LMFDB - Embedded newform 336.6.a.a.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 21)
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)

$f(q)$

$=$

$q-9.00000 q^{3} -106.000 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q-9.00000 q^{3} -106.000 q^{5} +49.0000 q^{7} +81.0000 q^{9} -92.0000 q^{11} +670.000 q^{13} +954.000 q^{15} -222.000 q^{17} +908.000 q^{19} -441.000 q^{21} +1176.00 q^{23} +8111.00 q^{25} -729.000 q^{27} +1118.00 q^{29} -3696.00 q^{31} +828.000 q^{33} -5194.00 q^{35} +4182.00 q^{37} -6030.00 q^{39} -6662.00 q^{41} +3700.00 q^{43} -8586.00 q^{45} +7056.00 q^{47} +2401.00 q^{49} +1998.00 q^{51} -37578.0 q^{53} +9752.00 q^{55} -8172.00 q^{57} -32700.0 q^{59} -10802.0 q^{61} +3969.00 q^{63} -71020.0 q^{65} -64996.0 q^{67} -10584.0 q^{69} +61320.0 q^{71} +38922.0 q^{73} -72999.0 q^{75} -4508.00 q^{77} +88096.0 q^{79} +6561.00 q^{81} -71892.0 q^{83} +23532.0 q^{85} -10062.0 q^{87} +111818. q^{89} +32830.0 q^{91} +33264.0 q^{93} -96248.0 q^{95} -150846. q^{97} -7452.00 q^{99} +O(q^{100})$

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
$5$	−106.000	−1.89619	−0.948093	−	0.317994i	$-0.896991\pi$
$5$	−106.000	−1.89619	−0.948093	+	0.317994i	$0.896991\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−92.0000	−0.229248	−0.114624	−	0.993409i	$-0.536566\pi$
$11$	−92.0000	−0.229248	−0.114624	+	0.993409i	$0.536566\pi$
$12$	0	0
$13$	670.000	1.09955	0.549777	−	0.835312i	$-0.314713\pi$
$13$	670.000	1.09955	0.549777	+	0.835312i	$0.314713\pi$
$14$	0	0
$15$	954.000	1.09476
$16$	0	0
$17$	−222.000	−0.186308	−0.0931538	−	0.995652i	$-0.529695\pi$
$17$	−222.000	−0.186308	−0.0931538	+	0.995652i	$0.529695\pi$
$18$	0	0
$19$	908.000	0.577035	0.288517	−	0.957475i	$-0.406838\pi$
$19$	908.000	0.577035	0.288517	+	0.957475i	$0.406838\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	0	0
$23$	1176.00	0.463541	0.231770	−	0.972771i	$-0.425548\pi$
$23$	1176.00	0.463541	0.231770	+	0.972771i	$0.425548\pi$
$24$	0	0
$25$	8111.00	2.59552
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	1118.00	0.246858	0.123429	−	0.992353i	$-0.460611\pi$
$29$	1118.00	0.246858	0.123429	+	0.992353i	$0.460611\pi$
$30$	0	0
$31$	−3696.00	−0.690761	−0.345380	−	0.938463i	$-0.612250\pi$
$31$	−3696.00	−0.690761	−0.345380	+	0.938463i	$0.612250\pi$
$32$	0	0
$33$	828.000	0.132357
$34$	0	0
$35$	−5194.00	−0.716691
$36$	0	0
$37$	4182.00	0.502203	0.251102	−	0.967961i	$-0.419207\pi$
$37$	4182.00	0.502203	0.251102	+	0.967961i	$0.419207\pi$
$38$	0	0
$39$	−6030.00	−0.634828
$40$	0	0
$41$	−6662.00	−0.618935	−0.309467	−	0.950910i	$-0.600151\pi$
$41$	−6662.00	−0.618935	−0.309467	+	0.950910i	$0.600151\pi$
$42$	0	0
$43$	3700.00	0.305162	0.152581	−	0.988291i	$-0.451241\pi$
$43$	3700.00	0.305162	0.152581	+	0.988291i	$0.451241\pi$
$44$	0	0
$45$	−8586.00	−0.632062
$46$	0	0
$47$	7056.00	0.465923	0.232961	−	0.972486i	$-0.425158\pi$
$47$	7056.00	0.465923	0.232961	+	0.972486i	$0.425158\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	1998.00	0.107565
$52$	0	0
$53$	−37578.0	−1.83757	−0.918785	−	0.394758i	$-0.870828\pi$
$53$	−37578.0	−1.83757	−0.918785	+	0.394758i	$0.870828\pi$
$54$	0	0
$55$	9752.00	0.434697
$56$	0	0
$57$	−8172.00	−0.333151
$58$	0	0
$59$	−32700.0	−1.22298	−0.611488	−	0.791254i	$-0.709429\pi$
$59$	−32700.0	−1.22298	−0.611488	+	0.791254i	$0.709429\pi$
$60$	0	0
$61$	−10802.0	−0.371689	−0.185844	−	0.982579i	$-0.559502\pi$
$61$	−10802.0	−0.371689	−0.185844	+	0.982579i	$0.559502\pi$
$62$	0	0
$63$	3969.00	0.125988
$64$	0	0
$65$	−71020.0	−2.08496
$66$	0	0
$67$	−64996.0	−1.76889	−0.884443	−	0.466649i	$-0.845461\pi$
$67$	−64996.0	−1.76889	−0.884443	+	0.466649i	$0.845461\pi$
$68$	0	0
$69$	−10584.0	−0.267625
$70$	0	0
$71$	61320.0	1.44363	0.721816	−	0.692085i	$-0.243308\pi$
$71$	61320.0	1.44363	0.721816	+	0.692085i	$0.243308\pi$
$72$	0	0
$73$	38922.0	0.854846	0.427423	−	0.904052i	$-0.359421\pi$
$73$	38922.0	0.854846	0.427423	+	0.904052i	$0.359421\pi$
$74$	0	0
$75$	−72999.0	−1.49852
$76$	0	0
$77$	−4508.00	−0.0866477
$78$	0	0
$79$	88096.0	1.58814	0.794069	−	0.607827i	$-0.207959\pi$
$79$	88096.0	1.58814	0.794069	+	0.607827i	$0.207959\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−71892.0	−1.14547	−0.572737	−	0.819739i	$-0.694118\pi$
$83$	−71892.0	−1.14547	−0.572737	+	0.819739i	$0.694118\pi$
$84$	0	0
$85$	23532.0	0.353274
$86$	0	0
$87$	−10062.0	−0.142523
$88$	0	0
$89$	111818.	1.49636	0.748181	−	0.663495i	$-0.230927\pi$
$89$	111818.	1.49636	0.748181	+	0.663495i	$0.230927\pi$
$90$	0	0
$91$	32830.0	0.415592
$92$	0	0
$93$	33264.0	0.398811
$94$	0	0
$95$	−96248.0	−1.09416
$96$	0	0
$97$	−150846.	−1.62781	−0.813906	−	0.580996i	$-0.802663\pi$
$97$	−150846.	−1.62781	−0.813906	+	0.580996i	$0.802663\pi$
$98$	0	0
$99$	−7452.00	−0.0764161
$100$	0	0
$101$	−137354.	−1.33979	−0.669897	−	0.742454i	$-0.733662\pi$
$101$	−137354.	−1.33979	−0.669897	+	0.742454i	$0.733662\pi$
$102$	0	0
$103$	−28760.0	−0.267113	−0.133557	−	0.991041i	$-0.542640\pi$
$103$	−28760.0	−0.267113	−0.133557	+	0.991041i	$0.542640\pi$
$104$	0	0
$105$	46746.0	0.413782
$106$	0	0
$107$	−22556.0	−0.190460	−0.0952298	−	0.995455i	$-0.530359\pi$
$107$	−22556.0	−0.190460	−0.0952298	+	0.995455i	$0.530359\pi$
$108$	0	0
$109$	19998.0	0.161221	0.0806103	−	0.996746i	$-0.474313\pi$
$109$	19998.0	0.161221	0.0806103	+	0.996746i	$0.474313\pi$
$110$	0	0
$111$	−37638.0	−0.289947
$112$	0	0
$113$	17906.0	0.131918	0.0659588	−	0.997822i	$-0.478989\pi$
$113$	17906.0	0.131918	0.0659588	+	0.997822i	$0.478989\pi$
$114$	0	0
$115$	−124656.	−0.878959
$116$	0	0
$117$	54270.0	0.366518
$118$	0	0
$119$	−10878.0	−0.0704177
$120$	0	0
$121$	−152587.	−0.947445
$122$	0	0
$123$	59958.0	0.357342
$124$	0	0
$125$	−528516.	−3.02540
$126$	0	0
$127$	−66864.0	−0.367860	−0.183930	−	0.982939i	$-0.558882\pi$
$127$	−66864.0	−0.367860	−0.183930	+	0.982939i	$0.558882\pi$
$128$	0	0
$129$	−33300.0	−0.176185
$130$	0	0
$131$	−153764.	−0.782846	−0.391423	−	0.920211i	$-0.628017\pi$
$131$	−153764.	−0.782846	−0.391423	+	0.920211i	$0.628017\pi$
$132$	0	0
$133$	44492.0	0.218099
$134$	0	0
$135$	77274.0	0.364921
$136$	0	0
$137$	255978.	1.16520	0.582601	−	0.812758i	$-0.302035\pi$
$137$	255978.	1.16520	0.582601	+	0.812758i	$0.302035\pi$
$138$	0	0
$139$	−282924.	−1.24203	−0.621016	−	0.783798i	$-0.713280\pi$
$139$	−282924.	−1.24203	−0.621016	+	0.783798i	$0.713280\pi$
$140$	0	0
$141$	−63504.0	−0.269001
$142$	0	0
$143$	−61640.0	−0.252071
$144$	0	0
$145$	−118508.	−0.468088
$146$	0	0
$147$	−21609.0	−0.0824786
$148$	0	0
$149$	408054.	1.50575	0.752873	−	0.658165i	$-0.228667\pi$
$149$	408054.	1.50575	0.752873	+	0.658165i	$0.228667\pi$
$150$	0	0
$151$	−362504.	−1.29381	−0.646905	−	0.762571i	$-0.723937\pi$
$151$	−362504.	−1.29381	−0.646905	+	0.762571i	$0.723937\pi$
$152$	0	0
$153$	−17982.0	−0.0621025
$154$	0	0
$155$	391776.	1.30981
$156$	0	0
$157$	−152786.	−0.494691	−0.247346	−	0.968927i	$-0.579558\pi$
$157$	−152786.	−0.494691	−0.247346	+	0.968927i	$0.579558\pi$
$158$	0	0
$159$	338202.	1.06092
$160$	0	0
$161$	57624.0	0.175202
$162$	0	0
$163$	150428.	0.443465	0.221733	−	0.975107i	$-0.428829\pi$
$163$	150428.	0.443465	0.221733	+	0.975107i	$0.428829\pi$
$164$	0	0
$165$	−87768.0	−0.250973
$166$	0	0
$167$	7288.00	0.0202217	0.0101108	−	0.999949i	$-0.496782\pi$
$167$	7288.00	0.0202217	0.0101108	+	0.999949i	$0.496782\pi$
$168$	0	0
$169$	77607.0	0.209018
$170$	0	0
$171$	73548.0	0.192345
$172$	0	0
$173$	−289154.	−0.734537	−0.367269	−	0.930115i	$-0.619707\pi$
$173$	−289154.	−0.734537	−0.367269	+	0.930115i	$0.619707\pi$
$174$	0	0
$175$	397439.	0.981014
$176$	0	0
$177$	294300.	0.706085
$178$	0	0
$179$	−199492.	−0.465364	−0.232682	−	0.972553i	$-0.574750\pi$
$179$	−199492.	−0.465364	−0.232682	+	0.972553i	$0.574750\pi$
$180$	0	0
$181$	240550.	0.545769	0.272885	−	0.962047i	$-0.412022\pi$
$181$	240550.	0.545769	0.272885	+	0.962047i	$0.412022\pi$
$182$	0	0
$183$	97218.0	0.214595
$184$	0	0
$185$	−443292.	−0.952271
$186$	0	0
$187$	20424.0	0.0427107
$188$	0	0
$189$	−35721.0	−0.0727393
$190$	0	0
$191$	−290384.	−0.575956	−0.287978	−	0.957637i	$-0.592983\pi$
$191$	−290384.	−0.575956	−0.287978	+	0.957637i	$0.592983\pi$
$192$	0	0
$193$	−171454.	−0.331325	−0.165663	−	0.986182i	$-0.552976\pi$
$193$	−171454.	−0.331325	−0.165663	+	0.986182i	$0.552976\pi$
$194$	0	0
$195$	639180.	1.20375
$196$	0	0
$197$	401990.	0.737989	0.368994	−	0.929432i	$-0.379702\pi$
$197$	401990.	0.737989	0.368994	+	0.929432i	$0.379702\pi$
$198$	0	0
$199$	259176.	0.463940	0.231970	−	0.972723i	$-0.425483\pi$
$199$	259176.	0.463940	0.231970	+	0.972723i	$0.425483\pi$
$200$	0	0
$201$	584964.	1.02127
$202$	0	0
$203$	54782.0	0.0933035
$204$	0	0
$205$	706172.	1.17362
$206$	0	0
$207$	95256.0	0.154514
$208$	0	0
$209$	−83536.0	−0.132284
$210$	0	0
$211$	1.19179e6	1.84286	0.921431	−	0.388542i	$-0.127021\pi$
$211$	1.19179e6	1.84286	0.921431	+	0.388542i	$0.127021\pi$
$212$	0	0
$213$	−551880.	−0.833481
$214$	0	0
$215$	−392200.	−0.578644
$216$	0	0
$217$	−181104.	−0.261083
$218$	0	0
$219$	−350298.	−0.493546
$220$	0	0
$221$	−148740.	−0.204855
$222$	0	0
$223$	218384.	0.294075	0.147038	−	0.989131i	$-0.453026\pi$
$223$	218384.	0.294075	0.147038	+	0.989131i	$0.453026\pi$
$224$	0	0
$225$	656991.	0.865173
$226$	0	0
$227$	−582852.	−0.750747	−0.375374	−	0.926874i	$-0.622486\pi$
$227$	−582852.	−0.750747	−0.375374	+	0.926874i	$0.622486\pi$
$228$	0	0
$229$	961046.	1.21103	0.605516	−	0.795833i	$-0.292967\pi$
$229$	961046.	1.21103	0.605516	+	0.795833i	$0.292967\pi$
$230$	0	0
$231$	40572.0	0.0500261
$232$	0	0
$233$	605994.	0.731271	0.365636	−	0.930758i	$-0.380852\pi$
$233$	605994.	0.731271	0.365636	+	0.930758i	$0.380852\pi$
$234$	0	0
$235$	−747936.	−0.883476
$236$	0	0
$237$	−792864.	−0.916912
$238$	0	0
$239$	−1.17014e6	−1.32509	−0.662544	−	0.749023i	$-0.730523\pi$
$239$	−1.17014e6	−1.32509	−0.662544	+	0.749023i	$0.730523\pi$
$240$	0	0
$241$	−1.23691e6	−1.37181	−0.685907	−	0.727689i	$-0.740595\pi$
$241$	−1.23691e6	−1.37181	−0.685907	+	0.727689i	$0.740595\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−254506.	−0.270884
$246$	0	0
$247$	608360.	0.634480
$248$	0	0
$249$	647028.	0.661340
$250$	0	0
$251$	−959708.	−0.961512	−0.480756	−	0.876854i	$-0.659638\pi$
$251$	−959708.	−0.961512	−0.480756	+	0.876854i	$0.659638\pi$
$252$	0	0
$253$	−108192.	−0.106266
$254$	0	0
$255$	−211788.	−0.203963
$256$	0	0
$257$	−1.21259e6	−1.14520	−0.572600	−	0.819835i	$-0.694065\pi$
$257$	−1.21259e6	−1.14520	−0.572600	+	0.819835i	$0.694065\pi$
$258$	0	0
$259$	204918.	0.189815
$260$	0	0
$261$	90558.0	0.0822859
$262$	0	0
$263$	1.25274e6	1.11679	0.558397	−	0.829574i	$-0.311417\pi$
$263$	1.25274e6	1.11679	0.558397	+	0.829574i	$0.311417\pi$
$264$	0	0
$265$	3.98327e6	3.48437
$266$	0	0
$267$	−1.00636e6	−0.863925
$268$	0	0
$269$	−136866.	−0.115323	−0.0576614	−	0.998336i	$-0.518364\pi$
$269$	−136866.	−0.115323	−0.0576614	+	0.998336i	$0.518364\pi$
$270$	0	0
$271$	960896.	0.794791	0.397396	−	0.917647i	$-0.369914\pi$
$271$	960896.	0.794791	0.397396	+	0.917647i	$0.369914\pi$
$272$	0	0
$273$	−295470.	−0.239942
$274$	0	0
$275$	−746212.	−0.595019
$276$	0	0
$277$	905830.	0.709328	0.354664	−	0.934994i	$-0.384595\pi$
$277$	905830.	0.709328	0.354664	+	0.934994i	$0.384595\pi$
$278$	0	0
$279$	−299376.	−0.230254
$280$	0	0
$281$	−33062.0	−0.0249783	−0.0124892	−	0.999922i	$-0.503976\pi$
$281$	−33062.0	−0.0249783	−0.0124892	+	0.999922i	$0.503976\pi$
$282$	0	0
$283$	863588.	0.640974	0.320487	−	0.947253i	$-0.396153\pi$
$283$	863588.	0.640974	0.320487	+	0.947253i	$0.396153\pi$
$284$	0	0
$285$	866232.	0.631716
$286$	0	0
$287$	−326438.	−0.233935
$288$	0	0
$289$	−1.37057e6	−0.965289
$290$	0	0
$291$	1.35761e6	0.939818
$292$	0	0
$293$	−1.33755e6	−0.910206	−0.455103	−	0.890439i	$-0.650398\pi$
$293$	−1.33755e6	−0.910206	−0.455103	+	0.890439i	$0.650398\pi$
$294$	0	0
$295$	3.46620e6	2.31899
$296$	0	0
$297$	67068.0	0.0441189
$298$	0	0
$299$	787920.	0.509688
$300$	0	0
$301$	181300.	0.115340
$302$	0	0
$303$	1.23619e6	0.773530
$304$	0	0
$305$	1.14501e6	0.704791
$306$	0	0
$307$	1.32820e6	0.804301	0.402151	−	0.915573i	$-0.368263\pi$
$307$	1.32820e6	0.804301	0.402151	+	0.915573i	$0.368263\pi$
$308$	0	0
$309$	258840.	0.154218
$310$	0	0
$311$	665832.	0.390359	0.195179	−	0.980768i	$-0.437471\pi$
$311$	665832.	0.390359	0.195179	+	0.980768i	$0.437471\pi$
$312$	0	0
$313$	−3.09021e6	−1.78290	−0.891451	−	0.453116i	$-0.850312\pi$
$313$	−3.09021e6	−1.78290	−0.891451	+	0.453116i	$0.850312\pi$
$314$	0	0
$315$	−420714.	−0.238897
$316$	0	0
$317$	−974178.	−0.544490	−0.272245	−	0.962228i	$-0.587766\pi$
$317$	−974178.	−0.544490	−0.272245	+	0.962228i	$0.587766\pi$
$318$	0	0
$319$	−102856.	−0.0565917
$320$	0	0
$321$	203004.	0.109962
$322$	0	0
$323$	−201576.	−0.107506
$324$	0	0
$325$	5.43437e6	2.85391
$326$	0	0
$327$	−179982.	−0.0930807
$328$	0	0
$329$	345744.	0.176102
$330$	0	0
$331$	−781772.	−0.392202	−0.196101	−	0.980584i	$-0.562828\pi$
$331$	−781772.	−0.392202	−0.196101	+	0.980584i	$0.562828\pi$
$332$	0	0
$333$	338742.	0.167401
$334$	0	0
$335$	6.88958e6	3.35413
$336$	0	0
$337$	348754.	0.167280	0.0836401	−	0.996496i	$-0.473345\pi$
$337$	348754.	0.167280	0.0836401	+	0.996496i	$0.473345\pi$
$338$	0	0
$339$	−161154.	−0.0761626
$340$	0	0
$341$	340032.	0.158356
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	1.12190e6	0.507467
$346$	0	0
$347$	−2.50625e6	−1.11738	−0.558690	−	0.829376i	$-0.688696\pi$
$347$	−2.50625e6	−1.11738	−0.558690	+	0.829376i	$0.688696\pi$
$348$	0	0
$349$	3.05861e6	1.34419	0.672094	−	0.740466i	$-0.265395\pi$
$349$	3.05861e6	1.34419	0.672094	+	0.740466i	$0.265395\pi$
$350$	0	0
$351$	−488430.	−0.211609
$352$	0	0
$353$	−3.49291e6	−1.49194	−0.745969	−	0.665981i	$-0.768013\pi$
$353$	−3.49291e6	−1.49194	−0.745969	+	0.665981i	$0.768013\pi$
$354$	0	0
$355$	−6.49992e6	−2.73739
$356$	0	0
$357$	97902.0	0.0406557
$358$	0	0
$359$	−2.12034e6	−0.868301	−0.434150	−	0.900840i	$-0.642951\pi$
$359$	−2.12034e6	−0.868301	−0.434150	+	0.900840i	$0.642951\pi$
$360$	0	0
$361$	−1.65163e6	−0.667031
$362$	0	0
$363$	1.37328e6	0.547008
$364$	0	0
$365$	−4.12573e6	−1.62095
$366$	0	0
$367$	−746592.	−0.289346	−0.144673	−	0.989479i	$-0.546213\pi$
$367$	−746592.	−0.289346	−0.144673	+	0.989479i	$0.546213\pi$
$368$	0	0
$369$	−539622.	−0.206312
$370$	0	0
$371$	−1.84132e6	−0.694536
$372$	0	0
$373$	−939034.	−0.349469	−0.174735	−	0.984616i	$-0.555907\pi$
$373$	−939034.	−0.349469	−0.174735	+	0.984616i	$0.555907\pi$
$374$	0	0
$375$	4.75664e6	1.74672
$376$	0	0
$377$	749060.	0.271433
$378$	0	0
$379$	−5.16534e6	−1.84714	−0.923572	−	0.383424i	$-0.874745\pi$
$379$	−5.16534e6	−1.84714	−0.923572	+	0.383424i	$0.874745\pi$
$380$	0	0
$381$	601776.	0.212384
$382$	0	0
$383$	−400512.	−0.139514	−0.0697571	−	0.997564i	$-0.522222\pi$
$383$	−400512.	−0.139514	−0.0697571	+	0.997564i	$0.522222\pi$
$384$	0	0
$385$	477848.	0.164300
$386$	0	0
$387$	299700.	0.101721
$388$	0	0
$389$	306822.	0.102805	0.0514023	−	0.998678i	$-0.483631\pi$
$389$	306822.	0.102805	0.0514023	+	0.998678i	$0.483631\pi$
$390$	0	0
$391$	−261072.	−0.0863611
$392$	0	0
$393$	1.38388e6	0.451976
$394$	0	0
$395$	−9.33818e6	−3.01141
$396$	0	0
$397$	−3.83421e6	−1.22095	−0.610477	−	0.792034i	$-0.709022\pi$
$397$	−3.83421e6	−1.22095	−0.610477	+	0.792034i	$0.709022\pi$
$398$	0	0
$399$	−400428.	−0.125919
$400$	0	0
$401$	−3.29355e6	−1.02283	−0.511415	−	0.859334i	$-0.670878\pi$
$401$	−3.29355e6	−1.02283	−0.511415	+	0.859334i	$0.670878\pi$
$402$	0	0
$403$	−2.47632e6	−0.759529
$404$	0	0
$405$	−695466.	−0.210687
$406$	0	0
$407$	−384744.	−0.115129
$408$	0	0
$409$	−1.35473e6	−0.400445	−0.200223	−	0.979750i	$-0.564167\pi$
$409$	−1.35473e6	−0.400445	−0.200223	+	0.979750i	$0.564167\pi$
$410$	0	0
$411$	−2.30380e6	−0.672730
$412$	0	0
$413$	−1.60230e6	−0.462241
$414$	0	0
$415$	7.62055e6	2.17203
$416$	0	0
$417$	2.54632e6	0.717088
$418$	0	0
$419$	−5.08199e6	−1.41416	−0.707080	−	0.707134i	$-0.749988\pi$
$419$	−5.08199e6	−1.41416	−0.707080	+	0.707134i	$0.749988\pi$
$420$	0	0
$421$	628022.	0.172691	0.0863455	−	0.996265i	$-0.472481\pi$
$421$	628022.	0.172691	0.0863455	+	0.996265i	$0.472481\pi$
$422$	0	0
$423$	571536.	0.155308
$424$	0	0
$425$	−1.80064e6	−0.483565
$426$	0	0
$427$	−529298.	−0.140485
$428$	0	0
$429$	554760.	0.145533
$430$	0	0
$431$	3.00086e6	0.778132	0.389066	−	0.921210i	$-0.372798\pi$
$431$	3.00086e6	0.778132	0.389066	+	0.921210i	$0.372798\pi$
$432$	0	0
$433$	1.21496e6	0.311417	0.155709	−	0.987803i	$-0.450234\pi$
$433$	1.21496e6	0.311417	0.155709	+	0.987803i	$0.450234\pi$
$434$	0	0
$435$	1.06657e6	0.270251
$436$	0	0
$437$	1.06781e6	0.267479
$438$	0	0
$439$	−4.00654e6	−0.992219	−0.496110	−	0.868260i	$-0.665239\pi$
$439$	−4.00654e6	−0.992219	−0.496110	+	0.868260i	$0.665239\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	5.44751e6	1.31883	0.659415	−	0.751779i	$-0.270804\pi$
$443$	5.44751e6	1.31883	0.659415	+	0.751779i	$0.270804\pi$
$444$	0	0
$445$	−1.18527e7	−2.83738
$446$	0	0
$447$	−3.67249e6	−0.869343
$448$	0	0
$449$	−1.81577e6	−0.425056	−0.212528	−	0.977155i	$-0.568170\pi$
$449$	−1.81577e6	−0.425056	−0.212528	+	0.977155i	$0.568170\pi$
$450$	0	0
$451$	612904.	0.141890
$452$	0	0
$453$	3.26254e6	0.746981
$454$	0	0
$455$	−3.47998e6	−0.788040
$456$	0	0
$457$	−5.30082e6	−1.18728	−0.593639	−	0.804731i	$-0.702309\pi$
$457$	−5.30082e6	−1.18728	−0.593639	+	0.804731i	$0.702309\pi$
$458$	0	0
$459$	161838.	0.0358549
$460$	0	0
$461$	3.20381e6	0.702124	0.351062	−	0.936352i	$-0.385821\pi$
$461$	3.20381e6	0.702124	0.351062	+	0.936352i	$0.385821\pi$
$462$	0	0
$463$	1.11853e6	0.242490	0.121245	−	0.992623i	$-0.461311\pi$
$463$	1.11853e6	0.242490	0.121245	+	0.992623i	$0.461311\pi$
$464$	0	0
$465$	−3.52598e6	−0.756220
$466$	0	0
$467$	3.85134e6	0.817184	0.408592	−	0.912717i	$-0.366020\pi$
$467$	3.85134e6	0.817184	0.408592	+	0.912717i	$0.366020\pi$
$468$	0	0
$469$	−3.18480e6	−0.668576
$470$	0	0
$471$	1.37507e6	0.285610
$472$	0	0
$473$	−340400.	−0.0699579
$474$	0	0
$475$	7.36479e6	1.49770
$476$	0	0
$477$	−3.04382e6	−0.612523
$478$	0	0
$479$	−1.43536e6	−0.285839	−0.142920	−	0.989734i	$-0.545649\pi$
$479$	−1.43536e6	−0.285839	−0.142920	+	0.989734i	$0.545649\pi$
$480$	0	0
$481$	2.80194e6	0.552200
$482$	0	0
$483$	−518616.	−0.101153
$484$	0	0
$485$	1.59897e7	3.08664
$486$	0	0
$487$	−4.61097e6	−0.880987	−0.440494	−	0.897756i	$-0.645197\pi$
$487$	−4.61097e6	−0.880987	−0.440494	+	0.897756i	$0.645197\pi$
$488$	0	0
$489$	−1.35385e6	−0.256035
$490$	0	0
$491$	−7.40518e6	−1.38622	−0.693110	−	0.720832i	$-0.743760\pi$
$491$	−7.40518e6	−1.38622	−0.693110	+	0.720832i	$0.743760\pi$
$492$	0	0
$493$	−248196.	−0.0459915
$494$	0	0
$495$	789912.	0.144899
$496$	0	0
$497$	3.00468e6	0.545641
$498$	0	0
$499$	−3.93432e6	−0.707325	−0.353662	−	0.935373i	$-0.615064\pi$
$499$	−3.93432e6	−0.707325	−0.353662	+	0.935373i	$0.615064\pi$
$500$	0	0
$501$	−65592.0	−0.0116750
$502$	0	0
$503$	−3.40975e6	−0.600901	−0.300450	−	0.953797i	$-0.597137\pi$
$503$	−3.40975e6	−0.600901	−0.300450	+	0.953797i	$0.597137\pi$
$504$	0	0
$505$	1.45595e7	2.54050
$506$	0	0
$507$	−698463.	−0.120677
$508$	0	0
$509$	−7.72383e6	−1.32141	−0.660706	−	0.750645i	$-0.729743\pi$
$509$	−7.72383e6	−1.32141	−0.660706	+	0.750645i	$0.729743\pi$
$510$	0	0
$511$	1.90718e6	0.323102
$512$	0	0
$513$	−661932.	−0.111050
$514$	0	0
$515$	3.04856e6	0.506497
$516$	0	0
$517$	−649152.	−0.106812
$518$	0	0
$519$	2.60239e6	0.424085
$520$	0	0
$521$	−4.77658e6	−0.770944	−0.385472	−	0.922719i	$-0.625961\pi$
$521$	−4.77658e6	−0.770944	−0.385472	+	0.922719i	$0.625961\pi$
$522$	0	0
$523$	9.28754e6	1.48473	0.742363	−	0.669998i	$-0.233705\pi$
$523$	9.28754e6	1.48473	0.742363	+	0.669998i	$0.233705\pi$
$524$	0	0
$525$	−3.57695e6	−0.566389
$526$	0	0
$527$	820512.	0.128694
$528$	0	0
$529$	−5.05337e6	−0.785130
$530$	0	0
$531$	−2.64870e6	−0.407658
$532$	0	0
$533$	−4.46354e6	−0.680552
$534$	0	0
$535$	2.39094e6	0.361147
$536$	0	0
$537$	1.79543e6	0.268678
$538$	0	0
$539$	−220892.	−0.0327498
$540$	0	0
$541$	7.72917e6	1.13538	0.567688	−	0.823244i	$-0.307838\pi$
$541$	7.72917e6	1.13538	0.567688	+	0.823244i	$0.307838\pi$
$542$	0	0
$543$	−2.16495e6	−0.315100
$544$	0	0
$545$	−2.11979e6	−0.305704
$546$	0	0
$547$	8.60361e6	1.22945	0.614727	−	0.788740i	$-0.289266\pi$
$547$	8.60361e6	1.22945	0.614727	+	0.788740i	$0.289266\pi$
$548$	0	0
$549$	−874962.	−0.123896
$550$	0	0
$551$	1.01514e6	0.142445
$552$	0	0
$553$	4.31670e6	0.600260
$554$	0	0
$555$	3.98963e6	0.549794
$556$	0	0
$557$	−1.77723e6	−0.242721	−0.121360	−	0.992609i	$-0.538726\pi$
$557$	−1.77723e6	−0.242721	−0.121360	+	0.992609i	$0.538726\pi$
$558$	0	0
$559$	2.47900e6	0.335542
$560$	0	0
$561$	−183816.	−0.0246590
$562$	0	0
$563$	−2.68860e6	−0.357482	−0.178741	−	0.983896i	$-0.557202\pi$
$563$	−2.68860e6	−0.357482	−0.178741	+	0.983896i	$0.557202\pi$
$564$	0	0
$565$	−1.89804e6	−0.250140
$566$	0	0
$567$	321489.	0.0419961
$568$	0	0
$569$	5.32630e6	0.689675	0.344838	−	0.938662i	$-0.387934\pi$
$569$	5.32630e6	0.689675	0.344838	+	0.938662i	$0.387934\pi$
$570$	0	0
$571$	−1.33992e7	−1.71984	−0.859921	−	0.510427i	$-0.829487\pi$
$571$	−1.33992e7	−1.71984	−0.859921	+	0.510427i	$0.829487\pi$
$572$	0	0
$573$	2.61346e6	0.332528
$574$	0	0
$575$	9.53854e6	1.20313
$576$	0	0
$577$	−1.10502e6	−0.138176	−0.0690878	−	0.997611i	$-0.522009\pi$
$577$	−1.10502e6	−0.138176	−0.0690878	+	0.997611i	$0.522009\pi$
$578$	0	0
$579$	1.54309e6	0.191291
$580$	0	0
$581$	−3.52271e6	−0.432949
$582$	0	0
$583$	3.45718e6	0.421260
$584$	0	0
$585$	−5.75262e6	−0.694986
$586$	0	0
$587$	−5.97288e6	−0.715465	−0.357732	−	0.933824i	$-0.616450\pi$
$587$	−5.97288e6	−0.715465	−0.357732	+	0.933824i	$0.616450\pi$
$588$	0	0
$589$	−3.35597e6	−0.398593
$590$	0	0
$591$	−3.61791e6	−0.426078
$592$	0	0
$593$	−1.11945e7	−1.30728	−0.653639	−	0.756807i	$-0.726758\pi$
$593$	−1.11945e7	−1.30728	−0.653639	+	0.756807i	$0.726758\pi$
$594$	0	0
$595$	1.15307e6	0.133525
$596$	0	0
$597$	−2.33258e6	−0.267856
$598$	0	0
$599$	−1.09055e7	−1.24187	−0.620937	−	0.783860i	$-0.713248\pi$
$599$	−1.09055e7	−1.24187	−0.620937	+	0.783860i	$0.713248\pi$
$600$	0	0
$601$	7.39737e6	0.835394	0.417697	−	0.908586i	$-0.362837\pi$
$601$	7.39737e6	0.835394	0.417697	+	0.908586i	$0.362837\pi$
$602$	0	0
$603$	−5.26468e6	−0.589628
$604$	0	0
$605$	1.61742e7	1.79653
$606$	0	0
$607$	−7.13355e6	−0.785840	−0.392920	−	0.919573i	$-0.628535\pi$
$607$	−7.13355e6	−0.785840	−0.392920	+	0.919573i	$0.628535\pi$
$608$	0	0
$609$	−493038.	−0.0538688
$610$	0	0
$611$	4.72752e6	0.512307
$612$	0	0
$613$	−1.71264e7	−1.84083	−0.920416	−	0.390939i	$-0.872150\pi$
$613$	−1.71264e7	−1.84083	−0.920416	+	0.390939i	$0.872150\pi$
$614$	0	0
$615$	−6.35555e6	−0.677587
$616$	0	0
$617$	2.29924e6	0.243149	0.121574	−	0.992582i	$-0.461206\pi$
$617$	2.29924e6	0.243149	0.121574	+	0.992582i	$0.461206\pi$
$618$	0	0
$619$	1.85176e6	0.194249	0.0971245	−	0.995272i	$-0.469035\pi$
$619$	1.85176e6	0.194249	0.0971245	+	0.995272i	$0.469035\pi$
$620$	0	0
$621$	−857304.	−0.0892084
$622$	0	0
$623$	5.47908e6	0.565572
$624$	0	0
$625$	3.06758e7	3.14120
$626$	0	0
$627$	751824.	0.0763743
$628$	0	0
$629$	−928404.	−0.0935643
$630$	0	0
$631$	−9.25978e6	−0.925822	−0.462911	−	0.886405i	$-0.653195\pi$
$631$	−9.25978e6	−0.925822	−0.462911	+	0.886405i	$0.653195\pi$
$632$	0	0
$633$	−1.07261e7	−1.06398
$634$	0	0
$635$	7.08758e6	0.697532
$636$	0	0
$637$	1.60867e6	0.157079
$638$	0	0
$639$	4.96692e6	0.481210
$640$	0	0
$641$	1.79419e7	1.72474	0.862369	−	0.506280i	$-0.168980\pi$
$641$	1.79419e7	1.72474	0.862369	+	0.506280i	$0.168980\pi$
$642$	0	0
$643$	−6.70020e6	−0.639087	−0.319544	−	0.947572i	$-0.603530\pi$
$643$	−6.70020e6	−0.639087	−0.319544	+	0.947572i	$0.603530\pi$
$644$	0	0
$645$	3.52980e6	0.334080
$646$	0	0
$647$	1.12549e7	1.05701	0.528507	−	0.848929i	$-0.322752\pi$
$647$	1.12549e7	1.05701	0.528507	+	0.848929i	$0.322752\pi$
$648$	0	0
$649$	3.00840e6	0.280365
$650$	0	0
$651$	1.62994e6	0.150736
$652$	0	0
$653$	1.31704e7	1.20869	0.604347	−	0.796721i	$-0.293434\pi$
$653$	1.31704e7	1.20869	0.604347	+	0.796721i	$0.293434\pi$
$654$	0	0
$655$	1.62990e7	1.48442
$656$	0	0
$657$	3.15268e6	0.284949
$658$	0	0
$659$	1.43453e7	1.28676	0.643380	−	0.765547i	$-0.277532\pi$
$659$	1.43453e7	1.28676	0.643380	+	0.765547i	$0.277532\pi$
$660$	0	0
$661$	−14138.0	−0.00125859	−0.000629295	−	1.00000i	$-0.500200\pi$
$661$	−14138.0	−0.00125859	−0.000629295		1.00000i	$0.500200\pi$
$662$	0	0
$663$	1.33866e6	0.118273
$664$	0	0
$665$	−4.71615e6	−0.413555
$666$	0	0
$667$	1.31477e6	0.114429
$668$	0	0
$669$	−1.96546e6	−0.169784
$670$	0	0
$671$	993784.	0.0852090
$672$	0	0
$673$	1.37787e7	1.17266	0.586329	−	0.810073i	$-0.300573\pi$
$673$	1.37787e7	1.17266	0.586329	+	0.810073i	$0.300573\pi$
$674$	0	0
$675$	−5.91292e6	−0.499508
$676$	0	0
$677$	1.26155e7	1.05787	0.528936	−	0.848662i	$-0.322591\pi$
$677$	1.26155e7	1.05787	0.528936	+	0.848662i	$0.322591\pi$
$678$	0	0
$679$	−7.39145e6	−0.615255
$680$	0	0
$681$	5.24567e6	0.433444
$682$	0	0
$683$	1.08656e6	0.0891258	0.0445629	−	0.999007i	$-0.485810\pi$
$683$	1.08656e6	0.0891258	0.0445629	+	0.999007i	$0.485810\pi$
$684$	0	0
$685$	−2.71337e7	−2.20944
$686$	0	0
$687$	−8.64941e6	−0.699189
$688$	0	0
$689$	−2.51773e7	−2.02051
$690$	0	0
$691$	−1.91229e7	−1.52356	−0.761780	−	0.647836i	$-0.775674\pi$
$691$	−1.91229e7	−1.52356	−0.761780	+	0.647836i	$0.775674\pi$
$692$	0	0
$693$	−365148.	−0.0288826
$694$	0	0
$695$	2.99899e7	2.35512
$696$	0	0
$697$	1.47896e6	0.115312
$698$	0	0
$699$	−5.45395e6	−0.422200
$700$	0	0
$701$	1.15000e7	0.883897	0.441948	−	0.897040i	$-0.354287\pi$
$701$	1.15000e7	0.883897	0.441948	+	0.897040i	$0.354287\pi$
$702$	0	0
$703$	3.79726e6	0.289789
$704$	0	0
$705$	6.73142e6	0.510075
$706$	0	0
$707$	−6.73035e6	−0.506394
$708$	0	0
$709$	6.97551e6	0.521147	0.260574	−	0.965454i	$-0.416088\pi$
$709$	6.97551e6	0.521147	0.260574	+	0.965454i	$0.416088\pi$
$710$	0	0
$711$	7.13578e6	0.529380
$712$	0	0
$713$	−4.34650e6	−0.320196
$714$	0	0
$715$	6.53384e6	0.477973
$716$	0	0
$717$	1.05313e7	0.765040
$718$	0	0
$719$	1.44264e6	0.104072	0.0520362	−	0.998645i	$-0.483429\pi$
$719$	1.44264e6	0.104072	0.0520362	+	0.998645i	$0.483429\pi$
$720$	0	0
$721$	−1.40924e6	−0.100959
$722$	0	0
$723$	1.11322e7	0.792018
$724$	0	0
$725$	9.06810e6	0.640724
$726$	0	0
$727$	1.90334e7	1.33561	0.667807	−	0.744334i	$-0.267233\pi$
$727$	1.90334e7	1.33561	0.667807	+	0.744334i	$0.267233\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−821400.	−0.0568540
$732$	0	0
$733$	5.45585e6	0.375062	0.187531	−	0.982259i	$-0.439952\pi$
$733$	5.45585e6	0.375062	0.187531	+	0.982259i	$0.439952\pi$
$734$	0	0
$735$	2.29055e6	0.156395
$736$	0	0
$737$	5.97963e6	0.405514
$738$	0	0
$739$	−7.85197e6	−0.528893	−0.264446	−	0.964400i	$-0.585189\pi$
$739$	−7.85197e6	−0.528893	−0.264446	+	0.964400i	$0.585189\pi$
$740$	0	0
$741$	−5.47524e6	−0.366317
$742$	0	0
$743$	1.48695e7	0.988154	0.494077	−	0.869418i	$-0.335506\pi$
$743$	1.48695e7	0.988154	0.494077	+	0.869418i	$0.335506\pi$
$744$	0	0
$745$	−4.32537e7	−2.85518
$746$	0	0
$747$	−5.82325e6	−0.381825
$748$	0	0
$749$	−1.10524e6	−0.0719869
$750$	0	0
$751$	2.51309e7	1.62595	0.812977	−	0.582295i	$-0.197845\pi$
$751$	2.51309e7	1.62595	0.812977	+	0.582295i	$0.197845\pi$
$752$	0	0
$753$	8.63737e6	0.555129
$754$	0	0
$755$	3.84254e7	2.45330
$756$	0	0
$757$	−1.97874e7	−1.25501	−0.627507	−	0.778611i	$-0.715925\pi$
$757$	−1.97874e7	−1.25501	−0.627507	+	0.778611i	$0.715925\pi$
$758$	0	0
$759$	973728.	0.0613526
$760$	0	0
$761$	−1.49642e7	−0.936678	−0.468339	−	0.883549i	$-0.655147\pi$
$761$	−1.49642e7	−0.936678	−0.468339	+	0.883549i	$0.655147\pi$
$762$	0	0
$763$	979902.	0.0609356
$764$	0	0
$765$	1.90609e6	0.117758
$766$	0	0
$767$	−2.19090e7	−1.34473
$768$	0	0
$769$	5.06419e6	0.308812	0.154406	−	0.988007i	$-0.450654\pi$
$769$	5.06419e6	0.308812	0.154406	+	0.988007i	$0.450654\pi$
$770$	0	0
$771$	1.09133e7	0.661181
$772$	0	0
$773$	2.63025e7	1.58324	0.791622	−	0.611011i	$-0.209237\pi$
$773$	2.63025e7	1.58324	0.791622	+	0.611011i	$0.209237\pi$
$774$	0	0
$775$	−2.99783e7	−1.79288
$776$	0	0
$777$	−1.84426e6	−0.109590
$778$	0	0
$779$	−6.04910e6	−0.357147
$780$	0	0
$781$	−5.64144e6	−0.330950
$782$	0	0
$783$	−815022.	−0.0475078
$784$	0	0
$785$	1.61953e7	0.938027
$786$	0	0
$787$	−1.00525e7	−0.578545	−0.289273	−	0.957247i	$-0.593413\pi$
$787$	−1.00525e7	−0.578545	−0.289273	+	0.957247i	$0.593413\pi$
$788$	0	0
$789$	−1.12747e7	−0.644781
$790$	0	0
$791$	877394.	0.0498601
$792$	0	0
$793$	−7.23734e6	−0.408692
$794$	0	0
$795$	−3.58494e7	−2.01170
$796$	0	0
$797$	2.78516e7	1.55312	0.776560	−	0.630044i	$-0.216963\pi$
$797$	2.78516e7	1.55312	0.776560	+	0.630044i	$0.216963\pi$
$798$	0	0
$799$	−1.56643e6	−0.0868050
$800$	0	0
$801$	9.05726e6	0.498787
$802$	0	0
$803$	−3.58082e6	−0.195972
$804$	0	0
$805$	−6.10814e6	−0.332215
$806$	0	0
$807$	1.23179e6	0.0665816
$808$	0	0
$809$	−1.96936e7	−1.05792	−0.528961	−	0.848646i	$-0.677418\pi$
$809$	−1.96936e7	−1.05792	−0.528961	+	0.848646i	$0.677418\pi$
$810$	0	0
$811$	−5.05617e6	−0.269942	−0.134971	−	0.990850i	$-0.543094\pi$
$811$	−5.05617e6	−0.269942	−0.134971	+	0.990850i	$0.543094\pi$
$812$	0	0
$813$	−8.64806e6	−0.458873
$814$	0	0
$815$	−1.59454e7	−0.840893
$816$	0	0
$817$	3.35960e6	0.176089
$818$	0	0
$819$	2.65923e6	0.138531
$820$	0	0
$821$	−2.82324e6	−0.146181	−0.0730904	−	0.997325i	$-0.523286\pi$
$821$	−2.82324e6	−0.146181	−0.0730904	+	0.997325i	$0.523286\pi$
$822$	0	0
$823$	−2.54741e7	−1.31099	−0.655494	−	0.755201i	$-0.727539\pi$
$823$	−2.54741e7	−1.31099	−0.655494	+	0.755201i	$0.727539\pi$
$824$	0	0
$825$	6.71591e6	0.343534
$826$	0	0
$827$	1.75616e7	0.892893	0.446446	−	0.894810i	$-0.352689\pi$
$827$	1.75616e7	0.892893	0.446446	+	0.894810i	$0.352689\pi$
$828$	0	0
$829$	9.14728e6	0.462280	0.231140	−	0.972920i	$-0.425754\pi$
$829$	9.14728e6	0.462280	0.231140	+	0.972920i	$0.425754\pi$
$830$	0	0
$831$	−8.15247e6	−0.409531
$832$	0	0
$833$	−533022.	−0.0266154
$834$	0	0
$835$	−772528.	−0.0383441
$836$	0	0
$837$	2.69438e6	0.132937
$838$	0	0
$839$	1.09891e7	0.538961	0.269481	−	0.963006i	$-0.413148\pi$
$839$	1.09891e7	0.538961	0.269481	+	0.963006i	$0.413148\pi$
$840$	0	0
$841$	−1.92612e7	−0.939061
$842$	0	0
$843$	297558.	0.0144212
$844$	0	0
$845$	−8.22634e6	−0.396337
$846$	0	0
$847$	−7.47676e6	−0.358101
$848$	0	0
$849$	−7.77229e6	−0.370067
$850$	0	0
$851$	4.91803e6	0.232792
$852$	0	0
$853$	−1.34854e7	−0.634586	−0.317293	−	0.948328i	$-0.602774\pi$
$853$	−1.34854e7	−0.634586	−0.317293	+	0.948328i	$0.602774\pi$
$854$	0	0
$855$	−7.79609e6	−0.364722
$856$	0	0
$857$	1.07032e7	0.497808	0.248904	−	0.968528i	$-0.419930\pi$
$857$	1.07032e7	0.497808	0.248904	+	0.968528i	$0.419930\pi$
$858$	0	0
$859$	1.33747e7	0.618446	0.309223	−	0.950990i	$-0.399931\pi$
$859$	1.33747e7	0.618446	0.309223	+	0.950990i	$0.399931\pi$
$860$	0	0
$861$	2.93794e6	0.135063
$862$	0	0
$863$	−1.99768e7	−0.913058	−0.456529	−	0.889708i	$-0.650908\pi$
$863$	−1.99768e7	−0.913058	−0.456529	+	0.889708i	$0.650908\pi$
$864$	0	0
$865$	3.06503e7	1.39282
$866$	0	0
$867$	1.23352e7	0.557310
$868$	0	0
$869$	−8.10483e6	−0.364078
$870$	0	0
$871$	−4.35473e7	−1.94498
$872$	0	0
$873$	−1.22185e7	−0.542604
$874$	0	0
$875$	−2.58973e7	−1.14349
$876$	0	0
$877$	8.81107e6	0.386838	0.193419	−	0.981116i	$-0.438042\pi$
$877$	8.81107e6	0.386838	0.193419	+	0.981116i	$0.438042\pi$
$878$	0	0
$879$	1.20379e7	0.525508
$880$	0	0
$881$	4.01078e7	1.74096	0.870481	−	0.492202i	$-0.163808\pi$
$881$	4.01078e7	1.74096	0.870481	+	0.492202i	$0.163808\pi$
$882$	0	0
$883$	−1.49664e7	−0.645976	−0.322988	−	0.946403i	$-0.604687\pi$
$883$	−1.49664e7	−0.645976	−0.322988	+	0.946403i	$0.604687\pi$
$884$	0	0
$885$	−3.11958e7	−1.33887
$886$	0	0
$887$	575144.	0.0245453	0.0122726	−	0.999925i	$-0.496093\pi$
$887$	575144.	0.0245453	0.0122726	+	0.999925i	$0.496093\pi$
$888$	0	0
$889$	−3.27634e6	−0.139038
$890$	0	0
$891$	−603612.	−0.0254720
$892$	0	0
$893$	6.40685e6	0.268854
$894$	0	0
$895$	2.11462e7	0.882417
$896$	0	0
$897$	−7.09128e6	−0.294268
$898$	0	0
$899$	−4.13213e6	−0.170520
$900$	0	0
$901$	8.34232e6	0.342353
$902$	0	0
$903$	−1.63170e6	−0.0665918
$904$	0	0
$905$	−2.54983e7	−1.03488
$906$	0	0
$907$	3.33367e7	1.34557	0.672783	−	0.739840i	$-0.265099\pi$
$907$	3.33367e7	1.34557	0.672783	+	0.739840i	$0.265099\pi$
$908$	0	0
$909$	−1.11257e7	−0.446598
$910$	0	0
$911$	−2.17451e7	−0.868090	−0.434045	−	0.900891i	$-0.642914\pi$
$911$	−2.17451e7	−0.868090	−0.434045	+	0.900891i	$0.642914\pi$
$912$	0	0
$913$	6.61406e6	0.262598
$914$	0	0
$915$	−1.03051e7	−0.406911
$916$	0	0
$917$	−7.53444e6	−0.295888
$918$	0	0
$919$	4.03986e6	0.157789	0.0788947	−	0.996883i	$-0.474861\pi$
$919$	4.03986e6	0.157789	0.0788947	+	0.996883i	$0.474861\pi$
$920$	0	0
$921$	−1.19538e7	−0.464364
$922$	0	0
$923$	4.10844e7	1.58735
$924$	0	0
$925$	3.39202e7	1.30348
$926$	0	0
$927$	−2.32956e6	−0.0890378
$928$	0	0
$929$	7.69679e6	0.292597	0.146299	−	0.989240i	$-0.453264\pi$
$929$	7.69679e6	0.292597	0.146299	+	0.989240i	$0.453264\pi$
$930$	0	0
$931$	2.18011e6	0.0824335
$932$	0	0
$933$	−5.99249e6	−0.225374
$934$	0	0
$935$	−2.16494e6	−0.0809874
$936$	0	0
$937$	−453558.	−0.0168766	−0.00843828	−	0.999964i	$-0.502686\pi$
$937$	−453558.	−0.0168766	−0.00843828	+	0.999964i	$0.502686\pi$
$938$	0	0
$939$	2.78119e7	1.02936
$940$	0	0
$941$	−2.43852e7	−0.897745	−0.448873	−	0.893596i	$-0.648174\pi$
$941$	−2.43852e7	−0.897745	−0.448873	+	0.893596i	$0.648174\pi$
$942$	0	0
$943$	−7.83451e6	−0.286901
$944$	0	0
$945$	3.78643e6	0.137927
$946$	0	0
$947$	2.18745e7	0.792615	0.396308	−	0.918118i	$-0.370291\pi$
$947$	2.18745e7	0.792615	0.396308	+	0.918118i	$0.370291\pi$
$948$	0	0
$949$	2.60777e7	0.939949
$950$	0	0
$951$	8.76760e6	0.314362
$952$	0	0
$953$	−3.93319e7	−1.40286	−0.701428	−	0.712741i	$-0.747454\pi$
$953$	−3.93319e7	−1.40286	−0.701428	+	0.712741i	$0.747454\pi$
$954$	0	0
$955$	3.07807e7	1.09212
$956$	0	0
$957$	925704.	0.0326732
$958$	0	0
$959$	1.25429e7	0.440405
$960$	0	0
$961$	−1.49687e7	−0.522849
$962$	0	0
$963$	−1.82704e6	−0.0634865
$964$	0	0
$965$	1.81741e7	0.628254
$966$	0	0
$967$	−3.85234e7	−1.32483	−0.662413	−	0.749139i	$-0.730468\pi$
$967$	−3.85234e7	−1.32483	−0.662413	+	0.749139i	$0.730468\pi$
$968$	0	0
$969$	1.81418e6	0.0620686
$970$	0	0
$971$	−2.43543e7	−0.828947	−0.414473	−	0.910061i	$-0.636034\pi$
$971$	−2.43543e7	−0.828947	−0.414473	+	0.910061i	$0.636034\pi$
$972$	0	0
$973$	−1.38633e7	−0.469444
$974$	0	0
$975$	−4.89093e7	−1.64771
$976$	0	0
$977$	1.62534e7	0.544763	0.272382	−	0.962189i	$-0.412189\pi$
$977$	1.62534e7	0.544763	0.272382	+	0.962189i	$0.412189\pi$
$978$	0	0
$979$	−1.02873e7	−0.343038
$980$	0	0
$981$	1.61984e6	0.0537402
$982$	0	0
$983$	252456.	0.00833301	0.00416650	−	0.999991i	$-0.498674\pi$
$983$	252456.	0.00833301	0.00416650	+	0.999991i	$0.498674\pi$
$984$	0	0
$985$	−4.26109e7	−1.39936
$986$	0	0
$987$	−3.11170e6	−0.101673
$988$	0	0
$989$	4.35120e6	0.141455
$990$	0	0
$991$	2.49728e7	0.807761	0.403880	−	0.914812i	$-0.367661\pi$
$991$	2.49728e7	0.807761	0.403880	+	0.914812i	$0.367661\pi$
$992$	0	0
$993$	7.03595e6	0.226438
$994$	0	0
$995$	−2.74727e7	−0.879717
$996$	0	0
$997$	−3.60983e7	−1.15013	−0.575067	−	0.818106i	$-0.695024\pi$
$997$	−3.60983e7	−1.15013	−0.575067	+	0.818106i	$0.695024\pi$
$998$	0	0
$999$	−3.04868e6	−0.0966491

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.a.a.1.1		1
3.2	odd	2		1008.6.a.bc.1.1		1
4.3	odd	2		21.6.a.d.1.1	✓	1
12.11	even	2		63.6.a.a.1.1		1
20.3	even	4		525.6.d.a.274.1		2
20.7	even	4		525.6.d.a.274.2		2
20.19	odd	2		525.6.a.a.1.1		1
28.3	even	6		147.6.e.b.79.1		2
28.11	odd	6		147.6.e.a.79.1		2
28.19	even	6		147.6.e.b.67.1		2
28.23	odd	6		147.6.e.a.67.1		2
28.27	even	2		147.6.a.g.1.1		1
84.83	odd	2		441.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.d.1.1	✓	1	4.3	odd	2
63.6.a.a.1.1		1	12.11	even	2
147.6.a.g.1.1		1	28.27	even	2
147.6.e.a.67.1		2	28.23	odd	6
147.6.e.a.79.1		2	28.11	odd	6
147.6.e.b.67.1		2	28.19	even	6
147.6.e.b.79.1		2	28.3	even	6
336.6.a.a.1.1		1	1.1	even	1	trivial
441.6.a.b.1.1		1	84.83	odd	2
525.6.a.a.1.1		1	20.19	odd	2
525.6.d.a.274.1		2	20.3	even	4
525.6.d.a.274.2		2	20.7	even	4
1008.6.a.bc.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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