LMFDB - Embedded newform 336.6.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("336.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.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:	$53.8889634572$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} +6.00000 q^{5} -49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} +6.00000 q^{5} -49.0000 q^{7} +81.0000 q^{9} +108.000 q^{11} -346.000 q^{13} +54.0000 q^{15} -1398.00 q^{17} +1012.00 q^{19} -441.000 q^{21} +1536.00 q^{23} -3089.00 q^{25} +729.000 q^{27} -3762.00 q^{29} +736.000 q^{31} +972.000 q^{33} -294.000 q^{35} +2054.00 q^{37} -3114.00 q^{39} -15534.0 q^{41} -11036.0 q^{43} +486.000 q^{45} -4560.00 q^{47} +2401.00 q^{49} -12582.0 q^{51} -7962.00 q^{53} +648.000 q^{55} +9108.00 q^{57} +7020.00 q^{59} +26870.0 q^{61} -3969.00 q^{63} -2076.00 q^{65} -52148.0 q^{67} +13824.0 q^{69} +2544.00 q^{71} -9766.00 q^{73} -27801.0 q^{75} -5292.00 q^{77} -68672.0 q^{79} +6561.00 q^{81} +61668.0 q^{83} -8388.00 q^{85} -33858.0 q^{87} -41454.0 q^{89} +16954.0 q^{91} +6624.00 q^{93} +6072.00 q^{95} -111262. q^{97} +8748.00 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	6.00000	0.107331	0.0536656	−	0.998559i	$-0.482909\pi$
$5$	6.00000	0.107331	0.0536656	+	0.998559i	$0.482909\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	108.000	0.269118	0.134559	−	0.990906i	$-0.457038\pi$
$11$	108.000	0.269118	0.134559	+	0.990906i	$0.457038\pi$
$12$	0	0
$13$	−346.000	−0.567829	−0.283915	−	0.958850i	$-0.591633\pi$
$13$	−346.000	−0.567829	−0.283915	+	0.958850i	$0.591633\pi$
$14$	0	0
$15$	54.0000	0.0619677
$16$	0	0
$17$	−1398.00	−1.17323	−0.586617	−	0.809864i	$-0.699541\pi$
$17$	−1398.00	−1.17323	−0.586617	+	0.809864i	$0.699541\pi$
$18$	0	0
$19$	1012.00	0.643127	0.321563	−	0.946888i	$-0.395792\pi$
$19$	1012.00	0.643127	0.321563	+	0.946888i	$0.395792\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	0	0
$23$	1536.00	0.605441	0.302720	−	0.953079i	$-0.402105\pi$
$23$	1536.00	0.605441	0.302720	+	0.953079i	$0.402105\pi$
$24$	0	0
$25$	−3089.00	−0.988480
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−3762.00	−0.830661	−0.415330	−	0.909671i	$-0.636334\pi$
$29$	−3762.00	−0.830661	−0.415330	+	0.909671i	$0.636334\pi$
$30$	0	0
$31$	736.000	0.137554	0.0687771	−	0.997632i	$-0.478090\pi$
$31$	736.000	0.137554	0.0687771	+	0.997632i	$0.478090\pi$
$32$	0	0
$33$	972.000	0.155375
$34$	0	0
$35$	−294.000	−0.0405674
$36$	0	0
$37$	2054.00	0.246659	0.123329	−	0.992366i	$-0.460643\pi$
$37$	2054.00	0.246659	0.123329	+	0.992366i	$0.460643\pi$
$38$	0	0
$39$	−3114.00	−0.327836
$40$	0	0
$41$	−15534.0	−1.44319	−0.721595	−	0.692315i	$-0.756591\pi$
$41$	−15534.0	−1.44319	−0.721595	+	0.692315i	$0.756591\pi$
$42$	0	0
$43$	−11036.0	−0.910208	−0.455104	−	0.890438i	$-0.650398\pi$
$43$	−11036.0	−0.910208	−0.455104	+	0.890438i	$0.650398\pi$
$44$	0	0
$45$	486.000	0.0357771
$46$	0	0
$47$	−4560.00	−0.301107	−0.150553	−	0.988602i	$-0.548106\pi$
$47$	−4560.00	−0.301107	−0.150553	+	0.988602i	$0.548106\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−12582.0	−0.677367
$52$	0	0
$53$	−7962.00	−0.389343	−0.194672	−	0.980868i	$-0.562364\pi$
$53$	−7962.00	−0.389343	−0.194672	+	0.980868i	$0.562364\pi$
$54$	0	0
$55$	648.000	0.0288847
$56$	0	0
$57$	9108.00	0.371309
$58$	0	0
$59$	7020.00	0.262547	0.131274	−	0.991346i	$-0.458093\pi$
$59$	7020.00	0.262547	0.131274	+	0.991346i	$0.458093\pi$
$60$	0	0
$61$	26870.0	0.924577	0.462288	−	0.886730i	$-0.347028\pi$
$61$	26870.0	0.924577	0.462288	+	0.886730i	$0.347028\pi$
$62$	0	0
$63$	−3969.00	−0.125988
$64$	0	0
$65$	−2076.00	−0.0609458
$66$	0	0
$67$	−52148.0	−1.41922	−0.709612	−	0.704593i	$-0.751130\pi$
$67$	−52148.0	−1.41922	−0.709612	+	0.704593i	$0.751130\pi$
$68$	0	0
$69$	13824.0	0.349551
$70$	0	0
$71$	2544.00	0.0598923	0.0299462	−	0.999552i	$-0.490466\pi$
$71$	2544.00	0.0598923	0.0299462	+	0.999552i	$0.490466\pi$
$72$	0	0
$73$	−9766.00	−0.214491	−0.107246	−	0.994233i	$-0.534203\pi$
$73$	−9766.00	−0.214491	−0.107246	+	0.994233i	$0.534203\pi$
$74$	0	0
$75$	−27801.0	−0.570699
$76$	0	0
$77$	−5292.00	−0.101717
$78$	0	0
$79$	−68672.0	−1.23798	−0.618988	−	0.785401i	$-0.712457\pi$
$79$	−68672.0	−1.23798	−0.618988	+	0.785401i	$0.712457\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	61668.0	0.982573	0.491286	−	0.870998i	$-0.336527\pi$
$83$	61668.0	0.982573	0.491286	+	0.870998i	$0.336527\pi$
$84$	0	0
$85$	−8388.00	−0.125925
$86$	0	0
$87$	−33858.0	−0.479582
$88$	0	0
$89$	−41454.0	−0.554742	−0.277371	−	0.960763i	$-0.589463\pi$
$89$	−41454.0	−0.554742	−0.277371	+	0.960763i	$0.589463\pi$
$90$	0	0
$91$	16954.0	0.214619
$92$	0	0
$93$	6624.00	0.0794169
$94$	0	0
$95$	6072.00	0.0690276
$96$	0	0
$97$	−111262.	−1.20065	−0.600327	−	0.799755i	$-0.704963\pi$
$97$	−111262.	−1.20065	−0.600327	+	0.799755i	$0.704963\pi$
$98$	0	0
$99$	8748.00	0.0897059
$100$	0	0
$101$	−180426.	−1.75993	−0.879966	−	0.475037i	$-0.842435\pi$
$101$	−180426.	−1.75993	−0.879966	+	0.475037i	$0.842435\pi$
$102$	0	0
$103$	−35912.0	−0.333539	−0.166769	−	0.985996i	$-0.553334\pi$
$103$	−35912.0	−0.333539	−0.166769	+	0.985996i	$0.553334\pi$
$104$	0	0
$105$	−2646.00	−0.0234216
$106$	0	0
$107$	30492.0	0.257470	0.128735	−	0.991679i	$-0.458908\pi$
$107$	30492.0	0.257470	0.128735	+	0.991679i	$0.458908\pi$
$108$	0	0
$109$	82382.0	0.664150	0.332075	−	0.943253i	$-0.392251\pi$
$109$	82382.0	0.664150	0.332075	+	0.943253i	$0.392251\pi$
$110$	0	0
$111$	18486.0	0.142408
$112$	0	0
$113$	−160398.	−1.18169	−0.590844	−	0.806786i	$-0.701205\pi$
$113$	−160398.	−1.18169	−0.590844	+	0.806786i	$0.701205\pi$
$114$	0	0
$115$	9216.00	0.0649827
$116$	0	0
$117$	−28026.0	−0.189276
$118$	0	0
$119$	68502.0	0.443441
$120$	0	0
$121$	−149387.	−0.927576
$122$	0	0
$123$	−139806.	−0.833226
$124$	0	0
$125$	−37284.0	−0.213426
$126$	0	0
$127$	80896.0	0.445059	0.222530	−	0.974926i	$-0.428569\pi$
$127$	80896.0	0.445059	0.222530	+	0.974926i	$0.428569\pi$
$128$	0	0
$129$	−99324.0	−0.525509
$130$	0	0
$131$	−173676.	−0.884223	−0.442111	−	0.896960i	$-0.645770\pi$
$131$	−173676.	−0.884223	−0.442111	+	0.896960i	$0.645770\pi$
$132$	0	0
$133$	−49588.0	−0.243079
$134$	0	0
$135$	4374.00	0.0206559
$136$	0	0
$137$	390426.	1.77720	0.888602	−	0.458679i	$-0.151677\pi$
$137$	390426.	1.77720	0.888602	+	0.458679i	$0.151677\pi$
$138$	0	0
$139$	−83204.0	−0.365264	−0.182632	−	0.983181i	$-0.558462\pi$
$139$	−83204.0	−0.365264	−0.182632	+	0.983181i	$0.558462\pi$
$140$	0	0
$141$	−41040.0	−0.173844
$142$	0	0
$143$	−37368.0	−0.152813
$144$	0	0
$145$	−22572.0	−0.0891559
$146$	0	0
$147$	21609.0	0.0824786
$148$	0	0
$149$	140358.	0.517931	0.258965	−	0.965887i	$-0.416618\pi$
$149$	140358.	0.517931	0.258965	+	0.965887i	$0.416618\pi$
$150$	0	0
$151$	−320360.	−1.14339	−0.571697	−	0.820465i	$-0.693715\pi$
$151$	−320360.	−1.14339	−0.571697	+	0.820465i	$0.693715\pi$
$152$	0	0
$153$	−113238.	−0.391078
$154$	0	0
$155$	4416.00	0.0147639
$156$	0	0
$157$	−158266.	−0.512435	−0.256217	−	0.966619i	$-0.582476\pi$
$157$	−158266.	−0.512435	−0.256217	+	0.966619i	$0.582476\pi$
$158$	0	0
$159$	−71658.0	−0.224787
$160$	0	0
$161$	−75264.0	−0.228835
$162$	0	0
$163$	−345476.	−1.01847	−0.509236	−	0.860627i	$-0.670072\pi$
$163$	−345476.	−1.01847	−0.509236	+	0.860627i	$0.670072\pi$
$164$	0	0
$165$	5832.00	0.0166766
$166$	0	0
$167$	20568.0	0.0570691	0.0285345	−	0.999593i	$-0.490916\pi$
$167$	20568.0	0.0570691	0.0285345	+	0.999593i	$0.490916\pi$
$168$	0	0
$169$	−251577.	−0.677570
$170$	0	0
$171$	81972.0	0.214376
$172$	0	0
$173$	732558.	1.86092	0.930458	−	0.366399i	$-0.119409\pi$
$173$	732558.	1.86092	0.930458	+	0.366399i	$0.119409\pi$
$174$	0	0
$175$	151361.	0.373610
$176$	0	0
$177$	63180.0	0.151582
$178$	0	0
$179$	−572220.	−1.33484	−0.667422	−	0.744680i	$-0.732602\pi$
$179$	−572220.	−1.33484	−0.667422	+	0.744680i	$0.732602\pi$
$180$	0	0
$181$	−352402.	−0.799543	−0.399772	−	0.916615i	$-0.630911\pi$
$181$	−352402.	−0.799543	−0.399772	+	0.916615i	$0.630911\pi$
$182$	0	0
$183$	241830.	0.533805
$184$	0	0
$185$	12324.0	0.0264742
$186$	0	0
$187$	−150984.	−0.315738
$188$	0	0
$189$	−35721.0	−0.0727393
$190$	0	0
$191$	18456.0	0.0366062	0.0183031	−	0.999832i	$-0.494174\pi$
$191$	18456.0	0.0366062	0.0183031	+	0.999832i	$0.494174\pi$
$192$	0	0
$193$	832322.	1.60841	0.804207	−	0.594349i	$-0.202590\pi$
$193$	832322.	1.60841	0.804207	+	0.594349i	$0.202590\pi$
$194$	0	0
$195$	−18684.0	−0.0351871
$196$	0	0
$197$	612438.	1.12434	0.562169	−	0.827023i	$-0.309967\pi$
$197$	612438.	1.12434	0.562169	+	0.827023i	$0.309967\pi$
$198$	0	0
$199$	501352.	0.897450	0.448725	−	0.893670i	$-0.351878\pi$
$199$	501352.	0.897450	0.448725	+	0.893670i	$0.351878\pi$
$200$	0	0
$201$	−469332.	−0.819389
$202$	0	0
$203$	184338.	0.313960
$204$	0	0
$205$	−93204.0	−0.154899
$206$	0	0
$207$	124416.	0.201814
$208$	0	0
$209$	109296.	0.173077
$210$	0	0
$211$	556588.	0.860652	0.430326	−	0.902673i	$-0.358399\pi$
$211$	556588.	0.860652	0.430326	+	0.902673i	$0.358399\pi$
$212$	0	0
$213$	22896.0	0.0345789
$214$	0	0
$215$	−66216.0	−0.0976938
$216$	0	0
$217$	−36064.0	−0.0519906
$218$	0	0
$219$	−87894.0	−0.123837
$220$	0	0
$221$	483708.	0.666197
$222$	0	0
$223$	1.25680e6	1.69240	0.846202	−	0.532862i	$-0.178884\pi$
$223$	1.25680e6	1.69240	0.846202	+	0.532862i	$0.178884\pi$
$224$	0	0
$225$	−250209.	−0.329493
$226$	0	0
$227$	700932.	0.902841	0.451420	−	0.892311i	$-0.350917\pi$
$227$	700932.	0.902841	0.451420	+	0.892311i	$0.350917\pi$
$228$	0	0
$229$	153374.	0.193269	0.0966347	−	0.995320i	$-0.469192\pi$
$229$	153374.	0.193269	0.0966347	+	0.995320i	$0.469192\pi$
$230$	0	0
$231$	−47628.0	−0.0587263
$232$	0	0
$233$	154266.	0.186157	0.0930787	−	0.995659i	$-0.470329\pi$
$233$	154266.	0.186157	0.0930787	+	0.995659i	$0.470329\pi$
$234$	0	0
$235$	−27360.0	−0.0323181
$236$	0	0
$237$	−618048.	−0.714745
$238$	0	0
$239$	926376.	1.04904	0.524521	−	0.851398i	$-0.324245\pi$
$239$	926376.	1.04904	0.524521	+	0.851398i	$0.324245\pi$
$240$	0	0
$241$	−1.05662e6	−1.17186	−0.585932	−	0.810360i	$-0.699271\pi$
$241$	−1.05662e6	−1.17186	−0.585932	+	0.810360i	$0.699271\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	14406.0	0.0153330
$246$	0	0
$247$	−350152.	−0.365186
$248$	0	0
$249$	555012.	0.567289
$250$	0	0
$251$	1.45984e6	1.46258	0.731290	−	0.682066i	$-0.238919\pi$
$251$	1.45984e6	1.46258	0.731290	+	0.682066i	$0.238919\pi$
$252$	0	0
$253$	165888.	0.162935
$254$	0	0
$255$	−75492.0	−0.0727027
$256$	0	0
$257$	1.57142e6	1.48409	0.742043	−	0.670353i	$-0.233857\pi$
$257$	1.57142e6	1.48409	0.742043	+	0.670353i	$0.233857\pi$
$258$	0	0
$259$	−100646.	−0.0932282
$260$	0	0
$261$	−304722.	−0.276887
$262$	0	0
$263$	1.46275e6	1.30401	0.652006	−	0.758214i	$-0.273928\pi$
$263$	1.46275e6	1.30401	0.652006	+	0.758214i	$0.273928\pi$
$264$	0	0
$265$	−47772.0	−0.0417887
$266$	0	0
$267$	−373086.	−0.320281
$268$	0	0
$269$	−230850.	−0.194513	−0.0972566	−	0.995259i	$-0.531007\pi$
$269$	−230850.	−0.194513	−0.0972566	+	0.995259i	$0.531007\pi$
$270$	0	0
$271$	574432.	0.475133	0.237567	−	0.971371i	$-0.423650\pi$
$271$	574432.	0.475133	0.237567	+	0.971371i	$0.423650\pi$
$272$	0	0
$273$	152586.	0.123910
$274$	0	0
$275$	−333612.	−0.266017
$276$	0	0
$277$	510950.	0.400110	0.200055	−	0.979785i	$-0.435888\pi$
$277$	510950.	0.400110	0.200055	+	0.979785i	$0.435888\pi$
$278$	0	0
$279$	59616.0	0.0458514
$280$	0	0
$281$	931146.	0.703480	0.351740	−	0.936098i	$-0.385590\pi$
$281$	931146.	0.703480	0.351740	+	0.936098i	$0.385590\pi$
$282$	0	0
$283$	−2.15560e6	−1.59994	−0.799969	−	0.600042i	$-0.795151\pi$
$283$	−2.15560e6	−1.59994	−0.799969	+	0.600042i	$0.795151\pi$
$284$	0	0
$285$	54648.0	0.0398531
$286$	0	0
$287$	761166.	0.545475
$288$	0	0
$289$	534547.	0.376479
$290$	0	0
$291$	−1.00136e6	−0.693197
$292$	0	0
$293$	962070.	0.654693	0.327346	−	0.944904i	$-0.393846\pi$
$293$	962070.	0.654693	0.327346	+	0.944904i	$0.393846\pi$
$294$	0	0
$295$	42120.0	0.0281795
$296$	0	0
$297$	78732.0	0.0517917
$298$	0	0
$299$	−531456.	−0.343787
$300$	0	0
$301$	540764.	0.344026
$302$	0	0
$303$	−1.62383e6	−1.01610
$304$	0	0
$305$	161220.	0.0992360
$306$	0	0
$307$	−1.71988e6	−1.04149	−0.520743	−	0.853714i	$-0.674345\pi$
$307$	−1.71988e6	−1.04149	−0.520743	+	0.853714i	$0.674345\pi$
$308$	0	0
$309$	−323208.	−0.192569
$310$	0	0
$311$	2.38121e6	1.39604	0.698018	−	0.716081i	$-0.254066\pi$
$311$	2.38121e6	1.39604	0.698018	+	0.716081i	$0.254066\pi$
$312$	0	0
$313$	−293494.	−0.169332	−0.0846659	−	0.996409i	$-0.526982\pi$
$313$	−293494.	−0.169332	−0.0846659	+	0.996409i	$0.526982\pi$
$314$	0	0
$315$	−23814.0	−0.0135225
$316$	0	0
$317$	−1.94642e6	−1.08790	−0.543949	−	0.839118i	$-0.683071\pi$
$317$	−1.94642e6	−1.08790	−0.543949	+	0.839118i	$0.683071\pi$
$318$	0	0
$319$	−406296.	−0.223545
$320$	0	0
$321$	274428.	0.148650
$322$	0	0
$323$	−1.41478e6	−0.754538
$324$	0	0
$325$	1.06879e6	0.561288
$326$	0	0
$327$	741438.	0.383447
$328$	0	0
$329$	223440.	0.113808
$330$	0	0
$331$	−1.25184e6	−0.628026	−0.314013	−	0.949419i	$-0.601674\pi$
$331$	−1.25184e6	−0.628026	−0.314013	+	0.949419i	$0.601674\pi$
$332$	0	0
$333$	166374.	0.0822195
$334$	0	0
$335$	−312888.	−0.152327
$336$	0	0
$337$	297458.	0.142676	0.0713380	−	0.997452i	$-0.477273\pi$
$337$	297458.	0.142676	0.0713380	+	0.997452i	$0.477273\pi$
$338$	0	0
$339$	−1.44358e6	−0.682248
$340$	0	0
$341$	79488.0	0.0370182
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	82944.0	0.0375178
$346$	0	0
$347$	3.40657e6	1.51878	0.759388	−	0.650638i	$-0.225498\pi$
$347$	3.40657e6	1.51878	0.759388	+	0.650638i	$0.225498\pi$
$348$	0	0
$349$	−420826.	−0.184943	−0.0924717	−	0.995715i	$-0.529477\pi$
$349$	−420826.	−0.184943	−0.0924717	+	0.995715i	$0.529477\pi$
$350$	0	0
$351$	−252234.	−0.109279
$352$	0	0
$353$	1.39435e6	0.595571	0.297786	−	0.954633i	$-0.403752\pi$
$353$	1.39435e6	0.595571	0.297786	+	0.954633i	$0.403752\pi$
$354$	0	0
$355$	15264.0	0.00642832
$356$	0	0
$357$	616518.	0.256021
$358$	0	0
$359$	3.55037e6	1.45391	0.726955	−	0.686686i	$-0.240935\pi$
$359$	3.55037e6	1.45391	0.726955	+	0.686686i	$0.240935\pi$
$360$	0	0
$361$	−1.45196e6	−0.586388
$362$	0	0
$363$	−1.34448e6	−0.535536
$364$	0	0
$365$	−58596.0	−0.0230216
$366$	0	0
$367$	391696.	0.151804	0.0759021	−	0.997115i	$-0.475816\pi$
$367$	391696.	0.151804	0.0759021	+	0.997115i	$0.475816\pi$
$368$	0	0
$369$	−1.25825e6	−0.481063
$370$	0	0
$371$	390138.	0.147158
$372$	0	0
$373$	−163834.	−0.0609722	−0.0304861	−	0.999535i	$-0.509706\pi$
$373$	−163834.	−0.0609722	−0.0304861	+	0.999535i	$0.509706\pi$
$374$	0	0
$375$	−335556.	−0.123222
$376$	0	0
$377$	1.30165e6	0.471674
$378$	0	0
$379$	−206156.	−0.0737221	−0.0368611	−	0.999320i	$-0.511736\pi$
$379$	−206156.	−0.0737221	−0.0368611	+	0.999320i	$0.511736\pi$
$380$	0	0
$381$	728064.	0.256955
$382$	0	0
$383$	−484176.	−0.168658	−0.0843289	−	0.996438i	$-0.526875\pi$
$383$	−484176.	−0.168658	−0.0843289	+	0.996438i	$0.526875\pi$
$384$	0	0
$385$	−31752.0	−0.0109174
$386$	0	0
$387$	−893916.	−0.303403
$388$	0	0
$389$	−4.17479e6	−1.39882	−0.699409	−	0.714722i	$-0.746553\pi$
$389$	−4.17479e6	−1.39882	−0.699409	+	0.714722i	$0.746553\pi$
$390$	0	0
$391$	−2.14733e6	−0.710324
$392$	0	0
$393$	−1.56308e6	−0.510506
$394$	0	0
$395$	−412032.	−0.132873
$396$	0	0
$397$	−4.39998e6	−1.40112	−0.700558	−	0.713595i	$-0.747066\pi$
$397$	−4.39998e6	−1.40112	−0.700558	+	0.713595i	$0.747066\pi$
$398$	0	0
$399$	−446292.	−0.140342
$400$	0	0
$401$	−5.45917e6	−1.69537	−0.847687	−	0.530497i	$-0.822005\pi$
$401$	−5.45917e6	−1.69537	−0.847687	+	0.530497i	$0.822005\pi$
$402$	0	0
$403$	−254656.	−0.0781072
$404$	0	0
$405$	39366.0	0.0119257
$406$	0	0
$407$	221832.	0.0663801
$408$	0	0
$409$	2.18307e6	0.645295	0.322648	−	0.946519i	$-0.395427\pi$
$409$	2.18307e6	0.645295	0.322648	+	0.946519i	$0.395427\pi$
$410$	0	0
$411$	3.51383e6	1.02607
$412$	0	0
$413$	−343980.	−0.0992334
$414$	0	0
$415$	370008.	0.105461
$416$	0	0
$417$	−748836.	−0.210885
$418$	0	0
$419$	−5.44561e6	−1.51535	−0.757673	−	0.652635i	$-0.773664\pi$
$419$	−5.44561e6	−1.51535	−0.757673	+	0.652635i	$0.773664\pi$
$420$	0	0
$421$	−4.83054e6	−1.32828	−0.664141	−	0.747607i	$-0.731203\pi$
$421$	−4.83054e6	−1.32828	−0.664141	+	0.747607i	$0.731203\pi$
$422$	0	0
$423$	−369360.	−0.100369
$424$	0	0
$425$	4.31842e6	1.15972
$426$	0	0
$427$	−1.31663e6	−0.349457
$428$	0	0
$429$	−336312.	−0.0882265
$430$	0	0
$431$	7.58182e6	1.96598	0.982992	−	0.183647i	$-0.0587904\pi$
$431$	7.58182e6	1.96598	0.982992	+	0.183647i	$0.0587904\pi$
$432$	0	0
$433$	−99838.0	−0.0255903	−0.0127952	−	0.999918i	$-0.504073\pi$
$433$	−99838.0	−0.0255903	−0.0127952	+	0.999918i	$0.504073\pi$
$434$	0	0
$435$	−203148.	−0.0514742
$436$	0	0
$437$	1.55443e6	0.389375
$438$	0	0
$439$	7.77690e6	1.92595	0.962976	−	0.269587i	$-0.0868873\pi$
$439$	7.77690e6	1.92595	0.962976	+	0.269587i	$0.0868873\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	−3.15488e6	−0.763790	−0.381895	−	0.924206i	$-0.624728\pi$
$443$	−3.15488e6	−0.763790	−0.381895	+	0.924206i	$0.624728\pi$
$444$	0	0
$445$	−248724.	−0.0595412
$446$	0	0
$447$	1.26322e6	0.299027
$448$	0	0
$449$	4.91450e6	1.15044	0.575219	−	0.817999i	$-0.304917\pi$
$449$	4.91450e6	1.15044	0.575219	+	0.817999i	$0.304917\pi$
$450$	0	0
$451$	−1.67767e6	−0.388388
$452$	0	0
$453$	−2.88324e6	−0.660139
$454$	0	0
$455$	101724.	0.0230354
$456$	0	0
$457$	−5.77001e6	−1.29237	−0.646183	−	0.763182i	$-0.723636\pi$
$457$	−5.77001e6	−1.29237	−0.646183	+	0.763182i	$0.723636\pi$
$458$	0	0
$459$	−1.01914e6	−0.225789
$460$	0	0
$461$	1.65272e6	0.362198	0.181099	−	0.983465i	$-0.442035\pi$
$461$	1.65272e6	0.362198	0.181099	+	0.983465i	$0.442035\pi$
$462$	0	0
$463$	−7.69146e6	−1.66746	−0.833731	−	0.552170i	$-0.813800\pi$
$463$	−7.69146e6	−1.66746	−0.833731	+	0.552170i	$0.813800\pi$
$464$	0	0
$465$	39744.0	0.00852392
$466$	0	0
$467$	−4.95880e6	−1.05217	−0.526083	−	0.850433i	$-0.676340\pi$
$467$	−4.95880e6	−1.05217	−0.526083	+	0.850433i	$0.676340\pi$
$468$	0	0
$469$	2.55525e6	0.536416
$470$	0	0
$471$	−1.42439e6	−0.295854
$472$	0	0
$473$	−1.19189e6	−0.244953
$474$	0	0
$475$	−3.12607e6	−0.635718
$476$	0	0
$477$	−644922.	−0.129781
$478$	0	0
$479$	1.46899e6	0.292537	0.146268	−	0.989245i	$-0.453274\pi$
$479$	1.46899e6	0.292537	0.146268	+	0.989245i	$0.453274\pi$
$480$	0	0
$481$	−710684.	−0.140060
$482$	0	0
$483$	−677376.	−0.132118
$484$	0	0
$485$	−667572.	−0.128868
$486$	0	0
$487$	6.46679e6	1.23557	0.617784	−	0.786348i	$-0.288031\pi$
$487$	6.46679e6	1.23557	0.617784	+	0.786348i	$0.288031\pi$
$488$	0	0
$489$	−3.10928e6	−0.588015
$490$	0	0
$491$	−8.94479e6	−1.67443	−0.837214	−	0.546876i	$-0.815817\pi$
$491$	−8.94479e6	−1.67443	−0.837214	+	0.546876i	$0.815817\pi$
$492$	0	0
$493$	5.25928e6	0.974560
$494$	0	0
$495$	52488.0	0.00962824
$496$	0	0
$497$	−124656.	−0.0226372
$498$	0	0
$499$	−1.05052e7	−1.88866	−0.944329	−	0.329004i	$-0.893287\pi$
$499$	−1.05052e7	−1.88866	−0.944329	+	0.329004i	$0.893287\pi$
$500$	0	0
$501$	185112.	0.0329488
$502$	0	0
$503$	−7.97993e6	−1.40630	−0.703152	−	0.711040i	$-0.748225\pi$
$503$	−7.97993e6	−1.40630	−0.703152	+	0.711040i	$0.748225\pi$
$504$	0	0
$505$	−1.08256e6	−0.188896
$506$	0	0
$507$	−2.26419e6	−0.391195
$508$	0	0
$509$	8.13425e6	1.39163	0.695814	−	0.718222i	$-0.255044\pi$
$509$	8.13425e6	1.39163	0.695814	+	0.718222i	$0.255044\pi$
$510$	0	0
$511$	478534.	0.0810701
$512$	0	0
$513$	737748.	0.123770
$514$	0	0
$515$	−215472.	−0.0357992
$516$	0	0
$517$	−492480.	−0.0810331
$518$	0	0
$519$	6.59302e6	1.07440
$520$	0	0
$521$	1.76800e6	0.285357	0.142678	−	0.989769i	$-0.454429\pi$
$521$	1.76800e6	0.285357	0.142678	+	0.989769i	$0.454429\pi$
$522$	0	0
$523$	4.07211e6	0.650976	0.325488	−	0.945546i	$-0.394471\pi$
$523$	4.07211e6	0.650976	0.325488	+	0.945546i	$0.394471\pi$
$524$	0	0
$525$	1.36225e6	0.215704
$526$	0	0
$527$	−1.02893e6	−0.161383
$528$	0	0
$529$	−4.07705e6	−0.633442
$530$	0	0
$531$	568620.	0.0875157
$532$	0	0
$533$	5.37476e6	0.819486
$534$	0	0
$535$	182952.	0.0276346
$536$	0	0
$537$	−5.14998e6	−0.770672
$538$	0	0
$539$	259308.	0.0384454
$540$	0	0
$541$	−4.60904e6	−0.677045	−0.338522	−	0.940958i	$-0.609927\pi$
$541$	−4.60904e6	−0.677045	−0.338522	+	0.940958i	$0.609927\pi$
$542$	0	0
$543$	−3.17162e6	−0.461616
$544$	0	0
$545$	494292.	0.0712840
$546$	0	0
$547$	5.81091e6	0.830378	0.415189	−	0.909735i	$-0.363715\pi$
$547$	5.81091e6	0.830378	0.415189	+	0.909735i	$0.363715\pi$
$548$	0	0
$549$	2.17647e6	0.308192
$550$	0	0
$551$	−3.80714e6	−0.534220
$552$	0	0
$553$	3.36493e6	0.467911
$554$	0	0
$555$	110916.	0.0152849
$556$	0	0
$557$	−9.59949e6	−1.31102	−0.655511	−	0.755185i	$-0.727547\pi$
$557$	−9.59949e6	−1.31102	−0.655511	+	0.755185i	$0.727547\pi$
$558$	0	0
$559$	3.81846e6	0.516843
$560$	0	0
$561$	−1.35886e6	−0.182291
$562$	0	0
$563$	3.21152e6	0.427012	0.213506	−	0.976942i	$-0.431512\pi$
$563$	3.21152e6	0.427012	0.213506	+	0.976942i	$0.431512\pi$
$564$	0	0
$565$	−962388.	−0.126832
$566$	0	0
$567$	−321489.	−0.0419961
$568$	0	0
$569$	−9.23573e6	−1.19589	−0.597944	−	0.801538i	$-0.704016\pi$
$569$	−9.23573e6	−1.19589	−0.597944	+	0.801538i	$0.704016\pi$
$570$	0	0
$571$	−2.81683e6	−0.361551	−0.180776	−	0.983524i	$-0.557861\pi$
$571$	−2.81683e6	−0.361551	−0.180776	+	0.983524i	$0.557861\pi$
$572$	0	0
$573$	166104.	0.0211346
$574$	0	0
$575$	−4.74470e6	−0.598466
$576$	0	0
$577$	4.13415e6	0.516947	0.258474	−	0.966018i	$-0.416780\pi$
$577$	4.13415e6	0.516947	0.258474	+	0.966018i	$0.416780\pi$
$578$	0	0
$579$	7.49090e6	0.928619
$580$	0	0
$581$	−3.02173e6	−0.371378
$582$	0	0
$583$	−859896.	−0.104779
$584$	0	0
$585$	−168156.	−0.0203153
$586$	0	0
$587$	4.38155e6	0.524847	0.262423	−	0.964953i	$-0.415478\pi$
$587$	4.38155e6	0.524847	0.262423	+	0.964953i	$0.415478\pi$
$588$	0	0
$589$	744832.	0.0884647
$590$	0	0
$591$	5.51194e6	0.649136
$592$	0	0
$593$	1.22971e7	1.43604	0.718020	−	0.696023i	$-0.245049\pi$
$593$	1.22971e7	1.43604	0.718020	+	0.696023i	$0.245049\pi$
$594$	0	0
$595$	411012.	0.0475951
$596$	0	0
$597$	4.51217e6	0.518143
$598$	0	0
$599$	9.20928e6	1.04872	0.524359	−	0.851497i	$-0.324305\pi$
$599$	9.20928e6	1.04872	0.524359	+	0.851497i	$0.324305\pi$
$600$	0	0
$601$	−1.63394e7	−1.84522	−0.922612	−	0.385730i	$-0.873950\pi$
$601$	−1.63394e7	−1.84522	−0.922612	+	0.385730i	$0.873950\pi$
$602$	0	0
$603$	−4.22399e6	−0.473074
$604$	0	0
$605$	−896322.	−0.0995579
$606$	0	0
$607$	3.73082e6	0.410991	0.205495	−	0.978658i	$-0.434119\pi$
$607$	3.73082e6	0.410991	0.205495	+	0.978658i	$0.434119\pi$
$608$	0	0
$609$	1.65904e6	0.181265
$610$	0	0
$611$	1.57776e6	0.170977
$612$	0	0
$613$	1.46503e7	1.57469	0.787346	−	0.616511i	$-0.211455\pi$
$613$	1.46503e7	1.57469	0.787346	+	0.616511i	$0.211455\pi$
$614$	0	0
$615$	−838836.	−0.0894312
$616$	0	0
$617$	−4.33527e6	−0.458462	−0.229231	−	0.973372i	$-0.573621\pi$
$617$	−4.33527e6	−0.458462	−0.229231	+	0.973372i	$0.573621\pi$
$618$	0	0
$619$	1.79095e7	1.87869	0.939346	−	0.342970i	$-0.111433\pi$
$619$	1.79095e7	1.87869	0.939346	+	0.342970i	$0.111433\pi$
$620$	0	0
$621$	1.11974e6	0.116517
$622$	0	0
$623$	2.03125e6	0.209673
$624$	0	0
$625$	9.42942e6	0.965573
$626$	0	0
$627$	983664.	0.0999259
$628$	0	0
$629$	−2.87149e6	−0.289388
$630$	0	0
$631$	7.13615e6	0.713495	0.356747	−	0.934201i	$-0.383886\pi$
$631$	7.13615e6	0.713495	0.356747	+	0.934201i	$0.383886\pi$
$632$	0	0
$633$	5.00929e6	0.496898
$634$	0	0
$635$	485376.	0.0477688
$636$	0	0
$637$	−830746.	−0.0811185
$638$	0	0
$639$	206064.	0.0199641
$640$	0	0
$641$	−2.47025e6	−0.237463	−0.118732	−	0.992926i	$-0.537883\pi$
$641$	−2.47025e6	−0.237463	−0.118732	+	0.992926i	$0.537883\pi$
$642$	0	0
$643$	−1.13717e7	−1.08467	−0.542337	−	0.840161i	$-0.682460\pi$
$643$	−1.13717e7	−1.08467	−0.542337	+	0.840161i	$0.682460\pi$
$644$	0	0
$645$	−595944.	−0.0564035
$646$	0	0
$647$	1.69812e7	1.59480	0.797402	−	0.603449i	$-0.206207\pi$
$647$	1.69812e7	1.59480	0.797402	+	0.603449i	$0.206207\pi$
$648$	0	0
$649$	758160.	0.0706560
$650$	0	0
$651$	−324576.	−0.0300168
$652$	0	0
$653$	−1.56724e7	−1.43831	−0.719157	−	0.694847i	$-0.755472\pi$
$653$	−1.56724e7	−1.43831	−0.719157	+	0.694847i	$0.755472\pi$
$654$	0	0
$655$	−1.04206e6	−0.0949047
$656$	0	0
$657$	−791046.	−0.0714971
$658$	0	0
$659$	−2.85424e6	−0.256021	−0.128011	−	0.991773i	$-0.540859\pi$
$659$	−2.85424e6	−0.256021	−0.128011	+	0.991773i	$0.540859\pi$
$660$	0	0
$661$	1.56396e7	1.39226	0.696131	−	0.717915i	$-0.254903\pi$
$661$	1.56396e7	1.39226	0.696131	+	0.717915i	$0.254903\pi$
$662$	0	0
$663$	4.35337e6	0.384629
$664$	0	0
$665$	−297528.	−0.0260900
$666$	0	0
$667$	−5.77843e6	−0.502916
$668$	0	0
$669$	1.13112e7	0.977110
$670$	0	0
$671$	2.90196e6	0.248820
$672$	0	0
$673$	−3.01209e6	−0.256349	−0.128174	−	0.991752i	$-0.540912\pi$
$673$	−3.01209e6	−0.256349	−0.128174	+	0.991752i	$0.540912\pi$
$674$	0	0
$675$	−2.25188e6	−0.190233
$676$	0	0
$677$	8.98733e6	0.753632	0.376816	−	0.926288i	$-0.377019\pi$
$677$	8.98733e6	0.753632	0.376816	+	0.926288i	$0.377019\pi$
$678$	0	0
$679$	5.45184e6	0.453804
$680$	0	0
$681$	6.30839e6	0.521255
$682$	0	0
$683$	−7.13932e6	−0.585605	−0.292803	−	0.956173i	$-0.594588\pi$
$683$	−7.13932e6	−0.585605	−0.292803	+	0.956173i	$0.594588\pi$
$684$	0	0
$685$	2.34256e6	0.190750
$686$	0	0
$687$	1.38037e6	0.111584
$688$	0	0
$689$	2.75485e6	0.221080
$690$	0	0
$691$	1.55104e7	1.23575	0.617873	−	0.786278i	$-0.287995\pi$
$691$	1.55104e7	1.23575	0.617873	+	0.786278i	$0.287995\pi$
$692$	0	0
$693$	−428652.	−0.0339056
$694$	0	0
$695$	−499224.	−0.0392043
$696$	0	0
$697$	2.17165e7	1.69320
$698$	0	0
$699$	1.38839e6	0.107478
$700$	0	0
$701$	−2.22948e7	−1.71359	−0.856797	−	0.515654i	$-0.827549\pi$
$701$	−2.22948e7	−1.71359	−0.856797	+	0.515654i	$0.827549\pi$
$702$	0	0
$703$	2.07865e6	0.158633
$704$	0	0
$705$	−246240.	−0.0186589
$706$	0	0
$707$	8.84087e6	0.665191
$708$	0	0
$709$	2534.00	0.000189318 0	9.46588e−5	−	1.00000i	$-0.499970\pi$
$709$	2534.00	0.000189318 0	9.46588e−5		1.00000i	$0.499970\pi$
$710$	0	0
$711$	−5.56243e6	−0.412658
$712$	0	0
$713$	1.13050e6	0.0832809
$714$	0	0
$715$	−224208.	−0.0164016
$716$	0	0
$717$	8.33738e6	0.605664
$718$	0	0
$719$	−1.66728e7	−1.20278	−0.601390	−	0.798955i	$-0.705386\pi$
$719$	−1.66728e7	−1.20278	−0.601390	+	0.798955i	$0.705386\pi$
$720$	0	0
$721$	1.75969e6	0.126066
$722$	0	0
$723$	−9.50960e6	−0.676576
$724$	0	0
$725$	1.16208e7	0.821092
$726$	0	0
$727$	940648.	0.0660072	0.0330036	−	0.999455i	$-0.489493\pi$
$727$	940648.	0.0660072	0.0330036	+	0.999455i	$0.489493\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.54283e7	1.06789
$732$	0	0
$733$	2.76783e7	1.90274	0.951369	−	0.308053i	$-0.0996773\pi$
$733$	2.76783e7	1.90274	0.951369	+	0.308053i	$0.0996773\pi$
$734$	0	0
$735$	129654.	0.00885253
$736$	0	0
$737$	−5.63198e6	−0.381938
$738$	0	0
$739$	−1.00351e7	−0.675947	−0.337973	−	0.941156i	$-0.609741\pi$
$739$	−1.00351e7	−0.675947	−0.337973	+	0.941156i	$0.609741\pi$
$740$	0	0
$741$	−3.15137e6	−0.210840
$742$	0	0
$743$	−1.67202e7	−1.11114	−0.555572	−	0.831469i	$-0.687501\pi$
$743$	−1.67202e7	−1.11114	−0.555572	+	0.831469i	$0.687501\pi$
$744$	0	0
$745$	842148.	0.0555901
$746$	0	0
$747$	4.99511e6	0.327524
$748$	0	0
$749$	−1.49411e6	−0.0973145
$750$	0	0
$751$	−9.81728e6	−0.635172	−0.317586	−	0.948229i	$-0.602872\pi$
$751$	−9.81728e6	−0.635172	−0.317586	+	0.948229i	$0.602872\pi$
$752$	0	0
$753$	1.31385e7	0.844421
$754$	0	0
$755$	−1.92216e6	−0.122722
$756$	0	0
$757$	−2.72948e6	−0.173117	−0.0865587	−	0.996247i	$-0.527587\pi$
$757$	−2.72948e6	−0.173117	−0.0865587	+	0.996247i	$0.527587\pi$
$758$	0	0
$759$	1.49299e6	0.0940704
$760$	0	0
$761$	2.78176e6	0.174124	0.0870619	−	0.996203i	$-0.472252\pi$
$761$	2.78176e6	0.174124	0.0870619	+	0.996203i	$0.472252\pi$
$762$	0	0
$763$	−4.03672e6	−0.251025
$764$	0	0
$765$	−679428.	−0.0419749
$766$	0	0
$767$	−2.42892e6	−0.149082
$768$	0	0
$769$	5.04331e6	0.307539	0.153769	−	0.988107i	$-0.450859\pi$
$769$	5.04331e6	0.307539	0.153769	+	0.988107i	$0.450859\pi$
$770$	0	0
$771$	1.41428e7	0.856837
$772$	0	0
$773$	−1.49824e7	−0.901845	−0.450923	−	0.892563i	$-0.648905\pi$
$773$	−1.49824e7	−0.901845	−0.450923	+	0.892563i	$0.648905\pi$
$774$	0	0
$775$	−2.27350e6	−0.135969
$776$	0	0
$777$	−905814.	−0.0538253
$778$	0	0
$779$	−1.57204e7	−0.928154
$780$	0	0
$781$	274752.	0.0161181
$782$	0	0
$783$	−2.74250e6	−0.159861
$784$	0	0
$785$	−949596.	−0.0550003
$786$	0	0
$787$	−4.82851e6	−0.277892	−0.138946	−	0.990300i	$-0.544371\pi$
$787$	−4.82851e6	−0.277892	−0.138946	+	0.990300i	$0.544371\pi$
$788$	0	0
$789$	1.31648e7	0.752871
$790$	0	0
$791$	7.85950e6	0.446636
$792$	0	0
$793$	−9.29702e6	−0.525002
$794$	0	0
$795$	−429948.	−0.0241267
$796$	0	0
$797$	−2.22125e7	−1.23866	−0.619331	−	0.785130i	$-0.712596\pi$
$797$	−2.22125e7	−1.23866	−0.619331	+	0.785130i	$0.712596\pi$
$798$	0	0
$799$	6.37488e6	0.353269
$800$	0	0
$801$	−3.35777e6	−0.184914
$802$	0	0
$803$	−1.05473e6	−0.0577234
$804$	0	0
$805$	−451584.	−0.0245612
$806$	0	0
$807$	−2.07765e6	−0.112302
$808$	0	0
$809$	1.34312e7	0.721512	0.360756	−	0.932660i	$-0.382519\pi$
$809$	1.34312e7	0.721512	0.360756	+	0.932660i	$0.382519\pi$
$810$	0	0
$811$	1.99673e7	1.06602	0.533012	−	0.846107i	$-0.321060\pi$
$811$	1.99673e7	1.06602	0.533012	+	0.846107i	$0.321060\pi$
$812$	0	0
$813$	5.16989e6	0.274318
$814$	0	0
$815$	−2.07286e6	−0.109314
$816$	0	0
$817$	−1.11684e7	−0.585379
$818$	0	0
$819$	1.37327e6	0.0715398
$820$	0	0
$821$	−2.90063e7	−1.50188	−0.750939	−	0.660372i	$-0.770399\pi$
$821$	−2.90063e7	−1.50188	−0.750939	+	0.660372i	$0.770399\pi$
$822$	0	0
$823$	−2.38801e7	−1.22896	−0.614478	−	0.788934i	$-0.710634\pi$
$823$	−2.38801e7	−1.22896	−0.614478	+	0.788934i	$0.710634\pi$
$824$	0	0
$825$	−3.00251e6	−0.153585
$826$	0	0
$827$	9.58724e6	0.487450	0.243725	−	0.969844i	$-0.421631\pi$
$827$	9.58724e6	0.487450	0.243725	+	0.969844i	$0.421631\pi$
$828$	0	0
$829$	−2.66835e7	−1.34852	−0.674260	−	0.738494i	$-0.735537\pi$
$829$	−2.66835e7	−1.34852	−0.674260	+	0.738494i	$0.735537\pi$
$830$	0	0
$831$	4.59855e6	0.231003
$832$	0	0
$833$	−3.35660e6	−0.167605
$834$	0	0
$835$	123408.	0.00612530
$836$	0	0
$837$	536544.	0.0264723
$838$	0	0
$839$	−5.41870e6	−0.265760	−0.132880	−	0.991132i	$-0.542423\pi$
$839$	−5.41870e6	−0.265760	−0.132880	+	0.991132i	$0.542423\pi$
$840$	0	0
$841$	−6.35850e6	−0.310002
$842$	0	0
$843$	8.38031e6	0.406155
$844$	0	0
$845$	−1.50946e6	−0.0727244
$846$	0	0
$847$	7.31996e6	0.350591
$848$	0	0
$849$	−1.94004e7	−0.923724
$850$	0	0
$851$	3.15494e6	0.149337
$852$	0	0
$853$	−2.02323e7	−0.952079	−0.476040	−	0.879424i	$-0.657928\pi$
$853$	−2.02323e7	−0.952079	−0.476040	+	0.879424i	$0.657928\pi$
$854$	0	0
$855$	491832.	0.0230092
$856$	0	0
$857$	7.16071e6	0.333046	0.166523	−	0.986038i	$-0.446746\pi$
$857$	7.16071e6	0.333046	0.166523	+	0.986038i	$0.446746\pi$
$858$	0	0
$859$	1.24840e7	0.577258	0.288629	−	0.957441i	$-0.406801\pi$
$859$	1.24840e7	0.577258	0.288629	+	0.957441i	$0.406801\pi$
$860$	0	0
$861$	6.85049e6	0.314930
$862$	0	0
$863$	−3.32731e7	−1.52078	−0.760391	−	0.649466i	$-0.774993\pi$
$863$	−3.32731e7	−1.52078	−0.760391	+	0.649466i	$0.774993\pi$
$864$	0	0
$865$	4.39535e6	0.199734
$866$	0	0
$867$	4.81092e6	0.217361
$868$	0	0
$869$	−7.41658e6	−0.333161
$870$	0	0
$871$	1.80432e7	0.805876
$872$	0	0
$873$	−9.01222e6	−0.400218
$874$	0	0
$875$	1.82692e6	0.0806675
$876$	0	0
$877$	4.10248e6	0.180114	0.0900570	−	0.995937i	$-0.471295\pi$
$877$	4.10248e6	0.180114	0.0900570	+	0.995937i	$0.471295\pi$
$878$	0	0
$879$	8.65863e6	0.377987
$880$	0	0
$881$	−2.99636e6	−0.130063	−0.0650315	−	0.997883i	$-0.520715\pi$
$881$	−2.99636e6	−0.130063	−0.0650315	+	0.997883i	$0.520715\pi$
$882$	0	0
$883$	1.07514e7	0.464049	0.232024	−	0.972710i	$-0.425465\pi$
$883$	1.07514e7	0.464049	0.232024	+	0.972710i	$0.425465\pi$
$884$	0	0
$885$	379080.	0.0162694
$886$	0	0
$887$	−2.04970e7	−0.874744	−0.437372	−	0.899281i	$-0.644091\pi$
$887$	−2.04970e7	−0.874744	−0.437372	+	0.899281i	$0.644091\pi$
$888$	0	0
$889$	−3.96390e6	−0.168217
$890$	0	0
$891$	708588.	0.0299020
$892$	0	0
$893$	−4.61472e6	−0.193650
$894$	0	0
$895$	−3.43332e6	−0.143270
$896$	0	0
$897$	−4.78310e6	−0.198485
$898$	0	0
$899$	−2.76883e6	−0.114261
$900$	0	0
$901$	1.11309e7	0.456791
$902$	0	0
$903$	4.86688e6	0.198624
$904$	0	0
$905$	−2.11441e6	−0.0858160
$906$	0	0
$907$	1.97520e7	0.797245	0.398623	−	0.917115i	$-0.369488\pi$
$907$	1.97520e7	0.797245	0.398623	+	0.917115i	$0.369488\pi$
$908$	0	0
$909$	−1.46145e7	−0.586644
$910$	0	0
$911$	−2.48511e7	−0.992087	−0.496044	−	0.868298i	$-0.665214\pi$
$911$	−2.48511e7	−0.992087	−0.496044	+	0.868298i	$0.665214\pi$
$912$	0	0
$913$	6.66014e6	0.264428
$914$	0	0
$915$	1.45098e6	0.0572939
$916$	0	0
$917$	8.51012e6	0.334205
$918$	0	0
$919$	−1.54864e7	−0.604869	−0.302435	−	0.953170i	$-0.597799\pi$
$919$	−1.54864e7	−0.604869	−0.302435	+	0.953170i	$0.597799\pi$
$920$	0	0
$921$	−1.54790e7	−0.601302
$922$	0	0
$923$	−880224.	−0.0340086
$924$	0	0
$925$	−6.34481e6	−0.243817
$926$	0	0
$927$	−2.90887e6	−0.111180
$928$	0	0
$929$	1.73849e7	0.660895	0.330447	−	0.943824i	$-0.392800\pi$
$929$	1.73849e7	0.660895	0.330447	+	0.943824i	$0.392800\pi$
$930$	0	0
$931$	2.42981e6	0.0918752
$932$	0	0
$933$	2.14309e7	0.806001
$934$	0	0
$935$	−905904.	−0.0338886
$936$	0	0
$937$	−2.72897e7	−1.01543	−0.507714	−	0.861526i	$-0.669509\pi$
$937$	−2.72897e7	−1.01543	−0.507714	+	0.861526i	$0.669509\pi$
$938$	0	0
$939$	−2.64145e6	−0.0977637
$940$	0	0
$941$	−1.20400e7	−0.443256	−0.221628	−	0.975131i	$-0.571137\pi$
$941$	−1.20400e7	−0.443256	−0.221628	+	0.975131i	$0.571137\pi$
$942$	0	0
$943$	−2.38602e7	−0.873766
$944$	0	0
$945$	−214326.	−0.00780720
$946$	0	0
$947$	−3.82808e7	−1.38710	−0.693548	−	0.720411i	$-0.743953\pi$
$947$	−3.82808e7	−1.38710	−0.693548	+	0.720411i	$0.743953\pi$
$948$	0	0
$949$	3.37904e6	0.121794
$950$	0	0
$951$	−1.75178e7	−0.628098
$952$	0	0
$953$	1.83655e7	0.655043	0.327521	−	0.944844i	$-0.393787\pi$
$953$	1.83655e7	0.655043	0.327521	+	0.944844i	$0.393787\pi$
$954$	0	0
$955$	110736.	0.00392899
$956$	0	0
$957$	−3.65666e6	−0.129064
$958$	0	0
$959$	−1.91309e7	−0.671720
$960$	0	0
$961$	−2.80875e7	−0.981079
$962$	0	0
$963$	2.46985e6	0.0858233
$964$	0	0
$965$	4.99393e6	0.172633
$966$	0	0
$967$	−1.83781e7	−0.632024	−0.316012	−	0.948755i	$-0.602344\pi$
$967$	−1.83781e7	−0.632024	−0.316012	+	0.948755i	$0.602344\pi$
$968$	0	0
$969$	−1.27330e7	−0.435633
$970$	0	0
$971$	1.90795e7	0.649411	0.324706	−	0.945815i	$-0.394735\pi$
$971$	1.90795e7	0.649411	0.324706	+	0.945815i	$0.394735\pi$
$972$	0	0
$973$	4.07700e6	0.138057
$974$	0	0
$975$	9.61915e6	0.324060
$976$	0	0
$977$	3.37407e7	1.13088	0.565442	−	0.824788i	$-0.308706\pi$
$977$	3.37407e7	1.13088	0.565442	+	0.824788i	$0.308706\pi$
$978$	0	0
$979$	−4.47703e6	−0.149291
$980$	0	0
$981$	6.67294e6	0.221383
$982$	0	0
$983$	−1.56535e7	−0.516689	−0.258344	−	0.966053i	$-0.583177\pi$
$983$	−1.56535e7	−0.516689	−0.258344	+	0.966053i	$0.583177\pi$
$984$	0	0
$985$	3.67463e6	0.120677
$986$	0	0
$987$	2.01096e6	0.0657068
$988$	0	0
$989$	−1.69513e7	−0.551077
$990$	0	0
$991$	1.10122e7	0.356198	0.178099	−	0.984013i	$-0.443005\pi$
$991$	1.10122e7	0.356198	0.178099	+	0.984013i	$0.443005\pi$
$992$	0	0
$993$	−1.12665e7	−0.362591
$994$	0	0
$995$	3.00811e6	0.0963244
$996$	0	0
$997$	2.38356e7	0.759431	0.379716	−	0.925103i	$-0.376022\pi$
$997$	2.38356e7	0.759431	0.379716	+	0.925103i	$0.376022\pi$
$998$	0	0
$999$	1.49737e6	0.0474695

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.a.n.1.1		1
3.2	odd	2		1008.6.a.o.1.1		1
4.3	odd	2		84.6.a.a.1.1	✓	1
12.11	even	2		252.6.a.b.1.1		1
28.3	even	6		588.6.i.b.373.1		2
28.11	odd	6		588.6.i.f.373.1		2
28.19	even	6		588.6.i.b.361.1		2
28.23	odd	6		588.6.i.f.361.1		2
28.27	even	2		588.6.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.a.a.1.1	✓	1	4.3	odd	2
252.6.a.b.1.1		1	12.11	even	2
336.6.a.n.1.1		1	1.1	even	1	trivial
588.6.a.e.1.1		1	28.27	even	2
588.6.i.b.361.1		2	28.19	even	6
588.6.i.b.373.1		2	28.3	even	6
588.6.i.f.361.1		2	28.23	odd	6
588.6.i.f.373.1		2	28.11	odd	6
1008.6.a.o.1.1		1	3.2	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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