−0.118085
$945$	0	0
$946$	−9.17450e9	−0.352341
$947$	−1.11841e10	−0.427933	−0.213966	−	0.976841i	$-0.568638\pi$
$947$	−1.11841e10	−0.427933	−0.213966	+	0.976841i	$0.568638\pi$
$948$	0	0
$949$	−1.01129e10	−0.384101
$950$	1.29529e10	0.490154
$951$	0	0
$952$	5.33046e9	0.200233
$953$	5.19835e9	0.194554	0.0972770	−	0.995257i	$-0.468987\pi$
$953$	5.19835e9	0.194554	0.0972770	+	0.995257i	$0.468987\pi$
$954$	0	0
$955$	−1.86510e9	−0.0692931
$956$	−1.36371e10	−0.504799
$957$	0	0
$958$	1.47233e10	0.541036
$959$	3.02183e10	1.10638
$960$	0	0
$961$	−2.64136e10	−0.960053
$962$	−1.10915e9	−0.0401677
$963$	0	0
$964$	1.64662e10	0.592001
$965$	−1.75786e10	−0.629708
$966$	0	0
$967$	−2.69243e8	−0.00957529	−0.00478765	−	0.999989i	$-0.501524\pi$
$967$	−2.69243e8	−0.00957529	−0.00478765	+	0.999989i	$0.501524\pi$
$968$	−1.75468e10	−0.621776
$969$	0	0
$970$	3.70283e9	0.130267
$971$	−4.37283e9	−0.153284	−0.0766418	−	0.997059i	$-0.524420\pi$
$971$	−4.37283e9	−0.153284	−0.0766418	+	0.997059i	$0.524420\pi$
$972$	0	0
$973$	−2.08740e10	−0.726457
$974$	3.41341e9	0.118367
$975$	0	0
$976$	−6.80189e9	−0.234183
$977$	3.74991e10	1.28644	0.643220	−	0.765681i	$-0.277598\pi$
$977$	3.74991e10	1.28644	0.643220	+	0.765681i	$0.277598\pi$
$978$	0	0
$979$	4.39276e10	1.49623
$980$	1.21131e10	0.411114
$981$	0	0
$982$	4.86038e8	0.0163787
$983$	3.06190e10	1.02814	0.514071	−	0.857748i	$-0.328137\pi$
$983$	3.06190e10	1.02814	0.514071	+	0.857748i	$0.328137\pi$
$984$	0	0
$985$	−9.48808e9	−0.316338
$986$	−2.19901e9	−0.0730562
$987$	0	0
$988$	−6.04913e9	−0.199546
$989$	6.48296e9	0.213102
$990$	0	0
$991$	−1.72703e10	−0.563693	−0.281847	−	0.959459i	$-0.590947\pi$
$991$	−1.72703e10	−0.563693	−0.281847	+	0.959459i	$0.590947\pi$
$992$	−1.08632e9	−0.0353320
$993$	0	0
$994$	−7.20692e10	−2.32754
$995$	−8.68363e9	−0.279461
$996$	0	0
$997$	−3.75077e9	−0.119864	−0.0599319	−	0.998202i	$-0.519088\pi$
$997$	−3.75077e9	−0.119864	−0.0599319	+	0.998202i	$0.519088\pi$
$998$	−2.59670e10	−0.826923
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.8.a.a.1.1		1
3.2	odd	2		6.8.a.a.1.1	✓	1
4.3	odd	2		144.8.a.h.1.1		1
5.2	odd	4		450.8.c.a.199.1		2
5.3	odd	4		450.8.c.a.199.2		2
5.4	even	2		450.8.a.ba.1.1		1
8.3	odd	2		576.8.a.i.1.1		1
8.5	even	2		576.8.a.h.1.1		1
9.2	odd	6		162.8.c.d.109.1		2
9.4	even	3		162.8.c.i.55.1		2
9.5	odd	6		162.8.c.d.55.1		2
9.7	even	3		162.8.c.i.109.1		2
12.11	even	2		48.8.a.b.1.1		1
15.2	even	4		150.8.c.k.49.2		2
15.8	even	4		150.8.c.k.49.1		2
15.14	odd	2		150.8.a.e.1.1		1
21.2	odd	6		294.8.e.c.67.1		2
21.5	even	6		294.8.e.d.67.1		2
21.11	odd	6		294.8.e.c.79.1		2
21.17	even	6		294.8.e.d.79.1		2
21.20	even	2		294.8.a.l.1.1		1
24.5	odd	2		192.8.a.f.1.1		1
24.11	even	2		192.8.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.8.a.a.1.1	✓	1	3.2	odd	2
18.8.a.a.1.1		1	1.1	even	1	trivial
48.8.a.b.1.1		1	12.11	even	2
144.8.a.h.1.1		1	4.3	odd	2
150.8.a.e.1.1		1	15.14	odd	2
150.8.c.k.49.1		2	15.8	even	4
150.8.c.k.49.2		2	15.2	even	4
162.8.c.d.55.1		2	9.5	odd	6
162.8.c.d.109.1		2	9.2	odd	6
162.8.c.i.55.1		2	9.4	even	3
162.8.c.i.109.1		2	9.7	even	3
192.8.a.f.1.1		1	24.5	odd	2
192.8.a.n.1.1		1	24.11	even	2
294.8.a.l.1.1		1	21.20	even	2
294.8.e.c.67.1		2	21.2	odd	6
294.8.e.c.79.1		2	21.11	odd	6
294.8.e.d.67.1		2	21.5	even	6
294.8.e.d.79.1		2	21.17	even	6
450.8.a.ba.1.1		1	5.4	even	2
450.8.c.a.199.1		2	5.2	odd	4
450.8.c.a.199.2		2	5.3	odd	4
576.8.a.h.1.1		1	8.5	even	2
576.8.a.i.1.1		1	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more