LMFDB - Embedded newform 152.6.a.c.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [152,6,Mod(1,152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("152.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 152 = 2^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	152.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$24.3783406116$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 1012x^{4} + 2938x^{3} + 184643x^{2} - 1214504x + 729856 $ x^6 - 2x^5 - 1012x^4 + 2938x^3 + 184643x^2 - 1214504*x + 729856
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$11.3428$ of defining polynomial
Character	$\chi$	$=$	152.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+10.3428 q^{3} -9.61696 q^{5} +11.2548 q^{7} -136.027 q^{9} -268.762 q^{11} +306.020 q^{13} -99.4658 q^{15} -2226.33 q^{17} -361.000 q^{19} +116.406 q^{21} +2982.97 q^{23} -3032.51 q^{25} -3920.19 q^{27} +4066.96 q^{29} -7466.54 q^{31} -2779.74 q^{33} -108.237 q^{35} -11964.4 q^{37} +3165.09 q^{39} +14361.0 q^{41} +10897.4 q^{43} +1308.17 q^{45} -21817.4 q^{47} -16680.3 q^{49} -23026.4 q^{51} -12723.3 q^{53} +2584.67 q^{55} -3733.73 q^{57} -21285.7 q^{59} +2651.42 q^{61} -1530.96 q^{63} -2942.98 q^{65} -32724.6 q^{67} +30852.1 q^{69} -71179.1 q^{71} +73779.2 q^{73} -31364.5 q^{75} -3024.87 q^{77} -48764.6 q^{79} -7490.85 q^{81} +81985.2 q^{83} +21410.5 q^{85} +42063.5 q^{87} +59375.2 q^{89} +3444.20 q^{91} -77224.6 q^{93} +3471.72 q^{95} +106151. q^{97} +36559.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 25 q^{5} - 161 q^{7} + 572 q^{9} - 205 q^{11} - 382 q^{13} - 950 q^{15} - 65 q^{17} - 2166 q^{19} - 6006 q^{21} - 5584 q^{23} - 3111 q^{25} - 6874 q^{27} - 9972 q^{29} - 7140 q^{31} - 3774 q^{33}+ \cdots + 49107 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 - 25 * q^5 - 161 * q^7 + 572 * q^9 - 205 * q^11 - 382 * q^13 - 950 * q^15 - 65 * q^17 - 2166 * q^19 - 6006 * q^21 - 5584 * q^23 - 3111 * q^25 - 6874 * q^27 - 9972 * q^29 - 7140 * q^31 - 3774 * q^33 - 14979 * q^35 - 7872 * q^37 - 46448 * q^39 - 12476 * q^41 - 45829 * q^43 - 41933 * q^45 - 61259 * q^47 + 5315 * q^49 - 72572 * q^51 + 13382 * q^53 - 55199 * q^55 + 1444 * q^57 - 56812 * q^59 + 2559 * q^61 - 97953 * q^63 + 37452 * q^65 - 36678 * q^67 - 12888 * q^69 + 14934 * q^71 + 43567 * q^73 - 57842 * q^75 + 36359 * q^77 - 155476 * q^79 + 199946 * q^81 - 149856 * q^83 + 112601 * q^85 - 95180 * q^87 + 176022 * q^89 + 76214 * q^91 + 434620 * q^93 + 9025 * q^95 + 117190 * q^97 + 49107 * q^99

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
$3$	10.3428	0.663488	0.331744	−	0.943369i	$-0.392363\pi$
$3$	10.3428	0.663488	0.331744	+	0.943369i	$0.392363\pi$
$4$	0	0
$5$	−9.61696	−0.172033	−0.0860167	−	0.996294i	$-0.527414\pi$
$5$	−9.61696	−0.172033	−0.0860167	+	0.996294i	$0.527414\pi$
$6$	0	0
$7$	11.2548	0.0868148	0.0434074	−	0.999057i	$-0.486179\pi$
$7$	11.2548	0.0868148	0.0434074	+	0.999057i	$0.486179\pi$
$8$	0	0
$9$	−136.027	−0.559784
$10$	0	0
$11$	−268.762	−0.669709	−0.334854	−	0.942270i	$-0.608687\pi$
$11$	−268.762	−0.669709	−0.334854	+	0.942270i	$0.608687\pi$
$12$	0	0
$13$	306.020	0.502217	0.251108	−	0.967959i	$-0.419205\pi$
$13$	306.020	0.502217	0.251108	+	0.967959i	$0.419205\pi$
$14$	0	0
$15$	−99.4658	−0.114142
$16$	0	0
$17$	−2226.33	−1.86839	−0.934195	−	0.356762i	$-0.883881\pi$
$17$	−2226.33	−1.86839	−0.934195	+	0.356762i	$0.883881\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	0	0
$21$	116.406	0.0576005
$22$	0	0
$23$	2982.97	1.17579	0.587894	−	0.808938i	$-0.299957\pi$
$23$	2982.97	1.17579	0.587894	+	0.808938i	$0.299957\pi$
$24$	0	0
$25$	−3032.51	−0.970405
$26$	0	0
$27$	−3920.19	−1.03490
$28$	0	0
$29$	4066.96	0.897996	0.448998	−	0.893533i	$-0.351781\pi$
$29$	4066.96	0.897996	0.448998	+	0.893533i	$0.351781\pi$
$30$	0	0
$31$	−7466.54	−1.39545	−0.697727	−	0.716364i	$-0.745805\pi$
$31$	−7466.54	−1.39545	−0.697727	+	0.716364i	$0.745805\pi$
$32$	0	0
$33$	−2779.74	−0.444344
$34$	0	0
$35$	−108.237	−0.0149350
$36$	0	0
$37$	−11964.4	−1.43676	−0.718381	−	0.695649i	$-0.755117\pi$
$37$	−11964.4	−1.43676	−0.718381	+	0.695649i	$0.755117\pi$
$38$	0	0
$39$	3165.09	0.333215
$40$	0	0
$41$	14361.0	1.33421	0.667106	−	0.744963i	$-0.267533\pi$
$41$	14361.0	1.33421	0.667106	+	0.744963i	$0.267533\pi$
$42$	0	0
$43$	10897.4	0.898774	0.449387	−	0.893337i	$-0.351642\pi$
$43$	10897.4	0.898774	0.449387	+	0.893337i	$0.351642\pi$
$44$	0	0
$45$	1308.17	0.0963015
$46$	0	0
$47$	−21817.4	−1.44065	−0.720326	−	0.693636i	$-0.756008\pi$
$47$	−21817.4	−1.44065	−0.720326	+	0.693636i	$0.756008\pi$
$48$	0	0
$49$	−16680.3	−0.992463
$50$	0	0
$51$	−23026.4	−1.23965
$52$	0	0
$53$	−12723.3	−0.622172	−0.311086	−	0.950382i	$-0.600693\pi$
$53$	−12723.3	−0.622172	−0.311086	+	0.950382i	$0.600693\pi$
$54$	0	0
$55$	2584.67	0.115212
$56$	0	0
$57$	−3733.73	−0.152215
$58$	0	0
$59$	−21285.7	−0.796082	−0.398041	−	0.917368i	$-0.630310\pi$
$59$	−21285.7	−0.796082	−0.398041	+	0.917368i	$0.630310\pi$
$60$	0	0
$61$	2651.42	0.0912333	0.0456166	−	0.998959i	$-0.485475\pi$
$61$	2651.42	0.0912333	0.0456166	+	0.998959i	$0.485475\pi$
$62$	0	0
$63$	−1530.96	−0.0485975
$64$	0	0
$65$	−2942.98	−0.0863981
$66$	0	0
$67$	−32724.6	−0.890611	−0.445305	−	0.895379i	$-0.646905\pi$
$67$	−32724.6	−0.890611	−0.445305	+	0.895379i	$0.646905\pi$
$68$	0	0
$69$	30852.1	0.780121
$70$	0	0
$71$	−71179.1	−1.67574	−0.837870	−	0.545869i	$-0.816199\pi$
$71$	−71179.1	−1.67574	−0.837870	+	0.545869i	$0.816199\pi$
$72$	0	0
$73$	73779.2	1.62042	0.810209	−	0.586141i	$-0.199354\pi$
$73$	73779.2	1.62042	0.810209	+	0.586141i	$0.199354\pi$
$74$	0	0
$75$	−31364.5	−0.643852
$76$	0	0
$77$	−3024.87	−0.0581406
$78$	0	0
$79$	−48764.6	−0.879097	−0.439548	−	0.898219i	$-0.644861\pi$
$79$	−48764.6	−0.879097	−0.439548	+	0.898219i	$0.644861\pi$
$80$	0	0
$81$	−7490.85	−0.126858
$82$	0	0
$83$	81985.2	1.30629	0.653146	−	0.757232i	$-0.273449\pi$
$83$	81985.2	1.30629	0.653146	+	0.757232i	$0.273449\pi$
$84$	0	0
$85$	21410.5	0.321425
$86$	0	0
$87$	42063.5	0.595810
$88$	0	0
$89$	59375.2	0.794566	0.397283	−	0.917696i	$-0.369953\pi$
$89$	59375.2	0.794566	0.397283	+	0.917696i	$0.369953\pi$
$90$	0	0
$91$	3444.20	0.0435998
$92$	0	0
$93$	−77224.6	−0.925866
$94$	0	0
$95$	3471.72	0.0394672
$96$	0	0
$97$	106151.	1.14550	0.572751	−	0.819730i	$-0.305876\pi$
$97$	106151.	1.14550	0.572751	+	0.819730i	$0.305876\pi$
$98$	0	0
$99$	36559.0	0.374892
$100$	0	0
$101$	153867.	1.50087	0.750435	−	0.660944i	$-0.229844\pi$
$101$	153867.	1.50087	0.750435	+	0.660944i	$0.229844\pi$
$102$	0	0
$103$	−57044.2	−0.529808	−0.264904	−	0.964275i	$-0.585340\pi$
$103$	−57044.2	−0.529808	−0.264904	+	0.964275i	$0.585340\pi$
$104$	0	0
$105$	−1119.47	−0.00990921
$106$	0	0
$107$	72467.5	0.611904	0.305952	−	0.952047i	$-0.401025\pi$
$107$	72467.5	0.611904	0.305952	+	0.952047i	$0.401025\pi$
$108$	0	0
$109$	188088.	1.51633	0.758166	−	0.652062i	$-0.226096\pi$
$109$	188088.	1.51633	0.758166	+	0.652062i	$0.226096\pi$
$110$	0	0
$111$	−123744.	−0.953275
$112$	0	0
$113$	80426.7	0.592521	0.296261	−	0.955107i	$-0.404260\pi$
$113$	80426.7	0.592521	0.296261	+	0.955107i	$0.404260\pi$
$114$	0	0
$115$	−28687.1	−0.202275
$116$	0	0
$117$	−41627.1	−0.281133
$118$	0	0
$119$	−25057.0	−0.162204
$120$	0	0
$121$	−88818.1	−0.551490
$122$	0	0
$123$	148532.	0.885233
$124$	0	0
$125$	59216.5	0.338975
$126$	0	0
$127$	123385.	0.678820	0.339410	−	0.940639i	$-0.389773\pi$
$127$	123385.	0.678820	0.339410	+	0.940639i	$0.389773\pi$
$128$	0	0
$129$	112709.	0.596326
$130$	0	0
$131$	165155.	0.840841	0.420420	−	0.907329i	$-0.361883\pi$
$131$	165155.	0.840841	0.420420	+	0.907329i	$0.361883\pi$
$132$	0	0
$133$	−4062.99	−0.0199167
$134$	0	0
$135$	37700.3	0.178037
$136$	0	0
$137$	125821.	0.572733	0.286367	−	0.958120i	$-0.407553\pi$
$137$	125821.	0.572733	0.286367	+	0.958120i	$0.407553\pi$
$138$	0	0
$139$	−291448.	−1.27945	−0.639726	−	0.768603i	$-0.720952\pi$
$139$	−291448.	−1.27945	−0.639726	+	0.768603i	$0.720952\pi$
$140$	0	0
$141$	−225652.	−0.955855
$142$	0	0
$143$	−82246.5	−0.336339
$144$	0	0
$145$	−39111.7	−0.154485
$146$	0	0
$147$	−172521.	−0.658487
$148$	0	0
$149$	−272325.	−1.00490	−0.502448	−	0.864607i	$-0.667567\pi$
$149$	−272325.	−1.00490	−0.502448	+	0.864607i	$0.667567\pi$
$150$	0	0
$151$	276941.	0.988428	0.494214	−	0.869340i	$-0.335456\pi$
$151$	276941.	0.988428	0.494214	+	0.869340i	$0.335456\pi$
$152$	0	0
$153$	302842.	1.04589
$154$	0	0
$155$	71805.4	0.240065
$156$	0	0
$157$	40951.6	0.132594	0.0662968	−	0.997800i	$-0.478882\pi$
$157$	40951.6	0.132594	0.0662968	+	0.997800i	$0.478882\pi$
$158$	0	0
$159$	−131594.	−0.412803
$160$	0	0
$161$	33572.8	0.102076
$162$	0	0
$163$	9129.77	0.0269148	0.0134574	−	0.999909i	$-0.495716\pi$
$163$	9129.77	0.0269148	0.0134574	+	0.999909i	$0.495716\pi$
$164$	0	0
$165$	26732.6	0.0764419
$166$	0	0
$167$	−132505.	−0.367656	−0.183828	−	0.982958i	$-0.558849\pi$
$167$	−132505.	−0.367656	−0.183828	+	0.982958i	$0.558849\pi$
$168$	0	0
$169$	−277645.	−0.747778
$170$	0	0
$171$	49105.9	0.128423
$172$	0	0
$173$	−532802.	−1.35348	−0.676738	−	0.736224i	$-0.736607\pi$
$173$	−532802.	−1.35348	−0.676738	+	0.736224i	$0.736607\pi$
$174$	0	0
$175$	−34130.4	−0.0842454
$176$	0	0
$177$	−220153.	−0.528191
$178$	0	0
$179$	57633.0	0.134443	0.0672216	−	0.997738i	$-0.478587\pi$
$179$	57633.0	0.134443	0.0672216	+	0.997738i	$0.478587\pi$
$180$	0	0
$181$	22465.0	0.0509695	0.0254848	−	0.999675i	$-0.491887\pi$
$181$	22465.0	0.0509695	0.0254848	+	0.999675i	$0.491887\pi$
$182$	0	0
$183$	27422.9	0.0605322
$184$	0	0
$185$	115061.	0.247171
$186$	0	0
$187$	598353.	1.25128
$188$	0	0
$189$	−44121.0	−0.0898444
$190$	0	0
$191$	−66820.2	−0.132533	−0.0662665	−	0.997802i	$-0.521109\pi$
$191$	−66820.2	−0.132533	−0.0662665	+	0.997802i	$0.521109\pi$
$192$	0	0
$193$	−515238.	−0.995668	−0.497834	−	0.867272i	$-0.665871\pi$
$193$	−515238.	−0.995668	−0.497834	+	0.867272i	$0.665871\pi$
$194$	0	0
$195$	−30438.5	−0.0573241
$196$	0	0
$197$	924019.	1.69635	0.848175	−	0.529716i	$-0.177702\pi$
$197$	924019.	1.69635	0.848175	+	0.529716i	$0.177702\pi$
$198$	0	0
$199$	−801572.	−1.43486	−0.717430	−	0.696630i	$-0.754682\pi$
$199$	−801572.	−1.43486	−0.717430	+	0.696630i	$0.754682\pi$
$200$	0	0
$201$	−338463.	−0.590909
$202$	0	0
$203$	45772.9	0.0779593
$204$	0	0
$205$	−138109.	−0.229529
$206$	0	0
$207$	−405766.	−0.658187
$208$	0	0
$209$	97023.0	0.153642
$210$	0	0
$211$	−805969.	−1.24627	−0.623135	−	0.782114i	$-0.714141\pi$
$211$	−805969.	−1.24627	−0.623135	+	0.782114i	$0.714141\pi$
$212$	0	0
$213$	−736188.	−1.11183
$214$	0	0
$215$	−104800.	−0.154619
$216$	0	0
$217$	−84034.6	−0.121146
$218$	0	0
$219$	763080.	1.07513
$220$	0	0
$221$	−681302.	−0.938337
$222$	0	0
$223$	1.06583e6	1.43524	0.717622	−	0.696433i	$-0.245231\pi$
$223$	1.06583e6	1.43524	0.717622	+	0.696433i	$0.245231\pi$
$224$	0	0
$225$	412505.	0.543217
$226$	0	0
$227$	−1.37166e6	−1.76677	−0.883386	−	0.468647i	$-0.844742\pi$
$227$	−1.37166e6	−1.76677	−0.883386	+	0.468647i	$0.844742\pi$
$228$	0	0
$229$	−1.19996e6	−1.51210	−0.756049	−	0.654515i	$-0.772873\pi$
$229$	−1.19996e6	−1.51210	−0.756049	+	0.654515i	$0.772873\pi$
$230$	0	0
$231$	−31285.5	−0.0385756
$232$	0	0
$233$	769601.	0.928701	0.464350	−	0.885652i	$-0.346288\pi$
$233$	769601.	0.928701	0.464350	+	0.885652i	$0.346288\pi$
$234$	0	0
$235$	209817.	0.247840
$236$	0	0
$237$	−504360.	−0.583270
$238$	0	0
$239$	1.60410e6	1.81651	0.908253	−	0.418421i	$-0.137416\pi$
$239$	1.60410e6	1.81651	0.908253	+	0.418421i	$0.137416\pi$
$240$	0	0
$241$	−900329.	−0.998525	−0.499262	−	0.866451i	$-0.666396\pi$
$241$	−900329.	−0.998525	−0.499262	+	0.866451i	$0.666396\pi$
$242$	0	0
$243$	875129.	0.950729
$244$	0	0
$245$	160414.	0.170737
$246$	0	0
$247$	−110473.	−0.115216
$248$	0	0
$249$	847953.	0.866709
$250$	0	0
$251$	−274625.	−0.275141	−0.137571	−	0.990492i	$-0.543929\pi$
$251$	−274625.	−0.275141	−0.137571	+	0.990492i	$0.543929\pi$
$252$	0	0
$253$	−801708.	−0.787435
$254$	0	0
$255$	221444.	0.213262
$256$	0	0
$257$	−863177.	−0.815205	−0.407603	−	0.913159i	$-0.633635\pi$
$257$	−863177.	−0.815205	−0.407603	+	0.913159i	$0.633635\pi$
$258$	0	0
$259$	−134657.	−0.124732
$260$	0	0
$261$	−553218.	−0.502684
$262$	0	0
$263$	−749008.	−0.667725	−0.333862	−	0.942622i	$-0.608352\pi$
$263$	−749008.	−0.667725	−0.333862	+	0.942622i	$0.608352\pi$
$264$	0	0
$265$	122359.	0.107034
$266$	0	0
$267$	614103.	0.527185
$268$	0	0
$269$	−138260.	−0.116497	−0.0582486	−	0.998302i	$-0.518552\pi$
$269$	−138260.	−0.116497	−0.0582486	+	0.998302i	$0.518552\pi$
$270$	0	0
$271$	210916.	0.174456	0.0872282	−	0.996188i	$-0.472199\pi$
$271$	210916.	0.174456	0.0872282	+	0.996188i	$0.472199\pi$
$272$	0	0
$273$	35622.5	0.0289280
$274$	0	0
$275$	815024.	0.649888
$276$	0	0
$277$	−552166.	−0.432384	−0.216192	−	0.976351i	$-0.569364\pi$
$277$	−552166.	−0.432384	−0.216192	+	0.976351i	$0.569364\pi$
$278$	0	0
$279$	1.01565e6	0.781152
$280$	0	0
$281$	1.47154e6	1.11175	0.555875	−	0.831266i	$-0.312383\pi$
$281$	1.47154e6	1.11175	0.555875	+	0.831266i	$0.312383\pi$
$282$	0	0
$283$	−297044.	−0.220472	−0.110236	−	0.993905i	$-0.535161\pi$
$283$	−297044.	−0.220472	−0.110236	+	0.993905i	$0.535161\pi$
$284$	0	0
$285$	35907.2	0.0261860
$286$	0	0
$287$	161630.	0.115829
$288$	0	0
$289$	3.53670e6	2.49088
$290$	0	0
$291$	1.09790e6	0.760026
$292$	0	0
$293$	−787646.	−0.535997	−0.267998	−	0.963419i	$-0.586362\pi$
$293$	−787646.	−0.535997	−0.267998	+	0.963419i	$0.586362\pi$
$294$	0	0
$295$	204704.	0.136953
$296$	0	0
$297$	1.05360e6	0.693080
$298$	0	0
$299$	912848.	0.590501
$300$	0	0
$301$	122648.	0.0780269
$302$	0	0
$303$	1.59141e6	0.995809
$304$	0	0
$305$	−25498.5	−0.0156952
$306$	0	0
$307$	−1.17593e6	−0.712093	−0.356046	−	0.934468i	$-0.615875\pi$
$307$	−1.17593e6	−0.712093	−0.356046	+	0.934468i	$0.615875\pi$
$308$	0	0
$309$	−589994.	−0.351521
$310$	0	0
$311$	2.40819e6	1.41186	0.705928	−	0.708284i	$-0.250530\pi$
$311$	2.40819e6	1.41186	0.705928	+	0.708284i	$0.250530\pi$
$312$	0	0
$313$	−431998.	−0.249242	−0.124621	−	0.992204i	$-0.539771\pi$
$313$	−431998.	−0.249242	−0.124621	+	0.992204i	$0.539771\pi$
$314$	0	0
$315$	14723.2	0.00836039
$316$	0	0
$317$	390232.	0.218109	0.109055	−	0.994036i	$-0.465218\pi$
$317$	390232.	0.218109	0.109055	+	0.994036i	$0.465218\pi$
$318$	0	0
$319$	−1.09304e6	−0.601396
$320$	0	0
$321$	749513.	0.405991
$322$	0	0
$323$	803706.	0.428638
$324$	0	0
$325$	−928010.	−0.487354
$326$	0	0
$327$	1.94534e6	1.00607
$328$	0	0
$329$	−245551.	−0.125070
$330$	0	0
$331$	−3.26030e6	−1.63564	−0.817819	−	0.575475i	$-0.804817\pi$
$331$	−3.26030e6	−1.63564	−0.817819	+	0.575475i	$0.804817\pi$
$332$	0	0
$333$	1.62748e6	0.804277
$334$	0	0
$335$	314711.	0.153215
$336$	0	0
$337$	355953.	0.170733	0.0853665	−	0.996350i	$-0.472794\pi$
$337$	355953.	0.170733	0.0853665	+	0.996350i	$0.472794\pi$
$338$	0	0
$339$	831833.	0.393131
$340$	0	0
$341$	2.00672e6	0.934547
$342$	0	0
$343$	−376894.	−0.172975
$344$	0	0
$345$	−296703.	−0.134207
$346$	0	0
$347$	−3.24862e6	−1.44835	−0.724177	−	0.689614i	$-0.757780\pi$
$347$	−3.24862e6	−1.44835	−0.724177	+	0.689614i	$0.757780\pi$
$348$	0	0
$349$	−2.81147e6	−1.23558	−0.617788	−	0.786344i	$-0.711971\pi$
$349$	−2.81147e6	−1.23558	−0.617788	+	0.786344i	$0.711971\pi$
$350$	0	0
$351$	−1.19966e6	−0.519743
$352$	0	0
$353$	4.28147e6	1.82876	0.914380	−	0.404858i	$-0.132679\pi$
$353$	4.28147e6	1.82876	0.914380	+	0.404858i	$0.132679\pi$
$354$	0	0
$355$	684527.	0.288283
$356$	0	0
$357$	−259158.	−0.107620
$358$	0	0
$359$	−1.06670e6	−0.436824	−0.218412	−	0.975857i	$-0.570088\pi$
$359$	−1.06670e6	−0.436824	−0.218412	+	0.975857i	$0.570088\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	−918623.	−0.365907
$364$	0	0
$365$	−709532.	−0.278766
$366$	0	0
$367$	−1.70462e6	−0.660638	−0.330319	−	0.943869i	$-0.607156\pi$
$367$	−1.70462e6	−0.660638	−0.330319	+	0.943869i	$0.607156\pi$
$368$	0	0
$369$	−1.95349e6	−0.746870
$370$	0	0
$371$	−143199.	−0.0540137
$372$	0	0
$373$	2.08821e6	0.777145	0.388572	−	0.921418i	$-0.372968\pi$
$373$	2.08821e6	0.777145	0.388572	+	0.921418i	$0.372968\pi$
$374$	0	0
$375$	612462.	0.224906
$376$	0	0
$377$	1.24457e6	0.450989
$378$	0	0
$379$	−349590.	−0.125015	−0.0625073	−	0.998045i	$-0.519910\pi$
$379$	−349590.	−0.125015	−0.0625073	+	0.998045i	$0.519910\pi$
$380$	0	0
$381$	1.27614e6	0.450389
$382$	0	0
$383$	−4.04499e6	−1.40903	−0.704516	−	0.709688i	$-0.748836\pi$
$383$	−4.04499e6	−1.40903	−0.704516	+	0.709688i	$0.748836\pi$
$384$	0	0
$385$	29090.0	0.0100021
$386$	0	0
$387$	−1.48234e6	−0.503119
$388$	0	0
$389$	−537591.	−0.180127	−0.0900633	−	0.995936i	$-0.528707\pi$
$389$	−537591.	−0.180127	−0.0900633	+	0.995936i	$0.528707\pi$
$390$	0	0
$391$	−6.64108e6	−2.19683
$392$	0	0
$393$	1.70816e6	0.557888
$394$	0	0
$395$	468967.	0.151234
$396$	0	0
$397$	−4.66458e6	−1.48537	−0.742687	−	0.669638i	$-0.766449\pi$
$397$	−4.66458e6	−1.48537	−0.742687	+	0.669638i	$0.766449\pi$
$398$	0	0
$399$	−42022.5	−0.0132145
$400$	0	0
$401$	4.06136e6	1.26128	0.630638	−	0.776077i	$-0.282793\pi$
$401$	4.06136e6	1.26128	0.630638	+	0.776077i	$0.282793\pi$
$402$	0	0
$403$	−2.28491e6	−0.700820
$404$	0	0
$405$	72039.2	0.0218239
$406$	0	0
$407$	3.21556e6	0.962213
$408$	0	0
$409$	−57984.7	−0.0171398	−0.00856989	−	0.999963i	$-0.502728\pi$
$409$	−57984.7	−0.0171398	−0.00856989	+	0.999963i	$0.502728\pi$
$410$	0	0
$411$	1.30134e6	0.380001
$412$	0	0
$413$	−239567.	−0.0691117
$414$	0	0
$415$	−788448.	−0.224726
$416$	0	0
$417$	−3.01437e6	−0.848901
$418$	0	0
$419$	−321540.	−0.0894745	−0.0447373	−	0.998999i	$-0.514245\pi$
$419$	−321540.	−0.0894745	−0.0447373	+	0.998999i	$0.514245\pi$
$420$	0	0
$421$	−5.02391e6	−1.38145	−0.690727	−	0.723115i	$-0.742709\pi$
$421$	−5.02391e6	−1.38145	−0.690727	+	0.723115i	$0.742709\pi$
$422$	0	0
$423$	2.96777e6	0.806454
$424$	0	0
$425$	6.75138e6	1.81309
$426$	0	0
$427$	29841.2	0.00792039
$428$	0	0
$429$	−850655.	−0.223157
$430$	0	0
$431$	−7.10877e6	−1.84332	−0.921661	−	0.387996i	$-0.873168\pi$
$431$	−7.10877e6	−1.84332	−0.921661	+	0.387996i	$0.873168\pi$
$432$	0	0
$433$	977473.	0.250545	0.125272	−	0.992122i	$-0.460020\pi$
$433$	977473.	0.250545	0.125272	+	0.992122i	$0.460020\pi$
$434$	0	0
$435$	−404523.	−0.102499
$436$	0	0
$437$	−1.07685e6	−0.269744
$438$	0	0
$439$	769414.	0.190546	0.0952728	−	0.995451i	$-0.469628\pi$
$439$	769414.	0.190546	0.0952728	+	0.995451i	$0.469628\pi$
$440$	0	0
$441$	2.26898e6	0.555565
$442$	0	0
$443$	−7.00188e6	−1.69514	−0.847570	−	0.530684i	$-0.821935\pi$
$443$	−7.00188e6	−1.69514	−0.847570	+	0.530684i	$0.821935\pi$
$444$	0	0
$445$	−571009.	−0.136692
$446$	0	0
$447$	−2.81659e6	−0.666737
$448$	0	0
$449$	−2.04512e6	−0.478744	−0.239372	−	0.970928i	$-0.576942\pi$
$449$	−2.04512e6	−0.478744	−0.239372	+	0.970928i	$0.576942\pi$
$450$	0	0
$451$	−3.85969e6	−0.893533
$452$	0	0
$453$	2.86433e6	0.655810
$454$	0	0
$455$	−33122.7	−0.00750063
$456$	0	0
$457$	1.84235e6	0.412649	0.206324	−	0.978484i	$-0.433850\pi$
$457$	1.84235e6	0.412649	0.206324	+	0.978484i	$0.433850\pi$
$458$	0	0
$459$	8.72764e6	1.93359
$460$	0	0
$461$	−4.45112e6	−0.975477	−0.487738	−	0.872990i	$-0.662178\pi$
$461$	−4.45112e6	−0.975477	−0.487738	+	0.872990i	$0.662178\pi$
$462$	0	0
$463$	7.27488e6	1.57715	0.788575	−	0.614938i	$-0.210819\pi$
$463$	7.27488e6	1.57715	0.788575	+	0.614938i	$0.210819\pi$
$464$	0	0
$465$	742666.	0.159280
$466$	0	0
$467$	6.99111e6	1.48338	0.741692	−	0.670740i	$-0.234023\pi$
$467$	6.99111e6	1.48338	0.741692	+	0.670740i	$0.234023\pi$
$468$	0	0
$469$	−368310.	−0.0773182
$470$	0	0
$471$	423553.	0.0879742
$472$	0	0
$473$	−2.92880e6	−0.601917
$474$	0	0
$475$	1.09474e6	0.222626
$476$	0	0
$477$	1.73072e6	0.348282
$478$	0	0
$479$	−2.22665e6	−0.443419	−0.221709	−	0.975113i	$-0.571164\pi$
$479$	−2.22665e6	−0.443419	−0.221709	+	0.975113i	$0.571164\pi$
$480$	0	0
$481$	−3.66133e6	−0.721567
$482$	0	0
$483$	347235.	0.0677260
$484$	0	0
$485$	−1.02085e6	−0.197064
$486$	0	0
$487$	−2.18483e6	−0.417442	−0.208721	−	0.977975i	$-0.566930\pi$
$487$	−2.18483e6	−0.417442	−0.208721	+	0.977975i	$0.566930\pi$
$488$	0	0
$489$	94426.9	0.0178576
$490$	0	0
$491$	−1.65873e6	−0.310508	−0.155254	−	0.987875i	$-0.549620\pi$
$491$	−1.65873e6	−0.310508	−0.155254	+	0.987875i	$0.549620\pi$
$492$	0	0
$493$	−9.05440e6	−1.67781
$494$	0	0
$495$	−351586.	−0.0644939
$496$	0	0
$497$	−801108.	−0.145479
$498$	0	0
$499$	−5.16603e6	−0.928764	−0.464382	−	0.885635i	$-0.653724\pi$
$499$	−5.16603e6	−0.928764	−0.464382	+	0.885635i	$0.653724\pi$
$500$	0	0
$501$	−1.37047e6	−0.243935
$502$	0	0
$503$	−2.93883e6	−0.517910	−0.258955	−	0.965889i	$-0.583378\pi$
$503$	−2.93883e6	−0.517910	−0.258955	+	0.965889i	$0.583378\pi$
$504$	0	0
$505$	−1.47974e6	−0.258200
$506$	0	0
$507$	−2.87161e6	−0.496142
$508$	0	0
$509$	6.03442e6	1.03238	0.516192	−	0.856473i	$-0.327349\pi$
$509$	6.03442e6	1.03238	0.516192	+	0.856473i	$0.327349\pi$
$510$	0	0
$511$	830372.	0.140676
$512$	0	0
$513$	1.41519e6	0.237422
$514$	0	0
$515$	548591.	0.0911446
$516$	0	0
$517$	5.86369e6	0.964817
$518$	0	0
$519$	−5.51064e6	−0.898015
$520$	0	0
$521$	5.87340e6	0.947972	0.473986	−	0.880532i	$-0.342815\pi$
$521$	5.87340e6	0.947972	0.473986	+	0.880532i	$0.342815\pi$
$522$	0	0
$523$	5.67148e6	0.906655	0.453327	−	0.891344i	$-0.350237\pi$
$523$	5.67148e6	0.906655	0.453327	+	0.891344i	$0.350237\pi$
$524$	0	0
$525$	−353002.	−0.0558958
$526$	0	0
$527$	1.66230e7	2.60725
$528$	0	0
$529$	2.46175e6	0.382477
$530$	0	0
$531$	2.89544e6	0.445634
$532$	0	0
$533$	4.39475e6	0.670064
$534$	0	0
$535$	−696916.	−0.105268
$536$	0	0
$537$	596084.	0.0892014
$538$	0	0
$539$	4.48304e6	0.664661
$540$	0	0
$541$	−1.08967e7	−1.60067	−0.800336	−	0.599552i	$-0.795346\pi$
$541$	−1.08967e7	−1.60067	−0.800336	+	0.599552i	$0.795346\pi$
$542$	0	0
$543$	232350.	0.0338177
$544$	0	0
$545$	−1.80883e6	−0.260860
$546$	0	0
$547$	−5.29622e6	−0.756829	−0.378414	−	0.925636i	$-0.623531\pi$
$547$	−5.29622e6	−0.756829	−0.378414	+	0.925636i	$0.623531\pi$
$548$	0	0
$549$	−360665.	−0.0510709
$550$	0	0
$551$	−1.46817e6	−0.206014
$552$	0	0
$553$	−548836.	−0.0763185
$554$	0	0
$555$	1.19004e6	0.163995
$556$	0	0
$557$	2.84427e6	0.388447	0.194224	−	0.980957i	$-0.437781\pi$
$557$	2.84427e6	0.388447	0.194224	+	0.980957i	$0.437781\pi$
$558$	0	0
$559$	3.33481e6	0.451380
$560$	0	0
$561$	6.18862e6	0.830207
$562$	0	0
$563$	8.21754e6	1.09262	0.546312	−	0.837582i	$-0.316031\pi$
$563$	8.21754e6	1.09262	0.546312	+	0.837582i	$0.316031\pi$
$564$	0	0
$565$	−773460.	−0.101933
$566$	0	0
$567$	−84308.2	−0.0110132
$568$	0	0
$569$	6.27293e6	0.812251	0.406125	−	0.913817i	$-0.366880\pi$
$569$	6.27293e6	0.812251	0.406125	+	0.913817i	$0.366880\pi$
$570$	0	0
$571$	1.50316e7	1.92937	0.964685	−	0.263404i	$-0.0848454\pi$
$571$	1.50316e7	1.92937	0.964685	+	0.263404i	$0.0848454\pi$
$572$	0	0
$573$	−691104.	−0.0879340
$574$	0	0
$575$	−9.04589e6	−1.14099
$576$	0	0
$577$	−1.29397e7	−1.61802	−0.809011	−	0.587794i	$-0.799997\pi$
$577$	−1.29397e7	−1.61802	−0.809011	+	0.587794i	$0.799997\pi$
$578$	0	0
$579$	−5.32898e6	−0.660614
$580$	0	0
$581$	922729.	0.113405
$582$	0	0
$583$	3.41954e6	0.416674
$584$	0	0
$585$	400326.	0.0483642
$586$	0	0
$587$	7.51154e6	0.899775	0.449887	−	0.893085i	$-0.351464\pi$
$587$	7.51154e6	0.899775	0.449887	+	0.893085i	$0.351464\pi$
$588$	0	0
$589$	2.69542e6	0.320139
$590$	0	0
$591$	9.55690e6	1.12551
$592$	0	0
$593$	−2.62097e6	−0.306073	−0.153036	−	0.988221i	$-0.548905\pi$
$593$	−2.62097e6	−0.306073	−0.153036	+	0.988221i	$0.548905\pi$
$594$	0	0
$595$	240972.	0.0279045
$596$	0	0
$597$	−8.29046e6	−0.952012
$598$	0	0
$599$	4.02540e6	0.458397	0.229199	−	0.973380i	$-0.426389\pi$
$599$	4.02540e6	0.458397	0.229199	+	0.973380i	$0.426389\pi$
$600$	0	0
$601$	1.27237e7	1.43691	0.718454	−	0.695575i	$-0.244850\pi$
$601$	1.27237e7	1.43691	0.718454	+	0.695575i	$0.244850\pi$
$602$	0	0
$603$	4.45145e6	0.498550
$604$	0	0
$605$	854159.	0.0948747
$606$	0	0
$607$	6.17632e6	0.680390	0.340195	−	0.940355i	$-0.389507\pi$
$607$	6.17632e6	0.680390	0.340195	+	0.940355i	$0.389507\pi$
$608$	0	0
$609$	473418.	0.0517251
$610$	0	0
$611$	−6.67657e6	−0.723520
$612$	0	0
$613$	−7.72670e6	−0.830506	−0.415253	−	0.909706i	$-0.636307\pi$
$613$	−7.72670e6	−0.830506	−0.415253	+	0.909706i	$0.636307\pi$
$614$	0	0
$615$	−1.42843e6	−0.152290
$616$	0	0
$617$	3.96955e6	0.419787	0.209893	−	0.977724i	$-0.432688\pi$
$617$	3.96955e6	0.419787	0.209893	+	0.977724i	$0.432688\pi$
$618$	0	0
$619$	1.55633e6	0.163258	0.0816289	−	0.996663i	$-0.473988\pi$
$619$	1.55633e6	0.163258	0.0816289	+	0.996663i	$0.473988\pi$
$620$	0	0
$621$	−1.16938e7	−1.21682
$622$	0	0
$623$	668257.	0.0689801
$624$	0	0
$625$	8.90712e6	0.912089
$626$	0	0
$627$	1.00349e6	0.101939
$628$	0	0
$629$	2.66366e7	2.68443
$630$	0	0
$631$	−9.70032e6	−0.969868	−0.484934	−	0.874551i	$-0.661156\pi$
$631$	−9.70032e6	−0.969868	−0.484934	+	0.874551i	$0.661156\pi$
$632$	0	0
$633$	−8.33594e6	−0.826885
$634$	0	0
$635$	−1.18659e6	−0.116780
$636$	0	0
$637$	−5.10451e6	−0.498432
$638$	0	0
$639$	9.68232e6	0.938052
$640$	0	0
$641$	−6.33483e6	−0.608962	−0.304481	−	0.952518i	$-0.598483\pi$
$641$	−6.33483e6	−0.608962	−0.304481	+	0.952518i	$0.598483\pi$
$642$	0	0
$643$	−3.07299e6	−0.293112	−0.146556	−	0.989202i	$-0.546819\pi$
$643$	−3.07299e6	−0.293112	−0.146556	+	0.989202i	$0.546819\pi$
$644$	0	0
$645$	−1.08392e6	−0.102588
$646$	0	0
$647$	−5.74018e6	−0.539094	−0.269547	−	0.962987i	$-0.586874\pi$
$647$	−5.74018e6	−0.539094	−0.269547	+	0.962987i	$0.586874\pi$
$648$	0	0
$649$	5.72078e6	0.533143
$650$	0	0
$651$	−869149.	−0.0803789
$652$	0	0
$653$	−8.69840e6	−0.798282	−0.399141	−	0.916889i	$-0.630692\pi$
$653$	−8.69840e6	−0.798282	−0.399141	+	0.916889i	$0.630692\pi$
$654$	0	0
$655$	−1.58829e6	−0.144653
$656$	0	0
$657$	−1.00360e7	−0.907084
$658$	0	0
$659$	−1.11634e6	−0.100135	−0.0500674	−	0.998746i	$-0.515944\pi$
$659$	−1.11634e6	−0.100135	−0.0500674	+	0.998746i	$0.515944\pi$
$660$	0	0
$661$	1.46309e6	0.130247	0.0651233	−	0.997877i	$-0.479256\pi$
$661$	1.46309e6	0.130247	0.0651233	+	0.997877i	$0.479256\pi$
$662$	0	0
$663$	−7.04654e6	−0.622576
$664$	0	0
$665$	39073.6	0.00342633
$666$	0	0
$667$	1.21316e7	1.05585
$668$	0	0
$669$	1.10236e7	0.952266
$670$	0	0
$671$	−712599.	−0.0610997
$672$	0	0
$673$	−4.69133e6	−0.399262	−0.199631	−	0.979871i	$-0.563974\pi$
$673$	−4.69133e6	−0.399262	−0.199631	+	0.979871i	$0.563974\pi$
$674$	0	0
$675$	1.18880e7	1.00427
$676$	0	0
$677$	−847754.	−0.0710883	−0.0355441	−	0.999368i	$-0.511316\pi$
$677$	−847754.	−0.0710883	−0.0355441	+	0.999368i	$0.511316\pi$
$678$	0	0
$679$	1.19471e6	0.0994464
$680$	0	0
$681$	−1.41867e7	−1.17223
$682$	0	0
$683$	−1.91086e7	−1.56739	−0.783695	−	0.621146i	$-0.786667\pi$
$683$	−1.91086e7	−1.56739	−0.783695	+	0.621146i	$0.786667\pi$
$684$	0	0
$685$	−1.21002e6	−0.0985292
$686$	0	0
$687$	−1.24109e7	−1.00326
$688$	0	0
$689$	−3.89358e6	−0.312465
$690$	0	0
$691$	−3.39649e6	−0.270605	−0.135302	−	0.990804i	$-0.543201\pi$
$691$	−3.39649e6	−0.270605	−0.135302	+	0.990804i	$0.543201\pi$
$692$	0	0
$693$	411465.	0.0325462
$694$	0	0
$695$	2.80284e6	0.220108
$696$	0	0
$697$	−3.19723e7	−2.49283
$698$	0	0
$699$	7.95979e6	0.616182
$700$	0	0
$701$	−1.80641e6	−0.138842	−0.0694209	−	0.997587i	$-0.522115\pi$
$701$	−1.80641e6	−0.138842	−0.0694209	+	0.997587i	$0.522115\pi$
$702$	0	0
$703$	4.31913e6	0.329616
$704$	0	0
$705$	2.17009e6	0.164439
$706$	0	0
$707$	1.73175e6	0.130298
$708$	0	0
$709$	−1.59078e7	−1.18849	−0.594243	−	0.804285i	$-0.702548\pi$
$709$	−1.59078e7	−1.18849	−0.594243	+	0.804285i	$0.702548\pi$
$710$	0	0
$711$	6.63332e6	0.492104
$712$	0	0
$713$	−2.22725e7	−1.64076
$714$	0	0
$715$	790961.	0.0578615
$716$	0	0
$717$	1.65908e7	1.20523
$718$	0	0
$719$	4.47853e6	0.323082	0.161541	−	0.986866i	$-0.448354\pi$
$719$	4.47853e6	0.323082	0.161541	+	0.986866i	$0.448354\pi$
$720$	0	0
$721$	−642022.	−0.0459951
$722$	0	0
$723$	−9.31188e6	−0.662509
$724$	0	0
$725$	−1.23331e7	−0.871420
$726$	0	0
$727$	−2.27370e7	−1.59550	−0.797750	−	0.602988i	$-0.793977\pi$
$727$	−2.27370e7	−1.59550	−0.797750	+	0.602988i	$0.793977\pi$
$728$	0	0
$729$	1.08715e7	0.757655
$730$	0	0
$731$	−2.42612e7	−1.67926
$732$	0	0
$733$	6.47613e6	0.445201	0.222600	−	0.974910i	$-0.428546\pi$
$733$	6.47613e6	0.445201	0.222600	+	0.974910i	$0.428546\pi$
$734$	0	0
$735$	1.65912e6	0.113282
$736$	0	0
$737$	8.79514e6	0.596450
$738$	0	0
$739$	995404.	0.0670484	0.0335242	−	0.999438i	$-0.489327\pi$
$739$	995404.	0.0670484	0.0335242	+	0.999438i	$0.489327\pi$
$740$	0	0
$741$	−1.14260e6	−0.0764447
$742$	0	0
$743$	7.89367e6	0.524574	0.262287	−	0.964990i	$-0.415523\pi$
$743$	7.89367e6	0.524574	0.262287	+	0.964990i	$0.415523\pi$
$744$	0	0
$745$	2.61894e6	0.172876
$746$	0	0
$747$	−1.11522e7	−0.731241
$748$	0	0
$749$	815608.	0.0531223
$750$	0	0
$751$	4.00975e6	0.259428	0.129714	−	0.991551i	$-0.458594\pi$
$751$	4.00975e6	0.259428	0.129714	+	0.991551i	$0.458594\pi$
$752$	0	0
$753$	−2.84038e6	−0.182553
$754$	0	0
$755$	−2.66333e6	−0.170043
$756$	0	0
$757$	2.18293e7	1.38452	0.692260	−	0.721648i	$-0.256615\pi$
$757$	2.18293e7	1.38452	0.692260	+	0.721648i	$0.256615\pi$
$758$	0	0
$759$	−8.29187e6	−0.522454
$760$	0	0
$761$	2.37089e7	1.48405	0.742027	−	0.670370i	$-0.233865\pi$
$761$	2.37089e7	1.48405	0.742027	+	0.670370i	$0.233865\pi$
$762$	0	0
$763$	2.11689e6	0.131640
$764$	0	0
$765$	−2.91242e6	−0.179929
$766$	0	0
$767$	−6.51385e6	−0.399806
$768$	0	0
$769$	−2.45681e7	−1.49815	−0.749075	−	0.662485i	$-0.769502\pi$
$769$	−2.45681e7	−1.49815	−0.749075	+	0.662485i	$0.769502\pi$
$770$	0	0
$771$	−8.92763e6	−0.540879
$772$	0	0
$773$	−1.94815e6	−0.117266	−0.0586331	−	0.998280i	$-0.518674\pi$
$773$	−1.94815e6	−0.117266	−0.0586331	+	0.998280i	$0.518674\pi$
$774$	0	0
$775$	2.26424e7	1.35415
$776$	0	0
$777$	−1.39272e6	−0.0827583
$778$	0	0
$779$	−5.18432e6	−0.306089
$780$	0	0
$781$	1.91302e7	1.12226
$782$	0	0
$783$	−1.59432e7	−0.929334
$784$	0	0
$785$	−393830.	−0.0228105
$786$	0	0
$787$	1.28507e7	0.739586	0.369793	−	0.929114i	$-0.379429\pi$
$787$	1.28507e7	0.739586	0.369793	+	0.929114i	$0.379429\pi$
$788$	0	0
$789$	−7.74681e6	−0.443027
$790$	0	0
$791$	905188.	0.0514396
$792$	0	0
$793$	811386.	0.0458189
$794$	0	0
$795$	1.26553e6	0.0710159
$796$	0	0
$797$	−1.78593e7	−0.995909	−0.497954	−	0.867203i	$-0.665915\pi$
$797$	−1.78593e7	−0.995909	−0.497954	+	0.867203i	$0.665915\pi$
$798$	0	0
$799$	4.85729e7	2.69170
$800$	0	0
$801$	−8.07666e6	−0.444785
$802$	0	0
$803$	−1.98290e7	−1.08521
$804$	0	0
$805$	−322868.	−0.0175604
$806$	0	0
$807$	−1.42999e6	−0.0772945
$808$	0	0
$809$	−7.75903e6	−0.416808	−0.208404	−	0.978043i	$-0.566827\pi$
$809$	−7.75903e6	−0.416808	−0.208404	+	0.978043i	$0.566827\pi$
$810$	0	0
$811$	−2.75388e7	−1.47026	−0.735128	−	0.677929i	$-0.762878\pi$
$811$	−2.75388e7	−1.47026	−0.735128	+	0.677929i	$0.762878\pi$
$812$	0	0
$813$	2.18145e6	0.115750
$814$	0	0
$815$	−87800.6	−0.00463024
$816$	0	0
$817$	−3.93395e6	−0.206193
$818$	0	0
$819$	−468506.	−0.0244065
$820$	0	0
$821$	2.39140e7	1.23821	0.619105	−	0.785308i	$-0.287496\pi$
$821$	2.39140e7	1.23821	0.619105	+	0.785308i	$0.287496\pi$
$822$	0	0
$823$	−2.33465e7	−1.20149	−0.600747	−	0.799439i	$-0.705130\pi$
$823$	−2.33465e7	−1.20149	−0.600747	+	0.799439i	$0.705130\pi$
$824$	0	0
$825$	8.42959e6	0.431193
$826$	0	0
$827$	1.83932e7	0.935178	0.467589	−	0.883946i	$-0.345123\pi$
$827$	1.83932e7	0.935178	0.467589	+	0.883946i	$0.345123\pi$
$828$	0	0
$829$	1.57573e7	0.796337	0.398168	−	0.917312i	$-0.369646\pi$
$829$	1.57573e7	0.796337	0.398168	+	0.917312i	$0.369646\pi$
$830$	0	0
$831$	−5.71091e6	−0.286882
$832$	0	0
$833$	3.71360e7	1.85431
$834$	0	0
$835$	1.27430e6	0.0632491
$836$	0	0
$837$	2.92702e7	1.44415
$838$	0	0
$839$	1.90793e7	0.935743	0.467872	−	0.883796i	$-0.345021\pi$
$839$	1.90793e7	0.935743	0.467872	+	0.883796i	$0.345021\pi$
$840$	0	0
$841$	−3.97101e6	−0.193603
$842$	0	0
$843$	1.52198e7	0.737633
$844$	0	0
$845$	2.67010e6	0.128643
$846$	0	0
$847$	−999632.	−0.0478775
$848$	0	0
$849$	−3.07225e6	−0.146281
$850$	0	0
$851$	−3.56893e7	−1.68933
$852$	0	0
$853$	−1.44587e7	−0.680386	−0.340193	−	0.940356i	$-0.610492\pi$
$853$	−1.44587e7	−0.680386	−0.340193	+	0.940356i	$0.610492\pi$
$854$	0	0
$855$	−472249.	−0.0220931
$856$	0	0
$857$	3.07370e7	1.42958	0.714791	−	0.699338i	$-0.246522\pi$
$857$	3.07370e7	1.42958	0.714791	+	0.699338i	$0.246522\pi$
$858$	0	0
$859$	−4.64956e6	−0.214995	−0.107498	−	0.994205i	$-0.534284\pi$
$859$	−4.64956e6	−0.214995	−0.107498	+	0.994205i	$0.534284\pi$
$860$	0	0
$861$	1.67170e6	0.0768513
$862$	0	0
$863$	−2.24867e7	−1.02777	−0.513887	−	0.857858i	$-0.671795\pi$
$863$	−2.24867e7	−1.02777	−0.513887	+	0.857858i	$0.671795\pi$
$864$	0	0
$865$	5.12394e6	0.232843
$866$	0	0
$867$	3.65792e7	1.65267
$868$	0	0
$869$	1.31061e7	0.588739
$870$	0	0
$871$	−1.00144e7	−0.447280
$872$	0	0
$873$	−1.44395e7	−0.641233
$874$	0	0
$875$	666472.	0.0294281
$876$	0	0
$877$	7.63556e6	0.335229	0.167615	−	0.985853i	$-0.446394\pi$
$877$	7.63556e6	0.335229	0.167615	+	0.985853i	$0.446394\pi$
$878$	0	0
$879$	−8.14643e6	−0.355627
$880$	0	0
$881$	7.76916e6	0.337236	0.168618	−	0.985681i	$-0.446070\pi$
$881$	7.76916e6	0.337236	0.168618	+	0.985681i	$0.446070\pi$
$882$	0	0
$883$	−1.01591e7	−0.438484	−0.219242	−	0.975671i	$-0.570358\pi$
$883$	−1.01591e7	−0.438484	−0.219242	+	0.975671i	$0.570358\pi$
$884$	0	0
$885$	2.11720e6	0.0908664
$886$	0	0
$887$	−4.30063e7	−1.83537	−0.917684	−	0.397311i	$-0.869944\pi$
$887$	−4.30063e7	−1.83537	−0.917684	+	0.397311i	$0.869944\pi$
$888$	0	0
$889$	1.38868e6	0.0589316
$890$	0	0
$891$	2.01326e6	0.0849581
$892$	0	0
$893$	7.87609e6	0.330508
$894$	0	0
$895$	−554254.	−0.0231287
$896$	0	0
$897$	9.44136e6	0.391790
$898$	0	0
$899$	−3.03661e7	−1.25311
$900$	0	0
$901$	2.83263e7	1.16246
$902$	0	0
$903$	1.26852e6	0.0517699
$904$	0	0
$905$	−216045.	−0.00876845
$906$	0	0
$907$	3.60530e7	1.45520	0.727600	−	0.686001i	$-0.240636\pi$
$907$	3.60530e7	1.45520	0.727600	+	0.686001i	$0.240636\pi$
$908$	0	0
$909$	−2.09302e7	−0.840163
$910$	0	0
$911$	2.76790e7	1.10498	0.552490	−	0.833519i	$-0.313678\pi$
$911$	2.76790e7	1.10498	0.552490	+	0.833519i	$0.313678\pi$
$912$	0	0
$913$	−2.20345e7	−0.874835
$914$	0	0
$915$	−263725.	−0.0104136
$916$	0	0
$917$	1.85879e6	0.0729974
$918$	0	0
$919$	2.99352e7	1.16921	0.584607	−	0.811317i	$-0.301249\pi$
$919$	2.99352e7	1.16921	0.584607	+	0.811317i	$0.301249\pi$
$920$	0	0
$921$	−1.21624e7	−0.472465
$922$	0	0
$923$	−2.17822e7	−0.841585
$924$	0	0
$925$	3.62821e7	1.39424
$926$	0	0
$927$	7.75957e6	0.296578
$928$	0	0
$929$	2.33303e6	0.0886913	0.0443457	−	0.999016i	$-0.485880\pi$
$929$	2.33303e6	0.0886913	0.0443457	+	0.999016i	$0.485880\pi$
$930$	0	0
$931$	6.02160e6	0.227687
$932$	0	0
$933$	2.49073e7	0.936749
$934$	0	0
$935$	−5.75434e6	−0.215261
$936$	0	0
$937$	−3.65160e7	−1.35873	−0.679367	−	0.733799i	$-0.737746\pi$
$937$	−3.65160e7	−1.35873	−0.679367	+	0.733799i	$0.737746\pi$
$938$	0	0
$939$	−4.46805e6	−0.165369
$940$	0	0
$941$	4.88450e7	1.79823	0.899117	−	0.437709i	$-0.144210\pi$
$941$	4.88450e7	1.79823	0.899117	+	0.437709i	$0.144210\pi$
$942$	0	0
$943$	4.28384e7	1.56875
$944$	0	0
$945$	424310.	0.0154562
$946$	0	0
$947$	−1.28816e7	−0.466761	−0.233380	−	0.972385i	$-0.574979\pi$
$947$	−1.28816e7	−0.466761	−0.233380	+	0.972385i	$0.574979\pi$
$948$	0	0
$949$	2.25779e7	0.813801
$950$	0	0
$951$	4.03607e6	0.144713
$952$	0	0
$953$	−4.25843e7	−1.51886	−0.759430	−	0.650589i	$-0.774522\pi$
$953$	−4.25843e7	−1.51886	−0.759430	+	0.650589i	$0.774522\pi$
$954$	0	0
$955$	642607.	0.0228001
$956$	0	0
$957$	−1.13051e7	−0.399019
$958$	0	0
$959$	1.41609e6	0.0497217
$960$	0	0
$961$	2.71201e7	0.947290
$962$	0	0
$963$	−9.85756e6	−0.342534
$964$	0	0
$965$	4.95502e6	0.171288
$966$	0	0
$967$	1.49582e6	0.0514415	0.0257208	−	0.999669i	$-0.491812\pi$
$967$	1.49582e6	0.0514415	0.0257208	+	0.999669i	$0.491812\pi$
$968$	0	0
$969$	8.31253e6	0.284396
$970$	0	0
$971$	3.34637e7	1.13901	0.569504	−	0.821989i	$-0.307135\pi$
$971$	3.34637e7	1.13901	0.569504	+	0.821989i	$0.307135\pi$
$972$	0	0
$973$	−3.28020e6	−0.111075
$974$	0	0
$975$	−9.59818e6	−0.323353
$976$	0	0
$977$	−3.14364e7	−1.05365	−0.526826	−	0.849973i	$-0.676618\pi$
$977$	−3.14364e7	−1.05365	−0.526826	+	0.849973i	$0.676618\pi$
$978$	0	0
$979$	−1.59578e7	−0.532128
$980$	0	0
$981$	−2.55851e7	−0.848818
$982$	0	0
$983$	−4.32581e7	−1.42785	−0.713927	−	0.700220i	$-0.753085\pi$
$983$	−4.32581e7	−1.42785	−0.713927	+	0.700220i	$0.753085\pi$
$984$	0	0
$985$	−8.88625e6	−0.291829
$986$	0	0
$987$	−2.53968e6	−0.0829823
$988$	0	0
$989$	3.25065e7	1.05677
$990$	0	0
$991$	−2.59120e7	−0.838142	−0.419071	−	0.907954i	$-0.637644\pi$
$991$	−2.59120e7	−0.838142	−0.419071	+	0.907954i	$0.637644\pi$
$992$	0	0
$993$	−3.37204e7	−1.08523
$994$	0	0
$995$	7.70868e6	0.246844
$996$	0	0
$997$	1.89926e7	0.605127	0.302564	−	0.953129i	$-0.402158\pi$
$997$	1.89926e7	0.605127	0.302564	+	0.953129i	$0.402158\pi$
$998$	0	0
$999$	4.69025e7	1.48690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.6.a.c.1.5	✓	6
4.3	odd	2		304.6.a.n.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.6.a.c.1.5	✓	6	1.1	even	1	trivial
304.6.a.n.1.2		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 152 = 2^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	152.a (trivial)

\(f(q)\)	\(=\)	\(q+10.3428 q^{3} -9.61696 q^{5} +11.2548 q^{7} -136.027 q^{9} -268.762 q^{11} +306.020 q^{13} -99.4658 q^{15} -2226.33 q^{17} -361.000 q^{19} +116.406 q^{21} +2982.97 q^{23} -3032.51 q^{25} -3920.19 q^{27} +4066.96 q^{29} -7466.54 q^{31} -2779.74 q^{33} -108.237 q^{35} -11964.4 q^{37} +3165.09 q^{39} +14361.0 q^{41} +10897.4 q^{43} +1308.17 q^{45} -21817.4 q^{47} -16680.3 q^{49} -23026.4 q^{51} -12723.3 q^{53} +2584.67 q^{55} -3733.73 q^{57} -21285.7 q^{59} +2651.42 q^{61} -1530.96 q^{63} -2942.98 q^{65} -32724.6 q^{67} +30852.1 q^{69} -71179.1 q^{71} +73779.2 q^{73} -31364.5 q^{75} -3024.87 q^{77} -48764.6 q^{79} -7490.85 q^{81} +81985.2 q^{83} +21410.5 q^{85} +42063.5 q^{87} +59375.2 q^{89} +3444.20 q^{91} -77224.6 q^{93} +3471.72 q^{95} +106151. q^{97} +36559.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 25 q^{5} - 161 q^{7} + 572 q^{9} - 205 q^{11} - 382 q^{13} - 950 q^{15} - 65 q^{17} - 2166 q^{19} - 6006 q^{21} - 5584 q^{23} - 3111 q^{25} - 6874 q^{27} - 9972 q^{29} - 7140 q^{31} - 3774 q^{33}+ \cdots + 49107 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 - 25 * q^5 - 161 * q^7 + 572 * q^9 - 205 * q^11 - 382 * q^13 - 950 * q^15 - 65 * q^17 - 2166 * q^19 - 6006 * q^21 - 5584 * q^23 - 3111 * q^25 - 6874 * q^27 - 9972 * q^29 - 7140 * q^31 - 3774 * q^33 - 14979 * q^35 - 7872 * q^37 - 46448 * q^39 - 12476 * q^41 - 45829 * q^43 - 41933 * q^45 - 61259 * q^47 + 5315 * q^49 - 72572 * q^51 + 13382 * q^53 - 55199 * q^55 + 1444 * q^57 - 56812 * q^59 + 2559 * q^61 - 97953 * q^63 + 37452 * q^65 - 36678 * q^67 - 12888 * q^69 + 14934 * q^71 + 43567 * q^73 - 57842 * q^75 + 36359 * q^77 - 155476 * q^79 + 199946 * q^81 - 149856 * q^83 + 112601 * q^85 - 95180 * q^87 + 176022 * q^89 + 76214 * q^91 + 434620 * q^93 + 9025 * q^95 + 117190 * q^97 + 49107 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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