LMFDB - Embedded newform 8024.2.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.88993 q^{3} -1.71727 q^{5} -4.47538 q^{7} +5.35170 q^{9} +O(q^{10})$	$q-2.88993 q^{3} -1.71727 q^{5} -4.47538 q^{7} +5.35170 q^{9} -4.06786 q^{11} -4.73647 q^{13} +4.96278 q^{15} -1.00000 q^{17} -8.07202 q^{19} +12.9335 q^{21} +4.47945 q^{23} -2.05100 q^{25} -6.79624 q^{27} -1.34908 q^{29} +5.63309 q^{31} +11.7558 q^{33} +7.68542 q^{35} +9.06811 q^{37} +13.6881 q^{39} +8.49539 q^{41} +0.753759 q^{43} -9.19029 q^{45} -4.24429 q^{47} +13.0290 q^{49} +2.88993 q^{51} -1.55350 q^{53} +6.98559 q^{55} +23.3276 q^{57} -1.00000 q^{59} -8.96870 q^{61} -23.9509 q^{63} +8.13378 q^{65} +2.40539 q^{67} -12.9453 q^{69} -11.9331 q^{71} +6.99818 q^{73} +5.92724 q^{75} +18.2052 q^{77} -0.499053 q^{79} +3.58558 q^{81} -0.659380 q^{83} +1.71727 q^{85} +3.89876 q^{87} -4.70016 q^{89} +21.1975 q^{91} -16.2792 q^{93} +13.8618 q^{95} +13.0666 q^{97} -21.7699 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * 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
$3$	−2.88993	−1.66850	−0.834251	−	0.551385i	$-0.814100\pi$
$3$	−2.88993	−1.66850	−0.834251	+	0.551385i	$0.814100\pi$
$4$	0	0
$5$	−1.71727	−0.767985	−0.383992	−	0.923336i	$-0.625451\pi$
$5$	−1.71727	−0.767985	−0.383992	+	0.923336i	$0.625451\pi$
$6$	0	0
$7$	−4.47538	−1.69153	−0.845767	−	0.533552i	$-0.820857\pi$
$7$	−4.47538	−1.69153	−0.845767	+	0.533552i	$0.820857\pi$
$8$	0	0
$9$	5.35170	1.78390
$10$	0	0
$11$	−4.06786	−1.22650	−0.613252	−	0.789887i	$-0.710139\pi$
$11$	−4.06786	−1.22650	−0.613252	+	0.789887i	$0.710139\pi$
$12$	0	0
$13$	−4.73647	−1.31366	−0.656830	−	0.754038i	$-0.728103\pi$
$13$	−4.73647	−1.31366	−0.656830	+	0.754038i	$0.728103\pi$
$14$	0	0
$15$	4.96278	1.28138
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−8.07202	−1.85185	−0.925924	−	0.377710i	$-0.876712\pi$
$19$	−8.07202	−1.85185	−0.925924	+	0.377710i	$0.876712\pi$
$20$	0	0
$21$	12.9335	2.82233
$22$	0	0
$23$	4.47945	0.934030	0.467015	−	0.884249i	$-0.345329\pi$
$23$	4.47945	0.934030	0.467015	+	0.884249i	$0.345329\pi$
$24$	0	0
$25$	−2.05100	−0.410199
$26$	0	0
$27$	−6.79624	−1.30794
$28$	0	0
$29$	−1.34908	−0.250519	−0.125259	−	0.992124i	$-0.539976\pi$
$29$	−1.34908	−0.250519	−0.125259	+	0.992124i	$0.539976\pi$
$30$	0	0
$31$	5.63309	1.01173	0.505866	−	0.862612i	$-0.331173\pi$
$31$	5.63309	1.01173	0.505866	+	0.862612i	$0.331173\pi$
$32$	0	0
$33$	11.7558	2.04643
$34$	0	0
$35$	7.68542	1.29907
$36$	0	0
$37$	9.06811	1.49079	0.745394	−	0.666624i	$-0.232261\pi$
$37$	9.06811	1.49079	0.745394	+	0.666624i	$0.232261\pi$
$38$	0	0
$39$	13.6881	2.19185
$40$	0	0
$41$	8.49539	1.32676	0.663378	−	0.748284i	$-0.269122\pi$
$41$	8.49539	1.32676	0.663378	+	0.748284i	$0.269122\pi$
$42$	0	0
$43$	0.753759	0.114947	0.0574736	−	0.998347i	$-0.481696\pi$
$43$	0.753759	0.114947	0.0574736	+	0.998347i	$0.481696\pi$
$44$	0	0
$45$	−9.19029	−1.37001
$46$	0	0
$47$	−4.24429	−0.619094	−0.309547	−	0.950884i	$-0.600177\pi$
$47$	−4.24429	−0.619094	−0.309547	+	0.950884i	$0.600177\pi$
$48$	0	0
$49$	13.0290	1.86129
$50$	0	0
$51$	2.88993	0.404671
$52$	0	0
$53$	−1.55350	−0.213390	−0.106695	−	0.994292i	$-0.534027\pi$
$53$	−1.55350	−0.213390	−0.106695	+	0.994292i	$0.534027\pi$
$54$	0	0
$55$	6.98559	0.941937
$56$	0	0
$57$	23.3276	3.08981
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−8.96870	−1.14832	−0.574162	−	0.818741i	$-0.694672\pi$
$61$	−8.96870	−1.14832	−0.574162	+	0.818741i	$0.694672\pi$
$62$	0	0
$63$	−23.9509	−3.01753
$64$	0	0
$65$	8.13378	1.00887
$66$	0	0
$67$	2.40539	0.293865	0.146933	−	0.989146i	$-0.453060\pi$
$67$	2.40539	0.293865	0.146933	+	0.989146i	$0.453060\pi$
$68$	0	0
$69$	−12.9453	−1.55843
$70$	0	0
$71$	−11.9331	−1.41620	−0.708100	−	0.706112i	$-0.750447\pi$
$71$	−11.9331	−1.41620	−0.708100	+	0.706112i	$0.750447\pi$
$72$	0	0
$73$	6.99818	0.819075	0.409537	−	0.912293i	$-0.365690\pi$
$73$	6.99818	0.819075	0.409537	+	0.912293i	$0.365690\pi$
$74$	0	0
$75$	5.92724	0.684418
$76$	0	0
$77$	18.2052	2.07468
$78$	0	0
$79$	−0.499053	−0.0561479	−0.0280739	−	0.999606i	$-0.508937\pi$
$79$	−0.499053	−0.0561479	−0.0280739	+	0.999606i	$0.508937\pi$
$80$	0	0
$81$	3.58558	0.398397
$82$	0	0
$83$	−0.659380	−0.0723764	−0.0361882	−	0.999345i	$-0.511522\pi$
$83$	−0.659380	−0.0723764	−0.0361882	+	0.999345i	$0.511522\pi$
$84$	0	0
$85$	1.71727	0.186264
$86$	0	0
$87$	3.89876	0.417991
$88$	0	0
$89$	−4.70016	−0.498216	−0.249108	−	0.968476i	$-0.580137\pi$
$89$	−4.70016	−0.498216	−0.249108	+	0.968476i	$0.580137\pi$
$90$	0	0
$91$	21.1975	2.22210
$92$	0	0
$93$	−16.2792	−1.68808
$94$	0	0
$95$	13.8618	1.42219
$96$	0	0
$97$	13.0666	1.32671	0.663357	−	0.748303i	$-0.269131\pi$
$97$	13.0666	1.32671	0.663357	+	0.748303i	$0.269131\pi$
$98$	0	0
$99$	−21.7699	−2.18796
$100$	0	0
$101$	3.30043	0.328405	0.164203	−	0.986427i	$-0.447495\pi$
$101$	3.30043	0.328405	0.164203	+	0.986427i	$0.447495\pi$
$102$	0	0
$103$	9.80224	0.965843	0.482921	−	0.875664i	$-0.339576\pi$
$103$	9.80224	0.965843	0.482921	+	0.875664i	$0.339576\pi$
$104$	0	0
$105$	−22.2103	−2.16751
$106$	0	0
$107$	−7.77994	−0.752116	−0.376058	−	0.926596i	$-0.622721\pi$
$107$	−7.77994	−0.752116	−0.376058	+	0.926596i	$0.622721\pi$
$108$	0	0
$109$	−4.58102	−0.438782	−0.219391	−	0.975637i	$-0.570407\pi$
$109$	−4.58102	−0.438782	−0.219391	+	0.975637i	$0.570407\pi$
$110$	0	0
$111$	−26.2062	−2.48738
$112$	0	0
$113$	16.7672	1.57733	0.788663	−	0.614826i	$-0.210774\pi$
$113$	16.7672	1.57733	0.788663	+	0.614826i	$0.210774\pi$
$114$	0	0
$115$	−7.69241	−0.717321
$116$	0	0
$117$	−25.3482	−2.34344
$118$	0	0
$119$	4.47538	0.410257
$120$	0	0
$121$	5.54745	0.504314
$122$	0	0
$123$	−24.5511	−2.21370
$124$	0	0
$125$	12.1084	1.08301
$126$	0	0
$127$	−19.1287	−1.69739	−0.848697	−	0.528879i	$-0.822613\pi$
$127$	−19.1287	−1.69739	−0.848697	+	0.528879i	$0.822613\pi$
$128$	0	0
$129$	−2.17831	−0.191789
$130$	0	0
$131$	−0.475014	−0.0415021	−0.0207511	−	0.999785i	$-0.506606\pi$
$131$	−0.475014	−0.0415021	−0.0207511	+	0.999785i	$0.506606\pi$
$132$	0	0
$133$	36.1254	3.13247
$134$	0	0
$135$	11.6710	1.00448
$136$	0	0
$137$	−5.28095	−0.451182	−0.225591	−	0.974222i	$-0.572431\pi$
$137$	−5.28095	−0.451182	−0.225591	+	0.974222i	$0.572431\pi$
$138$	0	0
$139$	−1.83085	−0.155290	−0.0776452	−	0.996981i	$-0.524740\pi$
$139$	−1.83085	−0.155290	−0.0776452	+	0.996981i	$0.524740\pi$
$140$	0	0
$141$	12.2657	1.03296
$142$	0	0
$143$	19.2673	1.61121
$144$	0	0
$145$	2.31674	0.192395
$146$	0	0
$147$	−37.6530	−3.10557
$148$	0	0
$149$	8.76962	0.718435	0.359218	−	0.933254i	$-0.383044\pi$
$149$	8.76962	0.718435	0.359218	+	0.933254i	$0.383044\pi$
$150$	0	0
$151$	14.1799	1.15394	0.576971	−	0.816765i	$-0.304235\pi$
$151$	14.1799	1.15394	0.576971	+	0.816765i	$0.304235\pi$
$152$	0	0
$153$	−5.35170	−0.432659
$154$	0	0
$155$	−9.67351	−0.776995
$156$	0	0
$157$	1.82382	0.145556	0.0727782	−	0.997348i	$-0.476813\pi$
$157$	1.82382	0.145556	0.0727782	+	0.997348i	$0.476813\pi$
$158$	0	0
$159$	4.48951	0.356041
$160$	0	0
$161$	−20.0472	−1.57994
$162$	0	0
$163$	10.9175	0.855127	0.427564	−	0.903985i	$-0.359372\pi$
$163$	10.9175	0.855127	0.427564	+	0.903985i	$0.359372\pi$
$164$	0	0
$165$	−20.1879	−1.57162
$166$	0	0
$167$	4.30460	0.333100	0.166550	−	0.986033i	$-0.446737\pi$
$167$	4.30460	0.333100	0.166550	+	0.986033i	$0.446737\pi$
$168$	0	0
$169$	9.43416	0.725705
$170$	0	0
$171$	−43.1990	−3.30351
$172$	0	0
$173$	−19.3909	−1.47427	−0.737133	−	0.675748i	$-0.763821\pi$
$173$	−19.3909	−1.47427	−0.737133	+	0.675748i	$0.763821\pi$
$174$	0	0
$175$	9.17899	0.693866
$176$	0	0
$177$	2.88993	0.217220
$178$	0	0
$179$	1.97367	0.147519	0.0737594	−	0.997276i	$-0.476500\pi$
$179$	1.97367	0.147519	0.0737594	+	0.997276i	$0.476500\pi$
$180$	0	0
$181$	10.1504	0.754475	0.377237	−	0.926117i	$-0.376874\pi$
$181$	10.1504	0.754475	0.377237	+	0.926117i	$0.376874\pi$
$182$	0	0
$183$	25.9189	1.91598
$184$	0	0
$185$	−15.5724	−1.14490
$186$	0	0
$187$	4.06786	0.297471
$188$	0	0
$189$	30.4158	2.21242
$190$	0	0
$191$	−8.66394	−0.626901	−0.313450	−	0.949605i	$-0.601485\pi$
$191$	−8.66394	−0.626901	−0.313450	+	0.949605i	$0.601485\pi$
$192$	0	0
$193$	18.8228	1.35490	0.677449	−	0.735570i	$-0.263085\pi$
$193$	18.8228	1.35490	0.677449	+	0.735570i	$0.263085\pi$
$194$	0	0
$195$	−23.5061	−1.68330
$196$	0	0
$197$	14.3221	1.02041	0.510205	−	0.860053i	$-0.329569\pi$
$197$	14.3221	1.02041	0.510205	+	0.860053i	$0.329569\pi$
$198$	0	0
$199$	0.793266	0.0562332	0.0281166	−	0.999605i	$-0.491049\pi$
$199$	0.793266	0.0562332	0.0281166	+	0.999605i	$0.491049\pi$
$200$	0	0
$201$	−6.95142	−0.490315
$202$	0	0
$203$	6.03766	0.423761
$204$	0	0
$205$	−14.5888	−1.01893
$206$	0	0
$207$	23.9727	1.66622
$208$	0	0
$209$	32.8358	2.27130
$210$	0	0
$211$	5.57426	0.383748	0.191874	−	0.981420i	$-0.438544\pi$
$211$	5.57426	0.383748	0.191874	+	0.981420i	$0.438544\pi$
$212$	0	0
$213$	34.4859	2.36293
$214$	0	0
$215$	−1.29440	−0.0882776
$216$	0	0
$217$	−25.2102	−1.71138
$218$	0	0
$219$	−20.2243	−1.36663
$220$	0	0
$221$	4.73647	0.318610
$222$	0	0
$223$	15.6461	1.04774	0.523869	−	0.851799i	$-0.324488\pi$
$223$	15.6461	1.04774	0.523869	+	0.851799i	$0.324488\pi$
$224$	0	0
$225$	−10.9763	−0.731754
$226$	0	0
$227$	−3.54310	−0.235164	−0.117582	−	0.993063i	$-0.537514\pi$
$227$	−3.54310	−0.235164	−0.117582	+	0.993063i	$0.537514\pi$
$228$	0	0
$229$	17.2044	1.13690	0.568451	−	0.822717i	$-0.307543\pi$
$229$	17.2044	1.13690	0.568451	+	0.822717i	$0.307543\pi$
$230$	0	0
$231$	−52.6118	−3.46160
$232$	0	0
$233$	18.0935	1.18534	0.592672	−	0.805444i	$-0.298073\pi$
$233$	18.0935	1.18534	0.592672	+	0.805444i	$0.298073\pi$
$234$	0	0
$235$	7.28858	0.475455
$236$	0	0
$237$	1.44223	0.0936829
$238$	0	0
$239$	7.99926	0.517429	0.258714	−	0.965954i	$-0.416701\pi$
$239$	7.99926	0.517429	0.258714	+	0.965954i	$0.416701\pi$
$240$	0	0
$241$	−30.9463	−1.99343	−0.996715	−	0.0809913i	$-0.974191\pi$
$241$	−30.9463	−1.99343	−0.996715	+	0.0809913i	$0.974191\pi$
$242$	0	0
$243$	10.0267	0.643211
$244$	0	0
$245$	−22.3743	−1.42944
$246$	0	0
$247$	38.2329	2.43270
$248$	0	0
$249$	1.90556	0.120760
$250$	0	0
$251$	−4.54671	−0.286986	−0.143493	−	0.989651i	$-0.545833\pi$
$251$	−4.54671	−0.286986	−0.143493	+	0.989651i	$0.545833\pi$
$252$	0	0
$253$	−18.2218	−1.14559
$254$	0	0
$255$	−4.96278	−0.310781
$256$	0	0
$257$	4.76844	0.297447	0.148723	−	0.988879i	$-0.452484\pi$
$257$	4.76844	0.297447	0.148723	+	0.988879i	$0.452484\pi$
$258$	0	0
$259$	−40.5833	−2.52172
$260$	0	0
$261$	−7.21989	−0.446900
$262$	0	0
$263$	21.3309	1.31532	0.657661	−	0.753314i	$-0.271546\pi$
$263$	21.3309	1.31532	0.657661	+	0.753314i	$0.271546\pi$
$264$	0	0
$265$	2.66778	0.163880
$266$	0	0
$267$	13.5831	0.831275
$268$	0	0
$269$	−0.652086	−0.0397584	−0.0198792	−	0.999802i	$-0.506328\pi$
$269$	−0.652086	−0.0397584	−0.0198792	+	0.999802i	$0.506328\pi$
$270$	0	0
$271$	−32.3022	−1.96222	−0.981108	−	0.193460i	$-0.938029\pi$
$271$	−32.3022	−1.96222	−0.981108	+	0.193460i	$0.938029\pi$
$272$	0	0
$273$	−61.2593	−3.70758
$274$	0	0
$275$	8.34316	0.503111
$276$	0	0
$277$	14.3466	0.862006	0.431003	−	0.902350i	$-0.358160\pi$
$277$	14.3466	0.862006	0.431003	+	0.902350i	$0.358160\pi$
$278$	0	0
$279$	30.1466	1.80483
$280$	0	0
$281$	−31.0110	−1.84996	−0.924981	−	0.380014i	$-0.875919\pi$
$281$	−31.0110	−1.84996	−0.924981	+	0.380014i	$0.875919\pi$
$282$	0	0
$283$	6.55374	0.389579	0.194790	−	0.980845i	$-0.437598\pi$
$283$	6.55374	0.389579	0.194790	+	0.980845i	$0.437598\pi$
$284$	0	0
$285$	−40.0597	−2.37293
$286$	0	0
$287$	−38.0201	−2.24425
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−37.7616	−2.21363
$292$	0	0
$293$	−23.1873	−1.35462	−0.677308	−	0.735700i	$-0.736853\pi$
$293$	−23.1873	−1.35462	−0.677308	+	0.735700i	$0.736853\pi$
$294$	0	0
$295$	1.71727	0.0999831
$296$	0	0
$297$	27.6461	1.60419
$298$	0	0
$299$	−21.2168	−1.22700
$300$	0	0
$301$	−3.37336	−0.194437
$302$	0	0
$303$	−9.53801	−0.547944
$304$	0	0
$305$	15.4017	0.881896
$306$	0	0
$307$	−22.8589	−1.30462	−0.652312	−	0.757951i	$-0.726201\pi$
$307$	−22.8589	−1.30462	−0.652312	+	0.757951i	$0.726201\pi$
$308$	0	0
$309$	−28.3278	−1.61151
$310$	0	0
$311$	−34.0184	−1.92901	−0.964504	−	0.264068i	$-0.914936\pi$
$311$	−34.0184	−1.92901	−0.964504	+	0.264068i	$0.914936\pi$
$312$	0	0
$313$	−29.0574	−1.64242	−0.821210	−	0.570626i	$-0.806700\pi$
$313$	−29.0574	−1.64242	−0.821210	+	0.570626i	$0.806700\pi$
$314$	0	0
$315$	41.1300	2.31742
$316$	0	0
$317$	1.33493	0.0749770	0.0374885	−	0.999297i	$-0.488064\pi$
$317$	1.33493	0.0749770	0.0374885	+	0.999297i	$0.488064\pi$
$318$	0	0
$319$	5.48788	0.307262
$320$	0	0
$321$	22.4835	1.25491
$322$	0	0
$323$	8.07202	0.449139
$324$	0	0
$325$	9.71448	0.538863
$326$	0	0
$327$	13.2388	0.732109
$328$	0	0
$329$	18.9948	1.04722
$330$	0	0
$331$	4.56118	0.250705	0.125353	−	0.992112i	$-0.459994\pi$
$331$	4.56118	0.250705	0.125353	+	0.992112i	$0.459994\pi$
$332$	0	0
$333$	48.5298	2.65942
$334$	0	0
$335$	−4.13070	−0.225684
$336$	0	0
$337$	−4.51392	−0.245889	−0.122944	−	0.992414i	$-0.539234\pi$
$337$	−4.51392	−0.245889	−0.122944	+	0.992414i	$0.539234\pi$
$338$	0	0
$339$	−48.4560	−2.63177
$340$	0	0
$341$	−22.9146	−1.24089
$342$	0	0
$343$	−26.9822	−1.45690
$344$	0	0
$345$	22.2305	1.19685
$346$	0	0
$347$	−33.7414	−1.81133	−0.905666	−	0.423992i	$-0.860628\pi$
$347$	−33.7414	−1.81133	−0.905666	+	0.423992i	$0.860628\pi$
$348$	0	0
$349$	−14.8975	−0.797445	−0.398722	−	0.917072i	$-0.630546\pi$
$349$	−14.8975	−0.797445	−0.398722	+	0.917072i	$0.630546\pi$
$350$	0	0
$351$	32.1902	1.71819
$352$	0	0
$353$	7.47350	0.397774	0.198887	−	0.980022i	$-0.436267\pi$
$353$	7.47350	0.397774	0.198887	+	0.980022i	$0.436267\pi$
$354$	0	0
$355$	20.4923	1.08762
$356$	0	0
$357$	−12.9335	−0.684515
$358$	0	0
$359$	−14.1124	−0.744824	−0.372412	−	0.928068i	$-0.621469\pi$
$359$	−14.1124	−0.744824	−0.372412	+	0.928068i	$0.621469\pi$
$360$	0	0
$361$	46.1575	2.42934
$362$	0	0
$363$	−16.0317	−0.841448
$364$	0	0
$365$	−12.0177	−0.629037
$366$	0	0
$367$	28.5518	1.49039	0.745197	−	0.666845i	$-0.232356\pi$
$367$	28.5518	1.49039	0.745197	+	0.666845i	$0.232356\pi$
$368$	0	0
$369$	45.4647	2.36680
$370$	0	0
$371$	6.95251	0.360956
$372$	0	0
$373$	−6.44808	−0.333869	−0.166935	−	0.985968i	$-0.553387\pi$
$373$	−6.44808	−0.333869	−0.166935	+	0.985968i	$0.553387\pi$
$374$	0	0
$375$	−34.9925	−1.80701
$376$	0	0
$377$	6.38990	0.329097
$378$	0	0
$379$	21.2657	1.09235	0.546174	−	0.837672i	$-0.316084\pi$
$379$	21.2657	1.09235	0.546174	+	0.837672i	$0.316084\pi$
$380$	0	0
$381$	55.2805	2.83211
$382$	0	0
$383$	3.26860	0.167018	0.0835088	−	0.996507i	$-0.473387\pi$
$383$	3.26860	0.167018	0.0835088	+	0.996507i	$0.473387\pi$
$384$	0	0
$385$	−31.2632	−1.59332
$386$	0	0
$387$	4.03389	0.205054
$388$	0	0
$389$	28.4979	1.44490	0.722451	−	0.691422i	$-0.243015\pi$
$389$	28.4979	1.44490	0.722451	+	0.691422i	$0.243015\pi$
$390$	0	0
$391$	−4.47945	−0.226536
$392$	0	0
$393$	1.37276	0.0692464
$394$	0	0
$395$	0.857008	0.0431207
$396$	0	0
$397$	−2.05913	−0.103345	−0.0516724	−	0.998664i	$-0.516455\pi$
$397$	−2.05913	−0.103345	−0.0516724	+	0.998664i	$0.516455\pi$
$398$	0	0
$399$	−104.400	−5.22653
$400$	0	0
$401$	−7.52117	−0.375589	−0.187795	−	0.982208i	$-0.560134\pi$
$401$	−7.52117	−0.375589	−0.187795	+	0.982208i	$0.560134\pi$
$402$	0	0
$403$	−26.6810	−1.32907
$404$	0	0
$405$	−6.15739	−0.305963
$406$	0	0
$407$	−36.8878	−1.82846
$408$	0	0
$409$	0.412793	0.0204113	0.0102056	−	0.999948i	$-0.496751\pi$
$409$	0.412793	0.0204113	0.0102056	+	0.999948i	$0.496751\pi$
$410$	0	0
$411$	15.2616	0.752798
$412$	0	0
$413$	4.47538	0.220219
$414$	0	0
$415$	1.13233	0.0555840
$416$	0	0
$417$	5.29102	0.259102
$418$	0	0
$419$	2.68136	0.130993	0.0654966	−	0.997853i	$-0.479137\pi$
$419$	2.68136	0.130993	0.0654966	+	0.997853i	$0.479137\pi$
$420$	0	0
$421$	−5.87045	−0.286108	−0.143054	−	0.989715i	$-0.545692\pi$
$421$	−5.87045	−0.286108	−0.143054	+	0.989715i	$0.545692\pi$
$422$	0	0
$423$	−22.7142	−1.10440
$424$	0	0
$425$	2.05100	0.0994879
$426$	0	0
$427$	40.1384	1.94243
$428$	0	0
$429$	−55.6811	−2.68831
$430$	0	0
$431$	6.35986	0.306344	0.153172	−	0.988200i	$-0.451051\pi$
$431$	6.35986	0.306344	0.153172	+	0.988200i	$0.451051\pi$
$432$	0	0
$433$	−36.1286	−1.73623	−0.868115	−	0.496364i	$-0.834668\pi$
$433$	−36.1286	−1.73623	−0.868115	+	0.496364i	$0.834668\pi$
$434$	0	0
$435$	−6.69521	−0.321011
$436$	0	0
$437$	−36.1582	−1.72968
$438$	0	0
$439$	−19.2290	−0.917751	−0.458876	−	0.888501i	$-0.651748\pi$
$439$	−19.2290	−0.917751	−0.458876	+	0.888501i	$0.651748\pi$
$440$	0	0
$441$	69.7274	3.32035
$442$	0	0
$443$	40.3422	1.91671	0.958357	−	0.285574i	$-0.0921841\pi$
$443$	40.3422	1.91671	0.958357	+	0.285574i	$0.0921841\pi$
$444$	0	0
$445$	8.07143	0.382623
$446$	0	0
$447$	−25.3436	−1.19871
$448$	0	0
$449$	−38.1638	−1.80106	−0.900531	−	0.434791i	$-0.856822\pi$
$449$	−38.1638	−1.80106	−0.900531	+	0.434791i	$0.856822\pi$
$450$	0	0
$451$	−34.5580	−1.62727
$452$	0	0
$453$	−40.9788	−1.92535
$454$	0	0
$455$	−36.4018	−1.70654
$456$	0	0
$457$	−27.8331	−1.30198	−0.650989	−	0.759087i	$-0.725646\pi$
$457$	−27.8331	−1.30198	−0.650989	+	0.759087i	$0.725646\pi$
$458$	0	0
$459$	6.79624	0.317221
$460$	0	0
$461$	−18.5222	−0.862665	−0.431332	−	0.902193i	$-0.641956\pi$
$461$	−18.5222	−0.862665	−0.431332	+	0.902193i	$0.641956\pi$
$462$	0	0
$463$	−9.69466	−0.450549	−0.225275	−	0.974295i	$-0.572328\pi$
$463$	−9.69466	−0.450549	−0.225275	+	0.974295i	$0.572328\pi$
$464$	0	0
$465$	27.9558	1.29642
$466$	0	0
$467$	18.8171	0.870754	0.435377	−	0.900248i	$-0.356615\pi$
$467$	18.8171	0.870754	0.435377	+	0.900248i	$0.356615\pi$
$468$	0	0
$469$	−10.7650	−0.497084
$470$	0	0
$471$	−5.27070	−0.242861
$472$	0	0
$473$	−3.06618	−0.140983
$474$	0	0
$475$	16.5557	0.759627
$476$	0	0
$477$	−8.31387	−0.380666
$478$	0	0
$479$	24.6563	1.12657	0.563287	−	0.826261i	$-0.309536\pi$
$479$	24.6563	1.12657	0.563287	+	0.826261i	$0.309536\pi$
$480$	0	0
$481$	−42.9509	−1.95839
$482$	0	0
$483$	57.9351	2.63614
$484$	0	0
$485$	−22.4389	−1.01890
$486$	0	0
$487$	42.3588	1.91946	0.959729	−	0.280927i	$-0.0906418\pi$
$487$	42.3588	1.91946	0.959729	+	0.280927i	$0.0906418\pi$
$488$	0	0
$489$	−31.5509	−1.42678
$490$	0	0
$491$	7.39626	0.333789	0.166894	−	0.985975i	$-0.446626\pi$
$491$	7.39626	0.333789	0.166894	+	0.985975i	$0.446626\pi$
$492$	0	0
$493$	1.34908	0.0607597
$494$	0	0
$495$	37.3848	1.68032
$496$	0	0
$497$	53.4052	2.39555
$498$	0	0
$499$	16.2424	0.727111	0.363555	−	0.931573i	$-0.381563\pi$
$499$	16.2424	0.727111	0.363555	+	0.931573i	$0.381563\pi$
$500$	0	0
$501$	−12.4400	−0.555778
$502$	0	0
$503$	−28.3872	−1.26572	−0.632862	−	0.774265i	$-0.718120\pi$
$503$	−28.3872	−1.26572	−0.632862	+	0.774265i	$0.718120\pi$
$504$	0	0
$505$	−5.66772	−0.252210
$506$	0	0
$507$	−27.2641	−1.21084
$508$	0	0
$509$	19.1909	0.850621	0.425311	−	0.905047i	$-0.360165\pi$
$509$	19.1909	0.850621	0.425311	+	0.905047i	$0.360165\pi$
$510$	0	0
$511$	−31.3195	−1.38549
$512$	0	0
$513$	54.8594	2.42210
$514$	0	0
$515$	−16.8331	−0.741753
$516$	0	0
$517$	17.2652	0.759322
$518$	0	0
$519$	56.0384	2.45981
$520$	0	0
$521$	23.9137	1.04768	0.523840	−	0.851817i	$-0.324499\pi$
$521$	23.9137	1.04768	0.523840	+	0.851817i	$0.324499\pi$
$522$	0	0
$523$	30.2824	1.32416	0.662078	−	0.749435i	$-0.269675\pi$
$523$	30.2824	1.32416	0.662078	+	0.749435i	$0.269675\pi$
$524$	0	0
$525$	−26.5266	−1.15772
$526$	0	0
$527$	−5.63309	−0.245381
$528$	0	0
$529$	−2.93452	−0.127588
$530$	0	0
$531$	−5.35170	−0.232244
$532$	0	0
$533$	−40.2382	−1.74291
$534$	0	0
$535$	13.3602	0.577613
$536$	0	0
$537$	−5.70376	−0.246135
$538$	0	0
$539$	−53.0002	−2.28288
$540$	0	0
$541$	−38.2312	−1.64369	−0.821844	−	0.569713i	$-0.807055\pi$
$541$	−38.2312	−1.64369	−0.821844	+	0.569713i	$0.807055\pi$
$542$	0	0
$543$	−29.3340	−1.25884
$544$	0	0
$545$	7.86684	0.336978
$546$	0	0
$547$	19.9692	0.853820	0.426910	−	0.904294i	$-0.359602\pi$
$547$	19.9692	0.853820	0.426910	+	0.904294i	$0.359602\pi$
$548$	0	0
$549$	−47.9978	−2.04850
$550$	0	0
$551$	10.8898	0.463922
$552$	0	0
$553$	2.23345	0.0949761
$554$	0	0
$555$	45.0031	1.91027
$556$	0	0
$557$	31.7958	1.34723	0.673616	−	0.739081i	$-0.264740\pi$
$557$	31.7958	1.34723	0.673616	+	0.739081i	$0.264740\pi$
$558$	0	0
$559$	−3.57016	−0.151002
$560$	0	0
$561$	−11.7558	−0.496331
$562$	0	0
$563$	−33.0314	−1.39211	−0.696053	−	0.717990i	$-0.745062\pi$
$563$	−33.0314	−1.39211	−0.696053	+	0.717990i	$0.745062\pi$
$564$	0	0
$565$	−28.7937	−1.21136
$566$	0	0
$567$	−16.0468	−0.673903
$568$	0	0
$569$	6.22713	0.261055	0.130527	−	0.991445i	$-0.458333\pi$
$569$	6.22713	0.261055	0.130527	+	0.991445i	$0.458333\pi$
$570$	0	0
$571$	24.0822	1.00781	0.503905	−	0.863759i	$-0.331896\pi$
$571$	24.0822	1.00781	0.503905	+	0.863759i	$0.331896\pi$
$572$	0	0
$573$	25.0382	1.04599
$574$	0	0
$575$	−9.18733	−0.383138
$576$	0	0
$577$	37.2371	1.55020	0.775100	−	0.631838i	$-0.217699\pi$
$577$	37.2371	1.55020	0.775100	+	0.631838i	$0.217699\pi$
$578$	0	0
$579$	−54.3967	−2.26065
$580$	0	0
$581$	2.95098	0.122427
$582$	0	0
$583$	6.31942	0.261724
$584$	0	0
$585$	43.5296	1.79973
$586$	0	0
$587$	5.29699	0.218630	0.109315	−	0.994007i	$-0.465134\pi$
$587$	5.29699	0.218630	0.109315	+	0.994007i	$0.465134\pi$
$588$	0	0
$589$	−45.4704	−1.87357
$590$	0	0
$591$	−41.3900	−1.70256
$592$	0	0
$593$	−23.7883	−0.976869	−0.488435	−	0.872601i	$-0.662432\pi$
$593$	−23.7883	−0.976869	−0.488435	+	0.872601i	$0.662432\pi$
$594$	0	0
$595$	−7.68542	−0.315072
$596$	0	0
$597$	−2.29248	−0.0938251
$598$	0	0
$599$	17.4922	0.714712	0.357356	−	0.933968i	$-0.383678\pi$
$599$	17.4922	0.714712	0.357356	+	0.933968i	$0.383678\pi$
$600$	0	0
$601$	2.56533	0.104642	0.0523210	−	0.998630i	$-0.483338\pi$
$601$	2.56533	0.104642	0.0523210	+	0.998630i	$0.483338\pi$
$602$	0	0
$603$	12.8729	0.524226
$604$	0	0
$605$	−9.52645	−0.387305
$606$	0	0
$607$	−25.6850	−1.04252	−0.521261	−	0.853398i	$-0.674538\pi$
$607$	−25.6850	−1.04252	−0.521261	+	0.853398i	$0.674538\pi$
$608$	0	0
$609$	−17.4484	−0.707046
$610$	0	0
$611$	20.1030	0.813280
$612$	0	0
$613$	−43.0772	−1.73987	−0.869935	−	0.493166i	$-0.835839\pi$
$613$	−43.0772	−1.73987	−0.869935	+	0.493166i	$0.835839\pi$
$614$	0	0
$615$	42.1607	1.70008
$616$	0	0
$617$	38.7902	1.56164	0.780818	−	0.624758i	$-0.214803\pi$
$617$	38.7902	1.56164	0.780818	+	0.624758i	$0.214803\pi$
$618$	0	0
$619$	−31.9438	−1.28393	−0.641965	−	0.766734i	$-0.721881\pi$
$619$	−31.9438	−1.28393	−0.641965	+	0.766734i	$0.721881\pi$
$620$	0	0
$621$	−30.4434	−1.22165
$622$	0	0
$623$	21.0350	0.842750
$624$	0	0
$625$	−10.5384	−0.421537
$626$	0	0
$627$	−94.8932	−3.78967
$628$	0	0
$629$	−9.06811	−0.361569
$630$	0	0
$631$	39.2873	1.56400	0.782002	−	0.623276i	$-0.214199\pi$
$631$	39.2873	1.56400	0.782002	+	0.623276i	$0.214199\pi$
$632$	0	0
$633$	−16.1092	−0.640284
$634$	0	0
$635$	32.8490	1.30357
$636$	0	0
$637$	−61.7116	−2.44510
$638$	0	0
$639$	−63.8624	−2.52636
$640$	0	0
$641$	44.5326	1.75893	0.879467	−	0.475960i	$-0.157899\pi$
$641$	44.5326	1.75893	0.879467	+	0.475960i	$0.157899\pi$
$642$	0	0
$643$	−44.7336	−1.76412	−0.882061	−	0.471136i	$-0.843844\pi$
$643$	−44.7336	−1.76412	−0.882061	+	0.471136i	$0.843844\pi$
$644$	0	0
$645$	3.74074	0.147291
$646$	0	0
$647$	14.6161	0.574619	0.287310	−	0.957838i	$-0.407239\pi$
$647$	14.6161	0.574619	0.287310	+	0.957838i	$0.407239\pi$
$648$	0	0
$649$	4.06786	0.159677
$650$	0	0
$651$	72.8557	2.85544
$652$	0	0
$653$	30.9412	1.21082	0.605412	−	0.795913i	$-0.293009\pi$
$653$	30.9412	1.21082	0.605412	+	0.795913i	$0.293009\pi$
$654$	0	0
$655$	0.815725	0.0318730
$656$	0	0
$657$	37.4521	1.46115
$658$	0	0
$659$	−10.2649	−0.399862	−0.199931	−	0.979810i	$-0.564072\pi$
$659$	−10.2649	−0.399862	−0.199931	+	0.979810i	$0.564072\pi$
$660$	0	0
$661$	−5.59649	−0.217678	−0.108839	−	0.994059i	$-0.534713\pi$
$661$	−5.59649	−0.217678	−0.108839	+	0.994059i	$0.534713\pi$
$662$	0	0
$663$	−13.6881	−0.531601
$664$	0	0
$665$	−62.0369	−2.40569
$666$	0	0
$667$	−6.04316	−0.233992
$668$	0	0
$669$	−45.2160	−1.74815
$670$	0	0
$671$	36.4834	1.40843
$672$	0	0
$673$	8.81939	0.339962	0.169981	−	0.985447i	$-0.445629\pi$
$673$	8.81939	0.339962	0.169981	+	0.985447i	$0.445629\pi$
$674$	0	0
$675$	13.9391	0.536515
$676$	0	0
$677$	1.11855	0.0429893	0.0214947	−	0.999769i	$-0.493158\pi$
$677$	1.11855	0.0429893	0.0214947	+	0.999769i	$0.493158\pi$
$678$	0	0
$679$	−58.4781	−2.24418
$680$	0	0
$681$	10.2393	0.392371
$682$	0	0
$683$	47.8278	1.83008	0.915041	−	0.403361i	$-0.132158\pi$
$683$	47.8278	1.83008	0.915041	+	0.403361i	$0.132158\pi$
$684$	0	0
$685$	9.06879	0.346501
$686$	0	0
$687$	−49.7196	−1.89692
$688$	0	0
$689$	7.35812	0.280322
$690$	0	0
$691$	−16.4334	−0.625158	−0.312579	−	0.949892i	$-0.601193\pi$
$691$	−16.4334	−0.625158	−0.312579	+	0.949892i	$0.601193\pi$
$692$	0	0
$693$	97.4287	3.70101
$694$	0	0
$695$	3.14405	0.119261
$696$	0	0
$697$	−8.49539	−0.321786
$698$	0	0
$699$	−52.2889	−1.97775
$700$	0	0
$701$	2.88149	0.108832	0.0544162	−	0.998518i	$-0.482670\pi$
$701$	2.88149	0.108832	0.0544162	+	0.998518i	$0.482670\pi$
$702$	0	0
$703$	−73.1980	−2.76071
$704$	0	0
$705$	−21.0635	−0.793297
$706$	0	0
$707$	−14.7707	−0.555508
$708$	0	0
$709$	−21.2223	−0.797020	−0.398510	−	0.917164i	$-0.630473\pi$
$709$	−21.2223	−0.797020	−0.398510	+	0.917164i	$0.630473\pi$
$710$	0	0
$711$	−2.67078	−0.100162
$712$	0	0
$713$	25.2331	0.944988
$714$	0	0
$715$	−33.0871	−1.23739
$716$	0	0
$717$	−23.1173	−0.863331
$718$	0	0
$719$	−29.1647	−1.08766	−0.543830	−	0.839195i	$-0.683027\pi$
$719$	−29.1647	−1.08766	−0.543830	+	0.839195i	$0.683027\pi$
$720$	0	0
$721$	−43.8687	−1.63376
$722$	0	0
$723$	89.4328	3.32604
$724$	0	0
$725$	2.76697	0.102763
$726$	0	0
$727$	−30.7973	−1.14221	−0.571104	−	0.820877i	$-0.693485\pi$
$727$	−30.7973	−1.14221	−0.571104	+	0.820877i	$0.693485\pi$
$728$	0	0
$729$	−39.7331	−1.47160
$730$	0	0
$731$	−0.753759	−0.0278788
$732$	0	0
$733$	8.40796	0.310555	0.155277	−	0.987871i	$-0.450373\pi$
$733$	8.40796	0.310555	0.155277	+	0.987871i	$0.450373\pi$
$734$	0	0
$735$	64.6602	2.38503
$736$	0	0
$737$	−9.78479	−0.360427
$738$	0	0
$739$	25.7995	0.949049	0.474524	−	0.880242i	$-0.342620\pi$
$739$	25.7995	0.949049	0.474524	+	0.880242i	$0.342620\pi$
$740$	0	0
$741$	−110.490	−4.05897
$742$	0	0
$743$	22.8708	0.839047	0.419524	−	0.907744i	$-0.362197\pi$
$743$	22.8708	0.839047	0.419524	+	0.907744i	$0.362197\pi$
$744$	0	0
$745$	−15.0598	−0.551747
$746$	0	0
$747$	−3.52881	−0.129112
$748$	0	0
$749$	34.8182	1.27223
$750$	0	0
$751$	45.5859	1.66345	0.831727	−	0.555185i	$-0.187353\pi$
$751$	45.5859	1.66345	0.831727	+	0.555185i	$0.187353\pi$
$752$	0	0
$753$	13.1397	0.478836
$754$	0	0
$755$	−24.3506	−0.886210
$756$	0	0
$757$	6.92439	0.251671	0.125836	−	0.992051i	$-0.459839\pi$
$757$	6.92439	0.251671	0.125836	+	0.992051i	$0.459839\pi$
$758$	0	0
$759$	52.6596	1.91142
$760$	0	0
$761$	44.3616	1.60811	0.804053	−	0.594557i	$-0.202673\pi$
$761$	44.3616	1.60811	0.804053	+	0.594557i	$0.202673\pi$
$762$	0	0
$763$	20.5018	0.742216
$764$	0	0
$765$	9.19029	0.332276
$766$	0	0
$767$	4.73647	0.171024
$768$	0	0
$769$	37.1631	1.34014	0.670069	−	0.742299i	$-0.266265\pi$
$769$	37.1631	1.34014	0.670069	+	0.742299i	$0.266265\pi$
$770$	0	0
$771$	−13.7805	−0.496291
$772$	0	0
$773$	14.2175	0.511368	0.255684	−	0.966760i	$-0.417699\pi$
$773$	14.2175	0.511368	0.255684	+	0.966760i	$0.417699\pi$
$774$	0	0
$775$	−11.5534	−0.415012
$776$	0	0
$777$	117.283	4.20750
$778$	0	0
$779$	−68.5749	−2.45695
$780$	0	0
$781$	48.5422	1.73698
$782$	0	0
$783$	9.16870	0.327663
$784$	0	0
$785$	−3.13198	−0.111785
$786$	0	0
$787$	34.6933	1.23668	0.618342	−	0.785909i	$-0.287805\pi$
$787$	34.6933	1.23668	0.618342	+	0.785909i	$0.287805\pi$
$788$	0	0
$789$	−61.6449	−2.19462
$790$	0	0
$791$	−75.0396	−2.66810
$792$	0	0
$793$	42.4800	1.50851
$794$	0	0
$795$	−7.70969	−0.273434
$796$	0	0
$797$	−21.3133	−0.754956	−0.377478	−	0.926019i	$-0.623209\pi$
$797$	−21.3133	−0.754956	−0.377478	+	0.926019i	$0.623209\pi$
$798$	0	0
$799$	4.24429	0.150152
$800$	0	0
$801$	−25.1538	−0.888768
$802$	0	0
$803$	−28.4676	−1.00460
$804$	0	0
$805$	34.4265	1.21337
$806$	0	0
$807$	1.88448	0.0663370
$808$	0	0
$809$	−26.0844	−0.917078	−0.458539	−	0.888674i	$-0.651627\pi$
$809$	−26.0844	−0.917078	−0.458539	+	0.888674i	$0.651627\pi$
$810$	0	0
$811$	−0.830328	−0.0291568	−0.0145784	−	0.999894i	$-0.504641\pi$
$811$	−0.830328	−0.0291568	−0.0145784	+	0.999894i	$0.504641\pi$
$812$	0	0
$813$	93.3510	3.27396
$814$	0	0
$815$	−18.7483	−0.656725
$816$	0	0
$817$	−6.08435	−0.212865
$818$	0	0
$819$	113.443	3.96401
$820$	0	0
$821$	−20.1747	−0.704103	−0.352052	−	0.935981i	$-0.614516\pi$
$821$	−20.1747	−0.704103	−0.352052	+	0.935981i	$0.614516\pi$
$822$	0	0
$823$	8.83226	0.307873	0.153937	−	0.988081i	$-0.450805\pi$
$823$	8.83226	0.307873	0.153937	+	0.988081i	$0.450805\pi$
$824$	0	0
$825$	−24.1111	−0.839442
$826$	0	0
$827$	−20.8546	−0.725186	−0.362593	−	0.931948i	$-0.618108\pi$
$827$	−20.8546	−0.725186	−0.362593	+	0.931948i	$0.618108\pi$
$828$	0	0
$829$	−51.3201	−1.78242	−0.891210	−	0.453591i	$-0.850143\pi$
$829$	−51.3201	−1.78242	−0.891210	+	0.453591i	$0.850143\pi$
$830$	0	0
$831$	−41.4608	−1.43826
$832$	0	0
$833$	−13.0290	−0.451429
$834$	0	0
$835$	−7.39214	−0.255816
$836$	0	0
$837$	−38.2838	−1.32328
$838$	0	0
$839$	43.7431	1.51018	0.755089	−	0.655622i	$-0.227594\pi$
$839$	43.7431	1.51018	0.755089	+	0.655622i	$0.227594\pi$
$840$	0	0
$841$	−27.1800	−0.937240
$842$	0	0
$843$	89.6196	3.08666
$844$	0	0
$845$	−16.2010	−0.557330
$846$	0	0
$847$	−24.8269	−0.853064
$848$	0	0
$849$	−18.9399	−0.650014
$850$	0	0
$851$	40.6202	1.39244
$852$	0	0
$853$	−8.69461	−0.297698	−0.148849	−	0.988860i	$-0.547557\pi$
$853$	−8.69461	−0.297698	−0.148849	+	0.988860i	$0.547557\pi$
$854$	0	0
$855$	74.1842	2.53705
$856$	0	0
$857$	6.50397	0.222171	0.111086	−	0.993811i	$-0.464567\pi$
$857$	6.50397	0.222171	0.111086	+	0.993811i	$0.464567\pi$
$858$	0	0
$859$	−36.7116	−1.25258	−0.626292	−	0.779589i	$-0.715428\pi$
$859$	−36.7116	−1.25258	−0.626292	+	0.779589i	$0.715428\pi$
$860$	0	0
$861$	109.875	3.74454
$862$	0	0
$863$	38.9338	1.32532	0.662661	−	0.748920i	$-0.269427\pi$
$863$	38.9338	1.32532	0.662661	+	0.748920i	$0.269427\pi$
$864$	0	0
$865$	33.2994	1.13221
$866$	0	0
$867$	−2.88993	−0.0981472
$868$	0	0
$869$	2.03008	0.0688656
$870$	0	0
$871$	−11.3931	−0.386040
$872$	0	0
$873$	69.9286	2.36673
$874$	0	0
$875$	−54.1899	−1.83195
$876$	0	0
$877$	37.9779	1.28242	0.641211	−	0.767365i	$-0.278432\pi$
$877$	37.9779	1.28242	0.641211	+	0.767365i	$0.278432\pi$
$878$	0	0
$879$	67.0096	2.26018
$880$	0	0
$881$	7.62281	0.256819	0.128409	−	0.991721i	$-0.459013\pi$
$881$	7.62281	0.256819	0.128409	+	0.991721i	$0.459013\pi$
$882$	0	0
$883$	−23.8355	−0.802130	−0.401065	−	0.916050i	$-0.631360\pi$
$883$	−23.8355	−0.802130	−0.401065	+	0.916050i	$0.631360\pi$
$884$	0	0
$885$	−4.96278	−0.166822
$886$	0	0
$887$	18.3128	0.614885	0.307442	−	0.951567i	$-0.400527\pi$
$887$	18.3128	0.614885	0.307442	+	0.951567i	$0.400527\pi$
$888$	0	0
$889$	85.6081	2.87120
$890$	0	0
$891$	−14.5856	−0.488636
$892$	0	0
$893$	34.2600	1.14647
$894$	0	0
$895$	−3.38931	−0.113292
$896$	0	0
$897$	61.3151	2.04725
$898$	0	0
$899$	−7.59951	−0.253458
$900$	0	0
$901$	1.55350	0.0517546
$902$	0	0
$903$	9.74876	0.324419
$904$	0	0
$905$	−17.4310	−0.579425
$906$	0	0
$907$	6.30660	0.209407	0.104704	−	0.994503i	$-0.466611\pi$
$907$	6.30660	0.209407	0.104704	+	0.994503i	$0.466611\pi$
$908$	0	0
$909$	17.6629	0.585841
$910$	0	0
$911$	14.9693	0.495955	0.247977	−	0.968766i	$-0.420234\pi$
$911$	14.9693	0.495955	0.247977	+	0.968766i	$0.420234\pi$
$912$	0	0
$913$	2.68226	0.0887700
$914$	0	0
$915$	−44.5097	−1.47145
$916$	0	0
$917$	2.12587	0.0702023
$918$	0	0
$919$	−33.7560	−1.11351	−0.556754	−	0.830678i	$-0.687953\pi$
$919$	−33.7560	−1.11351	−0.556754	+	0.830678i	$0.687953\pi$
$920$	0	0
$921$	66.0605	2.17677
$922$	0	0
$923$	56.5208	1.86041
$924$	0	0
$925$	−18.5987	−0.611520
$926$	0	0
$927$	52.4586	1.72297
$928$	0	0
$929$	−31.4366	−1.03140	−0.515701	−	0.856769i	$-0.672468\pi$
$929$	−31.4366	−1.03140	−0.515701	+	0.856769i	$0.672468\pi$
$930$	0	0
$931$	−105.171	−3.44683
$932$	0	0
$933$	98.3109	3.21855
$934$	0	0
$935$	−6.98559	−0.228453
$936$	0	0
$937$	28.8399	0.942159	0.471079	−	0.882091i	$-0.343865\pi$
$937$	28.8399	0.942159	0.471079	+	0.882091i	$0.343865\pi$
$938$	0	0
$939$	83.9738	2.74038
$940$	0	0
$941$	54.4887	1.77628	0.888141	−	0.459571i	$-0.151997\pi$
$941$	54.4887	1.77628	0.888141	+	0.459571i	$0.151997\pi$
$942$	0	0
$943$	38.0547	1.23923
$944$	0	0
$945$	−52.2320	−1.69911
$946$	0	0
$947$	28.4472	0.924408	0.462204	−	0.886774i	$-0.347059\pi$
$947$	28.4472	0.924408	0.462204	+	0.886774i	$0.347059\pi$
$948$	0	0
$949$	−33.1467	−1.07599
$950$	0	0
$951$	−3.85785	−0.125099
$952$	0	0
$953$	44.2333	1.43286	0.716429	−	0.697660i	$-0.245775\pi$
$953$	44.2333	1.43286	0.716429	+	0.697660i	$0.245775\pi$
$954$	0	0
$955$	14.8783	0.481450
$956$	0	0
$957$	−15.8596	−0.512668
$958$	0	0
$959$	23.6342	0.763190
$960$	0	0
$961$	0.731652	0.0236017
$962$	0	0
$963$	−41.6359	−1.34170
$964$	0	0
$965$	−32.3238	−1.04054
$966$	0	0
$967$	−30.3358	−0.975534	−0.487767	−	0.872974i	$-0.662189\pi$
$967$	−30.3358	−0.975534	−0.487767	+	0.872974i	$0.662189\pi$
$968$	0	0
$969$	−23.3276	−0.749390
$970$	0	0
$971$	−28.4789	−0.913930	−0.456965	−	0.889485i	$-0.651064\pi$
$971$	−28.4789	−0.913930	−0.456965	+	0.889485i	$0.651064\pi$
$972$	0	0
$973$	8.19374	0.262679
$974$	0	0
$975$	−28.0742	−0.899093
$976$	0	0
$977$	−20.4121	−0.653040	−0.326520	−	0.945190i	$-0.605876\pi$
$977$	−20.4121	−0.653040	−0.326520	+	0.945190i	$0.605876\pi$
$978$	0	0
$979$	19.1196	0.611065
$980$	0	0
$981$	−24.5162	−0.782744
$982$	0	0
$983$	5.90253	0.188262	0.0941308	−	0.995560i	$-0.469993\pi$
$983$	5.90253	0.188262	0.0941308	+	0.995560i	$0.469993\pi$
$984$	0	0
$985$	−24.5949	−0.783660
$986$	0	0
$987$	−54.8937	−1.74729
$988$	0	0
$989$	3.37642	0.107364
$990$	0	0
$991$	−31.7212	−1.00766	−0.503828	−	0.863804i	$-0.668075\pi$
$991$	−31.7212	−1.00766	−0.503828	+	0.863804i	$0.668075\pi$
$992$	0	0
$993$	−13.1815	−0.418302
$994$	0	0
$995$	−1.36225	−0.0431862
$996$	0	0
$997$	25.2297	0.799032	0.399516	−	0.916726i	$-0.369178\pi$
$997$	25.2297	0.799032	0.399516	+	0.916726i	$0.369178\pi$
$998$	0	0
$999$	−61.6291	−1.94986

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.3	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.3	✓	23	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.88993 q^{3} -1.71727 q^{5} -4.47538 q^{7} +5.35170 q^{9} +O(q^{10})\)	\(q-2.88993 q^{3} -1.71727 q^{5} -4.47538 q^{7} +5.35170 q^{9} -4.06786 q^{11} -4.73647 q^{13} +4.96278 q^{15} -1.00000 q^{17} -8.07202 q^{19} +12.9335 q^{21} +4.47945 q^{23} -2.05100 q^{25} -6.79624 q^{27} -1.34908 q^{29} +5.63309 q^{31} +11.7558 q^{33} +7.68542 q^{35} +9.06811 q^{37} +13.6881 q^{39} +8.49539 q^{41} +0.753759 q^{43} -9.19029 q^{45} -4.24429 q^{47} +13.0290 q^{49} +2.88993 q^{51} -1.55350 q^{53} +6.98559 q^{55} +23.3276 q^{57} -1.00000 q^{59} -8.96870 q^{61} -23.9509 q^{63} +8.13378 q^{65} +2.40539 q^{67} -12.9453 q^{69} -11.9331 q^{71} +6.99818 q^{73} +5.92724 q^{75} +18.2052 q^{77} -0.499053 q^{79} +3.58558 q^{81} -0.659380 q^{83} +1.71727 q^{85} +3.89876 q^{87} -4.70016 q^{89} +21.1975 q^{91} -16.2792 q^{93} +13.8618 q^{95} +13.0666 q^{97} -21.7699 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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