LMFDB - Embedded newform 18.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,6,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, 6, names="a")

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$2.88690875663$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	18.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-4.00000 q^{2} +16.0000 q^{4} -96.0000 q^{5} -148.000 q^{7} -64.0000 q^{8} +384.000 q^{10} +384.000 q^{11} -334.000 q^{13} +592.000 q^{14} +256.000 q^{16} +576.000 q^{17} -664.000 q^{19} -1536.00 q^{20} -1536.00 q^{22} -3840.00 q^{23} +6091.00 q^{25} +1336.00 q^{26} -2368.00 q^{28} +96.0000 q^{29} -4564.00 q^{31} -1024.00 q^{32} -2304.00 q^{34} +14208.0 q^{35} +5798.00 q^{37} +2656.00 q^{38} +6144.00 q^{40} -6720.00 q^{41} -14872.0 q^{43} +6144.00 q^{44} +15360.0 q^{46} -19200.0 q^{47} +5097.00 q^{49} -24364.0 q^{50} -5344.00 q^{52} +7776.00 q^{53} -36864.0 q^{55} +9472.00 q^{56} -384.000 q^{58} -13056.0 q^{59} +42782.0 q^{61} +18256.0 q^{62} +4096.00 q^{64} +32064.0 q^{65} +36656.0 q^{67} +9216.00 q^{68} -56832.0 q^{70} +64512.0 q^{71} -16810.0 q^{73} -23192.0 q^{74} -10624.0 q^{76} -56832.0 q^{77} +28076.0 q^{79} -24576.0 q^{80} +26880.0 q^{82} -66432.0 q^{83} -55296.0 q^{85} +59488.0 q^{86} -24576.0 q^{88} -81792.0 q^{89} +49432.0 q^{91} -61440.0 q^{92} +76800.0 q^{94} +63744.0 q^{95} -29938.0 q^{97} -20388.0 q^{98} +O(q^{100})\)

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−96.0000	−1.71730	−0.858650	−	0.512562i	$-0.828696\pi$
$5$	−96.0000	−1.71730	−0.858650	+	0.512562i	$0.828696\pi$
$6$	0	0
$7$	−148.000	−1.14161	−0.570803	−	0.821087i	$-0.693368\pi$
$7$	−148.000	−1.14161	−0.570803	+	0.821087i	$0.693368\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	384.000	1.21431
$11$	384.000	0.956862	0.478431	−	0.878125i	$-0.341206\pi$
$11$	384.000	0.956862	0.478431	+	0.878125i	$0.341206\pi$
$12$	0	0
$13$	−334.000	−0.548136	−0.274068	−	0.961710i	$-0.588369\pi$
$13$	−334.000	−0.548136	−0.274068	+	0.961710i	$0.588369\pi$
$14$	592.000	0.807238
$15$	0	0
$16$	256.000	0.250000
$17$	576.000	0.483393	0.241696	−	0.970352i	$-0.422296\pi$
$17$	576.000	0.483393	0.241696	+	0.970352i	$0.422296\pi$
$18$	0	0
$19$	−664.000	−0.421972	−0.210986	−	0.977489i	$-0.567668\pi$
$19$	−664.000	−0.421972	−0.210986	+	0.977489i	$0.567668\pi$
$20$	−1536.00	−0.858650
$21$	0	0
$22$	−1536.00	−0.676604
$23$	−3840.00	−1.51360	−0.756801	−	0.653645i	$-0.773239\pi$
$23$	−3840.00	−1.51360	−0.756801	+	0.653645i	$0.773239\pi$
$24$	0	0
$25$	6091.00	1.94912
$26$	1336.00	0.387590
$27$	0	0
$28$	−2368.00	−0.570803
$29$	96.0000	0.0211971	0.0105985	−	0.999944i	$-0.496626\pi$
$29$	96.0000	0.0211971	0.0105985	+	0.999944i	$0.496626\pi$
$30$	0	0
$31$	−4564.00	−0.852985	−0.426493	−	0.904491i	$-0.640251\pi$
$31$	−4564.00	−0.852985	−0.426493	+	0.904491i	$0.640251\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−2304.00	−0.341810
$35$	14208.0	1.96048
$36$	0	0
$37$	5798.00	0.696264	0.348132	−	0.937446i	$-0.386816\pi$
$37$	5798.00	0.696264	0.348132	+	0.937446i	$0.386816\pi$
$38$	2656.00	0.298380
$39$	0	0
$40$	6144.00	0.607157
$41$	−6720.00	−0.624323	−0.312162	−	0.950029i	$-0.601053\pi$
$41$	−6720.00	−0.624323	−0.312162	+	0.950029i	$0.601053\pi$
$42$	0	0
$43$	−14872.0	−1.22659	−0.613293	−	0.789855i	$-0.710156\pi$
$43$	−14872.0	−1.22659	−0.613293	+	0.789855i	$0.710156\pi$
$44$	6144.00	0.478431
$45$	0	0
$46$	15360.0	1.07028
$47$	−19200.0	−1.26782	−0.633909	−	0.773408i	$-0.718550\pi$
$47$	−19200.0	−1.26782	−0.633909	+	0.773408i	$0.718550\pi$
$48$	0	0
$49$	5097.00	0.303266
$50$	−24364.0	−1.37824
$51$	0	0
$52$	−5344.00	−0.274068
$53$	7776.00	0.380248	0.190124	−	0.981760i	$-0.439111\pi$
$53$	7776.00	0.380248	0.190124	+	0.981760i	$0.439111\pi$
$54$	0	0
$55$	−36864.0	−1.64322
$56$	9472.00	0.403619
$57$	0	0
$58$	−384.000	−0.0149886
$59$	−13056.0	−0.488293	−0.244146	−	0.969738i	$-0.578508\pi$
$59$	−13056.0	−0.488293	−0.244146	+	0.969738i	$0.578508\pi$
$60$	0	0
$61$	42782.0	1.47210	0.736049	−	0.676929i	$-0.236689\pi$
$61$	42782.0	1.47210	0.736049	+	0.676929i	$0.236689\pi$
$62$	18256.0	0.603151
$63$	0	0
$64$	4096.00	0.125000
$65$	32064.0	0.941314
$66$	0	0
$67$	36656.0	0.997604	0.498802	−	0.866716i	$-0.333774\pi$
$67$	36656.0	0.997604	0.498802	+	0.866716i	$0.333774\pi$
$68$	9216.00	0.241696
$69$	0	0
$70$	−56832.0	−1.38627
$71$	64512.0	1.51878	0.759390	−	0.650636i	$-0.225498\pi$
$71$	64512.0	1.51878	0.759390	+	0.650636i	$0.225498\pi$
$72$	0	0
$73$	−16810.0	−0.369199	−0.184600	−	0.982814i	$-0.559099\pi$
$73$	−16810.0	−0.369199	−0.184600	+	0.982814i	$0.559099\pi$
$74$	−23192.0	−0.492333
$75$	0	0
$76$	−10624.0	−0.210986
$77$	−56832.0	−1.09236
$78$	0	0
$79$	28076.0	0.506136	0.253068	−	0.967448i	$-0.418560\pi$
$79$	28076.0	0.506136	0.253068	+	0.967448i	$0.418560\pi$
$80$	−24576.0	−0.429325
$81$	0	0
$82$	26880.0	0.441463
$83$	−66432.0	−1.05848	−0.529239	−	0.848473i	$-0.677523\pi$
$83$	−66432.0	−1.05848	−0.529239	+	0.848473i	$0.677523\pi$
$84$	0	0
$85$	−55296.0	−0.830131
$86$	59488.0	0.867328
$87$	0	0
$88$	−24576.0	−0.338302
$89$	−81792.0	−1.09455	−0.547275	−	0.836953i	$-0.684335\pi$
$89$	−81792.0	−1.09455	−0.547275	+	0.836953i	$0.684335\pi$
$90$	0	0
$91$	49432.0	0.625756
$92$	−61440.0	−0.756801
$93$	0	0
$94$	76800.0	0.896482
$95$	63744.0	0.724653
$96$	0	0
$97$	−29938.0	−0.323068	−0.161534	−	0.986867i	$-0.551644\pi$
$97$	−29938.0	−0.323068	−0.161534	+	0.986867i	$0.551644\pi$
$98$	−20388.0	−0.214442
$99$	0	0
$100$	97456.0	0.974560
$101$	178656.	1.74267	0.871333	−	0.490692i	$-0.163256\pi$
$101$	178656.	1.74267	0.871333	+	0.490692i	$0.163256\pi$
$102$	0	0
$103$	−115228.	−1.07020	−0.535100	−	0.844789i	$-0.679726\pi$
$103$	−115228.	−1.07020	−0.535100	+	0.844789i	$0.679726\pi$
$104$	21376.0	0.193795
$105$	0	0
$106$	−31104.0	−0.268876
$107$	−76032.0	−0.642003	−0.321001	−	0.947079i	$-0.604019\pi$
$107$	−76032.0	−0.642003	−0.321001	+	0.947079i	$0.604019\pi$
$108$	0	0
$109$	−231118.	−1.86323	−0.931617	−	0.363441i	$-0.881602\pi$
$109$	−231118.	−1.86323	−0.931617	+	0.363441i	$0.881602\pi$
$110$	147456.	1.16193
$111$	0	0
$112$	−37888.0	−0.285402
$113$	142464.	1.04956	0.524782	−	0.851237i	$-0.324147\pi$
$113$	142464.	1.04956	0.524782	+	0.851237i	$0.324147\pi$
$114$	0	0
$115$	368640.	2.59931
$116$	1536.00	0.0105985
$117$	0	0
$118$	52224.0	0.345275
$119$	−85248.0	−0.551845
$120$	0	0
$121$	−13595.0	−0.0844143
$122$	−171128.	−1.04093
$123$	0	0
$124$	−73024.0	−0.426493
$125$	−284736.	−1.62992
$126$	0	0
$127$	−988.000	−0.00543560	−0.00271780	−	0.999996i	$-0.500865\pi$
$127$	−988.000	−0.00543560	−0.00271780	+	0.999996i	$0.500865\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−128256.	−0.665609
$131$	224256.	1.14174	0.570868	−	0.821042i	$-0.306607\pi$
$131$	224256.	1.14174	0.570868	+	0.821042i	$0.306607\pi$
$132$	0	0
$133$	98272.0	0.481727
$134$	−146624.	−0.705412
$135$	0	0
$136$	−36864.0	−0.170905
$137$	−278976.	−1.26989	−0.634944	−	0.772558i	$-0.718977\pi$
$137$	−278976.	−1.26989	−0.634944	+	0.772558i	$0.718977\pi$
$138$	0	0
$139$	177200.	0.777905	0.388953	−	0.921258i	$-0.372837\pi$
$139$	177200.	0.777905	0.388953	+	0.921258i	$0.372837\pi$
$140$	227328.	0.980241
$141$	0	0
$142$	−258048.	−1.07394
$143$	−128256.	−0.524490
$144$	0	0
$145$	−9216.00	−0.0364018
$146$	67240.0	0.261063
$147$	0	0
$148$	92768.0	0.348132
$149$	236064.	0.871092	0.435546	−	0.900166i	$-0.356555\pi$
$149$	236064.	0.871092	0.435546	+	0.900166i	$0.356555\pi$
$150$	0	0
$151$	−482836.	−1.72329	−0.861643	−	0.507515i	$-0.830564\pi$
$151$	−482836.	−1.72329	−0.861643	+	0.507515i	$0.830564\pi$
$152$	42496.0	0.149190
$153$	0	0
$154$	227328.	0.772416
$155$	438144.	1.46483
$156$	0	0
$157$	381086.	1.23388	0.616941	−	0.787009i	$-0.288372\pi$
$157$	381086.	1.23388	0.616941	+	0.787009i	$0.288372\pi$
$158$	−112304.	−0.357892
$159$	0	0
$160$	98304.0	0.303579
$161$	568320.	1.72794
$162$	0	0
$163$	162920.	0.480292	0.240146	−	0.970737i	$-0.422805\pi$
$163$	162920.	0.480292	0.240146	+	0.970737i	$0.422805\pi$
$164$	−107520.	−0.312162
$165$	0	0
$166$	265728.	0.748457
$167$	−566016.	−1.57050	−0.785249	−	0.619180i	$-0.787465\pi$
$167$	−566016.	−1.57050	−0.785249	+	0.619180i	$0.787465\pi$
$168$	0	0
$169$	−259737.	−0.699547
$170$	221184.	0.586991
$171$	0	0
$172$	−237952.	−0.613293
$173$	−218208.	−0.554313	−0.277157	−	0.960825i	$-0.589392\pi$
$173$	−218208.	−0.554313	−0.277157	+	0.960825i	$0.589392\pi$
$174$	0	0
$175$	−901468.	−2.22513
$176$	98304.0	0.239216
$177$	0	0
$178$	327168.	0.773964
$179$	−412416.	−0.962062	−0.481031	−	0.876704i	$-0.659738\pi$
$179$	−412416.	−0.962062	−0.481031	+	0.876704i	$0.659738\pi$
$180$	0	0
$181$	−25558.0	−0.0579870	−0.0289935	−	0.999580i	$-0.509230\pi$
$181$	−25558.0	−0.0579870	−0.0289935	+	0.999580i	$0.509230\pi$
$182$	−197728.	−0.442476
$183$	0	0
$184$	245760.	0.535139
$185$	−556608.	−1.19569
$186$	0	0
$187$	221184.	0.462540
$188$	−307200.	−0.633909
$189$	0	0
$190$	−254976.	−0.512407
$191$	400128.	0.793625	0.396813	−	0.917900i	$-0.370116\pi$
$191$	400128.	0.793625	0.396813	+	0.917900i	$0.370116\pi$
$192$	0	0
$193$	699650.	1.35203	0.676017	−	0.736886i	$-0.263705\pi$
$193$	699650.	1.35203	0.676017	+	0.736886i	$0.263705\pi$
$194$	119752.	0.228443
$195$	0	0
$196$	81552.0	0.151633
$197$	406368.	0.746026	0.373013	−	0.927826i	$-0.378325\pi$
$197$	406368.	0.746026	0.373013	+	0.927826i	$0.378325\pi$
$198$	0	0
$199$	−361996.	−0.647994	−0.323997	−	0.946058i	$-0.605027\pi$
$199$	−361996.	−0.647994	−0.323997	+	0.946058i	$0.605027\pi$
$200$	−389824.	−0.689118
$201$	0	0
$202$	−714624.	−1.23225
$203$	−14208.0	−0.0241987
$204$	0	0
$205$	645120.	1.07215
$206$	460912.	0.756746
$207$	0	0
$208$	−85504.0	−0.137034
$209$	−254976.	−0.403770
$210$	0	0
$211$	151856.	0.234815	0.117407	−	0.993084i	$-0.462542\pi$
$211$	151856.	0.234815	0.117407	+	0.993084i	$0.462542\pi$
$212$	124416.	0.190124
$213$	0	0
$214$	304128.	0.453965
$215$	1.42771e6	2.10642
$216$	0	0
$217$	675472.	0.973774
$218$	924472.	1.31751
$219$	0	0
$220$	−589824.	−0.821610
$221$	−192384.	−0.264965
$222$	0	0
$223$	−1.09332e6	−1.47227	−0.736134	−	0.676836i	$-0.763351\pi$
$223$	−1.09332e6	−1.47227	−0.736134	+	0.676836i	$0.763351\pi$
$224$	151552.	0.201810
$225$	0	0
$226$	−569856.	−0.742154
$227$	−566400.	−0.729556	−0.364778	−	0.931095i	$-0.618855\pi$
$227$	−566400.	−0.729556	−0.364778	+	0.931095i	$0.618855\pi$
$228$	0	0
$229$	−587206.	−0.739949	−0.369974	−	0.929042i	$-0.620634\pi$
$229$	−587206.	−0.739949	−0.369974	+	0.929042i	$0.620634\pi$
$230$	−1.47456e6	−1.83799
$231$	0	0
$232$	−6144.00	−0.00749430
$233$	579456.	0.699247	0.349624	−	0.936890i	$-0.386309\pi$
$233$	579456.	0.699247	0.349624	+	0.936890i	$0.386309\pi$
$234$	0	0
$235$	1.84320e6	2.17722
$236$	−208896.	−0.244146
$237$	0	0
$238$	340992.	0.390213
$239$	−584448.	−0.661837	−0.330919	−	0.943659i	$-0.607359\pi$
$239$	−584448.	−0.661837	−0.330919	+	0.943659i	$0.607359\pi$
$240$	0	0
$241$	−414130.	−0.459298	−0.229649	−	0.973274i	$-0.573758\pi$
$241$	−414130.	−0.459298	−0.229649	+	0.973274i	$0.573758\pi$
$242$	54380.0	0.0596899
$243$	0	0
$244$	684512.	0.736049
$245$	−489312.	−0.520800
$246$	0	0
$247$	221776.	0.231298
$248$	292096.	0.301576
$249$	0	0
$250$	1.13894e6	1.15253
$251$	−1.89965e6	−1.90322	−0.951610	−	0.307309i	$-0.900571\pi$
$251$	−1.89965e6	−1.90322	−0.951610	+	0.307309i	$0.900571\pi$
$252$	0	0
$253$	−1.47456e6	−1.44831
$254$	3952.00	0.00384355
$255$	0	0
$256$	65536.0	0.0625000
$257$	447744.	0.422860	0.211430	−	0.977393i	$-0.432188\pi$
$257$	447744.	0.422860	0.211430	+	0.977393i	$0.432188\pi$
$258$	0	0
$259$	−858104.	−0.794860
$260$	513024.	0.470657
$261$	0	0
$262$	−897024.	−0.807330
$263$	−67584.0	−0.0602496	−0.0301248	−	0.999546i	$-0.509590\pi$
$263$	−67584.0	−0.0602496	−0.0301248	+	0.999546i	$0.509590\pi$
$264$	0	0
$265$	−746496.	−0.652999
$266$	−393088.	−0.340632
$267$	0	0
$268$	586496.	0.498802
$269$	564192.	0.475386	0.237693	−	0.971340i	$-0.423609\pi$
$269$	564192.	0.475386	0.237693	+	0.971340i	$0.423609\pi$
$270$	0	0
$271$	720308.	0.595792	0.297896	−	0.954598i	$-0.403715\pi$
$271$	720308.	0.595792	0.297896	+	0.954598i	$0.403715\pi$
$272$	147456.	0.120848
$273$	0	0
$274$	1.11590e6	0.897946
$275$	2.33894e6	1.86504
$276$	0	0
$277$	−141142.	−0.110524	−0.0552620	−	0.998472i	$-0.517599\pi$
$277$	−141142.	−0.110524	−0.0552620	+	0.998472i	$0.517599\pi$
$278$	−708800.	−0.550062
$279$	0	0
$280$	−909312.	−0.693135
$281$	584448.	0.441550	0.220775	−	0.975325i	$-0.429141\pi$
$281$	584448.	0.441550	0.220775	+	0.975325i	$0.429141\pi$
$282$	0	0
$283$	177056.	0.131415	0.0657074	−	0.997839i	$-0.479070\pi$
$283$	177056.	0.131415	0.0657074	+	0.997839i	$0.479070\pi$
$284$	1.03219e6	0.759390
$285$	0	0
$286$	513024.	0.370871
$287$	994560.	0.712732
$288$	0	0
$289$	−1.08808e6	−0.766331
$290$	36864.0	0.0257399
$291$	0	0
$292$	−268960.	−0.184600
$293$	−956832.	−0.651128	−0.325564	−	0.945520i	$-0.605554\pi$
$293$	−956832.	−0.651128	−0.325564	+	0.945520i	$0.605554\pi$
$294$	0	0
$295$	1.25338e6	0.838545
$296$	−371072.	−0.246166
$297$	0	0
$298$	−944256.	−0.615955
$299$	1.28256e6	0.829659
$300$	0	0
$301$	2.20106e6	1.40028
$302$	1.93134e6	1.21855
$303$	0	0
$304$	−169984.	−0.105493
$305$	−4.10707e6	−2.52803
$306$	0	0
$307$	2.88286e6	1.74573	0.872867	−	0.487958i	$-0.162258\pi$
$307$	2.88286e6	1.74573	0.872867	+	0.487958i	$0.162258\pi$
$308$	−909312.	−0.546180
$309$	0	0
$310$	−1.75258e6	−1.03579
$311$	−2.60045e6	−1.52457	−0.762285	−	0.647242i	$-0.775922\pi$
$311$	−2.60045e6	−1.52457	−0.762285	+	0.647242i	$0.775922\pi$
$312$	0	0
$313$	−2.58079e6	−1.48899	−0.744495	−	0.667628i	$-0.767310\pi$
$313$	−2.58079e6	−1.48899	−0.744495	+	0.667628i	$0.767310\pi$
$314$	−1.52434e6	−0.872487
$315$	0	0
$316$	449216.	0.253068
$317$	2.31101e6	1.29168	0.645838	−	0.763475i	$-0.276508\pi$
$317$	2.31101e6	1.29168	0.645838	+	0.763475i	$0.276508\pi$
$318$	0	0
$319$	36864.0	0.0202827
$320$	−393216.	−0.214663
$321$	0	0
$322$	−2.27328e6	−1.22184
$323$	−382464.	−0.203978
$324$	0	0
$325$	−2.03439e6	−1.06838
$326$	−651680.	−0.339618
$327$	0	0
$328$	430080.	0.220732
$329$	2.84160e6	1.44735
$330$	0	0
$331$	−637024.	−0.319585	−0.159792	−	0.987151i	$-0.551082\pi$
$331$	−637024.	−0.319585	−0.159792	+	0.987151i	$0.551082\pi$
$332$	−1.06291e6	−0.529239
$333$	0	0
$334$	2.26406e6	1.11051
$335$	−3.51898e6	−1.71319
$336$	0	0
$337$	3.38665e6	1.62441	0.812206	−	0.583371i	$-0.198267\pi$
$337$	3.38665e6	1.62441	0.812206	+	0.583371i	$0.198267\pi$
$338$	1.03895e6	0.494655
$339$	0	0
$340$	−884736.	−0.415065
$341$	−1.75258e6	−0.816189
$342$	0	0
$343$	1.73308e6	0.795396
$344$	951808.	0.433664
$345$	0	0
$346$	872832.	0.391959
$347$	2.77824e6	1.23864	0.619321	−	0.785138i	$-0.287408\pi$
$347$	2.77824e6	1.23864	0.619321	+	0.785138i	$0.287408\pi$
$348$	0	0
$349$	1.55536e6	0.683545	0.341772	−	0.939783i	$-0.388973\pi$
$349$	1.55536e6	0.683545	0.341772	+	0.939783i	$0.388973\pi$
$350$	3.60587e6	1.57340
$351$	0	0
$352$	−393216.	−0.169151
$353$	2.11776e6	0.904565	0.452283	−	0.891875i	$-0.350610\pi$
$353$	2.11776e6	0.904565	0.452283	+	0.891875i	$0.350610\pi$
$354$	0	0
$355$	−6.19315e6	−2.60820
$356$	−1.30867e6	−0.547275
$357$	0	0
$358$	1.64966e6	0.680280
$359$	2.17498e6	0.890673	0.445337	−	0.895363i	$-0.353084\pi$
$359$	2.17498e6	0.890673	0.445337	+	0.895363i	$0.353084\pi$
$360$	0	0
$361$	−2.03520e6	−0.821939
$362$	102232.	0.0410030
$363$	0	0
$364$	790912.	0.312878
$365$	1.61376e6	0.634026
$366$	0	0
$367$	1.05336e6	0.408235	0.204117	−	0.978946i	$-0.434568\pi$
$367$	1.05336e6	0.408235	0.204117	+	0.978946i	$0.434568\pi$
$368$	−983040.	−0.378400
$369$	0	0
$370$	2.22643e6	0.845483
$371$	−1.15085e6	−0.434093
$372$	0	0
$373$	−677098.	−0.251988	−0.125994	−	0.992031i	$-0.540212\pi$
$373$	−677098.	−0.251988	−0.125994	+	0.992031i	$0.540212\pi$
$374$	−884736.	−0.327065
$375$	0	0
$376$	1.22880e6	0.448241
$377$	−32064.0	−0.0116189
$378$	0	0
$379$	−5.10748e6	−1.82645	−0.913227	−	0.407452i	$-0.866418\pi$
$379$	−5.10748e6	−1.82645	−0.913227	+	0.407452i	$0.866418\pi$
$380$	1.01990e6	0.362327
$381$	0	0
$382$	−1.60051e6	−0.561178
$383$	1.63200e6	0.568491	0.284245	−	0.958752i	$-0.408257\pi$
$383$	1.63200e6	0.568491	0.284245	+	0.958752i	$0.408257\pi$
$384$	0	0
$385$	5.45587e6	1.87591
$386$	−2.79860e6	−0.956032
$387$	0	0
$388$	−479008.	−0.161534
$389$	−4.46563e6	−1.49627	−0.748133	−	0.663549i	$-0.769050\pi$
$389$	−4.46563e6	−1.49627	−0.748133	+	0.663549i	$0.769050\pi$
$390$	0	0
$391$	−2.21184e6	−0.731664
$392$	−326208.	−0.107221
$393$	0	0
$394$	−1.62547e6	−0.527520
$395$	−2.69530e6	−0.869188
$396$	0	0
$397$	−611026.	−0.194573	−0.0972867	−	0.995256i	$-0.531016\pi$
$397$	−611026.	−0.194573	−0.0972867	+	0.995256i	$0.531016\pi$
$398$	1.44798e6	0.458201
$399$	0	0
$400$	1.55930e6	0.487280
$401$	−6.09158e6	−1.89177	−0.945887	−	0.324496i	$-0.894805\pi$
$401$	−6.09158e6	−1.89177	−0.945887	+	0.324496i	$0.894805\pi$
$402$	0	0
$403$	1.52438e6	0.467552
$404$	2.85850e6	0.871333
$405$	0	0
$406$	56832.0	0.0171111
$407$	2.22643e6	0.666229
$408$	0	0
$409$	2.89108e6	0.854578	0.427289	−	0.904115i	$-0.359469\pi$
$409$	2.89108e6	0.854578	0.427289	+	0.904115i	$0.359469\pi$
$410$	−2.58048e6	−0.758125
$411$	0	0
$412$	−1.84365e6	−0.535100
$413$	1.93229e6	0.557438
$414$	0	0
$415$	6.37747e6	1.81773
$416$	342016.	0.0968976
$417$	0	0
$418$	1.01990e6	0.285508
$419$	−2.98406e6	−0.830373	−0.415186	−	0.909736i	$-0.636284\pi$
$419$	−2.98406e6	−0.830373	−0.415186	+	0.909736i	$0.636284\pi$
$420$	0	0
$421$	822074.	0.226051	0.113025	−	0.993592i	$-0.463946\pi$
$421$	822074.	0.226051	0.113025	+	0.993592i	$0.463946\pi$
$422$	−607424.	−0.166039
$423$	0	0
$424$	−497664.	−0.134438
$425$	3.50842e6	0.942191
$426$	0	0
$427$	−6.33174e6	−1.68056
$428$	−1.21651e6	−0.321001
$429$	0	0
$430$	−5.71085e6	−1.48946
$431$	6.32448e6	1.63995	0.819977	−	0.572397i	$-0.193986\pi$
$431$	6.32448e6	1.63995	0.819977	+	0.572397i	$0.193986\pi$
$432$	0	0
$433$	−851902.	−0.218358	−0.109179	−	0.994022i	$-0.534822\pi$
$433$	−851902.	−0.218358	−0.109179	+	0.994022i	$0.534822\pi$
$434$	−2.70189e6	−0.688562
$435$	0	0
$436$	−3.69789e6	−0.931617
$437$	2.54976e6	0.638698
$438$	0	0
$439$	−334732.	−0.0828964	−0.0414482	−	0.999141i	$-0.513197\pi$
$439$	−334732.	−0.0828964	−0.0414482	+	0.999141i	$0.513197\pi$
$440$	2.35930e6	0.580966
$441$	0	0
$442$	769536.	0.187358
$443$	−1.76218e6	−0.426619	−0.213309	−	0.976985i	$-0.568424\pi$
$443$	−1.76218e6	−0.426619	−0.213309	+	0.976985i	$0.568424\pi$
$444$	0	0
$445$	7.85203e6	1.87967
$446$	4.37330e6	1.04105
$447$	0	0
$448$	−606208.	−0.142701
$449$	6.51859e6	1.52594	0.762971	−	0.646433i	$-0.223740\pi$
$449$	6.51859e6	1.52594	0.762971	+	0.646433i	$0.223740\pi$
$450$	0	0
$451$	−2.58048e6	−0.597392
$452$	2.27942e6	0.524782
$453$	0	0
$454$	2.26560e6	0.515874
$455$	−4.74547e6	−1.07461
$456$	0	0
$457$	−92074.0	−0.0206227	−0.0103114	−	0.999947i	$-0.503282\pi$
$457$	−92074.0	−0.0206227	−0.0103114	+	0.999947i	$0.503282\pi$
$458$	2.34882e6	0.523223
$459$	0	0
$460$	5.89824e6	1.29965
$461$	257568.	0.0564468	0.0282234	−	0.999602i	$-0.491015\pi$
$461$	257568.	0.0564468	0.0282234	+	0.999602i	$0.491015\pi$
$462$	0	0
$463$	3.96228e6	0.858998	0.429499	−	0.903067i	$-0.358690\pi$
$463$	3.96228e6	0.858998	0.429499	+	0.903067i	$0.358690\pi$
$464$	24576.0	0.00529927
$465$	0	0
$466$	−2.31782e6	−0.494442
$467$	−3.48941e6	−0.740388	−0.370194	−	0.928954i	$-0.620709\pi$
$467$	−3.48941e6	−0.740388	−0.370194	+	0.928954i	$0.620709\pi$
$468$	0	0
$469$	−5.42509e6	−1.13887
$470$	−7.37280e6	−1.53953
$471$	0	0
$472$	835584.	0.172637
$473$	−5.71085e6	−1.17367
$474$	0	0
$475$	−4.04442e6	−0.822475
$476$	−1.36397e6	−0.275922
$477$	0	0
$478$	2.33779e6	0.467990
$479$	513024.	0.102164	0.0510821	−	0.998694i	$-0.483733\pi$
$479$	513024.	0.102164	0.0510821	+	0.998694i	$0.483733\pi$
$480$	0	0
$481$	−1.93653e6	−0.381647
$482$	1.65652e6	0.324772
$483$	0	0
$484$	−217520.	−0.0422071
$485$	2.87405e6	0.554804
$486$	0	0
$487$	4.14499e6	0.791956	0.395978	−	0.918260i	$-0.370406\pi$
$487$	4.14499e6	0.791956	0.395978	+	0.918260i	$0.370406\pi$
$488$	−2.73805e6	−0.520465
$489$	0	0
$490$	1.95725e6	0.368261
$491$	2.75866e6	0.516409	0.258205	−	0.966090i	$-0.416869\pi$
$491$	2.75866e6	0.516409	0.258205	+	0.966090i	$0.416869\pi$
$492$	0	0
$493$	55296.0	0.0102465
$494$	−887104.	−0.163552
$495$	0	0
$496$	−1.16838e6	−0.213246
$497$	−9.54778e6	−1.73385
$498$	0	0
$499$	660896.	0.118818	0.0594089	−	0.998234i	$-0.481078\pi$
$499$	660896.	0.118818	0.0594089	+	0.998234i	$0.481078\pi$
$500$	−4.55578e6	−0.814962
$501$	0	0
$502$	7.59859e6	1.34578
$503$	−944640.	−0.166474	−0.0832370	−	0.996530i	$-0.526526\pi$
$503$	−944640.	−0.166474	−0.0832370	+	0.996530i	$0.526526\pi$
$504$	0	0
$505$	−1.71510e7	−2.99268
$506$	5.89824e6	1.02411
$507$	0	0
$508$	−15808.0	−0.00271780
$509$	−7.83773e6	−1.34090	−0.670449	−	0.741956i	$-0.733898\pi$
$509$	−7.83773e6	−1.34090	−0.670449	+	0.741956i	$0.733898\pi$
$510$	0	0
$511$	2.48788e6	0.421480
$512$	−262144.	−0.0441942
$513$	0	0
$514$	−1.79098e6	−0.299007
$515$	1.10619e7	1.83785
$516$	0	0
$517$	−7.37280e6	−1.21313
$518$	3.43242e6	0.562051
$519$	0	0
$520$	−2.05210e6	−0.332805
$521$	3.29645e6	0.532049	0.266025	−	0.963966i	$-0.414290\pi$
$521$	3.29645e6	0.532049	0.266025	+	0.963966i	$0.414290\pi$
$522$	0	0
$523$	6.50238e6	1.03948	0.519742	−	0.854323i	$-0.326028\pi$
$523$	6.50238e6	1.03948	0.519742	+	0.854323i	$0.326028\pi$
$524$	3.58810e6	0.570868
$525$	0	0
$526$	270336.	0.0426029
$527$	−2.62886e6	−0.412327
$528$	0	0
$529$	8.30926e6	1.29099
$530$	2.98598e6	0.461740
$531$	0	0
$532$	1.57235e6	0.240863
$533$	2.24448e6	0.342214
$534$	0	0
$535$	7.29907e6	1.10251
$536$	−2.34598e6	−0.352706
$537$	0	0
$538$	−2.25677e6	−0.336149
$539$	1.95725e6	0.290184
$540$	0	0
$541$	9.82714e6	1.44356	0.721778	−	0.692124i	$-0.243325\pi$
$541$	9.82714e6	1.44356	0.721778	+	0.692124i	$0.243325\pi$
$542$	−2.88123e6	−0.421289
$543$	0	0
$544$	−589824.	−0.0854526
$545$	2.21873e7	3.19973
$546$	0	0
$547$	−3.42580e6	−0.489546	−0.244773	−	0.969580i	$-0.578713\pi$
$547$	−3.42580e6	−0.489546	−0.244773	+	0.969580i	$0.578713\pi$
$548$	−4.46362e6	−0.634944
$549$	0	0
$550$	−9.35578e6	−1.31878
$551$	−63744.0	−0.00894459
$552$	0	0
$553$	−4.15525e6	−0.577809
$554$	564568.	0.0781523
$555$	0	0
$556$	2.83520e6	0.388953
$557$	−6.43363e6	−0.878655	−0.439327	−	0.898327i	$-0.644783\pi$
$557$	−6.43363e6	−0.878655	−0.439327	+	0.898327i	$0.644783\pi$
$558$	0	0
$559$	4.96725e6	0.672336
$560$	3.63725e6	0.490120
$561$	0	0
$562$	−2.33779e6	−0.312223
$563$	753024.	0.100124	0.0500620	−	0.998746i	$-0.484058\pi$
$563$	753024.	0.100124	0.0500620	+	0.998746i	$0.484058\pi$
$564$	0	0
$565$	−1.36765e7	−1.80242
$566$	−708224.	−0.0929244
$567$	0	0
$568$	−4.12877e6	−0.536970
$569$	−1.11481e7	−1.44351	−0.721755	−	0.692148i	$-0.756664\pi$
$569$	−1.11481e7	−1.44351	−0.721755	+	0.692148i	$0.756664\pi$
$570$	0	0
$571$	191024.	0.0245187	0.0122594	−	0.999925i	$-0.496098\pi$
$571$	191024.	0.0245187	0.0122594	+	0.999925i	$0.496098\pi$
$572$	−2.05210e6	−0.262245
$573$	0	0
$574$	−3.97824e6	−0.503978
$575$	−2.33894e7	−2.95019
$576$	0	0
$577$	1.03722e7	1.29697	0.648486	−	0.761227i	$-0.275403\pi$
$577$	1.03722e7	1.29697	0.648486	+	0.761227i	$0.275403\pi$
$578$	4.35232e6	0.541878
$579$	0	0
$580$	−147456.	−0.0182009
$581$	9.83194e6	1.20837
$582$	0	0
$583$	2.98598e6	0.363845
$584$	1.07584e6	0.130532
$585$	0	0
$586$	3.82733e6	0.460417
$587$	−2.97062e6	−0.355838	−0.177919	−	0.984045i	$-0.556937\pi$
$587$	−2.97062e6	−0.355838	−0.177919	+	0.984045i	$0.556937\pi$
$588$	0	0
$589$	3.03050e6	0.359936
$590$	−5.01350e6	−0.592941
$591$	0	0
$592$	1.48429e6	0.174066
$593$	−7.55827e6	−0.882644	−0.441322	−	0.897349i	$-0.645490\pi$
$593$	−7.55827e6	−0.882644	−0.441322	+	0.897349i	$0.645490\pi$
$594$	0	0
$595$	8.18381e6	0.947683
$596$	3.77702e6	0.435546
$597$	0	0
$598$	−5.13024e6	−0.586658
$599$	6.69158e6	0.762012	0.381006	−	0.924573i	$-0.375578\pi$
$599$	6.69158e6	0.762012	0.381006	+	0.924573i	$0.375578\pi$
$600$	0	0
$601$	−3.20359e6	−0.361785	−0.180893	−	0.983503i	$-0.557899\pi$
$601$	−3.20359e6	−0.361785	−0.180893	+	0.983503i	$0.557899\pi$
$602$	−8.80422e6	−0.990147
$603$	0	0
$604$	−7.72538e6	−0.861643
$605$	1.30512e6	0.144965
$606$	0	0
$607$	−1.35585e7	−1.49362	−0.746809	−	0.665038i	$-0.768415\pi$
$607$	−1.35585e7	−1.49362	−0.746809	+	0.665038i	$0.768415\pi$
$608$	679936.	0.0745949
$609$	0	0
$610$	1.64283e7	1.78759
$611$	6.41280e6	0.694936
$612$	0	0
$613$	−1.07654e7	−1.15712	−0.578561	−	0.815639i	$-0.696385\pi$
$613$	−1.07654e7	−1.15712	−0.578561	+	0.815639i	$0.696385\pi$
$614$	−1.15315e7	−1.23442
$615$	0	0
$616$	3.63725e6	0.386208
$617$	−9.33504e6	−0.987196	−0.493598	−	0.869690i	$-0.664319\pi$
$617$	−9.33504e6	−0.987196	−0.493598	+	0.869690i	$0.664319\pi$
$618$	0	0
$619$	9.07664e6	0.952135	0.476067	−	0.879409i	$-0.342062\pi$
$619$	9.07664e6	0.952135	0.476067	+	0.879409i	$0.342062\pi$
$620$	7.01030e6	0.732416
$621$	0	0
$622$	1.04018e7	1.07803
$623$	1.21052e7	1.24955
$624$	0	0
$625$	8.30028e6	0.849949
$626$	1.03232e7	1.05288
$627$	0	0
$628$	6.09738e6	0.616941
$629$	3.33965e6	0.336569
$630$	0	0
$631$	−1.13367e7	−1.13348	−0.566741	−	0.823896i	$-0.691796\pi$
$631$	−1.13367e7	−1.13348	−0.566741	+	0.823896i	$0.691796\pi$
$632$	−1.79686e6	−0.178946
$633$	0	0
$634$	−9.24403e6	−0.913352
$635$	94848.0	0.00933456
$636$	0	0
$637$	−1.70240e6	−0.166231
$638$	−147456.	−0.0143420
$639$	0	0
$640$	1.57286e6	0.151789
$641$	1.55449e7	1.49432	0.747159	−	0.664646i	$-0.231418\pi$
$641$	1.55449e7	1.49432	0.747159	+	0.664646i	$0.231418\pi$
$642$	0	0
$643$	−8.80026e6	−0.839399	−0.419699	−	0.907663i	$-0.637864\pi$
$643$	−8.80026e6	−0.839399	−0.419699	+	0.907663i	$0.637864\pi$
$644$	9.09312e6	0.863969
$645$	0	0
$646$	1.52986e6	0.144235
$647$	1.08449e7	1.01851	0.509256	−	0.860615i	$-0.329921\pi$
$647$	1.08449e7	1.01851	0.509256	+	0.860615i	$0.329921\pi$
$648$	0	0
$649$	−5.01350e6	−0.467229
$650$	8.13758e6	0.755460
$651$	0	0
$652$	2.60672e6	0.240146
$653$	−9.88771e6	−0.907429	−0.453715	−	0.891147i	$-0.649901\pi$
$653$	−9.88771e6	−0.907429	−0.453715	+	0.891147i	$0.649901\pi$
$654$	0	0
$655$	−2.15286e7	−1.96070
$656$	−1.72032e6	−0.156081
$657$	0	0
$658$	−1.13664e7	−1.02343
$659$	−1.46150e7	−1.31095	−0.655476	−	0.755216i	$-0.727532\pi$
$659$	−1.46150e7	−1.31095	−0.655476	+	0.755216i	$0.727532\pi$
$660$	0	0
$661$	1.57792e7	1.40469	0.702347	−	0.711834i	$-0.252135\pi$
$661$	1.57792e7	1.40469	0.702347	+	0.711834i	$0.252135\pi$
$662$	2.54810e6	0.225980
$663$	0	0
$664$	4.25165e6	0.374229
$665$	−9.43411e6	−0.827269
$666$	0	0
$667$	−368640.	−0.0320840
$668$	−9.05626e6	−0.785249
$669$	0	0
$670$	1.40759e7	1.21140
$671$	1.64283e7	1.40859
$672$	0	0
$673$	6.64939e6	0.565906	0.282953	−	0.959134i	$-0.408686\pi$
$673$	6.64939e6	0.565906	0.282953	+	0.959134i	$0.408686\pi$
$674$	−1.35466e7	−1.14863
$675$	0	0
$676$	−4.15579e6	−0.349774
$677$	4.42550e6	0.371100	0.185550	−	0.982635i	$-0.440593\pi$
$677$	4.42550e6	0.371100	0.185550	+	0.982635i	$0.440593\pi$
$678$	0	0
$679$	4.43082e6	0.368816
$680$	3.53894e6	0.293495
$681$	0	0
$682$	7.01030e6	0.577133
$683$	−4.67827e6	−0.383737	−0.191869	−	0.981421i	$-0.561455\pi$
$683$	−4.67827e6	−0.383737	−0.191869	+	0.981421i	$0.561455\pi$
$684$	0	0
$685$	2.67817e7	2.18078
$686$	−6.93232e6	−0.562430
$687$	0	0
$688$	−3.80723e6	−0.306647
$689$	−2.59718e6	−0.208427
$690$	0	0
$691$	−5.54102e6	−0.441463	−0.220731	−	0.975335i	$-0.570844\pi$
$691$	−5.54102e6	−0.441463	−0.220731	+	0.975335i	$0.570844\pi$
$692$	−3.49133e6	−0.277157
$693$	0	0
$694$	−1.11130e7	−0.875853
$695$	−1.70112e7	−1.33590
$696$	0	0
$697$	−3.87072e6	−0.301793
$698$	−6.22143e6	−0.483339
$699$	0	0
$700$	−1.44235e7	−1.11256
$701$	−6.53443e6	−0.502242	−0.251121	−	0.967956i	$-0.580799\pi$
$701$	−6.53443e6	−0.502242	−0.251121	+	0.967956i	$0.580799\pi$
$702$	0	0
$703$	−3.84987e6	−0.293804
$704$	1.57286e6	0.119608
$705$	0	0
$706$	−8.47104e6	−0.639624
$707$	−2.64411e7	−1.98944
$708$	0	0
$709$	−3.86541e6	−0.288789	−0.144394	−	0.989520i	$-0.546123\pi$
$709$	−3.86541e6	−0.288789	−0.144394	+	0.989520i	$0.546123\pi$
$710$	2.47726e7	1.84428
$711$	0	0
$712$	5.23469e6	0.386982
$713$	1.75258e7	1.29108
$714$	0	0
$715$	1.23126e7	0.900708
$716$	−6.59866e6	−0.481031
$717$	0	0
$718$	−8.69990e6	−0.629801
$719$	4.80614e6	0.346717	0.173358	−	0.984859i	$-0.444538\pi$
$719$	4.80614e6	0.346717	0.173358	+	0.984859i	$0.444538\pi$
$720$	0	0
$721$	1.70537e7	1.22175
$722$	8.14081e6	0.581199
$723$	0	0
$724$	−408928.	−0.0289935
$725$	584736.	0.0413157
$726$	0	0
$727$	−1.90590e7	−1.33741	−0.668704	−	0.743529i	$-0.733150\pi$
$727$	−1.90590e7	−1.33741	−0.668704	+	0.743529i	$0.733150\pi$
$728$	−3.16365e6	−0.221238
$729$	0	0
$730$	−6.45504e6	−0.448324
$731$	−8.56627e6	−0.592923
$732$	0	0
$733$	5.69616e6	0.391582	0.195791	−	0.980646i	$-0.437273\pi$
$733$	5.69616e6	0.391582	0.195791	+	0.980646i	$0.437273\pi$
$734$	−4.21342e6	−0.288666
$735$	0	0
$736$	3.93216e6	0.267570
$737$	1.40759e7	0.954570
$738$	0	0
$739$	1.84902e7	1.24546	0.622730	−	0.782437i	$-0.286024\pi$
$739$	1.84902e7	1.24546	0.622730	+	0.782437i	$0.286024\pi$
$740$	−8.90573e6	−0.597847
$741$	0	0
$742$	4.60339e6	0.306950
$743$	9.90336e6	0.658128	0.329064	−	0.944308i	$-0.393267\pi$
$743$	9.90336e6	0.658128	0.329064	+	0.944308i	$0.393267\pi$
$744$	0	0
$745$	−2.26621e7	−1.49593
$746$	2.70839e6	0.178182
$747$	0	0
$748$	3.53894e6	0.231270
$749$	1.12527e7	0.732915
$750$	0	0
$751$	−9.05914e6	−0.586121	−0.293060	−	0.956094i	$-0.594674\pi$
$751$	−9.05914e6	−0.586121	−0.293060	+	0.956094i	$0.594674\pi$
$752$	−4.91520e6	−0.316954
$753$	0	0
$754$	128256.	0.00821579
$755$	4.63523e7	2.95940
$756$	0	0
$757$	−1.16677e7	−0.740022	−0.370011	−	0.929027i	$-0.620646\pi$
$757$	−1.16677e7	−0.740022	−0.370011	+	0.929027i	$0.620646\pi$
$758$	2.04299e7	1.29150
$759$	0	0
$760$	−4.07962e6	−0.256204
$761$	−1.27398e7	−0.797444	−0.398722	−	0.917072i	$-0.630546\pi$
$761$	−1.27398e7	−0.797444	−0.398722	+	0.917072i	$0.630546\pi$
$762$	0	0
$763$	3.42055e7	2.12708
$764$	6.40205e6	0.396813
$765$	0	0
$766$	−6.52800e6	−0.401983
$767$	4.36070e6	0.267651
$768$	0	0
$769$	1.06783e7	0.651156	0.325578	−	0.945515i	$-0.394441\pi$
$769$	1.06783e7	0.651156	0.325578	+	0.945515i	$0.394441\pi$
$770$	−2.18235e7	−1.32647
$771$	0	0
$772$	1.11944e7	0.676017
$773$	−9.18634e6	−0.552960	−0.276480	−	0.961020i	$-0.589168\pi$
$773$	−9.18634e6	−0.552960	−0.276480	+	0.961020i	$0.589168\pi$
$774$	0	0
$775$	−2.77993e7	−1.66257
$776$	1.91603e6	0.114222
$777$	0	0
$778$	1.78625e7	1.05802
$779$	4.46208e6	0.263447
$780$	0	0
$781$	2.47726e7	1.45326
$782$	8.84736e6	0.517365
$783$	0	0
$784$	1.30483e6	0.0758166
$785$	−3.65843e7	−2.11895
$786$	0	0
$787$	−9.28209e6	−0.534206	−0.267103	−	0.963668i	$-0.586066\pi$
$787$	−9.28209e6	−0.534206	−0.267103	+	0.963668i	$0.586066\pi$
$788$	6.50189e6	0.373013
$789$	0	0
$790$	1.07812e7	0.614609
$791$	−2.10847e7	−1.19819
$792$	0	0
$793$	−1.42892e7	−0.806909
$794$	2.44410e6	0.137584
$795$	0	0
$796$	−5.79194e6	−0.323997
$797$	21792.0	0.00121521	0.000607605	−	1.00000i	$-0.499807\pi$
$797$	21792.0	0.00121521	0.000607605		1.00000i	$0.499807\pi$
$798$	0	0
$799$	−1.10592e7	−0.612854
$800$	−6.23718e6	−0.344559
$801$	0	0
$802$	2.43663e7	1.33769
$803$	−6.45504e6	−0.353273
$804$	0	0
$805$	−5.45587e7	−2.96739
$806$	−6.09750e6	−0.330609
$807$	0	0
$808$	−1.14340e7	−0.616125
$809$	1.23085e7	0.661204	0.330602	−	0.943770i	$-0.392748\pi$
$809$	1.23085e7	0.661204	0.330602	+	0.943770i	$0.392748\pi$
$810$	0	0
$811$	−2.34636e7	−1.25269	−0.626343	−	0.779547i	$-0.715449\pi$
$811$	−2.34636e7	−1.25269	−0.626343	+	0.779547i	$0.715449\pi$
$812$	−227328.	−0.0120994
$813$	0	0
$814$	−8.90573e6	−0.471095
$815$	−1.56403e7	−0.824806
$816$	0	0
$817$	9.87501e6	0.517586
$818$	−1.15643e7	−0.604278
$819$	0	0
$820$	1.03219e7	0.536075
$821$	−1.44206e7	−0.746666	−0.373333	−	0.927697i	$-0.621785\pi$
$821$	−1.44206e7	−0.746666	−0.373333	+	0.927697i	$0.621785\pi$
$822$	0	0
$823$	3.43419e7	1.76736	0.883679	−	0.468093i	$-0.155059\pi$
$823$	3.43419e7	1.76736	0.883679	+	0.468093i	$0.155059\pi$
$824$	7.37459e6	0.378373
$825$	0	0
$826$	−7.72915e6	−0.394168
$827$	2.13327e7	1.08463	0.542316	−	0.840174i	$-0.317547\pi$
$827$	2.13327e7	1.08463	0.542316	+	0.840174i	$0.317547\pi$
$828$	0	0
$829$	2.63751e6	0.133293	0.0666465	−	0.997777i	$-0.478770\pi$
$829$	2.63751e6	0.133293	0.0666465	+	0.997777i	$0.478770\pi$
$830$	−2.55099e7	−1.28533
$831$	0	0
$832$	−1.36806e6	−0.0685170
$833$	2.93587e6	0.146597
$834$	0	0
$835$	5.43375e7	2.69702
$836$	−4.07962e6	−0.201885
$837$	0	0
$838$	1.19363e7	0.587162
$839$	−1.00577e7	−0.493282	−0.246641	−	0.969107i	$-0.579327\pi$
$839$	−1.00577e7	−0.493282	−0.246641	+	0.969107i	$0.579327\pi$
$840$	0	0
$841$	−2.05019e7	−0.999551
$842$	−3.28830e6	−0.159842
$843$	0	0
$844$	2.42970e6	0.117407
$845$	2.49348e7	1.20133
$846$	0	0
$847$	2.01206e6	0.0963679
$848$	1.99066e6	0.0950619
$849$	0	0
$850$	−1.40337e7	−0.666229
$851$	−2.22643e7	−1.05387
$852$	0	0
$853$	−2.30748e7	−1.08584	−0.542919	−	0.839785i	$-0.682681\pi$
$853$	−2.30748e7	−1.08584	−0.542919	+	0.839785i	$0.682681\pi$
$854$	2.53269e7	1.18833
$855$	0	0
$856$	4.86605e6	0.226982
$857$	3.51646e7	1.63551	0.817756	−	0.575565i	$-0.195218\pi$
$857$	3.51646e7	1.63551	0.817756	+	0.575565i	$0.195218\pi$
$858$	0	0
$859$	1.66022e7	0.767684	0.383842	−	0.923399i	$-0.374601\pi$
$859$	1.66022e7	0.767684	0.383842	+	0.923399i	$0.374601\pi$
$860$	2.28434e7	1.05321
$861$	0	0
$862$	−2.52979e7	−1.15962
$863$	−2.97009e7	−1.35751	−0.678754	−	0.734366i	$-0.737480\pi$
$863$	−2.97009e7	−1.35751	−0.678754	+	0.734366i	$0.737480\pi$
$864$	0	0
$865$	2.09480e7	0.951923
$866$	3.40761e6	0.154403
$867$	0	0
$868$	1.08076e7	0.486887
$869$	1.07812e7	0.484303
$870$	0	0
$871$	−1.22431e7	−0.546822
$872$	1.47916e7	0.658753
$873$	0	0
$874$	−1.01990e7	−0.451628
$875$	4.21409e7	1.86073
$876$	0	0
$877$	−1.17943e7	−0.517811	−0.258906	−	0.965903i	$-0.583362\pi$
$877$	−1.17943e7	−0.517811	−0.258906	+	0.965903i	$0.583362\pi$
$878$	1.33893e6	0.0586166
$879$	0	0
$880$	−9.43718e6	−0.410805
$881$	−2.10378e7	−0.913190	−0.456595	−	0.889675i	$-0.650931\pi$
$881$	−2.10378e7	−0.913190	−0.456595	+	0.889675i	$0.650931\pi$
$882$	0	0
$883$	2.12192e7	0.915855	0.457928	−	0.888990i	$-0.348592\pi$
$883$	2.12192e7	0.915855	0.457928	+	0.888990i	$0.348592\pi$
$884$	−3.07814e6	−0.132482
$885$	0	0
$886$	7.04870e6	0.301665
$887$	2.28818e7	0.976520	0.488260	−	0.872698i	$-0.337632\pi$
$887$	2.28818e7	0.976520	0.488260	+	0.872698i	$0.337632\pi$
$888$	0	0
$889$	146224.	0.00620532
$890$	−3.14081e7	−1.32913
$891$	0	0
$892$	−1.74932e7	−0.736134
$893$	1.27488e7	0.534984
$894$	0	0
$895$	3.95919e7	1.65215
$896$	2.42483e6	0.100905
$897$	0	0
$898$	−2.60744e7	−1.07900
$899$	−438144.	−0.0180808
$900$	0	0
$901$	4.47898e6	0.183809
$902$	1.03219e7	0.422420
$903$	0	0
$904$	−9.11770e6	−0.371077
$905$	2.45357e6	0.0995810
$906$	0	0
$907$	−9.68692e6	−0.390992	−0.195496	−	0.980705i	$-0.562632\pi$
$907$	−9.68692e6	−0.390992	−0.195496	+	0.980705i	$0.562632\pi$
$908$	−9.06240e6	−0.364778
$909$	0	0
$910$	1.89819e7	0.759864
$911$	9.38112e6	0.374506	0.187253	−	0.982312i	$-0.440042\pi$
$911$	9.38112e6	0.374506	0.187253	+	0.982312i	$0.440042\pi$
$912$	0	0
$913$	−2.55099e7	−1.01282
$914$	368296.	0.0145825
$915$	0	0
$916$	−9.39530e6	−0.369974
$917$	−3.31899e7	−1.30341
$918$	0	0
$919$	−4.21870e7	−1.64774	−0.823872	−	0.566775i	$-0.808191\pi$
$919$	−4.21870e7	−1.64774	−0.823872	+	0.566775i	$0.808191\pi$
$920$	−2.35930e7	−0.918994
$921$	0	0
$922$	−1.03027e6	−0.0399139
$923$	−2.15470e7	−0.832497
$924$	0	0
$925$	3.53156e7	1.35710
$926$	−1.58491e7	−0.607403
$927$	0	0
$928$	−98304.0	−0.00374715
$929$	−3.04556e7	−1.15779	−0.578893	−	0.815404i	$-0.696515\pi$
$929$	−3.04556e7	−1.15779	−0.578893	+	0.815404i	$0.696515\pi$
$930$	0	0
$931$	−3.38441e6	−0.127970
$932$	9.27130e6	0.349624
$933$	0	0
$934$	1.39576e7	0.523534
$935$	−2.12337e7	−0.794321
$936$	0	0
$937$	1.47847e7	0.550128	0.275064	−	0.961426i	$-0.411301\pi$
$937$	1.47847e7	0.550128	0.275064	+	0.961426i	$0.411301\pi$
$938$	2.17004e7	0.805304
$939$	0	0
$940$	2.94912e7	1.08861
$941$	−9.91997e6	−0.365205	−0.182602	−	0.983187i	$-0.558452\pi$
$941$	−9.91997e6	−0.365205	−0.182602	+	0.983187i	$0.558452\pi$
$942$	0	0
$943$	2.58048e7	0.944977
$944$	−3.34234e6	−0.122073
$945$	0	0
$946$	2.28434e7	0.829913
$947$	2.91610e6	0.105664	0.0528320	−	0.998603i	$-0.483175\pi$
$947$	2.91610e6	0.105664	0.0528320	+	0.998603i	$0.483175\pi$
$948$	0	0
$949$	5.61454e6	0.202371
$950$	1.61777e7	0.581578
$951$	0	0
$952$	5.45587e6	0.195107
$953$	1.40861e7	0.502410	0.251205	−	0.967934i	$-0.419173\pi$
$953$	1.40861e7	0.502410	0.251205	+	0.967934i	$0.419173\pi$
$954$	0	0
$955$	−3.84123e7	−1.36289
$956$	−9.35117e6	−0.330919
$957$	0	0
$958$	−2.05210e6	−0.0722410
$959$	4.12884e7	1.44971
$960$	0	0
$961$	−7.79906e6	−0.272417
$962$	7.74613e6	0.269865
$963$	0	0
$964$	−6.62608e6	−0.229649
$965$	−6.71664e7	−2.32185
$966$	0	0
$967$	1.51949e7	0.522553	0.261276	−	0.965264i	$-0.415857\pi$
$967$	1.51949e7	0.522553	0.261276	+	0.965264i	$0.415857\pi$
$968$	870080.	0.0298449
$969$	0	0
$970$	−1.14962e7	−0.392306
$971$	−5.61220e7	−1.91023	−0.955113	−	0.296240i	$-0.904267\pi$
$971$	−5.61220e7	−1.91023	−0.955113	+	0.296240i	$0.904267\pi$
$972$	0	0
$973$	−2.62256e7	−0.888062
$974$	−1.65800e7	−0.559997
$975$	0	0
$976$	1.09522e7	0.368024
$977$	3.45625e7	1.15843	0.579214	−	0.815176i	$-0.303360\pi$
$977$	3.45625e7	1.15843	0.579214	+	0.815176i	$0.303360\pi$
$978$	0	0
$979$	−3.14081e7	−1.04733
$980$	−7.82899e6	−0.260400
$981$	0	0
$982$	−1.10346e7	−0.365156
$983$	5.56385e7	1.83650	0.918252	−	0.395997i	$-0.129601\pi$
$983$	5.56385e7	1.83650	0.918252	+	0.395997i	$0.129601\pi$
$984$	0	0
$985$	−3.90113e7	−1.28115
$986$	−221184.	−0.00724538
$987$	0	0
$988$	3.54842e6	0.115649
$989$	5.71085e7	1.85656
$990$	0	0
$991$	−3.60028e7	−1.16453	−0.582267	−	0.812998i	$-0.697834\pi$
$991$	−3.60028e7	−1.16453	−0.582267	+	0.812998i	$0.697834\pi$
$992$	4.67354e6	0.150788
$993$	0	0
$994$	3.81911e7	1.22602
$995$	3.47516e7	1.11280
$996$	0	0
$997$	2.35811e7	0.751322	0.375661	−	0.926757i	$-0.377416\pi$
$997$	2.35811e7	0.751322	0.375661	+	0.926757i	$0.377416\pi$
$998$	−2.64358e6	−0.0840169
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.6.a.a.1.1	✓	1
3.2	odd	2		18.6.a.c.1.1	yes	1
4.3	odd	2		144.6.a.a.1.1		1
5.2	odd	4		450.6.c.m.199.1		2
5.3	odd	4		450.6.c.m.199.2		2
5.4	even	2		450.6.a.v.1.1		1
7.6	odd	2		882.6.a.k.1.1		1
8.3	odd	2		576.6.a.bi.1.1		1
8.5	even	2		576.6.a.bh.1.1		1
9.2	odd	6		162.6.c.a.109.1		2
9.4	even	3		162.6.c.l.55.1		2
9.5	odd	6		162.6.c.a.55.1		2
9.7	even	3		162.6.c.l.109.1		2
12.11	even	2		144.6.a.l.1.1		1
15.2	even	4		450.6.c.c.199.2		2
15.8	even	4		450.6.c.c.199.1		2
15.14	odd	2		450.6.a.k.1.1		1
21.20	even	2		882.6.a.l.1.1		1
24.5	odd	2		576.6.a.a.1.1		1
24.11	even	2		576.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.6.a.a.1.1	✓	1	1.1	even	1	trivial
18.6.a.c.1.1	yes	1	3.2	odd	2
144.6.a.a.1.1		1	4.3	odd	2
144.6.a.l.1.1		1	12.11	even	2
162.6.c.a.55.1		2	9.5	odd	6
162.6.c.a.109.1		2	9.2	odd	6
162.6.c.l.55.1		2	9.4	even	3
162.6.c.l.109.1		2	9.7	even	3
450.6.a.k.1.1		1	15.14	odd	2
450.6.a.v.1.1		1	5.4	even	2
450.6.c.c.199.1		2	15.8	even	4
450.6.c.c.199.2		2	15.2	even	4
450.6.c.m.199.1		2	5.2	odd	4
450.6.c.m.199.2		2	5.3	odd	4
576.6.a.a.1.1		1	24.5	odd	2
576.6.a.b.1.1		1	24.11	even	2
576.6.a.bh.1.1		1	8.5	even	2
576.6.a.bi.1.1		1	8.3	odd	2
882.6.a.k.1.1		1	7.6	odd	2
882.6.a.l.1.1		1	21.20	even	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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