LMFDB - Embedded newform 1008.6.a.by.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,6,Mod(1,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, 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("1008.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1008.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:	$161.666890371$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.358541904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 111x^{2} + 756 $ x^4 - 111*x^2 + 756
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-10.1838$ of defining polynomial
Character	$\chi$	$=$	1008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.37743 q^{5} -49.0000 q^{7} +O(q^{10})$	$q+4.37743 q^{5} -49.0000 q^{7} -484.447 q^{11} -553.368 q^{13} -2145.48 q^{17} -2471.37 q^{19} +1183.39 q^{23} -3105.84 q^{25} -4213.61 q^{29} +3494.10 q^{31} -214.494 q^{35} +13084.5 q^{37} -8997.77 q^{41} -8806.94 q^{43} +23310.8 q^{47} +2401.00 q^{49} +15744.0 q^{53} -2120.63 q^{55} +12167.6 q^{59} -8244.51 q^{61} -2422.33 q^{65} -46247.8 q^{67} -5660.21 q^{71} +58377.7 q^{73} +23737.9 q^{77} +23797.4 q^{79} -91737.4 q^{83} -9391.69 q^{85} -17754.8 q^{89} +27115.0 q^{91} -10818.2 q^{95} +120013. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 196 q^{7}+O(q^{10}) $ 4 * q - 196 * q^7	$ 4 q - 196 q^{7} + 872 q^{13} - 6800 q^{19} + 3004 q^{25} + 4720 q^{31} + 6056 q^{37} - 10544 q^{43} + 9604 q^{49} - 11568 q^{55} + 13304 q^{61} - 203504 q^{67} + 17528 q^{73} - 34400 q^{79} + 354288 q^{85} - 42728 q^{91} + 436856 q^{97}+O(q^{100}) $ 4 * q - 196 * q^7 + 872 * q^13 - 6800 * q^19 + 3004 * q^25 + 4720 * q^31 + 6056 * q^37 - 10544 * q^43 + 9604 * q^49 - 11568 * q^55 + 13304 * q^61 - 203504 * q^67 + 17528 * q^73 - 34400 * q^79 + 354288 * q^85 - 42728 * q^91 + 436856 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	4.37743	0.0783059	0.0391529	−	0.999233i	$-0.487534\pi$
$5$	4.37743	0.0783059	0.0391529	+	0.999233i	$0.487534\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−484.447	−1.20716	−0.603579	−	0.797303i	$-0.706259\pi$
$11$	−484.447	−1.20716	−0.603579	+	0.797303i	$0.706259\pi$
$12$	0	0
$13$	−553.368	−0.908145	−0.454073	−	0.890965i	$-0.650029\pi$
$13$	−553.368	−0.908145	−0.454073	+	0.890965i	$0.650029\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2145.48	−1.80054	−0.900268	−	0.435336i	$-0.856630\pi$
$17$	−2145.48	−1.80054	−0.900268	+	0.435336i	$0.856630\pi$
$18$	0	0
$19$	−2471.37	−1.57056	−0.785278	−	0.619144i	$-0.787480\pi$
$19$	−2471.37	−1.57056	−0.785278	+	0.619144i	$0.787480\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1183.39	0.466452	0.233226	−	0.972423i	$-0.425072\pi$
$23$	1183.39	0.466452	0.233226	+	0.972423i	$0.425072\pi$
$24$	0	0
$25$	−3105.84	−0.993868
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4213.61	−0.930378	−0.465189	−	0.885211i	$-0.654014\pi$
$29$	−4213.61	−0.930378	−0.465189	+	0.885211i	$0.654014\pi$
$30$	0	0
$31$	3494.10	0.653027	0.326514	−	0.945192i	$-0.394126\pi$
$31$	3494.10	0.653027	0.326514	+	0.945192i	$0.394126\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−214.494	−0.0295968
$36$	0	0
$37$	13084.5	1.57128	0.785639	−	0.618685i	$-0.212334\pi$
$37$	13084.5	1.57128	0.785639	+	0.618685i	$0.212334\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8997.77	−0.835940	−0.417970	−	0.908461i	$-0.637258\pi$
$41$	−8997.77	−0.835940	−0.417970	+	0.908461i	$0.637258\pi$
$42$	0	0
$43$	−8806.94	−0.726363	−0.363182	−	0.931718i	$-0.618310\pi$
$43$	−8806.94	−0.726363	−0.363182	+	0.931718i	$0.618310\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	23310.8	1.53926	0.769631	−	0.638488i	$-0.220440\pi$
$47$	23310.8	1.53926	0.769631	+	0.638488i	$0.220440\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	15744.0	0.769885	0.384943	−	0.922940i	$-0.374221\pi$
$53$	15744.0	0.769885	0.384943	+	0.922940i	$0.374221\pi$
$54$	0	0
$55$	−2120.63	−0.0945276
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12167.6	0.455065	0.227533	−	0.973770i	$-0.426934\pi$
$59$	12167.6	0.455065	0.227533	+	0.973770i	$0.426934\pi$
$60$	0	0
$61$	−8244.51	−0.283688	−0.141844	−	0.989889i	$-0.545303\pi$
$61$	−8244.51	−0.283688	−0.141844	+	0.989889i	$0.545303\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2422.33	−0.0711131
$66$	0	0
$67$	−46247.8	−1.25865	−0.629324	−	0.777143i	$-0.716668\pi$
$67$	−46247.8	−1.25865	−0.629324	+	0.777143i	$0.716668\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5660.21	−0.133256	−0.0666280	−	0.997778i	$-0.521224\pi$
$71$	−5660.21	−0.133256	−0.0666280	+	0.997778i	$0.521224\pi$
$72$	0	0
$73$	58377.7	1.28215	0.641077	−	0.767477i	$-0.278488\pi$
$73$	58377.7	1.28215	0.641077	+	0.767477i	$0.278488\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	23737.9	0.456263
$78$	0	0
$79$	23797.4	0.429005	0.214503	−	0.976723i	$-0.431187\pi$
$79$	23797.4	0.429005	0.214503	+	0.976723i	$0.431187\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−91737.4	−1.46168	−0.730838	−	0.682551i	$-0.760871\pi$
$83$	−91737.4	−1.46168	−0.730838	+	0.682551i	$0.760871\pi$
$84$	0	0
$85$	−9391.69	−0.140993
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−17754.8	−0.237597	−0.118798	−	0.992918i	$-0.537904\pi$
$89$	−17754.8	−0.237597	−0.118798	+	0.992918i	$0.537904\pi$
$90$	0	0
$91$	27115.0	0.343247
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−10818.2	−0.122984
$96$	0	0
$97$	120013.	1.29509	0.647544	−	0.762028i	$-0.275796\pi$
$97$	120013.	1.29509	0.647544	+	0.762028i	$0.275796\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	182460.	1.77977	0.889885	−	0.456185i	$-0.150785\pi$
$101$	182460.	1.77977	0.889885	+	0.456185i	$0.150785\pi$
$102$	0	0
$103$	−85121.4	−0.790580	−0.395290	−	0.918556i	$-0.629356\pi$
$103$	−85121.4	−0.790580	−0.395290	+	0.918556i	$0.629356\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−79185.8	−0.668633	−0.334317	−	0.942461i	$-0.608505\pi$
$107$	−79185.8	−0.668633	−0.334317	+	0.942461i	$0.608505\pi$
$108$	0	0
$109$	−50413.1	−0.406422	−0.203211	−	0.979135i	$-0.565138\pi$
$109$	−50413.1	−0.406422	−0.203211	+	0.979135i	$0.565138\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−134497.	−0.990869	−0.495435	−	0.868645i	$-0.664991\pi$
$113$	−134497.	−0.990869	−0.495435	+	0.868645i	$0.664991\pi$
$114$	0	0
$115$	5180.20	0.0365260
$116$	0	0
$117$	0	0
$118$	0	0
$119$	105128.	0.680539
$120$	0	0
$121$	73637.5	0.457231
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−27275.1	−0.156132
$126$	0	0
$127$	11788.2	0.0648544	0.0324272	−	0.999474i	$-0.489676\pi$
$127$	11788.2	0.0648544	0.0324272	+	0.999474i	$0.489676\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−95377.3	−0.485587	−0.242793	−	0.970078i	$-0.578064\pi$
$131$	−95377.3	−0.485587	−0.242793	+	0.970078i	$0.578064\pi$
$132$	0	0
$133$	121097.	0.593614
$134$	0	0
$135$	0	0
$136$	0	0
$137$	193814.	0.882234	0.441117	−	0.897450i	$-0.354582\pi$
$137$	193814.	0.882234	0.441117	+	0.897450i	$0.354582\pi$
$138$	0	0
$139$	411417.	1.80611	0.903057	−	0.429521i	$-0.141318\pi$
$139$	411417.	1.80611	0.903057	+	0.429521i	$0.141318\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	268077.	1.09628
$144$	0	0
$145$	−18444.8	−0.0728541
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−454988.	−1.67893	−0.839467	−	0.543410i	$-0.817133\pi$
$149$	−454988.	−1.67893	−0.839467	+	0.543410i	$0.817133\pi$
$150$	0	0
$151$	37368.7	0.133372	0.0666862	−	0.997774i	$-0.478757\pi$
$151$	37368.7	0.133372	0.0666862	+	0.997774i	$0.478757\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	15295.2	0.0511359
$156$	0	0
$157$	−19273.6	−0.0624042	−0.0312021	−	0.999513i	$-0.509934\pi$
$157$	−19273.6	−0.0624042	−0.0312021	+	0.999513i	$0.509934\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−57986.0	−0.176302
$162$	0	0
$163$	−126487.	−0.372887	−0.186443	−	0.982466i	$-0.559696\pi$
$163$	−126487.	−0.372887	−0.186443	+	0.982466i	$0.559696\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−502926.	−1.39545	−0.697723	−	0.716367i	$-0.745803\pi$
$167$	−502926.	−1.39545	−0.697723	+	0.716367i	$0.745803\pi$
$168$	0	0
$169$	−65077.3	−0.175272
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−240846.	−0.611821	−0.305911	−	0.952060i	$-0.598961\pi$
$173$	−240846.	−0.611821	−0.305911	+	0.952060i	$0.598961\pi$
$174$	0	0
$175$	152186.	0.375647
$176$	0	0
$177$	0	0
$178$	0	0
$179$	533167.	1.24374	0.621872	−	0.783119i	$-0.286373\pi$
$179$	533167.	1.24374	0.621872	+	0.783119i	$0.286373\pi$
$180$	0	0
$181$	149768.	0.339800	0.169900	−	0.985461i	$-0.445656\pi$
$181$	149768.	0.339800	0.169900	+	0.985461i	$0.445656\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	57276.6	0.123040
$186$	0	0
$187$	1.03937e6	2.17353
$188$	0	0
$189$	0	0
$190$	0	0
$191$	641565.	1.27250	0.636249	−	0.771483i	$-0.280485\pi$
$191$	641565.	1.27250	0.636249	+	0.771483i	$0.280485\pi$
$192$	0	0
$193$	−36977.2	−0.0714563	−0.0357281	−	0.999362i	$-0.511375\pi$
$193$	−36977.2	−0.0714563	−0.0357281	+	0.999362i	$0.511375\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−612050.	−1.12363	−0.561813	−	0.827264i	$-0.689896\pi$
$197$	−612050.	−1.12363	−0.561813	+	0.827264i	$0.689896\pi$
$198$	0	0
$199$	77852.3	0.139360	0.0696801	−	0.997569i	$-0.477802\pi$
$199$	77852.3	0.139360	0.0696801	+	0.997569i	$0.477802\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	206467.	0.351650
$204$	0	0
$205$	−39387.1	−0.0654591
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.19725e6	1.89591
$210$	0	0
$211$	372155.	0.575464	0.287732	−	0.957711i	$-0.407099\pi$
$211$	372155.	0.575464	0.287732	+	0.957711i	$0.407099\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−38551.8	−0.0568785
$216$	0	0
$217$	−171211.	−0.246821
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.18724e6	1.63515
$222$	0	0
$223$	−184749.	−0.248783	−0.124391	−	0.992233i	$-0.539698\pi$
$223$	−184749.	−0.248783	−0.124391	+	0.992233i	$0.539698\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−579671.	−0.746650	−0.373325	−	0.927701i	$-0.621782\pi$
$227$	−579671.	−0.746650	−0.373325	+	0.927701i	$0.621782\pi$
$228$	0	0
$229$	1.01195e6	1.27518	0.637590	−	0.770376i	$-0.279932\pi$
$229$	1.01195e6	1.27518	0.637590	+	0.770376i	$0.279932\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23960.0	0.0289132	0.0144566	−	0.999895i	$-0.495398\pi$
$233$	23960.0	0.0289132	0.0144566	+	0.999895i	$0.495398\pi$
$234$	0	0
$235$	102042.	0.120533
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−335949.	−0.380433	−0.190217	−	0.981742i	$-0.560919\pi$
$239$	−335949.	−0.380433	−0.190217	+	0.981742i	$0.560919\pi$
$240$	0	0
$241$	310646.	0.344527	0.172264	−	0.985051i	$-0.444892\pi$
$241$	310646.	0.344527	0.172264	+	0.985051i	$0.444892\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	10510.2	0.0111866
$246$	0	0
$247$	1.36757e6	1.42629
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.35811e6	−1.36067	−0.680334	−	0.732902i	$-0.738165\pi$
$251$	−1.35811e6	−1.36067	−0.680334	+	0.732902i	$0.738165\pi$
$252$	0	0
$253$	−573288.	−0.563082
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.78802e6	−1.68865	−0.844324	−	0.535833i	$-0.819998\pi$
$257$	−1.78802e6	−1.68865	−0.844324	+	0.535833i	$0.819998\pi$
$258$	0	0
$259$	−641141.	−0.593888
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.87628e6	−1.67266	−0.836331	−	0.548225i	$-0.815304\pi$
$263$	−1.87628e6	−1.67266	−0.836331	+	0.548225i	$0.815304\pi$
$264$	0	0
$265$	68918.4	0.0602866
$266$	0	0
$267$	0	0
$268$	0	0
$269$	324303.	0.273256	0.136628	−	0.990622i	$-0.456374\pi$
$269$	324303.	0.273256	0.136628	+	0.990622i	$0.456374\pi$
$270$	0	0
$271$	1.20859e6	0.999671	0.499835	−	0.866120i	$-0.333394\pi$
$271$	1.20859e6	0.999671	0.499835	+	0.866120i	$0.333394\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.50461e6	1.19976
$276$	0	0
$277$	2.38986e6	1.87143	0.935715	−	0.352757i	$-0.114756\pi$
$277$	2.38986e6	1.87143	0.935715	+	0.352757i	$0.114756\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−702548.	−0.530775	−0.265387	−	0.964142i	$-0.585500\pi$
$281$	−702548.	−0.530775	−0.265387	+	0.964142i	$0.585500\pi$
$282$	0	0
$283$	306653.	0.227605	0.113802	−	0.993503i	$-0.463697\pi$
$283$	306653.	0.227605	0.113802	+	0.993503i	$0.463697\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	440891.	0.315956
$288$	0	0
$289$	3.18322e6	2.24193
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.05123e6	0.715369	0.357684	−	0.933843i	$-0.383566\pi$
$293$	1.05123e6	0.715369	0.357684	+	0.933843i	$0.383566\pi$
$294$	0	0
$295$	53262.7	0.0356343
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−654848.	−0.423607
$300$	0	0
$301$	431540.	0.274540
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−36089.8	−0.0222144
$306$	0	0
$307$	1.81985e6	1.10202	0.551011	−	0.834498i	$-0.314242\pi$
$307$	1.81985e6	1.10202	0.551011	+	0.834498i	$0.314242\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.56125e6	−0.915317	−0.457658	−	0.889128i	$-0.651312\pi$
$311$	−1.56125e6	−0.915317	−0.457658	+	0.889128i	$0.651312\pi$
$312$	0	0
$313$	−1.21687e6	−0.702074	−0.351037	−	0.936362i	$-0.614171\pi$
$313$	−1.21687e6	−0.702074	−0.351037	+	0.936362i	$0.614171\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.84405e6	1.03068	0.515341	−	0.856985i	$-0.327665\pi$
$317$	1.84405e6	1.03068	0.515341	+	0.856985i	$0.327665\pi$
$318$	0	0
$319$	2.04127e6	1.12311
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.30227e6	2.82784
$324$	0	0
$325$	1.71867e6	0.902577
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.14223e6	−0.581787
$330$	0	0
$331$	−1.55327e6	−0.779249	−0.389624	−	0.920974i	$-0.627395\pi$
$331$	−1.55327e6	−0.779249	−0.389624	+	0.920974i	$0.627395\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−202447.	−0.0985595
$336$	0	0
$337$	3.02648e6	1.45165	0.725826	−	0.687878i	$-0.241458\pi$
$337$	3.02648e6	1.45165	0.725826	+	0.687878i	$0.241458\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.69271e6	−0.788307
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−387651.	−0.172829	−0.0864145	−	0.996259i	$-0.527541\pi$
$347$	−387651.	−0.172829	−0.0864145	+	0.996259i	$0.527541\pi$
$348$	0	0
$349$	421998.	0.185458	0.0927292	−	0.995691i	$-0.470441\pi$
$349$	421998.	0.185458	0.0927292	+	0.995691i	$0.470441\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−93956.1	−0.0401317	−0.0200659	−	0.999799i	$-0.506388\pi$
$353$	−93956.1	−0.0401317	−0.0200659	+	0.999799i	$0.506388\pi$
$354$	0	0
$355$	−24777.2	−0.0104347
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.75361e6	1.53714	0.768569	−	0.639767i	$-0.220969\pi$
$359$	3.75361e6	1.53714	0.768569	+	0.639767i	$0.220969\pi$
$360$	0	0
$361$	3.63156e6	1.46665
$362$	0	0
$363$	0	0
$364$	0	0
$365$	255545.	0.100400
$366$	0	0
$367$	−649772.	−0.251823	−0.125912	−	0.992041i	$-0.540186\pi$
$367$	−649772.	−0.251823	−0.125912	+	0.992041i	$0.540186\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−771457.	−0.290989
$372$	0	0
$373$	−1.85971e6	−0.692107	−0.346053	−	0.938215i	$-0.612478\pi$
$373$	−1.85971e6	−0.692107	−0.346053	+	0.938215i	$0.612478\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.33168e6	0.844919
$378$	0	0
$379$	2.92330e6	1.04538	0.522691	−	0.852522i	$-0.324928\pi$
$379$	2.92330e6	1.04538	0.522691	+	0.852522i	$0.324928\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.12372e6	1.43645	0.718227	−	0.695808i	$-0.244954\pi$
$383$	4.12372e6	1.43645	0.718227	+	0.695808i	$0.244954\pi$
$384$	0	0
$385$	103911.	0.0357281
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.06677e6	−0.692499	−0.346249	−	0.938142i	$-0.612545\pi$
$389$	−2.06677e6	−0.692499	−0.346249	+	0.938142i	$0.612545\pi$
$390$	0	0
$391$	−2.53893e6	−0.839864
$392$	0	0
$393$	0	0
$394$	0	0
$395$	104172.	0.0335936
$396$	0	0
$397$	−2.76772e6	−0.881345	−0.440673	−	0.897668i	$-0.645260\pi$
$397$	−2.76772e6	−0.881345	−0.440673	+	0.897668i	$0.645260\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.84543e6	−1.50477	−0.752387	−	0.658722i	$-0.771097\pi$
$401$	−4.84543e6	−1.50477	−0.752387	+	0.658722i	$0.771097\pi$
$402$	0	0
$403$	−1.93352e6	−0.593044
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.33875e6	−1.89678
$408$	0	0
$409$	459910.	0.135946	0.0679728	−	0.997687i	$-0.478347\pi$
$409$	459910.	0.135946	0.0679728	+	0.997687i	$0.478347\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−596211.	−0.171999
$414$	0	0
$415$	−401574.	−0.114458
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.18785e6	−1.16535	−0.582675	−	0.812706i	$-0.697994\pi$
$419$	−4.18785e6	−1.16535	−0.582675	+	0.812706i	$0.697994\pi$
$420$	0	0
$421$	−1.76702e6	−0.485888	−0.242944	−	0.970040i	$-0.578113\pi$
$421$	−1.76702e6	−0.485888	−0.242944	+	0.970040i	$0.578113\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.66351e6	1.78950
$426$	0	0
$427$	403981.	0.107224
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−56325.0	−0.0146052	−0.00730261	−	0.999973i	$-0.502325\pi$
$431$	−56325.0	−0.0146052	−0.00730261	+	0.999973i	$0.502325\pi$
$432$	0	0
$433$	3.20825e6	0.822333	0.411167	−	0.911560i	$-0.365121\pi$
$433$	3.20825e6	0.822333	0.411167	+	0.911560i	$0.365121\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.92459e6	−0.732590
$438$	0	0
$439$	4.92795e6	1.22041	0.610204	−	0.792244i	$-0.291087\pi$
$439$	4.92795e6	1.22041	0.610204	+	0.792244i	$0.291087\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.43549e6	−1.31592	−0.657960	−	0.753053i	$-0.728580\pi$
$443$	−5.43549e6	−1.31592	−0.657960	+	0.753053i	$0.728580\pi$
$444$	0	0
$445$	−77720.4	−0.0186052
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−631641.	−0.147861	−0.0739307	−	0.997263i	$-0.523554\pi$
$449$	−631641.	−0.147861	−0.0739307	+	0.997263i	$0.523554\pi$
$450$	0	0
$451$	4.35894e6	1.00911
$452$	0	0
$453$	0	0
$454$	0	0
$455$	118694.	0.0268782
$456$	0	0
$457$	4.76464e6	1.06719	0.533593	−	0.845742i	$-0.320842\pi$
$457$	4.76464e6	1.06719	0.533593	+	0.845742i	$0.320842\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.50320e6	−0.329432	−0.164716	−	0.986341i	$-0.552671\pi$
$461$	−1.50320e6	−0.329432	−0.164716	+	0.986341i	$0.552671\pi$
$462$	0	0
$463$	7.70530e6	1.67046	0.835232	−	0.549897i	$-0.185333\pi$
$463$	7.70530e6	1.67046	0.835232	+	0.549897i	$0.185333\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.62260e6	−0.768650	−0.384325	−	0.923198i	$-0.625566\pi$
$467$	−3.62260e6	−0.768650	−0.384325	+	0.923198i	$0.625566\pi$
$468$	0	0
$469$	2.26614e6	0.475724
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.26649e6	0.876836
$474$	0	0
$475$	7.67567e6	1.56093
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.01413e6	0.201956	0.100978	−	0.994889i	$-0.467803\pi$
$479$	1.01413e6	0.201956	0.100978	+	0.994889i	$0.467803\pi$
$480$	0	0
$481$	−7.24055e6	−1.42695
$482$	0	0
$483$	0	0
$484$	0	0
$485$	525349.	0.101413
$486$	0	0
$487$	−3.74179e6	−0.714919	−0.357460	−	0.933929i	$-0.616357\pi$
$487$	−3.74179e6	−0.714919	−0.357460	+	0.933929i	$0.616357\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.90839e6	0.544439	0.272220	−	0.962235i	$-0.412242\pi$
$491$	2.90839e6	0.544439	0.272220	+	0.962235i	$0.412242\pi$
$492$	0	0
$493$	9.04021e6	1.67518
$494$	0	0
$495$	0	0
$496$	0	0
$497$	277350.	0.0503661
$498$	0	0
$499$	−2.77770e6	−0.499384	−0.249692	−	0.968325i	$-0.580329\pi$
$499$	−2.77770e6	−0.499384	−0.249692	+	0.968325i	$0.580329\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.89116e6	1.39066	0.695330	−	0.718691i	$-0.255258\pi$
$503$	7.89116e6	1.39066	0.695330	+	0.718691i	$0.255258\pi$
$504$	0	0
$505$	798706.	0.139366
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.80211e6	0.992639	0.496319	−	0.868140i	$-0.334684\pi$
$509$	5.80211e6	0.992639	0.496319	+	0.868140i	$0.334684\pi$
$510$	0	0
$511$	−2.86051e6	−0.484609
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−372613.	−0.0619071
$516$	0	0
$517$	−1.12928e7	−1.85813
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.21254e6	0.679907	0.339953	−	0.940442i	$-0.389589\pi$
$521$	4.21254e6	0.679907	0.339953	+	0.940442i	$0.389589\pi$
$522$	0	0
$523$	−417124.	−0.0666824	−0.0333412	−	0.999444i	$-0.510615\pi$
$523$	−417124.	−0.0666824	−0.0333412	+	0.999444i	$0.510615\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.49652e6	−1.17580
$528$	0	0
$529$	−5.03594e6	−0.782422
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.97907e6	0.759155
$534$	0	0
$535$	−346631.	−0.0523579
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.16316e6	−0.172451
$540$	0	0
$541$	−7.10039e6	−1.04301	−0.521506	−	0.853248i	$-0.674629\pi$
$541$	−7.10039e6	−1.04301	−0.521506	+	0.853248i	$0.674629\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−220680.	−0.0318252
$546$	0	0
$547$	−4.06802e6	−0.581320	−0.290660	−	0.956826i	$-0.593875\pi$
$547$	−4.06802e6	−0.581320	−0.290660	+	0.956826i	$0.593875\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.04134e7	1.46121
$552$	0	0
$553$	−1.16607e6	−0.162149
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.01198e6	−1.09421	−0.547107	−	0.837063i	$-0.684271\pi$
$557$	−8.01198e6	−1.09421	−0.547107	+	0.837063i	$0.684271\pi$
$558$	0	0
$559$	4.87348e6	0.659644
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.63920e6	1.01573	0.507863	−	0.861438i	$-0.330436\pi$
$563$	7.63920e6	1.01573	0.507863	+	0.861438i	$0.330436\pi$
$564$	0	0
$565$	−588751.	−0.0775909
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.40994e6	−0.959475	−0.479738	−	0.877412i	$-0.659268\pi$
$569$	−7.40994e6	−0.959475	−0.479738	+	0.877412i	$0.659268\pi$
$570$	0	0
$571$	−8.02564e6	−1.03012	−0.515062	−	0.857153i	$-0.672231\pi$
$571$	−8.02564e6	−1.03012	−0.515062	+	0.857153i	$0.672231\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.67541e6	−0.463592
$576$	0	0
$577$	−2.61646e6	−0.327171	−0.163585	−	0.986529i	$-0.552306\pi$
$577$	−2.61646e6	−0.327171	−0.163585	+	0.986529i	$0.552306\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.49513e6	0.552462
$582$	0	0
$583$	−7.62714e6	−0.929374
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.25603e6	0.869168	0.434584	−	0.900631i	$-0.356895\pi$
$587$	7.25603e6	0.869168	0.434584	+	0.900631i	$0.356895\pi$
$588$	0	0
$589$	−8.63521e6	−1.02562
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.28647e6	−0.383789	−0.191895	−	0.981416i	$-0.561463\pi$
$593$	−3.28647e6	−0.383789	−0.191895	+	0.981416i	$0.561463\pi$
$594$	0	0
$595$	460193.	0.0532902
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.89563e6	0.785248	0.392624	−	0.919699i	$-0.371567\pi$
$599$	6.89563e6	0.785248	0.392624	+	0.919699i	$0.371567\pi$
$600$	0	0
$601$	1.41795e7	1.60130	0.800652	−	0.599129i	$-0.204486\pi$
$601$	1.41795e7	1.60130	0.800652	+	0.599129i	$0.204486\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	322343.	0.0358039
$606$	0	0
$607$	1.51165e7	1.66526	0.832628	−	0.553833i	$-0.186835\pi$
$607$	1.51165e7	1.66526	0.832628	+	0.553833i	$0.186835\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.28995e7	−1.39787
$612$	0	0
$613$	1.73358e6	0.186334	0.0931670	−	0.995650i	$-0.470301\pi$
$613$	1.73358e6	0.186334	0.0931670	+	0.995650i	$0.470301\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.84592e6	−0.195209	−0.0976044	−	0.995225i	$-0.531118\pi$
$617$	−1.84592e6	−0.195209	−0.0976044	+	0.995225i	$0.531118\pi$
$618$	0	0
$619$	−7.09387e6	−0.744143	−0.372072	−	0.928204i	$-0.621352\pi$
$619$	−7.09387e6	−0.744143	−0.372072	+	0.928204i	$0.621352\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	869984.	0.0898031
$624$	0	0
$625$	9.58635e6	0.981642
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.80725e7	−2.82914
$630$	0	0
$631$	−4.56566e6	−0.456489	−0.228245	−	0.973604i	$-0.573299\pi$
$631$	−4.56566e6	−0.456489	−0.228245	+	0.973604i	$0.573299\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	51602.2	0.00507848
$636$	0	0
$637$	−1.32864e6	−0.129735
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.71992e6	−0.645980	−0.322990	−	0.946402i	$-0.604688\pi$
$641$	−6.71992e6	−0.645980	−0.322990	+	0.946402i	$0.604688\pi$
$642$	0	0
$643$	−6.26958e6	−0.598013	−0.299007	−	0.954251i	$-0.596655\pi$
$643$	−6.26958e6	−0.598013	−0.299007	+	0.954251i	$0.596655\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.53026e7	−1.43716	−0.718580	−	0.695444i	$-0.755208\pi$
$647$	−1.53026e7	−1.43716	−0.718580	+	0.695444i	$0.755208\pi$
$648$	0	0
$649$	−5.89454e6	−0.549336
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.24070e7	1.13863	0.569315	−	0.822119i	$-0.307208\pi$
$653$	1.24070e7	1.13863	0.569315	+	0.822119i	$0.307208\pi$
$654$	0	0
$655$	−417508.	−0.0380243
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.98958e6	0.537258	0.268629	−	0.963244i	$-0.413430\pi$
$659$	5.98958e6	0.537258	0.268629	+	0.963244i	$0.413430\pi$
$660$	0	0
$661$	−1.05199e7	−0.936501	−0.468250	−	0.883596i	$-0.655115\pi$
$661$	−1.05199e7	−0.936501	−0.468250	+	0.883596i	$0.655115\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	530094.	0.0464835
$666$	0	0
$667$	−4.98633e6	−0.433977
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.99403e6	0.342456
$672$	0	0
$673$	−1.54559e7	−1.31539	−0.657696	−	0.753283i	$-0.728469\pi$
$673$	−1.54559e7	−1.31539	−0.657696	+	0.753283i	$0.728469\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.77865e6	0.652278	0.326139	−	0.945322i	$-0.394252\pi$
$677$	7.77865e6	0.652278	0.326139	+	0.945322i	$0.394252\pi$
$678$	0	0
$679$	−5.88064e6	−0.489497
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.97000e6	0.571717	0.285858	−	0.958272i	$-0.407721\pi$
$683$	6.97000e6	0.571717	0.285858	+	0.958272i	$0.407721\pi$
$684$	0	0
$685$	848408.	0.0690841
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.71223e6	−0.699168
$690$	0	0
$691$	−7.26502e6	−0.578818	−0.289409	−	0.957206i	$-0.593459\pi$
$691$	−7.26502e6	−0.578818	−0.289409	+	0.957206i	$0.593459\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.80095e6	0.141429
$696$	0	0
$697$	1.93045e7	1.50514
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.62528e7	1.24921	0.624603	−	0.780942i	$-0.285261\pi$
$701$	1.62528e7	1.24921	0.624603	+	0.780942i	$0.285261\pi$
$702$	0	0
$703$	−3.23366e7	−2.46778
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.94053e6	−0.672690
$708$	0	0
$709$	−2.83296e6	−0.211654	−0.105827	−	0.994385i	$-0.533749\pi$
$709$	−2.83296e6	−0.211654	−0.105827	+	0.994385i	$0.533749\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.13488e6	0.304606
$714$	0	0
$715$	1.17349e6	0.0858448
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.69761e7	−1.22466	−0.612330	−	0.790603i	$-0.709767\pi$
$719$	−1.69761e7	−1.22466	−0.612330	+	0.790603i	$0.709767\pi$
$720$	0	0
$721$	4.17095e6	0.298811
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.30868e7	0.924673
$726$	0	0
$727$	−3.94087e6	−0.276539	−0.138269	−	0.990395i	$-0.544154\pi$
$727$	−3.94087e6	−0.276539	−0.138269	+	0.990395i	$0.544154\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.88951e7	1.30784
$732$	0	0
$733$	−2.24246e7	−1.54158	−0.770788	−	0.637092i	$-0.780137\pi$
$733$	−2.24246e7	−1.54158	−0.770788	+	0.637092i	$0.780137\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.24046e7	1.51939
$738$	0	0
$739$	−3.83318e6	−0.258195	−0.129097	−	0.991632i	$-0.541208\pi$
$739$	−3.83318e6	−0.258195	−0.129097	+	0.991632i	$0.541208\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.19211e7	−1.45676	−0.728382	−	0.685171i	$-0.759728\pi$
$743$	−2.19211e7	−1.45676	−0.728382	+	0.685171i	$0.759728\pi$
$744$	0	0
$745$	−1.99168e6	−0.131470
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.88010e6	0.252720
$750$	0	0
$751$	−1.01492e7	−0.656649	−0.328324	−	0.944565i	$-0.606484\pi$
$751$	−1.01492e7	−0.656649	−0.328324	+	0.944565i	$0.606484\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	163579.	0.0104438
$756$	0	0
$757$	−1.00602e7	−0.638065	−0.319033	−	0.947744i	$-0.603358\pi$
$757$	−1.00602e7	−0.638065	−0.319033	+	0.947744i	$0.603358\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.10530e7	−0.691858	−0.345929	−	0.938261i	$-0.612436\pi$
$761$	−1.10530e7	−0.691858	−0.345929	+	0.938261i	$0.612436\pi$
$762$	0	0
$763$	2.47024e6	0.153613
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.73314e6	−0.413265
$768$	0	0
$769$	−2.14168e7	−1.30598	−0.652992	−	0.757365i	$-0.726486\pi$
$769$	−2.14168e7	−1.30598	−0.652992	+	0.757365i	$0.726486\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.72787e6	−0.525363	−0.262682	−	0.964883i	$-0.584607\pi$
$773$	−8.72787e6	−0.525363	−0.262682	+	0.964883i	$0.584607\pi$
$774$	0	0
$775$	−1.08521e7	−0.649023
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.22368e7	1.31289
$780$	0	0
$781$	2.74207e6	0.160861
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−84368.9	−0.00488662
$786$	0	0
$787$	5.67464e6	0.326589	0.163295	−	0.986577i	$-0.447788\pi$
$787$	5.67464e6	0.326589	0.163295	+	0.986577i	$0.447788\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.59035e6	0.374513
$792$	0	0
$793$	4.56225e6	0.257630
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.29046e7	−0.719614	−0.359807	−	0.933027i	$-0.617158\pi$
$797$	−1.29046e7	−0.719614	−0.359807	+	0.933027i	$0.617158\pi$
$798$	0	0
$799$	−5.00129e7	−2.77150
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.82809e7	−1.54776
$804$	0	0
$805$	−253830.	−0.0138055
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.13372e6	0.168341	0.0841703	−	0.996451i	$-0.473176\pi$
$809$	3.13372e6	0.168341	0.0841703	+	0.996451i	$0.473176\pi$
$810$	0	0
$811$	6.34606e6	0.338807	0.169403	−	0.985547i	$-0.445816\pi$
$811$	6.34606e6	0.338807	0.169403	+	0.985547i	$0.445816\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−553688.	−0.0291992
$816$	0	0
$817$	2.17652e7	1.14079
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.10483e7	0.572057	0.286029	−	0.958221i	$-0.407665\pi$
$821$	1.10483e7	0.572057	0.286029	+	0.958221i	$0.407665\pi$
$822$	0	0
$823$	2.54462e7	1.30955	0.654776	−	0.755823i	$-0.272763\pi$
$823$	2.54462e7	1.30955	0.654776	+	0.755823i	$0.272763\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.97845e7	−1.00591	−0.502957	−	0.864311i	$-0.667755\pi$
$827$	−1.97845e7	−1.00591	−0.502957	+	0.864311i	$0.667755\pi$
$828$	0	0
$829$	−1.63861e7	−0.828113	−0.414057	−	0.910251i	$-0.635888\pi$
$829$	−1.63861e7	−0.828113	−0.414057	+	0.910251i	$0.635888\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.15129e6	−0.257219
$834$	0	0
$835$	−2.20153e6	−0.109272
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.32045e7	1.13807	0.569033	−	0.822314i	$-0.307317\pi$
$839$	2.32045e7	1.13807	0.569033	+	0.822314i	$0.307317\pi$
$840$	0	0
$841$	−2.75663e6	−0.134397
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−284871.	−0.0137248
$846$	0	0
$847$	−3.60824e6	−0.172817
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.54840e7	0.732927
$852$	0	0
$853$	−9.73136e6	−0.457932	−0.228966	−	0.973434i	$-0.573534\pi$
$853$	−9.73136e6	−0.457932	−0.228966	+	0.973434i	$0.573534\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.95773e6	0.370115	0.185058	−	0.982728i	$-0.440753\pi$
$857$	7.95773e6	0.370115	0.185058	+	0.982728i	$0.440753\pi$
$858$	0	0
$859$	1.16773e7	0.539959	0.269979	−	0.962866i	$-0.412983\pi$
$859$	1.16773e7	0.539959	0.269979	+	0.962866i	$0.412983\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.55138e7	0.709076	0.354538	−	0.935042i	$-0.384638\pi$
$863$	1.55138e7	0.709076	0.354538	+	0.935042i	$0.384638\pi$
$864$	0	0
$865$	−1.05429e6	−0.0479092
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.15286e7	−0.517877
$870$	0	0
$871$	2.55920e7	1.14303
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.33648e6	0.0590122
$876$	0	0
$877$	−3.22314e7	−1.41508	−0.707539	−	0.706674i	$-0.750195\pi$
$877$	−3.22314e7	−1.41508	−0.707539	+	0.706674i	$0.750195\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.73867e7	1.62284	0.811422	−	0.584461i	$-0.198694\pi$
$881$	3.73867e7	1.62284	0.811422	+	0.584461i	$0.198694\pi$
$882$	0	0
$883$	1.77381e7	0.765605	0.382803	−	0.923830i	$-0.374959\pi$
$883$	1.77381e7	0.765605	0.382803	+	0.923830i	$0.374959\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.06714e7	−0.455420	−0.227710	−	0.973729i	$-0.573124\pi$
$887$	−1.06714e7	−0.455420	−0.227710	+	0.973729i	$0.573124\pi$
$888$	0	0
$889$	−577624.	−0.0245127
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.76096e7	−2.41750
$894$	0	0
$895$	2.33390e6	0.0973924
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.47228e7	−0.607562
$900$	0	0
$901$	−3.37785e7	−1.38621
$902$	0	0
$903$	0	0
$904$	0	0
$905$	655600.	0.0266083
$906$	0	0
$907$	−4.14860e7	−1.67449	−0.837247	−	0.546825i	$-0.815836\pi$
$907$	−4.14860e7	−1.67449	−0.837247	+	0.546825i	$0.815836\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.09389e7	1.23512	0.617560	−	0.786524i	$-0.288121\pi$
$911$	3.09389e7	1.23512	0.617560	+	0.786524i	$0.288121\pi$
$912$	0	0
$913$	4.44419e7	1.76448
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.67349e6	0.183535
$918$	0	0
$919$	189002.	0.00738207	0.00369104	−	0.999993i	$-0.498825\pi$
$919$	189002.	0.00738207	0.00369104	+	0.999993i	$0.498825\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.13218e6	0.121016
$924$	0	0
$925$	−4.06384e7	−1.56164
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4.47673e7	−1.70185	−0.850925	−	0.525287i	$-0.823958\pi$
$929$	−4.47673e7	−1.70185	−0.850925	+	0.525287i	$0.823958\pi$
$930$	0	0
$931$	−5.93375e6	−0.224365
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.54977e6	0.170200
$936$	0	0
$937$	4.66700e7	1.73656	0.868279	−	0.496076i	$-0.165226\pi$
$937$	4.66700e7	1.73656	0.868279	+	0.496076i	$0.165226\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.52351e6	0.0929033	0.0464517	−	0.998921i	$-0.485209\pi$
$941$	2.52351e6	0.0929033	0.0464517	+	0.998921i	$0.485209\pi$
$942$	0	0
$943$	−1.06478e7	−0.389926
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.19512e7	0.433048	0.216524	−	0.976277i	$-0.430528\pi$
$947$	1.19512e7	0.433048	0.216524	+	0.976277i	$0.430528\pi$
$948$	0	0
$949$	−3.23043e7	−1.16438
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.62572e7	−0.936519	−0.468260	−	0.883591i	$-0.655119\pi$
$953$	−2.62572e7	−0.936519	−0.468260	+	0.883591i	$0.655119\pi$
$954$	0	0
$955$	2.80841e6	0.0996442
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.49689e6	−0.333453
$960$	0	0
$961$	−1.64204e7	−0.573555
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−161865.	−0.00559545
$966$	0	0
$967$	2.87742e7	0.989550	0.494775	−	0.869021i	$-0.335251\pi$
$967$	2.87742e7	0.989550	0.494775	+	0.869021i	$0.335251\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.96258e7	0.668003	0.334002	−	0.942573i	$-0.391601\pi$
$971$	1.96258e7	0.668003	0.334002	+	0.942573i	$0.391601\pi$
$972$	0	0
$973$	−2.01594e7	−0.682647
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.29251e7	1.77389	0.886943	−	0.461879i	$-0.152825\pi$
$977$	5.29251e7	1.77389	0.886943	+	0.461879i	$0.152825\pi$
$978$	0	0
$979$	8.60124e6	0.286817
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.27685e7	1.08161	0.540807	−	0.841147i	$-0.318119\pi$
$983$	3.27685e7	1.08161	0.540807	+	0.841147i	$0.318119\pi$
$984$	0	0
$985$	−2.67921e6	−0.0879865
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.04220e7	−0.338814
$990$	0	0
$991$	1.27978e7	0.413952	0.206976	−	0.978346i	$-0.433638\pi$
$991$	1.27978e7	0.413952	0.206976	+	0.978346i	$0.433638\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	340793.	0.0109127
$996$	0	0
$997$	−3.90508e7	−1.24421	−0.622103	−	0.782936i	$-0.713721\pi$
$997$	−3.90508e7	−1.24421	−0.622103	+	0.782936i	$0.713721\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.6.a.by.1.3		4
3.2	odd	2	inner	1008.6.a.by.1.2		4
4.3	odd	2		63.6.a.h.1.4	yes	4
12.11	even	2		63.6.a.h.1.1	✓	4
28.27	even	2		441.6.a.x.1.4		4
84.83	odd	2		441.6.a.x.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.a.h.1.1	✓	4	12.11	even	2
63.6.a.h.1.4	yes	4	4.3	odd	2
441.6.a.x.1.1		4	84.83	odd	2
441.6.a.x.1.4		4	28.27	even	2
1008.6.a.by.1.2		4	3.2	odd	2	inner
1008.6.a.by.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

\(f(q)\)	\(=\)	\(q+4.37743 q^{5} -49.0000 q^{7} +O(q^{10})\)	\(q+4.37743 q^{5} -49.0000 q^{7} -484.447 q^{11} -553.368 q^{13} -2145.48 q^{17} -2471.37 q^{19} +1183.39 q^{23} -3105.84 q^{25} -4213.61 q^{29} +3494.10 q^{31} -214.494 q^{35} +13084.5 q^{37} -8997.77 q^{41} -8806.94 q^{43} +23310.8 q^{47} +2401.00 q^{49} +15744.0 q^{53} -2120.63 q^{55} +12167.6 q^{59} -8244.51 q^{61} -2422.33 q^{65} -46247.8 q^{67} -5660.21 q^{71} +58377.7 q^{73} +23737.9 q^{77} +23797.4 q^{79} -91737.4 q^{83} -9391.69 q^{85} -17754.8 q^{89} +27115.0 q^{91} -10818.2 q^{95} +120013. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 196 q^{7}+O(q^{10}) \) 4 * q - 196 * q^7	\( 4 q - 196 q^{7} + 872 q^{13} - 6800 q^{19} + 3004 q^{25} + 4720 q^{31} + 6056 q^{37} - 10544 q^{43} + 9604 q^{49} - 11568 q^{55} + 13304 q^{61} - 203504 q^{67} + 17528 q^{73} - 34400 q^{79} + 354288 q^{85} - 42728 q^{91} + 436856 q^{97}+O(q^{100}) \) 4 * q - 196 * q^7 + 872 * q^13 - 6800 * q^19 + 3004 * q^25 + 4720 * q^31 + 6056 * q^37 - 10544 * q^43 + 9604 * q^49 - 11568 * q^55 + 13304 * q^61 - 203504 * q^67 + 17528 * q^73 - 34400 * q^79 + 354288 * q^85 - 42728 * q^91 + 436856 * q^97

Introduction

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