LMFDB - Embedded newform 392.6.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,6,Mod(1,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, 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("392.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	392.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:	$62.8704573667$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 200x^{3} - 99x^{2} + 5803x - 3615 $ x^5 - 200x^3 - 99x^2 + 5803*x - 3615
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 7 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$13.2682$ of defining polynomial
Character	$\chi$	$=$	392.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.5365 q^{3} +79.9429 q^{5} +515.257 q^{9} +O(q^{10})$	$q-27.5365 q^{3} +79.9429 q^{5} +515.257 q^{9} +455.152 q^{11} +581.059 q^{13} -2201.35 q^{15} +2375.39 q^{17} +732.193 q^{19} -593.164 q^{23} +3265.86 q^{25} -7497.01 q^{27} -930.360 q^{29} +624.232 q^{31} -12533.3 q^{33} -4077.82 q^{37} -16000.3 q^{39} -6674.36 q^{41} -8956.88 q^{43} +41191.2 q^{45} +10280.9 q^{47} -65409.9 q^{51} +13843.9 q^{53} +36386.1 q^{55} -20162.0 q^{57} +10723.2 q^{59} +22151.0 q^{61} +46451.5 q^{65} -20042.4 q^{67} +16333.6 q^{69} -35780.7 q^{71} -55858.6 q^{73} -89930.4 q^{75} -55116.7 q^{79} +81233.6 q^{81} +87650.6 q^{83} +189896. q^{85} +25618.8 q^{87} +75763.0 q^{89} -17189.2 q^{93} +58533.6 q^{95} -32072.1 q^{97} +234520. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 81 q^{5} + 390 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 81 * q^5 + 390 * q^9	$ 5 q - 5 q^{3} + 81 q^{5} + 390 q^{9} + 361 q^{11} + 342 q^{13} - 1049 q^{15} + 1809 q^{17} + 1277 q^{19} + 911 q^{23} + 3940 q^{25} - 4751 q^{27} + 5442 q^{29} + 2187 q^{31} - 5553 q^{33} - 8181 q^{37} - 3422 q^{39} + 16578 q^{41} + 6332 q^{43} + 41310 q^{45} + 16101 q^{47} - 67865 q^{51} + 16047 q^{53} + 45629 q^{55} + 22347 q^{57} + 71027 q^{59} + 31093 q^{61} - 64370 q^{65} + 47981 q^{67} + 137249 q^{69} + 22512 q^{71} + 123333 q^{73} - 45460 q^{75} - 212481 q^{79} + 52917 q^{81} + 87460 q^{83} + 222141 q^{85} + 318070 q^{87} + 129045 q^{89} - 252835 q^{93} + 300417 q^{95} + 328274 q^{97} + 249798 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 81 * q^5 + 390 * q^9 + 361 * q^11 + 342 * q^13 - 1049 * q^15 + 1809 * q^17 + 1277 * q^19 + 911 * q^23 + 3940 * q^25 - 4751 * q^27 + 5442 * q^29 + 2187 * q^31 - 5553 * q^33 - 8181 * q^37 - 3422 * q^39 + 16578 * q^41 + 6332 * q^43 + 41310 * q^45 + 16101 * q^47 - 67865 * q^51 + 16047 * q^53 + 45629 * q^55 + 22347 * q^57 + 71027 * q^59 + 31093 * q^61 - 64370 * q^65 + 47981 * q^67 + 137249 * q^69 + 22512 * q^71 + 123333 * q^73 - 45460 * q^75 - 212481 * q^79 + 52917 * q^81 + 87460 * q^83 + 222141 * q^85 + 318070 * q^87 + 129045 * q^89 - 252835 * q^93 + 300417 * q^95 + 328274 * q^97 + 249798 * q^99

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
$3$	−27.5365	−1.76647	−0.883233	−	0.468935i	$-0.844638\pi$
$3$	−27.5365	−1.76647	−0.883233	+	0.468935i	$0.844638\pi$
$4$	0	0
$5$	79.9429	1.43006	0.715031	−	0.699093i	$-0.246413\pi$
$5$	79.9429	1.43006	0.715031	+	0.699093i	$0.246413\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	515.257	2.12040
$10$	0	0
$11$	455.152	1.13416	0.567080	−	0.823663i	$-0.308073\pi$
$11$	455.152	1.13416	0.567080	+	0.823663i	$0.308073\pi$
$12$	0	0
$13$	581.059	0.953591	0.476795	−	0.879014i	$-0.341798\pi$
$13$	581.059	0.953591	0.476795	+	0.879014i	$0.341798\pi$
$14$	0	0
$15$	−2201.35	−2.52615
$16$	0	0
$17$	2375.39	1.99348	0.996742	−	0.0806600i	$-0.0257028\pi$
$17$	2375.39	1.99348	0.996742	+	0.0806600i	$0.0257028\pi$
$18$	0	0
$19$	732.193	0.465309	0.232654	−	0.972559i	$-0.425259\pi$
$19$	732.193	0.465309	0.232654	+	0.972559i	$0.425259\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−593.164	−0.233806	−0.116903	−	0.993143i	$-0.537297\pi$
$23$	−593.164	−0.233806	−0.116903	+	0.993143i	$0.537297\pi$
$24$	0	0
$25$	3265.86	1.04508
$26$	0	0
$27$	−7497.01	−1.97915
$28$	0	0
$29$	−930.360	−0.205426	−0.102713	−	0.994711i	$-0.532752\pi$
$29$	−930.360	−0.205426	−0.102713	+	0.994711i	$0.532752\pi$
$30$	0	0
$31$	624.232	0.116665	0.0583327	−	0.998297i	$-0.481422\pi$
$31$	624.232	0.116665	0.0583327	+	0.998297i	$0.481422\pi$
$32$	0	0
$33$	−12533.3	−2.00346
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4077.82	−0.489693	−0.244847	−	0.969562i	$-0.578738\pi$
$37$	−4077.82	−0.489693	−0.244847	+	0.969562i	$0.578738\pi$
$38$	0	0
$39$	−16000.3	−1.68449
$40$	0	0
$41$	−6674.36	−0.620083	−0.310042	−	0.950723i	$-0.600343\pi$
$41$	−6674.36	−0.620083	−0.310042	+	0.950723i	$0.600343\pi$
$42$	0	0
$43$	−8956.88	−0.738730	−0.369365	−	0.929284i	$-0.620425\pi$
$43$	−8956.88	−0.738730	−0.369365	+	0.929284i	$0.620425\pi$
$44$	0	0
$45$	41191.2	3.03230
$46$	0	0
$47$	10280.9	0.678873	0.339436	−	0.940629i	$-0.389764\pi$
$47$	10280.9	0.678873	0.339436	+	0.940629i	$0.389764\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−65409.9	−3.52142
$52$	0	0
$53$	13843.9	0.676970	0.338485	−	0.940972i	$-0.390086\pi$
$53$	13843.9	0.676970	0.338485	+	0.940972i	$0.390086\pi$
$54$	0	0
$55$	36386.1	1.62192
$56$	0	0
$57$	−20162.0	−0.821952
$58$	0	0
$59$	10723.2	0.401045	0.200522	−	0.979689i	$-0.435736\pi$
$59$	10723.2	0.401045	0.200522	+	0.979689i	$0.435736\pi$
$60$	0	0
$61$	22151.0	0.762199	0.381099	−	0.924534i	$-0.375546\pi$
$61$	22151.0	0.762199	0.381099	+	0.924534i	$0.375546\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	46451.5	1.36369
$66$	0	0
$67$	−20042.4	−0.545460	−0.272730	−	0.962091i	$-0.587927\pi$
$67$	−20042.4	−0.545460	−0.272730	+	0.962091i	$0.587927\pi$
$68$	0	0
$69$	16333.6	0.413010
$70$	0	0
$71$	−35780.7	−0.842370	−0.421185	−	0.906975i	$-0.638386\pi$
$71$	−35780.7	−0.842370	−0.421185	+	0.906975i	$0.638386\pi$
$72$	0	0
$73$	−55858.6	−1.22683	−0.613413	−	0.789763i	$-0.710204\pi$
$73$	−55858.6	−1.22683	−0.613413	+	0.789763i	$0.710204\pi$
$74$	0	0
$75$	−89930.4	−1.84609
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−55116.7	−0.993608	−0.496804	−	0.867863i	$-0.665493\pi$
$79$	−55116.7	−0.993608	−0.496804	+	0.867863i	$0.665493\pi$
$80$	0	0
$81$	81233.6	1.37570
$82$	0	0
$83$	87650.6	1.39656	0.698280	−	0.715825i	$-0.253949\pi$
$83$	87650.6	1.39656	0.698280	+	0.715825i	$0.253949\pi$
$84$	0	0
$85$	189896.	2.85080
$86$	0	0
$87$	25618.8	0.362878
$88$	0	0
$89$	75763.0	1.01387	0.506935	−	0.861985i	$-0.330779\pi$
$89$	75763.0	1.01387	0.506935	+	0.861985i	$0.330779\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−17189.2	−0.206085
$94$	0	0
$95$	58533.6	0.665420
$96$	0	0
$97$	−32072.1	−0.346097	−0.173048	−	0.984913i	$-0.555362\pi$
$97$	−32072.1	−0.346097	−0.173048	+	0.984913i	$0.555362\pi$
$98$	0	0
$99$	234520.	2.40487
$100$	0	0
$101$	187860.	1.83245	0.916223	−	0.400669i	$-0.131222\pi$
$101$	187860.	1.83245	0.916223	+	0.400669i	$0.131222\pi$
$102$	0	0
$103$	160995.	1.49527	0.747633	−	0.664112i	$-0.231190\pi$
$103$	160995.	1.49527	0.747633	+	0.664112i	$0.231190\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	153336.	1.29475	0.647375	−	0.762172i	$-0.275867\pi$
$107$	153336.	1.29475	0.647375	+	0.762172i	$0.275867\pi$
$108$	0	0
$109$	−42063.0	−0.339105	−0.169552	−	0.985521i	$-0.554232\pi$
$109$	−42063.0	−0.339105	−0.169552	+	0.985521i	$0.554232\pi$
$110$	0	0
$111$	112289.	0.865026
$112$	0	0
$113$	92809.7	0.683750	0.341875	−	0.939745i	$-0.388938\pi$
$113$	92809.7	0.683750	0.341875	+	0.939745i	$0.388938\pi$
$114$	0	0
$115$	−47419.2	−0.334357
$116$	0	0
$117$	299395.	2.02199
$118$	0	0
$119$	0	0
$120$	0	0
$121$	46112.2	0.286320
$122$	0	0
$123$	183788.	1.09536
$124$	0	0
$125$	11261.1	0.0644622
$126$	0	0
$127$	−59778.1	−0.328877	−0.164438	−	0.986387i	$-0.552581\pi$
$127$	−59778.1	−0.328877	−0.164438	+	0.986387i	$0.552581\pi$
$128$	0	0
$129$	246641.	1.30494
$130$	0	0
$131$	−261722.	−1.33249	−0.666243	−	0.745735i	$-0.732099\pi$
$131$	−261722.	−1.33249	−0.666243	+	0.745735i	$0.732099\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−599332.	−2.83031
$136$	0	0
$137$	51875.5	0.236135	0.118068	−	0.993006i	$-0.462330\pi$
$137$	51875.5	0.236135	0.118068	+	0.993006i	$0.462330\pi$
$138$	0	0
$139$	−299390.	−1.31432	−0.657159	−	0.753752i	$-0.728242\pi$
$139$	−299390.	−1.31432	−0.657159	+	0.753752i	$0.728242\pi$
$140$	0	0
$141$	−283101.	−1.19920
$142$	0	0
$143$	264470.	1.08152
$144$	0	0
$145$	−74375.6	−0.293772
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−281682.	−1.03943	−0.519713	−	0.854341i	$-0.673961\pi$
$149$	−281682.	−1.03943	−0.519713	+	0.854341i	$0.673961\pi$
$150$	0	0
$151$	222399.	0.793761	0.396881	−	0.917870i	$-0.370093\pi$
$151$	222399.	0.793761	0.396881	+	0.917870i	$0.370093\pi$
$152$	0	0
$153$	1.22394e6	4.22698
$154$	0	0
$155$	49902.9	0.166839
$156$	0	0
$157$	59674.9	0.193216	0.0966078	−	0.995323i	$-0.469201\pi$
$157$	59674.9	0.193216	0.0966078	+	0.995323i	$0.469201\pi$
$158$	0	0
$159$	−381213.	−1.19584
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−378987.	−1.11726	−0.558631	−	0.829416i	$-0.688673\pi$
$163$	−378987.	−1.11726	−0.558631	+	0.829416i	$0.688673\pi$
$164$	0	0
$165$	−1.00195e6	−2.86507
$166$	0	0
$167$	143609.	0.398465	0.199233	−	0.979952i	$-0.436155\pi$
$167$	143609.	0.398465	0.199233	+	0.979952i	$0.436155\pi$
$168$	0	0
$169$	−33663.2	−0.0906648
$170$	0	0
$171$	377268.	0.986641
$172$	0	0
$173$	−774133.	−1.96653	−0.983265	−	0.182181i	$-0.941684\pi$
$173$	−774133.	−1.96653	−0.983265	+	0.182181i	$0.941684\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−295278.	−0.708432
$178$	0	0
$179$	466552.	1.08835	0.544173	−	0.838973i	$-0.316843\pi$
$179$	466552.	1.08835	0.544173	+	0.838973i	$0.316843\pi$
$180$	0	0
$181$	−575239.	−1.30512	−0.652562	−	0.757736i	$-0.726306\pi$
$181$	−575239.	−1.30512	−0.652562	+	0.757736i	$0.726306\pi$
$182$	0	0
$183$	−609960.	−1.34640
$184$	0	0
$185$	−325993.	−0.700292
$186$	0	0
$187$	1.08116e6	2.26093
$188$	0	0
$189$	0	0
$190$	0	0
$191$	280046.	0.555451	0.277726	−	0.960660i	$-0.410419\pi$
$191$	280046.	0.555451	0.277726	+	0.960660i	$0.410419\pi$
$192$	0	0
$193$	28761.8	0.0555805	0.0277902	−	0.999614i	$-0.491153\pi$
$193$	28761.8	0.0555805	0.0277902	+	0.999614i	$0.491153\pi$
$194$	0	0
$195$	−1.27911e6	−2.40892
$196$	0	0
$197$	−174264.	−0.319921	−0.159960	−	0.987123i	$-0.551137\pi$
$197$	−174264.	−0.319921	−0.159960	+	0.987123i	$0.551137\pi$
$198$	0	0
$199$	654406.	1.17143	0.585713	−	0.810519i	$-0.300815\pi$
$199$	654406.	1.17143	0.585713	+	0.810519i	$0.300815\pi$
$200$	0	0
$201$	551898.	0.963537
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−533568.	−0.886757
$206$	0	0
$207$	−305632.	−0.495762
$208$	0	0
$209$	333259.	0.527735
$210$	0	0
$211$	−610071.	−0.943354	−0.471677	−	0.881771i	$-0.656351\pi$
$211$	−610071.	−0.943354	−0.471677	+	0.881771i	$0.656351\pi$
$212$	0	0
$213$	985274.	1.48802
$214$	0	0
$215$	−716039.	−1.05643
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.53815e6	2.16714
$220$	0	0
$221$	1.38024e6	1.90097
$222$	0	0
$223$	−862752.	−1.16178	−0.580890	−	0.813982i	$-0.697295\pi$
$223$	−862752.	−1.16178	−0.580890	+	0.813982i	$0.697295\pi$
$224$	0	0
$225$	1.68276e6	2.21598
$226$	0	0
$227$	26778.5	0.0344923	0.0172461	−	0.999851i	$-0.494510\pi$
$227$	26778.5	0.0344923	0.0172461	+	0.999851i	$0.494510\pi$
$228$	0	0
$229$	812554.	1.02391	0.511957	−	0.859011i	$-0.328921\pi$
$229$	812554.	1.02391	0.511957	+	0.859011i	$0.328921\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.25004e6	1.50846	0.754228	−	0.656612i	$-0.228011\pi$
$233$	1.25004e6	1.50846	0.754228	+	0.656612i	$0.228011\pi$
$234$	0	0
$235$	821888.	0.970830
$236$	0	0
$237$	1.51772e6	1.75517
$238$	0	0
$239$	1.59397e6	1.80504	0.902518	−	0.430653i	$-0.141717\pi$
$239$	1.59397e6	1.80504	0.902518	+	0.430653i	$0.141717\pi$
$240$	0	0
$241$	−1.57916e6	−1.75139	−0.875696	−	0.482864i	$-0.839597\pi$
$241$	−1.57916e6	−1.75139	−0.875696	+	0.482864i	$0.839597\pi$
$242$	0	0
$243$	−415115.	−0.450975
$244$	0	0
$245$	0	0
$246$	0	0
$247$	425447.	0.443714
$248$	0	0
$249$	−2.41359e6	−2.46697
$250$	0	0
$251$	−1.47838e6	−1.48116	−0.740581	−	0.671967i	$-0.765450\pi$
$251$	−1.47838e6	−1.48116	−0.740581	+	0.671967i	$0.765450\pi$
$252$	0	0
$253$	−269980.	−0.265173
$254$	0	0
$255$	−5.22905e6	−5.03585
$256$	0	0
$257$	1.04962e6	0.991286	0.495643	−	0.868526i	$-0.334933\pi$
$257$	1.04962e6	0.991286	0.495643	+	0.868526i	$0.334933\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−479375.	−0.435586
$262$	0	0
$263$	159846.	0.142499	0.0712497	−	0.997459i	$-0.477301\pi$
$263$	159846.	0.142499	0.0712497	+	0.997459i	$0.477301\pi$
$264$	0	0
$265$	1.10672e6	0.968109
$266$	0	0
$267$	−2.08624e6	−1.79096
$268$	0	0
$269$	−226510.	−0.190856	−0.0954280	−	0.995436i	$-0.530422\pi$
$269$	−226510.	−0.190856	−0.0954280	+	0.995436i	$0.530422\pi$
$270$	0	0
$271$	−1.50643e6	−1.24602	−0.623011	−	0.782213i	$-0.714091\pi$
$271$	−1.50643e6	−1.24602	−0.623011	+	0.782213i	$0.714091\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.48646e6	1.18528
$276$	0	0
$277$	−1.91164e6	−1.49695	−0.748474	−	0.663164i	$-0.769213\pi$
$277$	−1.91164e6	−1.49695	−0.748474	+	0.663164i	$0.769213\pi$
$278$	0	0
$279$	321640.	0.247377
$280$	0	0
$281$	1.17933e6	0.890987	0.445493	−	0.895285i	$-0.353028\pi$
$281$	1.17933e6	0.890987	0.445493	+	0.895285i	$0.353028\pi$
$282$	0	0
$283$	353544.	0.262408	0.131204	−	0.991355i	$-0.458116\pi$
$283$	353544.	0.262408	0.131204	+	0.991355i	$0.458116\pi$
$284$	0	0
$285$	−1.61181e6	−1.17544
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4.22262e6	2.97398
$290$	0	0
$291$	883151.	0.611368
$292$	0	0
$293$	−61933.0	−0.0421457	−0.0210728	−	0.999778i	$-0.506708\pi$
$293$	−61933.0	−0.0421457	−0.0210728	+	0.999778i	$0.506708\pi$
$294$	0	0
$295$	857240.	0.573519
$296$	0	0
$297$	−3.41228e6	−2.24467
$298$	0	0
$299$	−344663.	−0.222955
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−5.17301e6	−3.23695
$304$	0	0
$305$	1.77081e6	1.08999
$306$	0	0
$307$	1.90020e6	1.15068	0.575338	−	0.817915i	$-0.304870\pi$
$307$	1.90020e6	1.15068	0.575338	+	0.817915i	$0.304870\pi$
$308$	0	0
$309$	−4.43323e6	−2.64134
$310$	0	0
$311$	2.63063e6	1.54226	0.771131	−	0.636676i	$-0.219691\pi$
$311$	2.63063e6	1.54226	0.771131	+	0.636676i	$0.219691\pi$
$312$	0	0
$313$	2.76044e6	1.59264	0.796319	−	0.604877i	$-0.206778\pi$
$313$	2.76044e6	1.59264	0.796319	+	0.604877i	$0.206778\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	621404.	0.347317	0.173658	−	0.984806i	$-0.444441\pi$
$317$	621404.	0.347317	0.173658	+	0.984806i	$0.444441\pi$
$318$	0	0
$319$	−423455.	−0.232986
$320$	0	0
$321$	−4.22234e6	−2.28713
$322$	0	0
$323$	1.73924e6	0.927586
$324$	0	0
$325$	1.89766e6	0.996575
$326$	0	0
$327$	1.15827e6	0.599017
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−878316.	−0.440637	−0.220318	−	0.975428i	$-0.570710\pi$
$331$	−878316.	−0.440637	−0.220318	+	0.975428i	$0.570710\pi$
$332$	0	0
$333$	−2.10113e6	−1.03835
$334$	0	0
$335$	−1.60225e6	−0.780042
$336$	0	0
$337$	1.72499e6	0.827392	0.413696	−	0.910415i	$-0.364238\pi$
$337$	1.72499e6	0.827392	0.413696	+	0.910415i	$0.364238\pi$
$338$	0	0
$339$	−2.55565e6	−1.20782
$340$	0	0
$341$	284121.	0.132317
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.30576e6	0.590629
$346$	0	0
$347$	−925574.	−0.412655	−0.206328	−	0.978483i	$-0.566151\pi$
$347$	−925574.	−0.412655	−0.206328	+	0.978483i	$0.566151\pi$
$348$	0	0
$349$	−3.27452e6	−1.43908	−0.719539	−	0.694452i	$-0.755647\pi$
$349$	−3.27452e6	−1.43908	−0.719539	+	0.694452i	$0.755647\pi$
$350$	0	0
$351$	−4.35621e6	−1.88730
$352$	0	0
$353$	1.90831e6	0.815103	0.407551	−	0.913182i	$-0.366383\pi$
$353$	1.90831e6	0.815103	0.407551	+	0.913182i	$0.366383\pi$
$354$	0	0
$355$	−2.86041e6	−1.20464
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.44900e6	0.593379	0.296690	−	0.954974i	$-0.404117\pi$
$359$	1.44900e6	0.593379	0.296690	+	0.954974i	$0.404117\pi$
$360$	0	0
$361$	−1.93999e6	−0.783488
$362$	0	0
$363$	−1.26977e6	−0.505775
$364$	0	0
$365$	−4.46550e6	−1.75444
$366$	0	0
$367$	1.87979e6	0.728524	0.364262	−	0.931296i	$-0.381321\pi$
$367$	1.87979e6	0.728524	0.364262	+	0.931296i	$0.381321\pi$
$368$	0	0
$369$	−3.43901e6	−1.31482
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4.50861e6	1.67792	0.838959	−	0.544194i	$-0.183165\pi$
$373$	4.50861e6	1.67792	0.838959	+	0.544194i	$0.183165\pi$
$374$	0	0
$375$	−310091.	−0.113870
$376$	0	0
$377$	−540594.	−0.195893
$378$	0	0
$379$	−2.33971e6	−0.836688	−0.418344	−	0.908289i	$-0.637389\pi$
$379$	−2.33971e6	−0.836688	−0.418344	+	0.908289i	$0.637389\pi$
$380$	0	0
$381$	1.64608e6	0.580949
$382$	0	0
$383$	−1.71181e6	−0.596293	−0.298146	−	0.954520i	$-0.596368\pi$
$383$	−1.71181e6	−0.596293	−0.298146	+	0.954520i	$0.596368\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.61510e6	−1.56640
$388$	0	0
$389$	1.07773e6	0.361108	0.180554	−	0.983565i	$-0.442211\pi$
$389$	1.07773e6	0.361108	0.180554	+	0.983565i	$0.442211\pi$
$390$	0	0
$391$	−1.40900e6	−0.466088
$392$	0	0
$393$	7.20691e6	2.35379
$394$	0	0
$395$	−4.40619e6	−1.42092
$396$	0	0
$397$	288083.	0.0917363	0.0458682	−	0.998948i	$-0.485395\pi$
$397$	288083.	0.0917363	0.0458682	+	0.998948i	$0.485395\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.46431e6	−1.69697	−0.848485	−	0.529220i	$-0.822485\pi$
$401$	−5.46431e6	−1.69697	−0.848485	+	0.529220i	$0.822485\pi$
$402$	0	0
$403$	362716.	0.111251
$404$	0	0
$405$	6.49405e6	1.96733
$406$	0	0
$407$	−1.85603e6	−0.555391
$408$	0	0
$409$	2.45675e6	0.726195	0.363098	−	0.931751i	$-0.381719\pi$
$409$	2.45675e6	0.726195	0.363098	+	0.931751i	$0.381719\pi$
$410$	0	0
$411$	−1.42847e6	−0.417125
$412$	0	0
$413$	0	0
$414$	0	0
$415$	7.00704e6	1.99717
$416$	0	0
$417$	8.24414e6	2.32170
$418$	0	0
$419$	−267685.	−0.0744884	−0.0372442	−	0.999306i	$-0.511858\pi$
$419$	−267685.	−0.0744884	−0.0372442	+	0.999306i	$0.511858\pi$
$420$	0	0
$421$	−1.11613e6	−0.306908	−0.153454	−	0.988156i	$-0.549040\pi$
$421$	−1.11613e6	−0.306908	−0.153454	+	0.988156i	$0.549040\pi$
$422$	0	0
$423$	5.29733e6	1.43948
$424$	0	0
$425$	7.75770e6	2.08334
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−7.28257e6	−1.91048
$430$	0	0
$431$	1.56045e6	0.404629	0.202314	−	0.979321i	$-0.435154\pi$
$431$	1.56045e6	0.404629	0.202314	+	0.979321i	$0.435154\pi$
$432$	0	0
$433$	2.91249e6	0.746525	0.373263	−	0.927726i	$-0.378239\pi$
$433$	2.91249e6	0.746525	0.373263	+	0.927726i	$0.378239\pi$
$434$	0	0
$435$	2.04804e6	0.518938
$436$	0	0
$437$	−434310.	−0.108792
$438$	0	0
$439$	1.10541e6	0.273754	0.136877	−	0.990588i	$-0.456293\pi$
$439$	1.10541e6	0.273754	0.136877	+	0.990588i	$0.456293\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.78806e6	0.432885	0.216443	−	0.976295i	$-0.430555\pi$
$443$	1.78806e6	0.432885	0.216443	+	0.976295i	$0.430555\pi$
$444$	0	0
$445$	6.05671e6	1.44990
$446$	0	0
$447$	7.75654e6	1.83611
$448$	0	0
$449$	−4.24314e6	−0.993279	−0.496639	−	0.867957i	$-0.665433\pi$
$449$	−4.24314e6	−0.993279	−0.496639	+	0.867957i	$0.665433\pi$
$450$	0	0
$451$	−3.03785e6	−0.703274
$452$	0	0
$453$	−6.12408e6	−1.40215
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.65975e6	0.371750	0.185875	−	0.982573i	$-0.440488\pi$
$457$	1.65975e6	0.371750	0.185875	+	0.982573i	$0.440488\pi$
$458$	0	0
$459$	−1.78083e7	−3.94540
$460$	0	0
$461$	−8.34155e6	−1.82808	−0.914038	−	0.405628i	$-0.867053\pi$
$461$	−8.34155e6	−1.82808	−0.914038	+	0.405628i	$0.867053\pi$
$462$	0	0
$463$	487990.	0.105793	0.0528967	−	0.998600i	$-0.483155\pi$
$463$	487990.	0.105793	0.0528967	+	0.998600i	$0.483155\pi$
$464$	0	0
$465$	−1.37415e6	−0.294715
$466$	0	0
$467$	5.98913e6	1.27078	0.635392	−	0.772190i	$-0.280839\pi$
$467$	5.98913e6	1.27078	0.635392	+	0.772190i	$0.280839\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.64324e6	−0.341309
$472$	0	0
$473$	−4.07674e6	−0.837838
$474$	0	0
$475$	2.39124e6	0.486283
$476$	0	0
$477$	7.13318e6	1.43545
$478$	0	0
$479$	3.97080e6	0.790751	0.395375	−	0.918520i	$-0.370615\pi$
$479$	3.97080e6	0.790751	0.395375	+	0.918520i	$0.370615\pi$
$480$	0	0
$481$	−2.36946e6	−0.466967
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.56393e6	−0.494940
$486$	0	0
$487$	7.63822e6	1.45938	0.729692	−	0.683775i	$-0.239663\pi$
$487$	7.63822e6	1.45938	0.729692	+	0.683775i	$0.239663\pi$
$488$	0	0
$489$	1.04360e7	1.97361
$490$	0	0
$491$	8.30768e6	1.55516	0.777581	−	0.628782i	$-0.216446\pi$
$491$	8.30768e6	1.55516	0.777581	+	0.628782i	$0.216446\pi$
$492$	0	0
$493$	−2.20997e6	−0.409514
$494$	0	0
$495$	1.87482e7	3.43912
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−5.70382e6	−1.02545	−0.512725	−	0.858553i	$-0.671364\pi$
$499$	−5.70382e6	−1.02545	−0.512725	+	0.858553i	$0.671364\pi$
$500$	0	0
$501$	−3.95448e6	−0.703875
$502$	0	0
$503$	−58117.5	−0.0102421	−0.00512103	−	0.999987i	$-0.501630\pi$
$503$	−58117.5	−0.0102421	−0.00512103	+	0.999987i	$0.501630\pi$
$504$	0	0
$505$	1.50181e7	2.62051
$506$	0	0
$507$	926966.	0.160156
$508$	0	0
$509$	−1.76812e6	−0.302494	−0.151247	−	0.988496i	$-0.548329\pi$
$509$	−1.76812e6	−0.302494	−0.151247	+	0.988496i	$0.548329\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.48925e6	−0.920916
$514$	0	0
$515$	1.28704e7	2.13832
$516$	0	0
$517$	4.67939e6	0.769950
$518$	0	0
$519$	2.13169e7	3.47381
$520$	0	0
$521$	−7.94082e6	−1.28165	−0.640827	−	0.767685i	$-0.721408\pi$
$521$	−7.94082e6	−1.28165	−0.640827	+	0.767685i	$0.721408\pi$
$522$	0	0
$523$	−7.76839e6	−1.24187	−0.620936	−	0.783862i	$-0.713247\pi$
$523$	−7.76839e6	−1.24187	−0.620936	+	0.783862i	$0.713247\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.48280e6	0.232571
$528$	0	0
$529$	−6.08450e6	−0.945335
$530$	0	0
$531$	5.52519e6	0.850376
$532$	0	0
$533$	−3.87820e6	−0.591306
$534$	0	0
$535$	1.22581e7	1.85157
$536$	0	0
$537$	−1.28472e7	−1.92253
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.88500e6	−0.717582	−0.358791	−	0.933418i	$-0.616811\pi$
$541$	−4.88500e6	−0.717582	−0.358791	+	0.933418i	$0.616811\pi$
$542$	0	0
$543$	1.58400e7	2.30546
$544$	0	0
$545$	−3.36264e6	−0.484941
$546$	0	0
$547$	−8.12743e6	−1.16141	−0.580704	−	0.814115i	$-0.697223\pi$
$547$	−8.12743e6	−1.16141	−0.580704	+	0.814115i	$0.697223\pi$
$548$	0	0
$549$	1.14135e7	1.61617
$550$	0	0
$551$	−681202.	−0.0955866
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.97670e6	1.23704
$556$	0	0
$557$	−6.09229e6	−0.832037	−0.416018	−	0.909356i	$-0.636575\pi$
$557$	−6.09229e6	−0.832037	−0.416018	+	0.909356i	$0.636575\pi$
$558$	0	0
$559$	−5.20448e6	−0.704446
$560$	0	0
$561$	−2.97714e7	−3.99386
$562$	0	0
$563$	3.34382e6	0.444602	0.222301	−	0.974978i	$-0.428643\pi$
$563$	3.34382e6	0.444602	0.222301	+	0.974978i	$0.428643\pi$
$564$	0	0
$565$	7.41948e6	0.977805
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.93388e6	1.02732	0.513659	−	0.857994i	$-0.328290\pi$
$569$	7.93388e6	1.02732	0.513659	+	0.857994i	$0.328290\pi$
$570$	0	0
$571$	−5.99862e6	−0.769948	−0.384974	−	0.922928i	$-0.625790\pi$
$571$	−5.99862e6	−0.769948	−0.384974	+	0.922928i	$0.625790\pi$
$572$	0	0
$573$	−7.71148e6	−0.981185
$574$	0	0
$575$	−1.93719e6	−0.244345
$576$	0	0
$577$	−3.42929e6	−0.428810	−0.214405	−	0.976745i	$-0.568781\pi$
$577$	−3.42929e6	−0.428810	−0.214405	+	0.976745i	$0.568781\pi$
$578$	0	0
$579$	−791997.	−0.0981810
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.30108e6	0.767792
$584$	0	0
$585$	2.39345e7	2.89158
$586$	0	0
$587$	−4.04291e6	−0.484282	−0.242141	−	0.970241i	$-0.577850\pi$
$587$	−4.04291e6	−0.484282	−0.242141	+	0.970241i	$0.577850\pi$
$588$	0	0
$589$	457058.	0.0542855
$590$	0	0
$591$	4.79862e6	0.565129
$592$	0	0
$593$	1.46937e6	0.171591	0.0857956	−	0.996313i	$-0.472657\pi$
$593$	1.46937e6	0.171591	0.0857956	+	0.996313i	$0.472657\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.80200e7	−2.06928
$598$	0	0
$599$	−9.35233e6	−1.06501	−0.532504	−	0.846427i	$-0.678749\pi$
$599$	−9.35233e6	−1.06501	−0.532504	+	0.846427i	$0.678749\pi$
$600$	0	0
$601$	4.19824e6	0.474112	0.237056	−	0.971496i	$-0.423818\pi$
$601$	4.19824e6	0.474112	0.237056	+	0.971496i	$0.423818\pi$
$602$	0	0
$603$	−1.03270e7	−1.15659
$604$	0	0
$605$	3.68634e6	0.409456
$606$	0	0
$607$	1.21401e7	1.33737	0.668683	−	0.743548i	$-0.266859\pi$
$607$	1.21401e7	1.33737	0.668683	+	0.743548i	$0.266859\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.97384e6	0.647367
$612$	0	0
$613$	6.99084e6	0.751412	0.375706	−	0.926739i	$-0.377400\pi$
$613$	6.99084e6	0.751412	0.375706	+	0.926739i	$0.377400\pi$
$614$	0	0
$615$	1.46926e7	1.56643
$616$	0	0
$617$	−9.72855e6	−1.02881	−0.514405	−	0.857547i	$-0.671987\pi$
$617$	−9.72855e6	−1.02881	−0.514405	+	0.857547i	$0.671987\pi$
$618$	0	0
$619$	8.66808e6	0.909277	0.454639	−	0.890676i	$-0.349768\pi$
$619$	8.66808e6	0.909277	0.454639	+	0.890676i	$0.349768\pi$
$620$	0	0
$621$	4.44695e6	0.462736
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−9.30558e6	−0.952892
$626$	0	0
$627$	−9.17677e6	−0.932226
$628$	0	0
$629$	−9.68642e6	−0.976195
$630$	0	0
$631$	4.83557e6	0.483476	0.241738	−	0.970342i	$-0.422283\pi$
$631$	4.83557e6	0.483476	0.241738	+	0.970342i	$0.422283\pi$
$632$	0	0
$633$	1.67992e7	1.66640
$634$	0	0
$635$	−4.77884e6	−0.470314
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.84363e7	−1.78616
$640$	0	0
$641$	−1.78219e7	−1.71320	−0.856601	−	0.515980i	$-0.827428\pi$
$641$	−1.78219e7	−1.71320	−0.856601	+	0.515980i	$0.827428\pi$
$642$	0	0
$643$	1.58095e7	1.50796	0.753980	−	0.656897i	$-0.228132\pi$
$643$	1.58095e7	1.50796	0.753980	+	0.656897i	$0.228132\pi$
$644$	0	0
$645$	1.97172e7	1.86615
$646$	0	0
$647$	−786375.	−0.0738532	−0.0369266	−	0.999318i	$-0.511757\pi$
$647$	−786375.	−0.0738532	−0.0369266	+	0.999318i	$0.511757\pi$
$648$	0	0
$649$	4.88067e6	0.454849
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.18512e6	0.108762	0.0543812	−	0.998520i	$-0.482681\pi$
$653$	1.18512e6	0.108762	0.0543812	+	0.998520i	$0.482681\pi$
$654$	0	0
$655$	−2.09228e7	−1.90554
$656$	0	0
$657$	−2.87815e7	−2.60136
$658$	0	0
$659$	−2.81475e6	−0.252479	−0.126240	−	0.992000i	$-0.540291\pi$
$659$	−2.81475e6	−0.252479	−0.126240	+	0.992000i	$0.540291\pi$
$660$	0	0
$661$	−295576.	−0.0263127	−0.0131564	−	0.999913i	$-0.504188\pi$
$661$	−295576.	−0.0263127	−0.0131564	+	0.999913i	$0.504188\pi$
$662$	0	0
$663$	−3.80070e7	−3.35799
$664$	0	0
$665$	0	0
$666$	0	0
$667$	551856.	0.0480298
$668$	0	0
$669$	2.37571e7	2.05224
$670$	0	0
$671$	1.00821e7	0.864456
$672$	0	0
$673$	−3.29325e6	−0.280277	−0.140138	−	0.990132i	$-0.544755\pi$
$673$	−3.29325e6	−0.280277	−0.140138	+	0.990132i	$0.544755\pi$
$674$	0	0
$675$	−2.44842e7	−2.06836
$676$	0	0
$677$	−1.49842e7	−1.25650	−0.628248	−	0.778013i	$-0.716228\pi$
$677$	−1.49842e7	−1.25650	−0.628248	+	0.778013i	$0.716228\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−737385.	−0.0609294
$682$	0	0
$683$	2.74204e6	0.224917	0.112458	−	0.993656i	$-0.464128\pi$
$683$	2.74204e6	0.224917	0.112458	+	0.993656i	$0.464128\pi$
$684$	0	0
$685$	4.14708e6	0.337688
$686$	0	0
$687$	−2.23749e7	−1.80871
$688$	0	0
$689$	8.04413e6	0.645552
$690$	0	0
$691$	1.95290e7	1.55591	0.777955	−	0.628321i	$-0.216257\pi$
$691$	1.95290e7	1.55591	0.777955	+	0.628321i	$0.216257\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.39341e7	−1.87955
$696$	0	0
$697$	−1.58542e7	−1.23613
$698$	0	0
$699$	−3.44216e7	−2.66464
$700$	0	0
$701$	7.97976e6	0.613331	0.306665	−	0.951817i	$-0.400787\pi$
$701$	7.97976e6	0.613331	0.306665	+	0.951817i	$0.400787\pi$
$702$	0	0
$703$	−2.98575e6	−0.227859
$704$	0	0
$705$	−2.26319e7	−1.71494
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8.20987e6	0.613368	0.306684	−	0.951811i	$-0.400781\pi$
$709$	8.20987e6	0.613368	0.306684	+	0.951811i	$0.400781\pi$
$710$	0	0
$711$	−2.83993e7	−2.10685
$712$	0	0
$713$	−370272.	−0.0272770
$714$	0	0
$715$	2.11425e7	1.54665
$716$	0	0
$717$	−4.38924e7	−3.18853
$718$	0	0
$719$	−9.88944e6	−0.713427	−0.356714	−	0.934214i	$-0.616103\pi$
$719$	−9.88944e6	−0.713427	−0.356714	+	0.934214i	$0.616103\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	4.34845e7	3.09377
$724$	0	0
$725$	−3.03843e6	−0.214686
$726$	0	0
$727$	1.40610e7	0.986691	0.493345	−	0.869834i	$-0.335774\pi$
$727$	1.40610e7	0.986691	0.493345	+	0.869834i	$0.335774\pi$
$728$	0	0
$729$	−8.30898e6	−0.579067
$730$	0	0
$731$	−2.12761e7	−1.47265
$732$	0	0
$733$	2.26419e7	1.55651	0.778257	−	0.627946i	$-0.216104\pi$
$733$	2.26419e7	1.55651	0.778257	+	0.627946i	$0.216104\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.12234e6	−0.618640
$738$	0	0
$739$	1.94357e7	1.30915	0.654576	−	0.755996i	$-0.272847\pi$
$739$	1.94357e7	1.30915	0.654576	+	0.755996i	$0.272847\pi$
$740$	0	0
$741$	−1.17153e7	−0.783806
$742$	0	0
$743$	6.54899e6	0.435214	0.217607	−	0.976037i	$-0.430175\pi$
$743$	6.54899e6	0.435214	0.217607	+	0.976037i	$0.430175\pi$
$744$	0	0
$745$	−2.25185e7	−1.48644
$746$	0	0
$747$	4.51626e7	2.96127
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.01572e7	−0.657163	−0.328582	−	0.944476i	$-0.606571\pi$
$751$	−1.01572e7	−0.657163	−0.328582	+	0.944476i	$0.606571\pi$
$752$	0	0
$753$	4.07094e7	2.61642
$754$	0	0
$755$	1.77792e7	1.13513
$756$	0	0
$757$	−8.02731e6	−0.509132	−0.254566	−	0.967055i	$-0.581933\pi$
$757$	−8.02731e6	−0.509132	−0.254566	+	0.967055i	$0.581933\pi$
$758$	0	0
$759$	7.43429e6	0.468419
$760$	0	0
$761$	5.31336e6	0.332589	0.166294	−	0.986076i	$-0.446820\pi$
$761$	5.31336e6	0.332589	0.166294	+	0.986076i	$0.446820\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	9.78451e7	6.04485
$766$	0	0
$767$	6.23079e6	0.382433
$768$	0	0
$769$	6.32376e6	0.385620	0.192810	−	0.981236i	$-0.438240\pi$
$769$	6.32376e6	0.385620	0.192810	+	0.981236i	$0.438240\pi$
$770$	0	0
$771$	−2.89028e7	−1.75107
$772$	0	0
$773$	−3.03024e6	−0.182402	−0.0912009	−	0.995833i	$-0.529071\pi$
$773$	−3.03024e6	−0.182402	−0.0912009	+	0.995833i	$0.529071\pi$
$774$	0	0
$775$	2.03866e6	0.121924
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.88692e6	−0.288530
$780$	0	0
$781$	−1.62856e7	−0.955383
$782$	0	0
$783$	6.97491e6	0.406569
$784$	0	0
$785$	4.77058e6	0.276310
$786$	0	0
$787$	2.44622e6	0.140786	0.0703930	−	0.997519i	$-0.477575\pi$
$787$	2.44622e6	0.140786	0.0703930	+	0.997519i	$0.477575\pi$
$788$	0	0
$789$	−4.40160e6	−0.251720
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.28710e7	0.726826
$794$	0	0
$795$	−3.04752e7	−1.71013
$796$	0	0
$797$	6.16846e6	0.343978	0.171989	−	0.985099i	$-0.444981\pi$
$797$	6.16846e6	0.343978	0.171989	+	0.985099i	$0.444981\pi$
$798$	0	0
$799$	2.44212e7	1.35332
$800$	0	0
$801$	3.90374e7	2.14981
$802$	0	0
$803$	−2.54241e7	−1.39142
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.23727e6	0.337140
$808$	0	0
$809$	3.08989e7	1.65986	0.829930	−	0.557868i	$-0.188380\pi$
$809$	3.08989e7	1.65986	0.829930	+	0.557868i	$0.188380\pi$
$810$	0	0
$811$	8.16479e6	0.435906	0.217953	−	0.975959i	$-0.430062\pi$
$811$	8.16479e6	0.435906	0.217953	+	0.975959i	$0.430062\pi$
$812$	0	0
$813$	4.14817e7	2.20105
$814$	0	0
$815$	−3.02973e7	−1.59775
$816$	0	0
$817$	−6.55816e6	−0.343737
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.77929e7	0.921272	0.460636	−	0.887589i	$-0.347621\pi$
$821$	1.77929e7	0.921272	0.460636	+	0.887589i	$0.347621\pi$
$822$	0	0
$823$	2.09125e7	1.07623	0.538116	−	0.842871i	$-0.319136\pi$
$823$	2.09125e7	1.07623	0.538116	+	0.842871i	$0.319136\pi$
$824$	0	0
$825$	−4.09320e7	−2.09376
$826$	0	0
$827$	−2.17379e7	−1.10523	−0.552617	−	0.833435i	$-0.686371\pi$
$827$	−2.17379e7	−1.10523	−0.552617	+	0.833435i	$0.686371\pi$
$828$	0	0
$829$	−3.51536e7	−1.77657	−0.888287	−	0.459288i	$-0.848105\pi$
$829$	−3.51536e7	−1.77657	−0.888287	+	0.459288i	$0.848105\pi$
$830$	0	0
$831$	5.26398e7	2.64431
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.14805e7	0.569830
$836$	0	0
$837$	−4.67988e6	−0.230898
$838$	0	0
$839$	−8.68616e6	−0.426013	−0.213007	−	0.977051i	$-0.568326\pi$
$839$	−8.68616e6	−0.426013	−0.213007	+	0.977051i	$0.568326\pi$
$840$	0	0
$841$	−1.96456e7	−0.957800
$842$	0	0
$843$	−3.24747e7	−1.57390
$844$	0	0
$845$	−2.69113e6	−0.129656
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9.73536e6	−0.463535
$850$	0	0
$851$	2.41882e6	0.114493
$852$	0	0
$853$	−125857.	−0.00592247	−0.00296124	−	0.999996i	$-0.500943\pi$
$853$	−125857.	−0.00592247	−0.00296124	+	0.999996i	$0.500943\pi$
$854$	0	0
$855$	3.01599e7	1.41096
$856$	0	0
$857$	1.49143e7	0.693669	0.346834	−	0.937926i	$-0.387257\pi$
$857$	1.49143e7	0.693669	0.346834	+	0.937926i	$0.387257\pi$
$858$	0	0
$859$	−4.06866e7	−1.88134	−0.940672	−	0.339317i	$-0.889804\pi$
$859$	−4.06866e7	−1.88134	−0.940672	+	0.339317i	$0.889804\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.14409e7	−1.43704	−0.718518	−	0.695508i	$-0.755179\pi$
$863$	−3.14409e7	−1.43704	−0.718518	+	0.695508i	$0.755179\pi$
$864$	0	0
$865$	−6.18865e7	−2.81226
$866$	0	0
$867$	−1.16276e8	−5.25343
$868$	0	0
$869$	−2.50864e7	−1.12691
$870$	0	0
$871$	−1.16458e7	−0.520146
$872$	0	0
$873$	−1.65254e7	−0.733864
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.71647e7	−1.19263	−0.596315	−	0.802751i	$-0.703369\pi$
$877$	−2.71647e7	−1.19263	−0.596315	+	0.802751i	$0.703369\pi$
$878$	0	0
$879$	1.70542e6	0.0744489
$880$	0	0
$881$	−1.03892e6	−0.0450966	−0.0225483	−	0.999746i	$-0.507178\pi$
$881$	−1.03892e6	−0.0450966	−0.0225483	+	0.999746i	$0.507178\pi$
$882$	0	0
$883$	2.29747e7	0.991625	0.495812	−	0.868430i	$-0.334870\pi$
$883$	2.29747e7	0.991625	0.495812	+	0.868430i	$0.334870\pi$
$884$	0	0
$885$	−2.36054e7	−1.01310
$886$	0	0
$887$	−3.42677e7	−1.46243	−0.731217	−	0.682145i	$-0.761047\pi$
$887$	−3.42677e7	−1.46243	−0.731217	+	0.682145i	$0.761047\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.69736e7	1.56026
$892$	0	0
$893$	7.52763e6	0.315885
$894$	0	0
$895$	3.72975e7	1.55640
$896$	0	0
$897$	9.49081e6	0.393842
$898$	0	0
$899$	−580761.	−0.0239661
$900$	0	0
$901$	3.28847e7	1.34953
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.59862e7	−1.86641
$906$	0	0
$907$	−3.29823e7	−1.33126	−0.665630	−	0.746282i	$-0.731837\pi$
$907$	−3.29823e7	−1.33126	−0.665630	+	0.746282i	$0.731837\pi$
$908$	0	0
$909$	9.67963e7	3.88552
$910$	0	0
$911$	−1.04563e7	−0.417429	−0.208714	−	0.977977i	$-0.566928\pi$
$911$	−1.04563e7	−0.417429	−0.208714	+	0.977977i	$0.566928\pi$
$912$	0	0
$913$	3.98943e7	1.58392
$914$	0	0
$915$	−4.87619e7	−1.92543
$916$	0	0
$917$	0	0
$918$	0	0
$919$	6.06514e6	0.236893	0.118446	−	0.992960i	$-0.462209\pi$
$919$	6.06514e6	0.236893	0.118446	+	0.992960i	$0.462209\pi$
$920$	0	0
$921$	−5.23248e7	−2.03263
$922$	0	0
$923$	−2.07907e7	−0.803276
$924$	0	0
$925$	−1.33176e7	−0.511767
$926$	0	0
$927$	8.29537e7	3.17056
$928$	0	0
$929$	3.43259e7	1.30492	0.652459	−	0.757824i	$-0.273737\pi$
$929$	3.43259e7	1.30492	0.652459	+	0.757824i	$0.273737\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−7.24382e7	−2.72435
$934$	0	0
$935$	8.64313e7	3.23327
$936$	0	0
$937$	1.21642e7	0.452621	0.226310	−	0.974055i	$-0.427334\pi$
$937$	1.21642e7	0.452621	0.226310	+	0.974055i	$0.427334\pi$
$938$	0	0
$939$	−7.60127e7	−2.81334
$940$	0	0
$941$	2.28376e7	0.840770	0.420385	−	0.907346i	$-0.361895\pi$
$941$	2.28376e7	0.840770	0.420385	+	0.907346i	$0.361895\pi$
$942$	0	0
$943$	3.95899e6	0.144979
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.84801e7	0.669623	0.334811	−	0.942285i	$-0.391327\pi$
$947$	1.84801e7	0.669623	0.334811	+	0.942285i	$0.391327\pi$
$948$	0	0
$949$	−3.24571e7	−1.16989
$950$	0	0
$951$	−1.71113e7	−0.613523
$952$	0	0
$953$	−1.03848e7	−0.370394	−0.185197	−	0.982701i	$-0.559292\pi$
$953$	−1.03848e7	−0.370394	−0.185197	+	0.982701i	$0.559292\pi$
$954$	0	0
$955$	2.23877e7	0.794329
$956$	0	0
$957$	1.16605e7	0.411562
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.82395e7	−0.986389
$962$	0	0
$963$	7.90077e7	2.74539
$964$	0	0
$965$	2.29930e6	0.0794835
$966$	0	0
$967$	−5.47893e7	−1.88421	−0.942105	−	0.335318i	$-0.891156\pi$
$967$	−5.47893e7	−1.88421	−0.942105	+	0.335318i	$0.891156\pi$
$968$	0	0
$969$	−4.78926e7	−1.63855
$970$	0	0
$971$	1.66991e7	0.568387	0.284194	−	0.958767i	$-0.408274\pi$
$971$	1.66991e7	0.568387	0.284194	+	0.958767i	$0.408274\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−5.22549e7	−1.76042
$976$	0	0
$977$	−2.99527e7	−1.00392	−0.501961	−	0.864890i	$-0.667388\pi$
$977$	−2.99527e7	−1.00392	−0.501961	+	0.864890i	$0.667388\pi$
$978$	0	0
$979$	3.44836e7	1.14989
$980$	0	0
$981$	−2.16733e7	−0.719038
$982$	0	0
$983$	5.42575e7	1.79092	0.895460	−	0.445141i	$-0.146847\pi$
$983$	5.42575e7	1.79092	0.895460	+	0.445141i	$0.146847\pi$
$984$	0	0
$985$	−1.39312e7	−0.457506
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.31290e6	0.172719
$990$	0	0
$991$	−5.93896e7	−1.92099	−0.960497	−	0.278290i	$-0.910232\pi$
$991$	−5.93896e7	−1.92099	−0.960497	+	0.278290i	$0.910232\pi$
$992$	0	0
$993$	2.41857e7	0.778370
$994$	0	0
$995$	5.23151e7	1.67521
$996$	0	0
$997$	2.67971e7	0.853786	0.426893	−	0.904302i	$-0.359608\pi$
$997$	2.67971e7	0.853786	0.426893	+	0.904302i	$0.359608\pi$
$998$	0	0
$999$	3.05715e7	0.969176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.6.a.j.1.1		5
4.3	odd	2		784.6.a.bl.1.5		5
7.2	even	3		392.6.i.o.361.5		10
7.3	odd	6		56.6.i.b.9.1	✓	10
7.4	even	3		392.6.i.o.177.5		10
7.5	odd	6		56.6.i.b.25.1	yes	10
7.6	odd	2		392.6.a.k.1.5		5
21.5	even	6		504.6.s.b.361.2		10
21.17	even	6		504.6.s.b.289.2		10
28.3	even	6		112.6.i.f.65.5		10
28.19	even	6		112.6.i.f.81.5		10
28.27	even	2		784.6.a.bk.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.i.b.9.1	✓	10	7.3	odd	6
56.6.i.b.25.1	yes	10	7.5	odd	6
112.6.i.f.65.5		10	28.3	even	6
112.6.i.f.81.5		10	28.19	even	6
392.6.a.j.1.1		5	1.1	even	1	trivial
392.6.a.k.1.5		5	7.6	odd	2
392.6.i.o.177.5		10	7.4	even	3
392.6.i.o.361.5		10	7.2	even	3
504.6.s.b.289.2		10	21.17	even	6
504.6.s.b.361.2		10	21.5	even	6
784.6.a.bk.1.1		5	28.27	even	2
784.6.a.bl.1.5		5	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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

\(f(q)\)	\(=\)	\(q-27.5365 q^{3} +79.9429 q^{5} +515.257 q^{9} +O(q^{10})\)	\(q-27.5365 q^{3} +79.9429 q^{5} +515.257 q^{9} +455.152 q^{11} +581.059 q^{13} -2201.35 q^{15} +2375.39 q^{17} +732.193 q^{19} -593.164 q^{23} +3265.86 q^{25} -7497.01 q^{27} -930.360 q^{29} +624.232 q^{31} -12533.3 q^{33} -4077.82 q^{37} -16000.3 q^{39} -6674.36 q^{41} -8956.88 q^{43} +41191.2 q^{45} +10280.9 q^{47} -65409.9 q^{51} +13843.9 q^{53} +36386.1 q^{55} -20162.0 q^{57} +10723.2 q^{59} +22151.0 q^{61} +46451.5 q^{65} -20042.4 q^{67} +16333.6 q^{69} -35780.7 q^{71} -55858.6 q^{73} -89930.4 q^{75} -55116.7 q^{79} +81233.6 q^{81} +87650.6 q^{83} +189896. q^{85} +25618.8 q^{87} +75763.0 q^{89} -17189.2 q^{93} +58533.6 q^{95} -32072.1 q^{97} +234520. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 81 q^{5} + 390 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 81 * q^5 + 390 * q^9	\( 5 q - 5 q^{3} + 81 q^{5} + 390 q^{9} + 361 q^{11} + 342 q^{13} - 1049 q^{15} + 1809 q^{17} + 1277 q^{19} + 911 q^{23} + 3940 q^{25} - 4751 q^{27} + 5442 q^{29} + 2187 q^{31} - 5553 q^{33} - 8181 q^{37} - 3422 q^{39} + 16578 q^{41} + 6332 q^{43} + 41310 q^{45} + 16101 q^{47} - 67865 q^{51} + 16047 q^{53} + 45629 q^{55} + 22347 q^{57} + 71027 q^{59} + 31093 q^{61} - 64370 q^{65} + 47981 q^{67} + 137249 q^{69} + 22512 q^{71} + 123333 q^{73} - 45460 q^{75} - 212481 q^{79} + 52917 q^{81} + 87460 q^{83} + 222141 q^{85} + 318070 q^{87} + 129045 q^{89} - 252835 q^{93} + 300417 q^{95} + 328274 q^{97} + 249798 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 81 * q^5 + 390 * q^9 + 361 * q^11 + 342 * q^13 - 1049 * q^15 + 1809 * q^17 + 1277 * q^19 + 911 * q^23 + 3940 * q^25 - 4751 * q^27 + 5442 * q^29 + 2187 * q^31 - 5553 * q^33 - 8181 * q^37 - 3422 * q^39 + 16578 * q^41 + 6332 * q^43 + 41310 * q^45 + 16101 * q^47 - 67865 * q^51 + 16047 * q^53 + 45629 * q^55 + 22347 * q^57 + 71027 * q^59 + 31093 * q^61 - 64370 * q^65 + 47981 * q^67 + 137249 * q^69 + 22512 * q^71 + 123333 * q^73 - 45460 * q^75 - 212481 * q^79 + 52917 * q^81 + 87460 * q^83 + 222141 * q^85 + 318070 * q^87 + 129045 * q^89 - 252835 * q^93 + 300417 * q^95 + 328274 * q^97 + 249798 * q^99

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