LMFDB - Embedded newform 768.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,6,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("768.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	768.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:	$123.174773616$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 384)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	768.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-9.00000 q^{3} +24.0000 q^{5} -68.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q-9.00000 q^{3} +24.0000 q^{5} -68.0000 q^{7} +81.0000 q^{9} +452.000 q^{11} +1052.00 q^{13} -216.000 q^{15} -418.000 q^{17} +2492.00 q^{19} +612.000 q^{21} +2392.00 q^{23} -2549.00 q^{25} -729.000 q^{27} -1520.00 q^{29} +1948.00 q^{31} -4068.00 q^{33} -1632.00 q^{35} -8972.00 q^{37} -9468.00 q^{39} +15174.0 q^{41} -2684.00 q^{43} +1944.00 q^{45} +10744.0 q^{47} -12183.0 q^{49} +3762.00 q^{51} -8048.00 q^{53} +10848.0 q^{55} -22428.0 q^{57} -13356.0 q^{59} -19260.0 q^{61} -5508.00 q^{63} +25248.0 q^{65} +36588.0 q^{67} -21528.0 q^{69} -63832.0 q^{71} +14106.0 q^{73} +22941.0 q^{75} -30736.0 q^{77} +98908.0 q^{79} +6561.00 q^{81} +63292.0 q^{83} -10032.0 q^{85} +13680.0 q^{87} -7014.00 q^{89} -71536.0 q^{91} -17532.0 q^{93} +59808.0 q^{95} +80830.0 q^{97} +36612.0 q^{99} +O(q^{100})$

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
$5$	24.0000	0.429325	0.214663	−	0.976688i	$-0.431135\pi$
$5$	24.0000	0.429325	0.214663	+	0.976688i	$0.431135\pi$
$6$	0	0
$7$	−68.0000	−0.524522	−0.262261	−	0.964997i	$-0.584468\pi$
$7$	−68.0000	−0.524522	−0.262261	+	0.964997i	$0.584468\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	452.000	1.12631	0.563153	−	0.826352i	$-0.309588\pi$
$11$	452.000	1.12631	0.563153	+	0.826352i	$0.309588\pi$
$12$	0	0
$13$	1052.00	1.72646	0.863232	−	0.504808i	$-0.168437\pi$
$13$	1052.00	1.72646	0.863232	+	0.504808i	$0.168437\pi$
$14$	0	0
$15$	−216.000	−0.247871
$16$	0	0
$17$	−418.000	−0.350795	−0.175398	−	0.984498i	$-0.556121\pi$
$17$	−418.000	−0.350795	−0.175398	+	0.984498i	$0.556121\pi$
$18$	0	0
$19$	2492.00	1.58367	0.791834	−	0.610737i	$-0.209127\pi$
$19$	2492.00	1.58367	0.791834	+	0.610737i	$0.209127\pi$
$20$	0	0
$21$	612.000	0.302833
$22$	0	0
$23$	2392.00	0.942848	0.471424	−	0.881907i	$-0.343740\pi$
$23$	2392.00	0.942848	0.471424	+	0.881907i	$0.343740\pi$
$24$	0	0
$25$	−2549.00	−0.815680
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−1520.00	−0.335621	−0.167810	−	0.985819i	$-0.553670\pi$
$29$	−1520.00	−0.335621	−0.167810	+	0.985819i	$0.553670\pi$
$30$	0	0
$31$	1948.00	0.364070	0.182035	−	0.983292i	$-0.441732\pi$
$31$	1948.00	0.364070	0.182035	+	0.983292i	$0.441732\pi$
$32$	0	0
$33$	−4068.00	−0.650274
$34$	0	0
$35$	−1632.00	−0.225190
$36$	0	0
$37$	−8972.00	−1.07742	−0.538710	−	0.842491i	$-0.681088\pi$
$37$	−8972.00	−1.07742	−0.538710	+	0.842491i	$0.681088\pi$
$38$	0	0
$39$	−9468.00	−0.996774
$40$	0	0
$41$	15174.0	1.40974	0.704872	−	0.709334i	$-0.251004\pi$
$41$	15174.0	1.40974	0.704872	+	0.709334i	$0.251004\pi$
$42$	0	0
$43$	−2684.00	−0.221366	−0.110683	−	0.993856i	$-0.535304\pi$
$43$	−2684.00	−0.221366	−0.110683	+	0.993856i	$0.535304\pi$
$44$	0	0
$45$	1944.00	0.143108
$46$	0	0
$47$	10744.0	0.709449	0.354725	−	0.934971i	$-0.384575\pi$
$47$	10744.0	0.709449	0.354725	+	0.934971i	$0.384575\pi$
$48$	0	0
$49$	−12183.0	−0.724877
$50$	0	0
$51$	3762.00	0.202532
$52$	0	0
$53$	−8048.00	−0.393549	−0.196774	−	0.980449i	$-0.563047\pi$
$53$	−8048.00	−0.393549	−0.196774	+	0.980449i	$0.563047\pi$
$54$	0	0
$55$	10848.0	0.483552
$56$	0	0
$57$	−22428.0	−0.914331
$58$	0	0
$59$	−13356.0	−0.499513	−0.249756	−	0.968309i	$-0.580350\pi$
$59$	−13356.0	−0.499513	−0.249756	+	0.968309i	$0.580350\pi$
$60$	0	0
$61$	−19260.0	−0.662722	−0.331361	−	0.943504i	$-0.607508\pi$
$61$	−19260.0	−0.662722	−0.331361	+	0.943504i	$0.607508\pi$
$62$	0	0
$63$	−5508.00	−0.174841
$64$	0	0
$65$	25248.0	0.741214
$66$	0	0
$67$	36588.0	0.995753	0.497877	−	0.867248i	$-0.334113\pi$
$67$	36588.0	0.995753	0.497877	+	0.867248i	$0.334113\pi$
$68$	0	0
$69$	−21528.0	−0.544353
$70$	0	0
$71$	−63832.0	−1.50277	−0.751385	−	0.659864i	$-0.770614\pi$
$71$	−63832.0	−1.50277	−0.751385	+	0.659864i	$0.770614\pi$
$72$	0	0
$73$	14106.0	0.309811	0.154905	−	0.987929i	$-0.450493\pi$
$73$	14106.0	0.309811	0.154905	+	0.987929i	$0.450493\pi$
$74$	0	0
$75$	22941.0	0.470933
$76$	0	0
$77$	−30736.0	−0.590773
$78$	0	0
$79$	98908.0	1.78305	0.891525	−	0.452971i	$-0.149636\pi$
$79$	98908.0	1.78305	0.891525	+	0.452971i	$0.149636\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	63292.0	1.00845	0.504224	−	0.863573i	$-0.331779\pi$
$83$	63292.0	1.00845	0.504224	+	0.863573i	$0.331779\pi$
$84$	0	0
$85$	−10032.0	−0.150605
$86$	0	0
$87$	13680.0	0.193771
$88$	0	0
$89$	−7014.00	−0.0938622	−0.0469311	−	0.998898i	$-0.514944\pi$
$89$	−7014.00	−0.0938622	−0.0469311	+	0.998898i	$0.514944\pi$
$90$	0	0
$91$	−71536.0	−0.905568
$92$	0	0
$93$	−17532.0	−0.210196
$94$	0	0
$95$	59808.0	0.679908
$96$	0	0
$97$	80830.0	0.872255	0.436127	−	0.899885i	$-0.356350\pi$
$97$	80830.0	0.872255	0.436127	+	0.899885i	$0.356350\pi$
$98$	0	0
$99$	36612.0	0.375436
$100$	0	0
$101$	56944.0	0.555449	0.277725	−	0.960661i	$-0.410420\pi$
$101$	56944.0	0.555449	0.277725	+	0.960661i	$0.410420\pi$
$102$	0	0
$103$	−2044.00	−0.0189840	−0.00949200	−	0.999955i	$-0.503021\pi$
$103$	−2044.00	−0.0189840	−0.00949200	+	0.999955i	$0.503021\pi$
$104$	0	0
$105$	14688.0	0.130014
$106$	0	0
$107$	−233692.	−1.97326	−0.986630	−	0.162975i	$-0.947891\pi$
$107$	−233692.	−1.97326	−0.986630	+	0.162975i	$0.947891\pi$
$108$	0	0
$109$	49148.0	0.396223	0.198111	−	0.980179i	$-0.436519\pi$
$109$	49148.0	0.396223	0.198111	+	0.980179i	$0.436519\pi$
$110$	0	0
$111$	80748.0	0.622049
$112$	0	0
$113$	−104206.	−0.767709	−0.383854	−	0.923394i	$-0.625404\pi$
$113$	−104206.	−0.767709	−0.383854	+	0.923394i	$0.625404\pi$
$114$	0	0
$115$	57408.0	0.404788
$116$	0	0
$117$	85212.0	0.575488
$118$	0	0
$119$	28424.0	0.184000
$120$	0	0
$121$	43253.0	0.268567
$122$	0	0
$123$	−136566.	−0.813916
$124$	0	0
$125$	−136176.	−0.779517
$126$	0	0
$127$	228836.	1.25897	0.629485	−	0.777013i	$-0.283266\pi$
$127$	228836.	1.25897	0.629485	+	0.777013i	$0.283266\pi$
$128$	0	0
$129$	24156.0	0.127806
$130$	0	0
$131$	−80148.0	−0.408051	−0.204026	−	0.978966i	$-0.565403\pi$
$131$	−80148.0	−0.408051	−0.204026	+	0.978966i	$0.565403\pi$
$132$	0	0
$133$	−169456.	−0.830669
$134$	0	0
$135$	−17496.0	−0.0826236
$136$	0	0
$137$	−292218.	−1.33017	−0.665083	−	0.746770i	$-0.731604\pi$
$137$	−292218.	−1.33017	−0.665083	+	0.746770i	$0.731604\pi$
$138$	0	0
$139$	−43276.0	−0.189981	−0.0949905	−	0.995478i	$-0.530282\pi$
$139$	−43276.0	−0.189981	−0.0949905	+	0.995478i	$0.530282\pi$
$140$	0	0
$141$	−96696.0	−0.409601
$142$	0	0
$143$	475504.	1.94453
$144$	0	0
$145$	−36480.0	−0.144090
$146$	0	0
$147$	109647.	0.418508
$148$	0	0
$149$	−170376.	−0.628699	−0.314349	−	0.949307i	$-0.601786\pi$
$149$	−170376.	−0.628699	−0.314349	+	0.949307i	$0.601786\pi$
$150$	0	0
$151$	−360948.	−1.28826	−0.644128	−	0.764918i	$-0.722780\pi$
$151$	−360948.	−1.28826	−0.644128	+	0.764918i	$0.722780\pi$
$152$	0	0
$153$	−33858.0	−0.116932
$154$	0	0
$155$	46752.0	0.156304
$156$	0	0
$157$	112180.	0.363217	0.181609	−	0.983371i	$-0.441870\pi$
$157$	112180.	0.363217	0.181609	+	0.983371i	$0.441870\pi$
$158$	0	0
$159$	72432.0	0.227215
$160$	0	0
$161$	−162656.	−0.494545
$162$	0	0
$163$	410476.	1.21009	0.605047	−	0.796190i	$-0.293155\pi$
$163$	410476.	1.21009	0.605047	+	0.796190i	$0.293155\pi$
$164$	0	0
$165$	−97632.0	−0.279179
$166$	0	0
$167$	561120.	1.55691	0.778457	−	0.627698i	$-0.216003\pi$
$167$	561120.	1.55691	0.778457	+	0.627698i	$0.216003\pi$
$168$	0	0
$169$	735411.	1.98068
$170$	0	0
$171$	201852.	0.527889
$172$	0	0
$173$	693912.	1.76274	0.881372	−	0.472423i	$-0.156621\pi$
$173$	693912.	1.76274	0.881372	+	0.472423i	$0.156621\pi$
$174$	0	0
$175$	173332.	0.427842
$176$	0	0
$177$	120204.	0.288394
$178$	0	0
$179$	474076.	1.10590	0.552949	−	0.833215i	$-0.313502\pi$
$179$	474076.	1.10590	0.552949	+	0.833215i	$0.313502\pi$
$180$	0	0
$181$	184124.	0.417748	0.208874	−	0.977943i	$-0.433020\pi$
$181$	184124.	0.417748	0.208874	+	0.977943i	$0.433020\pi$
$182$	0	0
$183$	173340.	0.382623
$184$	0	0
$185$	−215328.	−0.462563
$186$	0	0
$187$	−188936.	−0.395103
$188$	0	0
$189$	49572.0	0.100944
$190$	0	0
$191$	598672.	1.18742	0.593712	−	0.804678i	$-0.297662\pi$
$191$	598672.	1.18742	0.593712	+	0.804678i	$0.297662\pi$
$192$	0	0
$193$	−644754.	−1.24595	−0.622975	−	0.782242i	$-0.714076\pi$
$193$	−644754.	−1.24595	−0.622975	+	0.782242i	$0.714076\pi$
$194$	0	0
$195$	−227232.	−0.427940
$196$	0	0
$197$	1.02970e6	1.89036	0.945178	−	0.326556i	$-0.105888\pi$
$197$	1.02970e6	1.89036	0.945178	+	0.326556i	$0.105888\pi$
$198$	0	0
$199$	939524.	1.68180	0.840902	−	0.541188i	$-0.182025\pi$
$199$	939524.	1.68180	0.840902	+	0.541188i	$0.182025\pi$
$200$	0	0
$201$	−329292.	−0.574898
$202$	0	0
$203$	103360.	0.176040
$204$	0	0
$205$	364176.	0.605239
$206$	0	0
$207$	193752.	0.314283
$208$	0	0
$209$	1.12638e6	1.78370
$210$	0	0
$211$	−955300.	−1.47718	−0.738590	−	0.674154i	$-0.764508\pi$
$211$	−955300.	−1.47718	−0.738590	+	0.674154i	$0.764508\pi$
$212$	0	0
$213$	574488.	0.867625
$214$	0	0
$215$	−64416.0	−0.0950381
$216$	0	0
$217$	−132464.	−0.190963
$218$	0	0
$219$	−126954.	−0.178869
$220$	0	0
$221$	−439736.	−0.605636
$222$	0	0
$223$	141556.	0.190619	0.0953095	−	0.995448i	$-0.469616\pi$
$223$	141556.	0.190619	0.0953095	+	0.995448i	$0.469616\pi$
$224$	0	0
$225$	−206469.	−0.271893
$226$	0	0
$227$	−188420.	−0.242696	−0.121348	−	0.992610i	$-0.538722\pi$
$227$	−188420.	−0.242696	−0.121348	+	0.992610i	$0.538722\pi$
$228$	0	0
$229$	−1.12810e6	−1.42154	−0.710770	−	0.703425i	$-0.751653\pi$
$229$	−1.12810e6	−1.42154	−0.710770	+	0.703425i	$0.751653\pi$
$230$	0	0
$231$	276624.	0.341083
$232$	0	0
$233$	1.06542e6	1.28567	0.642836	−	0.766004i	$-0.277758\pi$
$233$	1.06542e6	1.28567	0.642836	+	0.766004i	$0.277758\pi$
$234$	0	0
$235$	257856.	0.304584
$236$	0	0
$237$	−890172.	−1.02944
$238$	0	0
$239$	−103104.	−0.116756	−0.0583782	−	0.998295i	$-0.518593\pi$
$239$	−103104.	−0.116756	−0.0583782	+	0.998295i	$0.518593\pi$
$240$	0	0
$241$	−78338.0	−0.0868820	−0.0434410	−	0.999056i	$-0.513832\pi$
$241$	−78338.0	−0.0868820	−0.0434410	+	0.999056i	$0.513832\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−292392.	−0.311208
$246$	0	0
$247$	2.62158e6	2.73414
$248$	0	0
$249$	−569628.	−0.582228
$250$	0	0
$251$	664116.	0.665365	0.332682	−	0.943039i	$-0.392046\pi$
$251$	664116.	0.665365	0.332682	+	0.943039i	$0.392046\pi$
$252$	0	0
$253$	1.08118e6	1.06194
$254$	0	0
$255$	90288.0	0.0869520
$256$	0	0
$257$	406786.	0.384179	0.192089	−	0.981377i	$-0.438474\pi$
$257$	406786.	0.384179	0.192089	+	0.981377i	$0.438474\pi$
$258$	0	0
$259$	610096.	0.565131
$260$	0	0
$261$	−123120.	−0.111874
$262$	0	0
$263$	1.54261e6	1.37520	0.687601	−	0.726089i	$-0.258664\pi$
$263$	1.54261e6	1.37520	0.687601	+	0.726089i	$0.258664\pi$
$264$	0	0
$265$	−193152.	−0.168960
$266$	0	0
$267$	63126.0	0.0541914
$268$	0	0
$269$	−1.24286e6	−1.04723	−0.523616	−	0.851954i	$-0.675417\pi$
$269$	−1.24286e6	−1.04723	−0.523616	+	0.851954i	$0.675417\pi$
$270$	0	0
$271$	1.99631e6	1.65122	0.825609	−	0.564243i	$-0.190832\pi$
$271$	1.99631e6	1.65122	0.825609	+	0.564243i	$0.190832\pi$
$272$	0	0
$273$	643824.	0.522830
$274$	0	0
$275$	−1.15215e6	−0.918706
$276$	0	0
$277$	−685652.	−0.536914	−0.268457	−	0.963292i	$-0.586514\pi$
$277$	−685652.	−0.536914	−0.268457	+	0.963292i	$0.586514\pi$
$278$	0	0
$279$	157788.	0.121357
$280$	0	0
$281$	−1.71485e6	−1.29557	−0.647786	−	0.761823i	$-0.724305\pi$
$281$	−1.71485e6	−1.29557	−0.647786	+	0.761823i	$0.724305\pi$
$282$	0	0
$283$	1.30685e6	0.969975	0.484987	−	0.874521i	$-0.338824\pi$
$283$	1.30685e6	0.969975	0.484987	+	0.874521i	$0.338824\pi$
$284$	0	0
$285$	−538272.	−0.392545
$286$	0	0
$287$	−1.03183e6	−0.739442
$288$	0	0
$289$	−1.24513e6	−0.876943
$290$	0	0
$291$	−727470.	−0.503596
$292$	0	0
$293$	2.23570e6	1.52140	0.760701	−	0.649103i	$-0.224856\pi$
$293$	2.23570e6	1.52140	0.760701	+	0.649103i	$0.224856\pi$
$294$	0	0
$295$	−320544.	−0.214453
$296$	0	0
$297$	−329508.	−0.216758
$298$	0	0
$299$	2.51638e6	1.62779
$300$	0	0
$301$	182512.	0.116111
$302$	0	0
$303$	−512496.	−0.320689
$304$	0	0
$305$	−462240.	−0.284523
$306$	0	0
$307$	−2.09039e6	−1.26585	−0.632923	−	0.774215i	$-0.718145\pi$
$307$	−2.09039e6	−1.26585	−0.632923	+	0.774215i	$0.718145\pi$
$308$	0	0
$309$	18396.0	0.0109604
$310$	0	0
$311$	2.63149e6	1.54277	0.771384	−	0.636370i	$-0.219565\pi$
$311$	2.63149e6	1.54277	0.771384	+	0.636370i	$0.219565\pi$
$312$	0	0
$313$	2.37074e6	1.36780	0.683901	−	0.729575i	$-0.260282\pi$
$313$	2.37074e6	1.36780	0.683901	+	0.729575i	$0.260282\pi$
$314$	0	0
$315$	−132192.	−0.0750635
$316$	0	0
$317$	1.13917e6	0.636707	0.318353	−	0.947972i	$-0.396870\pi$
$317$	1.13917e6	0.636707	0.318353	+	0.947972i	$0.396870\pi$
$318$	0	0
$319$	−687040.	−0.378012
$320$	0	0
$321$	2.10323e6	1.13926
$322$	0	0
$323$	−1.04166e6	−0.555543
$324$	0	0
$325$	−2.68155e6	−1.40824
$326$	0	0
$327$	−442332.	−0.228759
$328$	0	0
$329$	−730592.	−0.372122
$330$	0	0
$331$	−263132.	−0.132009	−0.0660045	−	0.997819i	$-0.521025\pi$
$331$	−263132.	−0.132009	−0.0660045	+	0.997819i	$0.521025\pi$
$332$	0	0
$333$	−726732.	−0.359140
$334$	0	0
$335$	878112.	0.427502
$336$	0	0
$337$	−3.67507e6	−1.76275	−0.881375	−	0.472417i	$-0.843382\pi$
$337$	−3.67507e6	−1.76275	−0.881375	+	0.472417i	$0.843382\pi$
$338$	0	0
$339$	937854.	0.443237
$340$	0	0
$341$	880496.	0.410054
$342$	0	0
$343$	1.97132e6	0.904736
$344$	0	0
$345$	−516672.	−0.233705
$346$	0	0
$347$	−3.76033e6	−1.67650	−0.838248	−	0.545289i	$-0.816420\pi$
$347$	−3.76033e6	−1.67650	−0.838248	+	0.545289i	$0.816420\pi$
$348$	0	0
$349$	−348556.	−0.153182	−0.0765912	−	0.997063i	$-0.524404\pi$
$349$	−348556.	−0.153182	−0.0765912	+	0.997063i	$0.524404\pi$
$350$	0	0
$351$	−766908.	−0.332258
$352$	0	0
$353$	−1822.00	−0.000778237 0	−0.000389118	−	1.00000i	$-0.500124\pi$
$353$	−1822.00	−0.000778237 0	−0.000389118		1.00000i	$0.500124\pi$
$354$	0	0
$355$	−1.53197e6	−0.645177
$356$	0	0
$357$	−255816.	−0.106232
$358$	0	0
$359$	1.16762e6	0.478153	0.239077	−	0.971001i	$-0.423155\pi$
$359$	1.16762e6	0.478153	0.239077	+	0.971001i	$0.423155\pi$
$360$	0	0
$361$	3.73396e6	1.50800
$362$	0	0
$363$	−389277.	−0.155057
$364$	0	0
$365$	338544.	0.133010
$366$	0	0
$367$	−2.54202e6	−0.985176	−0.492588	−	0.870263i	$-0.663949\pi$
$367$	−2.54202e6	−0.985176	−0.492588	+	0.870263i	$0.663949\pi$
$368$	0	0
$369$	1.22909e6	0.469915
$370$	0	0
$371$	547264.	0.206425
$372$	0	0
$373$	−4.32100e6	−1.60810	−0.804049	−	0.594563i	$-0.797325\pi$
$373$	−4.32100e6	−1.60810	−0.804049	+	0.594563i	$0.797325\pi$
$374$	0	0
$375$	1.22558e6	0.450054
$376$	0	0
$377$	−1.59904e6	−0.579437
$378$	0	0
$379$	301636.	0.107866	0.0539331	−	0.998545i	$-0.482824\pi$
$379$	301636.	0.107866	0.0539331	+	0.998545i	$0.482824\pi$
$380$	0	0
$381$	−2.05952e6	−0.726866
$382$	0	0
$383$	3.63416e6	1.26592	0.632961	−	0.774183i	$-0.281839\pi$
$383$	3.63416e6	1.26592	0.632961	+	0.774183i	$0.281839\pi$
$384$	0	0
$385$	−737664.	−0.253634
$386$	0	0
$387$	−217404.	−0.0737887
$388$	0	0
$389$	1.16982e6	0.391962	0.195981	−	0.980608i	$-0.437211\pi$
$389$	1.16982e6	0.391962	0.195981	+	0.980608i	$0.437211\pi$
$390$	0	0
$391$	−999856.	−0.330747
$392$	0	0
$393$	721332.	0.235588
$394$	0	0
$395$	2.37379e6	0.765508
$396$	0	0
$397$	3.24688e6	1.03393	0.516964	−	0.856007i	$-0.327062\pi$
$397$	3.24688e6	1.03393	0.516964	+	0.856007i	$0.327062\pi$
$398$	0	0
$399$	1.52510e6	0.479587
$400$	0	0
$401$	726910.	0.225746	0.112873	−	0.993609i	$-0.463995\pi$
$401$	726910.	0.225746	0.112873	+	0.993609i	$0.463995\pi$
$402$	0	0
$403$	2.04930e6	0.628553
$404$	0	0
$405$	157464.	0.0477028
$406$	0	0
$407$	−4.05534e6	−1.21351
$408$	0	0
$409$	−3.31001e6	−0.978410	−0.489205	−	0.872169i	$-0.662713\pi$
$409$	−3.31001e6	−0.978410	−0.489205	+	0.872169i	$0.662713\pi$
$410$	0	0
$411$	2.62996e6	0.767971
$412$	0	0
$413$	908208.	0.262005
$414$	0	0
$415$	1.51901e6	0.432952
$416$	0	0
$417$	389484.	0.109686
$418$	0	0
$419$	3.59486e6	1.00034	0.500169	−	0.865928i	$-0.333271\pi$
$419$	3.59486e6	1.00034	0.500169	+	0.865928i	$0.333271\pi$
$420$	0	0
$421$	−2.47356e6	−0.680168	−0.340084	−	0.940395i	$-0.610456\pi$
$421$	−2.47356e6	−0.680168	−0.340084	+	0.940395i	$0.610456\pi$
$422$	0	0
$423$	870264.	0.236483
$424$	0	0
$425$	1.06548e6	0.286137
$426$	0	0
$427$	1.30968e6	0.347613
$428$	0	0
$429$	−4.27954e6	−1.12267
$430$	0	0
$431$	−4.38838e6	−1.13792	−0.568959	−	0.822366i	$-0.692654\pi$
$431$	−4.38838e6	−1.13792	−0.568959	+	0.822366i	$0.692654\pi$
$432$	0	0
$433$	2.02273e6	0.518465	0.259232	−	0.965815i	$-0.416530\pi$
$433$	2.02273e6	0.518465	0.259232	+	0.965815i	$0.416530\pi$
$434$	0	0
$435$	328320.	0.0831906
$436$	0	0
$437$	5.96086e6	1.49316
$438$	0	0
$439$	−6.09352e6	−1.50906	−0.754530	−	0.656265i	$-0.772135\pi$
$439$	−6.09352e6	−1.50906	−0.754530	+	0.656265i	$0.772135\pi$
$440$	0	0
$441$	−986823.	−0.241626
$442$	0	0
$443$	−3.21155e6	−0.777508	−0.388754	−	0.921342i	$-0.627094\pi$
$443$	−3.21155e6	−0.777508	−0.388754	+	0.921342i	$0.627094\pi$
$444$	0	0
$445$	−168336.	−0.0402974
$446$	0	0
$447$	1.53338e6	0.362980
$448$	0	0
$449$	4.59653e6	1.07600	0.538002	−	0.842944i	$-0.319179\pi$
$449$	4.59653e6	1.07600	0.538002	+	0.842944i	$0.319179\pi$
$450$	0	0
$451$	6.85865e6	1.58780
$452$	0	0
$453$	3.24853e6	0.743775
$454$	0	0
$455$	−1.71686e6	−0.388783
$456$	0	0
$457$	5.40316e6	1.21020	0.605100	−	0.796149i	$-0.293133\pi$
$457$	5.40316e6	1.21020	0.605100	+	0.796149i	$0.293133\pi$
$458$	0	0
$459$	304722.	0.0675106
$460$	0	0
$461$	−114312.	−0.0250518	−0.0125259	−	0.999922i	$-0.503987\pi$
$461$	−114312.	−0.0250518	−0.0125259	+	0.999922i	$0.503987\pi$
$462$	0	0
$463$	−533324.	−0.115622	−0.0578108	−	0.998328i	$-0.518412\pi$
$463$	−533324.	−0.115622	−0.0578108	+	0.998328i	$0.518412\pi$
$464$	0	0
$465$	−420768.	−0.0902423
$466$	0	0
$467$	−3.79890e6	−0.806057	−0.403028	−	0.915187i	$-0.632042\pi$
$467$	−3.79890e6	−0.806057	−0.403028	+	0.915187i	$0.632042\pi$
$468$	0	0
$469$	−2.48798e6	−0.522295
$470$	0	0
$471$	−1.00962e6	−0.209703
$472$	0	0
$473$	−1.21317e6	−0.249326
$474$	0	0
$475$	−6.35211e6	−1.29177
$476$	0	0
$477$	−651888.	−0.131183
$478$	0	0
$479$	757816.	0.150912	0.0754562	−	0.997149i	$-0.475959\pi$
$479$	757816.	0.150912	0.0754562	+	0.997149i	$0.475959\pi$
$480$	0	0
$481$	−9.43854e6	−1.86013
$482$	0	0
$483$	1.46390e6	0.285525
$484$	0	0
$485$	1.93992e6	0.374481
$486$	0	0
$487$	−3.23823e6	−0.618707	−0.309353	−	0.950947i	$-0.600113\pi$
$487$	−3.23823e6	−0.618707	−0.309353	+	0.950947i	$0.600113\pi$
$488$	0	0
$489$	−3.69428e6	−0.698648
$490$	0	0
$491$	−1.76228e6	−0.329892	−0.164946	−	0.986303i	$-0.552745\pi$
$491$	−1.76228e6	−0.329892	−0.164946	+	0.986303i	$0.552745\pi$
$492$	0	0
$493$	635360.	0.117734
$494$	0	0
$495$	878688.	0.161184
$496$	0	0
$497$	4.34058e6	0.788236
$498$	0	0
$499$	3.04406e6	0.547270	0.273635	−	0.961834i	$-0.411774\pi$
$499$	3.04406e6	0.547270	0.273635	+	0.961834i	$0.411774\pi$
$500$	0	0
$501$	−5.05008e6	−0.898885
$502$	0	0
$503$	−1.07469e7	−1.89393	−0.946967	−	0.321332i	$-0.895870\pi$
$503$	−1.07469e7	−1.89393	−0.946967	+	0.321332i	$0.895870\pi$
$504$	0	0
$505$	1.36666e6	0.238468
$506$	0	0
$507$	−6.61870e6	−1.14354
$508$	0	0
$509$	1.00302e7	1.71599	0.857994	−	0.513660i	$-0.171711\pi$
$509$	1.00302e7	1.71599	0.857994	+	0.513660i	$0.171711\pi$
$510$	0	0
$511$	−959208.	−0.162503
$512$	0	0
$513$	−1.81667e6	−0.304777
$514$	0	0
$515$	−49056.0	−0.00815031
$516$	0	0
$517$	4.85629e6	0.799058
$518$	0	0
$519$	−6.24521e6	−1.01772
$520$	0	0
$521$	−3.76262e6	−0.607290	−0.303645	−	0.952785i	$-0.598204\pi$
$521$	−3.76262e6	−0.607290	−0.303645	+	0.952785i	$0.598204\pi$
$522$	0	0
$523$	2.55458e6	0.408381	0.204190	−	0.978931i	$-0.434544\pi$
$523$	2.55458e6	0.408381	0.204190	+	0.978931i	$0.434544\pi$
$524$	0	0
$525$	−1.55999e6	−0.247015
$526$	0	0
$527$	−814264.	−0.127714
$528$	0	0
$529$	−714679.	−0.111038
$530$	0	0
$531$	−1.08184e6	−0.166504
$532$	0	0
$533$	1.59630e7	2.43387
$534$	0	0
$535$	−5.60861e6	−0.847170
$536$	0	0
$537$	−4.26668e6	−0.638491
$538$	0	0
$539$	−5.50672e6	−0.816433
$540$	0	0
$541$	−5.38768e6	−0.791424	−0.395712	−	0.918375i	$-0.629502\pi$
$541$	−5.38768e6	−0.791424	−0.395712	+	0.918375i	$0.629502\pi$
$542$	0	0
$543$	−1.65712e6	−0.241187
$544$	0	0
$545$	1.17955e6	0.170108
$546$	0	0
$547$	−5.66807e6	−0.809966	−0.404983	−	0.914324i	$-0.632723\pi$
$547$	−5.66807e6	−0.809966	−0.404983	+	0.914324i	$0.632723\pi$
$548$	0	0
$549$	−1.56006e6	−0.220907
$550$	0	0
$551$	−3.78784e6	−0.531511
$552$	0	0
$553$	−6.72574e6	−0.935250
$554$	0	0
$555$	1.93795e6	0.267061
$556$	0	0
$557$	1.08279e7	1.47879	0.739393	−	0.673274i	$-0.235112\pi$
$557$	1.08279e7	1.47879	0.739393	+	0.673274i	$0.235112\pi$
$558$	0	0
$559$	−2.82357e6	−0.382181
$560$	0	0
$561$	1.70042e6	0.228113
$562$	0	0
$563$	7.46747e6	0.992893	0.496446	−	0.868067i	$-0.334638\pi$
$563$	7.46747e6	0.992893	0.496446	+	0.868067i	$0.334638\pi$
$564$	0	0
$565$	−2.50094e6	−0.329597
$566$	0	0
$567$	−446148.	−0.0582802
$568$	0	0
$569$	7.03461e6	0.910877	0.455438	−	0.890267i	$-0.349483\pi$
$569$	7.03461e6	0.910877	0.455438	+	0.890267i	$0.349483\pi$
$570$	0	0
$571$	−1.49185e7	−1.91485	−0.957425	−	0.288683i	$-0.906783\pi$
$571$	−1.49185e7	−1.91485	−0.957425	+	0.288683i	$0.906783\pi$
$572$	0	0
$573$	−5.38805e6	−0.685559
$574$	0	0
$575$	−6.09721e6	−0.769062
$576$	0	0
$577$	4.10498e6	0.513300	0.256650	−	0.966504i	$-0.417381\pi$
$577$	4.10498e6	0.513300	0.256650	+	0.966504i	$0.417381\pi$
$578$	0	0
$579$	5.80279e6	0.719350
$580$	0	0
$581$	−4.30386e6	−0.528953
$582$	0	0
$583$	−3.63770e6	−0.443256
$584$	0	0
$585$	2.04509e6	0.247071
$586$	0	0
$587$	1.16215e6	0.139209	0.0696043	−	0.997575i	$-0.477826\pi$
$587$	1.16215e6	0.139209	0.0696043	+	0.997575i	$0.477826\pi$
$588$	0	0
$589$	4.85442e6	0.576566
$590$	0	0
$591$	−9.26726e6	−1.09140
$592$	0	0
$593$	−6.78104e6	−0.791880	−0.395940	−	0.918276i	$-0.629581\pi$
$593$	−6.78104e6	−0.791880	−0.395940	+	0.918276i	$0.629581\pi$
$594$	0	0
$595$	682176.	0.0789958
$596$	0	0
$597$	−8.45572e6	−0.970990
$598$	0	0
$599$	5.29236e6	0.602674	0.301337	−	0.953518i	$-0.402567\pi$
$599$	5.29236e6	0.602674	0.301337	+	0.953518i	$0.402567\pi$
$600$	0	0
$601$	1.46032e7	1.64916	0.824579	−	0.565747i	$-0.191412\pi$
$601$	1.46032e7	1.64916	0.824579	+	0.565747i	$0.191412\pi$
$602$	0	0
$603$	2.96363e6	0.331918
$604$	0	0
$605$	1.03807e6	0.115303
$606$	0	0
$607$	2.03232e6	0.223883	0.111942	−	0.993715i	$-0.464293\pi$
$607$	2.03232e6	0.223883	0.111942	+	0.993715i	$0.464293\pi$
$608$	0	0
$609$	−930240.	−0.101637
$610$	0	0
$611$	1.13027e7	1.22484
$612$	0	0
$613$	−1.75995e7	−1.89169	−0.945843	−	0.324624i	$-0.894762\pi$
$613$	−1.75995e7	−1.89169	−0.945843	+	0.324624i	$0.894762\pi$
$614$	0	0
$615$	−3.27758e6	−0.349435
$616$	0	0
$617$	−3.14133e6	−0.332201	−0.166101	−	0.986109i	$-0.553118\pi$
$617$	−3.14133e6	−0.332201	−0.166101	+	0.986109i	$0.553118\pi$
$618$	0	0
$619$	1.62433e7	1.70391	0.851955	−	0.523615i	$-0.175417\pi$
$619$	1.62433e7	1.70391	0.851955	+	0.523615i	$0.175417\pi$
$620$	0	0
$621$	−1.74377e6	−0.181451
$622$	0	0
$623$	476952.	0.0492328
$624$	0	0
$625$	4.69740e6	0.481014
$626$	0	0
$627$	−1.01375e7	−1.02982
$628$	0	0
$629$	3.75030e6	0.377954
$630$	0	0
$631$	−3.51189e6	−0.351130	−0.175565	−	0.984468i	$-0.556175\pi$
$631$	−3.51189e6	−0.351130	−0.175565	+	0.984468i	$0.556175\pi$
$632$	0	0
$633$	8.59770e6	0.852851
$634$	0	0
$635$	5.49206e6	0.540507
$636$	0	0
$637$	−1.28165e7	−1.25147
$638$	0	0
$639$	−5.17039e6	−0.500923
$640$	0	0
$641$	−1.68831e7	−1.62295	−0.811477	−	0.584385i	$-0.801336\pi$
$641$	−1.68831e7	−1.62295	−0.811477	+	0.584385i	$0.801336\pi$
$642$	0	0
$643$	−1.23824e6	−0.118108	−0.0590539	−	0.998255i	$-0.518808\pi$
$643$	−1.23824e6	−0.118108	−0.0590539	+	0.998255i	$0.518808\pi$
$644$	0	0
$645$	579744.	0.0548703
$646$	0	0
$647$	−1.72670e7	−1.62165	−0.810824	−	0.585290i	$-0.800981\pi$
$647$	−1.72670e7	−1.62165	−0.810824	+	0.585290i	$0.800981\pi$
$648$	0	0
$649$	−6.03691e6	−0.562604
$650$	0	0
$651$	1.19218e6	0.110252
$652$	0	0
$653$	−4.00614e6	−0.367657	−0.183828	−	0.982958i	$-0.558849\pi$
$653$	−4.00614e6	−0.367657	−0.183828	+	0.982958i	$0.558849\pi$
$654$	0	0
$655$	−1.92355e6	−0.175187
$656$	0	0
$657$	1.14259e6	0.103270
$658$	0	0
$659$	−1.46889e7	−1.31758	−0.658788	−	0.752329i	$-0.728931\pi$
$659$	−1.46889e7	−1.31758	−0.658788	+	0.752329i	$0.728931\pi$
$660$	0	0
$661$	−6.13185e6	−0.545868	−0.272934	−	0.962033i	$-0.587994\pi$
$661$	−6.13185e6	−0.545868	−0.272934	+	0.962033i	$0.587994\pi$
$662$	0	0
$663$	3.95762e6	0.349664
$664$	0	0
$665$	−4.06694e6	−0.356627
$666$	0	0
$667$	−3.63584e6	−0.316439
$668$	0	0
$669$	−1.27400e6	−0.110054
$670$	0	0
$671$	−8.70552e6	−0.746429
$672$	0	0
$673$	2.29363e7	1.95203	0.976013	−	0.217714i	$-0.0698600\pi$
$673$	2.29363e7	1.95203	0.976013	+	0.217714i	$0.0698600\pi$
$674$	0	0
$675$	1.85822e6	0.156978
$676$	0	0
$677$	1.79097e7	1.50181	0.750907	−	0.660408i	$-0.229617\pi$
$677$	1.79097e7	1.50181	0.750907	+	0.660408i	$0.229617\pi$
$678$	0	0
$679$	−5.49644e6	−0.457517
$680$	0	0
$681$	1.69578e6	0.140121
$682$	0	0
$683$	3.45928e6	0.283749	0.141875	−	0.989885i	$-0.454687\pi$
$683$	3.45928e6	0.283749	0.141875	+	0.989885i	$0.454687\pi$
$684$	0	0
$685$	−7.01323e6	−0.571073
$686$	0	0
$687$	1.01529e7	0.820726
$688$	0	0
$689$	−8.46650e6	−0.679447
$690$	0	0
$691$	−1.23985e7	−0.987815	−0.493908	−	0.869514i	$-0.664432\pi$
$691$	−1.23985e7	−0.987815	−0.493908	+	0.869514i	$0.664432\pi$
$692$	0	0
$693$	−2.48962e6	−0.196924
$694$	0	0
$695$	−1.03862e6	−0.0815636
$696$	0	0
$697$	−6.34273e6	−0.494532
$698$	0	0
$699$	−9.58876e6	−0.742283
$700$	0	0
$701$	2.26502e7	1.74091	0.870455	−	0.492247i	$-0.163824\pi$
$701$	2.26502e7	1.74091	0.870455	+	0.492247i	$0.163824\pi$
$702$	0	0
$703$	−2.23582e7	−1.70627
$704$	0	0
$705$	−2.32070e6	−0.175852
$706$	0	0
$707$	−3.87219e6	−0.291346
$708$	0	0
$709$	1.14521e7	0.855598	0.427799	−	0.903874i	$-0.359289\pi$
$709$	1.14521e7	0.855598	0.427799	+	0.903874i	$0.359289\pi$
$710$	0	0
$711$	8.01155e6	0.594350
$712$	0	0
$713$	4.65962e6	0.343262
$714$	0	0
$715$	1.14121e7	0.834834
$716$	0	0
$717$	927936.	0.0674093
$718$	0	0
$719$	−1.55789e7	−1.12387	−0.561933	−	0.827183i	$-0.689942\pi$
$719$	−1.55789e7	−1.12387	−0.561933	+	0.827183i	$0.689942\pi$
$720$	0	0
$721$	138992.	0.00995753
$722$	0	0
$723$	705042.	0.0501614
$724$	0	0
$725$	3.87448e6	0.273759
$726$	0	0
$727$	2.34665e7	1.64669	0.823347	−	0.567539i	$-0.192104\pi$
$727$	2.34665e7	1.64669	0.823347	+	0.567539i	$0.192104\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.12191e6	0.0776543
$732$	0	0
$733$	−1.62922e7	−1.12001	−0.560003	−	0.828490i	$-0.689200\pi$
$733$	−1.62922e7	−1.12001	−0.560003	+	0.828490i	$0.689200\pi$
$734$	0	0
$735$	2.63153e6	0.179676
$736$	0	0
$737$	1.65378e7	1.12152
$738$	0	0
$739$	−129476.	−0.00872124	−0.00436062	−	0.999990i	$-0.501388\pi$
$739$	−129476.	−0.00872124	−0.00436062	+	0.999990i	$0.501388\pi$
$740$	0	0
$741$	−2.35943e7	−1.57856
$742$	0	0
$743$	1.87209e7	1.24410	0.622048	−	0.782979i	$-0.286301\pi$
$743$	1.87209e7	1.24410	0.622048	+	0.782979i	$0.286301\pi$
$744$	0	0
$745$	−4.08902e6	−0.269916
$746$	0	0
$747$	5.12665e6	0.336149
$748$	0	0
$749$	1.58911e7	1.03502
$750$	0	0
$751$	−6.65407e6	−0.430514	−0.215257	−	0.976557i	$-0.569059\pi$
$751$	−6.65407e6	−0.430514	−0.215257	+	0.976557i	$0.569059\pi$
$752$	0	0
$753$	−5.97704e6	−0.384148
$754$	0	0
$755$	−8.66275e6	−0.553081
$756$	0	0
$757$	1.44773e7	0.918222	0.459111	−	0.888379i	$-0.348168\pi$
$757$	1.44773e7	0.918222	0.459111	+	0.888379i	$0.348168\pi$
$758$	0	0
$759$	−9.73066e6	−0.613109
$760$	0	0
$761$	−3.99291e6	−0.249935	−0.124968	−	0.992161i	$-0.539883\pi$
$761$	−3.99291e6	−0.249935	−0.124968	+	0.992161i	$0.539883\pi$
$762$	0	0
$763$	−3.34206e6	−0.207828
$764$	0	0
$765$	−812592.	−0.0502018
$766$	0	0
$767$	−1.40505e7	−0.862390
$768$	0	0
$769$	−5.55643e6	−0.338829	−0.169414	−	0.985545i	$-0.554188\pi$
$769$	−5.55643e6	−0.338829	−0.169414	+	0.985545i	$0.554188\pi$
$770$	0	0
$771$	−3.66107e6	−0.221806
$772$	0	0
$773$	329984.	0.0198630	0.00993148	−	0.999951i	$-0.496839\pi$
$773$	329984.	0.0198630	0.00993148	+	0.999951i	$0.496839\pi$
$774$	0	0
$775$	−4.96545e6	−0.296964
$776$	0	0
$777$	−5.49086e6	−0.326278
$778$	0	0
$779$	3.78136e7	2.23257
$780$	0	0
$781$	−2.88521e7	−1.69258
$782$	0	0
$783$	1.10808e6	0.0645902
$784$	0	0
$785$	2.69232e6	0.155938
$786$	0	0
$787$	3.36500e6	0.193664	0.0968320	−	0.995301i	$-0.469129\pi$
$787$	3.36500e6	0.193664	0.0968320	+	0.995301i	$0.469129\pi$
$788$	0	0
$789$	−1.38835e7	−0.793973
$790$	0	0
$791$	7.08601e6	0.402680
$792$	0	0
$793$	−2.02615e7	−1.14417
$794$	0	0
$795$	1.73837e6	0.0975492
$796$	0	0
$797$	7.80702e6	0.435351	0.217675	−	0.976021i	$-0.430153\pi$
$797$	7.80702e6	0.435351	0.217675	+	0.976021i	$0.430153\pi$
$798$	0	0
$799$	−4.49099e6	−0.248872
$800$	0	0
$801$	−568134.	−0.0312874
$802$	0	0
$803$	6.37591e6	0.348942
$804$	0	0
$805$	−3.90374e6	−0.212320
$806$	0	0
$807$	1.11858e7	0.604620
$808$	0	0
$809$	1.27675e7	0.685857	0.342928	−	0.939362i	$-0.388581\pi$
$809$	1.27675e7	0.685857	0.342928	+	0.939362i	$0.388581\pi$
$810$	0	0
$811$	2.62901e7	1.40359	0.701796	−	0.712378i	$-0.252382\pi$
$811$	2.62901e7	1.40359	0.701796	+	0.712378i	$0.252382\pi$
$812$	0	0
$813$	−1.79668e7	−0.953331
$814$	0	0
$815$	9.85142e6	0.519523
$816$	0	0
$817$	−6.68853e6	−0.350571
$818$	0	0
$819$	−5.79442e6	−0.301856
$820$	0	0
$821$	−1.94367e7	−1.00639	−0.503194	−	0.864173i	$-0.667842\pi$
$821$	−1.94367e7	−1.00639	−0.503194	+	0.864173i	$0.667842\pi$
$822$	0	0
$823$	−1.31027e7	−0.674314	−0.337157	−	0.941448i	$-0.609465\pi$
$823$	−1.31027e7	−0.674314	−0.337157	+	0.941448i	$0.609465\pi$
$824$	0	0
$825$	1.03693e7	0.530415
$826$	0	0
$827$	−1.81039e7	−0.920467	−0.460234	−	0.887798i	$-0.652234\pi$
$827$	−1.81039e7	−0.920467	−0.460234	+	0.887798i	$0.652234\pi$
$828$	0	0
$829$	−1.73353e7	−0.876083	−0.438041	−	0.898955i	$-0.644328\pi$
$829$	−1.73353e7	−0.876083	−0.438041	+	0.898955i	$0.644328\pi$
$830$	0	0
$831$	6.17087e6	0.309987
$832$	0	0
$833$	5.09249e6	0.254283
$834$	0	0
$835$	1.34669e7	0.668422
$836$	0	0
$837$	−1.42009e6	−0.0700653
$838$	0	0
$839$	2.80058e7	1.37355	0.686773	−	0.726872i	$-0.259027\pi$
$839$	2.80058e7	1.37355	0.686773	+	0.726872i	$0.259027\pi$
$840$	0	0
$841$	−1.82007e7	−0.887359
$842$	0	0
$843$	1.54337e7	0.747998
$844$	0	0
$845$	1.76499e7	0.850354
$846$	0	0
$847$	−2.94120e6	−0.140869
$848$	0	0
$849$	−1.17617e7	−0.560015
$850$	0	0
$851$	−2.14610e7	−1.01584
$852$	0	0
$853$	1.97280e6	0.0928349	0.0464175	−	0.998922i	$-0.485220\pi$
$853$	1.97280e6	0.0928349	0.0464175	+	0.998922i	$0.485220\pi$
$854$	0	0
$855$	4.84445e6	0.226636
$856$	0	0
$857$	2.00624e7	0.933104	0.466552	−	0.884494i	$-0.345496\pi$
$857$	2.00624e7	0.933104	0.466552	+	0.884494i	$0.345496\pi$
$858$	0	0
$859$	−3.38191e7	−1.56379	−0.781896	−	0.623409i	$-0.785747\pi$
$859$	−3.38191e7	−1.56379	−0.781896	+	0.623409i	$0.785747\pi$
$860$	0	0
$861$	9.28649e6	0.426917
$862$	0	0
$863$	2.63424e7	1.20401	0.602003	−	0.798494i	$-0.294370\pi$
$863$	2.63424e7	1.20401	0.602003	+	0.798494i	$0.294370\pi$
$864$	0	0
$865$	1.66539e7	0.756790
$866$	0	0
$867$	1.12062e7	0.506303
$868$	0	0
$869$	4.47064e7	2.00826
$870$	0	0
$871$	3.84906e7	1.71913
$872$	0	0
$873$	6.54723e6	0.290752
$874$	0	0
$875$	9.25997e6	0.408874
$876$	0	0
$877$	−1.59896e7	−0.702002	−0.351001	−	0.936375i	$-0.614159\pi$
$877$	−1.59896e7	−0.702002	−0.351001	+	0.936375i	$0.614159\pi$
$878$	0	0
$879$	−2.01213e7	−0.878381
$880$	0	0
$881$	−2.29900e7	−0.997929	−0.498965	−	0.866622i	$-0.666286\pi$
$881$	−2.29900e7	−0.997929	−0.498965	+	0.866622i	$0.666286\pi$
$882$	0	0
$883$	−2.06949e7	−0.893226	−0.446613	−	0.894727i	$-0.647370\pi$
$883$	−2.06949e7	−0.893226	−0.446613	+	0.894727i	$0.647370\pi$
$884$	0	0
$885$	2.88490e6	0.123815
$886$	0	0
$887$	6.40267e6	0.273245	0.136622	−	0.990623i	$-0.456375\pi$
$887$	6.40267e6	0.273245	0.136622	+	0.990623i	$0.456375\pi$
$888$	0	0
$889$	−1.55608e7	−0.660357
$890$	0	0
$891$	2.96557e6	0.125145
$892$	0	0
$893$	2.67740e7	1.12353
$894$	0	0
$895$	1.13778e7	0.474790
$896$	0	0
$897$	−2.26475e7	−0.939806
$898$	0	0
$899$	−2.96096e6	−0.122189
$900$	0	0
$901$	3.36406e6	0.138055
$902$	0	0
$903$	−1.64261e6	−0.0670370
$904$	0	0
$905$	4.41898e6	0.179350
$906$	0	0
$907$	8.72660e6	0.352230	0.176115	−	0.984370i	$-0.443647\pi$
$907$	8.72660e6	0.352230	0.176115	+	0.984370i	$0.443647\pi$
$908$	0	0
$909$	4.61246e6	0.185150
$910$	0	0
$911$	3.54100e7	1.41361	0.706805	−	0.707408i	$-0.250136\pi$
$911$	3.54100e7	1.41361	0.706805	+	0.707408i	$0.250136\pi$
$912$	0	0
$913$	2.86080e7	1.13582
$914$	0	0
$915$	4.16016e6	0.164270
$916$	0	0
$917$	5.45006e6	0.214032
$918$	0	0
$919$	−3.58698e6	−0.140101	−0.0700503	−	0.997543i	$-0.522316\pi$
$919$	−3.58698e6	−0.140101	−0.0700503	+	0.997543i	$0.522316\pi$
$920$	0	0
$921$	1.88135e7	0.730837
$922$	0	0
$923$	−6.71513e7	−2.59448
$924$	0	0
$925$	2.28696e7	0.878830
$926$	0	0
$927$	−165564.	−0.00632800
$928$	0	0
$929$	2.86328e7	1.08849	0.544245	−	0.838926i	$-0.316816\pi$
$929$	2.86328e7	1.08849	0.544245	+	0.838926i	$0.316816\pi$
$930$	0	0
$931$	−3.03600e7	−1.14796
$932$	0	0
$933$	−2.36834e7	−0.890717
$934$	0	0
$935$	−4.53446e6	−0.169628
$936$	0	0
$937$	−2.36737e7	−0.880882	−0.440441	−	0.897782i	$-0.645178\pi$
$937$	−2.36737e7	−0.880882	−0.440441	+	0.897782i	$0.645178\pi$
$938$	0	0
$939$	−2.13367e7	−0.789701
$940$	0	0
$941$	4.63182e7	1.70521	0.852604	−	0.522558i	$-0.175022\pi$
$941$	4.63182e7	1.70521	0.852604	+	0.522558i	$0.175022\pi$
$942$	0	0
$943$	3.62962e7	1.32917
$944$	0	0
$945$	1.18973e6	0.0433379
$946$	0	0
$947$	6.76081e6	0.244976	0.122488	−	0.992470i	$-0.460913\pi$
$947$	6.76081e6	0.244976	0.122488	+	0.992470i	$0.460913\pi$
$948$	0	0
$949$	1.48395e7	0.534877
$950$	0	0
$951$	−1.02525e7	−0.367603
$952$	0	0
$953$	4.57286e7	1.63101	0.815503	−	0.578753i	$-0.196460\pi$
$953$	4.57286e7	1.63101	0.815503	+	0.578753i	$0.196460\pi$
$954$	0	0
$955$	1.43681e7	0.509791
$956$	0	0
$957$	6.18336e6	0.218245
$958$	0	0
$959$	1.98708e7	0.697701
$960$	0	0
$961$	−2.48344e7	−0.867453
$962$	0	0
$963$	−1.89291e7	−0.657753
$964$	0	0
$965$	−1.54741e7	−0.534918
$966$	0	0
$967$	2.68735e7	0.924182	0.462091	−	0.886833i	$-0.347099\pi$
$967$	2.68735e7	0.924182	0.462091	+	0.886833i	$0.347099\pi$
$968$	0	0
$969$	9.37490e6	0.320743
$970$	0	0
$971$	3.29846e7	1.12270	0.561349	−	0.827579i	$-0.310282\pi$
$971$	3.29846e7	1.12270	0.561349	+	0.827579i	$0.310282\pi$
$972$	0	0
$973$	2.94277e6	0.0996492
$974$	0	0
$975$	2.41339e7	0.813049
$976$	0	0
$977$	−2.04331e7	−0.684853	−0.342427	−	0.939545i	$-0.611249\pi$
$977$	−2.04331e7	−0.684853	−0.342427	+	0.939545i	$0.611249\pi$
$978$	0	0
$979$	−3.17033e6	−0.105718
$980$	0	0
$981$	3.98099e6	0.132074
$982$	0	0
$983$	2.54622e6	0.0840452	0.0420226	−	0.999117i	$-0.486620\pi$
$983$	2.54622e6	0.0840452	0.0420226	+	0.999117i	$0.486620\pi$
$984$	0	0
$985$	2.47127e7	0.811577
$986$	0	0
$987$	6.57533e6	0.214845
$988$	0	0
$989$	−6.42013e6	−0.208715
$990$	0	0
$991$	4.23354e7	1.36937	0.684683	−	0.728841i	$-0.259941\pi$
$991$	4.23354e7	1.36937	0.684683	+	0.728841i	$0.259941\pi$
$992$	0	0
$993$	2.36819e6	0.0762155
$994$	0	0
$995$	2.25486e7	0.722040
$996$	0	0
$997$	4.50922e6	0.143669	0.0718346	−	0.997417i	$-0.477115\pi$
$997$	4.50922e6	0.143669	0.0718346	+	0.997417i	$0.477115\pi$
$998$	0	0
$999$	6.54059e6	0.207350

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.6.a.d.1.1		1
4.3	odd	2		768.6.a.j.1.1		1
8.3	odd	2		768.6.a.c.1.1		1
8.5	even	2		768.6.a.i.1.1		1
16.3	odd	4		384.6.d.b.193.2	yes	2
16.5	even	4		384.6.d.e.193.2	yes	2
16.11	odd	4		384.6.d.b.193.1	✓	2
16.13	even	4		384.6.d.e.193.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.d.b.193.1	✓	2	16.11	odd	4
384.6.d.b.193.2	yes	2	16.3	odd	4
384.6.d.e.193.1	yes	2	16.13	even	4
384.6.d.e.193.2	yes	2	16.5	even	4
768.6.a.c.1.1		1	8.3	odd	2
768.6.a.d.1.1		1	1.1	even	1	trivial
768.6.a.i.1.1		1	8.5	even	2
768.6.a.j.1.1		1	4.3	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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