LMFDB - Embedded newform 6028.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$0$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.97866 q^{3} -3.46603 q^{5} +0.700745 q^{7} +5.87240 q^{9} +O(q^{10})$	$q-2.97866 q^{3} -3.46603 q^{5} +0.700745 q^{7} +5.87240 q^{9} +1.00000 q^{11} +0.506038 q^{13} +10.3241 q^{15} +0.704097 q^{17} +1.86130 q^{19} -2.08728 q^{21} +0.949603 q^{23} +7.01335 q^{25} -8.55590 q^{27} +4.63773 q^{29} +3.79278 q^{31} -2.97866 q^{33} -2.42880 q^{35} -2.62166 q^{37} -1.50731 q^{39} +12.6681 q^{41} -4.51354 q^{43} -20.3539 q^{45} +8.04814 q^{47} -6.50896 q^{49} -2.09726 q^{51} -4.49541 q^{53} -3.46603 q^{55} -5.54419 q^{57} -4.94384 q^{59} -12.9830 q^{61} +4.11505 q^{63} -1.75394 q^{65} +4.81273 q^{67} -2.82854 q^{69} -12.4570 q^{71} +11.4023 q^{73} -20.8904 q^{75} +0.700745 q^{77} -8.47461 q^{79} +7.86789 q^{81} +6.22743 q^{83} -2.44042 q^{85} -13.8142 q^{87} -4.96261 q^{89} +0.354604 q^{91} -11.2974 q^{93} -6.45133 q^{95} +10.6418 q^{97} +5.87240 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * 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
$3$	−2.97866	−1.71973	−0.859864	−	0.510523i	$-0.829452\pi$
$3$	−2.97866	−1.71973	−0.859864	+	0.510523i	$0.829452\pi$
$4$	0	0
$5$	−3.46603	−1.55005	−0.775027	−	0.631928i	$-0.782264\pi$
$5$	−3.46603	−1.55005	−0.775027	+	0.631928i	$0.782264\pi$
$6$	0	0
$7$	0.700745	0.264857	0.132428	−	0.991193i	$-0.457723\pi$
$7$	0.700745	0.264857	0.132428	+	0.991193i	$0.457723\pi$
$8$	0	0
$9$	5.87240	1.95747
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.506038	0.140350	0.0701749	−	0.997535i	$-0.477644\pi$
$13$	0.506038	0.140350	0.0701749	+	0.997535i	$0.477644\pi$
$14$	0	0
$15$	10.3241	2.66567
$16$	0	0
$17$	0.704097	0.170769	0.0853843	−	0.996348i	$-0.472788\pi$
$17$	0.704097	0.170769	0.0853843	+	0.996348i	$0.472788\pi$
$18$	0	0
$19$	1.86130	0.427013	0.213506	−	0.976942i	$-0.431512\pi$
$19$	1.86130	0.427013	0.213506	+	0.976942i	$0.431512\pi$
$20$	0	0
$21$	−2.08728	−0.455481
$22$	0	0
$23$	0.949603	0.198006	0.0990030	−	0.995087i	$-0.468435\pi$
$23$	0.949603	0.198006	0.0990030	+	0.995087i	$0.468435\pi$
$24$	0	0
$25$	7.01335	1.40267
$26$	0	0
$27$	−8.55590	−1.64658
$28$	0	0
$29$	4.63773	0.861205	0.430603	−	0.902542i	$-0.358301\pi$
$29$	4.63773	0.861205	0.430603	+	0.902542i	$0.358301\pi$
$30$	0	0
$31$	3.79278	0.681204	0.340602	−	0.940208i	$-0.389369\pi$
$31$	3.79278	0.681204	0.340602	+	0.940208i	$0.389369\pi$
$32$	0	0
$33$	−2.97866	−0.518518
$34$	0	0
$35$	−2.42880	−0.410542
$36$	0	0
$37$	−2.62166	−0.430998	−0.215499	−	0.976504i	$-0.569138\pi$
$37$	−2.62166	−0.430998	−0.215499	+	0.976504i	$0.569138\pi$
$38$	0	0
$39$	−1.50731	−0.241363
$40$	0	0
$41$	12.6681	1.97843	0.989214	−	0.146478i	$-0.0467938\pi$
$41$	12.6681	1.97843	0.989214	+	0.146478i	$0.0467938\pi$
$42$	0	0
$43$	−4.51354	−0.688309	−0.344155	−	0.938913i	$-0.611834\pi$
$43$	−4.51354	−0.688309	−0.344155	+	0.938913i	$0.611834\pi$
$44$	0	0
$45$	−20.3539	−3.03418
$46$	0	0
$47$	8.04814	1.17394	0.586971	−	0.809608i	$-0.300320\pi$
$47$	8.04814	1.17394	0.586971	+	0.809608i	$0.300320\pi$
$48$	0	0
$49$	−6.50896	−0.929851
$50$	0	0
$51$	−2.09726	−0.293676
$52$	0	0
$53$	−4.49541	−0.617492	−0.308746	−	0.951145i	$-0.599909\pi$
$53$	−4.49541	−0.617492	−0.308746	+	0.951145i	$0.599909\pi$
$54$	0	0
$55$	−3.46603	−0.467359
$56$	0	0
$57$	−5.54419	−0.734346
$58$	0	0
$59$	−4.94384	−0.643633	−0.321817	−	0.946802i	$-0.604293\pi$
$59$	−4.94384	−0.643633	−0.321817	+	0.946802i	$0.604293\pi$
$60$	0	0
$61$	−12.9830	−1.66230	−0.831151	−	0.556047i	$-0.812318\pi$
$61$	−12.9830	−1.66230	−0.831151	+	0.556047i	$0.812318\pi$
$62$	0	0
$63$	4.11505	0.518448
$64$	0	0
$65$	−1.75394	−0.217550
$66$	0	0
$67$	4.81273	0.587968	0.293984	−	0.955810i	$-0.405019\pi$
$67$	4.81273	0.587968	0.293984	+	0.955810i	$0.405019\pi$
$68$	0	0
$69$	−2.82854	−0.340517
$70$	0	0
$71$	−12.4570	−1.47837	−0.739184	−	0.673503i	$-0.764789\pi$
$71$	−12.4570	−1.47837	−0.739184	+	0.673503i	$0.764789\pi$
$72$	0	0
$73$	11.4023	1.33454	0.667271	−	0.744815i	$-0.267462\pi$
$73$	11.4023	1.33454	0.667271	+	0.744815i	$0.267462\pi$
$74$	0	0
$75$	−20.8904	−2.41221
$76$	0	0
$77$	0.700745	0.0798573
$78$	0	0
$79$	−8.47461	−0.953468	−0.476734	−	0.879048i	$-0.658179\pi$
$79$	−8.47461	−0.953468	−0.476734	+	0.879048i	$0.658179\pi$
$80$	0	0
$81$	7.86789	0.874210
$82$	0	0
$83$	6.22743	0.683549	0.341774	−	0.939782i	$-0.388972\pi$
$83$	6.22743	0.683549	0.341774	+	0.939782i	$0.388972\pi$
$84$	0	0
$85$	−2.44042	−0.264701
$86$	0	0
$87$	−13.8142	−1.48104
$88$	0	0
$89$	−4.96261	−0.526036	−0.263018	−	0.964791i	$-0.584718\pi$
$89$	−4.96261	−0.526036	−0.263018	+	0.964791i	$0.584718\pi$
$90$	0	0
$91$	0.354604	0.0371726
$92$	0	0
$93$	−11.2974	−1.17149
$94$	0	0
$95$	−6.45133	−0.661893
$96$	0	0
$97$	10.6418	1.08051	0.540257	−	0.841500i	$-0.318327\pi$
$97$	10.6418	1.08051	0.540257	+	0.841500i	$0.318327\pi$
$98$	0	0
$99$	5.87240	0.590198
$100$	0	0
$101$	−3.50109	−0.348372	−0.174186	−	0.984713i	$-0.555729\pi$
$101$	−3.50109	−0.348372	−0.174186	+	0.984713i	$0.555729\pi$
$102$	0	0
$103$	6.82384	0.672373	0.336186	−	0.941795i	$-0.390863\pi$
$103$	6.82384	0.672373	0.336186	+	0.941795i	$0.390863\pi$
$104$	0	0
$105$	7.23456	0.706021
$106$	0	0
$107$	15.5362	1.50194	0.750971	−	0.660335i	$-0.229586\pi$
$107$	15.5362	1.50194	0.750971	+	0.660335i	$0.229586\pi$
$108$	0	0
$109$	−9.35868	−0.896399	−0.448200	−	0.893934i	$-0.647935\pi$
$109$	−9.35868	−0.896399	−0.448200	+	0.893934i	$0.647935\pi$
$110$	0	0
$111$	7.80903	0.741200
$112$	0	0
$113$	−19.2376	−1.80972	−0.904858	−	0.425713i	$-0.860023\pi$
$113$	−19.2376	−1.80972	−0.904858	+	0.425713i	$0.860023\pi$
$114$	0	0
$115$	−3.29135	−0.306920
$116$	0	0
$117$	2.97166	0.274730
$118$	0	0
$119$	0.493392	0.0452292
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−37.7340	−3.40236
$124$	0	0
$125$	−6.97831	−0.624159
$126$	0	0
$127$	−1.24295	−0.110294	−0.0551470	−	0.998478i	$-0.517563\pi$
$127$	−1.24295	−0.110294	−0.0551470	+	0.998478i	$0.517563\pi$
$128$	0	0
$129$	13.4443	1.18370
$130$	0	0
$131$	1.52519	0.133256	0.0666281	−	0.997778i	$-0.478776\pi$
$131$	1.52519	0.133256	0.0666281	+	0.997778i	$0.478776\pi$
$132$	0	0
$133$	1.30430	0.113097
$134$	0	0
$135$	29.6550	2.55229
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	−8.58465	−0.728140	−0.364070	−	0.931372i	$-0.618613\pi$
$139$	−8.58465	−0.728140	−0.364070	+	0.931372i	$0.618613\pi$
$140$	0	0
$141$	−23.9727	−2.01886
$142$	0	0
$143$	0.506038	0.0423170
$144$	0	0
$145$	−16.0745	−1.33492
$146$	0	0
$147$	19.3880	1.59909
$148$	0	0
$149$	1.87899	0.153933	0.0769664	−	0.997034i	$-0.475477\pi$
$149$	1.87899	0.153933	0.0769664	+	0.997034i	$0.475477\pi$
$150$	0	0
$151$	−13.9760	−1.13735	−0.568676	−	0.822562i	$-0.692544\pi$
$151$	−13.9760	−1.13735	−0.568676	+	0.822562i	$0.692544\pi$
$152$	0	0
$153$	4.13474	0.334274
$154$	0	0
$155$	−13.1459	−1.05590
$156$	0	0
$157$	−13.2912	−1.06076	−0.530378	−	0.847761i	$-0.677950\pi$
$157$	−13.2912	−1.06076	−0.530378	+	0.847761i	$0.677950\pi$
$158$	0	0
$159$	13.3903	1.06192
$160$	0	0
$161$	0.665429	0.0524432
$162$	0	0
$163$	9.42828	0.738480	0.369240	−	0.929334i	$-0.379618\pi$
$163$	9.42828	0.738480	0.369240	+	0.929334i	$0.379618\pi$
$164$	0	0
$165$	10.3241	0.803731
$166$	0	0
$167$	14.5176	1.12340	0.561702	−	0.827340i	$-0.310147\pi$
$167$	14.5176	1.12340	0.561702	+	0.827340i	$0.310147\pi$
$168$	0	0
$169$	−12.7439	−0.980302
$170$	0	0
$171$	10.9303	0.835863
$172$	0	0
$173$	−4.56081	−0.346752	−0.173376	−	0.984856i	$-0.555468\pi$
$173$	−4.56081	−0.346752	−0.173376	+	0.984856i	$0.555468\pi$
$174$	0	0
$175$	4.91456	0.371506
$176$	0	0
$177$	14.7260	1.10687
$178$	0	0
$179$	−0.132234	−0.00988362	−0.00494181	−	0.999988i	$-0.501573\pi$
$179$	−0.132234	−0.00988362	−0.00494181	+	0.999988i	$0.501573\pi$
$180$	0	0
$181$	20.8028	1.54626	0.773131	−	0.634246i	$-0.218689\pi$
$181$	20.8028	1.54626	0.773131	+	0.634246i	$0.218689\pi$
$182$	0	0
$183$	38.6719	2.85871
$184$	0	0
$185$	9.08675	0.668071
$186$	0	0
$187$	0.704097	0.0514887
$188$	0	0
$189$	−5.99550	−0.436108
$190$	0	0
$191$	−3.17565	−0.229782	−0.114891	−	0.993378i	$-0.536652\pi$
$191$	−3.17565	−0.229782	−0.114891	+	0.993378i	$0.536652\pi$
$192$	0	0
$193$	15.1389	1.08972	0.544860	−	0.838527i	$-0.316583\pi$
$193$	15.1389	1.08972	0.544860	+	0.838527i	$0.316583\pi$
$194$	0	0
$195$	5.22439	0.374127
$196$	0	0
$197$	−6.16844	−0.439483	−0.219742	−	0.975558i	$-0.570521\pi$
$197$	−6.16844	−0.439483	−0.219742	+	0.975558i	$0.570521\pi$
$198$	0	0
$199$	−15.3011	−1.08467	−0.542333	−	0.840163i	$-0.682459\pi$
$199$	−15.3011	−1.08467	−0.542333	+	0.840163i	$0.682459\pi$
$200$	0	0
$201$	−14.3355	−1.01115
$202$	0	0
$203$	3.24987	0.228096
$204$	0	0
$205$	−43.9080	−3.06667
$206$	0	0
$207$	5.57645	0.387590
$208$	0	0
$209$	1.86130	0.128749
$210$	0	0
$211$	−16.8049	−1.15690	−0.578448	−	0.815719i	$-0.696341\pi$
$211$	−16.8049	−1.15690	−0.578448	+	0.815719i	$0.696341\pi$
$212$	0	0
$213$	37.1050	2.54239
$214$	0	0
$215$	15.6441	1.06692
$216$	0	0
$217$	2.65777	0.180421
$218$	0	0
$219$	−33.9637	−2.29505
$220$	0	0
$221$	0.356300	0.0239673
$222$	0	0
$223$	27.9126	1.86917	0.934584	−	0.355743i	$-0.115772\pi$
$223$	27.9126	1.86917	0.934584	+	0.355743i	$0.115772\pi$
$224$	0	0
$225$	41.1852	2.74568
$226$	0	0
$227$	−5.16147	−0.342579	−0.171289	−	0.985221i	$-0.554793\pi$
$227$	−5.16147	−0.342579	−0.171289	+	0.985221i	$0.554793\pi$
$228$	0	0
$229$	13.4634	0.889687	0.444843	−	0.895608i	$-0.353259\pi$
$229$	13.4634	0.889687	0.444843	+	0.895608i	$0.353259\pi$
$230$	0	0
$231$	−2.08728	−0.137333
$232$	0	0
$233$	30.1451	1.97487	0.987437	−	0.158014i	$-0.0505092\pi$
$233$	30.1451	1.97487	0.987437	+	0.158014i	$0.0505092\pi$
$234$	0	0
$235$	−27.8951	−1.81967
$236$	0	0
$237$	25.2430	1.63971
$238$	0	0
$239$	6.31688	0.408605	0.204302	−	0.978908i	$-0.434507\pi$
$239$	6.31688	0.408605	0.204302	+	0.978908i	$0.434507\pi$
$240$	0	0
$241$	−5.98883	−0.385775	−0.192887	−	0.981221i	$-0.561785\pi$
$241$	−5.98883	−0.385775	−0.192887	+	0.981221i	$0.561785\pi$
$242$	0	0
$243$	2.23195	0.143180
$244$	0	0
$245$	22.5602	1.44132
$246$	0	0
$247$	0.941891	0.0599311
$248$	0	0
$249$	−18.5494	−1.17552
$250$	0	0
$251$	27.2032	1.71705	0.858525	−	0.512772i	$-0.171382\pi$
$251$	27.2032	1.71705	0.858525	+	0.512772i	$0.171382\pi$
$252$	0	0
$253$	0.949603	0.0597010
$254$	0	0
$255$	7.26918	0.455213
$256$	0	0
$257$	−12.8279	−0.800185	−0.400092	−	0.916475i	$-0.631022\pi$
$257$	−12.8279	−0.800185	−0.400092	+	0.916475i	$0.631022\pi$
$258$	0	0
$259$	−1.83711	−0.114153
$260$	0	0
$261$	27.2346	1.68578
$262$	0	0
$263$	24.1278	1.48778	0.743892	−	0.668300i	$-0.232978\pi$
$263$	24.1278	1.48778	0.743892	+	0.668300i	$0.232978\pi$
$264$	0	0
$265$	15.5812	0.957146
$266$	0	0
$267$	14.7819	0.904638
$268$	0	0
$269$	−4.45993	−0.271927	−0.135963	−	0.990714i	$-0.543413\pi$
$269$	−4.45993	−0.271927	−0.135963	+	0.990714i	$0.543413\pi$
$270$	0	0
$271$	10.0402	0.609897	0.304948	−	0.952369i	$-0.401361\pi$
$271$	10.0402	0.609897	0.304948	+	0.952369i	$0.401361\pi$
$272$	0	0
$273$	−1.05624	−0.0639267
$274$	0	0
$275$	7.01335	0.422921
$276$	0	0
$277$	−12.7169	−0.764082	−0.382041	−	0.924145i	$-0.624779\pi$
$277$	−12.7169	−0.764082	−0.382041	+	0.924145i	$0.624779\pi$
$278$	0	0
$279$	22.2727	1.33343
$280$	0	0
$281$	25.7896	1.53848	0.769241	−	0.638959i	$-0.220635\pi$
$281$	25.7896	1.53848	0.769241	+	0.638959i	$0.220635\pi$
$282$	0	0
$283$	4.92649	0.292849	0.146425	−	0.989222i	$-0.453223\pi$
$283$	4.92649	0.292849	0.146425	+	0.989222i	$0.453223\pi$
$284$	0	0
$285$	19.2163	1.13828
$286$	0	0
$287$	8.87712	0.524000
$288$	0	0
$289$	−16.5042	−0.970838
$290$	0	0
$291$	−31.6984	−1.85819
$292$	0	0
$293$	−23.1704	−1.35363	−0.676814	−	0.736154i	$-0.736640\pi$
$293$	−23.1704	−1.35363	−0.676814	+	0.736154i	$0.736640\pi$
$294$	0	0
$295$	17.1355	0.997667
$296$	0	0
$297$	−8.55590	−0.496464
$298$	0	0
$299$	0.480536	0.0277901
$300$	0	0
$301$	−3.16284	−0.182303
$302$	0	0
$303$	10.4286	0.599105
$304$	0	0
$305$	44.9994	2.57666
$306$	0	0
$307$	18.3149	1.04529	0.522645	−	0.852551i	$-0.324946\pi$
$307$	18.3149	1.04529	0.522645	+	0.852551i	$0.324946\pi$
$308$	0	0
$309$	−20.3259	−1.15630
$310$	0	0
$311$	21.5401	1.22142	0.610712	−	0.791853i	$-0.290883\pi$
$311$	21.5401	1.22142	0.610712	+	0.791853i	$0.290883\pi$
$312$	0	0
$313$	4.07413	0.230284	0.115142	−	0.993349i	$-0.463268\pi$
$313$	4.07413	0.230284	0.115142	+	0.993349i	$0.463268\pi$
$314$	0	0
$315$	−14.2629	−0.803623
$316$	0	0
$317$	21.3470	1.19897	0.599484	−	0.800387i	$-0.295373\pi$
$317$	21.3470	1.19897	0.599484	+	0.800387i	$0.295373\pi$
$318$	0	0
$319$	4.63773	0.259663
$320$	0	0
$321$	−46.2770	−2.58293
$322$	0	0
$323$	1.31054	0.0729204
$324$	0	0
$325$	3.54902	0.196864
$326$	0	0
$327$	27.8763	1.54156
$328$	0	0
$329$	5.63969	0.310926
$330$	0	0
$331$	−0.861587	−0.0473571	−0.0236786	−	0.999720i	$-0.507538\pi$
$331$	−0.861587	−0.0473571	−0.0236786	+	0.999720i	$0.507538\pi$
$332$	0	0
$333$	−15.3954	−0.843665
$334$	0	0
$335$	−16.6810	−0.911382
$336$	0	0
$337$	−18.4029	−1.00247	−0.501235	−	0.865311i	$-0.667121\pi$
$337$	−18.4029	−1.00247	−0.501235	+	0.865311i	$0.667121\pi$
$338$	0	0
$339$	57.3021	3.11222
$340$	0	0
$341$	3.79278	0.205391
$342$	0	0
$343$	−9.46633	−0.511134
$344$	0	0
$345$	9.80381	0.527819
$346$	0	0
$347$	−0.900978	−0.0483670	−0.0241835	−	0.999708i	$-0.507699\pi$
$347$	−0.900978	−0.0483670	−0.0241835	+	0.999708i	$0.507699\pi$
$348$	0	0
$349$	4.22470	0.226143	0.113072	−	0.993587i	$-0.463931\pi$
$349$	4.22470	0.226143	0.113072	+	0.993587i	$0.463931\pi$
$350$	0	0
$351$	−4.32961	−0.231098
$352$	0	0
$353$	8.25216	0.439218	0.219609	−	0.975588i	$-0.429522\pi$
$353$	8.25216	0.439218	0.219609	+	0.975588i	$0.429522\pi$
$354$	0	0
$355$	43.1762	2.29155
$356$	0	0
$357$	−1.46965	−0.0777820
$358$	0	0
$359$	3.99669	0.210937	0.105469	−	0.994423i	$-0.466366\pi$
$359$	3.99669	0.210937	0.105469	+	0.994423i	$0.466366\pi$
$360$	0	0
$361$	−15.5355	−0.817660
$362$	0	0
$363$	−2.97866	−0.156339
$364$	0	0
$365$	−39.5208	−2.06861
$366$	0	0
$367$	−1.38680	−0.0723902	−0.0361951	−	0.999345i	$-0.511524\pi$
$367$	−1.38680	−0.0723902	−0.0361951	+	0.999345i	$0.511524\pi$
$368$	0	0
$369$	74.3923	3.87271
$370$	0	0
$371$	−3.15013	−0.163547
$372$	0	0
$373$	−17.8770	−0.925636	−0.462818	−	0.886453i	$-0.653162\pi$
$373$	−17.8770	−0.925636	−0.462818	+	0.886453i	$0.653162\pi$
$374$	0	0
$375$	20.7860	1.07338
$376$	0	0
$377$	2.34687	0.120870
$378$	0	0
$379$	10.8452	0.557081	0.278541	−	0.960424i	$-0.410149\pi$
$379$	10.8452	0.557081	0.278541	+	0.960424i	$0.410149\pi$
$380$	0	0
$381$	3.70232	0.189676
$382$	0	0
$383$	−5.50180	−0.281129	−0.140564	−	0.990072i	$-0.544892\pi$
$383$	−5.50180	−0.281129	−0.140564	+	0.990072i	$0.544892\pi$
$384$	0	0
$385$	−2.42880	−0.123783
$386$	0	0
$387$	−26.5053	−1.34734
$388$	0	0
$389$	5.72639	0.290340	0.145170	−	0.989407i	$-0.453627\pi$
$389$	5.72639	0.290340	0.145170	+	0.989407i	$0.453627\pi$
$390$	0	0
$391$	0.668613	0.0338132
$392$	0	0
$393$	−4.54301	−0.229164
$394$	0	0
$395$	29.3732	1.47793
$396$	0	0
$397$	−24.8519	−1.24728	−0.623640	−	0.781712i	$-0.714347\pi$
$397$	−24.8519	−1.24728	−0.623640	+	0.781712i	$0.714347\pi$
$398$	0	0
$399$	−3.88506	−0.194496
$400$	0	0
$401$	19.7026	0.983903	0.491952	−	0.870623i	$-0.336284\pi$
$401$	19.7026	0.983903	0.491952	+	0.870623i	$0.336284\pi$
$402$	0	0
$403$	1.91929	0.0956068
$404$	0	0
$405$	−27.2703	−1.35507
$406$	0	0
$407$	−2.62166	−0.129951
$408$	0	0
$409$	−15.0735	−0.745337	−0.372668	−	0.927965i	$-0.621557\pi$
$409$	−15.0735	−0.745337	−0.372668	+	0.927965i	$0.621557\pi$
$410$	0	0
$411$	−2.97866	−0.146926
$412$	0	0
$413$	−3.46437	−0.170470
$414$	0	0
$415$	−21.5844	−1.05954
$416$	0	0
$417$	25.5707	1.25220
$418$	0	0
$419$	−26.8317	−1.31082	−0.655408	−	0.755275i	$-0.727503\pi$
$419$	−26.8317	−1.31082	−0.655408	+	0.755275i	$0.727503\pi$
$420$	0	0
$421$	−0.388448	−0.0189318	−0.00946589	−	0.999955i	$-0.503013\pi$
$421$	−0.388448	−0.0189318	−0.00946589	+	0.999955i	$0.503013\pi$
$422$	0	0
$423$	47.2619	2.29795
$424$	0	0
$425$	4.93808	0.239532
$426$	0	0
$427$	−9.09776	−0.440272
$428$	0	0
$429$	−1.50731	−0.0727738
$430$	0	0
$431$	24.0175	1.15688	0.578442	−	0.815724i	$-0.303661\pi$
$431$	24.0175	1.15688	0.578442	+	0.815724i	$0.303661\pi$
$432$	0	0
$433$	−16.1300	−0.775159	−0.387580	−	0.921836i	$-0.626689\pi$
$433$	−16.1300	−0.775159	−0.387580	+	0.921836i	$0.626689\pi$
$434$	0	0
$435$	47.8805	2.29569
$436$	0	0
$437$	1.76750	0.0845510
$438$	0	0
$439$	20.1625	0.962304	0.481152	−	0.876637i	$-0.340218\pi$
$439$	20.1625	0.962304	0.481152	+	0.876637i	$0.340218\pi$
$440$	0	0
$441$	−38.2232	−1.82015
$442$	0	0
$443$	9.13624	0.434076	0.217038	−	0.976163i	$-0.430360\pi$
$443$	9.13624	0.434076	0.217038	+	0.976163i	$0.430360\pi$
$444$	0	0
$445$	17.2005	0.815384
$446$	0	0
$447$	−5.59687	−0.264723
$448$	0	0
$449$	−15.7346	−0.742560	−0.371280	−	0.928521i	$-0.621081\pi$
$449$	−15.7346	−0.742560	−0.371280	+	0.928521i	$0.621081\pi$
$450$	0	0
$451$	12.6681	0.596518
$452$	0	0
$453$	41.6298	1.95594
$454$	0	0
$455$	−1.22907	−0.0576195
$456$	0	0
$457$	−16.5285	−0.773169	−0.386585	−	0.922254i	$-0.626345\pi$
$457$	−16.5285	−0.773169	−0.386585	+	0.922254i	$0.626345\pi$
$458$	0	0
$459$	−6.02418	−0.281185
$460$	0	0
$461$	1.41514	0.0659099	0.0329549	−	0.999457i	$-0.489508\pi$
$461$	1.41514	0.0659099	0.0329549	+	0.999457i	$0.489508\pi$
$462$	0	0
$463$	14.5975	0.678401	0.339201	−	0.940714i	$-0.389843\pi$
$463$	14.5975	0.678401	0.339201	+	0.940714i	$0.389843\pi$
$464$	0	0
$465$	39.1571	1.81587
$466$	0	0
$467$	34.4176	1.59266	0.796329	−	0.604863i	$-0.206772\pi$
$467$	34.4176	1.59266	0.796329	+	0.604863i	$0.206772\pi$
$468$	0	0
$469$	3.37249	0.155727
$470$	0	0
$471$	39.5900	1.82421
$472$	0	0
$473$	−4.51354	−0.207533
$474$	0	0
$475$	13.0540	0.598957
$476$	0	0
$477$	−26.3988	−1.20872
$478$	0	0
$479$	12.8434	0.586829	0.293414	−	0.955985i	$-0.405208\pi$
$479$	12.8434	0.586829	0.293414	+	0.955985i	$0.405208\pi$
$480$	0	0
$481$	−1.32666	−0.0604905
$482$	0	0
$483$	−1.98209	−0.0901881
$484$	0	0
$485$	−36.8849	−1.67486
$486$	0	0
$487$	1.99306	0.0903140	0.0451570	−	0.998980i	$-0.485621\pi$
$487$	1.99306	0.0903140	0.0451570	+	0.998980i	$0.485621\pi$
$488$	0	0
$489$	−28.0836	−1.26998
$490$	0	0
$491$	20.5423	0.927062	0.463531	−	0.886081i	$-0.346582\pi$
$491$	20.5423	0.927062	0.463531	+	0.886081i	$0.346582\pi$
$492$	0	0
$493$	3.26541	0.147067
$494$	0	0
$495$	−20.3539	−0.914840
$496$	0	0
$497$	−8.72915	−0.391556
$498$	0	0
$499$	−28.9112	−1.29425	−0.647123	−	0.762386i	$-0.724028\pi$
$499$	−28.9112	−1.29425	−0.647123	+	0.762386i	$0.724028\pi$
$500$	0	0
$501$	−43.2429	−1.93195
$502$	0	0
$503$	29.6891	1.32377	0.661885	−	0.749605i	$-0.269757\pi$
$503$	29.6891	1.32377	0.661885	+	0.749605i	$0.269757\pi$
$504$	0	0
$505$	12.1349	0.539995
$506$	0	0
$507$	37.9598	1.68585
$508$	0	0
$509$	−27.7096	−1.22821	−0.614103	−	0.789226i	$-0.710482\pi$
$509$	−27.7096	−1.22821	−0.614103	+	0.789226i	$0.710482\pi$
$510$	0	0
$511$	7.99013	0.353462
$512$	0	0
$513$	−15.9251	−0.703112
$514$	0	0
$515$	−23.6516	−1.04221
$516$	0	0
$517$	8.04814	0.353957
$518$	0	0
$519$	13.5851	0.596320
$520$	0	0
$521$	16.5524	0.725175	0.362588	−	0.931950i	$-0.381893\pi$
$521$	16.5524	0.725175	0.362588	+	0.931950i	$0.381893\pi$
$522$	0	0
$523$	38.0287	1.66288	0.831439	−	0.555616i	$-0.187518\pi$
$523$	38.0287	1.66288	0.831439	+	0.555616i	$0.187518\pi$
$524$	0	0
$525$	−14.6388	−0.638890
$526$	0	0
$527$	2.67049	0.116328
$528$	0	0
$529$	−22.0983	−0.960794
$530$	0	0
$531$	−29.0322	−1.25989
$532$	0	0
$533$	6.41055	0.277672
$534$	0	0
$535$	−53.8489	−2.32809
$536$	0	0
$537$	0.393879	0.0169971
$538$	0	0
$539$	−6.50896	−0.280361
$540$	0	0
$541$	−3.55524	−0.152852	−0.0764258	−	0.997075i	$-0.524351\pi$
$541$	−3.55524	−0.152852	−0.0764258	+	0.997075i	$0.524351\pi$
$542$	0	0
$543$	−61.9645	−2.65915
$544$	0	0
$545$	32.4374	1.38947
$546$	0	0
$547$	38.6046	1.65061	0.825307	−	0.564684i	$-0.191002\pi$
$547$	38.6046	1.65061	0.825307	+	0.564684i	$0.191002\pi$
$548$	0	0
$549$	−76.2413	−3.25390
$550$	0	0
$551$	8.63223	0.367746
$552$	0	0
$553$	−5.93854	−0.252532
$554$	0	0
$555$	−27.0663	−1.14890
$556$	0	0
$557$	12.0151	0.509097	0.254548	−	0.967060i	$-0.418073\pi$
$557$	12.0151	0.509097	0.254548	+	0.967060i	$0.418073\pi$
$558$	0	0
$559$	−2.28403	−0.0966040
$560$	0	0
$561$	−2.09726	−0.0885466
$562$	0	0
$563$	−7.31365	−0.308234	−0.154117	−	0.988053i	$-0.549253\pi$
$563$	−7.31365	−0.308234	−0.154117	+	0.988053i	$0.549253\pi$
$564$	0	0
$565$	66.6779	2.80516
$566$	0	0
$567$	5.51338	0.231540
$568$	0	0
$569$	−39.9703	−1.67564	−0.837821	−	0.545944i	$-0.816171\pi$
$569$	−39.9703	−1.67564	−0.837821	+	0.545944i	$0.816171\pi$
$570$	0	0
$571$	6.52298	0.272978	0.136489	−	0.990642i	$-0.456418\pi$
$571$	6.52298	0.272978	0.136489	+	0.990642i	$0.456418\pi$
$572$	0	0
$573$	9.45918	0.395163
$574$	0	0
$575$	6.65990	0.277737
$576$	0	0
$577$	24.9867	1.04021	0.520105	−	0.854102i	$-0.325893\pi$
$577$	24.9867	1.04021	0.520105	+	0.854102i	$0.325893\pi$
$578$	0	0
$579$	−45.0935	−1.87402
$580$	0	0
$581$	4.36384	0.181042
$582$	0	0
$583$	−4.49541	−0.186181
$584$	0	0
$585$	−10.2999	−0.425846
$586$	0	0
$587$	44.2763	1.82748	0.913739	−	0.406301i	$-0.133182\pi$
$587$	44.2763	1.82748	0.913739	+	0.406301i	$0.133182\pi$
$588$	0	0
$589$	7.05953	0.290883
$590$	0	0
$591$	18.3737	0.755792
$592$	0	0
$593$	−27.5549	−1.13154	−0.565772	−	0.824562i	$-0.691422\pi$
$593$	−27.5549	−1.13154	−0.565772	+	0.824562i	$0.691422\pi$
$594$	0	0
$595$	−1.71011	−0.0701077
$596$	0	0
$597$	45.5768	1.86533
$598$	0	0
$599$	−13.9032	−0.568069	−0.284035	−	0.958814i	$-0.591673\pi$
$599$	−13.9032	−0.568069	−0.284035	+	0.958814i	$0.591673\pi$
$600$	0	0
$601$	25.7463	1.05021	0.525107	−	0.851036i	$-0.324025\pi$
$601$	25.7463	1.05021	0.525107	+	0.851036i	$0.324025\pi$
$602$	0	0
$603$	28.2622	1.15093
$604$	0	0
$605$	−3.46603	−0.140914
$606$	0	0
$607$	−4.39809	−0.178513	−0.0892564	−	0.996009i	$-0.528449\pi$
$607$	−4.39809	−0.178513	−0.0892564	+	0.996009i	$0.528449\pi$
$608$	0	0
$609$	−9.68024	−0.392263
$610$	0	0
$611$	4.07267	0.164762
$612$	0	0
$613$	4.45256	0.179837	0.0899187	−	0.995949i	$-0.471339\pi$
$613$	4.45256	0.179837	0.0899187	+	0.995949i	$0.471339\pi$
$614$	0	0
$615$	130.787	5.27384
$616$	0	0
$617$	−12.1165	−0.487790	−0.243895	−	0.969802i	$-0.578425\pi$
$617$	−12.1165	−0.487790	−0.243895	+	0.969802i	$0.578425\pi$
$618$	0	0
$619$	−25.7469	−1.03486	−0.517428	−	0.855727i	$-0.673111\pi$
$619$	−25.7469	−1.03486	−0.517428	+	0.855727i	$0.673111\pi$
$620$	0	0
$621$	−8.12471	−0.326033
$622$	0	0
$623$	−3.47752	−0.139324
$624$	0	0
$625$	−10.8797	−0.435188
$626$	0	0
$627$	−5.54419	−0.221414
$628$	0	0
$629$	−1.84590	−0.0736010
$630$	0	0
$631$	−28.1671	−1.12131	−0.560657	−	0.828048i	$-0.689451\pi$
$631$	−28.1671	−1.12131	−0.560657	+	0.828048i	$0.689451\pi$
$632$	0	0
$633$	50.0560	1.98955
$634$	0	0
$635$	4.30810	0.170962
$636$	0	0
$637$	−3.29378	−0.130504
$638$	0	0
$639$	−73.1522	−2.89386
$640$	0	0
$641$	11.0415	0.436112	0.218056	−	0.975936i	$-0.430028\pi$
$641$	11.0415	0.436112	0.218056	+	0.975936i	$0.430028\pi$
$642$	0	0
$643$	−10.3791	−0.409314	−0.204657	−	0.978834i	$-0.565608\pi$
$643$	−10.3791	−0.409314	−0.204657	+	0.978834i	$0.565608\pi$
$644$	0	0
$645$	−46.5983	−1.83481
$646$	0	0
$647$	−36.1274	−1.42032	−0.710158	−	0.704042i	$-0.751377\pi$
$647$	−36.1274	−1.42032	−0.710158	+	0.704042i	$0.751377\pi$
$648$	0	0
$649$	−4.94384	−0.194063
$650$	0	0
$651$	−7.91659	−0.310276
$652$	0	0
$653$	0.422842	0.0165471	0.00827354	−	0.999966i	$-0.497366\pi$
$653$	0.422842	0.0165471	0.00827354	+	0.999966i	$0.497366\pi$
$654$	0	0
$655$	−5.28634	−0.206554
$656$	0	0
$657$	66.9591	2.61232
$658$	0	0
$659$	2.67469	0.104191	0.0520955	−	0.998642i	$-0.483410\pi$
$659$	2.67469	0.104191	0.0520955	+	0.998642i	$0.483410\pi$
$660$	0	0
$661$	27.7368	1.07884	0.539419	−	0.842038i	$-0.318644\pi$
$661$	27.7368	1.07884	0.539419	+	0.842038i	$0.318644\pi$
$662$	0	0
$663$	−1.06130	−0.0412173
$664$	0	0
$665$	−4.52074	−0.175307
$666$	0	0
$667$	4.40401	0.170524
$668$	0	0
$669$	−83.1422	−3.21446
$670$	0	0
$671$	−12.9830	−0.501203
$672$	0	0
$673$	5.42165	0.208989	0.104495	−	0.994525i	$-0.466677\pi$
$673$	5.42165	0.208989	0.104495	+	0.994525i	$0.466677\pi$
$674$	0	0
$675$	−60.0055	−2.30961
$676$	0	0
$677$	−16.0677	−0.617533	−0.308766	−	0.951138i	$-0.599916\pi$
$677$	−16.0677	−0.617533	−0.308766	+	0.951138i	$0.599916\pi$
$678$	0	0
$679$	7.45721	0.286181
$680$	0	0
$681$	15.3742	0.589142
$682$	0	0
$683$	−37.8609	−1.44871	−0.724354	−	0.689429i	$-0.757862\pi$
$683$	−37.8609	−1.44871	−0.724354	+	0.689429i	$0.757862\pi$
$684$	0	0
$685$	−3.46603	−0.132430
$686$	0	0
$687$	−40.1029	−1.53002
$688$	0	0
$689$	−2.27485	−0.0866648
$690$	0	0
$691$	−29.9530	−1.13946	−0.569732	−	0.821830i	$-0.692953\pi$
$691$	−29.9530	−1.13946	−0.569732	+	0.821830i	$0.692953\pi$
$692$	0	0
$693$	4.11505	0.156318
$694$	0	0
$695$	29.7546	1.12866
$696$	0	0
$697$	8.91959	0.337853
$698$	0	0
$699$	−89.7920	−3.39625
$700$	0	0
$701$	0.534019	0.0201696	0.0100848	−	0.999949i	$-0.496790\pi$
$701$	0.534019	0.0201696	0.0100848	+	0.999949i	$0.496790\pi$
$702$	0	0
$703$	−4.87971	−0.184042
$704$	0	0
$705$	83.0899	3.12935
$706$	0	0
$707$	−2.45337	−0.0922685
$708$	0	0
$709$	29.7463	1.11714	0.558572	−	0.829456i	$-0.311349\pi$
$709$	29.7463	1.11714	0.558572	+	0.829456i	$0.311349\pi$
$710$	0	0
$711$	−49.7663	−1.86638
$712$	0	0
$713$	3.60164	0.134882
$714$	0	0
$715$	−1.75394	−0.0655937
$716$	0	0
$717$	−18.8158	−0.702689
$718$	0	0
$719$	−7.06893	−0.263627	−0.131813	−	0.991275i	$-0.542080\pi$
$719$	−7.06893	−0.263627	−0.131813	+	0.991275i	$0.542080\pi$
$720$	0	0
$721$	4.78177	0.178082
$722$	0	0
$723$	17.8387	0.663428
$724$	0	0
$725$	32.5260	1.20799
$726$	0	0
$727$	3.81918	0.141645	0.0708227	−	0.997489i	$-0.477438\pi$
$727$	3.81918	0.141645	0.0708227	+	0.997489i	$0.477438\pi$
$728$	0	0
$729$	−30.2519	−1.12044
$730$	0	0
$731$	−3.17797	−0.117542
$732$	0	0
$733$	−6.50491	−0.240264	−0.120132	−	0.992758i	$-0.538332\pi$
$733$	−6.50491	−0.240264	−0.120132	+	0.992758i	$0.538332\pi$
$734$	0	0
$735$	−67.1992	−2.47868
$736$	0	0
$737$	4.81273	0.177279
$738$	0	0
$739$	32.6149	1.19976	0.599879	−	0.800091i	$-0.295216\pi$
$739$	32.6149	1.19976	0.599879	+	0.800091i	$0.295216\pi$
$740$	0	0
$741$	−2.80557	−0.103065
$742$	0	0
$743$	11.9637	0.438905	0.219453	−	0.975623i	$-0.429573\pi$
$743$	11.9637	0.438905	0.219453	+	0.975623i	$0.429573\pi$
$744$	0	0
$745$	−6.51263	−0.238604
$746$	0	0
$747$	36.5699	1.33802
$748$	0	0
$749$	10.8869	0.397799
$750$	0	0
$751$	−16.0398	−0.585302	−0.292651	−	0.956219i	$-0.594537\pi$
$751$	−16.0398	−0.585302	−0.292651	+	0.956219i	$0.594537\pi$
$752$	0	0
$753$	−81.0289	−2.95286
$754$	0	0
$755$	48.4413	1.76296
$756$	0	0
$757$	7.10881	0.258374	0.129187	−	0.991620i	$-0.458763\pi$
$757$	7.10881	0.258374	0.129187	+	0.991620i	$0.458763\pi$
$758$	0	0
$759$	−2.82854	−0.102670
$760$	0	0
$761$	16.8896	0.612247	0.306123	−	0.951992i	$-0.400968\pi$
$761$	16.8896	0.612247	0.306123	+	0.951992i	$0.400968\pi$
$762$	0	0
$763$	−6.55805	−0.237417
$764$	0	0
$765$	−14.3311	−0.518143
$766$	0	0
$767$	−2.50177	−0.0903338
$768$	0	0
$769$	27.8598	1.00465	0.502325	−	0.864679i	$-0.332478\pi$
$769$	27.8598	1.00465	0.502325	+	0.864679i	$0.332478\pi$
$770$	0	0
$771$	38.2100	1.37610
$772$	0	0
$773$	−35.4747	−1.27594	−0.637969	−	0.770062i	$-0.720225\pi$
$773$	−35.4747	−1.27594	−0.637969	+	0.770062i	$0.720225\pi$
$774$	0	0
$775$	26.6001	0.955504
$776$	0	0
$777$	5.47214	0.196312
$778$	0	0
$779$	23.5792	0.844814
$780$	0	0
$781$	−12.4570	−0.445745
$782$	0	0
$783$	−39.6800	−1.41805
$784$	0	0
$785$	46.0678	1.64423
$786$	0	0
$787$	1.86703	0.0665524	0.0332762	−	0.999446i	$-0.489406\pi$
$787$	1.86703	0.0665524	0.0332762	+	0.999446i	$0.489406\pi$
$788$	0	0
$789$	−71.8684	−2.55858
$790$	0	0
$791$	−13.4806	−0.479315
$792$	0	0
$793$	−6.56989	−0.233304
$794$	0	0
$795$	−46.4111	−1.64603
$796$	0	0
$797$	−3.79437	−0.134403	−0.0672017	−	0.997739i	$-0.521407\pi$
$797$	−3.79437	−0.134403	−0.0672017	+	0.997739i	$0.521407\pi$
$798$	0	0
$799$	5.66667	0.200472
$800$	0	0
$801$	−29.1424	−1.02970
$802$	0	0
$803$	11.4023	0.402380
$804$	0	0
$805$	−2.30640	−0.0812898
$806$	0	0
$807$	13.2846	0.467641
$808$	0	0
$809$	28.0672	0.986790	0.493395	−	0.869805i	$-0.335756\pi$
$809$	28.0672	0.986790	0.493395	+	0.869805i	$0.335756\pi$
$810$	0	0
$811$	0.693408	0.0243489	0.0121744	−	0.999926i	$-0.496125\pi$
$811$	0.693408	0.0243489	0.0121744	+	0.999926i	$0.496125\pi$
$812$	0	0
$813$	−29.9062	−1.04886
$814$	0	0
$815$	−32.6787	−1.14468
$816$	0	0
$817$	−8.40108	−0.293917
$818$	0	0
$819$	2.08237	0.0727640
$820$	0	0
$821$	27.8107	0.970600	0.485300	−	0.874348i	$-0.338710\pi$
$821$	27.8107	0.970600	0.485300	+	0.874348i	$0.338710\pi$
$822$	0	0
$823$	−16.1401	−0.562609	−0.281304	−	0.959619i	$-0.590767\pi$
$823$	−16.1401	−0.562609	−0.281304	+	0.959619i	$0.590767\pi$
$824$	0	0
$825$	−20.8904	−0.727309
$826$	0	0
$827$	19.8425	0.689991	0.344996	−	0.938604i	$-0.387880\pi$
$827$	19.8425	0.689991	0.344996	+	0.938604i	$0.387880\pi$
$828$	0	0
$829$	−38.5384	−1.33849	−0.669247	−	0.743040i	$-0.733383\pi$
$829$	−38.5384	−1.33849	−0.669247	+	0.743040i	$0.733383\pi$
$830$	0	0
$831$	37.8792	1.31401
$832$	0	0
$833$	−4.58294	−0.158789
$834$	0	0
$835$	−50.3183	−1.74134
$836$	0	0
$837$	−32.4507	−1.12166
$838$	0	0
$839$	51.8736	1.79088	0.895438	−	0.445186i	$-0.146862\pi$
$839$	51.8736	1.79088	0.895438	+	0.445186i	$0.146862\pi$
$840$	0	0
$841$	−7.49144	−0.258325
$842$	0	0
$843$	−76.8185	−2.64577
$844$	0	0
$845$	44.1708	1.51952
$846$	0	0
$847$	0.700745	0.0240779
$848$	0	0
$849$	−14.6743	−0.503622
$850$	0	0
$851$	−2.48954	−0.0853402
$852$	0	0
$853$	−9.81091	−0.335919	−0.167960	−	0.985794i	$-0.553718\pi$
$853$	−9.81091	−0.335919	−0.167960	+	0.985794i	$0.553718\pi$
$854$	0	0
$855$	−37.8848	−1.29563
$856$	0	0
$857$	3.85741	0.131767	0.0658833	−	0.997827i	$-0.479014\pi$
$857$	3.85741	0.131767	0.0658833	+	0.997827i	$0.479014\pi$
$858$	0	0
$859$	−20.9795	−0.715812	−0.357906	−	0.933758i	$-0.616509\pi$
$859$	−20.9795	−0.715812	−0.357906	+	0.933758i	$0.616509\pi$
$860$	0	0
$861$	−26.4419	−0.901137
$862$	0	0
$863$	−22.5227	−0.766682	−0.383341	−	0.923607i	$-0.625227\pi$
$863$	−22.5227	−0.766682	−0.383341	+	0.923607i	$0.625227\pi$
$864$	0	0
$865$	15.8079	0.537485
$866$	0	0
$867$	49.1605	1.66958
$868$	0	0
$869$	−8.47461	−0.287481
$870$	0	0
$871$	2.43542	0.0825211
$872$	0	0
$873$	62.4931	2.11507
$874$	0	0
$875$	−4.89002	−0.165313
$876$	0	0
$877$	7.65780	0.258586	0.129293	−	0.991606i	$-0.458729\pi$
$877$	7.65780	0.258586	0.129293	+	0.991606i	$0.458729\pi$
$878$	0	0
$879$	69.0166	2.32787
$880$	0	0
$881$	−12.2127	−0.411457	−0.205729	−	0.978609i	$-0.565956\pi$
$881$	−12.2127	−0.411457	−0.205729	+	0.978609i	$0.565956\pi$
$882$	0	0
$883$	7.86360	0.264631	0.132316	−	0.991208i	$-0.457759\pi$
$883$	7.86360	0.264631	0.132316	+	0.991208i	$0.457759\pi$
$884$	0	0
$885$	−51.0407	−1.71572
$886$	0	0
$887$	−32.6163	−1.09515	−0.547574	−	0.836757i	$-0.684449\pi$
$887$	−32.6163	−1.09515	−0.547574	+	0.836757i	$0.684449\pi$
$888$	0	0
$889$	−0.870990	−0.0292121
$890$	0	0
$891$	7.86789	0.263584
$892$	0	0
$893$	14.9800	0.501288
$894$	0	0
$895$	0.458326	0.0153201
$896$	0	0
$897$	−1.43135	−0.0477914
$898$	0	0
$899$	17.5899	0.586657
$900$	0	0
$901$	−3.16520	−0.105448
$902$	0	0
$903$	9.42102	0.313512
$904$	0	0
$905$	−72.1032	−2.39679
$906$	0	0
$907$	−10.2549	−0.340510	−0.170255	−	0.985400i	$-0.554459\pi$
$907$	−10.2549	−0.340510	−0.170255	+	0.985400i	$0.554459\pi$
$908$	0	0
$909$	−20.5598	−0.681926
$910$	0	0
$911$	36.0031	1.19284	0.596418	−	0.802674i	$-0.296590\pi$
$911$	36.0031	1.19284	0.596418	+	0.802674i	$0.296590\pi$
$912$	0	0
$913$	6.22743	0.206098
$914$	0	0
$915$	−134.038	−4.43115
$916$	0	0
$917$	1.06877	0.0352938
$918$	0	0
$919$	−11.0258	−0.363706	−0.181853	−	0.983326i	$-0.558210\pi$
$919$	−11.0258	−0.363706	−0.181853	+	0.983326i	$0.558210\pi$
$920$	0	0
$921$	−54.5539	−1.79761
$922$	0	0
$923$	−6.30370	−0.207489
$924$	0	0
$925$	−18.3866	−0.604548
$926$	0	0
$927$	40.0723	1.31615
$928$	0	0
$929$	45.5171	1.49337	0.746683	−	0.665180i	$-0.231645\pi$
$929$	45.5171	1.49337	0.746683	+	0.665180i	$0.231645\pi$
$930$	0	0
$931$	−12.1152	−0.397058
$932$	0	0
$933$	−64.1605	−2.10052
$934$	0	0
$935$	−2.44042	−0.0798103
$936$	0	0
$937$	−8.50142	−0.277729	−0.138865	−	0.990311i	$-0.544345\pi$
$937$	−8.50142	−0.277729	−0.138865	+	0.990311i	$0.544345\pi$
$938$	0	0
$939$	−12.1355	−0.396026
$940$	0	0
$941$	43.0618	1.40378	0.701888	−	0.712288i	$-0.252341\pi$
$941$	43.0618	1.40378	0.701888	+	0.712288i	$0.252341\pi$
$942$	0	0
$943$	12.0297	0.391741
$944$	0	0
$945$	20.7806	0.675992
$946$	0	0
$947$	8.32888	0.270652	0.135326	−	0.990801i	$-0.456792\pi$
$947$	8.32888	0.270652	0.135326	+	0.990801i	$0.456792\pi$
$948$	0	0
$949$	5.77002	0.187303
$950$	0	0
$951$	−63.5854	−2.06190
$952$	0	0
$953$	−33.5380	−1.08640	−0.543202	−	0.839602i	$-0.682788\pi$
$953$	−33.5380	−1.08640	−0.543202	+	0.839602i	$0.682788\pi$
$954$	0	0
$955$	11.0069	0.356175
$956$	0	0
$957$	−13.8142	−0.446550
$958$	0	0
$959$	0.700745	0.0226282
$960$	0	0
$961$	−16.6148	−0.535961
$962$	0	0
$963$	91.2348	2.94000
$964$	0	0
$965$	−52.4717	−1.68912
$966$	0	0
$967$	16.7398	0.538315	0.269157	−	0.963096i	$-0.413255\pi$
$967$	16.7398	0.538315	0.269157	+	0.963096i	$0.413255\pi$
$968$	0	0
$969$	−3.90365	−0.125403
$970$	0	0
$971$	43.2979	1.38950	0.694748	−	0.719253i	$-0.255516\pi$
$971$	43.2979	1.38950	0.694748	+	0.719253i	$0.255516\pi$
$972$	0	0
$973$	−6.01565	−0.192853
$974$	0	0
$975$	−10.5713	−0.338553
$976$	0	0
$977$	−22.7659	−0.728345	−0.364172	−	0.931332i	$-0.618648\pi$
$977$	−22.7659	−0.728345	−0.364172	+	0.931332i	$0.618648\pi$
$978$	0	0
$979$	−4.96261	−0.158606
$980$	0	0
$981$	−54.9579	−1.75467
$982$	0	0
$983$	23.3088	0.743435	0.371717	−	0.928346i	$-0.378769\pi$
$983$	23.3088	0.743435	0.371717	+	0.928346i	$0.378769\pi$
$984$	0	0
$985$	21.3800	0.681223
$986$	0	0
$987$	−16.7987	−0.534709
$988$	0	0
$989$	−4.28608	−0.136289
$990$	0	0
$991$	−43.7152	−1.38866	−0.694330	−	0.719657i	$-0.744299\pi$
$991$	−43.7152	−1.38866	−0.694330	+	0.719657i	$0.744299\pi$
$992$	0	0
$993$	2.56637	0.0814414
$994$	0	0
$995$	53.0341	1.68129
$996$	0	0
$997$	−2.70775	−0.0857552	−0.0428776	−	0.999080i	$-0.513653\pi$
$997$	−2.70775	−0.0857552	−0.0428776	+	0.999080i	$0.513653\pi$
$998$	0	0
$999$	22.4307	0.709675

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.f.1.1	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.f.1.1	✓	29	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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-2.97866 q^{3} -3.46603 q^{5} +0.700745 q^{7} +5.87240 q^{9} +O(q^{10})\)	\(q-2.97866 q^{3} -3.46603 q^{5} +0.700745 q^{7} +5.87240 q^{9} +1.00000 q^{11} +0.506038 q^{13} +10.3241 q^{15} +0.704097 q^{17} +1.86130 q^{19} -2.08728 q^{21} +0.949603 q^{23} +7.01335 q^{25} -8.55590 q^{27} +4.63773 q^{29} +3.79278 q^{31} -2.97866 q^{33} -2.42880 q^{35} -2.62166 q^{37} -1.50731 q^{39} +12.6681 q^{41} -4.51354 q^{43} -20.3539 q^{45} +8.04814 q^{47} -6.50896 q^{49} -2.09726 q^{51} -4.49541 q^{53} -3.46603 q^{55} -5.54419 q^{57} -4.94384 q^{59} -12.9830 q^{61} +4.11505 q^{63} -1.75394 q^{65} +4.81273 q^{67} -2.82854 q^{69} -12.4570 q^{71} +11.4023 q^{73} -20.8904 q^{75} +0.700745 q^{77} -8.47461 q^{79} +7.86789 q^{81} +6.22743 q^{83} -2.44042 q^{85} -13.8142 q^{87} -4.96261 q^{89} +0.354604 q^{91} -11.2974 q^{93} -6.45133 q^{95} +10.6418 q^{97} +5.87240 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * q^99

Introduction

L-functions

Modular forms

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Database

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