LMFDB - Embedded newform 968.6.a.p.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,6,Mod(1,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, 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("968.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	968.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:	$155.251537579$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 2635 x^{14} + 10644 x^{13} + 2721739 x^{12} - 11107836 x^{11} + \cdots + 53\!\cdots\!20 $ x^16 - 4x^15 - 2635x^14 + 10644x^13 + 2721739x^12 - 11107836x^11 - 1388443249x^10 + 5594180788x^9 + 362606893936x^8 - 1331049359080x^7 - 46300877961632x^6 + 122447405593024x^5 + 2801116058142464x^4 - 3524304397180928x^3 - 71736438677110016x^2 + 11226318210301440*x + 533879212527569920
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{26}\cdot 11^{8} $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$10.0842$ of defining polynomial
Character	$\chi$	$=$	968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.7022 q^{3} -24.8620 q^{5} +76.4943 q^{7} -106.058 q^{9} -426.611 q^{13} -290.940 q^{15} +2220.08 q^{17} +2015.93 q^{19} +895.154 q^{21} +2677.09 q^{23} -2506.88 q^{25} -4084.75 q^{27} -5149.86 q^{29} +484.684 q^{31} -1901.80 q^{35} +5927.86 q^{37} -4992.30 q^{39} -12402.8 q^{41} -134.028 q^{43} +2636.81 q^{45} -6980.30 q^{47} -10955.6 q^{49} +25979.9 q^{51} -7559.84 q^{53} +23590.8 q^{57} +10060.8 q^{59} +17801.9 q^{61} -8112.84 q^{63} +10606.4 q^{65} +37247.9 q^{67} +31327.9 q^{69} +69878.2 q^{71} -3565.04 q^{73} -29336.1 q^{75} +72627.1 q^{79} -22028.6 q^{81} +55957.6 q^{83} -55195.6 q^{85} -60264.7 q^{87} +22393.0 q^{89} -32633.3 q^{91} +5671.88 q^{93} -50119.9 q^{95} -15592.4 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{3} + 81 q^{5} - 47 q^{7} + 1416 q^{9} + 859 q^{13} - 738 q^{15} + 1226 q^{17} + 616 q^{19} + 1141 q^{21} + 2258 q^{23} + 10307 q^{25} + 564 q^{27} + 1613 q^{29} + 18511 q^{31} - 23544 q^{35}+ \cdots + 171314 q^{97}+O(q^{100}) $ 16 * q + 12 * q^3 + 81 * q^5 - 47 * q^7 + 1416 * q^9 + 859 * q^13 - 738 * q^15 + 1226 * q^17 + 616 * q^19 + 1141 * q^21 + 2258 * q^23 + 10307 * q^25 + 564 * q^27 + 1613 * q^29 + 18511 * q^31 - 23544 * q^35 + 8553 * q^37 + 12020 * q^39 + 25514 * q^41 - 18931 * q^43 + 23278 * q^45 + 38781 * q^47 + 94523 * q^49 - 68584 * q^51 + 779 * q^53 + 19293 * q^57 + 50316 * q^59 + 53153 * q^61 - 54911 * q^63 + 51517 * q^65 + 10491 * q^67 - 95356 * q^69 - 91677 * q^71 + 97660 * q^73 - 3771 * q^75 - 96003 * q^79 + 120400 * q^81 + 4552 * q^83 - 129679 * q^85 + 77219 * q^87 + 6697 * q^89 + 217380 * q^91 + 132586 * q^93 - 54168 * q^95 + 171314 * q^97

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
$3$	11.7022	0.750698	0.375349	−	0.926884i	$-0.377523\pi$
$3$	11.7022	0.750698	0.375349	+	0.926884i	$0.377523\pi$
$4$	0	0
$5$	−24.8620	−0.444744	−0.222372	−	0.974962i	$-0.571380\pi$
$5$	−24.8620	−0.444744	−0.222372	+	0.974962i	$0.571380\pi$
$6$	0	0
$7$	76.4943	0.590044	0.295022	−	0.955490i	$-0.404673\pi$
$7$	76.4943	0.590044	0.295022	+	0.955490i	$0.404673\pi$
$8$	0	0
$9$	−106.058	−0.436453
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−426.611	−0.700122	−0.350061	−	0.936727i	$-0.613839\pi$
$13$	−426.611	−0.700122	−0.350061	+	0.936727i	$0.613839\pi$
$14$	0	0
$15$	−290.940	−0.333869
$16$	0	0
$17$	2220.08	1.86314	0.931572	−	0.363557i	$-0.118438\pi$
$17$	2220.08	1.86314	0.931572	+	0.363557i	$0.118438\pi$
$18$	0	0
$19$	2015.93	1.28112	0.640561	−	0.767907i	$-0.278702\pi$
$19$	2015.93	1.28112	0.640561	+	0.767907i	$0.278702\pi$
$20$	0	0
$21$	895.154	0.442945
$22$	0	0
$23$	2677.09	1.05522	0.527611	−	0.849486i	$-0.323088\pi$
$23$	2677.09	1.05522	0.527611	+	0.849486i	$0.323088\pi$
$24$	0	0
$25$	−2506.88	−0.802203
$26$	0	0
$27$	−4084.75	−1.07834
$28$	0	0
$29$	−5149.86	−1.13710	−0.568552	−	0.822647i	$-0.692496\pi$
$29$	−5149.86	−1.13710	−0.568552	+	0.822647i	$0.692496\pi$
$30$	0	0
$31$	484.684	0.0905846	0.0452923	−	0.998974i	$-0.485578\pi$
$31$	484.684	0.0905846	0.0452923	+	0.998974i	$0.485578\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1901.80	−0.262419
$36$	0	0
$37$	5927.86	0.711859	0.355929	−	0.934513i	$-0.384164\pi$
$37$	5927.86	0.711859	0.355929	+	0.934513i	$0.384164\pi$
$38$	0	0
$39$	−4992.30	−0.525580
$40$	0	0
$41$	−12402.8	−1.15228	−0.576142	−	0.817349i	$-0.695443\pi$
$41$	−12402.8	−1.15228	−0.576142	+	0.817349i	$0.695443\pi$
$42$	0	0
$43$	−134.028	−0.0110541	−0.00552706	−	0.999985i	$-0.501759\pi$
$43$	−134.028	−0.0110541	−0.00552706	+	0.999985i	$0.501759\pi$
$44$	0	0
$45$	2636.81	0.194110
$46$	0	0
$47$	−6980.30	−0.460924	−0.230462	−	0.973081i	$-0.574024\pi$
$47$	−6980.30	−0.460924	−0.230462	+	0.973081i	$0.574024\pi$
$48$	0	0
$49$	−10955.6	−0.651848
$50$	0	0
$51$	25979.9	1.39866
$52$	0	0
$53$	−7559.84	−0.369678	−0.184839	−	0.982769i	$-0.559176\pi$
$53$	−7559.84	−0.369678	−0.184839	+	0.982769i	$0.559176\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	23590.8	0.961736
$58$	0	0
$59$	10060.8	0.376274	0.188137	−	0.982143i	$-0.439755\pi$
$59$	10060.8	0.376274	0.188137	+	0.982143i	$0.439755\pi$
$60$	0	0
$61$	17801.9	0.612551	0.306276	−	0.951943i	$-0.400917\pi$
$61$	17801.9	0.612551	0.306276	+	0.951943i	$0.400917\pi$
$62$	0	0
$63$	−8112.84	−0.257526
$64$	0	0
$65$	10606.4	0.311375
$66$	0	0
$67$	37247.9	1.01371	0.506856	−	0.862031i	$-0.330808\pi$
$67$	37247.9	1.01371	0.506856	+	0.862031i	$0.330808\pi$
$68$	0	0
$69$	31327.9	0.792153
$70$	0	0
$71$	69878.2	1.64511	0.822557	−	0.568683i	$-0.192547\pi$
$71$	69878.2	1.64511	0.822557	+	0.568683i	$0.192547\pi$
$72$	0	0
$73$	−3565.04	−0.0782991	−0.0391496	−	0.999233i	$-0.512465\pi$
$73$	−3565.04	−0.0782991	−0.0391496	+	0.999233i	$0.512465\pi$
$74$	0	0
$75$	−29336.1	−0.602212
$76$	0	0
$77$	0	0
$78$	0	0
$79$	72627.1	1.30927	0.654637	−	0.755943i	$-0.272821\pi$
$79$	72627.1	1.30927	0.654637	+	0.755943i	$0.272821\pi$
$80$	0	0
$81$	−22028.6	−0.373056
$82$	0	0
$83$	55957.6	0.891588	0.445794	−	0.895136i	$-0.352921\pi$
$83$	55957.6	0.891588	0.445794	+	0.895136i	$0.352921\pi$
$84$	0	0
$85$	−55195.6	−0.828623
$86$	0	0
$87$	−60264.7	−0.853621
$88$	0	0
$89$	22393.0	0.299666	0.149833	−	0.988711i	$-0.452126\pi$
$89$	22393.0	0.299666	0.149833	+	0.988711i	$0.452126\pi$
$90$	0	0
$91$	−32633.3	−0.413103
$92$	0	0
$93$	5671.88	0.0680017
$94$	0	0
$95$	−50119.9	−0.569772
$96$	0	0
$97$	−15592.4	−0.168261	−0.0841307	−	0.996455i	$-0.526811\pi$
$97$	−15592.4	−0.168261	−0.0841307	+	0.996455i	$0.526811\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−203552.	−1.98551	−0.992756	−	0.120148i	$-0.961663\pi$
$101$	−203552.	−1.98551	−0.992756	+	0.120148i	$0.961663\pi$
$102$	0	0
$103$	173593.	1.61227	0.806137	−	0.591729i	$-0.201554\pi$
$103$	173593.	1.61227	0.806137	+	0.591729i	$0.201554\pi$
$104$	0	0
$105$	−22255.3	−0.196997
$106$	0	0
$107$	−50598.2	−0.427244	−0.213622	−	0.976916i	$-0.568526\pi$
$107$	−50598.2	−0.427244	−0.213622	+	0.976916i	$0.568526\pi$
$108$	0	0
$109$	−149343.	−1.20398	−0.601989	−	0.798504i	$-0.705625\pi$
$109$	−149343.	−1.20398	−0.601989	+	0.798504i	$0.705625\pi$
$110$	0	0
$111$	69369.2	0.534391
$112$	0	0
$113$	32622.5	0.240338	0.120169	−	0.992753i	$-0.461656\pi$
$113$	32622.5	0.240338	0.120169	+	0.992753i	$0.461656\pi$
$114$	0	0
$115$	−66557.8	−0.469304
$116$	0	0
$117$	45245.6	0.305570
$118$	0	0
$119$	169824.	1.09934
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−145140.	−0.865018
$124$	0	0
$125$	140020.	0.801519
$126$	0	0
$127$	140111.	0.770836	0.385418	−	0.922742i	$-0.374057\pi$
$127$	140111.	0.770836	0.385418	+	0.922742i	$0.374057\pi$
$128$	0	0
$129$	−1568.43	−0.00829831
$130$	0	0
$131$	376094.	1.91478	0.957389	−	0.288801i	$-0.0932566\pi$
$131$	376094.	1.91478	0.957389	+	0.288801i	$0.0932566\pi$
$132$	0	0
$133$	154207.	0.755919
$134$	0	0
$135$	101555.	0.479586
$136$	0	0
$137$	314053.	1.42956	0.714779	−	0.699351i	$-0.246527\pi$
$137$	314053.	1.42956	0.714779	+	0.699351i	$0.246527\pi$
$138$	0	0
$139$	48494.4	0.212890	0.106445	−	0.994319i	$-0.466053\pi$
$139$	48494.4	0.212890	0.106445	+	0.994319i	$0.466053\pi$
$140$	0	0
$141$	−81685.0	−0.346014
$142$	0	0
$143$	0	0
$144$	0	0
$145$	128035.	0.505720
$146$	0	0
$147$	−128205.	−0.489341
$148$	0	0
$149$	400395.	1.47749	0.738743	−	0.673988i	$-0.235420\pi$
$149$	400395.	1.47749	0.738743	+	0.673988i	$0.235420\pi$
$150$	0	0
$151$	9760.17	0.0348349	0.0174175	−	0.999848i	$-0.494456\pi$
$151$	9760.17	0.0348349	0.0174175	+	0.999848i	$0.494456\pi$
$152$	0	0
$153$	−235457.	−0.813175
$154$	0	0
$155$	−12050.2	−0.0402870
$156$	0	0
$157$	424485.	1.37440	0.687201	−	0.726467i	$-0.258839\pi$
$157$	424485.	1.37440	0.687201	+	0.726467i	$0.258839\pi$
$158$	0	0
$159$	−88466.9	−0.277516
$160$	0	0
$161$	204782.	0.622627
$162$	0	0
$163$	−169692.	−0.500256	−0.250128	−	0.968213i	$-0.580473\pi$
$163$	−169692.	−0.500256	−0.250128	+	0.968213i	$0.580473\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	130059.	0.360868	0.180434	−	0.983587i	$-0.442250\pi$
$167$	130059.	0.360868	0.180434	+	0.983587i	$0.442250\pi$
$168$	0	0
$169$	−189296.	−0.509829
$170$	0	0
$171$	−213805.	−0.559150
$172$	0	0
$173$	56920.9	0.144596	0.0722980	−	0.997383i	$-0.476967\pi$
$173$	56920.9	0.144596	0.0722980	+	0.997383i	$0.476967\pi$
$174$	0	0
$175$	−191762.	−0.473335
$176$	0	0
$177$	117734.	0.282468
$178$	0	0
$179$	322128.	0.751443	0.375722	−	0.926733i	$-0.377395\pi$
$179$	322128.	0.751443	0.375722	+	0.926733i	$0.377395\pi$
$180$	0	0
$181$	−277300.	−0.629148	−0.314574	−	0.949233i	$-0.601862\pi$
$181$	−277300.	−0.629148	−0.314574	+	0.949233i	$0.601862\pi$
$182$	0	0
$183$	208322.	0.459841
$184$	0	0
$185$	−147378.	−0.316595
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−312461.	−0.636269
$190$	0	0
$191$	861485.	1.70869	0.854347	−	0.519703i	$-0.173957\pi$
$191$	861485.	1.70869	0.854347	+	0.519703i	$0.173957\pi$
$192$	0	0
$193$	−619165.	−1.19650	−0.598250	−	0.801309i	$-0.704137\pi$
$193$	−619165.	−1.19650	−0.598250	+	0.801309i	$0.704137\pi$
$194$	0	0
$195$	124118.	0.233749
$196$	0	0
$197$	156011.	0.286412	0.143206	−	0.989693i	$-0.454259\pi$
$197$	156011.	0.286412	0.143206	+	0.989693i	$0.454259\pi$
$198$	0	0
$199$	679607.	1.21654	0.608269	−	0.793731i	$-0.291864\pi$
$199$	679607.	1.21654	0.608269	+	0.793731i	$0.291864\pi$
$200$	0	0
$201$	435883.	0.760991
$202$	0	0
$203$	−393935.	−0.670941
$204$	0	0
$205$	308358.	0.512472
$206$	0	0
$207$	−283927.	−0.460555
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−618384.	−0.956207	−0.478104	−	0.878303i	$-0.658676\pi$
$211$	−618384.	−0.956207	−0.478104	+	0.878303i	$0.658676\pi$
$212$	0	0
$213$	817730.	1.23498
$214$	0	0
$215$	3332.20	0.00491626
$216$	0	0
$217$	37075.6	0.0534489
$218$	0	0
$219$	−41718.8	−0.0587790
$220$	0	0
$221$	−947111.	−1.30443
$222$	0	0
$223$	−34281.6	−0.0461635	−0.0230817	−	0.999734i	$-0.507348\pi$
$223$	−34281.6	−0.0461635	−0.0230817	+	0.999734i	$0.507348\pi$
$224$	0	0
$225$	265875.	0.350124
$226$	0	0
$227$	673696.	0.867759	0.433880	−	0.900971i	$-0.357144\pi$
$227$	673696.	0.867759	0.433880	+	0.900971i	$0.357144\pi$
$228$	0	0
$229$	915382.	1.15349	0.576745	−	0.816925i	$-0.304323\pi$
$229$	915382.	1.15349	0.576745	+	0.816925i	$0.304323\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.44829e6	−1.74769	−0.873846	−	0.486203i	$-0.838382\pi$
$233$	−1.44829e6	−1.74769	−0.873846	+	0.486203i	$0.838382\pi$
$234$	0	0
$235$	173544.	0.204993
$236$	0	0
$237$	849898.	0.982870
$238$	0	0
$239$	44021.6	0.0498507	0.0249254	−	0.999689i	$-0.492065\pi$
$239$	44021.6	0.0498507	0.0249254	+	0.999689i	$0.492065\pi$
$240$	0	0
$241$	−840694.	−0.932385	−0.466193	−	0.884683i	$-0.654375\pi$
$241$	−840694.	−0.932385	−0.466193	+	0.884683i	$0.654375\pi$
$242$	0	0
$243$	734812.	0.798290
$244$	0	0
$245$	272378.	0.289906
$246$	0	0
$247$	−860017.	−0.896943
$248$	0	0
$249$	654829.	0.669313
$250$	0	0
$251$	286935.	0.287474	0.143737	−	0.989616i	$-0.454088\pi$
$251$	286935.	0.287474	0.143737	+	0.989616i	$0.454088\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−645911.	−0.622045
$256$	0	0
$257$	−165772.	−0.156559	−0.0782796	−	0.996931i	$-0.524943\pi$
$257$	−165772.	−0.156559	−0.0782796	+	0.996931i	$0.524943\pi$
$258$	0	0
$259$	453448.	0.420028
$260$	0	0
$261$	546184.	0.496292
$262$	0	0
$263$	−31735.7	−0.0282917	−0.0141458	−	0.999900i	$-0.504503\pi$
$263$	−31735.7	−0.0282917	−0.0141458	+	0.999900i	$0.504503\pi$
$264$	0	0
$265$	187952.	0.164412
$266$	0	0
$267$	262048.	0.224959
$268$	0	0
$269$	−2.00216e6	−1.68701	−0.843505	−	0.537121i	$-0.819512\pi$
$269$	−2.00216e6	−1.68701	−0.843505	+	0.537121i	$0.819512\pi$
$270$	0	0
$271$	1.86820e6	1.54526	0.772628	−	0.634860i	$-0.218942\pi$
$271$	1.86820e6	1.54526	0.772628	+	0.634860i	$0.218942\pi$
$272$	0	0
$273$	−381883.	−0.310115
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−552831.	−0.432906	−0.216453	−	0.976293i	$-0.569449\pi$
$277$	−552831.	−0.432906	−0.216453	+	0.976293i	$0.569449\pi$
$278$	0	0
$279$	−51404.6	−0.0395359
$280$	0	0
$281$	1.84184e6	1.39151	0.695755	−	0.718280i	$-0.255070\pi$
$281$	1.84184e6	1.39151	0.695755	+	0.718280i	$0.255070\pi$
$282$	0	0
$283$	156886.	0.116445	0.0582223	−	0.998304i	$-0.481457\pi$
$283$	156886.	0.116445	0.0582223	+	0.998304i	$0.481457\pi$
$284$	0	0
$285$	−586514.	−0.427727
$286$	0	0
$287$	−948744.	−0.679899
$288$	0	0
$289$	3.50890e6	2.47131
$290$	0	0
$291$	−182466.	−0.126314
$292$	0	0
$293$	700750.	0.476863	0.238432	−	0.971159i	$-0.423367\pi$
$293$	700750.	0.476863	0.238432	+	0.971159i	$0.423367\pi$
$294$	0	0
$295$	−250132.	−0.167346
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.14208e6	−0.738784
$300$	0	0
$301$	−10252.4	−0.00652242
$302$	0	0
$303$	−2.38201e6	−1.49052
$304$	0	0
$305$	−442591.	−0.272429
$306$	0	0
$307$	−1.06175e6	−0.642946	−0.321473	−	0.946919i	$-0.604178\pi$
$307$	−1.06175e6	−0.642946	−0.321473	+	0.946919i	$0.604178\pi$
$308$	0	0
$309$	2.03142e6	1.21033
$310$	0	0
$311$	2.72522e6	1.59772	0.798860	−	0.601517i	$-0.205437\pi$
$311$	2.72522e6	1.59772	0.798860	+	0.601517i	$0.205437\pi$
$312$	0	0
$313$	2.31705e6	1.33683	0.668414	−	0.743790i	$-0.266974\pi$
$313$	2.31705e6	1.33683	0.668414	+	0.743790i	$0.266974\pi$
$314$	0	0
$315$	201701.	0.114533
$316$	0	0
$317$	−2.79942e6	−1.56466	−0.782331	−	0.622863i	$-0.785970\pi$
$317$	−2.79942e6	−1.56466	−0.782331	+	0.622863i	$0.785970\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−592111.	−0.320731
$322$	0	0
$323$	4.47552e6	2.38692
$324$	0	0
$325$	1.06946e6	0.561640
$326$	0	0
$327$	−1.74764e6	−0.903823
$328$	0	0
$329$	−533953.	−0.271965
$330$	0	0
$331$	−1.00178e6	−0.502577	−0.251289	−	0.967912i	$-0.580854\pi$
$331$	−1.00178e6	−0.502577	−0.251289	+	0.967912i	$0.580854\pi$
$332$	0	0
$333$	−628698.	−0.310693
$334$	0	0
$335$	−926056.	−0.450843
$336$	0	0
$337$	2.63039e6	1.26167	0.630835	−	0.775917i	$-0.282712\pi$
$337$	2.63039e6	1.26167	0.630835	+	0.775917i	$0.282712\pi$
$338$	0	0
$339$	381756.	0.180421
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−2.12368e6	−0.974663
$344$	0	0
$345$	−778874.	−0.352305
$346$	0	0
$347$	−1.84175e6	−0.821121	−0.410560	−	0.911833i	$-0.634667\pi$
$347$	−1.84175e6	−0.821121	−0.410560	+	0.911833i	$0.634667\pi$
$348$	0	0
$349$	−974194.	−0.428136	−0.214068	−	0.976819i	$-0.568671\pi$
$349$	−974194.	−0.428136	−0.214068	+	0.976819i	$0.568671\pi$
$350$	0	0
$351$	1.74260e6	0.754971
$352$	0	0
$353$	3.34261e6	1.42774	0.713870	−	0.700278i	$-0.246941\pi$
$353$	3.34261e6	1.42774	0.713870	+	0.700278i	$0.246941\pi$
$354$	0	0
$355$	−1.73731e6	−0.731655
$356$	0	0
$357$	1.98731e6	0.825270
$358$	0	0
$359$	−1.42048e6	−0.581699	−0.290850	−	0.956769i	$-0.593938\pi$
$359$	−1.42048e6	−0.581699	−0.290850	+	0.956769i	$0.593938\pi$
$360$	0	0
$361$	1.58786e6	0.641276
$362$	0	0
$363$	0	0
$364$	0	0
$365$	88633.8	0.0348231
$366$	0	0
$367$	2.22232e6	0.861273	0.430636	−	0.902525i	$-0.358289\pi$
$367$	2.22232e6	0.861273	0.430636	+	0.902525i	$0.358289\pi$
$368$	0	0
$369$	1.31542e6	0.502918
$370$	0	0
$371$	−578285.	−0.218126
$372$	0	0
$373$	1.35037e6	0.502552	0.251276	−	0.967915i	$-0.419150\pi$
$373$	1.35037e6	0.502552	0.251276	+	0.967915i	$0.419150\pi$
$374$	0	0
$375$	1.63854e6	0.601699
$376$	0	0
$377$	2.19699e6	0.796112
$378$	0	0
$379$	2.42875e6	0.868529	0.434265	−	0.900785i	$-0.357008\pi$
$379$	2.42875e6	0.868529	0.434265	+	0.900785i	$0.357008\pi$
$380$	0	0
$381$	1.63961e6	0.578665
$382$	0	0
$383$	138625.	0.0482885	0.0241443	−	0.999708i	$-0.492314\pi$
$383$	138625.	0.0482885	0.0241443	+	0.999708i	$0.492314\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	14214.8	0.00482461
$388$	0	0
$389$	−27344.7	−0.00916218	−0.00458109	−	0.999990i	$-0.501458\pi$
$389$	−27344.7	−0.00916218	−0.00458109	+	0.999990i	$0.501458\pi$
$390$	0	0
$391$	5.94336e6	1.96603
$392$	0	0
$393$	4.40114e6	1.43742
$394$	0	0
$395$	−1.80565e6	−0.582293
$396$	0	0
$397$	1.84875e6	0.588711	0.294356	−	0.955696i	$-0.404895\pi$
$397$	1.84875e6	0.588711	0.294356	+	0.955696i	$0.404895\pi$
$398$	0	0
$399$	1.80456e6	0.567466
$400$	0	0
$401$	−5.19332e6	−1.61281	−0.806407	−	0.591361i	$-0.798591\pi$
$401$	−5.19332e6	−1.61281	−0.806407	+	0.591361i	$0.798591\pi$
$402$	0	0
$403$	−206772.	−0.0634203
$404$	0	0
$405$	547674.	0.165915
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.37849e6	0.703061	0.351530	−	0.936176i	$-0.385661\pi$
$409$	2.37849e6	0.703061	0.351530	+	0.936176i	$0.385661\pi$
$410$	0	0
$411$	3.67512e6	1.07317
$412$	0	0
$413$	769598.	0.222018
$414$	0	0
$415$	−1.39122e6	−0.396529
$416$	0	0
$417$	567492.	0.159816
$418$	0	0
$419$	3.99504e6	1.11170	0.555848	−	0.831284i	$-0.312394\pi$
$419$	3.99504e6	1.11170	0.555848	+	0.831284i	$0.312394\pi$
$420$	0	0
$421$	5.01278e6	1.37839	0.689197	−	0.724574i	$-0.257964\pi$
$421$	5.01278e6	1.37839	0.689197	+	0.724574i	$0.257964\pi$
$422$	0	0
$423$	740317.	0.201172
$424$	0	0
$425$	−5.56548e6	−1.49462
$426$	0	0
$427$	1.36175e6	0.361432
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.12741e6	−0.810945	−0.405472	−	0.914107i	$-0.632893\pi$
$431$	−3.12741e6	−0.810945	−0.405472	+	0.914107i	$0.632893\pi$
$432$	0	0
$433$	2.35722e6	0.604200	0.302100	−	0.953276i	$-0.402312\pi$
$433$	2.35722e6	0.604200	0.302100	+	0.953276i	$0.402312\pi$
$434$	0	0
$435$	1.49830e6	0.379643
$436$	0	0
$437$	5.39682e6	1.35187
$438$	0	0
$439$	−4.25642e6	−1.05410	−0.527051	−	0.849833i	$-0.676703\pi$
$439$	−4.25642e6	−1.05410	−0.527051	+	0.849833i	$0.676703\pi$
$440$	0	0
$441$	1.16193e6	0.284501
$442$	0	0
$443$	437693.	0.105965	0.0529823	−	0.998595i	$-0.483127\pi$
$443$	437693.	0.105965	0.0529823	+	0.998595i	$0.483127\pi$
$444$	0	0
$445$	−556734.	−0.133275
$446$	0	0
$447$	4.68551e6	1.10914
$448$	0	0
$449$	−137602.	−0.0322114	−0.0161057	−	0.999870i	$-0.505127\pi$
$449$	−137602.	−0.0322114	−0.0161057	+	0.999870i	$0.505127\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	114216.	0.0261505
$454$	0	0
$455$	811329.	0.183725
$456$	0	0
$457$	6.88672e6	1.54249	0.771244	−	0.636539i	$-0.219635\pi$
$457$	6.88672e6	1.54249	0.771244	+	0.636539i	$0.219635\pi$
$458$	0	0
$459$	−9.06848e6	−2.00911
$460$	0	0
$461$	−2.87490e6	−0.630042	−0.315021	−	0.949085i	$-0.602012\pi$
$461$	−2.87490e6	−0.630042	−0.315021	+	0.949085i	$0.602012\pi$
$462$	0	0
$463$	−3.45632e6	−0.749311	−0.374655	−	0.927164i	$-0.622239\pi$
$463$	−3.45632e6	−0.749311	−0.374655	+	0.927164i	$0.622239\pi$
$464$	0	0
$465$	−141014.	−0.0302433
$466$	0	0
$467$	6.51025e6	1.38135	0.690677	−	0.723163i	$-0.257312\pi$
$467$	6.51025e6	1.38135	0.690677	+	0.723163i	$0.257312\pi$
$468$	0	0
$469$	2.84925e6	0.598135
$470$	0	0
$471$	4.96742e6	1.03176
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−5.05369e6	−1.02772
$476$	0	0
$477$	801782.	0.161347
$478$	0	0
$479$	−8.85367e6	−1.76313	−0.881565	−	0.472062i	$-0.843510\pi$
$479$	−8.85367e6	−1.76313	−0.881565	+	0.472062i	$0.843510\pi$
$480$	0	0
$481$	−2.52889e6	−0.498388
$482$	0	0
$483$	2.39641e6	0.467405
$484$	0	0
$485$	387659.	0.0748333
$486$	0	0
$487$	−9.09168e6	−1.73709	−0.868543	−	0.495613i	$-0.834943\pi$
$487$	−9.09168e6	−1.73709	−0.868543	+	0.495613i	$0.834943\pi$
$488$	0	0
$489$	−1.98577e6	−0.375541
$490$	0	0
$491$	2.94251e6	0.550825	0.275412	−	0.961326i	$-0.411186\pi$
$491$	2.94251e6	0.550825	0.275412	+	0.961326i	$0.411186\pi$
$492$	0	0
$493$	−1.14331e7	−2.11859
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.34529e6	0.970689
$498$	0	0
$499$	−1.24935e6	−0.224613	−0.112306	−	0.993674i	$-0.535824\pi$
$499$	−1.24935e6	−0.224613	−0.112306	+	0.993674i	$0.535824\pi$
$500$	0	0
$501$	1.52197e6	0.270903
$502$	0	0
$503$	−4.12279e6	−0.726559	−0.363280	−	0.931680i	$-0.618343\pi$
$503$	−4.12279e6	−0.726559	−0.363280	+	0.931680i	$0.618343\pi$
$504$	0	0
$505$	5.06071e6	0.883045
$506$	0	0
$507$	−2.21518e6	−0.382727
$508$	0	0
$509$	−474613.	−0.0811980	−0.0405990	−	0.999176i	$-0.512927\pi$
$509$	−474613.	−0.0811980	−0.0405990	+	0.999176i	$0.512927\pi$
$510$	0	0
$511$	−272705.	−0.0461999
$512$	0	0
$513$	−8.23457e6	−1.38149
$514$	0	0
$515$	−4.31586e6	−0.717050
$516$	0	0
$517$	0	0
$518$	0	0
$519$	666101.	0.108548
$520$	0	0
$521$	−6.86718e6	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$521$	−6.86718e6	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$522$	0	0
$523$	−1.02467e7	−1.63806	−0.819031	−	0.573749i	$-0.805488\pi$
$523$	−1.02467e7	−1.63806	−0.819031	+	0.573749i	$0.805488\pi$
$524$	0	0
$525$	−2.24405e6	−0.355331
$526$	0	0
$527$	1.07604e6	0.168772
$528$	0	0
$529$	730479.	0.113493
$530$	0	0
$531$	−1.06703e6	−0.164226
$532$	0	0
$533$	5.29117e6	0.806740
$534$	0	0
$535$	1.25797e6	0.190014
$536$	0	0
$537$	3.76962e6	0.564107
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.97514e6	1.31840	0.659202	−	0.751966i	$-0.270894\pi$
$541$	8.97514e6	1.31840	0.659202	+	0.751966i	$0.270894\pi$
$542$	0	0
$543$	−3.24502e6	−0.472300
$544$	0	0
$545$	3.71296e6	0.535462
$546$	0	0
$547$	1.24955e6	0.178561	0.0892803	−	0.996007i	$-0.471543\pi$
$547$	1.24955e6	0.178561	0.0892803	+	0.996007i	$0.471543\pi$
$548$	0	0
$549$	−1.88804e6	−0.267350
$550$	0	0
$551$	−1.03817e7	−1.45677
$552$	0	0
$553$	5.55556e6	0.772530
$554$	0	0
$555$	−1.72465e6	−0.237667
$556$	0	0
$557$	9.63185e6	1.31544	0.657721	−	0.753261i	$-0.271520\pi$
$557$	9.63185e6	1.31544	0.657721	+	0.753261i	$0.271520\pi$
$558$	0	0
$559$	57177.9	0.00773924
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.13873e7	−1.51409	−0.757043	−	0.653365i	$-0.773357\pi$
$563$	−1.13873e7	−1.51409	−0.757043	+	0.653365i	$0.773357\pi$
$564$	0	0
$565$	−811060.	−0.106889
$566$	0	0
$567$	−1.68506e6	−0.220119
$568$	0	0
$569$	5.18852e6	0.671836	0.335918	−	0.941891i	$-0.390954\pi$
$569$	5.18852e6	0.671836	0.335918	+	0.941891i	$0.390954\pi$
$570$	0	0
$571$	2.66568e6	0.342151	0.171076	−	0.985258i	$-0.445276\pi$
$571$	2.66568e6	0.342151	0.171076	+	0.985258i	$0.445276\pi$
$572$	0	0
$573$	1.00813e7	1.28271
$574$	0	0
$575$	−6.71116e6	−0.846502
$576$	0	0
$577$	−4.56726e6	−0.571105	−0.285553	−	0.958363i	$-0.592177\pi$
$577$	−4.56726e6	−0.571105	−0.285553	+	0.958363i	$0.592177\pi$
$578$	0	0
$579$	−7.24560e6	−0.898210
$580$	0	0
$581$	4.28044e6	0.526076
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.12489e6	−0.135901
$586$	0	0
$587$	−1.24925e7	−1.49642	−0.748210	−	0.663462i	$-0.769086\pi$
$587$	−1.24925e7	−1.49642	−0.748210	+	0.663462i	$0.769086\pi$
$588$	0	0
$589$	977087.	0.116050
$590$	0	0
$591$	1.82568e6	0.215009
$592$	0	0
$593$	−1.01646e7	−1.18700	−0.593502	−	0.804832i	$-0.702255\pi$
$593$	−1.01646e7	−1.18700	−0.593502	+	0.804832i	$0.702255\pi$
$594$	0	0
$595$	−4.22215e6	−0.488924
$596$	0	0
$597$	7.95292e6	0.913252
$598$	0	0
$599$	1.18348e6	0.134770	0.0673851	−	0.997727i	$-0.478534\pi$
$599$	1.18348e6	0.134770	0.0673851	+	0.997727i	$0.478534\pi$
$600$	0	0
$601$	−1.29891e7	−1.46687	−0.733436	−	0.679759i	$-0.762085\pi$
$601$	−1.29891e7	−1.46687	−0.733436	+	0.679759i	$0.762085\pi$
$602$	0	0
$603$	−3.95044e6	−0.442438
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.07058e7	1.17936	0.589680	−	0.807637i	$-0.299254\pi$
$607$	1.07058e7	1.17936	0.589680	+	0.807637i	$0.299254\pi$
$608$	0	0
$609$	−4.60991e6	−0.503674
$610$	0	0
$611$	2.97787e6	0.322703
$612$	0	0
$613$	1.74997e7	1.88096	0.940481	−	0.339846i	$-0.110375\pi$
$613$	1.74997e7	1.88096	0.940481	+	0.339846i	$0.110375\pi$
$614$	0	0
$615$	3.60847e6	0.384712
$616$	0	0
$617$	−4.51576e6	−0.477549	−0.238775	−	0.971075i	$-0.576746\pi$
$617$	−4.51576e6	−0.477549	−0.238775	+	0.971075i	$0.576746\pi$
$618$	0	0
$619$	−8.89914e6	−0.933516	−0.466758	−	0.884385i	$-0.654578\pi$
$619$	−8.89914e6	−0.933516	−0.466758	+	0.884385i	$0.654578\pi$
$620$	0	0
$621$	−1.09353e7	−1.13789
$622$	0	0
$623$	1.71294e6	0.176816
$624$	0	0
$625$	4.35285e6	0.445731
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.31603e7	1.32630
$630$	0	0
$631$	1.96157e7	1.96124	0.980619	−	0.195926i	$-0.0627712\pi$
$631$	1.96157e7	1.96124	0.980619	+	0.195926i	$0.0627712\pi$
$632$	0	0
$633$	−7.23646e6	−0.717823
$634$	0	0
$635$	−3.48343e6	−0.342825
$636$	0	0
$637$	4.67379e6	0.456374
$638$	0	0
$639$	−7.41115e6	−0.718015
$640$	0	0
$641$	−1.94247e7	−1.86728	−0.933640	−	0.358213i	$-0.883386\pi$
$641$	−1.94247e7	−1.86728	−0.933640	+	0.358213i	$0.883386\pi$
$642$	0	0
$643$	−1.25381e7	−1.19592	−0.597961	−	0.801525i	$-0.704022\pi$
$643$	−1.25381e7	−1.19592	−0.597961	+	0.801525i	$0.704022\pi$
$644$	0	0
$645$	38994.1	0.00369063
$646$	0	0
$647$	8.50738e6	0.798979	0.399489	−	0.916738i	$-0.369187\pi$
$647$	8.50738e6	0.798979	0.399489	+	0.916738i	$0.369187\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	433867.	0.0401240
$652$	0	0
$653$	−1.85022e7	−1.69801	−0.849007	−	0.528381i	$-0.822799\pi$
$653$	−1.85022e7	−1.69801	−0.849007	+	0.528381i	$0.822799\pi$
$654$	0	0
$655$	−9.35044e6	−0.851587
$656$	0	0
$657$	378101.	0.0341739
$658$	0	0
$659$	3.09897e6	0.277974	0.138987	−	0.990294i	$-0.455615\pi$
$659$	3.09897e6	0.277974	0.138987	+	0.990294i	$0.455615\pi$
$660$	0	0
$661$	−1.64845e7	−1.46748	−0.733738	−	0.679432i	$-0.762226\pi$
$661$	−1.64845e7	−1.46748	−0.733738	+	0.679432i	$0.762226\pi$
$662$	0	0
$663$	−1.10833e7	−0.979232
$664$	0	0
$665$	−3.83389e6	−0.336190
$666$	0	0
$667$	−1.37866e7	−1.19990
$668$	0	0
$669$	−401171.	−0.0346548
$670$	0	0
$671$	0	0
$672$	0	0
$673$	3.66540e6	0.311949	0.155975	−	0.987761i	$-0.450148\pi$
$673$	3.66540e6	0.311949	0.155975	+	0.987761i	$0.450148\pi$
$674$	0	0
$675$	1.02400e7	0.865049
$676$	0	0
$677$	−5.96764e6	−0.500416	−0.250208	−	0.968192i	$-0.580499\pi$
$677$	−5.96764e6	−0.500416	−0.250208	+	0.968192i	$0.580499\pi$
$678$	0	0
$679$	−1.19273e6	−0.0992816
$680$	0	0
$681$	7.88374e6	0.651425
$682$	0	0
$683$	−1.65497e7	−1.35750	−0.678749	−	0.734371i	$-0.737477\pi$
$683$	−1.65497e7	−1.35750	−0.678749	+	0.734371i	$0.737477\pi$
$684$	0	0
$685$	−7.80798e6	−0.635788
$686$	0	0
$687$	1.07120e7	0.865922
$688$	0	0
$689$	3.22511e6	0.258819
$690$	0	0
$691$	1.57316e7	1.25336	0.626682	−	0.779275i	$-0.284413\pi$
$691$	1.57316e7	1.25336	0.626682	+	0.779275i	$0.284413\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.20567e6	−0.0946814
$696$	0	0
$697$	−2.75352e7	−2.14687
$698$	0	0
$699$	−1.69482e7	−1.31199
$700$	0	0
$701$	2.47771e7	1.90439	0.952194	−	0.305494i	$-0.0988215\pi$
$701$	2.47771e7	1.90439	0.952194	+	0.305494i	$0.0988215\pi$
$702$	0	0
$703$	1.19501e7	0.911979
$704$	0	0
$705$	2.03085e6	0.153888
$706$	0	0
$707$	−1.55706e7	−1.17154
$708$	0	0
$709$	1.94898e7	1.45610	0.728051	−	0.685523i	$-0.240426\pi$
$709$	1.94898e7	1.45610	0.728051	+	0.685523i	$0.240426\pi$
$710$	0	0
$711$	−7.70269e6	−0.571437
$712$	0	0
$713$	1.29754e6	0.0955868
$714$	0	0
$715$	0	0
$716$	0	0
$717$	515151.	0.0374228
$718$	0	0
$719$	−376567.	−0.0271656	−0.0135828	−	0.999908i	$-0.504324\pi$
$719$	−376567.	−0.0271656	−0.0135828	+	0.999908i	$0.504324\pi$
$720$	0	0
$721$	1.32789e7	0.951312
$722$	0	0
$723$	−9.83799e6	−0.699940
$724$	0	0
$725$	1.29101e7	0.912187
$726$	0	0
$727$	1.23377e7	0.865760	0.432880	−	0.901452i	$-0.357497\pi$
$727$	1.23377e7	0.865760	0.432880	+	0.901452i	$0.357497\pi$
$728$	0	0
$729$	1.39519e7	0.972330
$730$	0	0
$731$	−297553.	−0.0205954
$732$	0	0
$733$	−1.73757e7	−1.19449	−0.597244	−	0.802059i	$-0.703738\pi$
$733$	−1.73757e7	−1.19449	−0.597244	+	0.802059i	$0.703738\pi$
$734$	0	0
$735$	3.18743e6	0.217632
$736$	0	0
$737$	0	0
$738$	0	0
$739$	2.86908e6	0.193255	0.0966276	−	0.995321i	$-0.469194\pi$
$739$	2.86908e6	0.193255	0.0966276	+	0.995321i	$0.469194\pi$
$740$	0	0
$741$	−1.00641e7	−0.673333
$742$	0	0
$743$	−8.22217e6	−0.546405	−0.273202	−	0.961957i	$-0.588083\pi$
$743$	−8.22217e6	−0.546405	−0.273202	+	0.961957i	$0.588083\pi$
$744$	0	0
$745$	−9.95461e6	−0.657103
$746$	0	0
$747$	−5.93476e6	−0.389136
$748$	0	0
$749$	−3.87048e6	−0.252092
$750$	0	0
$751$	9.55661e6	0.618307	0.309153	−	0.951012i	$-0.399954\pi$
$751$	9.55661e6	0.618307	0.309153	+	0.951012i	$0.399954\pi$
$752$	0	0
$753$	3.35777e6	0.215806
$754$	0	0
$755$	−242657.	−0.0154926
$756$	0	0
$757$	−2.34735e7	−1.48880	−0.744402	−	0.667732i	$-0.767265\pi$
$757$	−2.34735e7	−1.48880	−0.744402	+	0.667732i	$0.767265\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.40528e7	0.879634	0.439817	−	0.898087i	$-0.355043\pi$
$761$	1.40528e7	0.879634	0.439817	+	0.898087i	$0.355043\pi$
$762$	0	0
$763$	−1.14239e7	−0.710400
$764$	0	0
$765$	5.85393e6	0.361655
$766$	0	0
$767$	−4.29207e6	−0.263438
$768$	0	0
$769$	−2.45947e7	−1.49977	−0.749887	−	0.661566i	$-0.769892\pi$
$769$	−2.45947e7	−1.49977	−0.749887	+	0.661566i	$0.769892\pi$
$770$	0	0
$771$	−1.93990e6	−0.117529
$772$	0	0
$773$	−3.40980e6	−0.205249	−0.102624	−	0.994720i	$-0.532724\pi$
$773$	−3.40980e6	−0.205249	−0.102624	+	0.994720i	$0.532724\pi$
$774$	0	0
$775$	−1.21505e6	−0.0726672
$776$	0	0
$777$	5.30635e6	0.315314
$778$	0	0
$779$	−2.50031e7	−1.47622
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.10359e7	1.22619
$784$	0	0
$785$	−1.05535e7	−0.611257
$786$	0	0
$787$	−2.70222e7	−1.55519	−0.777595	−	0.628766i	$-0.783560\pi$
$787$	−2.70222e7	−1.55519	−0.777595	+	0.628766i	$0.783560\pi$
$788$	0	0
$789$	−371378.	−0.0212385
$790$	0	0
$791$	2.49544e6	0.141810
$792$	0	0
$793$	−7.59450e6	−0.428861
$794$	0	0
$795$	2.19946e6	0.123424
$796$	0	0
$797$	1.32102e7	0.736654	0.368327	−	0.929696i	$-0.379931\pi$
$797$	1.32102e7	0.736654	0.368327	+	0.929696i	$0.379931\pi$
$798$	0	0
$799$	−1.54968e7	−0.858768
$800$	0	0
$801$	−2.37496e6	−0.130790
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−5.09129e6	−0.276910
$806$	0	0
$807$	−2.34297e7	−1.26643
$808$	0	0
$809$	1.26721e7	0.680735	0.340368	−	0.940292i	$-0.389448\pi$
$809$	1.26721e7	0.680735	0.340368	+	0.940292i	$0.389448\pi$
$810$	0	0
$811$	1.27417e7	0.680262	0.340131	−	0.940378i	$-0.389529\pi$
$811$	1.27417e7	0.680262	0.340131	+	0.940378i	$0.389529\pi$
$812$	0	0
$813$	2.18621e7	1.16002
$814$	0	0
$815$	4.21888e6	0.222486
$816$	0	0
$817$	−270191.	−0.0141617
$818$	0	0
$819$	3.46103e6	0.180300
$820$	0	0
$821$	−2.86210e7	−1.48193	−0.740963	−	0.671546i	$-0.765631\pi$
$821$	−2.86210e7	−1.48193	−0.740963	+	0.671546i	$0.765631\pi$
$822$	0	0
$823$	6.12271e6	0.315097	0.157549	−	0.987511i	$-0.449641\pi$
$823$	6.12271e6	0.315097	0.157549	+	0.987511i	$0.449641\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.26024e7	1.14919	0.574593	−	0.818440i	$-0.305161\pi$
$827$	2.26024e7	1.14919	0.574593	+	0.818440i	$0.305161\pi$
$828$	0	0
$829$	8.18153e6	0.413474	0.206737	−	0.978397i	$-0.433716\pi$
$829$	8.18153e6	0.413474	0.206737	+	0.978397i	$0.433716\pi$
$830$	0	0
$831$	−6.46935e6	−0.324981
$832$	0	0
$833$	−2.43223e7	−1.21449
$834$	0	0
$835$	−3.23351e6	−0.160494
$836$	0	0
$837$	−1.97981e6	−0.0976812
$838$	0	0
$839$	−3.40904e7	−1.67197	−0.835983	−	0.548756i	$-0.815102\pi$
$839$	−3.40904e7	−1.67197	−0.835983	+	0.548756i	$0.815102\pi$
$840$	0	0
$841$	6.00986e6	0.293005
$842$	0	0
$843$	2.15536e7	1.04460
$844$	0	0
$845$	4.70627e6	0.226743
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.83592e6	0.0874147
$850$	0	0
$851$	1.58694e7	0.751169
$852$	0	0
$853$	1.77945e7	0.837361	0.418680	−	0.908134i	$-0.362493\pi$
$853$	1.77945e7	0.837361	0.418680	+	0.908134i	$0.362493\pi$
$854$	0	0
$855$	5.31562e6	0.248679
$856$	0	0
$857$	184993.	0.00860404	0.00430202	−	0.999991i	$-0.498631\pi$
$857$	184993.	0.00860404	0.00430202	+	0.999991i	$0.498631\pi$
$858$	0	0
$859$	−1.38415e7	−0.640031	−0.320015	−	0.947412i	$-0.603688\pi$
$859$	−1.38415e7	−0.640031	−0.320015	+	0.947412i	$0.603688\pi$
$860$	0	0
$861$	−1.11024e7	−0.510398
$862$	0	0
$863$	−1.89425e7	−0.865787	−0.432893	−	0.901445i	$-0.642507\pi$
$863$	−1.89425e7	−0.865787	−0.432893	+	0.901445i	$0.642507\pi$
$864$	0	0
$865$	−1.41516e6	−0.0643082
$866$	0	0
$867$	4.10619e7	1.85520
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−1.58904e7	−0.709722
$872$	0	0
$873$	1.65370e6	0.0734382
$874$	0	0
$875$	1.07107e7	0.472931
$876$	0	0
$877$	−5.68223e6	−0.249471	−0.124735	−	0.992190i	$-0.539808\pi$
$877$	−5.68223e6	−0.249471	−0.124735	+	0.992190i	$0.539808\pi$
$878$	0	0
$879$	8.20033e6	0.357980
$880$	0	0
$881$	4.81474e6	0.208994	0.104497	−	0.994525i	$-0.466677\pi$
$881$	4.81474e6	0.208994	0.104497	+	0.994525i	$0.466677\pi$
$882$	0	0
$883$	−3.36855e7	−1.45392	−0.726962	−	0.686678i	$-0.759068\pi$
$883$	−3.36855e7	−1.45392	−0.726962	+	0.686678i	$0.759068\pi$
$884$	0	0
$885$	−2.92710e6	−0.125626
$886$	0	0
$887$	−1.63501e7	−0.697768	−0.348884	−	0.937166i	$-0.613439\pi$
$887$	−1.63501e7	−0.697768	−0.348884	+	0.937166i	$0.613439\pi$
$888$	0	0
$889$	1.07177e7	0.454827
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.40718e7	−0.590500
$894$	0	0
$895$	−8.00874e6	−0.334200
$896$	0	0
$897$	−1.33648e7	−0.554604
$898$	0	0
$899$	−2.49605e6	−0.103004
$900$	0	0
$901$	−1.67835e7	−0.688763
$902$	0	0
$903$	−119976.	−0.00489637
$904$	0	0
$905$	6.89421e6	0.279810
$906$	0	0
$907$	4.47189e7	1.80498	0.902490	−	0.430710i	$-0.141737\pi$
$907$	4.47189e7	1.80498	0.902490	+	0.430710i	$0.141737\pi$
$908$	0	0
$909$	2.15884e7	0.866582
$910$	0	0
$911$	−2.19624e7	−0.876767	−0.438383	−	0.898788i	$-0.644449\pi$
$911$	−2.19624e7	−0.876767	−0.438383	+	0.898788i	$0.644449\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−5.17929e6	−0.204512
$916$	0	0
$917$	2.87691e7	1.12980
$918$	0	0
$919$	2.70003e7	1.05458	0.527290	−	0.849686i	$-0.323208\pi$
$919$	2.70003e7	1.05458	0.527290	+	0.849686i	$0.323208\pi$
$920$	0	0
$921$	−1.24248e7	−0.482658
$922$	0	0
$923$	−2.98108e7	−1.15178
$924$	0	0
$925$	−1.48605e7	−0.571055
$926$	0	0
$927$	−1.84109e7	−0.703682
$928$	0	0
$929$	−4.39700e6	−0.167154	−0.0835770	−	0.996501i	$-0.526634\pi$
$929$	−4.39700e6	−0.167154	−0.0835770	+	0.996501i	$0.526634\pi$
$930$	0	0
$931$	−2.20857e7	−0.835098
$932$	0	0
$933$	3.18911e7	1.19940
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.77615e6	0.289345	0.144672	−	0.989480i	$-0.453787\pi$
$937$	7.77615e6	0.289345	0.144672	+	0.989480i	$0.453787\pi$
$938$	0	0
$939$	2.71147e7	1.00355
$940$	0	0
$941$	1.84363e7	0.678736	0.339368	−	0.940654i	$-0.389787\pi$
$941$	1.84363e7	0.678736	0.339368	+	0.940654i	$0.389787\pi$
$942$	0	0
$943$	−3.32034e7	−1.21592
$944$	0	0
$945$	7.76838e6	0.282977
$946$	0	0
$947$	1.27595e7	0.462338	0.231169	−	0.972914i	$-0.425745\pi$
$947$	1.27595e7	0.462338	0.231169	+	0.972914i	$0.425745\pi$
$948$	0	0
$949$	1.52088e6	0.0548190
$950$	0	0
$951$	−3.27595e7	−1.17459
$952$	0	0
$953$	4.12806e6	0.147236	0.0736179	−	0.997287i	$-0.476545\pi$
$953$	4.12806e6	0.147236	0.0736179	+	0.997287i	$0.476545\pi$
$954$	0	0
$955$	−2.14182e7	−0.759932
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.40233e7	0.843502
$960$	0	0
$961$	−2.83942e7	−0.991794
$962$	0	0
$963$	5.36635e6	0.186472
$964$	0	0
$965$	1.53936e7	0.532137
$966$	0	0
$967$	2.59567e7	0.892655	0.446328	−	0.894870i	$-0.352732\pi$
$967$	2.59567e7	0.892655	0.446328	+	0.894870i	$0.352732\pi$
$968$	0	0
$969$	5.23735e7	1.79185
$970$	0	0
$971$	3.47708e7	1.18350	0.591748	−	0.806123i	$-0.298438\pi$
$971$	3.47708e7	1.18350	0.591748	+	0.806123i	$0.298438\pi$
$972$	0	0
$973$	3.70955e6	0.125614
$974$	0	0
$975$	1.25151e7	0.421622
$976$	0	0
$977$	−2.27258e7	−0.761698	−0.380849	−	0.924637i	$-0.624368\pi$
$977$	−2.27258e7	−0.761698	−0.380849	+	0.924637i	$0.624368\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1.58390e7	0.525480
$982$	0	0
$983$	4.08225e7	1.34746	0.673730	−	0.738977i	$-0.264691\pi$
$983$	4.08225e7	1.34746	0.673730	+	0.738977i	$0.264691\pi$
$984$	0	0
$985$	−3.87875e6	−0.127380
$986$	0	0
$987$	−6.24844e6	−0.204164
$988$	0	0
$989$	−358805.	−0.0116646
$990$	0	0
$991$	5.50363e7	1.78019	0.890093	−	0.455779i	$-0.150639\pi$
$991$	5.50363e7	1.78019	0.890093	+	0.455779i	$0.150639\pi$
$992$	0	0
$993$	−1.17231e7	−0.377284
$994$	0	0
$995$	−1.68964e7	−0.541048
$996$	0	0
$997$	6.31626e6	0.201244	0.100622	−	0.994925i	$-0.467917\pi$
$997$	6.31626e6	0.201244	0.100622	+	0.994925i	$0.467917\pi$
$998$	0	0
$999$	−2.42139e7	−0.767627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.6.a.p.1.11		16
11.7	odd	10		88.6.i.b.49.3	yes	32
11.8	odd	10		88.6.i.b.9.3	✓	32
11.10	odd	2		968.6.a.q.1.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.6.i.b.9.3	✓	32	11.8	odd	10
88.6.i.b.49.3	yes	32	11.7	odd	10
968.6.a.p.1.11		16	1.1	even	1	trivial
968.6.a.q.1.11		16	11.10	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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)

\(f(q)\)	\(=\)	\(q+11.7022 q^{3} -24.8620 q^{5} +76.4943 q^{7} -106.058 q^{9} -426.611 q^{13} -290.940 q^{15} +2220.08 q^{17} +2015.93 q^{19} +895.154 q^{21} +2677.09 q^{23} -2506.88 q^{25} -4084.75 q^{27} -5149.86 q^{29} +484.684 q^{31} -1901.80 q^{35} +5927.86 q^{37} -4992.30 q^{39} -12402.8 q^{41} -134.028 q^{43} +2636.81 q^{45} -6980.30 q^{47} -10955.6 q^{49} +25979.9 q^{51} -7559.84 q^{53} +23590.8 q^{57} +10060.8 q^{59} +17801.9 q^{61} -8112.84 q^{63} +10606.4 q^{65} +37247.9 q^{67} +31327.9 q^{69} +69878.2 q^{71} -3565.04 q^{73} -29336.1 q^{75} +72627.1 q^{79} -22028.6 q^{81} +55957.6 q^{83} -55195.6 q^{85} -60264.7 q^{87} +22393.0 q^{89} -32633.3 q^{91} +5671.88 q^{93} -50119.9 q^{95} -15592.4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{3} + 81 q^{5} - 47 q^{7} + 1416 q^{9} + 859 q^{13} - 738 q^{15} + 1226 q^{17} + 616 q^{19} + 1141 q^{21} + 2258 q^{23} + 10307 q^{25} + 564 q^{27} + 1613 q^{29} + 18511 q^{31} - 23544 q^{35}+ \cdots + 171314 q^{97}+O(q^{100}) \) 16 * q + 12 * q^3 + 81 * q^5 - 47 * q^7 + 1416 * q^9 + 859 * q^13 - 738 * q^15 + 1226 * q^17 + 616 * q^19 + 1141 * q^21 + 2258 * q^23 + 10307 * q^25 + 564 * q^27 + 1613 * q^29 + 18511 * q^31 - 23544 * q^35 + 8553 * q^37 + 12020 * q^39 + 25514 * q^41 - 18931 * q^43 + 23278 * q^45 + 38781 * q^47 + 94523 * q^49 - 68584 * q^51 + 779 * q^53 + 19293 * q^57 + 50316 * q^59 + 53153 * q^61 - 54911 * q^63 + 51517 * q^65 + 10491 * q^67 - 95356 * q^69 - 91677 * q^71 + 97660 * q^73 - 3771 * q^75 - 96003 * q^79 + 120400 * q^81 + 4552 * q^83 - 129679 * q^85 + 77219 * q^87 + 6697 * q^89 + 217380 * q^91 + 132586 * q^93 - 54168 * q^95 + 171314 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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