LMFDB - Embedded newform 968.6.a.q.1.3

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			1.3
Root			$-25.8955$ 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)

$f(q)$	$=$	$q-24.2774 q^{3} -51.2310 q^{5} +25.3165 q^{7} +346.393 q^{9} +228.665 q^{13} +1243.76 q^{15} +1040.05 q^{17} +395.949 q^{19} -614.619 q^{21} -437.026 q^{23} -500.388 q^{25} -2510.12 q^{27} +2050.79 q^{29} +7467.26 q^{31} -1296.99 q^{35} +9584.87 q^{37} -5551.40 q^{39} -7727.93 q^{41} +19690.7 q^{43} -17746.0 q^{45} -12140.0 q^{47} -16166.1 q^{49} -25249.7 q^{51} -31880.3 q^{53} -9612.61 q^{57} -51200.6 q^{59} +44055.5 q^{61} +8769.45 q^{63} -11714.7 q^{65} -15640.4 q^{67} +10609.9 q^{69} -51353.9 q^{71} -69851.9 q^{73} +12148.1 q^{75} +104942. q^{79} -23234.4 q^{81} +72448.3 q^{83} -53282.8 q^{85} -49787.8 q^{87} -25728.0 q^{89} +5788.99 q^{91} -181286. q^{93} -20284.8 q^{95} -2875.42 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−24.2774	−1.55740	−0.778699	−	0.627398i	$-0.784120\pi$
$3$	−24.2774	−1.55740	−0.778699	+	0.627398i	$0.784120\pi$
$4$	0	0
$5$	−51.2310	−0.916447	−0.458224	−	0.888837i	$-0.651514\pi$
$5$	−51.2310	−0.916447	−0.458224	+	0.888837i	$0.651514\pi$
$6$	0	0
$7$	25.3165	0.195280	0.0976401	−	0.995222i	$-0.468871\pi$
$7$	25.3165	0.195280	0.0976401	+	0.995222i	$0.468871\pi$
$8$	0	0
$9$	346.393	1.42549
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	228.665	0.375268	0.187634	−	0.982239i	$-0.439918\pi$
$13$	228.665	0.375268	0.187634	+	0.982239i	$0.439918\pi$
$14$	0	0
$15$	1243.76	1.42727
$16$	0	0
$17$	1040.05	0.872835	0.436418	−	0.899744i	$-0.356247\pi$
$17$	1040.05	0.872835	0.436418	+	0.899744i	$0.356247\pi$
$18$	0	0
$19$	395.949	0.251626	0.125813	−	0.992054i	$-0.459846\pi$
$19$	395.949	0.251626	0.125813	+	0.992054i	$0.459846\pi$
$20$	0	0
$21$	−614.619	−0.304129
$22$	0	0
$23$	−437.026	−0.172261	−0.0861306	−	0.996284i	$-0.527450\pi$
$23$	−437.026	−0.172261	−0.0861306	+	0.996284i	$0.527450\pi$
$24$	0	0
$25$	−500.388	−0.160124
$26$	0	0
$27$	−2510.12	−0.662650
$28$	0	0
$29$	2050.79	0.452820	0.226410	−	0.974032i	$-0.427301\pi$
$29$	2050.79	0.452820	0.226410	+	0.974032i	$0.427301\pi$
$30$	0	0
$31$	7467.26	1.39559	0.697794	−	0.716298i	$-0.254165\pi$
$31$	7467.26	1.39559	0.697794	+	0.716298i	$0.254165\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1296.99	−0.178964
$36$	0	0
$37$	9584.87	1.15102	0.575509	−	0.817796i	$-0.304804\pi$
$37$	9584.87	1.15102	0.575509	+	0.817796i	$0.304804\pi$
$38$	0	0
$39$	−5551.40	−0.584441
$40$	0	0
$41$	−7727.93	−0.717965	−0.358983	−	0.933344i	$-0.616876\pi$
$41$	−7727.93	−0.717965	−0.358983	+	0.933344i	$0.616876\pi$
$42$	0	0
$43$	19690.7	1.62402	0.812009	−	0.583645i	$-0.198374\pi$
$43$	19690.7	1.62402	0.812009	+	0.583645i	$0.198374\pi$
$44$	0	0
$45$	−17746.0	−1.30638
$46$	0	0
$47$	−12140.0	−0.801632	−0.400816	−	0.916159i	$-0.631273\pi$
$47$	−12140.0	−0.801632	−0.400816	+	0.916159i	$0.631273\pi$
$48$	0	0
$49$	−16166.1	−0.961866
$50$	0	0
$51$	−25249.7	−1.35935
$52$	0	0
$53$	−31880.3	−1.55895	−0.779477	−	0.626431i	$-0.784515\pi$
$53$	−31880.3	−1.55895	−0.779477	+	0.626431i	$0.784515\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9612.61	−0.391881
$58$	0	0
$59$	−51200.6	−1.91490	−0.957448	−	0.288606i	$-0.906808\pi$
$59$	−51200.6	−1.91490	−0.957448	+	0.288606i	$0.906808\pi$
$60$	0	0
$61$	44055.5	1.51592	0.757959	−	0.652302i	$-0.226196\pi$
$61$	44055.5	1.51592	0.757959	+	0.652302i	$0.226196\pi$
$62$	0	0
$63$	8769.45	0.278369
$64$	0	0
$65$	−11714.7	−0.343913
$66$	0	0
$67$	−15640.4	−0.425657	−0.212829	−	0.977090i	$-0.568268\pi$
$67$	−15640.4	−0.425657	−0.212829	+	0.977090i	$0.568268\pi$
$68$	0	0
$69$	10609.9	0.268279
$70$	0	0
$71$	−51353.9	−1.20900	−0.604502	−	0.796604i	$-0.706628\pi$
$71$	−51353.9	−1.20900	−0.604502	+	0.796604i	$0.706628\pi$
$72$	0	0
$73$	−69851.9	−1.53416	−0.767080	−	0.641551i	$-0.778291\pi$
$73$	−69851.9	−1.53416	−0.767080	+	0.641551i	$0.778291\pi$
$74$	0	0
$75$	12148.1	0.249377
$76$	0	0
$77$	0	0
$78$	0	0
$79$	104942.	1.89182	0.945911	−	0.324425i	$-0.105171\pi$
$79$	104942.	1.89182	0.945911	+	0.324425i	$0.105171\pi$
$80$	0	0
$81$	−23234.4	−0.393476
$82$	0	0
$83$	72448.3	1.15434	0.577169	−	0.816625i	$-0.304157\pi$
$83$	72448.3	1.15434	0.577169	+	0.816625i	$0.304157\pi$
$84$	0	0
$85$	−53282.8	−0.799907
$86$	0	0
$87$	−49787.8	−0.705220
$88$	0	0
$89$	−25728.0	−0.344295	−0.172147	−	0.985071i	$-0.555071\pi$
$89$	−25728.0	−0.344295	−0.172147	+	0.985071i	$0.555071\pi$
$90$	0	0
$91$	5788.99	0.0732824
$92$	0	0
$93$	−181286.	−2.17349
$94$	0	0
$95$	−20284.8	−0.230602
$96$	0	0
$97$	−2875.42	−0.0310293	−0.0155146	−	0.999880i	$-0.504939\pi$
$97$	−2875.42	−0.0310293	−0.0155146	+	0.999880i	$0.504939\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	22960.6	0.223964	0.111982	−	0.993710i	$-0.464280\pi$
$101$	22960.6	0.223964	0.111982	+	0.993710i	$0.464280\pi$
$102$	0	0
$103$	−105365.	−0.978598	−0.489299	−	0.872116i	$-0.662747\pi$
$103$	−105365.	−0.978598	−0.489299	+	0.872116i	$0.662747\pi$
$104$	0	0
$105$	31487.5	0.278718
$106$	0	0
$107$	−65670.7	−0.554514	−0.277257	−	0.960796i	$-0.589425\pi$
$107$	−65670.7	−0.554514	−0.277257	+	0.960796i	$0.589425\pi$
$108$	0	0
$109$	−26662.1	−0.214945	−0.107473	−	0.994208i	$-0.534276\pi$
$109$	−26662.1	−0.214945	−0.107473	+	0.994208i	$0.534276\pi$
$110$	0	0
$111$	−232696.	−1.79259
$112$	0	0
$113$	132872.	0.978901	0.489451	−	0.872031i	$-0.337197\pi$
$113$	132872.	0.978901	0.489451	+	0.872031i	$0.337197\pi$
$114$	0	0
$115$	22389.2	0.157868
$116$	0	0
$117$	79208.0	0.534939
$118$	0	0
$119$	26330.4	0.170447
$120$	0	0
$121$	0	0
$122$	0	0
$123$	187614.	1.11816
$124$	0	0
$125$	185732.	1.06319
$126$	0	0
$127$	−134003.	−0.737232	−0.368616	−	0.929582i	$-0.620168\pi$
$127$	−134003.	−0.737232	−0.368616	+	0.929582i	$0.620168\pi$
$128$	0	0
$129$	−478040.	−2.52924
$130$	0	0
$131$	46383.6	0.236149	0.118074	−	0.993005i	$-0.462328\pi$
$131$	46383.6	0.236149	0.118074	+	0.993005i	$0.462328\pi$
$132$	0	0
$133$	10024.0	0.0491375
$134$	0	0
$135$	128596.	0.607284
$136$	0	0
$137$	−219983.	−1.00135	−0.500677	−	0.865634i	$-0.666915\pi$
$137$	−219983.	−1.00135	−0.500677	+	0.865634i	$0.666915\pi$
$138$	0	0
$139$	−68837.3	−0.302195	−0.151097	−	0.988519i	$-0.548281\pi$
$139$	−68837.3	−0.302195	−0.151097	+	0.988519i	$0.548281\pi$
$140$	0	0
$141$	294728.	1.24846
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−105064.	−0.414986
$146$	0	0
$147$	392471.	1.49801
$148$	0	0
$149$	291333.	1.07504	0.537519	−	0.843252i	$-0.319362\pi$
$149$	291333.	1.07504	0.537519	+	0.843252i	$0.319362\pi$
$150$	0	0
$151$	85539.5	0.305298	0.152649	−	0.988280i	$-0.451220\pi$
$151$	85539.5	0.305298	0.152649	+	0.988280i	$0.451220\pi$
$152$	0	0
$153$	360266.	1.24421
$154$	0	0
$155$	−382555.	−1.27898
$156$	0	0
$157$	109325.	0.353974	0.176987	−	0.984213i	$-0.443365\pi$
$157$	109325.	0.353974	0.176987	+	0.984213i	$0.443365\pi$
$158$	0	0
$159$	773972.	2.42791
$160$	0	0
$161$	−11063.9	−0.0336392
$162$	0	0
$163$	116176.	0.342490	0.171245	−	0.985229i	$-0.445221\pi$
$163$	116176.	0.342490	0.171245	+	0.985229i	$0.445221\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	320008.	0.887912	0.443956	−	0.896049i	$-0.353575\pi$
$167$	320008.	0.887912	0.443956	+	0.896049i	$0.353575\pi$
$168$	0	0
$169$	−319005.	−0.859174
$170$	0	0
$171$	137154.	0.358689
$172$	0	0
$173$	349469.	0.887756	0.443878	−	0.896087i	$-0.353602\pi$
$173$	349469.	0.887756	0.443878	+	0.896087i	$0.353602\pi$
$174$	0	0
$175$	−12668.1	−0.0312691
$176$	0	0
$177$	1.24302e6	2.98225
$178$	0	0
$179$	703286.	1.64059	0.820294	−	0.571942i	$-0.193810\pi$
$179$	703286.	1.64059	0.820294	+	0.571942i	$0.193810\pi$
$180$	0	0
$181$	−172161.	−0.390606	−0.195303	−	0.980743i	$-0.562569\pi$
$181$	−172161.	−0.390606	−0.195303	+	0.980743i	$0.562569\pi$
$182$	0	0
$183$	−1.06955e6	−2.36089
$184$	0	0
$185$	−491042.	−1.05485
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−63547.3	−0.129402
$190$	0	0
$191$	−93405.3	−0.185263	−0.0926313	−	0.995700i	$-0.529528\pi$
$191$	−93405.3	−0.185263	−0.0926313	+	0.995700i	$0.529528\pi$
$192$	0	0
$193$	−440885.	−0.851985	−0.425992	−	0.904727i	$-0.640075\pi$
$193$	−440885.	−0.851985	−0.425992	+	0.904727i	$0.640075\pi$
$194$	0	0
$195$	284404.	0.535610
$196$	0	0
$197$	−990541.	−1.81847	−0.909236	−	0.416280i	$-0.863334\pi$
$197$	−990541.	−1.81847	−0.909236	+	0.416280i	$0.863334\pi$
$198$	0	0
$199$	526698.	0.942820	0.471410	−	0.881914i	$-0.343745\pi$
$199$	526698.	0.942820	0.471410	+	0.881914i	$0.343745\pi$
$200$	0	0
$201$	379708.	0.662917
$202$	0	0
$203$	51918.7	0.0884267
$204$	0	0
$205$	395909.	0.657977
$206$	0	0
$207$	−151383.	−0.245556
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.28103e6	1.98086	0.990430	−	0.138018i	$-0.0440731\pi$
$211$	1.28103e6	1.98086	0.990430	+	0.138018i	$0.0440731\pi$
$212$	0	0
$213$	1.24674e6	1.88290
$214$	0	0
$215$	−1.00878e6	−1.48833
$216$	0	0
$217$	189045.	0.272531
$218$	0	0
$219$	1.69582e6	2.38930
$220$	0	0
$221$	237823.	0.327547
$222$	0	0
$223$	−374272.	−0.503993	−0.251997	−	0.967728i	$-0.581087\pi$
$223$	−374272.	−0.503993	−0.251997	+	0.967728i	$0.581087\pi$
$224$	0	0
$225$	−173331.	−0.228255
$226$	0	0
$227$	513515.	0.661437	0.330718	−	0.943730i	$-0.392709\pi$
$227$	513515.	0.661437	0.330718	+	0.943730i	$0.392709\pi$
$228$	0	0
$229$	−721422.	−0.909077	−0.454539	−	0.890727i	$-0.650196\pi$
$229$	−721422.	−0.909077	−0.454539	+	0.890727i	$0.650196\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−759805.	−0.916880	−0.458440	−	0.888725i	$-0.651592\pi$
$233$	−759805.	−0.916880	−0.458440	+	0.888725i	$0.651592\pi$
$234$	0	0
$235$	621945.	0.734653
$236$	0	0
$237$	−2.54771e6	−2.94632
$238$	0	0
$239$	1.67872e6	1.90100	0.950502	−	0.310718i	$-0.100570\pi$
$239$	1.67872e6	1.90100	0.950502	+	0.310718i	$0.100570\pi$
$240$	0	0
$241$	−476676.	−0.528666	−0.264333	−	0.964432i	$-0.585152\pi$
$241$	−476676.	−0.528666	−0.264333	+	0.964432i	$0.585152\pi$
$242$	0	0
$243$	1.17403e6	1.27545
$244$	0	0
$245$	828204.	0.881499
$246$	0	0
$247$	90539.6	0.0944270
$248$	0	0
$249$	−1.75886e6	−1.79776
$250$	0	0
$251$	−850177.	−0.851775	−0.425888	−	0.904776i	$-0.640038\pi$
$251$	−850177.	−0.851775	−0.425888	+	0.904776i	$0.640038\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.29357e6	1.24577
$256$	0	0
$257$	1.74706e6	1.64997	0.824985	−	0.565155i	$-0.191184\pi$
$257$	1.74706e6	1.64997	0.824985	+	0.565155i	$0.191184\pi$
$258$	0	0
$259$	242655.	0.224771
$260$	0	0
$261$	710378.	0.645488
$262$	0	0
$263$	477494.	0.425675	0.212838	−	0.977088i	$-0.431729\pi$
$263$	477494.	0.425675	0.212838	+	0.977088i	$0.431729\pi$
$264$	0	0
$265$	1.63326e6	1.42870
$266$	0	0
$267$	624609.	0.536204
$268$	0	0
$269$	−667486.	−0.562421	−0.281211	−	0.959646i	$-0.590736\pi$
$269$	−667486.	−0.562421	−0.281211	+	0.959646i	$0.590736\pi$
$270$	0	0
$271$	−374825.	−0.310031	−0.155016	−	0.987912i	$-0.549543\pi$
$271$	−374825.	−0.310031	−0.155016	+	0.987912i	$0.549543\pi$
$272$	0	0
$273$	−140542.	−0.114130
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−732728.	−0.573777	−0.286889	−	0.957964i	$-0.592621\pi$
$277$	−732728.	−0.573777	−0.286889	+	0.957964i	$0.592621\pi$
$278$	0	0
$279$	2.58661e6	1.98939
$280$	0	0
$281$	−247045.	−0.186643	−0.0933213	−	0.995636i	$-0.529748\pi$
$281$	−247045.	−0.186643	−0.0933213	+	0.995636i	$0.529748\pi$
$282$	0	0
$283$	1.63552e6	1.21392	0.606960	−	0.794732i	$-0.292389\pi$
$283$	1.63552e6	1.21392	0.606960	+	0.794732i	$0.292389\pi$
$284$	0	0
$285$	492463.	0.359138
$286$	0	0
$287$	−195644.	−0.140204
$288$	0	0
$289$	−338152.	−0.238159
$290$	0	0
$291$	69807.7	0.0483249
$292$	0	0
$293$	−1.20598e6	−0.820672	−0.410336	−	0.911935i	$-0.634588\pi$
$293$	−1.20598e6	−0.820672	−0.410336	+	0.911935i	$0.634588\pi$
$294$	0	0
$295$	2.62306e6	1.75490
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−99932.5	−0.0646441
$300$	0	0
$301$	498500.	0.317138
$302$	0	0
$303$	−557423.	−0.348802
$304$	0	0
$305$	−2.25701e6	−1.38926
$306$	0	0
$307$	1.99198e6	1.20625	0.603126	−	0.797646i	$-0.293921\pi$
$307$	1.99198e6	1.20625	0.603126	+	0.797646i	$0.293921\pi$
$308$	0	0
$309$	2.55800e6	1.52407
$310$	0	0
$311$	1.70723e6	1.00090	0.500451	−	0.865765i	$-0.333168\pi$
$311$	1.70723e6	1.00090	0.500451	+	0.865765i	$0.333168\pi$
$312$	0	0
$313$	2.38519e6	1.37614	0.688070	−	0.725644i	$-0.258458\pi$
$313$	2.38519e6	1.37614	0.688070	+	0.725644i	$0.258458\pi$
$314$	0	0
$315$	−449267.	−0.255111
$316$	0	0
$317$	1.16958e6	0.653704	0.326852	−	0.945076i	$-0.394012\pi$
$317$	1.16958e6	0.653704	0.326852	+	0.945076i	$0.394012\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	1.59432e6	0.863598
$322$	0	0
$323$	411807.	0.219628
$324$	0	0
$325$	−114421.	−0.0600895
$326$	0	0
$327$	647286.	0.334755
$328$	0	0
$329$	−307343.	−0.156543
$330$	0	0
$331$	2.07627e6	1.04163	0.520816	−	0.853669i	$-0.325628\pi$
$331$	2.07627e6	1.04163	0.520816	+	0.853669i	$0.325628\pi$
$332$	0	0
$333$	3.32013e6	1.64076
$334$	0	0
$335$	801271.	0.390092
$336$	0	0
$337$	−2.75353e6	−1.32073	−0.660367	−	0.750943i	$-0.729599\pi$
$337$	−2.75353e6	−1.32073	−0.660367	+	0.750943i	$0.729599\pi$
$338$	0	0
$339$	−3.22580e6	−1.52454
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−834762.	−0.383113
$344$	0	0
$345$	−543553.	−0.245864
$346$	0	0
$347$	3.57891e6	1.59561	0.797807	−	0.602913i	$-0.205994\pi$
$347$	3.57891e6	1.59561	0.797807	+	0.602913i	$0.205994\pi$
$348$	0	0
$349$	−2.23671e6	−0.982984	−0.491492	−	0.870882i	$-0.663548\pi$
$349$	−2.23671e6	−0.982984	−0.491492	+	0.870882i	$0.663548\pi$
$350$	0	0
$351$	−573976.	−0.248672
$352$	0	0
$353$	−1.45242e6	−0.620379	−0.310189	−	0.950675i	$-0.600392\pi$
$353$	−1.45242e6	−0.620379	−0.310189	+	0.950675i	$0.600392\pi$
$354$	0	0
$355$	2.63091e6	1.10799
$356$	0	0
$357$	−639234.	−0.265454
$358$	0	0
$359$	−2.44308e6	−1.00046	−0.500232	−	0.865891i	$-0.666752\pi$
$359$	−2.44308e6	−1.00046	−0.500232	+	0.865891i	$0.666752\pi$
$360$	0	0
$361$	−2.31932e6	−0.936685
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.57858e6	1.40598
$366$	0	0
$367$	841916.	0.326290	0.163145	−	0.986602i	$-0.447836\pi$
$367$	841916.	0.326290	0.163145	+	0.986602i	$0.447836\pi$
$368$	0	0
$369$	−2.67690e6	−1.02345
$370$	0	0
$371$	−807098.	−0.304433
$372$	0	0
$373$	2.81790e6	1.04870	0.524352	−	0.851501i	$-0.324307\pi$
$373$	2.81790e6	1.04870	0.524352	+	0.851501i	$0.324307\pi$
$374$	0	0
$375$	−4.50910e6	−1.65581
$376$	0	0
$377$	468943.	0.169929
$378$	0	0
$379$	−3.69372e6	−1.32089	−0.660443	−	0.750876i	$-0.729632\pi$
$379$	−3.69372e6	−1.32089	−0.660443	+	0.750876i	$0.729632\pi$
$380$	0	0
$381$	3.25324e6	1.14816
$382$	0	0
$383$	1.94993e6	0.679236	0.339618	−	0.940563i	$-0.389702\pi$
$383$	1.94993e6	0.679236	0.339618	+	0.940563i	$0.389702\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.82073e6	2.31501
$388$	0	0
$389$	−5.88126e6	−1.97059	−0.985296	−	0.170858i	$-0.945346\pi$
$389$	−5.88126e6	−1.97059	−0.985296	+	0.170858i	$0.945346\pi$
$390$	0	0
$391$	−454529.	−0.150356
$392$	0	0
$393$	−1.12607e6	−0.367778
$394$	0	0
$395$	−5.37626e6	−1.73376
$396$	0	0
$397$	−222102.	−0.0707254	−0.0353627	−	0.999375i	$-0.511259\pi$
$397$	−222102.	−0.0707254	−0.0353627	+	0.999375i	$0.511259\pi$
$398$	0	0
$399$	−243357.	−0.0765266
$400$	0	0
$401$	−2.81725e6	−0.874914	−0.437457	−	0.899239i	$-0.644121\pi$
$401$	−2.81725e6	−0.874914	−0.437457	+	0.899239i	$0.644121\pi$
$402$	0	0
$403$	1.70750e6	0.523720
$404$	0	0
$405$	1.19032e6	0.360600
$406$	0	0
$407$	0	0
$408$	0	0
$409$	4.11546e6	1.21649	0.608247	−	0.793748i	$-0.291873\pi$
$409$	4.11546e6	1.21649	0.608247	+	0.793748i	$0.291873\pi$
$410$	0	0
$411$	5.34062e6	1.55950
$412$	0	0
$413$	−1.29622e6	−0.373941
$414$	0	0
$415$	−3.71160e6	−1.05789
$416$	0	0
$417$	1.67119e6	0.470637
$418$	0	0
$419$	−3.25055e6	−0.904528	−0.452264	−	0.891884i	$-0.649383\pi$
$419$	−3.25055e6	−0.904528	−0.452264	+	0.891884i	$0.649383\pi$
$420$	0	0
$421$	5.31682e6	1.46200	0.730999	−	0.682379i	$-0.239054\pi$
$421$	5.31682e6	1.46200	0.730999	+	0.682379i	$0.239054\pi$
$422$	0	0
$423$	−4.20522e6	−1.14271
$424$	0	0
$425$	−520429.	−0.139762
$426$	0	0
$427$	1.11533e6	0.296029
$428$	0	0
$429$	0	0
$430$	0	0
$431$	925440.	0.239969	0.119985	−	0.992776i	$-0.461716\pi$
$431$	925440.	0.239969	0.119985	+	0.992776i	$0.461716\pi$
$432$	0	0
$433$	1.29866e6	0.332872	0.166436	−	0.986052i	$-0.446774\pi$
$433$	1.29866e6	0.332872	0.166436	+	0.986052i	$0.446774\pi$
$434$	0	0
$435$	2.55068e6	0.646297
$436$	0	0
$437$	−173040.	−0.0433453
$438$	0	0
$439$	−5.88898e6	−1.45841	−0.729203	−	0.684297i	$-0.760109\pi$
$439$	−5.88898e6	−1.45841	−0.729203	+	0.684297i	$0.760109\pi$
$440$	0	0
$441$	−5.59982e6	−1.37113
$442$	0	0
$443$	−3.75792e6	−0.909785	−0.454892	−	0.890546i	$-0.650322\pi$
$443$	−3.75792e6	−0.909785	−0.454892	+	0.890546i	$0.650322\pi$
$444$	0	0
$445$	1.31807e6	0.315528
$446$	0	0
$447$	−7.07281e6	−1.67426
$448$	0	0
$449$	3.35730e6	0.785912	0.392956	−	0.919557i	$-0.371453\pi$
$449$	3.35730e6	0.785912	0.392956	+	0.919557i	$0.371453\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−2.07668e6	−0.475470
$454$	0	0
$455$	−296576.	−0.0671595
$456$	0	0
$457$	3.44841e6	0.772374	0.386187	−	0.922420i	$-0.373792\pi$
$457$	3.44841e6	0.772374	0.386187	+	0.922420i	$0.373792\pi$
$458$	0	0
$459$	−2.61065e6	−0.578385
$460$	0	0
$461$	4.34027e6	0.951184	0.475592	−	0.879666i	$-0.342234\pi$
$461$	4.34027e6	0.951184	0.475592	+	0.879666i	$0.342234\pi$
$462$	0	0
$463$	3.20500e6	0.694825	0.347413	−	0.937712i	$-0.387060\pi$
$463$	3.20500e6	0.694825	0.347413	+	0.937712i	$0.387060\pi$
$464$	0	0
$465$	9.28745e6	1.99188
$466$	0	0
$467$	−236841.	−0.0502533	−0.0251267	−	0.999684i	$-0.507999\pi$
$467$	−236841.	−0.0502533	−0.0251267	+	0.999684i	$0.507999\pi$
$468$	0	0
$469$	−395959.	−0.0831224
$470$	0	0
$471$	−2.65414e6	−0.551278
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−198128.	−0.0402914
$476$	0	0
$477$	−1.10431e7	−2.22227
$478$	0	0
$479$	9.15886e6	1.82391	0.911953	−	0.410295i	$-0.134574\pi$
$479$	9.15886e6	1.82391	0.911953	+	0.410295i	$0.134574\pi$
$480$	0	0
$481$	2.19173e6	0.431940
$482$	0	0
$483$	268604.	0.0523896
$484$	0	0
$485$	147310.	0.0284367
$486$	0	0
$487$	−6.08400e6	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$487$	−6.08400e6	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$488$	0	0
$489$	−2.82045e6	−0.533392
$490$	0	0
$491$	−1.71572e6	−0.321176	−0.160588	−	0.987022i	$-0.551339\pi$
$491$	−1.71572e6	−0.321176	−0.160588	+	0.987022i	$0.551339\pi$
$492$	0	0
$493$	2.13292e6	0.395237
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.30010e6	−0.236094
$498$	0	0
$499$	4.03596e6	0.725598	0.362799	−	0.931867i	$-0.381821\pi$
$499$	4.03596e6	0.725598	0.362799	+	0.931867i	$0.381821\pi$
$500$	0	0
$501$	−7.76897e6	−1.38283
$502$	0	0
$503$	−2.97422e6	−0.524146	−0.262073	−	0.965048i	$-0.584406\pi$
$503$	−2.97422e6	−0.524146	−0.262073	+	0.965048i	$0.584406\pi$
$504$	0	0
$505$	−1.17629e6	−0.205252
$506$	0	0
$507$	7.74462e6	1.33807
$508$	0	0
$509$	912367.	0.156090	0.0780450	−	0.996950i	$-0.475132\pi$
$509$	912367.	0.156090	0.0780450	+	0.996950i	$0.475132\pi$
$510$	0	0
$511$	−1.76840e6	−0.299591
$512$	0	0
$513$	−993877.	−0.166740
$514$	0	0
$515$	5.39796e6	0.896833
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−8.48421e6	−1.38259
$520$	0	0
$521$	6.62217e6	1.06882	0.534412	−	0.845224i	$-0.320533\pi$
$521$	6.62217e6	1.06882	0.534412	+	0.845224i	$0.320533\pi$
$522$	0	0
$523$	939593.	0.150205	0.0751027	−	0.997176i	$-0.476072\pi$
$523$	939593.	0.150205	0.0751027	+	0.997176i	$0.476072\pi$
$524$	0	0
$525$	307548.	0.0486984
$526$	0	0
$527$	7.76633e6	1.21812
$528$	0	0
$529$	−6.24535e6	−0.970326
$530$	0	0
$531$	−1.77355e7	−2.72966
$532$	0	0
$533$	−1.76711e6	−0.269429
$534$	0	0
$535$	3.36437e6	0.508183
$536$	0	0
$537$	−1.70740e7	−2.55505
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5.64310e6	−0.828943	−0.414472	−	0.910062i	$-0.636034\pi$
$541$	−5.64310e6	−0.828943	−0.414472	+	0.910062i	$0.636034\pi$
$542$	0	0
$543$	4.17963e6	0.608329
$544$	0	0
$545$	1.36592e6	0.196986
$546$	0	0
$547$	7.06369e6	1.00940	0.504700	−	0.863295i	$-0.331603\pi$
$547$	7.06369e6	1.00940	0.504700	+	0.863295i	$0.331603\pi$
$548$	0	0
$549$	1.52605e7	2.16092
$550$	0	0
$551$	812006.	0.113941
$552$	0	0
$553$	2.65675e6	0.369435
$554$	0	0
$555$	1.19212e7	1.64282
$556$	0	0
$557$	−3.88072e6	−0.529998	−0.264999	−	0.964249i	$-0.585372\pi$
$557$	−3.88072e6	−0.529998	−0.264999	+	0.964249i	$0.585372\pi$
$558$	0	0
$559$	4.50258e6	0.609442
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.47325e6	−0.328849	−0.164425	−	0.986390i	$-0.552577\pi$
$563$	−2.47325e6	−0.328849	−0.164425	+	0.986390i	$0.552577\pi$
$564$	0	0
$565$	−6.80719e6	−0.897112
$566$	0	0
$567$	−588212.	−0.0768381
$568$	0	0
$569$	−1.29618e7	−1.67836	−0.839179	−	0.543855i	$-0.816964\pi$
$569$	−1.29618e7	−1.67836	−0.839179	+	0.543855i	$0.816964\pi$
$570$	0	0
$571$	−1.05228e7	−1.35065	−0.675323	−	0.737522i	$-0.735996\pi$
$571$	−1.05228e7	−1.35065	−0.675323	+	0.737522i	$0.735996\pi$
$572$	0	0
$573$	2.26764e6	0.288528
$574$	0	0
$575$	218683.	0.0275832
$576$	0	0
$577$	1.40291e7	1.75425	0.877123	−	0.480266i	$-0.159460\pi$
$577$	1.40291e7	1.75425	0.877123	+	0.480266i	$0.159460\pi$
$578$	0	0
$579$	1.07035e7	1.32688
$580$	0	0
$581$	1.83414e6	0.225419
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−4.05790e6	−0.490244
$586$	0	0
$587$	−5.34821e6	−0.640639	−0.320319	−	0.947310i	$-0.603790\pi$
$587$	−5.34821e6	−0.640639	−0.320319	+	0.947310i	$0.603790\pi$
$588$	0	0
$589$	2.95665e6	0.351166
$590$	0	0
$591$	2.40478e7	2.83208
$592$	0	0
$593$	−6.25465e6	−0.730409	−0.365205	−	0.930927i	$-0.619001\pi$
$593$	−6.25465e6	−0.730409	−0.365205	+	0.930927i	$0.619001\pi$
$594$	0	0
$595$	−1.34893e6	−0.156206
$596$	0	0
$597$	−1.27869e7	−1.46835
$598$	0	0
$599$	−1.62599e7	−1.85161	−0.925807	−	0.377997i	$-0.876613\pi$
$599$	−1.62599e7	−1.85161	−0.925807	+	0.377997i	$0.876613\pi$
$600$	0	0
$601$	3.97297e6	0.448672	0.224336	−	0.974512i	$-0.427979\pi$
$601$	3.97297e6	0.448672	0.224336	+	0.974512i	$0.427979\pi$
$602$	0	0
$603$	−5.41771e6	−0.606768
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4.26702e6	−0.470060	−0.235030	−	0.971988i	$-0.575519\pi$
$607$	−4.26702e6	−0.470060	−0.235030	+	0.971988i	$0.575519\pi$
$608$	0	0
$609$	−1.26045e6	−0.137716
$610$	0	0
$611$	−2.77600e6	−0.300827
$612$	0	0
$613$	1.64182e7	1.76472	0.882359	−	0.470577i	$-0.155954\pi$
$613$	1.64182e7	1.76472	0.882359	+	0.470577i	$0.155954\pi$
$614$	0	0
$615$	−9.61165e6	−1.02473
$616$	0	0
$617$	1.36165e7	1.43997	0.719985	−	0.693990i	$-0.244149\pi$
$617$	1.36165e7	1.43997	0.719985	+	0.693990i	$0.244149\pi$
$618$	0	0
$619$	−5.10598e6	−0.535615	−0.267807	−	0.963472i	$-0.586299\pi$
$619$	−5.10598e6	−0.535615	−0.267807	+	0.963472i	$0.586299\pi$
$620$	0	0
$621$	1.09699e6	0.114149
$622$	0	0
$623$	−651342.	−0.0672340
$624$	0	0
$625$	−7.95152e6	−0.814236
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.96875e6	1.00465
$630$	0	0
$631$	1.18290e7	1.18270	0.591349	−	0.806416i	$-0.298596\pi$
$631$	1.18290e7	1.18270	0.591349	+	0.806416i	$0.298596\pi$
$632$	0	0
$633$	−3.11001e7	−3.08498
$634$	0	0
$635$	6.86509e6	0.675634
$636$	0	0
$637$	−3.69662e6	−0.360957
$638$	0	0
$639$	−1.77886e7	−1.72342
$640$	0	0
$641$	1.16385e6	0.111880	0.0559401	−	0.998434i	$-0.482184\pi$
$641$	1.16385e6	0.111880	0.0559401	+	0.998434i	$0.482184\pi$
$642$	0	0
$643$	4.57835e6	0.436698	0.218349	−	0.975871i	$-0.429933\pi$
$643$	4.57835e6	0.436698	0.218349	+	0.975871i	$0.429933\pi$
$644$	0	0
$645$	2.44905e7	2.31792
$646$	0	0
$647$	9.90733e6	0.930456	0.465228	−	0.885191i	$-0.345972\pi$
$647$	9.90733e6	0.930456	0.465228	+	0.885191i	$0.345972\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−4.58952e6	−0.424439
$652$	0	0
$653$	−1.54281e7	−1.41589	−0.707947	−	0.706266i	$-0.750378\pi$
$653$	−1.54281e7	−1.41589	−0.707947	+	0.706266i	$0.750378\pi$
$654$	0	0
$655$	−2.37628e6	−0.216418
$656$	0	0
$657$	−2.41962e7	−2.18692
$658$	0	0
$659$	−4.51596e6	−0.405076	−0.202538	−	0.979274i	$-0.564919\pi$
$659$	−4.51596e6	−0.405076	−0.202538	+	0.979274i	$0.564919\pi$
$660$	0	0
$661$	1.85932e7	1.65520	0.827602	−	0.561316i	$-0.189705\pi$
$661$	1.85932e7	1.65520	0.827602	+	0.561316i	$0.189705\pi$
$662$	0	0
$663$	−5.77374e6	−0.510121
$664$	0	0
$665$	−513540.	−0.0450319
$666$	0	0
$667$	−896246.	−0.0780033
$668$	0	0
$669$	9.08635e6	0.784918
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1.08054e7	−0.919607	−0.459804	−	0.888021i	$-0.652080\pi$
$673$	−1.08054e7	−0.919607	−0.459804	+	0.888021i	$0.652080\pi$
$674$	0	0
$675$	1.25603e6	0.106106
$676$	0	0
$677$	5.16836e6	0.433392	0.216696	−	0.976239i	$-0.430472\pi$
$677$	5.16836e6	0.433392	0.216696	+	0.976239i	$0.430472\pi$
$678$	0	0
$679$	−72795.4	−0.00605940
$680$	0	0
$681$	−1.24668e7	−1.03012
$682$	0	0
$683$	4.88555e6	0.400739	0.200370	−	0.979720i	$-0.435786\pi$
$683$	4.88555e6	0.400739	0.200370	+	0.979720i	$0.435786\pi$
$684$	0	0
$685$	1.12699e7	0.917688
$686$	0	0
$687$	1.75143e7	1.41579
$688$	0	0
$689$	−7.28992e6	−0.585025
$690$	0	0
$691$	7.63827e6	0.608555	0.304278	−	0.952583i	$-0.401585\pi$
$691$	7.63827e6	0.608555	0.304278	+	0.952583i	$0.401585\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.52660e6	0.276946
$696$	0	0
$697$	−8.03744e6	−0.626665
$698$	0	0
$699$	1.84461e7	1.42795
$700$	0	0
$701$	9.86929e6	0.758562	0.379281	−	0.925282i	$-0.376171\pi$
$701$	9.86929e6	0.758562	0.379281	+	0.925282i	$0.376171\pi$
$702$	0	0
$703$	3.79512e6	0.289626
$704$	0	0
$705$	−1.50992e7	−1.14415
$706$	0	0
$707$	581280.	0.0437358
$708$	0	0
$709$	8.48410e6	0.633855	0.316928	−	0.948450i	$-0.397349\pi$
$709$	8.48410e6	0.633855	0.316928	+	0.948450i	$0.397349\pi$
$710$	0	0
$711$	3.63511e7	2.69677
$712$	0	0
$713$	−3.26339e6	−0.240406
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4.07549e7	−2.96062
$718$	0	0
$719$	−1.41418e7	−1.02019	−0.510095	−	0.860118i	$-0.670390\pi$
$719$	−1.41418e7	−1.02019	−0.510095	+	0.860118i	$0.670390\pi$
$720$	0	0
$721$	−2.66748e6	−0.191101
$722$	0	0
$723$	1.15725e7	0.823342
$724$	0	0
$725$	−1.02619e6	−0.0725075
$726$	0	0
$727$	1.03949e7	0.729429	0.364715	−	0.931119i	$-0.381167\pi$
$727$	1.03949e7	0.729429	0.364715	+	0.931119i	$0.381167\pi$
$728$	0	0
$729$	−2.28564e7	−1.59290
$730$	0	0
$731$	2.04794e7	1.41750
$732$	0	0
$733$	1.22788e7	0.844101	0.422050	−	0.906572i	$-0.361311\pi$
$733$	1.22788e7	0.844101	0.422050	+	0.906572i	$0.361311\pi$
$734$	0	0
$735$	−2.01066e7	−1.37284
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1.24896e7	0.841271	0.420636	−	0.907230i	$-0.361807\pi$
$739$	1.24896e7	0.841271	0.420636	+	0.907230i	$0.361807\pi$
$740$	0	0
$741$	−2.19807e6	−0.147060
$742$	0	0
$743$	2.45885e7	1.63403	0.817015	−	0.576616i	$-0.195627\pi$
$743$	2.45885e7	1.63403	0.817015	+	0.576616i	$0.195627\pi$
$744$	0	0
$745$	−1.49253e7	−0.985216
$746$	0	0
$747$	2.50956e7	1.64549
$748$	0	0
$749$	−1.66255e6	−0.108286
$750$	0	0
$751$	2.35137e6	0.152132	0.0760660	−	0.997103i	$-0.475764\pi$
$751$	2.35137e6	0.152132	0.0760660	+	0.997103i	$0.475764\pi$
$752$	0	0
$753$	2.06401e7	1.32655
$754$	0	0
$755$	−4.38227e6	−0.279790
$756$	0	0
$757$	2.12314e7	1.34660	0.673301	−	0.739368i	$-0.264876\pi$
$757$	2.12314e7	1.34660	0.673301	+	0.739368i	$0.264876\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.63419e7	−1.02292	−0.511459	−	0.859307i	$-0.670895\pi$
$761$	−1.63419e7	−1.02292	−0.511459	+	0.859307i	$0.670895\pi$
$762$	0	0
$763$	−674990.	−0.0419745
$764$	0	0
$765$	−1.84568e7	−1.14026
$766$	0	0
$767$	−1.17078e7	−0.718599
$768$	0	0
$769$	−9.03807e6	−0.551137	−0.275569	−	0.961281i	$-0.588866\pi$
$769$	−9.03807e6	−0.551137	−0.275569	+	0.961281i	$0.588866\pi$
$770$	0	0
$771$	−4.24142e7	−2.56966
$772$	0	0
$773$	−1.51026e7	−0.909082	−0.454541	−	0.890726i	$-0.650197\pi$
$773$	−1.51026e7	−0.909082	−0.454541	+	0.890726i	$0.650197\pi$
$774$	0	0
$775$	−3.73653e6	−0.223468
$776$	0	0
$777$	−5.89104e6	−0.350058
$778$	0	0
$779$	−3.05986e6	−0.180658
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−5.14771e6	−0.300061
$784$	0	0
$785$	−5.60084e6	−0.324399
$786$	0	0
$787$	4.40214e6	0.253353	0.126677	−	0.991944i	$-0.459569\pi$
$787$	4.40214e6	0.253353	0.126677	+	0.991944i	$0.459569\pi$
$788$	0	0
$789$	−1.15923e7	−0.662945
$790$	0	0
$791$	3.36386e6	0.191160
$792$	0	0
$793$	1.00740e7	0.568876
$794$	0	0
$795$	−3.96513e7	−2.22505
$796$	0	0
$797$	−2.35112e7	−1.31108	−0.655540	−	0.755160i	$-0.727559\pi$
$797$	−2.35112e7	−1.31108	−0.655540	+	0.755160i	$0.727559\pi$
$798$	0	0
$799$	−1.26262e7	−0.699692
$800$	0	0
$801$	−8.91199e6	−0.490787
$802$	0	0
$803$	0	0
$804$	0	0
$805$	566817.	0.0308285
$806$	0	0
$807$	1.62048e7	0.875913
$808$	0	0
$809$	3.02958e7	1.62746	0.813731	−	0.581242i	$-0.197433\pi$
$809$	3.02958e7	1.62746	0.813731	+	0.581242i	$0.197433\pi$
$810$	0	0
$811$	−9.66163e6	−0.515820	−0.257910	−	0.966169i	$-0.583034\pi$
$811$	−9.66163e6	−0.515820	−0.257910	+	0.966169i	$0.583034\pi$
$812$	0	0
$813$	9.09978e6	0.482842
$814$	0	0
$815$	−5.95181e6	−0.313874
$816$	0	0
$817$	7.79652e6	0.408644
$818$	0	0
$819$	2.00527e6	0.104463
$820$	0	0
$821$	3.23821e7	1.67667	0.838334	−	0.545156i	$-0.183530\pi$
$821$	3.23821e7	1.67667	0.838334	+	0.545156i	$0.183530\pi$
$822$	0	0
$823$	1.11591e7	0.574288	0.287144	−	0.957887i	$-0.407294\pi$
$823$	1.11591e7	0.574288	0.287144	+	0.957887i	$0.407294\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.60904e7	0.818093	0.409046	−	0.912514i	$-0.365861\pi$
$827$	1.60904e7	0.818093	0.409046	+	0.912514i	$0.365861\pi$
$828$	0	0
$829$	−3.16894e7	−1.60150	−0.800751	−	0.598997i	$-0.795566\pi$
$829$	−3.16894e7	−1.60150	−0.800751	+	0.598997i	$0.795566\pi$
$830$	0	0
$831$	1.77887e7	0.893599
$832$	0	0
$833$	−1.68135e7	−0.839550
$834$	0	0
$835$	−1.63943e7	−0.813725
$836$	0	0
$837$	−1.87437e7	−0.924787
$838$	0	0
$839$	−2.31362e6	−0.113472	−0.0567359	−	0.998389i	$-0.518069\pi$
$839$	−2.31362e6	−0.113472	−0.0567359	+	0.998389i	$0.518069\pi$
$840$	0	0
$841$	−1.63054e7	−0.794954
$842$	0	0
$843$	5.99762e6	0.290677
$844$	0	0
$845$	1.63429e7	0.787388
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.97062e7	−1.89056
$850$	0	0
$851$	−4.18883e6	−0.198276
$852$	0	0
$853$	4.69366e6	0.220871	0.110436	−	0.993883i	$-0.464775\pi$
$853$	4.69366e6	0.220871	0.110436	+	0.993883i	$0.464775\pi$
$854$	0	0
$855$	−7.02652e6	−0.328719
$856$	0	0
$857$	−3.06601e7	−1.42601	−0.713004	−	0.701160i	$-0.752666\pi$
$857$	−3.06601e7	−1.42601	−0.713004	+	0.701160i	$0.752666\pi$
$858$	0	0
$859$	−1.75826e7	−0.813018	−0.406509	−	0.913647i	$-0.633254\pi$
$859$	−1.75826e7	−0.813018	−0.406509	+	0.913647i	$0.633254\pi$
$860$	0	0
$861$	4.74973e6	0.218354
$862$	0	0
$863$	7.70340e6	0.352091	0.176046	−	0.984382i	$-0.443669\pi$
$863$	7.70340e6	0.352091	0.176046	+	0.984382i	$0.443669\pi$
$864$	0	0
$865$	−1.79036e7	−0.813582
$866$	0	0
$867$	8.20945e6	0.370908
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−3.57641e6	−0.159735
$872$	0	0
$873$	−996025.	−0.0442318
$874$	0	0
$875$	4.70208e6	0.207620
$876$	0	0
$877$	2.37050e7	1.04074	0.520368	−	0.853942i	$-0.325795\pi$
$877$	2.37050e7	1.04074	0.520368	+	0.853942i	$0.325795\pi$
$878$	0	0
$879$	2.92780e7	1.27811
$880$	0	0
$881$	−2.30169e6	−0.0999094	−0.0499547	−	0.998751i	$-0.515908\pi$
$881$	−2.30169e6	−0.0999094	−0.0499547	+	0.998751i	$0.515908\pi$
$882$	0	0
$883$	2.07113e7	0.893932	0.446966	−	0.894551i	$-0.352504\pi$
$883$	2.07113e7	0.893932	0.446966	+	0.894551i	$0.352504\pi$
$884$	0	0
$885$	−6.36811e7	−2.73308
$886$	0	0
$887$	−2.44090e7	−1.04170	−0.520849	−	0.853649i	$-0.674384\pi$
$887$	−2.44090e7	−1.04170	−0.520849	+	0.853649i	$0.674384\pi$
$888$	0	0
$889$	−3.39248e6	−0.143967
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.80683e6	−0.201711
$894$	0	0
$895$	−3.60300e7	−1.50351
$896$	0	0
$897$	2.42610e6	0.100677
$898$	0	0
$899$	1.53138e7	0.631950
$900$	0	0
$901$	−3.31572e7	−1.36071
$902$	0	0
$903$	−1.21023e7	−0.493910
$904$	0	0
$905$	8.81999e6	0.357970
$906$	0	0
$907$	−2.68654e7	−1.08436	−0.542182	−	0.840261i	$-0.682402\pi$
$907$	−2.68654e7	−1.08436	−0.542182	+	0.840261i	$0.682402\pi$
$908$	0	0
$909$	7.95338e6	0.319258
$910$	0	0
$911$	1.22198e7	0.487829	0.243914	−	0.969797i	$-0.421568\pi$
$911$	1.22198e7	0.487829	0.243914	+	0.969797i	$0.421568\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	5.47943e7	2.16363
$916$	0	0
$917$	1.17427e6	0.0461152
$918$	0	0
$919$	1.76834e7	0.690681	0.345340	−	0.938478i	$-0.387763\pi$
$919$	1.76834e7	0.690681	0.345340	+	0.938478i	$0.387763\pi$
$920$	0	0
$921$	−4.83600e7	−1.87861
$922$	0	0
$923$	−1.17428e7	−0.453700
$924$	0	0
$925$	−4.79616e6	−0.184306
$926$	0	0
$927$	−3.64978e7	−1.39498
$928$	0	0
$929$	−1.80890e7	−0.687664	−0.343832	−	0.939031i	$-0.611725\pi$
$929$	−1.80890e7	−0.687664	−0.343832	+	0.939031i	$0.611725\pi$
$930$	0	0
$931$	−6.40094e6	−0.242030
$932$	0	0
$933$	−4.14471e7	−1.55880
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.31784e7	1.97873	0.989364	−	0.145464i	$-0.0464674\pi$
$937$	5.31784e7	1.97873	0.989364	+	0.145464i	$0.0464674\pi$
$938$	0	0
$939$	−5.79063e7	−2.14320
$940$	0	0
$941$	−3.34713e7	−1.23225	−0.616124	−	0.787649i	$-0.711298\pi$
$941$	−3.34713e7	−1.23225	−0.616124	+	0.787649i	$0.711298\pi$
$942$	0	0
$943$	3.37730e6	0.123678
$944$	0	0
$945$	3.25559e6	0.118591
$946$	0	0
$947$	−3.89951e7	−1.41298	−0.706488	−	0.707725i	$-0.749721\pi$
$947$	−3.89951e7	−1.41298	−0.706488	+	0.707725i	$0.749721\pi$
$948$	0	0
$949$	−1.59727e7	−0.575721
$950$	0	0
$951$	−2.83943e7	−1.01808
$952$	0	0
$953$	−5.40079e6	−0.192631	−0.0963153	−	0.995351i	$-0.530706\pi$
$953$	−5.40079e6	−0.192631	−0.0963153	+	0.995351i	$0.530706\pi$
$954$	0	0
$955$	4.78524e6	0.169783
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.56919e6	−0.195544
$960$	0	0
$961$	2.71309e7	0.947667
$962$	0	0
$963$	−2.27479e7	−0.790452
$964$	0	0
$965$	2.25870e7	0.780799
$966$	0	0
$967$	−1.12876e7	−0.388183	−0.194091	−	0.980983i	$-0.562176\pi$
$967$	−1.12876e7	−0.388183	−0.194091	+	0.980983i	$0.562176\pi$
$968$	0	0
$969$	−9.99760e6	−0.342047
$970$	0	0
$971$	2.98554e7	1.01619	0.508096	−	0.861301i	$-0.330350\pi$
$971$	2.98554e7	1.01619	0.508096	+	0.861301i	$0.330350\pi$
$972$	0	0
$973$	−1.74272e6	−0.0590126
$974$	0	0
$975$	2.77786e6	0.0935833
$976$	0	0
$977$	−2.61437e7	−0.876256	−0.438128	−	0.898913i	$-0.644358\pi$
$977$	−2.61437e7	−0.876256	−0.438128	+	0.898913i	$0.644358\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−9.23556e6	−0.306401
$982$	0	0
$983$	2.66113e7	0.878381	0.439191	−	0.898394i	$-0.355265\pi$
$983$	2.66113e7	0.878381	0.439191	+	0.898394i	$0.355265\pi$
$984$	0	0
$985$	5.07464e7	1.66653
$986$	0	0
$987$	7.46148e6	0.243799
$988$	0	0
$989$	−8.60535e6	−0.279755
$990$	0	0
$991$	−9.05255e6	−0.292811	−0.146405	−	0.989225i	$-0.546770\pi$
$991$	−9.05255e6	−0.292811	−0.146405	+	0.989225i	$0.546770\pi$
$992$	0	0
$993$	−5.04065e7	−1.62223
$994$	0	0
$995$	−2.69832e7	−0.864045
$996$	0	0
$997$	1.69560e7	0.540240	0.270120	−	0.962827i	$-0.412937\pi$
$997$	1.69560e7	0.540240	0.270120	+	0.962827i	$0.412937\pi$
$998$	0	0
$999$	−2.40591e7	−0.762722

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.6.a.q.1.3		16
11.3	even	5		88.6.i.b.9.8	✓	32
11.4	even	5		88.6.i.b.49.8	yes	32
11.10	odd	2		968.6.a.p.1.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.6.i.b.9.8	✓	32	11.3	even	5
88.6.i.b.49.8	yes	32	11.4	even	5
968.6.a.p.1.3		16	11.10	odd	2
968.6.a.q.1.3		16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-24.2774 q^{3} -51.2310 q^{5} +25.3165 q^{7} +346.393 q^{9} +228.665 q^{13} +1243.76 q^{15} +1040.05 q^{17} +395.949 q^{19} -614.619 q^{21} -437.026 q^{23} -500.388 q^{25} -2510.12 q^{27} +2050.79 q^{29} +7467.26 q^{31} -1296.99 q^{35} +9584.87 q^{37} -5551.40 q^{39} -7727.93 q^{41} +19690.7 q^{43} -17746.0 q^{45} -12140.0 q^{47} -16166.1 q^{49} -25249.7 q^{51} -31880.3 q^{53} -9612.61 q^{57} -51200.6 q^{59} +44055.5 q^{61} +8769.45 q^{63} -11714.7 q^{65} -15640.4 q^{67} +10609.9 q^{69} -51353.9 q^{71} -69851.9 q^{73} +12148.1 q^{75} +104942. q^{79} -23234.4 q^{81} +72448.3 q^{83} -53282.8 q^{85} -49787.8 q^{87} -25728.0 q^{89} +5788.99 q^{91} -181286. q^{93} -20284.8 q^{95} -2875.42 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{3} + 81 q^{5} + 47 q^{7} + 1416 q^{9} - 859 q^{13} - 738 q^{15} - 1226 q^{17} - 616 q^{19} - 1141 q^{21} + 2258 q^{23} + 10307 q^{25} + 564 q^{27} - 1613 q^{29} + 18511 q^{31} + 23544 q^{35}+ \cdots + 171314 q^{97}+O(q^{100}) \) 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

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Database

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