LMFDB - Embedded newform 392.6.a.n.1.4

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 425x^{6} + 1312x^{5} + 56757x^{4} - 195610x^{3} - 2560079x^{2} + 6060020x + 37289602 $ x^8 - 2x^7 - 425x^6 + 1312x^5 + 56757x^4 - 195610x^3 - 2560079x^2 + 6060020*x + 37289602
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{10}\cdot 7^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$9.70194$ 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)

$f(q)$	$=$	$q-2.99351 q^{3} +103.023 q^{5} -234.039 q^{9} +O(q^{10})$	$q-2.99351 q^{3} +103.023 q^{5} -234.039 q^{9} +233.218 q^{11} +204.103 q^{13} -308.401 q^{15} +112.064 q^{17} +2487.29 q^{19} -3341.91 q^{23} +7488.72 q^{25} +1428.02 q^{27} -4559.48 q^{29} +8970.68 q^{31} -698.140 q^{33} +3784.12 q^{37} -610.987 q^{39} -3656.58 q^{41} +21173.5 q^{43} -24111.4 q^{45} -7790.19 q^{47} -335.464 q^{51} -9873.21 q^{53} +24026.8 q^{55} -7445.74 q^{57} -20731.3 q^{59} -44418.6 q^{61} +21027.3 q^{65} -10962.5 q^{67} +10004.1 q^{69} +57344.9 q^{71} -35016.3 q^{73} -22417.6 q^{75} +27054.9 q^{79} +52596.6 q^{81} +71394.3 q^{83} +11545.1 q^{85} +13648.9 q^{87} -6379.86 q^{89} -26853.9 q^{93} +256248. q^{95} +170004. q^{97} -54582.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 932 q^{9}+O(q^{10}) $ 8 * q + 932 * q^9	$ 8 q + 932 q^{9} + 1204 q^{11} + 864 q^{15} - 2368 q^{23} + 2188 q^{25} - 9684 q^{29} + 60 q^{37} + 16088 q^{39} + 43564 q^{43} + 76684 q^{51} + 94472 q^{53} + 61004 q^{57} + 41900 q^{65} + 136504 q^{67} + 192328 q^{71} - 29816 q^{79} + 457860 q^{81} + 111132 q^{85} - 166504 q^{93} + 670800 q^{95} + 857404 q^{99}+O(q^{100}) $ 8 * q + 932 * q^9 + 1204 * q^11 + 864 * q^15 - 2368 * q^23 + 2188 * q^25 - 9684 * q^29 + 60 * q^37 + 16088 * q^39 + 43564 * q^43 + 76684 * q^51 + 94472 * q^53 + 61004 * q^57 + 41900 * q^65 + 136504 * q^67 + 192328 * q^71 - 29816 * q^79 + 457860 * q^81 + 111132 * q^85 - 166504 * q^93 + 670800 * q^95 + 857404 * 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
$3$	−2.99351	−0.192034	−0.0960170	−	0.995380i	$-0.530610\pi$
$3$	−2.99351	−0.192034	−0.0960170	+	0.995380i	$0.530610\pi$
$4$	0	0
$5$	103.023	1.84293	0.921465	−	0.388462i	$-0.126993\pi$
$5$	103.023	1.84293	0.921465	+	0.388462i	$0.126993\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−234.039	−0.963123
$10$	0	0
$11$	233.218	0.581139	0.290569	−	0.956854i	$-0.406155\pi$
$11$	233.218	0.581139	0.290569	+	0.956854i	$0.406155\pi$
$12$	0	0
$13$	204.103	0.334959	0.167480	−	0.985876i	$-0.446437\pi$
$13$	204.103	0.334959	0.167480	+	0.985876i	$0.446437\pi$
$14$	0	0
$15$	−308.401	−0.353905
$16$	0	0
$17$	112.064	0.0940464	0.0470232	−	0.998894i	$-0.485027\pi$
$17$	112.064	0.0940464	0.0470232	+	0.998894i	$0.485027\pi$
$18$	0	0
$19$	2487.29	1.58068	0.790338	−	0.612671i	$-0.209905\pi$
$19$	2487.29	1.58068	0.790338	+	0.612671i	$0.209905\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3341.91	−1.31727	−0.658635	−	0.752462i	$-0.728866\pi$
$23$	−3341.91	−1.31727	−0.658635	+	0.752462i	$0.728866\pi$
$24$	0	0
$25$	7488.72	2.39639
$26$	0	0
$27$	1428.02	0.376986
$28$	0	0
$29$	−4559.48	−1.00675	−0.503374	−	0.864069i	$-0.667908\pi$
$29$	−4559.48	−1.00675	−0.503374	+	0.864069i	$0.667908\pi$
$30$	0	0
$31$	8970.68	1.67657	0.838284	−	0.545234i	$-0.183559\pi$
$31$	8970.68	1.67657	0.838284	+	0.545234i	$0.183559\pi$
$32$	0	0
$33$	−698.140	−0.111598
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3784.12	0.454423	0.227212	−	0.973845i	$-0.427039\pi$
$37$	3784.12	0.454423	0.227212	+	0.973845i	$0.427039\pi$
$38$	0	0
$39$	−610.987	−0.0643236
$40$	0	0
$41$	−3656.58	−0.339716	−0.169858	−	0.985469i	$-0.554331\pi$
$41$	−3656.58	−0.339716	−0.169858	+	0.985469i	$0.554331\pi$
$42$	0	0
$43$	21173.5	1.74631	0.873156	−	0.487441i	$-0.162069\pi$
$43$	21173.5	1.74631	0.873156	+	0.487441i	$0.162069\pi$
$44$	0	0
$45$	−24111.4	−1.77497
$46$	0	0
$47$	−7790.19	−0.514403	−0.257201	−	0.966358i	$-0.582800\pi$
$47$	−7790.19	−0.514403	−0.257201	+	0.966358i	$0.582800\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−335.464	−0.0180601
$52$	0	0
$53$	−9873.21	−0.482801	−0.241401	−	0.970426i	$-0.577607\pi$
$53$	−9873.21	−0.482801	−0.241401	+	0.970426i	$0.577607\pi$
$54$	0	0
$55$	24026.8	1.07100
$56$	0	0
$57$	−7445.74	−0.303544
$58$	0	0
$59$	−20731.3	−0.775348	−0.387674	−	0.921797i	$-0.626721\pi$
$59$	−20731.3	−0.775348	−0.387674	+	0.921797i	$0.626721\pi$
$60$	0	0
$61$	−44418.6	−1.52841	−0.764205	−	0.644973i	$-0.776869\pi$
$61$	−44418.6	−1.52841	−0.764205	+	0.644973i	$0.776869\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	21027.3	0.617306
$66$	0	0
$67$	−10962.5	−0.298346	−0.149173	−	0.988811i	$-0.547661\pi$
$67$	−10962.5	−0.298346	−0.149173	+	0.988811i	$0.547661\pi$
$68$	0	0
$69$	10004.1	0.252961
$70$	0	0
$71$	57344.9	1.35005	0.675024	−	0.737796i	$-0.264133\pi$
$71$	57344.9	1.35005	0.675024	+	0.737796i	$0.264133\pi$
$72$	0	0
$73$	−35016.3	−0.769065	−0.384533	−	0.923111i	$-0.625637\pi$
$73$	−35016.3	−0.769065	−0.384533	+	0.923111i	$0.625637\pi$
$74$	0	0
$75$	−22417.6	−0.460189
$76$	0	0
$77$	0	0
$78$	0	0
$79$	27054.9	0.487729	0.243865	−	0.969809i	$-0.421585\pi$
$79$	27054.9	0.487729	0.243865	+	0.969809i	$0.421585\pi$
$80$	0	0
$81$	52596.6	0.890729
$82$	0	0
$83$	71394.3	1.13754	0.568772	−	0.822495i	$-0.307419\pi$
$83$	71394.3	1.13754	0.568772	+	0.822495i	$0.307419\pi$
$84$	0	0
$85$	11545.1	0.173321
$86$	0	0
$87$	13648.9	0.193330
$88$	0	0
$89$	−6379.86	−0.0853761	−0.0426880	−	0.999088i	$-0.513592\pi$
$89$	−6379.86	−0.0853761	−0.0426880	+	0.999088i	$0.513592\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−26853.9	−0.321958
$94$	0	0
$95$	256248.	2.91307
$96$	0	0
$97$	170004.	1.83455	0.917277	−	0.398249i	$-0.130382\pi$
$97$	170004.	1.83455	0.917277	+	0.398249i	$0.130382\pi$
$98$	0	0
$99$	−54582.0	−0.559708
$100$	0	0
$101$	121270.	1.18291	0.591454	−	0.806339i	$-0.298554\pi$
$101$	121270.	1.18291	0.591454	+	0.806339i	$0.298554\pi$
$102$	0	0
$103$	153304.	1.42384	0.711920	−	0.702260i	$-0.247826\pi$
$103$	153304.	1.42384	0.711920	+	0.702260i	$0.247826\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17496.0	0.147733	0.0738667	−	0.997268i	$-0.476466\pi$
$107$	17496.0	0.147733	0.0738667	+	0.997268i	$0.476466\pi$
$108$	0	0
$109$	175066.	1.41136	0.705678	−	0.708532i	$-0.250642\pi$
$109$	175066.	1.41136	0.705678	+	0.708532i	$0.250642\pi$
$110$	0	0
$111$	−11327.8	−0.0872648
$112$	0	0
$113$	201895.	1.48741	0.743704	−	0.668509i	$-0.233067\pi$
$113$	201895.	1.48741	0.743704	+	0.668509i	$0.233067\pi$
$114$	0	0
$115$	−344293.	−2.42764
$116$	0	0
$117$	−47768.1	−0.322607
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−106661.	−0.662278
$122$	0	0
$123$	10946.0	0.0652370
$124$	0	0
$125$	449563.	2.57345
$126$	0	0
$127$	−88018.8	−0.484246	−0.242123	−	0.970246i	$-0.577844\pi$
$127$	−88018.8	−0.484246	−0.242123	+	0.970246i	$0.577844\pi$
$128$	0	0
$129$	−63383.2	−0.335351
$130$	0	0
$131$	−185808.	−0.945988	−0.472994	−	0.881066i	$-0.656827\pi$
$131$	−185808.	−0.945988	−0.472994	+	0.881066i	$0.656827\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	147119.	0.694760
$136$	0	0
$137$	−136671.	−0.622122	−0.311061	−	0.950390i	$-0.600684\pi$
$137$	−136671.	−0.622122	−0.311061	+	0.950390i	$0.600684\pi$
$138$	0	0
$139$	−9855.96	−0.0432675	−0.0216337	−	0.999766i	$-0.506887\pi$
$139$	−9855.96	−0.0432675	−0.0216337	+	0.999766i	$0.506887\pi$
$140$	0	0
$141$	23320.0	0.0987829
$142$	0	0
$143$	47600.5	0.194658
$144$	0	0
$145$	−469731.	−1.85536
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21476.6	−0.0792499	−0.0396250	−	0.999215i	$-0.512616\pi$
$149$	−21476.6	−0.0792499	−0.0396250	+	0.999215i	$0.512616\pi$
$150$	0	0
$151$	395842.	1.41279	0.706397	−	0.707816i	$-0.250319\pi$
$151$	395842.	1.41279	0.706397	+	0.707816i	$0.250319\pi$
$152$	0	0
$153$	−26227.2	−0.0905782
$154$	0	0
$155$	924186.	3.08980
$156$	0	0
$157$	−235385.	−0.762131	−0.381066	−	0.924548i	$-0.624443\pi$
$157$	−235385.	−0.762131	−0.381066	+	0.924548i	$0.624443\pi$
$158$	0	0
$159$	29555.6	0.0927143
$160$	0	0
$161$	0	0
$162$	0	0
$163$	642919.	1.89534	0.947670	−	0.319251i	$-0.103431\pi$
$163$	642919.	1.89534	0.947670	+	0.319251i	$0.103431\pi$
$164$	0	0
$165$	−71924.5	−0.205668
$166$	0	0
$167$	254622.	0.706489	0.353244	−	0.935531i	$-0.385078\pi$
$167$	254622.	0.706489	0.353244	+	0.935531i	$0.385078\pi$
$168$	0	0
$169$	−329635.	−0.887802
$170$	0	0
$171$	−582123.	−1.52238
$172$	0	0
$173$	−385736.	−0.979885	−0.489943	−	0.871755i	$-0.662982\pi$
$173$	−385736.	−0.979885	−0.489943	+	0.871755i	$0.662982\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	62059.5	0.148893
$178$	0	0
$179$	−629116.	−1.46757	−0.733784	−	0.679383i	$-0.762248\pi$
$179$	−629116.	−1.46757	−0.733784	+	0.679383i	$0.762248\pi$
$180$	0	0
$181$	−217330.	−0.493086	−0.246543	−	0.969132i	$-0.579295\pi$
$181$	−217330.	−0.493086	−0.246543	+	0.969132i	$0.579295\pi$
$182$	0	0
$183$	132968.	0.293507
$184$	0	0
$185$	389851.	0.837470
$186$	0	0
$187$	26135.2	0.0546540
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−810535.	−1.60764	−0.803820	−	0.594873i	$-0.797202\pi$
$191$	−810535.	−1.60764	−0.803820	+	0.594873i	$0.797202\pi$
$192$	0	0
$193$	686847.	1.32729	0.663646	−	0.748046i	$-0.269008\pi$
$193$	686847.	1.32729	0.663646	+	0.748046i	$0.269008\pi$
$194$	0	0
$195$	−62945.6	−0.118544
$196$	0	0
$197$	−560519.	−1.02902	−0.514511	−	0.857484i	$-0.672027\pi$
$197$	−560519.	−1.02902	−0.514511	+	0.857484i	$0.672027\pi$
$198$	0	0
$199$	354637.	0.634821	0.317410	−	0.948288i	$-0.397187\pi$
$199$	354637.	0.634821	0.317410	+	0.948288i	$0.397187\pi$
$200$	0	0
$201$	32816.3	0.0572927
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−376712.	−0.626072
$206$	0	0
$207$	782137.	1.26869
$208$	0	0
$209$	580080.	0.918592
$210$	0	0
$211$	494690.	0.764939	0.382469	−	0.923968i	$-0.375074\pi$
$211$	494690.	0.764939	0.382469	+	0.923968i	$0.375074\pi$
$212$	0	0
$213$	−171663.	−0.259255
$214$	0	0
$215$	2.18136e6	3.21833
$216$	0	0
$217$	0	0
$218$	0	0
$219$	104822.	0.147687
$220$	0	0
$221$	22872.6	0.0315017
$222$	0	0
$223$	−400114.	−0.538793	−0.269396	−	0.963029i	$-0.586824\pi$
$223$	−400114.	−0.538793	−0.269396	+	0.963029i	$0.586824\pi$
$224$	0	0
$225$	−1.75265e6	−2.30802
$226$	0	0
$227$	978600.	1.26049	0.630246	−	0.776395i	$-0.282954\pi$
$227$	978600.	1.26049	0.630246	+	0.776395i	$0.282954\pi$
$228$	0	0
$229$	−111952.	−0.141072	−0.0705362	−	0.997509i	$-0.522471\pi$
$229$	−111952.	−0.141072	−0.0705362	+	0.997509i	$0.522471\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−961292.	−1.16002	−0.580010	−	0.814609i	$-0.696951\pi$
$233$	−961292.	−1.16002	−0.580010	+	0.814609i	$0.696951\pi$
$234$	0	0
$235$	−802568.	−0.948008
$236$	0	0
$237$	−80989.4	−0.0936606
$238$	0	0
$239$	685567.	0.776346	0.388173	−	0.921586i	$-0.373106\pi$
$239$	685567.	0.776346	0.388173	+	0.921586i	$0.373106\pi$
$240$	0	0
$241$	161513.	0.179129	0.0895645	−	0.995981i	$-0.471452\pi$
$241$	161513.	0.179129	0.0895645	+	0.995981i	$0.471452\pi$
$242$	0	0
$243$	−504458.	−0.548037
$244$	0	0
$245$	0	0
$246$	0	0
$247$	507665.	0.529462
$248$	0	0
$249$	−213720.	−0.218447
$250$	0	0
$251$	−1.27926e6	−1.28166	−0.640832	−	0.767682i	$-0.721410\pi$
$251$	−1.27926e6	−1.28166	−0.640832	+	0.767682i	$0.721410\pi$
$252$	0	0
$253$	−779392.	−0.765517
$254$	0	0
$255$	−34560.5	−0.0332835
$256$	0	0
$257$	−791681.	−0.747683	−0.373841	−	0.927493i	$-0.621960\pi$
$257$	−791681.	−0.747683	−0.373841	+	0.927493i	$0.621960\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.06710e6	0.969621
$262$	0	0
$263$	−979574.	−0.873269	−0.436634	−	0.899639i	$-0.643830\pi$
$263$	−979574.	−0.873269	−0.436634	+	0.899639i	$0.643830\pi$
$264$	0	0
$265$	−1.01717e6	−0.889769
$266$	0	0
$267$	19098.2	0.0163951
$268$	0	0
$269$	1.88880e6	1.59150	0.795749	−	0.605627i	$-0.207077\pi$
$269$	1.88880e6	1.59150	0.795749	+	0.605627i	$0.207077\pi$
$270$	0	0
$271$	1.97599e6	1.63441	0.817206	−	0.576346i	$-0.195522\pi$
$271$	1.97599e6	1.63441	0.817206	+	0.576346i	$0.195522\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.74650e6	1.39263
$276$	0	0
$277$	−1.87467e6	−1.46800	−0.734000	−	0.679150i	$-0.762349\pi$
$277$	−1.87467e6	−1.46800	−0.734000	+	0.679150i	$0.762349\pi$
$278$	0	0
$279$	−2.09949e6	−1.61474
$280$	0	0
$281$	1.07738e6	0.813957	0.406978	−	0.913438i	$-0.366582\pi$
$281$	1.07738e6	0.813957	0.406978	+	0.913438i	$0.366582\pi$
$282$	0	0
$283$	−1.55890e6	−1.15705	−0.578525	−	0.815664i	$-0.696372\pi$
$283$	−1.55890e6	−1.15705	−0.578525	+	0.815664i	$0.696372\pi$
$284$	0	0
$285$	−767082.	−0.559409
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.40730e6	−0.991155
$290$	0	0
$291$	−508910.	−0.352297
$292$	0	0
$293$	838297.	0.570465	0.285232	−	0.958458i	$-0.407929\pi$
$293$	838297.	0.570465	0.285232	+	0.958458i	$0.407929\pi$
$294$	0	0
$295$	−2.13580e6	−1.42891
$296$	0	0
$297$	333040.	0.219081
$298$	0	0
$299$	−682095.	−0.441232
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−363024.	−0.227158
$304$	0	0
$305$	−4.57613e6	−2.81675
$306$	0	0
$307$	−1.35273e6	−0.819152	−0.409576	−	0.912276i	$-0.634323\pi$
$307$	−1.35273e6	−0.819152	−0.409576	+	0.912276i	$0.634323\pi$
$308$	0	0
$309$	−458919.	−0.273426
$310$	0	0
$311$	−1.05406e6	−0.617968	−0.308984	−	0.951067i	$-0.599989\pi$
$311$	−1.05406e6	−0.617968	−0.308984	+	0.951067i	$0.599989\pi$
$312$	0	0
$313$	−1.08030e6	−0.623283	−0.311642	−	0.950200i	$-0.600879\pi$
$313$	−1.08030e6	−0.623283	−0.311642	+	0.950200i	$0.600879\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.23089e6	−1.24689	−0.623446	−	0.781866i	$-0.714268\pi$
$317$	−2.23089e6	−1.24689	−0.623446	+	0.781866i	$0.714268\pi$
$318$	0	0
$319$	−1.06335e6	−0.585060
$320$	0	0
$321$	−52374.5	−0.0283698
$322$	0	0
$323$	278735.	0.148657
$324$	0	0
$325$	1.52847e6	0.802693
$326$	0	0
$327$	−524064.	−0.271029
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.63630e6	0.820906	0.410453	−	0.911882i	$-0.365371\pi$
$331$	1.63630e6	0.820906	0.410453	+	0.911882i	$0.365371\pi$
$332$	0	0
$333$	−885631.	−0.437666
$334$	0	0
$335$	−1.12938e6	−0.549831
$336$	0	0
$337$	−259966.	−0.124693	−0.0623466	−	0.998055i	$-0.519858\pi$
$337$	−259966.	−0.124693	−0.0623466	+	0.998055i	$0.519858\pi$
$338$	0	0
$339$	−604377.	−0.285633
$340$	0	0
$341$	2.09212e6	0.974318
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.03065e6	0.466189
$346$	0	0
$347$	3.67730e6	1.63948	0.819738	−	0.572738i	$-0.194119\pi$
$347$	3.67730e6	1.63948	0.819738	+	0.572738i	$0.194119\pi$
$348$	0	0
$349$	−2.71456e6	−1.19299	−0.596493	−	0.802618i	$-0.703440\pi$
$349$	−2.71456e6	−1.19299	−0.596493	+	0.802618i	$0.703440\pi$
$350$	0	0
$351$	291464.	0.126275
$352$	0	0
$353$	2.83639e6	1.21152	0.605758	−	0.795649i	$-0.292870\pi$
$353$	2.83639e6	1.21152	0.605758	+	0.795649i	$0.292870\pi$
$354$	0	0
$355$	5.90784e6	2.48804
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.06625e6	−0.846149	−0.423075	−	0.906095i	$-0.639049\pi$
$359$	−2.06625e6	−0.846149	−0.423075	+	0.906095i	$0.639049\pi$
$360$	0	0
$361$	3.71052e6	1.49854
$362$	0	0
$363$	319290.	0.127180
$364$	0	0
$365$	−3.60748e6	−1.41733
$366$	0	0
$367$	2.08490e6	0.808014	0.404007	−	0.914756i	$-0.367617\pi$
$367$	2.08490e6	0.808014	0.404007	+	0.914756i	$0.367617\pi$
$368$	0	0
$369$	855782.	0.327188
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.27052e6	−0.472836	−0.236418	−	0.971651i	$-0.575974\pi$
$373$	−1.27052e6	−0.472836	−0.236418	+	0.971651i	$0.575974\pi$
$374$	0	0
$375$	−1.34577e6	−0.494190
$376$	0	0
$377$	−930606.	−0.337219
$378$	0	0
$379$	−489277.	−0.174967	−0.0874836	−	0.996166i	$-0.527883\pi$
$379$	−489277.	−0.174967	−0.0874836	+	0.996166i	$0.527883\pi$
$380$	0	0
$381$	263486.	0.0929917
$382$	0	0
$383$	−1.46804e6	−0.511377	−0.255688	−	0.966759i	$-0.582302\pi$
$383$	−1.46804e6	−0.511377	−0.255688	+	0.966759i	$0.582302\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.95543e6	−1.68191
$388$	0	0
$389$	2.69883e6	0.904278	0.452139	−	0.891947i	$-0.350661\pi$
$389$	2.69883e6	0.904278	0.452139	+	0.891947i	$0.350661\pi$
$390$	0	0
$391$	−374506.	−0.123885
$392$	0	0
$393$	556218.	0.181662
$394$	0	0
$395$	2.78728e6	0.898851
$396$	0	0
$397$	−2.66315e6	−0.848045	−0.424022	−	0.905652i	$-0.639382\pi$
$397$	−2.66315e6	−0.848045	−0.424022	+	0.905652i	$0.639382\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.26161e6	0.702356	0.351178	−	0.936309i	$-0.385781\pi$
$401$	2.26161e6	0.702356	0.351178	+	0.936309i	$0.385781\pi$
$402$	0	0
$403$	1.83095e6	0.561582
$404$	0	0
$405$	5.41866e6	1.64155
$406$	0	0
$407$	882524.	0.264083
$408$	0	0
$409$	−3.86082e6	−1.14122	−0.570612	−	0.821220i	$-0.693294\pi$
$409$	−3.86082e6	−1.14122	−0.570612	+	0.821220i	$0.693294\pi$
$410$	0	0
$411$	409127.	0.119469
$412$	0	0
$413$	0	0
$414$	0	0
$415$	7.35525e6	2.09641
$416$	0	0
$417$	29503.9	0.00830883
$418$	0	0
$419$	1.46492e6	0.407643	0.203821	−	0.979008i	$-0.434664\pi$
$419$	1.46492e6	0.407643	0.203821	+	0.979008i	$0.434664\pi$
$420$	0	0
$421$	4.04748e6	1.11296	0.556480	−	0.830861i	$-0.312152\pi$
$421$	4.04748e6	1.11296	0.556480	+	0.830861i	$0.312152\pi$
$422$	0	0
$423$	1.82321e6	0.495433
$424$	0	0
$425$	839212.	0.225372
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−142493.	−0.0373809
$430$	0	0
$431$	−4.46535e6	−1.15788	−0.578939	−	0.815371i	$-0.696533\pi$
$431$	−4.46535e6	−1.15788	−0.578939	+	0.815371i	$0.696533\pi$
$432$	0	0
$433$	−940532.	−0.241076	−0.120538	−	0.992709i	$-0.538462\pi$
$433$	−940532.	−0.241076	−0.120538	+	0.992709i	$0.538462\pi$
$434$	0	0
$435$	1.40615e6	0.356293
$436$	0	0
$437$	−8.31230e6	−2.08218
$438$	0	0
$439$	−3.76966e6	−0.933558	−0.466779	−	0.884374i	$-0.654586\pi$
$439$	−3.76966e6	−0.933558	−0.466779	+	0.884374i	$0.654586\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.45966e6	−1.07967	−0.539837	−	0.841770i	$-0.681514\pi$
$443$	−4.45966e6	−1.07967	−0.539837	+	0.841770i	$0.681514\pi$
$444$	0	0
$445$	−657272.	−0.157342
$446$	0	0
$447$	64290.4	0.0152187
$448$	0	0
$449$	−4.77389e6	−1.11752	−0.558762	−	0.829328i	$-0.688723\pi$
$449$	−4.77389e6	−1.11752	−0.558762	+	0.829328i	$0.688723\pi$
$450$	0	0
$451$	−852779.	−0.197422
$452$	0	0
$453$	−1.18496e6	−0.271305
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.53352e6	−0.791439	−0.395719	−	0.918371i	$-0.629505\pi$
$457$	−3.53352e6	−0.791439	−0.395719	+	0.918371i	$0.629505\pi$
$458$	0	0
$459$	160029.	0.0354542
$460$	0	0
$461$	−1.21049e6	−0.265283	−0.132642	−	0.991164i	$-0.542346\pi$
$461$	−1.21049e6	−0.265283	−0.132642	+	0.991164i	$0.542346\pi$
$462$	0	0
$463$	6.23929e6	1.35264	0.676320	−	0.736608i	$-0.263574\pi$
$463$	6.23929e6	1.35264	0.676320	+	0.736608i	$0.263574\pi$
$464$	0	0
$465$	−2.76656e6	−0.593346
$466$	0	0
$467$	−6.35555e6	−1.34853	−0.674266	−	0.738489i	$-0.735540\pi$
$467$	−6.35555e6	−1.34853	−0.674266	+	0.738489i	$0.735540\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	704629.	0.146355
$472$	0	0
$473$	4.93804e6	1.01485
$474$	0	0
$475$	1.86266e7	3.78792
$476$	0	0
$477$	2.31071e6	0.464997
$478$	0	0
$479$	−5.87081e6	−1.16912	−0.584560	−	0.811350i	$-0.698733\pi$
$479$	−5.87081e6	−1.16912	−0.584560	+	0.811350i	$0.698733\pi$
$480$	0	0
$481$	772352.	0.152213
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.75143e7	3.38096
$486$	0	0
$487$	−6.44679e6	−1.23175	−0.615873	−	0.787845i	$-0.711197\pi$
$487$	−6.44679e6	−1.23175	−0.615873	+	0.787845i	$0.711197\pi$
$488$	0	0
$489$	−1.92459e6	−0.363970
$490$	0	0
$491$	−1.57412e6	−0.294668	−0.147334	−	0.989087i	$-0.547069\pi$
$491$	−1.57412e6	−0.294668	−0.147334	+	0.989087i	$0.547069\pi$
$492$	0	0
$493$	−510952.	−0.0946809
$494$	0	0
$495$	−5.62320e6	−1.03150
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1.05226e6	−0.189178	−0.0945891	−	0.995516i	$-0.530154\pi$
$499$	−1.05226e6	−0.189178	−0.0945891	+	0.995516i	$0.530154\pi$
$500$	0	0
$501$	−762216.	−0.135670
$502$	0	0
$503$	−5.52636e6	−0.973911	−0.486956	−	0.873427i	$-0.661893\pi$
$503$	−5.52636e6	−0.973911	−0.486956	+	0.873427i	$0.661893\pi$
$504$	0	0
$505$	1.24936e7	2.18002
$506$	0	0
$507$	986767.	0.170488
$508$	0	0
$509$	−1.83026e6	−0.313125	−0.156563	−	0.987668i	$-0.550041\pi$
$509$	−1.83026e6	−0.313125	−0.156563	+	0.987668i	$0.550041\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.55191e6	0.595893
$514$	0	0
$515$	1.57939e7	2.62404
$516$	0	0
$517$	−1.81681e6	−0.298939
$518$	0	0
$519$	1.15471e6	0.188171
$520$	0	0
$521$	−2.08624e6	−0.336721	−0.168360	−	0.985726i	$-0.553847\pi$
$521$	−2.08624e6	−0.336721	−0.168360	+	0.985726i	$0.553847\pi$
$522$	0	0
$523$	3.14540e6	0.502831	0.251416	−	0.967879i	$-0.419104\pi$
$523$	3.14540e6	0.502831	0.251416	+	0.967879i	$0.419104\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.00529e6	0.157675
$528$	0	0
$529$	4.73201e6	0.735202
$530$	0	0
$531$	4.85193e6	0.746755
$532$	0	0
$533$	−746321.	−0.113791
$534$	0	0
$535$	1.80249e6	0.272262
$536$	0	0
$537$	1.88327e6	0.281823
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.22660e7	−1.80182	−0.900908	−	0.434010i	$-0.857098\pi$
$541$	−1.22660e7	−1.80182	−0.900908	+	0.434010i	$0.857098\pi$
$542$	0	0
$543$	650579.	0.0946893
$544$	0	0
$545$	1.80359e7	2.60103
$546$	0	0
$547$	−3.94365e6	−0.563547	−0.281774	−	0.959481i	$-0.590923\pi$
$547$	−3.94365e6	−0.563547	−0.281774	+	0.959481i	$0.590923\pi$
$548$	0	0
$549$	1.03957e7	1.47205
$550$	0	0
$551$	−1.13408e7	−1.59134
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.16702e6	−0.160823
$556$	0	0
$557$	1.08224e7	1.47804	0.739020	−	0.673683i	$-0.235289\pi$
$557$	1.08224e7	1.47804	0.739020	+	0.673683i	$0.235289\pi$
$558$	0	0
$559$	4.32159e6	0.584943
$560$	0	0
$561$	−78236.1	−0.0104954
$562$	0	0
$563$	−7.41522e6	−0.985945	−0.492973	−	0.870045i	$-0.664090\pi$
$563$	−7.41522e6	−0.985945	−0.492973	+	0.870045i	$0.664090\pi$
$564$	0	0
$565$	2.07998e7	2.74119
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.42450e6	−0.961361	−0.480681	−	0.876896i	$-0.659610\pi$
$569$	−7.42450e6	−0.961361	−0.480681	+	0.876896i	$0.659610\pi$
$570$	0	0
$571$	1.43119e7	1.83699	0.918495	−	0.395431i	$-0.129405\pi$
$571$	1.43119e7	1.83699	0.918495	+	0.395431i	$0.129405\pi$
$572$	0	0
$573$	2.42635e6	0.308721
$574$	0	0
$575$	−2.50266e7	−3.15669
$576$	0	0
$577$	2.58041e6	0.322662	0.161331	−	0.986900i	$-0.448421\pi$
$577$	2.58041e6	0.322662	0.161331	+	0.986900i	$0.448421\pi$
$578$	0	0
$579$	−2.05609e6	−0.254885
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.30261e6	−0.280575
$584$	0	0
$585$	−4.92121e6	−0.594542
$586$	0	0
$587$	150470.	0.0180241	0.00901205	−	0.999959i	$-0.497131\pi$
$587$	150470.	0.0180241	0.00901205	+	0.999959i	$0.497131\pi$
$588$	0	0
$589$	2.23127e7	2.65011
$590$	0	0
$591$	1.67792e6	0.197607
$592$	0	0
$593$	4.08558e6	0.477109	0.238554	−	0.971129i	$-0.423326\pi$
$593$	4.08558e6	0.477109	0.238554	+	0.971129i	$0.423326\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.06161e6	−0.121907
$598$	0	0
$599$	−1.53310e6	−0.174583	−0.0872916	−	0.996183i	$-0.527821\pi$
$599$	−1.53310e6	−0.174583	−0.0872916	+	0.996183i	$0.527821\pi$
$600$	0	0
$601$	−1.39321e7	−1.57337	−0.786683	−	0.617357i	$-0.788203\pi$
$601$	−1.39321e7	−1.57337	−0.786683	+	0.617357i	$0.788203\pi$
$602$	0	0
$603$	2.56564e6	0.287344
$604$	0	0
$605$	−1.09885e7	−1.22053
$606$	0	0
$607$	2.58234e6	0.284473	0.142237	−	0.989833i	$-0.454571\pi$
$607$	2.58234e6	0.284473	0.142237	+	0.989833i	$0.454571\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.59000e6	−0.172304
$612$	0	0
$613$	2.13130e6	0.229083	0.114541	−	0.993418i	$-0.463460\pi$
$613$	2.13130e6	0.229083	0.114541	+	0.993418i	$0.463460\pi$
$614$	0	0
$615$	1.12769e6	0.120227
$616$	0	0
$617$	8.40268e6	0.888597	0.444299	−	0.895879i	$-0.353453\pi$
$617$	8.40268e6	0.888597	0.444299	+	0.895879i	$0.353453\pi$
$618$	0	0
$619$	−4.63076e6	−0.485765	−0.242882	−	0.970056i	$-0.578093\pi$
$619$	−4.63076e6	−0.485765	−0.242882	+	0.970056i	$0.578093\pi$
$620$	0	0
$621$	−4.77232e6	−0.496593
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.29130e7	2.34630
$626$	0	0
$627$	−1.73648e6	−0.176401
$628$	0	0
$629$	424062.	0.0427369
$630$	0	0
$631$	−2.75976e6	−0.275929	−0.137964	−	0.990437i	$-0.544056\pi$
$631$	−2.75976e6	−0.275929	−0.137964	+	0.990437i	$0.544056\pi$
$632$	0	0
$633$	−1.48086e6	−0.146894
$634$	0	0
$635$	−9.06795e6	−0.892432
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.34209e7	−1.30026
$640$	0	0
$641$	−6.10130e6	−0.586513	−0.293256	−	0.956034i	$-0.594739\pi$
$641$	−6.10130e6	−0.586513	−0.293256	+	0.956034i	$0.594739\pi$
$642$	0	0
$643$	−813800.	−0.0776230	−0.0388115	−	0.999247i	$-0.512357\pi$
$643$	−813800.	−0.0776230	−0.0388115	+	0.999247i	$0.512357\pi$
$644$	0	0
$645$	−6.52992e6	−0.618029
$646$	0	0
$647$	7.39047e6	0.694082	0.347041	−	0.937850i	$-0.387186\pi$
$647$	7.39047e6	0.694082	0.347041	+	0.937850i	$0.387186\pi$
$648$	0	0
$649$	−4.83491e6	−0.450585
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.05462e6	−0.739200	−0.369600	−	0.929191i	$-0.620505\pi$
$653$	−8.05462e6	−0.739200	−0.369600	+	0.929191i	$0.620505\pi$
$654$	0	0
$655$	−1.91425e7	−1.74339
$656$	0	0
$657$	8.19518e6	0.740704
$658$	0	0
$659$	979390.	0.0878501	0.0439251	−	0.999035i	$-0.486014\pi$
$659$	979390.	0.0878501	0.0439251	+	0.999035i	$0.486014\pi$
$660$	0	0
$661$	−5.37078e6	−0.478117	−0.239058	−	0.971005i	$-0.576839\pi$
$661$	−5.37078e6	−0.478117	−0.239058	+	0.971005i	$0.576839\pi$
$662$	0	0
$663$	−68469.3	−0.00604940
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.52374e7	1.32616
$668$	0	0
$669$	1.19775e6	0.103467
$670$	0	0
$671$	−1.03592e7	−0.888218
$672$	0	0
$673$	3.95194e6	0.336336	0.168168	−	0.985758i	$-0.446215\pi$
$673$	3.95194e6	0.336336	0.168168	+	0.985758i	$0.446215\pi$
$674$	0	0
$675$	1.06941e7	0.903407
$676$	0	0
$677$	2.92078e6	0.244921	0.122461	−	0.992473i	$-0.460921\pi$
$677$	2.92078e6	0.244921	0.122461	+	0.992473i	$0.460921\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−2.92945e6	−0.242058
$682$	0	0
$683$	−9.77362e6	−0.801685	−0.400843	−	0.916147i	$-0.631283\pi$
$683$	−9.77362e6	−0.801685	−0.400843	+	0.916147i	$0.631283\pi$
$684$	0	0
$685$	−1.40803e7	−1.14653
$686$	0	0
$687$	335129.	0.0270907
$688$	0	0
$689$	−2.01516e6	−0.161719
$690$	0	0
$691$	−2.17317e7	−1.73140	−0.865701	−	0.500562i	$-0.833127\pi$
$691$	−2.17317e7	−1.73140	−0.865701	+	0.500562i	$0.833127\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.01539e6	−0.0797390
$696$	0	0
$697$	−409769.	−0.0319490
$698$	0	0
$699$	2.87764e6	0.222763
$700$	0	0
$701$	−1.79561e6	−0.138012	−0.0690060	−	0.997616i	$-0.521983\pi$
$701$	−1.79561e6	−0.138012	−0.0690060	+	0.997616i	$0.521983\pi$
$702$	0	0
$703$	9.41221e6	0.718296
$704$	0	0
$705$	2.40250e6	0.182050
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.38207e7	1.77967	0.889835	−	0.456283i	$-0.150819\pi$
$709$	2.38207e7	1.77967	0.889835	+	0.456283i	$0.150819\pi$
$710$	0	0
$711$	−6.33191e6	−0.469743
$712$	0	0
$713$	−2.99792e7	−2.20849
$714$	0	0
$715$	4.90394e6	0.358741
$716$	0	0
$717$	−2.05226e6	−0.149085
$718$	0	0
$719$	1.50880e7	1.08845	0.544225	−	0.838939i	$-0.316824\pi$
$719$	1.50880e7	1.08845	0.544225	+	0.838939i	$0.316824\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−483492.	−0.0343989
$724$	0	0
$725$	−3.41447e7	−2.41256
$726$	0	0
$727$	−1.89611e7	−1.33054	−0.665270	−	0.746603i	$-0.731684\pi$
$727$	−1.89611e7	−1.33054	−0.665270	+	0.746603i	$0.731684\pi$
$728$	0	0
$729$	−1.12709e7	−0.785487
$730$	0	0
$731$	2.37278e6	0.164234
$732$	0	0
$733$	590866.	0.0406190	0.0203095	−	0.999794i	$-0.493535\pi$
$733$	590866.	0.0406190	0.0203095	+	0.999794i	$0.493535\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.55664e6	−0.173381
$738$	0	0
$739$	−3.24251e6	−0.218409	−0.109204	−	0.994019i	$-0.534830\pi$
$739$	−3.24251e6	−0.218409	−0.109204	+	0.994019i	$0.534830\pi$
$740$	0	0
$741$	−1.51970e6	−0.101675
$742$	0	0
$743$	−1.10828e7	−0.736507	−0.368254	−	0.929725i	$-0.620044\pi$
$743$	−1.10828e7	−0.736507	−0.368254	+	0.929725i	$0.620044\pi$
$744$	0	0
$745$	−2.21258e6	−0.146052
$746$	0	0
$747$	−1.67090e7	−1.09559
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.77118e7	−1.79294	−0.896470	−	0.443105i	$-0.853877\pi$
$751$	−2.77118e7	−1.79294	−0.896470	+	0.443105i	$0.853877\pi$
$752$	0	0
$753$	3.82948e6	0.246123
$754$	0	0
$755$	4.07807e7	2.60368
$756$	0	0
$757$	−1.97155e7	−1.25046	−0.625228	−	0.780442i	$-0.714994\pi$
$757$	−1.97155e7	−1.25046	−0.625228	+	0.780442i	$0.714994\pi$
$758$	0	0
$759$	2.33312e6	0.147005
$760$	0	0
$761$	9.37292e6	0.586696	0.293348	−	0.956006i	$-0.405231\pi$
$761$	9.37292e6	0.586696	0.293348	+	0.956006i	$0.405231\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.70201e6	−0.166929
$766$	0	0
$767$	−4.23133e6	−0.259710
$768$	0	0
$769$	−3.37535e6	−0.205827	−0.102914	−	0.994690i	$-0.532817\pi$
$769$	−3.37535e6	−0.205827	−0.102914	+	0.994690i	$0.532817\pi$
$770$	0	0
$771$	2.36991e6	0.143581
$772$	0	0
$773$	2.10256e7	1.26561	0.632805	−	0.774311i	$-0.281903\pi$
$773$	2.10256e7	1.26561	0.632805	+	0.774311i	$0.281903\pi$
$774$	0	0
$775$	6.71789e7	4.01771
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.09498e6	−0.536980
$780$	0	0
$781$	1.33738e7	0.784564
$782$	0	0
$783$	−6.51104e6	−0.379530
$784$	0	0
$785$	−2.42501e7	−1.40455
$786$	0	0
$787$	−1.51047e7	−0.869312	−0.434656	−	0.900597i	$-0.643130\pi$
$787$	−1.51047e7	−0.869312	−0.434656	+	0.900597i	$0.643130\pi$
$788$	0	0
$789$	2.93237e6	0.167697
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9.06598e6	−0.511955
$794$	0	0
$795$	3.04490e6	0.170866
$796$	0	0
$797$	−2.75875e7	−1.53839	−0.769194	−	0.639015i	$-0.779342\pi$
$797$	−2.75875e7	−1.53839	−0.769194	+	0.639015i	$0.779342\pi$
$798$	0	0
$799$	−872996.	−0.0483777
$800$	0	0
$801$	1.49314e6	0.0822277
$802$	0	0
$803$	−8.16642e6	−0.446933
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.65416e6	−0.305622
$808$	0	0
$809$	6.78884e6	0.364690	0.182345	−	0.983235i	$-0.441631\pi$
$809$	6.78884e6	0.364690	0.182345	+	0.983235i	$0.441631\pi$
$810$	0	0
$811$	−2.77774e7	−1.48300	−0.741498	−	0.670956i	$-0.765884\pi$
$811$	−2.77774e7	−1.48300	−0.741498	+	0.670956i	$0.765884\pi$
$812$	0	0
$813$	−5.91516e6	−0.313863
$814$	0	0
$815$	6.62354e7	3.49298
$816$	0	0
$817$	5.26647e7	2.76035
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.28028e7	−0.662899	−0.331450	−	0.943473i	$-0.607538\pi$
$821$	−1.28028e7	−0.662899	−0.331450	+	0.943473i	$0.607538\pi$
$822$	0	0
$823$	1.16269e6	0.0598360	0.0299180	−	0.999552i	$-0.490475\pi$
$823$	1.16269e6	0.0598360	0.0299180	+	0.999552i	$0.490475\pi$
$824$	0	0
$825$	−5.22818e6	−0.267433
$826$	0	0
$827$	2.80372e7	1.42551	0.712756	−	0.701412i	$-0.247447\pi$
$827$	2.80372e7	1.42551	0.712756	+	0.701412i	$0.247447\pi$
$828$	0	0
$829$	−1.23475e7	−0.624013	−0.312006	−	0.950080i	$-0.601001\pi$
$829$	−1.23475e7	−0.624013	−0.312006	+	0.950080i	$0.601001\pi$
$830$	0	0
$831$	5.61186e6	0.281906
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.62319e7	1.30201
$836$	0	0
$837$	1.28103e7	0.632043
$838$	0	0
$839$	−1.34932e7	−0.661774	−0.330887	−	0.943670i	$-0.607348\pi$
$839$	−1.34932e7	−0.661774	−0.330887	+	0.943670i	$0.607348\pi$
$840$	0	0
$841$	277718.	0.0135398
$842$	0	0
$843$	−3.22514e6	−0.156307
$844$	0	0
$845$	−3.39599e7	−1.63616
$846$	0	0
$847$	0	0
$848$	0	0
$849$	4.66659e6	0.222193
$850$	0	0
$851$	−1.26462e7	−0.598599
$852$	0	0
$853$	2.66553e7	1.25433	0.627164	−	0.778887i	$-0.284216\pi$
$853$	2.66553e7	1.25433	0.627164	+	0.778887i	$0.284216\pi$
$854$	0	0
$855$	−5.99720e7	−2.80565
$856$	0	0
$857$	1.52888e7	0.711084	0.355542	−	0.934660i	$-0.384296\pi$
$857$	1.52888e7	0.711084	0.355542	+	0.934660i	$0.384296\pi$
$858$	0	0
$859$	−1.20277e6	−0.0556161	−0.0278081	−	0.999613i	$-0.508853\pi$
$859$	−1.20277e6	−0.0556161	−0.0278081	+	0.999613i	$0.508853\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.47643e6	0.433130	0.216565	−	0.976268i	$-0.430515\pi$
$863$	9.47643e6	0.433130	0.216565	+	0.976268i	$0.430515\pi$
$864$	0	0
$865$	−3.97397e7	−1.80586
$866$	0	0
$867$	4.21277e6	0.190336
$868$	0	0
$869$	6.30969e6	0.283438
$870$	0	0
$871$	−2.23747e6	−0.0999339
$872$	0	0
$873$	−3.97876e7	−1.76690
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.14328e7	−0.501941	−0.250971	−	0.967995i	$-0.580750\pi$
$877$	−1.14328e7	−0.501941	−0.250971	+	0.967995i	$0.580750\pi$
$878$	0	0
$879$	−2.50945e6	−0.109549
$880$	0	0
$881$	3.40559e7	1.47827	0.739133	−	0.673559i	$-0.235235\pi$
$881$	3.40559e7	1.47827	0.739133	+	0.673559i	$0.235235\pi$
$882$	0	0
$883$	2.46784e7	1.06516	0.532580	−	0.846380i	$-0.321223\pi$
$883$	2.46784e7	1.06516	0.532580	+	0.846380i	$0.321223\pi$
$884$	0	0
$885$	6.39355e6	0.274400
$886$	0	0
$887$	−1.74886e7	−0.746355	−0.373177	−	0.927760i	$-0.621732\pi$
$887$	−1.74886e7	−0.746355	−0.373177	+	0.927760i	$0.621732\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.22665e7	0.517637
$892$	0	0
$893$	−1.93765e7	−0.813104
$894$	0	0
$895$	−6.48134e7	−2.70463
$896$	0	0
$897$	2.04186e6	0.0847316
$898$	0	0
$899$	−4.09017e7	−1.68788
$900$	0	0
$901$	−1.10643e6	−0.0454057
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.23899e7	−0.908722
$906$	0	0
$907$	7.38158e6	0.297942	0.148971	−	0.988842i	$-0.452404\pi$
$907$	7.38158e6	0.297942	0.148971	+	0.988842i	$0.452404\pi$
$908$	0	0
$909$	−2.83819e7	−1.13929
$910$	0	0
$911$	−1.02807e7	−0.410419	−0.205210	−	0.978718i	$-0.565788\pi$
$911$	−1.02807e7	−0.410419	−0.205210	+	0.978718i	$0.565788\pi$
$912$	0	0
$913$	1.66504e7	0.661071
$914$	0	0
$915$	1.36987e7	0.540913
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5.15523e6	0.201353	0.100677	−	0.994919i	$-0.467899\pi$
$919$	5.15523e6	0.201353	0.100677	+	0.994919i	$0.467899\pi$
$920$	0	0
$921$	4.04941e6	0.157305
$922$	0	0
$923$	1.17043e7	0.452211
$924$	0	0
$925$	2.83382e7	1.08898
$926$	0	0
$927$	−3.58792e7	−1.37133
$928$	0	0
$929$	−1.66402e7	−0.632586	−0.316293	−	0.948662i	$-0.602438\pi$
$929$	−1.66402e7	−0.632586	−0.316293	+	0.948662i	$0.602438\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.15535e6	0.118671
$934$	0	0
$935$	2.69252e6	0.100723
$936$	0	0
$937$	6.77114e6	0.251949	0.125975	−	0.992033i	$-0.459794\pi$
$937$	6.77114e6	0.251949	0.125975	+	0.992033i	$0.459794\pi$
$938$	0	0
$939$	3.23391e6	0.119692
$940$	0	0
$941$	3.13904e7	1.15564	0.577820	−	0.816164i	$-0.303903\pi$
$941$	3.13904e7	1.15564	0.577820	+	0.816164i	$0.303903\pi$
$942$	0	0
$943$	1.22200e7	0.447497
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.51857e7	0.912598	0.456299	−	0.889827i	$-0.349175\pi$
$947$	2.51857e7	0.912598	0.456299	+	0.889827i	$0.349175\pi$
$948$	0	0
$949$	−7.14695e6	−0.257605
$950$	0	0
$951$	6.67819e6	0.239446
$952$	0	0
$953$	3.10156e7	1.10624	0.553118	−	0.833103i	$-0.313438\pi$
$953$	3.10156e7	1.10624	0.553118	+	0.833103i	$0.313438\pi$
$954$	0	0
$955$	−8.35037e7	−2.96277
$956$	0	0
$957$	3.18316e6	0.112351
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.18440e7	1.81088
$962$	0	0
$963$	−4.09474e6	−0.142285
$964$	0	0
$965$	7.07610e7	2.44611
$966$	0	0
$967$	3.71988e7	1.27927	0.639636	−	0.768678i	$-0.279085\pi$
$967$	3.71988e7	1.27927	0.639636	+	0.768678i	$0.279085\pi$
$968$	0	0
$969$	−834396.	−0.0285472
$970$	0	0
$971$	−1.36699e7	−0.465282	−0.232641	−	0.972563i	$-0.574737\pi$
$971$	−1.36699e7	−0.465282	−0.232641	+	0.972563i	$0.574737\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−4.57551e6	−0.154144
$976$	0	0
$977$	−1.70816e7	−0.572522	−0.286261	−	0.958152i	$-0.592412\pi$
$977$	−1.70816e7	−0.572522	−0.286261	+	0.958152i	$0.592412\pi$
$978$	0	0
$979$	−1.48790e6	−0.0496153
$980$	0	0
$981$	−4.09724e7	−1.35931
$982$	0	0
$983$	2.79261e7	0.921777	0.460889	−	0.887458i	$-0.347531\pi$
$983$	2.79261e7	0.921777	0.460889	+	0.887458i	$0.347531\pi$
$984$	0	0
$985$	−5.77463e7	−1.89642
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.07600e7	−2.30037
$990$	0	0
$991$	1.75234e6	0.0566807	0.0283403	−	0.999598i	$-0.490978\pi$
$991$	1.75234e6	0.0566807	0.0283403	+	0.999598i	$0.490978\pi$
$992$	0	0
$993$	−4.89829e6	−0.157642
$994$	0	0
$995$	3.65357e7	1.16993
$996$	0	0
$997$	−1.86648e7	−0.594683	−0.297341	−	0.954771i	$-0.596100\pi$
$997$	−1.86648e7	−0.594683	−0.297341	+	0.954771i	$0.596100\pi$
$998$	0	0
$999$	5.40381e6	0.171311

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.6.a.n.1.4	✓	8
4.3	odd	2		784.6.a.bo.1.5		8
7.2	even	3		392.6.i.r.361.5		16
7.3	odd	6		392.6.i.r.177.4		16
7.4	even	3		392.6.i.r.177.5		16
7.5	odd	6		392.6.i.r.361.4		16
7.6	odd	2	inner	392.6.a.n.1.5	yes	8
28.27	even	2		784.6.a.bo.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.6.a.n.1.4	✓	8	1.1	even	1	trivial
392.6.a.n.1.5	yes	8	7.6	odd	2	inner
392.6.i.r.177.4		16	7.3	odd	6
392.6.i.r.177.5		16	7.4	even	3
392.6.i.r.361.4		16	7.5	odd	6
392.6.i.r.361.5		16	7.2	even	3
784.6.a.bo.1.4		8	28.27	even	2
784.6.a.bo.1.5		8	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)

\(f(q)\)	\(=\)	\(q-2.99351 q^{3} +103.023 q^{5} -234.039 q^{9} +O(q^{10})\)	\(q-2.99351 q^{3} +103.023 q^{5} -234.039 q^{9} +233.218 q^{11} +204.103 q^{13} -308.401 q^{15} +112.064 q^{17} +2487.29 q^{19} -3341.91 q^{23} +7488.72 q^{25} +1428.02 q^{27} -4559.48 q^{29} +8970.68 q^{31} -698.140 q^{33} +3784.12 q^{37} -610.987 q^{39} -3656.58 q^{41} +21173.5 q^{43} -24111.4 q^{45} -7790.19 q^{47} -335.464 q^{51} -9873.21 q^{53} +24026.8 q^{55} -7445.74 q^{57} -20731.3 q^{59} -44418.6 q^{61} +21027.3 q^{65} -10962.5 q^{67} +10004.1 q^{69} +57344.9 q^{71} -35016.3 q^{73} -22417.6 q^{75} +27054.9 q^{79} +52596.6 q^{81} +71394.3 q^{83} +11545.1 q^{85} +13648.9 q^{87} -6379.86 q^{89} -26853.9 q^{93} +256248. q^{95} +170004. q^{97} -54582.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 932 q^{9}+O(q^{10}) \) 8 * q + 932 * q^9	\( 8 q + 932 q^{9} + 1204 q^{11} + 864 q^{15} - 2368 q^{23} + 2188 q^{25} - 9684 q^{29} + 60 q^{37} + 16088 q^{39} + 43564 q^{43} + 76684 q^{51} + 94472 q^{53} + 61004 q^{57} + 41900 q^{65} + 136504 q^{67} + 192328 q^{71} - 29816 q^{79} + 457860 q^{81} + 111132 q^{85} - 166504 q^{93} + 670800 q^{95} + 857404 q^{99}+O(q^{100}) \) 8 * q + 932 * q^9 + 1204 * q^11 + 864 * q^15 - 2368 * q^23 + 2188 * q^25 - 9684 * q^29 + 60 * q^37 + 16088 * q^39 + 43564 * q^43 + 76684 * q^51 + 94472 * q^53 + 61004 * q^57 + 41900 * q^65 + 136504 * q^67 + 192328 * q^71 - 29816 * q^79 + 457860 * q^81 + 111132 * q^85 - 166504 * q^93 + 670800 * q^95 + 857404 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more