LMFDB - Embedded newform 392.6.a.i.1.2

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:	$1$
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} - 167x^{3} - 387x^{2} + 1720x + 2340 $ x^5 - 167x^3 - 387x^2 + 1720*x + 2340
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			1.2
Root			$2.94609$ 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)

$f(q)$	$=$	$q-14.6817 q^{3} +72.1603 q^{5} -27.4480 q^{9} +O(q^{10})$	$q-14.6817 q^{3} +72.1603 q^{5} -27.4480 q^{9} -246.305 q^{11} +1130.90 q^{13} -1059.43 q^{15} -1644.26 q^{17} -1646.23 q^{19} -469.033 q^{23} +2082.11 q^{25} +3970.63 q^{27} +5816.06 q^{29} -5738.00 q^{31} +3616.17 q^{33} +435.509 q^{37} -16603.5 q^{39} -11216.3 q^{41} +6147.31 q^{43} -1980.66 q^{45} +14621.2 q^{47} +24140.6 q^{51} +26805.9 q^{53} -17773.4 q^{55} +24169.5 q^{57} -25054.0 q^{59} -15745.8 q^{61} +81605.8 q^{65} +20233.0 q^{67} +6886.20 q^{69} -69564.9 q^{71} -10481.2 q^{73} -30568.9 q^{75} -90425.7 q^{79} -51625.7 q^{81} -30880.2 q^{83} -118651. q^{85} -85389.5 q^{87} +16493.6 q^{89} +84243.5 q^{93} -118793. q^{95} -103206. q^{97} +6760.59 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 13 q^{3} - 31 q^{5} + 230 q^{9}+O(q^{10}) $ 5 * q - 13 * q^3 - 31 * q^5 + 230 * q^9	$ 5 q - 13 q^{3} - 31 q^{5} + 230 q^{9} - 351 q^{11} + 54 q^{13} + 607 q^{15} - 111 q^{17} - 1035 q^{19} + 3639 q^{23} + 1540 q^{25} - 3607 q^{27} - 734 q^{29} - 7677 q^{31} + 7439 q^{33} + 13595 q^{37} - 1406 q^{39} - 5310 q^{41} + 764 q^{43} - 38978 q^{45} - 6675 q^{47} + 20975 q^{51} - 30753 q^{53} - 28267 q^{55} - 14389 q^{57} - 87989 q^{59} - 19899 q^{61} + 119470 q^{65} - 33067 q^{67} - 100399 q^{69} - 108720 q^{71} - 141659 q^{73} - 108788 q^{75} + 118919 q^{79} - 143851 q^{81} - 211004 q^{83} - 143379 q^{85} - 302154 q^{87} + 55861 q^{89} + 410381 q^{93} - 26279 q^{95} - 135470 q^{97} - 300154 q^{99}+O(q^{100}) $ 5 * q - 13 * q^3 - 31 * q^5 + 230 * q^9 - 351 * q^11 + 54 * q^13 + 607 * q^15 - 111 * q^17 - 1035 * q^19 + 3639 * q^23 + 1540 * q^25 - 3607 * q^27 - 734 * q^29 - 7677 * q^31 + 7439 * q^33 + 13595 * q^37 - 1406 * q^39 - 5310 * q^41 + 764 * q^43 - 38978 * q^45 - 6675 * q^47 + 20975 * q^51 - 30753 * q^53 - 28267 * q^55 - 14389 * q^57 - 87989 * q^59 - 19899 * q^61 + 119470 * q^65 - 33067 * q^67 - 100399 * q^69 - 108720 * q^71 - 141659 * q^73 - 108788 * q^75 + 118919 * q^79 - 143851 * q^81 - 211004 * q^83 - 143379 * q^85 - 302154 * q^87 + 55861 * q^89 + 410381 * q^93 - 26279 * q^95 - 135470 * q^97 - 300154 * 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
$3$	−14.6817	−0.941831	−0.470915	−	0.882178i	$-0.656076\pi$
$3$	−14.6817	−0.941831	−0.470915	+	0.882178i	$0.656076\pi$
$4$	0	0
$5$	72.1603	1.29084	0.645421	−	0.763827i	$-0.276682\pi$
$5$	72.1603	1.29084	0.645421	+	0.763827i	$0.276682\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−27.4480	−0.112955
$10$	0	0
$11$	−246.305	−0.613750	−0.306875	−	0.951750i	$-0.599283\pi$
$11$	−246.305	−0.613750	−0.306875	+	0.951750i	$0.599283\pi$
$12$	0	0
$13$	1130.90	1.85594	0.927971	−	0.372652i	$-0.121552\pi$
$13$	1130.90	1.85594	0.927971	+	0.372652i	$0.121552\pi$
$14$	0	0
$15$	−1059.43	−1.21576
$16$	0	0
$17$	−1644.26	−1.37991	−0.689953	−	0.723855i	$-0.742369\pi$
$17$	−1644.26	−1.37991	−0.689953	+	0.723855i	$0.742369\pi$
$18$	0	0
$19$	−1646.23	−1.04618	−0.523091	−	0.852277i	$-0.675221\pi$
$19$	−1646.23	−1.04618	−0.523091	+	0.852277i	$0.675221\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−469.033	−0.184878	−0.0924388	−	0.995718i	$-0.529466\pi$
$23$	−469.033	−0.184878	−0.0924388	+	0.995718i	$0.529466\pi$
$24$	0	0
$25$	2082.11	0.666275
$26$	0	0
$27$	3970.63	1.04822
$28$	0	0
$29$	5816.06	1.28420	0.642102	−	0.766620i	$-0.278063\pi$
$29$	5816.06	1.28420	0.642102	+	0.766620i	$0.278063\pi$
$30$	0	0
$31$	−5738.00	−1.07240	−0.536199	−	0.844092i	$-0.680140\pi$
$31$	−5738.00	−1.07240	−0.536199	+	0.844092i	$0.680140\pi$
$32$	0	0
$33$	3616.17	0.578048
$34$	0	0
$35$	0	0
$36$	0	0
$37$	435.509	0.0522990	0.0261495	−	0.999658i	$-0.491675\pi$
$37$	435.509	0.0522990	0.0261495	+	0.999658i	$0.491675\pi$
$38$	0	0
$39$	−16603.5	−1.74798
$40$	0	0
$41$	−11216.3	−1.04205	−0.521027	−	0.853540i	$-0.674451\pi$
$41$	−11216.3	−1.04205	−0.521027	+	0.853540i	$0.674451\pi$
$42$	0	0
$43$	6147.31	0.507007	0.253504	−	0.967334i	$-0.418417\pi$
$43$	6147.31	0.507007	0.253504	+	0.967334i	$0.418417\pi$
$44$	0	0
$45$	−1980.66	−0.145807
$46$	0	0
$47$	14621.2	0.965472	0.482736	−	0.875766i	$-0.339643\pi$
$47$	14621.2	0.965472	0.482736	+	0.875766i	$0.339643\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	24140.6	1.29964
$52$	0	0
$53$	26805.9	1.31081	0.655405	−	0.755277i	$-0.272498\pi$
$53$	26805.9	1.31081	0.655405	+	0.755277i	$0.272498\pi$
$54$	0	0
$55$	−17773.4	−0.792254
$56$	0	0
$57$	24169.5	0.985327
$58$	0	0
$59$	−25054.0	−0.937018	−0.468509	−	0.883459i	$-0.655209\pi$
$59$	−25054.0	−0.937018	−0.468509	+	0.883459i	$0.655209\pi$
$60$	0	0
$61$	−15745.8	−0.541800	−0.270900	−	0.962607i	$-0.587321\pi$
$61$	−15745.8	−0.541800	−0.270900	+	0.962607i	$0.587321\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	81605.8	2.39573
$66$	0	0
$67$	20233.0	0.550646	0.275323	−	0.961352i	$-0.411215\pi$
$67$	20233.0	0.550646	0.275323	+	0.961352i	$0.411215\pi$
$68$	0	0
$69$	6886.20	0.174123
$70$	0	0
$71$	−69564.9	−1.63774	−0.818869	−	0.573980i	$-0.805399\pi$
$71$	−69564.9	−1.63774	−0.818869	+	0.573980i	$0.805399\pi$
$72$	0	0
$73$	−10481.2	−0.230200	−0.115100	−	0.993354i	$-0.536719\pi$
$73$	−10481.2	−0.230200	−0.115100	+	0.993354i	$0.536719\pi$
$74$	0	0
$75$	−30568.9	−0.627518
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−90425.7	−1.63014	−0.815068	−	0.579365i	$-0.803301\pi$
$79$	−90425.7	−1.63014	−0.815068	+	0.579365i	$0.803301\pi$
$80$	0	0
$81$	−51625.7	−0.874286
$82$	0	0
$83$	−30880.2	−0.492023	−0.246011	−	0.969267i	$-0.579120\pi$
$83$	−30880.2	−0.492023	−0.246011	+	0.969267i	$0.579120\pi$
$84$	0	0
$85$	−118651.	−1.78124
$86$	0	0
$87$	−85389.5	−1.20950
$88$	0	0
$89$	16493.6	0.220719	0.110359	−	0.993892i	$-0.464800\pi$
$89$	16493.6	0.220719	0.110359	+	0.993892i	$0.464800\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	84243.5	1.01002
$94$	0	0
$95$	−118793.	−1.35046
$96$	0	0
$97$	−103206.	−1.11372	−0.556860	−	0.830606i	$-0.687994\pi$
$97$	−103206.	−1.11372	−0.556860	+	0.830606i	$0.687994\pi$
$98$	0	0
$99$	6760.59	0.0693260
$100$	0	0
$101$	−45622.9	−0.445020	−0.222510	−	0.974930i	$-0.571425\pi$
$101$	−45622.9	−0.445020	−0.222510	+	0.974930i	$0.571425\pi$
$102$	0	0
$103$	−110143.	−1.02297	−0.511484	−	0.859293i	$-0.670904\pi$
$103$	−110143.	−1.02297	−0.511484	+	0.859293i	$0.670904\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4785.08	−0.0404045	−0.0202022	−	0.999796i	$-0.506431\pi$
$107$	−4785.08	−0.0404045	−0.0202022	+	0.999796i	$0.506431\pi$
$108$	0	0
$109$	−104843.	−0.845229	−0.422615	−	0.906309i	$-0.638888\pi$
$109$	−104843.	−0.845229	−0.422615	+	0.906309i	$0.638888\pi$
$110$	0	0
$111$	−6394.01	−0.0492568
$112$	0	0
$113$	−12359.7	−0.0910570	−0.0455285	−	0.998963i	$-0.514497\pi$
$113$	−12359.7	−0.0910570	−0.0455285	+	0.998963i	$0.514497\pi$
$114$	0	0
$115$	−33845.6	−0.238648
$116$	0	0
$117$	−31040.9	−0.209638
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−100385.	−0.623311
$122$	0	0
$123$	164674.	0.981438
$124$	0	0
$125$	−75255.4	−0.430787
$126$	0	0
$127$	−15641.1	−0.0860517	−0.0430258	−	0.999074i	$-0.513700\pi$
$127$	−15641.1	−0.0860517	−0.0430258	+	0.999074i	$0.513700\pi$
$128$	0	0
$129$	−90252.9	−0.477515
$130$	0	0
$131$	−87498.5	−0.445474	−0.222737	−	0.974879i	$-0.571499\pi$
$131$	−87498.5	−0.445474	−0.222737	+	0.974879i	$0.571499\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	286522.	1.35308
$136$	0	0
$137$	−351405.	−1.59958	−0.799791	−	0.600278i	$-0.795056\pi$
$137$	−351405.	−1.59958	−0.799791	+	0.600278i	$0.795056\pi$
$138$	0	0
$139$	−401624.	−1.76313	−0.881563	−	0.472067i	$-0.843508\pi$
$139$	−401624.	−1.76313	−0.881563	+	0.472067i	$0.843508\pi$
$140$	0	0
$141$	−214664.	−0.909311
$142$	0	0
$143$	−278545.	−1.13908
$144$	0	0
$145$	419688.	1.65770
$146$	0	0
$147$	0	0
$148$	0	0
$149$	442998.	1.63469	0.817347	−	0.576146i	$-0.195444\pi$
$149$	442998.	1.63469	0.817347	+	0.576146i	$0.195444\pi$
$150$	0	0
$151$	289123.	1.03191	0.515953	−	0.856617i	$-0.327438\pi$
$151$	289123.	1.03191	0.515953	+	0.856617i	$0.327438\pi$
$152$	0	0
$153$	45131.8	0.155867
$154$	0	0
$155$	−414055.	−1.38430
$156$	0	0
$157$	−563102.	−1.82322	−0.911608	−	0.411060i	$-0.865159\pi$
$157$	−563102.	−1.82322	−0.911608	+	0.411060i	$0.865159\pi$
$158$	0	0
$159$	−393555.	−1.23456
$160$	0	0
$161$	0	0
$162$	0	0
$163$	638572.	1.88253	0.941263	−	0.337674i	$-0.109640\pi$
$163$	638572.	1.88253	0.941263	+	0.337674i	$0.109640\pi$
$164$	0	0
$165$	260944.	0.746169
$166$	0	0
$167$	72016.8	0.199822	0.0999108	−	0.994996i	$-0.468144\pi$
$167$	72016.8	0.199822	0.0999108	+	0.994996i	$0.468144\pi$
$168$	0	0
$169$	907634.	2.44452
$170$	0	0
$171$	45185.9	0.118171
$172$	0	0
$173$	19019.4	0.0483150	0.0241575	−	0.999708i	$-0.492310\pi$
$173$	19019.4	0.0483150	0.0241575	+	0.999708i	$0.492310\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	367836.	0.882512
$178$	0	0
$179$	−109948.	−0.256482	−0.128241	−	0.991743i	$-0.540933\pi$
$179$	−109948.	−0.256482	−0.128241	+	0.991743i	$0.540933\pi$
$180$	0	0
$181$	781927.	1.77407	0.887033	−	0.461706i	$-0.152762\pi$
$181$	781927.	1.77407	0.887033	+	0.461706i	$0.152762\pi$
$182$	0	0
$183$	231175.	0.510284
$184$	0	0
$185$	31426.5	0.0675097
$186$	0	0
$187$	404990.	0.846916
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−877632.	−1.74072	−0.870360	−	0.492415i	$-0.836114\pi$
$191$	−877632.	−1.74072	−0.870360	+	0.492415i	$0.836114\pi$
$192$	0	0
$193$	2524.01	0.00487751	0.00243876	−	0.999997i	$-0.499224\pi$
$193$	2524.01	0.00487751	0.00243876	+	0.999997i	$0.499224\pi$
$194$	0	0
$195$	−1.19811e6	−2.25637
$196$	0	0
$197$	−375303.	−0.688996	−0.344498	−	0.938787i	$-0.611951\pi$
$197$	−375303.	−0.688996	−0.344498	+	0.938787i	$0.611951\pi$
$198$	0	0
$199$	−210532.	−0.376864	−0.188432	−	0.982086i	$-0.560341\pi$
$199$	−210532.	−0.376864	−0.188432	+	0.982086i	$0.560341\pi$
$200$	0	0
$201$	−297054.	−0.518616
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−809372.	−1.34513
$206$	0	0
$207$	12874.0	0.0208828
$208$	0	0
$209$	405476.	0.642094
$210$	0	0
$211$	323898.	0.500844	0.250422	−	0.968137i	$-0.419431\pi$
$211$	323898.	0.500844	0.250422	+	0.968137i	$0.419431\pi$
$212$	0	0
$213$	1.02133e6	1.54247
$214$	0	0
$215$	443592.	0.654466
$216$	0	0
$217$	0	0
$218$	0	0
$219$	153882.	0.216809
$220$	0	0
$221$	−1.85949e6	−2.56102
$222$	0	0
$223$	−413352.	−0.556618	−0.278309	−	0.960492i	$-0.589774\pi$
$223$	−413352.	−0.556618	−0.278309	+	0.960492i	$0.589774\pi$
$224$	0	0
$225$	−57149.8	−0.0752590
$226$	0	0
$227$	−623945.	−0.803677	−0.401839	−	0.915711i	$-0.631629\pi$
$227$	−623945.	−0.803677	−0.401839	+	0.915711i	$0.631629\pi$
$228$	0	0
$229$	−441658.	−0.556541	−0.278271	−	0.960503i	$-0.589761\pi$
$229$	−441658.	−0.556541	−0.278271	+	0.960503i	$0.589761\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−58664.3	−0.0707920	−0.0353960	−	0.999373i	$-0.511269\pi$
$233$	−58664.3	−0.0707920	−0.0353960	+	0.999373i	$0.511269\pi$
$234$	0	0
$235$	1.05507e6	1.24627
$236$	0	0
$237$	1.32760e6	1.53531
$238$	0	0
$239$	−633286.	−0.717142	−0.358571	−	0.933502i	$-0.616736\pi$
$239$	−633286.	−0.717142	−0.358571	+	0.933502i	$0.616736\pi$
$240$	0	0
$241$	−108888.	−0.120763	−0.0603817	−	0.998175i	$-0.519232\pi$
$241$	−108888.	−0.120763	−0.0603817	+	0.998175i	$0.519232\pi$
$242$	0	0
$243$	−206911.	−0.224785
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.86172e6	−1.94166
$248$	0	0
$249$	453374.	0.463402
$250$	0	0
$251$	48619.2	0.0487106	0.0243553	−	0.999703i	$-0.492247\pi$
$251$	48619.2	0.0487106	0.0243553	+	0.999703i	$0.492247\pi$
$252$	0	0
$253$	115525.	0.113469
$254$	0	0
$255$	1.74199e6	1.67763
$256$	0	0
$257$	591667.	0.558785	0.279393	−	0.960177i	$-0.409867\pi$
$257$	591667.	0.558785	0.279393	+	0.960177i	$0.409867\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−159639.	−0.145057
$262$	0	0
$263$	926427.	0.825889	0.412945	−	0.910756i	$-0.364500\pi$
$263$	926427.	0.825889	0.412945	+	0.910756i	$0.364500\pi$
$264$	0	0
$265$	1.93432e6	1.69205
$266$	0	0
$267$	−242153.	−0.207880
$268$	0	0
$269$	2.08829e6	1.75958	0.879792	−	0.475359i	$-0.157682\pi$
$269$	2.08829e6	1.75958	0.879792	+	0.475359i	$0.157682\pi$
$270$	0	0
$271$	−2.15806e6	−1.78501	−0.892503	−	0.451041i	$-0.851053\pi$
$271$	−2.15806e6	−1.78501	−0.892503	+	0.451041i	$0.851053\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−512833.	−0.408926
$276$	0	0
$277$	912365.	0.714446	0.357223	−	0.934019i	$-0.383724\pi$
$277$	912365.	0.714446	0.357223	+	0.934019i	$0.383724\pi$
$278$	0	0
$279$	157497.	0.121133
$280$	0	0
$281$	357409.	0.270022	0.135011	−	0.990844i	$-0.456893\pi$
$281$	357409.	0.270022	0.135011	+	0.990844i	$0.456893\pi$
$282$	0	0
$283$	2.23669e6	1.66012	0.830062	−	0.557671i	$-0.188305\pi$
$283$	2.23669e6	1.66012	0.830062	+	0.557671i	$0.188305\pi$
$284$	0	0
$285$	1.74408e6	1.27190
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.28375e6	0.904138
$290$	0	0
$291$	1.51524e6	1.04894
$292$	0	0
$293$	−2.18692e6	−1.48821	−0.744103	−	0.668065i	$-0.767123\pi$
$293$	−2.18692e6	−1.48821	−0.744103	+	0.668065i	$0.767123\pi$
$294$	0	0
$295$	−1.80791e6	−1.20954
$296$	0	0
$297$	−977986.	−0.643342
$298$	0	0
$299$	−530428.	−0.343122
$300$	0	0
$301$	0	0
$302$	0	0
$303$	669822.	0.419134
$304$	0	0
$305$	−1.13622e6	−0.699379
$306$	0	0
$307$	−56855.1	−0.0344289	−0.0172144	−	0.999852i	$-0.505480\pi$
$307$	−56855.1	−0.0344289	−0.0172144	+	0.999852i	$0.505480\pi$
$308$	0	0
$309$	1.61708e6	0.963463
$310$	0	0
$311$	−699329.	−0.409997	−0.204999	−	0.978762i	$-0.565719\pi$
$311$	−699329.	−0.409997	−0.204999	+	0.978762i	$0.565719\pi$
$312$	0	0
$313$	2.23809e6	1.29127	0.645636	−	0.763646i	$-0.276592\pi$
$313$	2.23809e6	1.29127	0.645636	+	0.763646i	$0.276592\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.66782e6	−1.49110	−0.745552	−	0.666447i	$-0.767814\pi$
$317$	−2.66782e6	−1.49110	−0.745552	+	0.666447i	$0.767814\pi$
$318$	0	0
$319$	−1.43252e6	−0.788179
$320$	0	0
$321$	70253.0	0.0380542
$322$	0	0
$323$	2.70684e6	1.44363
$324$	0	0
$325$	2.35465e6	1.23657
$326$	0	0
$327$	1.53928e6	0.796063
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.85083e6	−0.928530	−0.464265	−	0.885696i	$-0.653682\pi$
$331$	−1.85083e6	−0.928530	−0.464265	+	0.885696i	$0.653682\pi$
$332$	0	0
$333$	−11953.9	−0.00590742
$334$	0	0
$335$	1.46002e6	0.710798
$336$	0	0
$337$	1.10700e6	0.530972	0.265486	−	0.964115i	$-0.414468\pi$
$337$	1.10700e6	0.530972	0.265486	+	0.964115i	$0.414468\pi$
$338$	0	0
$339$	181462.	0.0857603
$340$	0	0
$341$	1.41330e6	0.658184
$342$	0	0
$343$	0	0
$344$	0	0
$345$	496910.	0.224766
$346$	0	0
$347$	2.22108e6	0.990241	0.495121	−	0.868824i	$-0.335124\pi$
$347$	2.22108e6	0.990241	0.495121	+	0.868824i	$0.335124\pi$
$348$	0	0
$349$	−114207.	−0.0501913	−0.0250957	−	0.999685i	$-0.507989\pi$
$349$	−114207.	−0.0501913	−0.0250957	+	0.999685i	$0.507989\pi$
$350$	0	0
$351$	4.49038e6	1.94543
$352$	0	0
$353$	610115.	0.260600	0.130300	−	0.991475i	$-0.458406\pi$
$353$	610115.	0.260600	0.130300	+	0.991475i	$0.458406\pi$
$354$	0	0
$355$	−5.01983e6	−2.11406
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.37887e6	−0.564658	−0.282329	−	0.959318i	$-0.591107\pi$
$359$	−1.37887e6	−0.564658	−0.282329	+	0.959318i	$0.591107\pi$
$360$	0	0
$361$	233989.	0.0944988
$362$	0	0
$363$	1.47382e6	0.587054
$364$	0	0
$365$	−756328.	−0.297152
$366$	0	0
$367$	531415.	0.205953	0.102977	−	0.994684i	$-0.467163\pi$
$367$	531415.	0.205953	0.102977	+	0.994684i	$0.467163\pi$
$368$	0	0
$369$	307865.	0.117705
$370$	0	0
$371$	0	0
$372$	0	0
$373$	661371.	0.246135	0.123067	−	0.992398i	$-0.460727\pi$
$373$	661371.	0.246135	0.123067	+	0.992398i	$0.460727\pi$
$374$	0	0
$375$	1.10488e6	0.405728
$376$	0	0
$377$	6.57736e6	2.38341
$378$	0	0
$379$	−3.58043e6	−1.28037	−0.640187	−	0.768219i	$-0.721143\pi$
$379$	−3.58043e6	−1.28037	−0.640187	+	0.768219i	$0.721143\pi$
$380$	0	0
$381$	229638.	0.0810461
$382$	0	0
$383$	2.64671e6	0.921956	0.460978	−	0.887412i	$-0.347499\pi$
$383$	2.64671e6	0.921956	0.460978	+	0.887412i	$0.347499\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−168732.	−0.0572689
$388$	0	0
$389$	−2.51493e6	−0.842660	−0.421330	−	0.906907i	$-0.638437\pi$
$389$	−2.51493e6	−0.842660	−0.421330	+	0.906907i	$0.638437\pi$
$390$	0	0
$391$	771215.	0.255113
$392$	0	0
$393$	1.28463e6	0.419561
$394$	0	0
$395$	−6.52514e6	−2.10425
$396$	0	0
$397$	−3.14870e6	−1.00266	−0.501331	−	0.865256i	$-0.667156\pi$
$397$	−3.14870e6	−1.00266	−0.501331	+	0.865256i	$0.667156\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.43733e6	0.756925	0.378462	−	0.925617i	$-0.376453\pi$
$401$	2.43733e6	0.756925	0.378462	+	0.925617i	$0.376453\pi$
$402$	0	0
$403$	−6.48908e6	−1.99031
$404$	0	0
$405$	−3.72533e6	−1.12857
$406$	0	0
$407$	−107268.	−0.0320985
$408$	0	0
$409$	−5.17668e6	−1.53018	−0.765091	−	0.643922i	$-0.777306\pi$
$409$	−5.17668e6	−1.53018	−0.765091	+	0.643922i	$0.777306\pi$
$410$	0	0
$411$	5.15922e6	1.50654
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−2.22833e6	−0.635124
$416$	0	0
$417$	5.89652e6	1.66057
$418$	0	0
$419$	−225726.	−0.0628125	−0.0314062	−	0.999507i	$-0.509999\pi$
$419$	−225726.	−0.0628125	−0.0314062	+	0.999507i	$0.509999\pi$
$420$	0	0
$421$	−2.79842e6	−0.769498	−0.384749	−	0.923021i	$-0.625712\pi$
$421$	−2.79842e6	−0.769498	−0.384749	+	0.923021i	$0.625712\pi$
$422$	0	0
$423$	−401324.	−0.109055
$424$	0	0
$425$	−3.42353e6	−0.919396
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.08951e6	1.07282
$430$	0	0
$431$	3.41262e6	0.884901	0.442450	−	0.896793i	$-0.354109\pi$
$431$	3.41262e6	0.884901	0.442450	+	0.896793i	$0.354109\pi$
$432$	0	0
$433$	−1.63239e6	−0.418412	−0.209206	−	0.977872i	$-0.567088\pi$
$433$	−1.63239e6	−0.418412	−0.209206	+	0.977872i	$0.567088\pi$
$434$	0	0
$435$	−6.16173e6	−1.56128
$436$	0	0
$437$	772139.	0.193416
$438$	0	0
$439$	170371.	0.0421924	0.0210962	−	0.999777i	$-0.493284\pi$
$439$	170371.	0.0421924	0.0210962	+	0.999777i	$0.493284\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	700975.	0.169704	0.0848522	−	0.996394i	$-0.472958\pi$
$443$	700975.	0.169704	0.0848522	+	0.996394i	$0.472958\pi$
$444$	0	0
$445$	1.19018e6	0.284913
$446$	0	0
$447$	−6.50396e6	−1.53960
$448$	0	0
$449$	1.56769e6	0.366983	0.183491	−	0.983021i	$-0.441260\pi$
$449$	1.56769e6	0.366983	0.183491	+	0.983021i	$0.441260\pi$
$450$	0	0
$451$	2.76263e6	0.639560
$452$	0	0
$453$	−4.24482e6	−0.971881
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.37156e6	0.531183	0.265591	−	0.964086i	$-0.414433\pi$
$457$	2.37156e6	0.531183	0.265591	+	0.964086i	$0.414433\pi$
$458$	0	0
$459$	−6.52877e6	−1.44644
$460$	0	0
$461$	22412.1	0.00491167	0.00245584	−	0.999997i	$-0.499218\pi$
$461$	22412.1	0.00491167	0.00245584	+	0.999997i	$0.499218\pi$
$462$	0	0
$463$	704311.	0.152691	0.0763453	−	0.997081i	$-0.475675\pi$
$463$	704311.	0.152691	0.0763453	+	0.997081i	$0.475675\pi$
$464$	0	0
$465$	6.07903e6	1.30377
$466$	0	0
$467$	1.32159e6	0.280418	0.140209	−	0.990122i	$-0.455223\pi$
$467$	1.32159e6	0.280418	0.140209	+	0.990122i	$0.455223\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	8.26729e6	1.71716
$472$	0	0
$473$	−1.51411e6	−0.311175
$474$	0	0
$475$	−3.42764e6	−0.697045
$476$	0	0
$477$	−735768.	−0.148062
$478$	0	0
$479$	9.12503e6	1.81717	0.908585	−	0.417700i	$-0.137164\pi$
$479$	9.12503e6	1.81717	0.908585	+	0.417700i	$0.137164\pi$
$480$	0	0
$481$	492516.	0.0970638
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.44738e6	−1.43764
$486$	0	0
$487$	6.35483e6	1.21418	0.607088	−	0.794635i	$-0.292338\pi$
$487$	6.35483e6	1.21418	0.607088	+	0.794635i	$0.292338\pi$
$488$	0	0
$489$	−9.37532e6	−1.77302
$490$	0	0
$491$	4.23640e6	0.793037	0.396518	−	0.918027i	$-0.370218\pi$
$491$	4.23640e6	0.793037	0.396518	+	0.918027i	$0.370218\pi$
$492$	0	0
$493$	−9.56313e6	−1.77208
$494$	0	0
$495$	487846.	0.0894890
$496$	0	0
$497$	0	0
$498$	0	0
$499$	6.32644e6	1.13739	0.568694	−	0.822550i	$-0.307449\pi$
$499$	6.32644e6	1.13739	0.568694	+	0.822550i	$0.307449\pi$
$500$	0	0
$501$	−1.05733e6	−0.188198
$502$	0	0
$503$	725474.	0.127850	0.0639251	−	0.997955i	$-0.479638\pi$
$503$	725474.	0.127850	0.0639251	+	0.997955i	$0.479638\pi$
$504$	0	0
$505$	−3.29216e6	−0.574451
$506$	0	0
$507$	−1.33256e7	−2.30232
$508$	0	0
$509$	34453.2	0.00589434	0.00294717	−	0.999996i	$-0.499062\pi$
$509$	34453.2	0.00589434	0.00294717	+	0.999996i	$0.499062\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−6.53659e6	−1.09662
$514$	0	0
$515$	−7.94792e6	−1.32049
$516$	0	0
$517$	−3.60128e6	−0.592558
$518$	0	0
$519$	−279237.	−0.0455046
$520$	0	0
$521$	−4.41567e6	−0.712692	−0.356346	−	0.934354i	$-0.615978\pi$
$521$	−4.41567e6	−0.712692	−0.356346	+	0.934354i	$0.615978\pi$
$522$	0	0
$523$	−206472.	−0.0330071	−0.0165035	−	0.999864i	$-0.505253\pi$
$523$	−206472.	−0.0330071	−0.0165035	+	0.999864i	$0.505253\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.43478e6	1.47981
$528$	0	0
$529$	−6.21635e6	−0.965820
$530$	0	0
$531$	687684.	0.105841
$532$	0	0
$533$	−1.26845e7	−1.93399
$534$	0	0
$535$	−345293.	−0.0521558
$536$	0	0
$537$	1.61423e6	0.241562
$538$	0	0
$539$	0	0
$540$	0	0
$541$	144952.	0.0212927	0.0106464	−	0.999943i	$-0.496611\pi$
$541$	144952.	0.0212927	0.0106464	+	0.999943i	$0.496611\pi$
$542$	0	0
$543$	−1.14800e7	−1.67087
$544$	0	0
$545$	−7.56552e6	−1.09106
$546$	0	0
$547$	−3.65203e6	−0.521874	−0.260937	−	0.965356i	$-0.584031\pi$
$547$	−3.65203e6	−0.521874	−0.260937	+	0.965356i	$0.584031\pi$
$548$	0	0
$549$	432191.	0.0611990
$550$	0	0
$551$	−9.57459e6	−1.34351
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−461394.	−0.0635827
$556$	0	0
$557$	8.55923e6	1.16895	0.584476	−	0.811411i	$-0.301300\pi$
$557$	8.55923e6	1.16895	0.584476	+	0.811411i	$0.301300\pi$
$558$	0	0
$559$	6.95197e6	0.940976
$560$	0	0
$561$	−5.94594e6	−0.797652
$562$	0	0
$563$	−3.81296e6	−0.506981	−0.253490	−	0.967338i	$-0.581579\pi$
$563$	−3.81296e6	−0.506981	−0.253490	+	0.967338i	$0.581579\pi$
$564$	0	0
$565$	−891883.	−0.117540
$566$	0	0
$567$	0	0
$568$	0	0
$569$	304441.	0.0394206	0.0197103	−	0.999806i	$-0.493726\pi$
$569$	304441.	0.0394206	0.0197103	+	0.999806i	$0.493726\pi$
$570$	0	0
$571$	9.70934e6	1.24623	0.623117	−	0.782129i	$-0.285866\pi$
$571$	9.70934e6	1.24623	0.623117	+	0.782129i	$0.285866\pi$
$572$	0	0
$573$	1.28851e7	1.63946
$574$	0	0
$575$	−976578.	−0.123179
$576$	0	0
$577$	−1.17345e7	−1.46732	−0.733660	−	0.679517i	$-0.762190\pi$
$577$	−1.17345e7	−1.46732	−0.733660	+	0.679517i	$0.762190\pi$
$578$	0	0
$579$	−37056.8	−0.00459379
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.60241e6	−0.804510
$584$	0	0
$585$	−2.23992e6	−0.270609
$586$	0	0
$587$	1.17417e7	1.40648	0.703241	−	0.710951i	$-0.251735\pi$
$587$	1.17417e7	1.40648	0.703241	+	0.710951i	$0.251735\pi$
$588$	0	0
$589$	9.44609e6	1.12192
$590$	0	0
$591$	5.51009e6	0.648918
$592$	0	0
$593$	−8.50643e6	−0.993369	−0.496684	−	0.867931i	$-0.665449\pi$
$593$	−8.50643e6	−0.993369	−0.496684	+	0.867931i	$0.665449\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.09096e6	0.354943
$598$	0	0
$599$	7.04802e6	0.802601	0.401301	−	0.915946i	$-0.368558\pi$
$599$	7.04802e6	0.802601	0.401301	+	0.915946i	$0.368558\pi$
$600$	0	0
$601$	−1.32634e7	−1.49785	−0.748923	−	0.662657i	$-0.769429\pi$
$601$	−1.32634e7	−1.49785	−0.748923	+	0.662657i	$0.769429\pi$
$602$	0	0
$603$	−555355.	−0.0621982
$604$	0	0
$605$	−7.24380e6	−0.804597
$606$	0	0
$607$	1.10341e7	1.21553	0.607765	−	0.794117i	$-0.292066\pi$
$607$	1.10341e7	1.21553	0.607765	+	0.794117i	$0.292066\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.65351e7	1.79186
$612$	0	0
$613$	−2.13671e6	−0.229665	−0.114832	−	0.993385i	$-0.536633\pi$
$613$	−2.13671e6	−0.229665	−0.114832	+	0.993385i	$0.536633\pi$
$614$	0	0
$615$	1.18829e7	1.26688
$616$	0	0
$617$	1.02769e7	1.08680	0.543399	−	0.839475i	$-0.317137\pi$
$617$	1.02769e7	1.08680	0.543399	+	0.839475i	$0.317137\pi$
$618$	0	0
$619$	−5.58320e6	−0.585675	−0.292838	−	0.956162i	$-0.594600\pi$
$619$	−5.58320e6	−0.585675	−0.292838	+	0.956162i	$0.594600\pi$
$620$	0	0
$621$	−1.86236e6	−0.193791
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.19370e7	−1.22235
$626$	0	0
$627$	−5.95307e6	−0.604744
$628$	0	0
$629$	−716092.	−0.0721676
$630$	0	0
$631$	1.34087e7	1.34064	0.670322	−	0.742071i	$-0.266156\pi$
$631$	1.34087e7	1.34064	0.670322	+	0.742071i	$0.266156\pi$
$632$	0	0
$633$	−4.75537e6	−0.471710
$634$	0	0
$635$	−1.12867e6	−0.111079
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.90942e6	0.184991
$640$	0	0
$641$	1.03725e6	0.0997103	0.0498551	−	0.998756i	$-0.484124\pi$
$641$	1.03725e6	0.0997103	0.0498551	+	0.998756i	$0.484124\pi$
$642$	0	0
$643$	−1.71227e7	−1.63322	−0.816608	−	0.577192i	$-0.804148\pi$
$643$	−1.71227e7	−1.63322	−0.816608	+	0.577192i	$0.804148\pi$
$644$	0	0
$645$	−6.51268e6	−0.616397
$646$	0	0
$647$	6.66382e6	0.625839	0.312919	−	0.949780i	$-0.398693\pi$
$647$	6.66382e6	0.625839	0.312919	+	0.949780i	$0.398693\pi$
$648$	0	0
$649$	6.17093e6	0.575094
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.95841e7	1.79730	0.898649	−	0.438669i	$-0.144550\pi$
$653$	1.95841e7	1.79730	0.898649	+	0.438669i	$0.144550\pi$
$654$	0	0
$655$	−6.31392e6	−0.575037
$656$	0	0
$657$	287689.	0.0260022
$658$	0	0
$659$	1.56939e7	1.40772	0.703862	−	0.710337i	$-0.251457\pi$
$659$	1.56939e7	1.40772	0.703862	+	0.710337i	$0.251457\pi$
$660$	0	0
$661$	1.85789e6	0.165393	0.0826964	−	0.996575i	$-0.473647\pi$
$661$	1.85789e6	0.165393	0.0826964	+	0.996575i	$0.473647\pi$
$662$	0	0
$663$	2.73005e7	2.41205
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.72793e6	−0.237420
$668$	0	0
$669$	6.06870e6	0.524240
$670$	0	0
$671$	3.87826e6	0.332530
$672$	0	0
$673$	−1.12319e7	−0.955907	−0.477953	−	0.878385i	$-0.658621\pi$
$673$	−1.12319e7	−0.955907	−0.477953	+	0.878385i	$0.658621\pi$
$674$	0	0
$675$	8.26729e6	0.698399
$676$	0	0
$677$	3.94349e6	0.330681	0.165340	−	0.986237i	$-0.447128\pi$
$677$	3.94349e6	0.330681	0.165340	+	0.986237i	$0.447128\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	9.16056e6	0.756928
$682$	0	0
$683$	−1.27174e7	−1.04315	−0.521577	−	0.853204i	$-0.674656\pi$
$683$	−1.27174e7	−1.04315	−0.521577	+	0.853204i	$0.674656\pi$
$684$	0	0
$685$	−2.53575e7	−2.06481
$686$	0	0
$687$	6.48429e6	0.524168
$688$	0	0
$689$	3.03146e7	2.43279
$690$	0	0
$691$	−8.31891e6	−0.662783	−0.331391	−	0.943493i	$-0.607518\pi$
$691$	−8.31891e6	−0.662783	−0.331391	+	0.943493i	$0.607518\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.89813e7	−2.27592
$696$	0	0
$697$	1.84426e7	1.43793
$698$	0	0
$699$	861291.	0.0666741
$700$	0	0
$701$	1.06600e7	0.819337	0.409669	−	0.912234i	$-0.365644\pi$
$701$	1.06600e7	0.819337	0.409669	+	0.912234i	$0.365644\pi$
$702$	0	0
$703$	−716950.	−0.0547143
$704$	0	0
$705$	−1.54903e7	−1.17378
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.24696e7	−0.931617	−0.465809	−	0.884886i	$-0.654236\pi$
$709$	−1.24696e7	−0.931617	−0.465809	+	0.884886i	$0.654236\pi$
$710$	0	0
$711$	2.48201e6	0.184132
$712$	0	0
$713$	2.69131e6	0.198262
$714$	0	0
$715$	−2.00999e7	−1.47038
$716$	0	0
$717$	9.29771e6	0.675426
$718$	0	0
$719$	34256.9	0.00247130	0.00123565	−	0.999999i	$-0.499607\pi$
$719$	34256.9	0.00247130	0.00123565	+	0.999999i	$0.499607\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.59865e6	0.113739
$724$	0	0
$725$	1.21097e7	0.855632
$726$	0	0
$727$	9.37823e6	0.658090	0.329045	−	0.944314i	$-0.393273\pi$
$727$	9.37823e6	0.658090	0.329045	+	0.944314i	$0.393273\pi$
$728$	0	0
$729$	1.55829e7	1.08600
$730$	0	0
$731$	−1.01078e7	−0.699622
$732$	0	0
$733$	−4.55584e6	−0.313190	−0.156595	−	0.987663i	$-0.550052\pi$
$733$	−4.55584e6	−0.313190	−0.156595	+	0.987663i	$0.550052\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.98348e6	−0.337959
$738$	0	0
$739$	2.44646e7	1.64788	0.823941	−	0.566675i	$-0.191770\pi$
$739$	2.44646e7	1.64788	0.823941	+	0.566675i	$0.191770\pi$
$740$	0	0
$741$	2.73332e7	1.82871
$742$	0	0
$743$	5.06680e6	0.336715	0.168357	−	0.985726i	$-0.446154\pi$
$743$	5.06680e6	0.336715	0.168357	+	0.985726i	$0.446154\pi$
$744$	0	0
$745$	3.19669e7	2.11013
$746$	0	0
$747$	847601.	0.0555764
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.12217e7	−1.37303	−0.686517	−	0.727114i	$-0.740861\pi$
$751$	−2.12217e7	−1.37303	−0.686517	+	0.727114i	$0.740861\pi$
$752$	0	0
$753$	−713813.	−0.0458772
$754$	0	0
$755$	2.08632e7	1.33203
$756$	0	0
$757$	−184262.	−0.0116868	−0.00584342	−	0.999983i	$-0.501860\pi$
$757$	−184262.	−0.0116868	−0.00584342	+	0.999983i	$0.501860\pi$
$758$	0	0
$759$	−1.69610e6	−0.106868
$760$	0	0
$761$	1.06343e7	0.665650	0.332825	−	0.942989i	$-0.391998\pi$
$761$	1.06343e7	0.665650	0.332825	+	0.942989i	$0.391998\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.25673e6	0.201200
$766$	0	0
$767$	−2.83335e7	−1.73905
$768$	0	0
$769$	5.46176e6	0.333056	0.166528	−	0.986037i	$-0.446744\pi$
$769$	5.46176e6	0.333056	0.166528	+	0.986037i	$0.446744\pi$
$770$	0	0
$771$	−8.68667e6	−0.526281
$772$	0	0
$773$	−1.68523e7	−1.01440	−0.507201	−	0.861828i	$-0.669320\pi$
$773$	−1.68523e7	−1.01440	−0.507201	+	0.861828i	$0.669320\pi$
$774$	0	0
$775$	−1.19471e7	−0.714511
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.84647e7	1.09018
$780$	0	0
$781$	1.71342e7	1.00516
$782$	0	0
$783$	2.30934e7	1.34612
$784$	0	0
$785$	−4.06336e7	−2.35349
$786$	0	0
$787$	1.33846e7	0.770317	0.385158	−	0.922851i	$-0.374147\pi$
$787$	1.33846e7	0.770317	0.385158	+	0.922851i	$0.374147\pi$
$788$	0	0
$789$	−1.36015e7	−0.777848
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1.78068e7	−1.00555
$794$	0	0
$795$	−2.83991e7	−1.59362
$796$	0	0
$797$	−1.88049e6	−0.104864	−0.0524319	−	0.998625i	$-0.516697\pi$
$797$	−1.88049e6	−0.104864	−0.0524319	+	0.998625i	$0.516697\pi$
$798$	0	0
$799$	−2.40412e7	−1.33226
$800$	0	0
$801$	−452716.	−0.0249313
$802$	0	0
$803$	2.58158e6	0.141285
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.06596e7	−1.65723
$808$	0	0
$809$	−1.39864e6	−0.0751337	−0.0375668	−	0.999294i	$-0.511961\pi$
$809$	−1.39864e6	−0.0751337	−0.0375668	+	0.999294i	$0.511961\pi$
$810$	0	0
$811$	−2.91052e7	−1.55388	−0.776941	−	0.629573i	$-0.783230\pi$
$811$	−2.91052e7	−1.55388	−0.776941	+	0.629573i	$0.783230\pi$
$812$	0	0
$813$	3.16839e7	1.68117
$814$	0	0
$815$	4.60796e7	2.43004
$816$	0	0
$817$	−1.01199e7	−0.530422
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.02666e7	−1.56713	−0.783566	−	0.621309i	$-0.786601\pi$
$821$	−3.02666e7	−1.56713	−0.783566	+	0.621309i	$0.786601\pi$
$822$	0	0
$823$	1.60589e7	0.826447	0.413224	−	0.910630i	$-0.364403\pi$
$823$	1.60589e7	0.826447	0.413224	+	0.910630i	$0.364403\pi$
$824$	0	0
$825$	7.52926e6	0.385139
$826$	0	0
$827$	−4.36135e6	−0.221747	−0.110873	−	0.993835i	$-0.535365\pi$
$827$	−4.36135e6	−0.221747	−0.110873	+	0.993835i	$0.535365\pi$
$828$	0	0
$829$	−2.43234e6	−0.122924	−0.0614622	−	0.998109i	$-0.519576\pi$
$829$	−2.43234e6	−0.122924	−0.0614622	+	0.998109i	$0.519576\pi$
$830$	0	0
$831$	−1.33951e7	−0.672887
$832$	0	0
$833$	0	0
$834$	0	0
$835$	5.19675e6	0.257938
$836$	0	0
$837$	−2.27835e7	−1.12410
$838$	0	0
$839$	2.83907e7	1.39243	0.696213	−	0.717836i	$-0.254867\pi$
$839$	2.83907e7	1.39243	0.696213	+	0.717836i	$0.254867\pi$
$840$	0	0
$841$	1.33154e7	0.649178
$842$	0	0
$843$	−5.24736e6	−0.254315
$844$	0	0
$845$	6.54951e7	3.15549
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.28384e7	−1.56356
$850$	0	0
$851$	−204268.	−0.00966890
$852$	0	0
$853$	−1.36423e7	−0.641972	−0.320986	−	0.947084i	$-0.604014\pi$
$853$	−1.36423e7	−0.641972	−0.320986	+	0.947084i	$0.604014\pi$
$854$	0	0
$855$	3.26063e6	0.152541
$856$	0	0
$857$	−1.26545e7	−0.588562	−0.294281	−	0.955719i	$-0.595080\pi$
$857$	−1.26545e7	−0.588562	−0.294281	+	0.955719i	$0.595080\pi$
$858$	0	0
$859$	8.39148e6	0.388021	0.194011	−	0.980999i	$-0.437850\pi$
$859$	8.39148e6	0.388021	0.194011	+	0.980999i	$0.437850\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.08837e7	−0.954511	−0.477256	−	0.878764i	$-0.658368\pi$
$863$	−2.08837e7	−0.954511	−0.477256	+	0.878764i	$0.658368\pi$
$864$	0	0
$865$	1.37245e6	0.0623671
$866$	0	0
$867$	−1.88476e7	−0.851545
$868$	0	0
$869$	2.22723e7	1.00050
$870$	0	0
$871$	2.28814e7	1.02197
$872$	0	0
$873$	2.83280e6	0.125800
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.22777e7	0.978074	0.489037	−	0.872263i	$-0.337348\pi$
$877$	2.22777e7	0.978074	0.489037	+	0.872263i	$0.337348\pi$
$878$	0	0
$879$	3.21076e7	1.40164
$880$	0	0
$881$	2.51345e7	1.09102	0.545508	−	0.838106i	$-0.316337\pi$
$881$	2.51345e7	1.09102	0.545508	+	0.838106i	$0.316337\pi$
$882$	0	0
$883$	3.23710e7	1.39719	0.698594	−	0.715518i	$-0.253809\pi$
$883$	3.23710e7	1.39719	0.698594	+	0.715518i	$0.253809\pi$
$884$	0	0
$885$	2.65431e7	1.13918
$886$	0	0
$887$	−2.96905e7	−1.26709	−0.633547	−	0.773704i	$-0.718402\pi$
$887$	−2.96905e7	−1.26709	−0.633547	+	0.773704i	$0.718402\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.27157e7	0.536593
$892$	0	0
$893$	−2.40700e7	−1.01006
$894$	0	0
$895$	−7.93391e6	−0.331078
$896$	0	0
$897$	7.78758e6	0.323163
$898$	0	0
$899$	−3.33725e7	−1.37718
$900$	0	0
$901$	−4.40759e7	−1.80879
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.64241e7	2.29004
$906$	0	0
$907$	3.81102e7	1.53824	0.769118	−	0.639107i	$-0.220696\pi$
$907$	3.81102e7	1.53824	0.769118	+	0.639107i	$0.220696\pi$
$908$	0	0
$909$	1.25226e6	0.0502672
$910$	0	0
$911$	3.69460e7	1.47493	0.737466	−	0.675385i	$-0.236022\pi$
$911$	3.69460e7	1.47493	0.737466	+	0.675385i	$0.236022\pi$
$912$	0	0
$913$	7.60595e6	0.301979
$914$	0	0
$915$	1.66816e7	0.658697
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.34431e7	0.915643	0.457822	−	0.889044i	$-0.348630\pi$
$919$	2.34431e7	0.915643	0.457822	+	0.889044i	$0.348630\pi$
$920$	0	0
$921$	834728.	0.0324262
$922$	0	0
$923$	−7.86707e7	−3.03955
$924$	0	0
$925$	906777.	0.0348455
$926$	0	0
$927$	3.02320e6	0.115549
$928$	0	0
$929$	3.13042e7	1.19004	0.595022	−	0.803710i	$-0.297143\pi$
$929$	3.13042e7	1.19004	0.595022	+	0.803710i	$0.297143\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.02673e7	0.386148
$934$	0	0
$935$	2.92242e7	1.09324
$936$	0	0
$937$	1.12939e7	0.420238	0.210119	−	0.977676i	$-0.432615\pi$
$937$	1.12939e7	0.420238	0.210119	+	0.977676i	$0.432615\pi$
$938$	0	0
$939$	−3.28590e7	−1.21616
$940$	0	0
$941$	1.71554e6	0.0631578	0.0315789	−	0.999501i	$-0.489946\pi$
$941$	1.71554e6	0.0631578	0.0315789	+	0.999501i	$0.489946\pi$
$942$	0	0
$943$	5.26082e6	0.192652
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.75601e7	1.36098	0.680489	−	0.732758i	$-0.261767\pi$
$947$	3.75601e7	1.36098	0.680489	+	0.732758i	$0.261767\pi$
$948$	0	0
$949$	−1.18532e7	−0.427237
$950$	0	0
$951$	3.91681e7	1.40437
$952$	0	0
$953$	−4.90008e7	−1.74771	−0.873857	−	0.486182i	$-0.838389\pi$
$953$	−4.90008e7	−1.74771	−0.873857	+	0.486182i	$0.838389\pi$
$954$	0	0
$955$	−6.33302e7	−2.24700
$956$	0	0
$957$	2.10319e7	0.742331
$958$	0	0
$959$	0	0
$960$	0	0
$961$	4.29545e6	0.150038
$962$	0	0
$963$	131341.	0.00456388
$964$	0	0
$965$	182134.	0.00629610
$966$	0	0
$967$	4.24075e7	1.45840	0.729199	−	0.684301i	$-0.239893\pi$
$967$	4.24075e7	1.45840	0.729199	+	0.684301i	$0.239893\pi$
$968$	0	0
$969$	−3.97410e7	−1.35966
$970$	0	0
$971$	2.81684e7	0.958770	0.479385	−	0.877605i	$-0.340860\pi$
$971$	2.81684e7	0.958770	0.479385	+	0.877605i	$0.340860\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.45702e7	−1.16464
$976$	0	0
$977$	−5.32664e7	−1.78532	−0.892662	−	0.450727i	$-0.851165\pi$
$977$	−5.32664e7	−1.78532	−0.892662	+	0.450727i	$0.851165\pi$
$978$	0	0
$979$	−4.06245e6	−0.135466
$980$	0	0
$981$	2.87774e6	0.0954728
$982$	0	0
$983$	2.89251e7	0.954754	0.477377	−	0.878699i	$-0.341588\pi$
$983$	2.89251e7	0.954754	0.477377	+	0.878699i	$0.341588\pi$
$984$	0	0
$985$	−2.70820e7	−0.889386
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.88329e6	−0.0937342
$990$	0	0
$991$	3.07125e7	0.993414	0.496707	−	0.867918i	$-0.334542\pi$
$991$	3.07125e7	0.993414	0.496707	+	0.867918i	$0.334542\pi$
$992$	0	0
$993$	2.71733e7	0.874518
$994$	0	0
$995$	−1.51920e7	−0.486473
$996$	0	0
$997$	715992.	0.0228124	0.0114062	−	0.999935i	$-0.496369\pi$
$997$	715992.	0.0228124	0.0114062	+	0.999935i	$0.496369\pi$
$998$	0	0
$999$	1.72925e6	0.0548206

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.6.a.i.1.2		5
4.3	odd	2		784.6.a.bm.1.4		5
7.2	even	3		392.6.i.p.361.4		10
7.3	odd	6		56.6.i.a.9.2	✓	10
7.4	even	3		392.6.i.p.177.4		10
7.5	odd	6		56.6.i.a.25.2	yes	10
7.6	odd	2		392.6.a.l.1.4		5
21.5	even	6		504.6.s.d.361.1		10
21.17	even	6		504.6.s.d.289.1		10
28.3	even	6		112.6.i.g.65.4		10
28.19	even	6		112.6.i.g.81.4		10
28.27	even	2		784.6.a.bj.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.i.a.9.2	✓	10	7.3	odd	6
56.6.i.a.25.2	yes	10	7.5	odd	6
112.6.i.g.65.4		10	28.3	even	6
112.6.i.g.81.4		10	28.19	even	6
392.6.a.i.1.2		5	1.1	even	1	trivial
392.6.a.l.1.4		5	7.6	odd	2
392.6.i.p.177.4		10	7.4	even	3
392.6.i.p.361.4		10	7.2	even	3
504.6.s.d.289.1		10	21.17	even	6
504.6.s.d.361.1		10	21.5	even	6
784.6.a.bj.1.2		5	28.27	even	2
784.6.a.bm.1.4		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)

\(f(q)\)	\(=\)	\(q-14.6817 q^{3} +72.1603 q^{5} -27.4480 q^{9} +O(q^{10})\)	\(q-14.6817 q^{3} +72.1603 q^{5} -27.4480 q^{9} -246.305 q^{11} +1130.90 q^{13} -1059.43 q^{15} -1644.26 q^{17} -1646.23 q^{19} -469.033 q^{23} +2082.11 q^{25} +3970.63 q^{27} +5816.06 q^{29} -5738.00 q^{31} +3616.17 q^{33} +435.509 q^{37} -16603.5 q^{39} -11216.3 q^{41} +6147.31 q^{43} -1980.66 q^{45} +14621.2 q^{47} +24140.6 q^{51} +26805.9 q^{53} -17773.4 q^{55} +24169.5 q^{57} -25054.0 q^{59} -15745.8 q^{61} +81605.8 q^{65} +20233.0 q^{67} +6886.20 q^{69} -69564.9 q^{71} -10481.2 q^{73} -30568.9 q^{75} -90425.7 q^{79} -51625.7 q^{81} -30880.2 q^{83} -118651. q^{85} -85389.5 q^{87} +16493.6 q^{89} +84243.5 q^{93} -118793. q^{95} -103206. q^{97} +6760.59 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 13 q^{3} - 31 q^{5} + 230 q^{9}+O(q^{10}) \) 5 * q - 13 * q^3 - 31 * q^5 + 230 * q^9	\( 5 q - 13 q^{3} - 31 q^{5} + 230 q^{9} - 351 q^{11} + 54 q^{13} + 607 q^{15} - 111 q^{17} - 1035 q^{19} + 3639 q^{23} + 1540 q^{25} - 3607 q^{27} - 734 q^{29} - 7677 q^{31} + 7439 q^{33} + 13595 q^{37} - 1406 q^{39} - 5310 q^{41} + 764 q^{43} - 38978 q^{45} - 6675 q^{47} + 20975 q^{51} - 30753 q^{53} - 28267 q^{55} - 14389 q^{57} - 87989 q^{59} - 19899 q^{61} + 119470 q^{65} - 33067 q^{67} - 100399 q^{69} - 108720 q^{71} - 141659 q^{73} - 108788 q^{75} + 118919 q^{79} - 143851 q^{81} - 211004 q^{83} - 143379 q^{85} - 302154 q^{87} + 55861 q^{89} + 410381 q^{93} - 26279 q^{95} - 135470 q^{97} - 300154 q^{99}+O(q^{100}) \) 5 * q - 13 * q^3 - 31 * q^5 + 230 * q^9 - 351 * q^11 + 54 * q^13 + 607 * q^15 - 111 * q^17 - 1035 * q^19 + 3639 * q^23 + 1540 * q^25 - 3607 * q^27 - 734 * q^29 - 7677 * q^31 + 7439 * q^33 + 13595 * q^37 - 1406 * q^39 - 5310 * q^41 + 764 * q^43 - 38978 * q^45 - 6675 * q^47 + 20975 * q^51 - 30753 * q^53 - 28267 * q^55 - 14389 * q^57 - 87989 * q^59 - 19899 * q^61 + 119470 * q^65 - 33067 * q^67 - 100399 * q^69 - 108720 * q^71 - 141659 * q^73 - 108788 * q^75 + 118919 * q^79 - 143851 * q^81 - 211004 * q^83 - 143379 * q^85 - 302154 * q^87 + 55861 * q^89 + 410381 * q^93 - 26279 * q^95 - 135470 * q^97 - 300154 * q^99

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