LMFDB - Embedded newform 336.6.a.i.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)

$f(q)$

$=$

$q-9.00000 q^{3} +94.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q-9.00000 q^{3} +94.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} -52.0000 q^{11} -770.000 q^{13} -846.000 q^{15} -2022.00 q^{17} -1732.00 q^{19} -441.000 q^{21} +576.000 q^{23} +5711.00 q^{25} -729.000 q^{27} +5518.00 q^{29} -6336.00 q^{31} +468.000 q^{33} +4606.00 q^{35} -7338.00 q^{37} +6930.00 q^{39} -3262.00 q^{41} -5420.00 q^{43} +7614.00 q^{45} -864.000 q^{47} +2401.00 q^{49} +18198.0 q^{51} +4182.00 q^{53} -4888.00 q^{55} +15588.0 q^{57} +11220.0 q^{59} -45602.0 q^{61} +3969.00 q^{63} -72380.0 q^{65} -1396.00 q^{67} -5184.00 q^{69} -18720.0 q^{71} +46362.0 q^{73} -51399.0 q^{75} -2548.00 q^{77} -97424.0 q^{79} +6561.00 q^{81} +81228.0 q^{83} -190068. q^{85} -49662.0 q^{87} -3182.00 q^{89} -37730.0 q^{91} +57024.0 q^{93} -162808. q^{95} +4914.00 q^{97} -4212.00 q^{99} +O(q^{100})$

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
$5$	94.0000	1.68152	0.840762	−	0.541406i	$-0.182108\pi$
$5$	94.0000	1.68152	0.840762	+	0.541406i	$0.182108\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−52.0000	−0.129575	−0.0647876	−	0.997899i	$-0.520637\pi$
$11$	−52.0000	−0.129575	−0.0647876	+	0.997899i	$0.520637\pi$
$12$	0	0
$13$	−770.000	−1.26367	−0.631833	−	0.775104i	$-0.717697\pi$
$13$	−770.000	−1.26367	−0.631833	+	0.775104i	$0.717697\pi$
$14$	0	0
$15$	−846.000	−0.970828
$16$	0	0
$17$	−2022.00	−1.69691	−0.848455	−	0.529267i	$-0.822467\pi$
$17$	−2022.00	−1.69691	−0.848455	+	0.529267i	$0.822467\pi$
$18$	0	0
$19$	−1732.00	−1.10069	−0.550344	−	0.834938i	$-0.685503\pi$
$19$	−1732.00	−1.10069	−0.550344	+	0.834938i	$0.685503\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	0	0
$23$	576.000	0.227040	0.113520	−	0.993536i	$-0.463787\pi$
$23$	576.000	0.227040	0.113520	+	0.993536i	$0.463787\pi$
$24$	0	0
$25$	5711.00	1.82752
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	5518.00	1.21839	0.609196	−	0.793020i	$-0.291492\pi$
$29$	5518.00	1.21839	0.609196	+	0.793020i	$0.291492\pi$
$30$	0	0
$31$	−6336.00	−1.18416	−0.592081	−	0.805879i	$-0.701693\pi$
$31$	−6336.00	−1.18416	−0.592081	+	0.805879i	$0.701693\pi$
$32$	0	0
$33$	468.000	0.0748102
$34$	0	0
$35$	4606.00	0.635556
$36$	0	0
$37$	−7338.00	−0.881198	−0.440599	−	0.897704i	$-0.645234\pi$
$37$	−7338.00	−0.881198	−0.440599	+	0.897704i	$0.645234\pi$
$38$	0	0
$39$	6930.00	0.729578
$40$	0	0
$41$	−3262.00	−0.303057	−0.151528	−	0.988453i	$-0.548420\pi$
$41$	−3262.00	−0.303057	−0.151528	+	0.988453i	$0.548420\pi$
$42$	0	0
$43$	−5420.00	−0.447021	−0.223511	−	0.974701i	$-0.571752\pi$
$43$	−5420.00	−0.447021	−0.223511	+	0.974701i	$0.571752\pi$
$44$	0	0
$45$	7614.00	0.560508
$46$	0	0
$47$	−864.000	−0.0570518	−0.0285259	−	0.999593i	$-0.509081\pi$
$47$	−864.000	−0.0570518	−0.0285259	+	0.999593i	$0.509081\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	18198.0	0.979712
$52$	0	0
$53$	4182.00	0.204500	0.102250	−	0.994759i	$-0.467396\pi$
$53$	4182.00	0.204500	0.102250	+	0.994759i	$0.467396\pi$
$54$	0	0
$55$	−4888.00	−0.217884
$56$	0	0
$57$	15588.0	0.635482
$58$	0	0
$59$	11220.0	0.419626	0.209813	−	0.977741i	$-0.432714\pi$
$59$	11220.0	0.419626	0.209813	+	0.977741i	$0.432714\pi$
$60$	0	0
$61$	−45602.0	−1.56913	−0.784566	−	0.620046i	$-0.787114\pi$
$61$	−45602.0	−1.56913	−0.784566	+	0.620046i	$0.787114\pi$
$62$	0	0
$63$	3969.00	0.125988
$64$	0	0
$65$	−72380.0	−2.12488
$66$	0	0
$67$	−1396.00	−0.0379925	−0.0189963	−	0.999820i	$-0.506047\pi$
$67$	−1396.00	−0.0379925	−0.0189963	+	0.999820i	$0.506047\pi$
$68$	0	0
$69$	−5184.00	−0.131082
$70$	0	0
$71$	−18720.0	−0.440717	−0.220359	−	0.975419i	$-0.570723\pi$
$71$	−18720.0	−0.440717	−0.220359	+	0.975419i	$0.570723\pi$
$72$	0	0
$73$	46362.0	1.01825	0.509126	−	0.860692i	$-0.329969\pi$
$73$	46362.0	1.01825	0.509126	+	0.860692i	$0.329969\pi$
$74$	0	0
$75$	−51399.0	−1.05512
$76$	0	0
$77$	−2548.00	−0.0489748
$78$	0	0
$79$	−97424.0	−1.75630	−0.878149	−	0.478387i	$-0.841222\pi$
$79$	−97424.0	−1.75630	−0.878149	+	0.478387i	$0.841222\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	81228.0	1.29423	0.647114	−	0.762394i	$-0.275976\pi$
$83$	81228.0	1.29423	0.647114	+	0.762394i	$0.275976\pi$
$84$	0	0
$85$	−190068.	−2.85339
$86$	0	0
$87$	−49662.0	−0.703438
$88$	0	0
$89$	−3182.00	−0.0425819	−0.0212910	−	0.999773i	$-0.506778\pi$
$89$	−3182.00	−0.0425819	−0.0212910	+	0.999773i	$0.506778\pi$
$90$	0	0
$91$	−37730.0	−0.477621
$92$	0	0
$93$	57024.0	0.683676
$94$	0	0
$95$	−162808.	−1.85083
$96$	0	0
$97$	4914.00	0.0530281	0.0265140	−	0.999648i	$-0.491559\pi$
$97$	4914.00	0.0530281	0.0265140	+	0.999648i	$0.491559\pi$
$98$	0	0
$99$	−4212.00	−0.0431917
$100$	0	0
$101$	−166354.	−1.62267	−0.811334	−	0.584583i	$-0.801258\pi$
$101$	−166354.	−1.62267	−0.811334	+	0.584583i	$0.801258\pi$
$102$	0	0
$103$	−157160.	−1.45965	−0.729825	−	0.683634i	$-0.760399\pi$
$103$	−157160.	−1.45965	−0.729825	+	0.683634i	$0.760399\pi$
$104$	0	0
$105$	−41454.0	−0.366938
$106$	0	0
$107$	6764.00	0.0571142	0.0285571	−	0.999592i	$-0.490909\pi$
$107$	6764.00	0.0571142	0.0285571	+	0.999592i	$0.490909\pi$
$108$	0	0
$109$	178398.	1.43821	0.719107	−	0.694899i	$-0.244551\pi$
$109$	178398.	1.43821	0.719107	+	0.694899i	$0.244551\pi$
$110$	0	0
$111$	66042.0	0.508760
$112$	0	0
$113$	−45134.0	−0.332512	−0.166256	−	0.986083i	$-0.553168\pi$
$113$	−45134.0	−0.332512	−0.166256	+	0.986083i	$0.553168\pi$
$114$	0	0
$115$	54144.0	0.381773
$116$	0	0
$117$	−62370.0	−0.421222
$118$	0	0
$119$	−99078.0	−0.641372
$120$	0	0
$121$	−158347.	−0.983210
$122$	0	0
$123$	29358.0	0.174970
$124$	0	0
$125$	243084.	1.39149
$126$	0	0
$127$	205056.	1.12814	0.564070	−	0.825727i	$-0.309235\pi$
$127$	205056.	1.12814	0.564070	+	0.825727i	$0.309235\pi$
$128$	0	0
$129$	48780.0	0.258088
$130$	0	0
$131$	−72964.0	−0.371476	−0.185738	−	0.982599i	$-0.559468\pi$
$131$	−72964.0	−0.371476	−0.185738	+	0.982599i	$0.559468\pi$
$132$	0	0
$133$	−84868.0	−0.416021
$134$	0	0
$135$	−68526.0	−0.323609
$136$	0	0
$137$	−94182.0	−0.428713	−0.214356	−	0.976756i	$-0.568765\pi$
$137$	−94182.0	−0.428713	−0.214356	+	0.976756i	$0.568765\pi$
$138$	0	0
$139$	47796.0	0.209824	0.104912	−	0.994482i	$-0.466544\pi$
$139$	47796.0	0.209824	0.104912	+	0.994482i	$0.466544\pi$
$140$	0	0
$141$	7776.00	0.0329389
$142$	0	0
$143$	40040.0	0.163740
$144$	0	0
$145$	518692.	2.04875
$146$	0	0
$147$	−21609.0	−0.0824786
$148$	0	0
$149$	−124266.	−0.458550	−0.229275	−	0.973362i	$-0.573636\pi$
$149$	−124266.	−0.458550	−0.229275	+	0.973362i	$0.573636\pi$
$150$	0	0
$151$	446296.	1.59287	0.796436	−	0.604723i	$-0.206716\pi$
$151$	446296.	1.59287	0.796436	+	0.604723i	$0.206716\pi$
$152$	0	0
$153$	−163782.	−0.565637
$154$	0	0
$155$	−595584.	−1.99119
$156$	0	0
$157$	−159746.	−0.517227	−0.258613	−	0.965981i	$-0.583266\pi$
$157$	−159746.	−0.517227	−0.258613	+	0.965981i	$0.583266\pi$
$158$	0	0
$159$	−37638.0	−0.118068
$160$	0	0
$161$	28224.0	0.0858132
$162$	0	0
$163$	−247252.	−0.728905	−0.364452	−	0.931222i	$-0.618744\pi$
$163$	−247252.	−0.728905	−0.364452	+	0.931222i	$0.618744\pi$
$164$	0	0
$165$	43992.0	0.125795
$166$	0	0
$167$	684488.	1.89922	0.949609	−	0.313438i	$-0.101481\pi$
$167$	684488.	1.89922	0.949609	+	0.313438i	$0.101481\pi$
$168$	0	0
$169$	221607.	0.596852
$170$	0	0
$171$	−140292.	−0.366896
$172$	0	0
$173$	−610474.	−1.55079	−0.775393	−	0.631479i	$-0.782448\pi$
$173$	−610474.	−1.55079	−0.775393	+	0.631479i	$0.782448\pi$
$174$	0	0
$175$	279839.	0.690738
$176$	0	0
$177$	−100980.	−0.242271
$178$	0	0
$179$	−662252.	−1.54487	−0.772433	−	0.635097i	$-0.780960\pi$
$179$	−662252.	−1.54487	−0.772433	+	0.635097i	$0.780960\pi$
$180$	0	0
$181$	154630.	0.350830	0.175415	−	0.984495i	$-0.443873\pi$
$181$	154630.	0.350830	0.175415	+	0.984495i	$0.443873\pi$
$182$	0	0
$183$	410418.	0.905938
$184$	0	0
$185$	−689772.	−1.48175
$186$	0	0
$187$	105144.	0.219877
$188$	0	0
$189$	−35721.0	−0.0727393
$190$	0	0
$191$	−486904.	−0.965739	−0.482870	−	0.875692i	$-0.660406\pi$
$191$	−486904.	−0.965739	−0.482870	+	0.875692i	$0.660406\pi$
$192$	0	0
$193$	620546.	1.19917	0.599585	−	0.800311i	$-0.295332\pi$
$193$	620546.	1.19917	0.599585	+	0.800311i	$0.295332\pi$
$194$	0	0
$195$	651420.	1.22680
$196$	0	0
$197$	−236570.	−0.434304	−0.217152	−	0.976138i	$-0.569677\pi$
$197$	−236570.	−0.434304	−0.217152	+	0.976138i	$0.569677\pi$
$198$	0	0
$199$	−82104.0	−0.146971	−0.0734855	−	0.997296i	$-0.523412\pi$
$199$	−82104.0	−0.146971	−0.0734855	+	0.997296i	$0.523412\pi$
$200$	0	0
$201$	12564.0	0.0219350
$202$	0	0
$203$	270382.	0.460509
$204$	0	0
$205$	−306628.	−0.509597
$206$	0	0
$207$	46656.0	0.0756801
$208$	0	0
$209$	90064.0	0.142622
$210$	0	0
$211$	−99892.0	−0.154463	−0.0772315	−	0.997013i	$-0.524608\pi$
$211$	−99892.0	−0.154463	−0.0772315	+	0.997013i	$0.524608\pi$
$212$	0	0
$213$	168480.	0.254448
$214$	0	0
$215$	−509480.	−0.751677
$216$	0	0
$217$	−310464.	−0.447571
$218$	0	0
$219$	−417258.	−0.587888
$220$	0	0
$221$	1.55694e6	2.14433
$222$	0	0
$223$	186704.	0.251415	0.125708	−	0.992067i	$-0.459880\pi$
$223$	186704.	0.251415	0.125708	+	0.992067i	$0.459880\pi$
$224$	0	0
$225$	462591.	0.609173
$226$	0	0
$227$	−336372.	−0.433267	−0.216633	−	0.976253i	$-0.569508\pi$
$227$	−336372.	−0.433267	−0.216633	+	0.976253i	$0.569508\pi$
$228$	0	0
$229$	−926314.	−1.16727	−0.583633	−	0.812018i	$-0.698369\pi$
$229$	−926314.	−1.16727	−0.583633	+	0.812018i	$0.698369\pi$
$230$	0	0
$231$	22932.0	0.0282756
$232$	0	0
$233$	1.25711e6	1.51700	0.758499	−	0.651675i	$-0.225933\pi$
$233$	1.25711e6	1.51700	0.758499	+	0.651675i	$0.225933\pi$
$234$	0	0
$235$	−81216.0	−0.0959339
$236$	0	0
$237$	876816.	1.01400
$238$	0	0
$239$	347016.	0.392966	0.196483	−	0.980507i	$-0.437048\pi$
$239$	347016.	0.392966	0.196483	+	0.980507i	$0.437048\pi$
$240$	0	0
$241$	99170.0	0.109986	0.0549930	−	0.998487i	$-0.482486\pi$
$241$	99170.0	0.109986	0.0549930	+	0.998487i	$0.482486\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	225694.	0.240218
$246$	0	0
$247$	1.33364e6	1.39090
$248$	0	0
$249$	−731052.	−0.747222
$250$	0	0
$251$	−344428.	−0.345076	−0.172538	−	0.985003i	$-0.555197\pi$
$251$	−344428.	−0.345076	−0.172538	+	0.985003i	$0.555197\pi$
$252$	0	0
$253$	−29952.0	−0.0294188
$254$	0	0
$255$	1.71061e6	1.64741
$256$	0	0
$257$	295130.	0.278728	0.139364	−	0.990241i	$-0.455494\pi$
$257$	295130.	0.278728	0.139364	+	0.990241i	$0.455494\pi$
$258$	0	0
$259$	−359562.	−0.333061
$260$	0	0
$261$	446958.	0.406130
$262$	0	0
$263$	1.27246e6	1.13437	0.567187	−	0.823589i	$-0.308032\pi$
$263$	1.27246e6	1.13437	0.567187	+	0.823589i	$0.308032\pi$
$264$	0	0
$265$	393108.	0.343872
$266$	0	0
$267$	28638.0	0.0245847
$268$	0	0
$269$	276774.	0.233209	0.116604	−	0.993178i	$-0.462799\pi$
$269$	276774.	0.233209	0.116604	+	0.993178i	$0.462799\pi$
$270$	0	0
$271$	1.28994e6	1.06695	0.533476	−	0.845815i	$-0.320885\pi$
$271$	1.28994e6	1.06695	0.533476	+	0.845815i	$0.320885\pi$
$272$	0	0
$273$	339570.	0.275755
$274$	0	0
$275$	−296972.	−0.236801
$276$	0	0
$277$	1.71655e6	1.34418	0.672089	−	0.740470i	$-0.265397\pi$
$277$	1.71655e6	1.34418	0.672089	+	0.740470i	$0.265397\pi$
$278$	0	0
$279$	−513216.	−0.394720
$280$	0	0
$281$	−1.47218e6	−1.11223	−0.556116	−	0.831104i	$-0.687709\pi$
$281$	−1.47218e6	−1.11223	−0.556116	+	0.831104i	$0.687709\pi$
$282$	0	0
$283$	−1.02881e6	−0.763607	−0.381804	−	0.924244i	$-0.624697\pi$
$283$	−1.02881e6	−0.763607	−0.381804	+	0.924244i	$0.624697\pi$
$284$	0	0
$285$	1.46527e6	1.06858
$286$	0	0
$287$	−159838.	−0.114545
$288$	0	0
$289$	2.66863e6	1.87950
$290$	0	0
$291$	−44226.0	−0.0306158
$292$	0	0
$293$	−1.18607e6	−0.807123	−0.403562	−	0.914952i	$-0.632228\pi$
$293$	−1.18607e6	−0.807123	−0.403562	+	0.914952i	$0.632228\pi$
$294$	0	0
$295$	1.05468e6	0.705612
$296$	0	0
$297$	37908.0	0.0249367
$298$	0	0
$299$	−443520.	−0.286903
$300$	0	0
$301$	−265580.	−0.168958
$302$	0	0
$303$	1.49719e6	0.936848
$304$	0	0
$305$	−4.28659e6	−2.63853
$306$	0	0
$307$	−1.51892e6	−0.919788	−0.459894	−	0.887974i	$-0.652113\pi$
$307$	−1.51892e6	−0.919788	−0.459894	+	0.887974i	$0.652113\pi$
$308$	0	0
$309$	1.41444e6	0.842730
$310$	0	0
$311$	−212808.	−0.124763	−0.0623817	−	0.998052i	$-0.519870\pi$
$311$	−212808.	−0.124763	−0.0623817	+	0.998052i	$0.519870\pi$
$312$	0	0
$313$	−1894.00	−0.00109275	−0.000546373	−	1.00000i	$-0.500174\pi$
$313$	−1894.00	−0.00109275	−0.000546373		1.00000i	$0.500174\pi$
$314$	0	0
$315$	373086.	0.211852
$316$	0	0
$317$	−1.57898e6	−0.882527	−0.441263	−	0.897378i	$-0.645470\pi$
$317$	−1.57898e6	−0.882527	−0.441263	+	0.897378i	$0.645470\pi$
$318$	0	0
$319$	−286936.	−0.157873
$320$	0	0
$321$	−60876.0	−0.0329749
$322$	0	0
$323$	3.50210e6	1.86777
$324$	0	0
$325$	−4.39747e6	−2.30938
$326$	0	0
$327$	−1.60558e6	−0.830354
$328$	0	0
$329$	−42336.0	−0.0215635
$330$	0	0
$331$	3.39471e6	1.70307	0.851535	−	0.524298i	$-0.175672\pi$
$331$	3.39471e6	1.70307	0.851535	+	0.524298i	$0.175672\pi$
$332$	0	0
$333$	−594378.	−0.293733
$334$	0	0
$335$	−131224.	−0.0638853
$336$	0	0
$337$	2.02731e6	0.972403	0.486201	−	0.873847i	$-0.338382\pi$
$337$	2.02731e6	0.972403	0.486201	+	0.873847i	$0.338382\pi$
$338$	0	0
$339$	406206.	0.191976
$340$	0	0
$341$	329472.	0.153438
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−487296.	−0.220417
$346$	0	0
$347$	−3.48885e6	−1.55546	−0.777730	−	0.628598i	$-0.783629\pi$
$347$	−3.48885e6	−1.55546	−0.777730	+	0.628598i	$0.783629\pi$
$348$	0	0
$349$	965566.	0.424344	0.212172	−	0.977232i	$-0.431946\pi$
$349$	965566.	0.424344	0.212172	+	0.977232i	$0.431946\pi$
$350$	0	0
$351$	561330.	0.243193
$352$	0	0
$353$	1.15393e6	0.492882	0.246441	−	0.969158i	$-0.420739\pi$
$353$	1.15393e6	0.492882	0.246441	+	0.969158i	$0.420739\pi$
$354$	0	0
$355$	−1.75968e6	−0.741076
$356$	0	0
$357$	891702.	0.370296
$358$	0	0
$359$	−1.61110e6	−0.659762	−0.329881	−	0.944022i	$-0.607009\pi$
$359$	−1.61110e6	−0.659762	−0.329881	+	0.944022i	$0.607009\pi$
$360$	0	0
$361$	523725.	0.211512
$362$	0	0
$363$	1.42512e6	0.567657
$364$	0	0
$365$	4.35803e6	1.71221
$366$	0	0
$367$	−3.67747e6	−1.42523	−0.712614	−	0.701557i	$-0.752489\pi$
$367$	−3.67747e6	−1.42523	−0.712614	+	0.701557i	$0.752489\pi$
$368$	0	0
$369$	−264222.	−0.101019
$370$	0	0
$371$	204918.	0.0772939
$372$	0	0
$373$	649766.	0.241816	0.120908	−	0.992664i	$-0.461419\pi$
$373$	649766.	0.241816	0.120908	+	0.992664i	$0.461419\pi$
$374$	0	0
$375$	−2.18776e6	−0.803379
$376$	0	0
$377$	−4.24886e6	−1.53964
$378$	0	0
$379$	−320700.	−0.114683	−0.0573417	−	0.998355i	$-0.518262\pi$
$379$	−320700.	−0.114683	−0.0573417	+	0.998355i	$0.518262\pi$
$380$	0	0
$381$	−1.84550e6	−0.651332
$382$	0	0
$383$	2.36189e6	0.822740	0.411370	−	0.911469i	$-0.365050\pi$
$383$	2.36189e6	0.822740	0.411370	+	0.911469i	$0.365050\pi$
$384$	0	0
$385$	−239512.	−0.0823522
$386$	0	0
$387$	−439020.	−0.149007
$388$	0	0
$389$	−3.53390e6	−1.18408	−0.592039	−	0.805910i	$-0.701677\pi$
$389$	−3.53390e6	−1.18408	−0.592039	+	0.805910i	$0.701677\pi$
$390$	0	0
$391$	−1.16467e6	−0.385267
$392$	0	0
$393$	656676.	0.214472
$394$	0	0
$395$	−9.15786e6	−2.95326
$396$	0	0
$397$	4.04811e6	1.28907	0.644534	−	0.764575i	$-0.277051\pi$
$397$	4.04811e6	1.28907	0.644534	+	0.764575i	$0.277051\pi$
$398$	0	0
$399$	763812.	0.240190
$400$	0	0
$401$	2.07645e6	0.644853	0.322426	−	0.946595i	$-0.395502\pi$
$401$	2.07645e6	0.644853	0.322426	+	0.946595i	$0.395502\pi$
$402$	0	0
$403$	4.87872e6	1.49638
$404$	0	0
$405$	616734.	0.186836
$406$	0	0
$407$	381576.	0.114181
$408$	0	0
$409$	2.57431e6	0.760945	0.380472	−	0.924792i	$-0.375761\pi$
$409$	2.57431e6	0.760945	0.380472	+	0.924792i	$0.375761\pi$
$410$	0	0
$411$	847638.	0.247517
$412$	0	0
$413$	549780.	0.158604
$414$	0	0
$415$	7.63543e6	2.17627
$416$	0	0
$417$	−430164.	−0.121142
$418$	0	0
$419$	−848148.	−0.236013	−0.118007	−	0.993013i	$-0.537650\pi$
$419$	−848148.	−0.236013	−0.118007	+	0.993013i	$0.537650\pi$
$420$	0	0
$421$	1.43682e6	0.395092	0.197546	−	0.980294i	$-0.436703\pi$
$421$	1.43682e6	0.395092	0.197546	+	0.980294i	$0.436703\pi$
$422$	0	0
$423$	−69984.0	−0.0190173
$424$	0	0
$425$	−1.15476e7	−3.10114
$426$	0	0
$427$	−2.23450e6	−0.593076
$428$	0	0
$429$	−360360.	−0.0945352
$430$	0	0
$431$	−2.35438e6	−0.610496	−0.305248	−	0.952273i	$-0.598739\pi$
$431$	−2.35438e6	−0.610496	−0.305248	+	0.952273i	$0.598739\pi$
$432$	0	0
$433$	−3.78808e6	−0.970955	−0.485478	−	0.874249i	$-0.661354\pi$
$433$	−3.78808e6	−0.970955	−0.485478	+	0.874249i	$0.661354\pi$
$434$	0	0
$435$	−4.66823e6	−1.18285
$436$	0	0
$437$	−997632.	−0.249900
$438$	0	0
$439$	3.64322e6	0.902245	0.451123	−	0.892462i	$-0.351024\pi$
$439$	3.64322e6	0.902245	0.451123	+	0.892462i	$0.351024\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	−2.48389e6	−0.601345	−0.300672	−	0.953728i	$-0.597211\pi$
$443$	−2.48389e6	−0.601345	−0.300672	+	0.953728i	$0.597211\pi$
$444$	0	0
$445$	−299108.	−0.0716025
$446$	0	0
$447$	1.11839e6	0.264744
$448$	0	0
$449$	−2.63177e6	−0.616074	−0.308037	−	0.951374i	$-0.599672\pi$
$449$	−2.63177e6	−0.616074	−0.308037	+	0.951374i	$0.599672\pi$
$450$	0	0
$451$	169624.	0.0392686
$452$	0	0
$453$	−4.01666e6	−0.919645
$454$	0	0
$455$	−3.54662e6	−0.803131
$456$	0	0
$457$	−1.16130e6	−0.260109	−0.130054	−	0.991507i	$-0.541515\pi$
$457$	−1.16130e6	−0.260109	−0.130054	+	0.991507i	$0.541515\pi$
$458$	0	0
$459$	1.47404e6	0.326571
$460$	0	0
$461$	2.81385e6	0.616663	0.308332	−	0.951279i	$-0.400229\pi$
$461$	2.81385e6	0.616663	0.308332	+	0.951279i	$0.400229\pi$
$462$	0	0
$463$	−6.84299e6	−1.48352	−0.741760	−	0.670665i	$-0.766009\pi$
$463$	−6.84299e6	−1.48352	−0.741760	+	0.670665i	$0.766009\pi$
$464$	0	0
$465$	5.36026e6	1.14962
$466$	0	0
$467$	−3.34314e6	−0.709353	−0.354676	−	0.934989i	$-0.615409\pi$
$467$	−3.34314e6	−0.709353	−0.354676	+	0.934989i	$0.615409\pi$
$468$	0	0
$469$	−68404.0	−0.0143598
$470$	0	0
$471$	1.43771e6	0.298621
$472$	0	0
$473$	281840.	0.0579228
$474$	0	0
$475$	−9.89145e6	−2.01153
$476$	0	0
$477$	338742.	0.0681668
$478$	0	0
$479$	4.28248e6	0.852818	0.426409	−	0.904530i	$-0.359778\pi$
$479$	4.28248e6	0.852818	0.426409	+	0.904530i	$0.359778\pi$
$480$	0	0
$481$	5.65026e6	1.11354
$482$	0	0
$483$	−254016.	−0.0495442
$484$	0	0
$485$	461916.	0.0891679
$486$	0	0
$487$	8.93175e6	1.70653	0.853266	−	0.521477i	$-0.174619\pi$
$487$	8.93175e6	1.70653	0.853266	+	0.521477i	$0.174619\pi$
$488$	0	0
$489$	2.22527e6	0.420833
$490$	0	0
$491$	−2.75306e6	−0.515361	−0.257681	−	0.966230i	$-0.582958\pi$
$491$	−2.75306e6	−0.515361	−0.257681	+	0.966230i	$0.582958\pi$
$492$	0	0
$493$	−1.11574e7	−2.06750
$494$	0	0
$495$	−395928.	−0.0726279
$496$	0	0
$497$	−917280.	−0.166575
$498$	0	0
$499$	−4.80408e6	−0.863693	−0.431846	−	0.901947i	$-0.642138\pi$
$499$	−4.80408e6	−0.863693	−0.431846	+	0.901947i	$0.642138\pi$
$500$	0	0
$501$	−6.16039e6	−1.09651
$502$	0	0
$503$	6.02465e6	1.06172	0.530862	−	0.847458i	$-0.321868\pi$
$503$	6.02465e6	1.06172	0.530862	+	0.847458i	$0.321868\pi$
$504$	0	0
$505$	−1.56373e7	−2.72855
$506$	0	0
$507$	−1.99446e6	−0.344593
$508$	0	0
$509$	−8.42987e6	−1.44220	−0.721101	−	0.692830i	$-0.756364\pi$
$509$	−8.42987e6	−1.44220	−0.721101	+	0.692830i	$0.756364\pi$
$510$	0	0
$511$	2.27174e6	0.384863
$512$	0	0
$513$	1.26263e6	0.211827
$514$	0	0
$515$	−1.47730e7	−2.45444
$516$	0	0
$517$	44928.0	0.00739249
$518$	0	0
$519$	5.49427e6	0.895347
$520$	0	0
$521$	9.25058e6	1.49305	0.746525	−	0.665357i	$-0.231721\pi$
$521$	9.25058e6	1.49305	0.746525	+	0.665357i	$0.231721\pi$
$522$	0	0
$523$	−5.84494e6	−0.934385	−0.467192	−	0.884156i	$-0.654734\pi$
$523$	−5.84494e6	−0.934385	−0.467192	+	0.884156i	$0.654734\pi$
$524$	0	0
$525$	−2.51855e6	−0.398798
$526$	0	0
$527$	1.28114e7	2.00942
$528$	0	0
$529$	−6.10457e6	−0.948453
$530$	0	0
$531$	908820.	0.139875
$532$	0	0
$533$	2.51174e6	0.382963
$534$	0	0
$535$	635816.	0.0960389
$536$	0	0
$537$	5.96027e6	0.891929
$538$	0	0
$539$	−124852.	−0.0185107
$540$	0	0
$541$	9.22533e6	1.35515	0.677577	−	0.735452i	$-0.263030\pi$
$541$	9.22533e6	1.35515	0.677577	+	0.735452i	$0.263030\pi$
$542$	0	0
$543$	−1.39167e6	−0.202552
$544$	0	0
$545$	1.67694e7	2.41839
$546$	0	0
$547$	6.44337e6	0.920757	0.460378	−	0.887723i	$-0.347714\pi$
$547$	6.44337e6	0.920757	0.460378	+	0.887723i	$0.347714\pi$
$548$	0	0
$549$	−3.69376e6	−0.523044
$550$	0	0
$551$	−9.55718e6	−1.34107
$552$	0	0
$553$	−4.77378e6	−0.663818
$554$	0	0
$555$	6.20795e6	0.855491
$556$	0	0
$557$	3.74213e6	0.511070	0.255535	−	0.966800i	$-0.417748\pi$
$557$	3.74213e6	0.511070	0.255535	+	0.966800i	$0.417748\pi$
$558$	0	0
$559$	4.17340e6	0.564886
$560$	0	0
$561$	−946296.	−0.126946
$562$	0	0
$563$	1.46384e7	1.94635	0.973176	−	0.230060i	$-0.0738923\pi$
$563$	1.46384e7	1.94635	0.973176	+	0.230060i	$0.0738923\pi$
$564$	0	0
$565$	−4.24260e6	−0.559127
$566$	0	0
$567$	321489.	0.0419961
$568$	0	0
$569$	1.41805e7	1.83616	0.918078	−	0.396400i	$-0.129741\pi$
$569$	1.41805e7	1.83616	0.918078	+	0.396400i	$0.129741\pi$
$570$	0	0
$571$	1.25160e6	0.160648	0.0803242	−	0.996769i	$-0.474404\pi$
$571$	1.25160e6	0.160648	0.0803242	+	0.996769i	$0.474404\pi$
$572$	0	0
$573$	4.38214e6	0.557570
$574$	0	0
$575$	3.28954e6	0.414921
$576$	0	0
$577$	5.94378e6	0.743230	0.371615	−	0.928387i	$-0.378804\pi$
$577$	5.94378e6	0.743230	0.371615	+	0.928387i	$0.378804\pi$
$578$	0	0
$579$	−5.58491e6	−0.692341
$580$	0	0
$581$	3.98017e6	0.489172
$582$	0	0
$583$	−217464.	−0.0264982
$584$	0	0
$585$	−5.86278e6	−0.708295
$586$	0	0
$587$	6.46192e6	0.774046	0.387023	−	0.922070i	$-0.373503\pi$
$587$	6.46192e6	0.774046	0.387023	+	0.922070i	$0.373503\pi$
$588$	0	0
$589$	1.09740e7	1.30339
$590$	0	0
$591$	2.12913e6	0.250746
$592$	0	0
$593$	−2.34605e6	−0.273969	−0.136984	−	0.990573i	$-0.543741\pi$
$593$	−2.34605e6	−0.273969	−0.136984	+	0.990573i	$0.543741\pi$
$594$	0	0
$595$	−9.31333e6	−1.07848
$596$	0	0
$597$	738936.	0.0848537
$598$	0	0
$599$	1.34959e7	1.53686	0.768432	−	0.639931i	$-0.221037\pi$
$599$	1.34959e7	1.53686	0.768432	+	0.639931i	$0.221037\pi$
$600$	0	0
$601$	3.87849e6	0.438002	0.219001	−	0.975725i	$-0.429720\pi$
$601$	3.87849e6	0.438002	0.219001	+	0.975725i	$0.429720\pi$
$602$	0	0
$603$	−113076.	−0.0126642
$604$	0	0
$605$	−1.48846e7	−1.65329
$606$	0	0
$607$	533488.	0.0587696	0.0293848	−	0.999568i	$-0.490645\pi$
$607$	533488.	0.0587696	0.0293848	+	0.999568i	$0.490645\pi$
$608$	0	0
$609$	−2.43344e6	−0.265875
$610$	0	0
$611$	665280.	0.0720944
$612$	0	0
$613$	5.14610e6	0.553130	0.276565	−	0.960995i	$-0.410804\pi$
$613$	5.14610e6	0.553130	0.276565	+	0.960995i	$0.410804\pi$
$614$	0	0
$615$	2.75965e6	0.294216
$616$	0	0
$617$	−2.37860e6	−0.251541	−0.125770	−	0.992059i	$-0.540140\pi$
$617$	−2.37860e6	−0.251541	−0.125770	+	0.992059i	$0.540140\pi$
$618$	0	0
$619$	−1.60023e7	−1.67863	−0.839317	−	0.543642i	$-0.817045\pi$
$619$	−1.60023e7	−1.67863	−0.839317	+	0.543642i	$0.817045\pi$
$620$	0	0
$621$	−419904.	−0.0436939
$622$	0	0
$623$	−155918.	−0.0160944
$624$	0	0
$625$	5.00302e6	0.512309
$626$	0	0
$627$	−810576.	−0.0823427
$628$	0	0
$629$	1.48374e7	1.49531
$630$	0	0
$631$	−1.23459e7	−1.23439	−0.617193	−	0.786812i	$-0.711730\pi$
$631$	−1.23459e7	−1.23439	−0.617193	+	0.786812i	$0.711730\pi$
$632$	0	0
$633$	899028.	0.0891793
$634$	0	0
$635$	1.92753e7	1.89699
$636$	0	0
$637$	−1.84877e6	−0.180524
$638$	0	0
$639$	−1.51632e6	−0.146906
$640$	0	0
$641$	−3.43755e6	−0.330449	−0.165224	−	0.986256i	$-0.552835\pi$
$641$	−3.43755e6	−0.330449	−0.165224	+	0.986256i	$0.552835\pi$
$642$	0	0
$643$	1.62191e7	1.54703	0.773515	−	0.633778i	$-0.218497\pi$
$643$	1.62191e7	1.54703	0.773515	+	0.633778i	$0.218497\pi$
$644$	0	0
$645$	4.58532e6	0.433981
$646$	0	0
$647$	−1.19929e7	−1.12632	−0.563160	−	0.826348i	$-0.690415\pi$
$647$	−1.19929e7	−1.12632	−0.563160	+	0.826348i	$0.690415\pi$
$648$	0	0
$649$	−583440.	−0.0543731
$650$	0	0
$651$	2.79418e6	0.258405
$652$	0	0
$653$	1.58009e6	0.145011	0.0725053	−	0.997368i	$-0.476901\pi$
$653$	1.58009e6	0.145011	0.0725053	+	0.997368i	$0.476901\pi$
$654$	0	0
$655$	−6.85862e6	−0.624645
$656$	0	0
$657$	3.75532e6	0.339417
$658$	0	0
$659$	−6.98358e6	−0.626419	−0.313209	−	0.949684i	$-0.601404\pi$
$659$	−6.98358e6	−0.626419	−0.313209	+	0.949684i	$0.601404\pi$
$660$	0	0
$661$	3.69602e6	0.329027	0.164513	−	0.986375i	$-0.447395\pi$
$661$	3.69602e6	0.329027	0.164513	+	0.986375i	$0.447395\pi$
$662$	0	0
$663$	−1.40125e7	−1.23803
$664$	0	0
$665$	−7.97759e6	−0.699548
$666$	0	0
$667$	3.17837e6	0.276624
$668$	0	0
$669$	−1.68034e6	−0.145155
$670$	0	0
$671$	2.37130e6	0.203320
$672$	0	0
$673$	1.84688e6	0.157182	0.0785908	−	0.996907i	$-0.474958\pi$
$673$	1.84688e6	0.157182	0.0785908	+	0.996907i	$0.474958\pi$
$674$	0	0
$675$	−4.16332e6	−0.351706
$676$	0	0
$677$	−7.68501e6	−0.644426	−0.322213	−	0.946667i	$-0.604427\pi$
$677$	−7.68501e6	−0.644426	−0.322213	+	0.946667i	$0.604427\pi$
$678$	0	0
$679$	240786.	0.0200427
$680$	0	0
$681$	3.02735e6	0.250147
$682$	0	0
$683$	−7.12180e6	−0.584168	−0.292084	−	0.956393i	$-0.594349\pi$
$683$	−7.12180e6	−0.584168	−0.292084	+	0.956393i	$0.594349\pi$
$684$	0	0
$685$	−8.85311e6	−0.720891
$686$	0	0
$687$	8.33683e6	0.673921
$688$	0	0
$689$	−3.22014e6	−0.258420
$690$	0	0
$691$	3.23787e6	0.257967	0.128983	−	0.991647i	$-0.458829\pi$
$691$	3.23787e6	0.257967	0.128983	+	0.991647i	$0.458829\pi$
$692$	0	0
$693$	−206388.	−0.0163249
$694$	0	0
$695$	4.49282e6	0.352823
$696$	0	0
$697$	6.59576e6	0.514260
$698$	0	0
$699$	−1.13140e7	−0.875839
$700$	0	0
$701$	−7.39163e6	−0.568127	−0.284063	−	0.958805i	$-0.591683\pi$
$701$	−7.39163e6	−0.568127	−0.284063	+	0.958805i	$0.591683\pi$
$702$	0	0
$703$	1.27094e7	0.969923
$704$	0	0
$705$	730944.	0.0553874
$706$	0	0
$707$	−8.15135e6	−0.613311
$708$	0	0
$709$	−5.33361e6	−0.398479	−0.199240	−	0.979951i	$-0.563847\pi$
$709$	−5.33361e6	−0.398479	−0.199240	+	0.979951i	$0.563847\pi$
$710$	0	0
$711$	−7.89134e6	−0.585433
$712$	0	0
$713$	−3.64954e6	−0.268852
$714$	0	0
$715$	3.76376e6	0.275332
$716$	0	0
$717$	−3.12314e6	−0.226879
$718$	0	0
$719$	1.14564e7	0.826468	0.413234	−	0.910625i	$-0.364399\pi$
$719$	1.14564e7	0.826468	0.413234	+	0.910625i	$0.364399\pi$
$720$	0	0
$721$	−7.70084e6	−0.551696
$722$	0	0
$723$	−892530.	−0.0635005
$724$	0	0
$725$	3.15133e7	2.22663
$726$	0	0
$727$	2.49540e7	1.75107	0.875536	−	0.483153i	$-0.160508\pi$
$727$	2.49540e7	1.75107	0.875536	+	0.483153i	$0.160508\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.09592e7	0.758555
$732$	0	0
$733$	−1.43398e7	−0.985789	−0.492894	−	0.870089i	$-0.664061\pi$
$733$	−1.43398e7	−0.985789	−0.492894	+	0.870089i	$0.664061\pi$
$734$	0	0
$735$	−2.03125e6	−0.138690
$736$	0	0
$737$	72592.0	0.00492289
$738$	0	0
$739$	−922932.	−0.0621668	−0.0310834	−	0.999517i	$-0.509896\pi$
$739$	−922932.	−0.0621668	−0.0310834	+	0.999517i	$0.509896\pi$
$740$	0	0
$741$	−1.20028e7	−0.803037
$742$	0	0
$743$	9.38995e6	0.624010	0.312005	−	0.950081i	$-0.398999\pi$
$743$	9.38995e6	0.624010	0.312005	+	0.950081i	$0.398999\pi$
$744$	0	0
$745$	−1.16810e7	−0.771062
$746$	0	0
$747$	6.57947e6	0.431409
$748$	0	0
$749$	331436.	0.0215871
$750$	0	0
$751$	408032.	0.0263994	0.0131997	−	0.999913i	$-0.495798\pi$
$751$	408032.	0.0263994	0.0131997	+	0.999913i	$0.495798\pi$
$752$	0	0
$753$	3.09985e6	0.199229
$754$	0	0
$755$	4.19518e7	2.67845
$756$	0	0
$757$	2.59605e7	1.64654	0.823271	−	0.567649i	$-0.192147\pi$
$757$	2.59605e7	1.64654	0.823271	+	0.567649i	$0.192147\pi$
$758$	0	0
$759$	269568.	0.0169849
$760$	0	0
$761$	1.83554e7	1.14895	0.574477	−	0.818521i	$-0.305206\pi$
$761$	1.83554e7	1.14895	0.574477	+	0.818521i	$0.305206\pi$
$762$	0	0
$763$	8.74150e6	0.543594
$764$	0	0
$765$	−1.53955e7	−0.951131
$766$	0	0
$767$	−8.63940e6	−0.530268
$768$	0	0
$769$	−747166.	−0.0455618	−0.0227809	−	0.999740i	$-0.507252\pi$
$769$	−747166.	−0.0455618	−0.0227809	+	0.999740i	$0.507252\pi$
$770$	0	0
$771$	−2.65617e6	−0.160924
$772$	0	0
$773$	2.02692e7	1.22008	0.610038	−	0.792372i	$-0.291154\pi$
$773$	2.02692e7	1.22008	0.610038	+	0.792372i	$0.291154\pi$
$774$	0	0
$775$	−3.61849e7	−2.16408
$776$	0	0
$777$	3.23606e6	0.192293
$778$	0	0
$779$	5.64978e6	0.333571
$780$	0	0
$781$	973440.	0.0571060
$782$	0	0
$783$	−4.02262e6	−0.234479
$784$	0	0
$785$	−1.50161e7	−0.869729
$786$	0	0
$787$	4.69982e6	0.270486	0.135243	−	0.990812i	$-0.456819\pi$
$787$	4.69982e6	0.270486	0.135243	+	0.990812i	$0.456819\pi$
$788$	0	0
$789$	−1.14522e7	−0.654931
$790$	0	0
$791$	−2.21157e6	−0.125678
$792$	0	0
$793$	3.51135e7	1.98286
$794$	0	0
$795$	−3.53797e6	−0.198535
$796$	0	0
$797$	584710.	0.0326058	0.0163029	−	0.999867i	$-0.494810\pi$
$797$	584710.	0.0326058	0.0163029	+	0.999867i	$0.494810\pi$
$798$	0	0
$799$	1.74701e6	0.0968117
$800$	0	0
$801$	−257742.	−0.0141940
$802$	0	0
$803$	−2.41082e6	−0.131940
$804$	0	0
$805$	2.65306e6	0.144297
$806$	0	0
$807$	−2.49097e6	−0.134643
$808$	0	0
$809$	−1.64013e7	−0.881061	−0.440531	−	0.897738i	$-0.645210\pi$
$809$	−1.64013e7	−0.881061	−0.440531	+	0.897738i	$0.645210\pi$
$810$	0	0
$811$	304948.	0.0162807	0.00814036	−	0.999967i	$-0.497409\pi$
$811$	304948.	0.0162807	0.00814036	+	0.999967i	$0.497409\pi$
$812$	0	0
$813$	−1.16094e7	−0.616005
$814$	0	0
$815$	−2.32417e7	−1.22567
$816$	0	0
$817$	9.38744e6	0.492031
$818$	0	0
$819$	−3.05613e6	−0.159207
$820$	0	0
$821$	3.43428e7	1.77819	0.889095	−	0.457722i	$-0.151335\pi$
$821$	3.43428e7	1.77819	0.889095	+	0.457722i	$0.151335\pi$
$822$	0	0
$823$	−1.56684e7	−0.806351	−0.403176	−	0.915123i	$-0.632094\pi$
$823$	−1.56684e7	−0.806351	−0.403176	+	0.915123i	$0.632094\pi$
$824$	0	0
$825$	2.67275e6	0.136717
$826$	0	0
$827$	2.96886e7	1.50948	0.754738	−	0.656026i	$-0.227764\pi$
$827$	2.96886e7	1.50948	0.754738	+	0.656026i	$0.227764\pi$
$828$	0	0
$829$	−2.30708e7	−1.16594	−0.582970	−	0.812494i	$-0.698110\pi$
$829$	−2.30708e7	−1.16594	−0.582970	+	0.812494i	$0.698110\pi$
$830$	0	0
$831$	−1.54490e7	−0.776062
$832$	0	0
$833$	−4.85482e6	−0.242416
$834$	0	0
$835$	6.43419e7	3.19358
$836$	0	0
$837$	4.61894e6	0.227892
$838$	0	0
$839$	−2.32642e7	−1.14100	−0.570498	−	0.821299i	$-0.693250\pi$
$839$	−2.32642e7	−1.14100	−0.570498	+	0.821299i	$0.693250\pi$
$840$	0	0
$841$	9.93718e6	0.484477
$842$	0	0
$843$	1.32496e7	0.642148
$844$	0	0
$845$	2.08311e7	1.00362
$846$	0	0
$847$	−7.75900e6	−0.371619
$848$	0	0
$849$	9.25931e6	0.440869
$850$	0	0
$851$	−4.22669e6	−0.200067
$852$	0	0
$853$	1.91515e7	0.901219	0.450610	−	0.892721i	$-0.351207\pi$
$853$	1.91515e7	0.901219	0.450610	+	0.892721i	$0.351207\pi$
$854$	0	0
$855$	−1.31874e7	−0.616944
$856$	0	0
$857$	−5.34683e6	−0.248682	−0.124341	−	0.992240i	$-0.539682\pi$
$857$	−5.34683e6	−0.248682	−0.124341	+	0.992240i	$0.539682\pi$
$858$	0	0
$859$	−3.95858e7	−1.83045	−0.915223	−	0.402948i	$-0.867986\pi$
$859$	−3.95858e7	−1.83045	−0.915223	+	0.402948i	$0.867986\pi$
$860$	0	0
$861$	1.43854e6	0.0661325
$862$	0	0
$863$	2.50284e7	1.14395	0.571973	−	0.820272i	$-0.306178\pi$
$863$	2.50284e7	1.14395	0.571973	+	0.820272i	$0.306178\pi$
$864$	0	0
$865$	−5.73846e7	−2.60768
$866$	0	0
$867$	−2.40176e7	−1.08513
$868$	0	0
$869$	5.06605e6	0.227573
$870$	0	0
$871$	1.07492e6	0.0480099
$872$	0	0
$873$	398034.	0.0176760
$874$	0	0
$875$	1.19111e7	0.525935
$876$	0	0
$877$	−5.02589e6	−0.220655	−0.110328	−	0.993895i	$-0.535190\pi$
$877$	−5.02589e6	−0.220655	−0.110328	+	0.993895i	$0.535190\pi$
$878$	0	0
$879$	1.06746e7	0.465993
$880$	0	0
$881$	−2.60490e7	−1.13071	−0.565356	−	0.824847i	$-0.691261\pi$
$881$	−2.60490e7	−1.13071	−0.565356	+	0.824847i	$0.691261\pi$
$882$	0	0
$883$	6.82462e6	0.294562	0.147281	−	0.989095i	$-0.452948\pi$
$883$	6.82462e6	0.294562	0.147281	+	0.989095i	$0.452948\pi$
$884$	0	0
$885$	−9.49212e6	−0.407385
$886$	0	0
$887$	−2.33835e7	−0.997931	−0.498965	−	0.866622i	$-0.666287\pi$
$887$	−2.33835e7	−0.997931	−0.498965	+	0.866622i	$0.666287\pi$
$888$	0	0
$889$	1.00477e7	0.426397
$890$	0	0
$891$	−341172.	−0.0143972
$892$	0	0
$893$	1.49645e6	0.0627961
$894$	0	0
$895$	−6.22517e7	−2.59773
$896$	0	0
$897$	3.99168e6	0.165644
$898$	0	0
$899$	−3.49620e7	−1.44277
$900$	0	0
$901$	−8.45600e6	−0.347019
$902$	0	0
$903$	2.39022e6	0.0975480
$904$	0	0
$905$	1.45352e7	0.589930
$906$	0	0
$907$	−3.95959e7	−1.59820	−0.799102	−	0.601196i	$-0.794691\pi$
$907$	−3.95959e7	−1.59820	−0.799102	+	0.601196i	$0.794691\pi$
$908$	0	0
$909$	−1.34747e7	−0.540890
$910$	0	0
$911$	4.67570e6	0.186660	0.0933300	−	0.995635i	$-0.470249\pi$
$911$	4.67570e6	0.186660	0.0933300	+	0.995635i	$0.470249\pi$
$912$	0	0
$913$	−4.22386e6	−0.167700
$914$	0	0
$915$	3.85793e7	1.52336
$916$	0	0
$917$	−3.57524e6	−0.140405
$918$	0	0
$919$	4.92594e6	0.192398	0.0961990	−	0.995362i	$-0.469331\pi$
$919$	4.92594e6	0.192398	0.0961990	+	0.995362i	$0.469331\pi$
$920$	0	0
$921$	1.36702e7	0.531040
$922$	0	0
$923$	1.44144e7	0.556919
$924$	0	0
$925$	−4.19073e7	−1.61041
$926$	0	0
$927$	−1.27300e7	−0.486550
$928$	0	0
$929$	−3.23688e7	−1.23052	−0.615258	−	0.788326i	$-0.710948\pi$
$929$	−3.23688e7	−1.23052	−0.615258	+	0.788326i	$0.710948\pi$
$930$	0	0
$931$	−4.15853e6	−0.157241
$932$	0	0
$933$	1.91527e6	0.0720321
$934$	0	0
$935$	9.88354e6	0.369729
$936$	0	0
$937$	−3.32337e7	−1.23660	−0.618301	−	0.785941i	$-0.712179\pi$
$937$	−3.32337e7	−1.23660	−0.618301	+	0.785941i	$0.712179\pi$
$938$	0	0
$939$	17046.0	0.000630897 0
$940$	0	0
$941$	−2.66426e7	−0.980852	−0.490426	−	0.871483i	$-0.663159\pi$
$941$	−2.66426e7	−0.980852	−0.490426	+	0.871483i	$0.663159\pi$
$942$	0	0
$943$	−1.87891e6	−0.0688061
$944$	0	0
$945$	−3.35777e6	−0.122313
$946$	0	0
$947$	−3.14663e7	−1.14017	−0.570086	−	0.821585i	$-0.693090\pi$
$947$	−3.14663e7	−1.14017	−0.570086	+	0.821585i	$0.693090\pi$
$948$	0	0
$949$	−3.56987e7	−1.28673
$950$	0	0
$951$	1.42108e7	0.509527
$952$	0	0
$953$	−1.34516e7	−0.479779	−0.239890	−	0.970800i	$-0.577111\pi$
$953$	−1.34516e7	−0.479779	−0.239890	+	0.970800i	$0.577111\pi$
$954$	0	0
$955$	−4.57690e7	−1.62391
$956$	0	0
$957$	2.58242e6	0.0911481
$958$	0	0
$959$	−4.61492e6	−0.162038
$960$	0	0
$961$	1.15157e7	0.402238
$962$	0	0
$963$	547884.	0.0190381
$964$	0	0
$965$	5.83313e7	2.01643
$966$	0	0
$967$	2.84963e7	0.979992	0.489996	−	0.871725i	$-0.336998\pi$
$967$	2.84963e7	0.979992	0.489996	+	0.871725i	$0.336998\pi$
$968$	0	0
$969$	−3.15189e7	−1.07836
$970$	0	0
$971$	−1.81858e7	−0.618990	−0.309495	−	0.950901i	$-0.600160\pi$
$971$	−1.81858e7	−0.618990	−0.309495	+	0.950901i	$0.600160\pi$
$972$	0	0
$973$	2.34200e6	0.0793059
$974$	0	0
$975$	3.95772e7	1.33332
$976$	0	0
$977$	3.20941e7	1.07569	0.537847	−	0.843042i	$-0.319238\pi$
$977$	3.20941e7	1.07569	0.537847	+	0.843042i	$0.319238\pi$
$978$	0	0
$979$	165464.	0.00551756
$980$	0	0
$981$	1.44502e7	0.479405
$982$	0	0
$983$	−1.56154e7	−0.515429	−0.257715	−	0.966221i	$-0.582969\pi$
$983$	−1.56154e7	−0.515429	−0.257715	+	0.966221i	$0.582969\pi$
$984$	0	0
$985$	−2.22376e7	−0.730293
$986$	0	0
$987$	381024.	0.0124497
$988$	0	0
$989$	−3.12192e6	−0.101492
$990$	0	0
$991$	−4.84499e7	−1.56714	−0.783572	−	0.621301i	$-0.786605\pi$
$991$	−4.84499e7	−1.56714	−0.783572	+	0.621301i	$0.786605\pi$
$992$	0	0
$993$	−3.05524e7	−0.983268
$994$	0	0
$995$	−7.71778e6	−0.247135
$996$	0	0
$997$	−4.54336e7	−1.44757	−0.723784	−	0.690027i	$-0.757599\pi$
$997$	−4.54336e7	−1.44757	−0.723784	+	0.690027i	$0.757599\pi$
$998$	0	0
$999$	5.34940e6	0.169587

Twists

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

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