LMFDB - Embedded newform 336.6.a.h.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 42)
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)

$f(q)$

$=$

$q-9.00000 q^{3} +26.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q-9.00000 q^{3} +26.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} -664.000 q^{11} +318.000 q^{13} -234.000 q^{15} +1582.00 q^{17} -236.000 q^{19} -441.000 q^{21} -2212.00 q^{23} -2449.00 q^{25} -729.000 q^{27} -4954.00 q^{29} +7128.00 q^{31} +5976.00 q^{33} +1274.00 q^{35} +4358.00 q^{37} -2862.00 q^{39} +10542.0 q^{41} +8452.00 q^{43} +2106.00 q^{45} -5352.00 q^{47} +2401.00 q^{49} -14238.0 q^{51} -33354.0 q^{53} -17264.0 q^{55} +2124.00 q^{57} +15436.0 q^{59} -36762.0 q^{61} +3969.00 q^{63} +8268.00 q^{65} -40972.0 q^{67} +19908.0 q^{69} +9092.00 q^{71} -73454.0 q^{73} +22041.0 q^{75} -32536.0 q^{77} -89400.0 q^{79} +6561.00 q^{81} +6428.00 q^{83} +41132.0 q^{85} +44586.0 q^{87} -122658. q^{89} +15582.0 q^{91} -64152.0 q^{93} -6136.00 q^{95} +21370.0 q^{97} -53784.0 q^{99} +O(q^{100})$

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
$5$	26.0000	0.465102	0.232551	−	0.972584i	$-0.425293\pi$
$5$	26.0000	0.465102	0.232551	+	0.972584i	$0.425293\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−664.000	−1.65457	−0.827287	−	0.561779i	$-0.810117\pi$
$11$	−664.000	−1.65457	−0.827287	+	0.561779i	$0.810117\pi$
$12$	0	0
$13$	318.000	0.521878	0.260939	−	0.965355i	$-0.415968\pi$
$13$	318.000	0.521878	0.260939	+	0.965355i	$0.415968\pi$
$14$	0	0
$15$	−234.000	−0.268527
$16$	0	0
$17$	1582.00	1.32765	0.663826	−	0.747887i	$-0.268932\pi$
$17$	1582.00	1.32765	0.663826	+	0.747887i	$0.268932\pi$
$18$	0	0
$19$	−236.000	−0.149978	−0.0749891	−	0.997184i	$-0.523892\pi$
$19$	−236.000	−0.149978	−0.0749891	+	0.997184i	$0.523892\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	0	0
$23$	−2212.00	−0.871898	−0.435949	−	0.899971i	$-0.643587\pi$
$23$	−2212.00	−0.871898	−0.435949	+	0.899971i	$0.643587\pi$
$24$	0	0
$25$	−2449.00	−0.783680
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−4954.00	−1.09386	−0.546929	−	0.837179i	$-0.684203\pi$
$29$	−4954.00	−1.09386	−0.546929	+	0.837179i	$0.684203\pi$
$30$	0	0
$31$	7128.00	1.33218	0.666091	−	0.745871i	$-0.267966\pi$
$31$	7128.00	1.33218	0.666091	+	0.745871i	$0.267966\pi$
$32$	0	0
$33$	5976.00	0.955269
$34$	0	0
$35$	1274.00	0.175792
$36$	0	0
$37$	4358.00	0.523339	0.261669	−	0.965158i	$-0.415727\pi$
$37$	4358.00	0.523339	0.261669	+	0.965158i	$0.415727\pi$
$38$	0	0
$39$	−2862.00	−0.301306
$40$	0	0
$41$	10542.0	0.979407	0.489704	−	0.871889i	$-0.337105\pi$
$41$	10542.0	0.979407	0.489704	+	0.871889i	$0.337105\pi$
$42$	0	0
$43$	8452.00	0.697089	0.348545	−	0.937292i	$-0.386676\pi$
$43$	8452.00	0.697089	0.348545	+	0.937292i	$0.386676\pi$
$44$	0	0
$45$	2106.00	0.155034
$46$	0	0
$47$	−5352.00	−0.353404	−0.176702	−	0.984264i	$-0.556543\pi$
$47$	−5352.00	−0.353404	−0.176702	+	0.984264i	$0.556543\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−14238.0	−0.766520
$52$	0	0
$53$	−33354.0	−1.63102	−0.815508	−	0.578746i	$-0.803542\pi$
$53$	−33354.0	−1.63102	−0.815508	+	0.578746i	$0.803542\pi$
$54$	0	0
$55$	−17264.0	−0.769546
$56$	0	0
$57$	2124.00	0.0865899
$58$	0	0
$59$	15436.0	0.577304	0.288652	−	0.957434i	$-0.406793\pi$
$59$	15436.0	0.577304	0.288652	+	0.957434i	$0.406793\pi$
$60$	0	0
$61$	−36762.0	−1.26495	−0.632477	−	0.774579i	$-0.717962\pi$
$61$	−36762.0	−1.26495	−0.632477	+	0.774579i	$0.717962\pi$
$62$	0	0
$63$	3969.00	0.125988
$64$	0	0
$65$	8268.00	0.242726
$66$	0	0
$67$	−40972.0	−1.11506	−0.557532	−	0.830155i	$-0.688252\pi$
$67$	−40972.0	−1.11506	−0.557532	+	0.830155i	$0.688252\pi$
$68$	0	0
$69$	19908.0	0.503390
$70$	0	0
$71$	9092.00	0.214049	0.107025	−	0.994256i	$-0.465868\pi$
$71$	9092.00	0.214049	0.107025	+	0.994256i	$0.465868\pi$
$72$	0	0
$73$	−73454.0	−1.61327	−0.806637	−	0.591047i	$-0.798715\pi$
$73$	−73454.0	−1.61327	−0.806637	+	0.591047i	$0.798715\pi$
$74$	0	0
$75$	22041.0	0.452458
$76$	0	0
$77$	−32536.0	−0.625370
$78$	0	0
$79$	−89400.0	−1.61165	−0.805823	−	0.592156i	$-0.798277\pi$
$79$	−89400.0	−1.61165	−0.805823	+	0.592156i	$0.798277\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	6428.00	0.102419	0.0512095	−	0.998688i	$-0.483692\pi$
$83$	6428.00	0.102419	0.0512095	+	0.998688i	$0.483692\pi$
$84$	0	0
$85$	41132.0	0.617494
$86$	0	0
$87$	44586.0	0.631539
$88$	0	0
$89$	−122658.	−1.64142	−0.820712	−	0.571342i	$-0.806423\pi$
$89$	−122658.	−1.64142	−0.820712	+	0.571342i	$0.806423\pi$
$90$	0	0
$91$	15582.0	0.197251
$92$	0	0
$93$	−64152.0	−0.769135
$94$	0	0
$95$	−6136.00	−0.0697552
$96$	0	0
$97$	21370.0	0.230608	0.115304	−	0.993330i	$-0.463216\pi$
$97$	21370.0	0.230608	0.115304	+	0.993330i	$0.463216\pi$
$98$	0	0
$99$	−53784.0	−0.551525
$100$	0	0
$101$	−36814.0	−0.359095	−0.179548	−	0.983749i	$-0.557463\pi$
$101$	−36814.0	−0.359095	−0.179548	+	0.983749i	$0.557463\pi$
$102$	0	0
$103$	−104528.	−0.970822	−0.485411	−	0.874286i	$-0.661330\pi$
$103$	−104528.	−0.970822	−0.485411	+	0.874286i	$0.661330\pi$
$104$	0	0
$105$	−11466.0	−0.101494
$106$	0	0
$107$	−214440.	−1.81070	−0.905350	−	0.424667i	$-0.860391\pi$
$107$	−214440.	−1.81070	−0.905350	+	0.424667i	$0.860391\pi$
$108$	0	0
$109$	28798.0	0.232165	0.116082	−	0.993240i	$-0.462966\pi$
$109$	28798.0	0.232165	0.116082	+	0.993240i	$0.462966\pi$
$110$	0	0
$111$	−39222.0	−0.302150
$112$	0	0
$113$	−56014.0	−0.412668	−0.206334	−	0.978482i	$-0.566153\pi$
$113$	−56014.0	−0.412668	−0.206334	+	0.978482i	$0.566153\pi$
$114$	0	0
$115$	−57512.0	−0.405521
$116$	0	0
$117$	25758.0	0.173959
$118$	0	0
$119$	77518.0	0.501805
$120$	0	0
$121$	279845.	1.73762
$122$	0	0
$123$	−94878.0	−0.565461
$124$	0	0
$125$	−144924.	−0.829593
$126$	0	0
$127$	−185400.	−1.02000	−0.510000	−	0.860174i	$-0.670355\pi$
$127$	−185400.	−1.02000	−0.510000	+	0.860174i	$0.670355\pi$
$128$	0	0
$129$	−76068.0	−0.402465
$130$	0	0
$131$	−64532.0	−0.328547	−0.164273	−	0.986415i	$-0.552528\pi$
$131$	−64532.0	−0.328547	−0.164273	+	0.986415i	$0.552528\pi$
$132$	0	0
$133$	−11564.0	−0.0566864
$134$	0	0
$135$	−18954.0	−0.0895089
$136$	0	0
$137$	152930.	0.696131	0.348066	−	0.937470i	$-0.386839\pi$
$137$	152930.	0.696131	0.348066	+	0.937470i	$0.386839\pi$
$138$	0	0
$139$	343460.	1.50778	0.753892	−	0.656998i	$-0.228174\pi$
$139$	343460.	1.50778	0.753892	+	0.656998i	$0.228174\pi$
$140$	0	0
$141$	48168.0	0.204038
$142$	0	0
$143$	−211152.	−0.863486
$144$	0	0
$145$	−128804.	−0.508756
$146$	0	0
$147$	−21609.0	−0.0824786
$148$	0	0
$149$	−174858.	−0.645238	−0.322619	−	0.946529i	$-0.604563\pi$
$149$	−174858.	−0.645238	−0.322619	+	0.946529i	$0.604563\pi$
$150$	0	0
$151$	452552.	1.61520	0.807600	−	0.589731i	$-0.200766\pi$
$151$	452552.	1.61520	0.807600	+	0.589731i	$0.200766\pi$
$152$	0	0
$153$	128142.	0.442551
$154$	0	0
$155$	185328.	0.619601
$156$	0	0
$157$	−499066.	−1.61588	−0.807940	−	0.589265i	$-0.799417\pi$
$157$	−499066.	−1.61588	−0.807940	+	0.589265i	$0.799417\pi$
$158$	0	0
$159$	300186.	0.941668
$160$	0	0
$161$	−108388.	−0.329546
$162$	0	0
$163$	475588.	1.40204	0.701022	−	0.713139i	$-0.252727\pi$
$163$	475588.	1.40204	0.701022	+	0.713139i	$0.252727\pi$
$164$	0	0
$165$	155376.	0.444298
$166$	0	0
$167$	−120224.	−0.333580	−0.166790	−	0.985992i	$-0.553340\pi$
$167$	−120224.	−0.333580	−0.166790	+	0.985992i	$0.553340\pi$
$168$	0	0
$169$	−270169.	−0.727644
$170$	0	0
$171$	−19116.0	−0.0499927
$172$	0	0
$173$	508874.	1.29269	0.646346	−	0.763045i	$-0.276296\pi$
$173$	508874.	1.29269	0.646346	+	0.763045i	$0.276296\pi$
$174$	0	0
$175$	−120001.	−0.296203
$176$	0	0
$177$	−138924.	−0.333307
$178$	0	0
$179$	−487560.	−1.13735	−0.568677	−	0.822561i	$-0.692544\pi$
$179$	−487560.	−1.13735	−0.568677	+	0.822561i	$0.692544\pi$
$180$	0	0
$181$	−544410.	−1.23518	−0.617589	−	0.786501i	$-0.711891\pi$
$181$	−544410.	−1.23518	−0.617589	+	0.786501i	$0.711891\pi$
$182$	0	0
$183$	330858.	0.730321
$184$	0	0
$185$	113308.	0.243406
$186$	0	0
$187$	−1.05045e6	−2.19670
$188$	0	0
$189$	−35721.0	−0.0727393
$190$	0	0
$191$	−376404.	−0.746570	−0.373285	−	0.927717i	$-0.621769\pi$
$191$	−376404.	−0.746570	−0.373285	+	0.927717i	$0.621769\pi$
$192$	0	0
$193$	844946.	1.63281	0.816405	−	0.577480i	$-0.195964\pi$
$193$	844946.	1.63281	0.816405	+	0.577480i	$0.195964\pi$
$194$	0	0
$195$	−74412.0	−0.140138
$196$	0	0
$197$	−492794.	−0.904690	−0.452345	−	0.891843i	$-0.649412\pi$
$197$	−492794.	−0.904690	−0.452345	+	0.891843i	$0.649412\pi$
$198$	0	0
$199$	914776.	1.63750	0.818751	−	0.574148i	$-0.194667\pi$
$199$	914776.	1.63750	0.818751	+	0.574148i	$0.194667\pi$
$200$	0	0
$201$	368748.	0.643783
$202$	0	0
$203$	−242746.	−0.413440
$204$	0	0
$205$	274092.	0.455524
$206$	0	0
$207$	−179172.	−0.290633
$208$	0	0
$209$	156704.	0.248150
$210$	0	0
$211$	−311780.	−0.482106	−0.241053	−	0.970512i	$-0.577493\pi$
$211$	−311780.	−0.482106	−0.241053	+	0.970512i	$0.577493\pi$
$212$	0	0
$213$	−81828.0	−0.123581
$214$	0	0
$215$	219752.	0.324218
$216$	0	0
$217$	349272.	0.503517
$218$	0	0
$219$	661086.	0.931425
$220$	0	0
$221$	503076.	0.692872
$222$	0	0
$223$	1.28776e6	1.73409	0.867047	−	0.498226i	$-0.166015\pi$
$223$	1.28776e6	1.73409	0.867047	+	0.498226i	$0.166015\pi$
$224$	0	0
$225$	−198369.	−0.261227
$226$	0	0
$227$	−1.28905e6	−1.66037	−0.830187	−	0.557485i	$-0.811766\pi$
$227$	−1.28905e6	−1.66037	−0.830187	+	0.557485i	$0.811766\pi$
$228$	0	0
$229$	678214.	0.854630	0.427315	−	0.904103i	$-0.359460\pi$
$229$	678214.	0.854630	0.427315	+	0.904103i	$0.359460\pi$
$230$	0	0
$231$	292824.	0.361058
$232$	0	0
$233$	−1.11731e6	−1.34829	−0.674146	−	0.738598i	$-0.735488\pi$
$233$	−1.11731e6	−1.34829	−0.674146	+	0.738598i	$0.735488\pi$
$234$	0	0
$235$	−139152.	−0.164369
$236$	0	0
$237$	804600.	0.930485
$238$	0	0
$239$	1.26196e6	1.42906	0.714528	−	0.699606i	$-0.246641\pi$
$239$	1.26196e6	1.42906	0.714528	+	0.699606i	$0.246641\pi$
$240$	0	0
$241$	948218.	1.05164	0.525818	−	0.850597i	$-0.323759\pi$
$241$	948218.	1.05164	0.525818	+	0.850597i	$0.323759\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	62426.0	0.0664432
$246$	0	0
$247$	−75048.0	−0.0782703
$248$	0	0
$249$	−57852.0	−0.0591317
$250$	0	0
$251$	486396.	0.487310	0.243655	−	0.969862i	$-0.421653\pi$
$251$	486396.	0.487310	0.243655	+	0.969862i	$0.421653\pi$
$252$	0	0
$253$	1.46877e6	1.44262
$254$	0	0
$255$	−370188.	−0.356510
$256$	0	0
$257$	−1.03910e6	−0.981349	−0.490675	−	0.871343i	$-0.663250\pi$
$257$	−1.03910e6	−0.981349	−0.490675	+	0.871343i	$0.663250\pi$
$258$	0	0
$259$	213542.	0.197803
$260$	0	0
$261$	−401274.	−0.364619
$262$	0	0
$263$	−1.35104e6	−1.20443	−0.602213	−	0.798335i	$-0.705714\pi$
$263$	−1.35104e6	−1.20443	−0.602213	+	0.798335i	$0.705714\pi$
$264$	0	0
$265$	−867204.	−0.758589
$266$	0	0
$267$	1.10392e6	0.947677
$268$	0	0
$269$	−1.11811e6	−0.942115	−0.471057	−	0.882103i	$-0.656128\pi$
$269$	−1.11811e6	−0.942115	−0.471057	+	0.882103i	$0.656128\pi$
$270$	0	0
$271$	190104.	0.157242	0.0786209	−	0.996905i	$-0.474948\pi$
$271$	190104.	0.157242	0.0786209	+	0.996905i	$0.474948\pi$
$272$	0	0
$273$	−140238.	−0.113883
$274$	0	0
$275$	1.62614e6	1.29666
$276$	0	0
$277$	−200506.	−0.157010	−0.0785051	−	0.996914i	$-0.525015\pi$
$277$	−200506.	−0.157010	−0.0785051	+	0.996914i	$0.525015\pi$
$278$	0	0
$279$	577368.	0.444061
$280$	0	0
$281$	1.09237e6	0.825285	0.412643	−	0.910893i	$-0.364606\pi$
$281$	1.09237e6	0.825285	0.412643	+	0.910893i	$0.364606\pi$
$282$	0	0
$283$	−1.81258e6	−1.34534	−0.672669	−	0.739944i	$-0.734852\pi$
$283$	−1.81258e6	−1.34534	−0.672669	+	0.739944i	$0.734852\pi$
$284$	0	0
$285$	55224.0	0.0402732
$286$	0	0
$287$	516558.	0.370181
$288$	0	0
$289$	1.08287e6	0.762659
$290$	0	0
$291$	−192330.	−0.133142
$292$	0	0
$293$	2.10031e6	1.42927	0.714634	−	0.699499i	$-0.246593\pi$
$293$	2.10031e6	1.42927	0.714634	+	0.699499i	$0.246593\pi$
$294$	0	0
$295$	401336.	0.268505
$296$	0	0
$297$	484056.	0.318423
$298$	0	0
$299$	−703416.	−0.455024
$300$	0	0
$301$	414148.	0.263475
$302$	0	0
$303$	331326.	0.207324
$304$	0	0
$305$	−955812.	−0.588333
$306$	0	0
$307$	1.64104e6	0.993743	0.496872	−	0.867824i	$-0.334482\pi$
$307$	1.64104e6	0.993743	0.496872	+	0.867824i	$0.334482\pi$
$308$	0	0
$309$	940752.	0.560504
$310$	0	0
$311$	945232.	0.554163	0.277081	−	0.960846i	$-0.410633\pi$
$311$	945232.	0.554163	0.277081	+	0.960846i	$0.410633\pi$
$312$	0	0
$313$	415354.	0.239639	0.119820	−	0.992796i	$-0.461768\pi$
$313$	415354.	0.239639	0.119820	+	0.992796i	$0.461768\pi$
$314$	0	0
$315$	103194.	0.0585974
$316$	0	0
$317$	1.18481e6	0.662220	0.331110	−	0.943592i	$-0.392577\pi$
$317$	1.18481e6	0.662220	0.331110	+	0.943592i	$0.392577\pi$
$318$	0	0
$319$	3.28946e6	1.80987
$320$	0	0
$321$	1.92996e6	1.04541
$322$	0	0
$323$	−373352.	−0.199119
$324$	0	0
$325$	−778782.	−0.408985
$326$	0	0
$327$	−259182.	−0.134040
$328$	0	0
$329$	−262248.	−0.133574
$330$	0	0
$331$	−1.37155e6	−0.688083	−0.344042	−	0.938954i	$-0.611796\pi$
$331$	−1.37155e6	−0.688083	−0.344042	+	0.938954i	$0.611796\pi$
$332$	0	0
$333$	352998.	0.174446
$334$	0	0
$335$	−1.06527e6	−0.518619
$336$	0	0
$337$	963522.	0.462154	0.231077	−	0.972935i	$-0.425775\pi$
$337$	963522.	0.462154	0.231077	+	0.972935i	$0.425775\pi$
$338$	0	0
$339$	504126.	0.238254
$340$	0	0
$341$	−4.73299e6	−2.20419
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	517608.	0.234128
$346$	0	0
$347$	−2.57731e6	−1.14906	−0.574531	−	0.818483i	$-0.694815\pi$
$347$	−2.57731e6	−1.14906	−0.574531	+	0.818483i	$0.694815\pi$
$348$	0	0
$349$	−3.06751e6	−1.34810	−0.674051	−	0.738684i	$-0.735447\pi$
$349$	−3.06751e6	−1.34810	−0.674051	+	0.738684i	$0.735447\pi$
$350$	0	0
$351$	−231822.	−0.100435
$352$	0	0
$353$	−3.10144e6	−1.32473	−0.662364	−	0.749182i	$-0.730447\pi$
$353$	−3.10144e6	−1.32473	−0.662364	+	0.749182i	$0.730447\pi$
$354$	0	0
$355$	236392.	0.0995547
$356$	0	0
$357$	−697662.	−0.289717
$358$	0	0
$359$	327508.	0.134118	0.0670588	−	0.997749i	$-0.478638\pi$
$359$	327508.	0.134118	0.0670588	+	0.997749i	$0.478638\pi$
$360$	0	0
$361$	−2.42040e6	−0.977507
$362$	0	0
$363$	−2.51860e6	−1.00321
$364$	0	0
$365$	−1.90980e6	−0.750337
$366$	0	0
$367$	2.86739e6	1.11128	0.555638	−	0.831424i	$-0.312474\pi$
$367$	2.86739e6	1.11128	0.555638	+	0.831424i	$0.312474\pi$
$368$	0	0
$369$	853902.	0.326469
$370$	0	0
$371$	−1.63435e6	−0.616466
$372$	0	0
$373$	3.58029e6	1.33244	0.666218	−	0.745757i	$-0.267912\pi$
$373$	3.58029e6	1.33244	0.666218	+	0.745757i	$0.267912\pi$
$374$	0	0
$375$	1.30432e6	0.478966
$376$	0	0
$377$	−1.57537e6	−0.570860
$378$	0	0
$379$	−1.64235e6	−0.587310	−0.293655	−	0.955912i	$-0.594872\pi$
$379$	−1.64235e6	−0.587310	−0.293655	+	0.955912i	$0.594872\pi$
$380$	0	0
$381$	1.66860e6	0.588898
$382$	0	0
$383$	2.05698e6	0.716527	0.358263	−	0.933621i	$-0.383369\pi$
$383$	2.05698e6	0.716527	0.358263	+	0.933621i	$0.383369\pi$
$384$	0	0
$385$	−845936.	−0.290861
$386$	0	0
$387$	684612.	0.232363
$388$	0	0
$389$	616142.	0.206446	0.103223	−	0.994658i	$-0.467084\pi$
$389$	616142.	0.206446	0.103223	+	0.994658i	$0.467084\pi$
$390$	0	0
$391$	−3.49938e6	−1.15758
$392$	0	0
$393$	580788.	0.189686
$394$	0	0
$395$	−2.32440e6	−0.749580
$396$	0	0
$397$	2.19212e6	0.698052	0.349026	−	0.937113i	$-0.386513\pi$
$397$	2.19212e6	0.698052	0.349026	+	0.937113i	$0.386513\pi$
$398$	0	0
$399$	104076.	0.0327279
$400$	0	0
$401$	3.28454e6	1.02003	0.510015	−	0.860165i	$-0.329640\pi$
$401$	3.28454e6	1.02003	0.510015	+	0.860165i	$0.329640\pi$
$402$	0	0
$403$	2.26670e6	0.695236
$404$	0	0
$405$	170586.	0.0516780
$406$	0	0
$407$	−2.89371e6	−0.865903
$408$	0	0
$409$	−3.61219e6	−1.06773	−0.533866	−	0.845569i	$-0.679261\pi$
$409$	−3.61219e6	−1.06773	−0.533866	+	0.845569i	$0.679261\pi$
$410$	0	0
$411$	−1.37637e6	−0.401912
$412$	0	0
$413$	756364.	0.218200
$414$	0	0
$415$	167128.	0.0476353
$416$	0	0
$417$	−3.09114e6	−0.870520
$418$	0	0
$419$	−5.41489e6	−1.50680	−0.753398	−	0.657564i	$-0.771587\pi$
$419$	−5.41489e6	−1.50680	−0.753398	+	0.657564i	$0.771587\pi$
$420$	0	0
$421$	3.60629e6	0.991644	0.495822	−	0.868424i	$-0.334867\pi$
$421$	3.60629e6	0.991644	0.495822	+	0.868424i	$0.334867\pi$
$422$	0	0
$423$	−433512.	−0.117801
$424$	0	0
$425$	−3.87432e6	−1.04045
$426$	0	0
$427$	−1.80134e6	−0.478107
$428$	0	0
$429$	1.90037e6	0.498534
$430$	0	0
$431$	2.78214e6	0.721416	0.360708	−	0.932679i	$-0.382535\pi$
$431$	2.78214e6	0.721416	0.360708	+	0.932679i	$0.382535\pi$
$432$	0	0
$433$	6.27619e6	1.60871	0.804353	−	0.594152i	$-0.202512\pi$
$433$	6.27619e6	1.60871	0.804353	+	0.594152i	$0.202512\pi$
$434$	0	0
$435$	1.15924e6	0.293730
$436$	0	0
$437$	522032.	0.130766
$438$	0	0
$439$	−641592.	−0.158890	−0.0794452	−	0.996839i	$-0.525315\pi$
$439$	−641592.	−0.158890	−0.0794452	+	0.996839i	$0.525315\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	−6.05546e6	−1.46601	−0.733006	−	0.680222i	$-0.761883\pi$
$443$	−6.05546e6	−1.46601	−0.733006	+	0.680222i	$0.761883\pi$
$444$	0	0
$445$	−3.18911e6	−0.763430
$446$	0	0
$447$	1.57372e6	0.372528
$448$	0	0
$449$	−5.16681e6	−1.20950	−0.604752	−	0.796414i	$-0.706728\pi$
$449$	−5.16681e6	−1.20950	−0.604752	+	0.796414i	$0.706728\pi$
$450$	0	0
$451$	−6.99989e6	−1.62050
$452$	0	0
$453$	−4.07297e6	−0.932536
$454$	0	0
$455$	405132.	0.0917420
$456$	0	0
$457$	−227798.	−0.0510222	−0.0255111	−	0.999675i	$-0.508121\pi$
$457$	−227798.	−0.0510222	−0.0255111	+	0.999675i	$0.508121\pi$
$458$	0	0
$459$	−1.15328e6	−0.255507
$460$	0	0
$461$	585146.	0.128237	0.0641183	−	0.997942i	$-0.479577\pi$
$461$	585146.	0.128237	0.0641183	+	0.997942i	$0.479577\pi$
$462$	0	0
$463$	3.41454e6	0.740251	0.370126	−	0.928982i	$-0.379315\pi$
$463$	3.41454e6	0.740251	0.370126	+	0.928982i	$0.379315\pi$
$464$	0	0
$465$	−1.66795e6	−0.357727
$466$	0	0
$467$	−716300.	−0.151986	−0.0759929	−	0.997108i	$-0.524213\pi$
$467$	−716300.	−0.151986	−0.0759929	+	0.997108i	$0.524213\pi$
$468$	0	0
$469$	−2.00763e6	−0.421455
$470$	0	0
$471$	4.49159e6	0.932928
$472$	0	0
$473$	−5.61213e6	−1.15339
$474$	0	0
$475$	577964.	0.117535
$476$	0	0
$477$	−2.70167e6	−0.543672
$478$	0	0
$479$	−5.24092e6	−1.04368	−0.521842	−	0.853042i	$-0.674755\pi$
$479$	−5.24092e6	−1.04368	−0.521842	+	0.853042i	$0.674755\pi$
$480$	0	0
$481$	1.38584e6	0.273119
$482$	0	0
$483$	975492.	0.190264
$484$	0	0
$485$	555620.	0.107256
$486$	0	0
$487$	−1.11702e6	−0.213421	−0.106710	−	0.994290i	$-0.534032\pi$
$487$	−1.11702e6	−0.213421	−0.106710	+	0.994290i	$0.534032\pi$
$488$	0	0
$489$	−4.28029e6	−0.809471
$490$	0	0
$491$	−1.34458e6	−0.251699	−0.125850	−	0.992049i	$-0.540166\pi$
$491$	−1.34458e6	−0.251699	−0.125850	+	0.992049i	$0.540166\pi$
$492$	0	0
$493$	−7.83723e6	−1.45226
$494$	0	0
$495$	−1.39838e6	−0.256515
$496$	0	0
$497$	445508.	0.0809030
$498$	0	0
$499$	6.54648e6	1.17695	0.588473	−	0.808517i	$-0.299729\pi$
$499$	6.54648e6	1.17695	0.588473	+	0.808517i	$0.299729\pi$
$500$	0	0
$501$	1.08202e6	0.192592
$502$	0	0
$503$	8.22050e6	1.44870	0.724350	−	0.689432i	$-0.242140\pi$
$503$	8.22050e6	1.44870	0.724350	+	0.689432i	$0.242140\pi$
$504$	0	0
$505$	−957164.	−0.167016
$506$	0	0
$507$	2.43152e6	0.420105
$508$	0	0
$509$	−5.11045e6	−0.874308	−0.437154	−	0.899387i	$-0.644013\pi$
$509$	−5.11045e6	−0.874308	−0.437154	+	0.899387i	$0.644013\pi$
$510$	0	0
$511$	−3.59925e6	−0.609760
$512$	0	0
$513$	172044.	0.0288633
$514$	0	0
$515$	−2.71773e6	−0.451531
$516$	0	0
$517$	3.55373e6	0.584733
$518$	0	0
$519$	−4.57987e6	−0.746336
$520$	0	0
$521$	9.69999e6	1.56559	0.782793	−	0.622282i	$-0.213794\pi$
$521$	9.69999e6	1.56559	0.782793	+	0.622282i	$0.213794\pi$
$522$	0	0
$523$	3.17295e6	0.507234	0.253617	−	0.967305i	$-0.418380\pi$
$523$	3.17295e6	0.507234	0.253617	+	0.967305i	$0.418380\pi$
$524$	0	0
$525$	1.08001e6	0.171013
$526$	0	0
$527$	1.12765e7	1.76867
$528$	0	0
$529$	−1.54340e6	−0.239794
$530$	0	0
$531$	1.25032e6	0.192435
$532$	0	0
$533$	3.35236e6	0.511131
$534$	0	0
$535$	−5.57544e6	−0.842160
$536$	0	0
$537$	4.38804e6	0.656651
$538$	0	0
$539$	−1.59426e6	−0.236368
$540$	0	0
$541$	−6.62575e6	−0.973289	−0.486644	−	0.873600i	$-0.661779\pi$
$541$	−6.62575e6	−0.973289	−0.486644	+	0.873600i	$0.661779\pi$
$542$	0	0
$543$	4.89969e6	0.713131
$544$	0	0
$545$	748748.	0.107980
$546$	0	0
$547$	−3.84707e6	−0.549745	−0.274873	−	0.961481i	$-0.588636\pi$
$547$	−3.84707e6	−0.549745	−0.274873	+	0.961481i	$0.588636\pi$
$548$	0	0
$549$	−2.97772e6	−0.421651
$550$	0	0
$551$	1.16914e6	0.164055
$552$	0	0
$553$	−4.38060e6	−0.609145
$554$	0	0
$555$	−1.01977e6	−0.140531
$556$	0	0
$557$	5.00176e6	0.683101	0.341550	−	0.939863i	$-0.389048\pi$
$557$	5.00176e6	0.683101	0.341550	+	0.939863i	$0.389048\pi$
$558$	0	0
$559$	2.68774e6	0.363795
$560$	0	0
$561$	9.45403e6	1.26826
$562$	0	0
$563$	−2.27772e6	−0.302852	−0.151426	−	0.988469i	$-0.548386\pi$
$563$	−2.27772e6	−0.302852	−0.151426	+	0.988469i	$0.548386\pi$
$564$	0	0
$565$	−1.45636e6	−0.191933
$566$	0	0
$567$	321489.	0.0419961
$568$	0	0
$569$	8.86979e6	1.14850	0.574252	−	0.818678i	$-0.305293\pi$
$569$	8.86979e6	1.14850	0.574252	+	0.818678i	$0.305293\pi$
$570$	0	0
$571$	−1.40102e7	−1.79826	−0.899132	−	0.437678i	$-0.855801\pi$
$571$	−1.40102e7	−1.79826	−0.899132	+	0.437678i	$0.855801\pi$
$572$	0	0
$573$	3.38764e6	0.431033
$574$	0	0
$575$	5.41719e6	0.683289
$576$	0	0
$577$	8.75327e6	1.09454	0.547269	−	0.836957i	$-0.315668\pi$
$577$	8.75327e6	1.09454	0.547269	+	0.836957i	$0.315668\pi$
$578$	0	0
$579$	−7.60451e6	−0.942703
$580$	0	0
$581$	314972.	0.0387108
$582$	0	0
$583$	2.21471e7	2.69864
$584$	0	0
$585$	669708.	0.0809088
$586$	0	0
$587$	1.06117e7	1.27113	0.635564	−	0.772048i	$-0.280768\pi$
$587$	1.06117e7	1.27113	0.635564	+	0.772048i	$0.280768\pi$
$588$	0	0
$589$	−1.68221e6	−0.199798
$590$	0	0
$591$	4.43515e6	0.522323
$592$	0	0
$593$	1.88552e6	0.220188	0.110094	−	0.993921i	$-0.464885\pi$
$593$	1.88552e6	0.220188	0.110094	+	0.993921i	$0.464885\pi$
$594$	0	0
$595$	2.01547e6	0.233391
$596$	0	0
$597$	−8.23298e6	−0.945413
$598$	0	0
$599$	−1.27256e7	−1.44915	−0.724573	−	0.689198i	$-0.757963\pi$
$599$	−1.27256e7	−1.44915	−0.724573	+	0.689198i	$0.757963\pi$
$600$	0	0
$601$	7.18846e6	0.811801	0.405900	−	0.913917i	$-0.366958\pi$
$601$	7.18846e6	0.811801	0.405900	+	0.913917i	$0.366958\pi$
$602$	0	0
$603$	−3.31873e6	−0.371688
$604$	0	0
$605$	7.27597e6	0.808170
$606$	0	0
$607$	−1.08494e7	−1.19519	−0.597593	−	0.801800i	$-0.703876\pi$
$607$	−1.08494e7	−1.19519	−0.597593	+	0.801800i	$0.703876\pi$
$608$	0	0
$609$	2.18471e6	0.238699
$610$	0	0
$611$	−1.70194e6	−0.184434
$612$	0	0
$613$	−4.90511e6	−0.527227	−0.263614	−	0.964628i	$-0.584914\pi$
$613$	−4.90511e6	−0.527227	−0.263614	+	0.964628i	$0.584914\pi$
$614$	0	0
$615$	−2.46683e6	−0.262997
$616$	0	0
$617$	2.58445e6	0.273310	0.136655	−	0.990619i	$-0.456365\pi$
$617$	2.58445e6	0.273310	0.136655	+	0.990619i	$0.456365\pi$
$618$	0	0
$619$	4.99336e6	0.523801	0.261901	−	0.965095i	$-0.415651\pi$
$619$	4.99336e6	0.523801	0.261901	+	0.965095i	$0.415651\pi$
$620$	0	0
$621$	1.61255e6	0.167797
$622$	0	0
$623$	−6.01024e6	−0.620400
$624$	0	0
$625$	3.88510e6	0.397834
$626$	0	0
$627$	−1.41034e6	−0.143269
$628$	0	0
$629$	6.89436e6	0.694812
$630$	0	0
$631$	1.18219e7	1.18199	0.590997	−	0.806674i	$-0.298735\pi$
$631$	1.18219e7	1.18199	0.590997	+	0.806674i	$0.298735\pi$
$632$	0	0
$633$	2.80602e6	0.278344
$634$	0	0
$635$	−4.82040e6	−0.474404
$636$	0	0
$637$	763518.	0.0745540
$638$	0	0
$639$	736452.	0.0713497
$640$	0	0
$641$	−5.47007e6	−0.525833	−0.262916	−	0.964819i	$-0.584684\pi$
$641$	−5.47007e6	−0.525833	−0.262916	+	0.964819i	$0.584684\pi$
$642$	0	0
$643$	−9.64934e6	−0.920386	−0.460193	−	0.887819i	$-0.652220\pi$
$643$	−9.64934e6	−0.920386	−0.460193	+	0.887819i	$0.652220\pi$
$644$	0	0
$645$	−1.97777e6	−0.187187
$646$	0	0
$647$	−292368.	−0.0274580	−0.0137290	−	0.999906i	$-0.504370\pi$
$647$	−292368.	−0.0274580	−0.0137290	+	0.999906i	$0.504370\pi$
$648$	0	0
$649$	−1.02495e7	−0.955193
$650$	0	0
$651$	−3.14345e6	−0.290706
$652$	0	0
$653$	6.94081e6	0.636982	0.318491	−	0.947926i	$-0.396824\pi$
$653$	6.94081e6	0.636982	0.318491	+	0.947926i	$0.396824\pi$
$654$	0	0
$655$	−1.67783e6	−0.152808
$656$	0	0
$657$	−5.94977e6	−0.537758
$658$	0	0
$659$	1.32912e7	1.19221	0.596104	−	0.802908i	$-0.296715\pi$
$659$	1.32912e7	1.19221	0.596104	+	0.802908i	$0.296715\pi$
$660$	0	0
$661$	2.05219e6	0.182690	0.0913448	−	0.995819i	$-0.470883\pi$
$661$	2.05219e6	0.182690	0.0913448	+	0.995819i	$0.470883\pi$
$662$	0	0
$663$	−4.52768e6	−0.400030
$664$	0	0
$665$	−300664.	−0.0263650
$666$	0	0
$667$	1.09582e7	0.953732
$668$	0	0
$669$	−1.15898e7	−1.00118
$670$	0	0
$671$	2.44100e7	2.09296
$672$	0	0
$673$	−1.57039e7	−1.33650	−0.668252	−	0.743935i	$-0.732957\pi$
$673$	−1.57039e7	−1.33650	−0.668252	+	0.743935i	$0.732957\pi$
$674$	0	0
$675$	1.78532e6	0.150819
$676$	0	0
$677$	−969534.	−0.0813002	−0.0406501	−	0.999173i	$-0.512943\pi$
$677$	−969534.	−0.0813002	−0.0406501	+	0.999173i	$0.512943\pi$
$678$	0	0
$679$	1.04713e6	0.0871618
$680$	0	0
$681$	1.16015e7	0.958617
$682$	0	0
$683$	1.49908e7	1.22962	0.614812	−	0.788673i	$-0.289232\pi$
$683$	1.49908e7	1.22962	0.614812	+	0.788673i	$0.289232\pi$
$684$	0	0
$685$	3.97618e6	0.323772
$686$	0	0
$687$	−6.10393e6	−0.493421
$688$	0	0
$689$	−1.06066e7	−0.851191
$690$	0	0
$691$	7.16038e6	0.570481	0.285240	−	0.958456i	$-0.407927\pi$
$691$	7.16038e6	0.570481	0.285240	+	0.958456i	$0.407927\pi$
$692$	0	0
$693$	−2.63542e6	−0.208457
$694$	0	0
$695$	8.92996e6	0.701274
$696$	0	0
$697$	1.66774e7	1.30031
$698$	0	0
$699$	1.00558e7	0.778437
$700$	0	0
$701$	−91834.0	−0.00705844	−0.00352922	−	0.999994i	$-0.501123\pi$
$701$	−91834.0	−0.00705844	−0.00352922	+	0.999994i	$0.501123\pi$
$702$	0	0
$703$	−1.02849e6	−0.0784894
$704$	0	0
$705$	1.25237e6	0.0948985
$706$	0	0
$707$	−1.80389e6	−0.135725
$708$	0	0
$709$	2.20981e7	1.65097	0.825487	−	0.564422i	$-0.190901\pi$
$709$	2.20981e7	1.65097	0.825487	+	0.564422i	$0.190901\pi$
$710$	0	0
$711$	−7.24140e6	−0.537216
$712$	0	0
$713$	−1.57671e7	−1.16153
$714$	0	0
$715$	−5.48995e6	−0.401609
$716$	0	0
$717$	−1.13576e7	−0.825066
$718$	0	0
$719$	−1.58388e7	−1.14262	−0.571308	−	0.820736i	$-0.693564\pi$
$719$	−1.58388e7	−1.14262	−0.571308	+	0.820736i	$0.693564\pi$
$720$	0	0
$721$	−5.12187e6	−0.366936
$722$	0	0
$723$	−8.53396e6	−0.607163
$724$	0	0
$725$	1.21323e7	0.857235
$726$	0	0
$727$	−6.31418e6	−0.443078	−0.221539	−	0.975151i	$-0.571108\pi$
$727$	−6.31418e6	−0.443078	−0.221539	+	0.975151i	$0.571108\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.33711e7	0.925492
$732$	0	0
$733$	6.93003e6	0.476404	0.238202	−	0.971216i	$-0.423442\pi$
$733$	6.93003e6	0.476404	0.238202	+	0.971216i	$0.423442\pi$
$734$	0	0
$735$	−561834.	−0.0383610
$736$	0	0
$737$	2.72054e7	1.84496
$738$	0	0
$739$	−1.42331e7	−0.958714	−0.479357	−	0.877620i	$-0.659130\pi$
$739$	−1.42331e7	−0.958714	−0.479357	+	0.877620i	$0.659130\pi$
$740$	0	0
$741$	675432.	0.0451894
$742$	0	0
$743$	5.94460e6	0.395048	0.197524	−	0.980298i	$-0.436710\pi$
$743$	5.94460e6	0.395048	0.197524	+	0.980298i	$0.436710\pi$
$744$	0	0
$745$	−4.54631e6	−0.300102
$746$	0	0
$747$	520668.	0.0341397
$748$	0	0
$749$	−1.05076e7	−0.684380
$750$	0	0
$751$	682752.	0.0441736	0.0220868	−	0.999756i	$-0.492969\pi$
$751$	682752.	0.0441736	0.0220868	+	0.999756i	$0.492969\pi$
$752$	0	0
$753$	−4.37756e6	−0.281349
$754$	0	0
$755$	1.17664e7	0.751233
$756$	0	0
$757$	1.46333e7	0.928116	0.464058	−	0.885805i	$-0.346393\pi$
$757$	1.46333e7	0.928116	0.464058	+	0.885805i	$0.346393\pi$
$758$	0	0
$759$	−1.32189e7	−0.832897
$760$	0	0
$761$	−1.16367e7	−0.728399	−0.364200	−	0.931321i	$-0.618657\pi$
$761$	−1.16367e7	−0.728399	−0.364200	+	0.931321i	$0.618657\pi$
$762$	0	0
$763$	1.41110e6	0.0877500
$764$	0	0
$765$	3.33169e6	0.205831
$766$	0	0
$767$	4.90865e6	0.301282
$768$	0	0
$769$	1.91472e7	1.16759	0.583793	−	0.811902i	$-0.301568\pi$
$769$	1.91472e7	1.16759	0.583793	+	0.811902i	$0.301568\pi$
$770$	0	0
$771$	9.35188e6	0.566582
$772$	0	0
$773$	−5.39261e6	−0.324601	−0.162301	−	0.986741i	$-0.551891\pi$
$773$	−5.39261e6	−0.324601	−0.162301	+	0.986741i	$0.551891\pi$
$774$	0	0
$775$	−1.74565e7	−1.04400
$776$	0	0
$777$	−1.92188e6	−0.114202
$778$	0	0
$779$	−2.48791e6	−0.146890
$780$	0	0
$781$	−6.03709e6	−0.354160
$782$	0	0
$783$	3.61147e6	0.210513
$784$	0	0
$785$	−1.29757e7	−0.751549
$786$	0	0
$787$	−3.04348e6	−0.175159	−0.0875796	−	0.996158i	$-0.527913\pi$
$787$	−3.04348e6	−0.175159	−0.0875796	+	0.996158i	$0.527913\pi$
$788$	0	0
$789$	1.21594e7	0.695376
$790$	0	0
$791$	−2.74469e6	−0.155974
$792$	0	0
$793$	−1.16903e7	−0.660151
$794$	0	0
$795$	7.80484e6	0.437972
$796$	0	0
$797$	2.29652e7	1.28063	0.640316	−	0.768111i	$-0.278803\pi$
$797$	2.29652e7	1.28063	0.640316	+	0.768111i	$0.278803\pi$
$798$	0	0
$799$	−8.46686e6	−0.469197
$800$	0	0
$801$	−9.93530e6	−0.547141
$802$	0	0
$803$	4.87735e7	2.66928
$804$	0	0
$805$	−2.81809e6	−0.153273
$806$	0	0
$807$	1.00630e7	0.543930
$808$	0	0
$809$	1.90787e7	1.02489	0.512445	−	0.858720i	$-0.328740\pi$
$809$	1.90787e7	1.02489	0.512445	+	0.858720i	$0.328740\pi$
$810$	0	0
$811$	−1.09414e7	−0.584147	−0.292074	−	0.956396i	$-0.594345\pi$
$811$	−1.09414e7	−0.584147	−0.292074	+	0.956396i	$0.594345\pi$
$812$	0	0
$813$	−1.71094e6	−0.0907836
$814$	0	0
$815$	1.23653e7	0.652094
$816$	0	0
$817$	−1.99467e6	−0.104548
$818$	0	0
$819$	1.26214e6	0.0657504
$820$	0	0
$821$	2.12594e7	1.10076	0.550380	−	0.834914i	$-0.314483\pi$
$821$	2.12594e7	1.10076	0.550380	+	0.834914i	$0.314483\pi$
$822$	0	0
$823$	1.42256e7	0.732103	0.366052	−	0.930595i	$-0.380709\pi$
$823$	1.42256e7	0.732103	0.366052	+	0.930595i	$0.380709\pi$
$824$	0	0
$825$	−1.46352e7	−0.748625
$826$	0	0
$827$	−2.76103e6	−0.140381	−0.0701904	−	0.997534i	$-0.522361\pi$
$827$	−2.76103e6	−0.140381	−0.0701904	+	0.997534i	$0.522361\pi$
$828$	0	0
$829$	−3.82147e7	−1.93127	−0.965637	−	0.259895i	$-0.916312\pi$
$829$	−3.82147e7	−1.93127	−0.965637	+	0.259895i	$0.916312\pi$
$830$	0	0
$831$	1.80455e6	0.0906499
$832$	0	0
$833$	3.79838e6	0.189665
$834$	0	0
$835$	−3.12582e6	−0.155149
$836$	0	0
$837$	−5.19631e6	−0.256378
$838$	0	0
$839$	−1.06044e7	−0.520094	−0.260047	−	0.965596i	$-0.583738\pi$
$839$	−1.06044e7	−0.520094	−0.260047	+	0.965596i	$0.583738\pi$
$840$	0	0
$841$	4.03097e6	0.196526
$842$	0	0
$843$	−9.83133e6	−0.476479
$844$	0	0
$845$	−7.02439e6	−0.338429
$846$	0	0
$847$	1.37124e7	0.656758
$848$	0	0
$849$	1.63132e7	0.776731
$850$	0	0
$851$	−9.63990e6	−0.456298
$852$	0	0
$853$	−4.07009e7	−1.91527	−0.957637	−	0.287977i	$-0.907017\pi$
$853$	−4.07009e7	−1.91527	−0.957637	+	0.287977i	$0.907017\pi$
$854$	0	0
$855$	−497016.	−0.0232517
$856$	0	0
$857$	−3.10120e7	−1.44237	−0.721187	−	0.692741i	$-0.756403\pi$
$857$	−3.10120e7	−1.44237	−0.721187	+	0.692741i	$0.756403\pi$
$858$	0	0
$859$	−1.09104e7	−0.504495	−0.252247	−	0.967663i	$-0.581170\pi$
$859$	−1.09104e7	−0.504495	−0.252247	+	0.967663i	$0.581170\pi$
$860$	0	0
$861$	−4.64902e6	−0.213724
$862$	0	0
$863$	−1.04089e7	−0.475751	−0.237875	−	0.971296i	$-0.576451\pi$
$863$	−1.04089e7	−0.475751	−0.237875	+	0.971296i	$0.576451\pi$
$864$	0	0
$865$	1.32307e7	0.601234
$866$	0	0
$867$	−9.74580e6	−0.440321
$868$	0	0
$869$	5.93616e7	2.66659
$870$	0	0
$871$	−1.30291e7	−0.581928
$872$	0	0
$873$	1.73097e6	0.0768695
$874$	0	0
$875$	−7.10128e6	−0.313557
$876$	0	0
$877$	1.64064e7	0.720299	0.360150	−	0.932895i	$-0.382726\pi$
$877$	1.64064e7	0.720299	0.360150	+	0.932895i	$0.382726\pi$
$878$	0	0
$879$	−1.89028e7	−0.825188
$880$	0	0
$881$	1.48577e7	0.644927	0.322464	−	0.946582i	$-0.395489\pi$
$881$	1.48577e7	0.644927	0.322464	+	0.946582i	$0.395489\pi$
$882$	0	0
$883$	2.72018e7	1.17407	0.587037	−	0.809560i	$-0.300294\pi$
$883$	2.72018e7	1.17407	0.587037	+	0.809560i	$0.300294\pi$
$884$	0	0
$885$	−3.61202e6	−0.155022
$886$	0	0
$887$	−2.71242e7	−1.15757	−0.578785	−	0.815480i	$-0.696473\pi$
$887$	−2.71242e7	−1.15757	−0.578785	+	0.815480i	$0.696473\pi$
$888$	0	0
$889$	−9.08460e6	−0.385524
$890$	0	0
$891$	−4.35650e6	−0.183842
$892$	0	0
$893$	1.26307e6	0.0530029
$894$	0	0
$895$	−1.26766e7	−0.528986
$896$	0	0
$897$	6.33074e6	0.262708
$898$	0	0
$899$	−3.53121e7	−1.45722
$900$	0	0
$901$	−5.27660e7	−2.16542
$902$	0	0
$903$	−3.72733e6	−0.152117
$904$	0	0
$905$	−1.41547e7	−0.574484
$906$	0	0
$907$	8.42269e6	0.339964	0.169982	−	0.985447i	$-0.445629\pi$
$907$	8.42269e6	0.339964	0.169982	+	0.985447i	$0.445629\pi$
$908$	0	0
$909$	−2.98193e6	−0.119698
$910$	0	0
$911$	−3.08637e7	−1.23212	−0.616060	−	0.787700i	$-0.711272\pi$
$911$	−3.08637e7	−1.23212	−0.616060	+	0.787700i	$0.711272\pi$
$912$	0	0
$913$	−4.26819e6	−0.169460
$914$	0	0
$915$	8.60231e6	0.339674
$916$	0	0
$917$	−3.16207e6	−0.124179
$918$	0	0
$919$	−4.93895e6	−0.192906	−0.0964531	−	0.995338i	$-0.530750\pi$
$919$	−4.93895e6	−0.192906	−0.0964531	+	0.995338i	$0.530750\pi$
$920$	0	0
$921$	−1.47694e7	−0.573738
$922$	0	0
$923$	2.89126e6	0.111707
$924$	0	0
$925$	−1.06727e7	−0.410130
$926$	0	0
$927$	−8.46677e6	−0.323607
$928$	0	0
$929$	5.62575e6	0.213866	0.106933	−	0.994266i	$-0.465897\pi$
$929$	5.62575e6	0.213866	0.106933	+	0.994266i	$0.465897\pi$
$930$	0	0
$931$	−566636.	−0.0214255
$932$	0	0
$933$	−8.50709e6	−0.319946
$934$	0	0
$935$	−2.73116e7	−1.02169
$936$	0	0
$937$	2.60073e7	0.967714	0.483857	−	0.875147i	$-0.339236\pi$
$937$	2.60073e7	0.967714	0.483857	+	0.875147i	$0.339236\pi$
$938$	0	0
$939$	−3.73819e6	−0.138356
$940$	0	0
$941$	3.02160e6	0.111241	0.0556203	−	0.998452i	$-0.482286\pi$
$941$	3.02160e6	0.111241	0.0556203	+	0.998452i	$0.482286\pi$
$942$	0	0
$943$	−2.33189e7	−0.853943
$944$	0	0
$945$	−928746.	−0.0338312
$946$	0	0
$947$	3.48282e7	1.26199	0.630995	−	0.775787i	$-0.282647\pi$
$947$	3.48282e7	1.26199	0.630995	+	0.775787i	$0.282647\pi$
$948$	0	0
$949$	−2.33584e7	−0.841932
$950$	0	0
$951$	−1.06633e7	−0.382333
$952$	0	0
$953$	−9.39009e6	−0.334917	−0.167459	−	0.985879i	$-0.553556\pi$
$953$	−9.39009e6	−0.334917	−0.167459	+	0.985879i	$0.553556\pi$
$954$	0	0
$955$	−9.78650e6	−0.347232
$956$	0	0
$957$	−2.96051e7	−1.04493
$958$	0	0
$959$	7.49357e6	0.263113
$960$	0	0
$961$	2.21792e7	0.774708
$962$	0	0
$963$	−1.73696e7	−0.603566
$964$	0	0
$965$	2.19686e7	0.759423
$966$	0	0
$967$	−1.44768e7	−0.497860	−0.248930	−	0.968521i	$-0.580079\pi$
$967$	−1.44768e7	−0.497860	−0.248930	+	0.968521i	$0.580079\pi$
$968$	0	0
$969$	3.36017e6	0.114961
$970$	0	0
$971$	−9.24976e6	−0.314834	−0.157417	−	0.987532i	$-0.550317\pi$
$971$	−9.24976e6	−0.314834	−0.157417	+	0.987532i	$0.550317\pi$
$972$	0	0
$973$	1.68295e7	0.569889
$974$	0	0
$975$	7.00904e6	0.236128
$976$	0	0
$977$	−4.97780e7	−1.66840	−0.834202	−	0.551459i	$-0.814071\pi$
$977$	−4.97780e7	−1.66840	−0.834202	+	0.551459i	$0.814071\pi$
$978$	0	0
$979$	8.14449e7	2.71586
$980$	0	0
$981$	2.33264e6	0.0773882
$982$	0	0
$983$	8.95601e6	0.295618	0.147809	−	0.989016i	$-0.452778\pi$
$983$	8.95601e6	0.295618	0.147809	+	0.989016i	$0.452778\pi$
$984$	0	0
$985$	−1.28126e7	−0.420773
$986$	0	0
$987$	2.36023e6	0.0771191
$988$	0	0
$989$	−1.86958e7	−0.607790
$990$	0	0
$991$	−2.62400e7	−0.848751	−0.424376	−	0.905486i	$-0.639506\pi$
$991$	−2.62400e7	−0.848751	−0.424376	+	0.905486i	$0.639506\pi$
$992$	0	0
$993$	1.23439e7	0.397265
$994$	0	0
$995$	2.37842e7	0.761606
$996$	0	0
$997$	2.80506e7	0.893727	0.446863	−	0.894602i	$-0.352541\pi$
$997$	2.80506e7	0.893727	0.446863	+	0.894602i	$0.352541\pi$
$998$	0	0
$999$	−3.17698e6	−0.100717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.a.h.1.1		1
3.2	odd	2		1008.6.a.j.1.1		1
4.3	odd	2		42.6.a.d.1.1	✓	1
12.11	even	2		126.6.a.i.1.1		1
20.3	even	4		1050.6.g.i.799.2		2
20.7	even	4		1050.6.g.i.799.1		2
20.19	odd	2		1050.6.a.k.1.1		1
28.3	even	6		294.6.e.p.79.1		2
28.11	odd	6		294.6.e.i.79.1		2
28.19	even	6		294.6.e.p.67.1		2
28.23	odd	6		294.6.e.i.67.1		2
28.27	even	2		294.6.a.b.1.1		1
84.83	odd	2		882.6.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.d.1.1	✓	1	4.3	odd	2
126.6.a.i.1.1		1	12.11	even	2
294.6.a.b.1.1		1	28.27	even	2
294.6.e.i.67.1		2	28.23	odd	6
294.6.e.i.79.1		2	28.11	odd	6
294.6.e.p.67.1		2	28.19	even	6
294.6.e.p.79.1		2	28.3	even	6
336.6.a.h.1.1		1	1.1	even	1	trivial
882.6.a.s.1.1		1	84.83	odd	2
1008.6.a.j.1.1		1	3.2	odd	2
1050.6.a.k.1.1		1	20.19	odd	2
1050.6.g.i.799.1		2	20.7	even	4
1050.6.g.i.799.2		2	20.3	even	4

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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