LMFDB - Embedded newform 8046.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 9x^{7} + 25x^{6} + 29x^{5} - 58x^{4} - 43x^{3} + 34x^{2} + 25x + 2 $ x^9 - 3x^8 - 9x^7 + 25x^6 + 29x^5 - 58x^4 - 43x^3 + 34x^2 + 25x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.89499$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.72525 q^{5} -2.60583 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.72525 q^{5} -2.60583 q^{7} -1.00000 q^{8} +1.72525 q^{10} +5.87208 q^{11} +3.97610 q^{13} +2.60583 q^{14} +1.00000 q^{16} -3.69002 q^{17} -0.972350 q^{19} -1.72525 q^{20} -5.87208 q^{22} +1.45252 q^{23} -2.02352 q^{25} -3.97610 q^{26} -2.60583 q^{28} -0.793752 q^{29} -5.90931 q^{31} -1.00000 q^{32} +3.69002 q^{34} +4.49571 q^{35} -1.91150 q^{37} +0.972350 q^{38} +1.72525 q^{40} -2.77935 q^{41} +0.964786 q^{43} +5.87208 q^{44} -1.45252 q^{46} +5.72077 q^{47} -0.209634 q^{49} +2.02352 q^{50} +3.97610 q^{52} +12.1545 q^{53} -10.1308 q^{55} +2.60583 q^{56} +0.793752 q^{58} -1.69551 q^{59} -8.91875 q^{61} +5.90931 q^{62} +1.00000 q^{64} -6.85976 q^{65} +1.13575 q^{67} -3.69002 q^{68} -4.49571 q^{70} +3.26677 q^{71} +3.64845 q^{73} +1.91150 q^{74} -0.972350 q^{76} -15.3017 q^{77} -6.18956 q^{79} -1.72525 q^{80} +2.77935 q^{82} -14.2544 q^{83} +6.36619 q^{85} -0.964786 q^{86} -5.87208 q^{88} +2.08402 q^{89} -10.3611 q^{91} +1.45252 q^{92} -5.72077 q^{94} +1.67754 q^{95} +4.60522 q^{97} +0.209634 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) $ 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8	$ 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8} - 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 9 q^{16} + q^{17} - 10 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 3 q^{25} + 8 q^{26} - 4 q^{28} + 4 q^{29} - 17 q^{31} - 9 q^{32} - q^{34} + 10 q^{35} - 11 q^{37} + 10 q^{38} - 4 q^{40} - 16 q^{43} + 4 q^{44} - 8 q^{46} + 7 q^{47} - 5 q^{49} + 3 q^{50} - 8 q^{52} + 12 q^{53} - 23 q^{55} + 4 q^{56} - 4 q^{58} + 6 q^{59} - 13 q^{61} + 17 q^{62} + 9 q^{64} - 24 q^{65} - 14 q^{67} + q^{68} - 10 q^{70} + 30 q^{71} - 12 q^{73} + 11 q^{74} - 10 q^{76} + 12 q^{77} - 35 q^{79} + 4 q^{80} + 5 q^{83} - 27 q^{85} + 16 q^{86} - 4 q^{88} + 23 q^{89} - 28 q^{91} + 8 q^{92} - 7 q^{94} + 32 q^{95} - 21 q^{97} + 5 q^{98}+O(q^{100}) $ 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8 - 4 * q^10 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 9 * q^16 + q^17 - 10 * q^19 + 4 * q^20 - 4 * q^22 + 8 * q^23 - 3 * q^25 + 8 * q^26 - 4 * q^28 + 4 * q^29 - 17 * q^31 - 9 * q^32 - q^34 + 10 * q^35 - 11 * q^37 + 10 * q^38 - 4 * q^40 - 16 * q^43 + 4 * q^44 - 8 * q^46 + 7 * q^47 - 5 * q^49 + 3 * q^50 - 8 * q^52 + 12 * q^53 - 23 * q^55 + 4 * q^56 - 4 * q^58 + 6 * q^59 - 13 * q^61 + 17 * q^62 + 9 * q^64 - 24 * q^65 - 14 * q^67 + q^68 - 10 * q^70 + 30 * q^71 - 12 * q^73 + 11 * q^74 - 10 * q^76 + 12 * q^77 - 35 * q^79 + 4 * q^80 + 5 * q^83 - 27 * q^85 + 16 * q^86 - 4 * q^88 + 23 * q^89 - 28 * q^91 + 8 * q^92 - 7 * q^94 + 32 * q^95 - 21 * q^97 + 5 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.72525	−0.771554	−0.385777	−	0.922592i	$-0.626067\pi$
$5$	−1.72525	−0.771554	−0.385777	+	0.922592i	$0.626067\pi$
$6$	0	0
$7$	−2.60583	−0.984912	−0.492456	−	0.870337i	$-0.663901\pi$
$7$	−2.60583	−0.984912	−0.492456	+	0.870337i	$0.663901\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.72525	0.545571
$11$	5.87208	1.77050	0.885249	−	0.465117i	$-0.153988\pi$
$11$	5.87208	1.77050	0.885249	+	0.465117i	$0.153988\pi$
$12$	0	0
$13$	3.97610	1.10277	0.551386	−	0.834250i	$-0.314099\pi$
$13$	3.97610	1.10277	0.551386	+	0.834250i	$0.314099\pi$
$14$	2.60583	0.696438
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.69002	−0.894961	−0.447480	−	0.894294i	$-0.647679\pi$
$17$	−3.69002	−0.894961	−0.447480	+	0.894294i	$0.647679\pi$
$18$	0	0
$19$	−0.972350	−0.223072	−0.111536	−	0.993760i	$-0.535577\pi$
$19$	−0.972350	−0.223072	−0.111536	+	0.993760i	$0.535577\pi$
$20$	−1.72525	−0.385777
$21$	0	0
$22$	−5.87208	−1.25193
$23$	1.45252	0.302871	0.151436	−	0.988467i	$-0.451610\pi$
$23$	1.45252	0.302871	0.151436	+	0.988467i	$0.451610\pi$
$24$	0	0
$25$	−2.02352	−0.404705
$26$	−3.97610	−0.779778
$27$	0	0
$28$	−2.60583	−0.492456
$29$	−0.793752	−0.147396	−0.0736980	−	0.997281i	$-0.523480\pi$
$29$	−0.793752	−0.147396	−0.0736980	+	0.997281i	$0.523480\pi$
$30$	0	0
$31$	−5.90931	−1.06134	−0.530672	−	0.847578i	$-0.678060\pi$
$31$	−5.90931	−1.06134	−0.530672	+	0.847578i	$0.678060\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.69002	0.632833
$35$	4.49571	0.759913
$36$	0	0
$37$	−1.91150	−0.314249	−0.157125	−	0.987579i	$-0.550222\pi$
$37$	−1.91150	−0.314249	−0.157125	+	0.987579i	$0.550222\pi$
$38$	0.972350	0.157736
$39$	0	0
$40$	1.72525	0.272785
$41$	−2.77935	−0.434061	−0.217030	−	0.976165i	$-0.569637\pi$
$41$	−2.77935	−0.434061	−0.217030	+	0.976165i	$0.569637\pi$
$42$	0	0
$43$	0.964786	0.147128	0.0735642	−	0.997290i	$-0.476563\pi$
$43$	0.964786	0.147128	0.0735642	+	0.997290i	$0.476563\pi$
$44$	5.87208	0.885249
$45$	0	0
$46$	−1.45252	−0.214162
$47$	5.72077	0.834461	0.417230	−	0.908801i	$-0.363001\pi$
$47$	5.72077	0.834461	0.417230	+	0.908801i	$0.363001\pi$
$48$	0	0
$49$	−0.209634	−0.0299478
$50$	2.02352	0.286169
$51$	0	0
$52$	3.97610	0.551386
$53$	12.1545	1.66955	0.834775	−	0.550592i	$-0.185598\pi$
$53$	12.1545	1.66955	0.834775	+	0.550592i	$0.185598\pi$
$54$	0	0
$55$	−10.1308	−1.36603
$56$	2.60583	0.348219
$57$	0	0
$58$	0.793752	0.104225
$59$	−1.69551	−0.220737	−0.110368	−	0.993891i	$-0.535203\pi$
$59$	−1.69551	−0.220737	−0.110368	+	0.993891i	$0.535203\pi$
$60$	0	0
$61$	−8.91875	−1.14193	−0.570965	−	0.820975i	$-0.693431\pi$
$61$	−8.91875	−1.14193	−0.570965	+	0.820975i	$0.693431\pi$
$62$	5.90931	0.750483
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.85976	−0.850849
$66$	0	0
$67$	1.13575	0.138754	0.0693770	−	0.997591i	$-0.477899\pi$
$67$	1.13575	0.138754	0.0693770	+	0.997591i	$0.477899\pi$
$68$	−3.69002	−0.447480
$69$	0	0
$70$	−4.49571	−0.537340
$71$	3.26677	0.387694	0.193847	−	0.981032i	$-0.437904\pi$
$71$	3.26677	0.387694	0.193847	+	0.981032i	$0.437904\pi$
$72$	0	0
$73$	3.64845	0.427018	0.213509	−	0.976941i	$-0.431511\pi$
$73$	3.64845	0.427018	0.213509	+	0.976941i	$0.431511\pi$
$74$	1.91150	0.222208
$75$	0	0
$76$	−0.972350	−0.111536
$77$	−15.3017	−1.74379
$78$	0	0
$79$	−6.18956	−0.696380	−0.348190	−	0.937424i	$-0.613204\pi$
$79$	−6.18956	−0.696380	−0.348190	+	0.937424i	$0.613204\pi$
$80$	−1.72525	−0.192888
$81$	0	0
$82$	2.77935	0.306927
$83$	−14.2544	−1.56462	−0.782309	−	0.622890i	$-0.785958\pi$
$83$	−14.2544	−1.56462	−0.782309	+	0.622890i	$0.785958\pi$
$84$	0	0
$85$	6.36619	0.690510
$86$	−0.964786	−0.104035
$87$	0	0
$88$	−5.87208	−0.625966
$89$	2.08402	0.220906	0.110453	−	0.993881i	$-0.464770\pi$
$89$	2.08402	0.220906	0.110453	+	0.993881i	$0.464770\pi$
$90$	0	0
$91$	−10.3611	−1.08613
$92$	1.45252	0.151436
$93$	0	0
$94$	−5.72077	−0.590053
$95$	1.67754	0.172112
$96$	0	0
$97$	4.60522	0.467589	0.233795	−	0.972286i	$-0.424886\pi$
$97$	4.60522	0.467589	0.233795	+	0.972286i	$0.424886\pi$
$98$	0.209634	0.0211763
$99$	0	0
$100$	−2.02352	−0.202352
$101$	6.36787	0.633627	0.316814	−	0.948488i	$-0.397387\pi$
$101$	6.36787	0.633627	0.316814	+	0.948488i	$0.397387\pi$
$102$	0	0
$103$	0.442994	0.0436495	0.0218248	−	0.999762i	$-0.493052\pi$
$103$	0.442994	0.0436495	0.0218248	+	0.999762i	$0.493052\pi$
$104$	−3.97610	−0.389889
$105$	0	0
$106$	−12.1545	−1.18055
$107$	1.20231	0.116232	0.0581158	−	0.998310i	$-0.481491\pi$
$107$	1.20231	0.116232	0.0581158	+	0.998310i	$0.481491\pi$
$108$	0	0
$109$	0.528131	0.0505858	0.0252929	−	0.999680i	$-0.491948\pi$
$109$	0.528131	0.0505858	0.0252929	+	0.999680i	$0.491948\pi$
$110$	10.1308	0.965932
$111$	0	0
$112$	−2.60583	−0.246228
$113$	−3.78400	−0.355969	−0.177984	−	0.984033i	$-0.556958\pi$
$113$	−3.78400	−0.355969	−0.177984	+	0.984033i	$0.556958\pi$
$114$	0	0
$115$	−2.50596	−0.233682
$116$	−0.793752	−0.0736980
$117$	0	0
$118$	1.69551	0.156084
$119$	9.61557	0.881458
$120$	0	0
$121$	23.4813	2.13466
$122$	8.91875	0.807466
$123$	0	0
$124$	−5.90931	−0.530672
$125$	12.1173	1.08381
$126$	0	0
$127$	−7.25200	−0.643511	−0.321755	−	0.946823i	$-0.604273\pi$
$127$	−7.25200	−0.643511	−0.321755	+	0.946823i	$0.604273\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	6.85976	0.601641
$131$	3.56564	0.311531	0.155765	−	0.987794i	$-0.450216\pi$
$131$	3.56564	0.311531	0.155765	+	0.987794i	$0.450216\pi$
$132$	0	0
$133$	2.53378	0.219707
$134$	−1.13575	−0.0981140
$135$	0	0
$136$	3.69002	0.316416
$137$	16.5706	1.41573	0.707863	−	0.706350i	$-0.249659\pi$
$137$	16.5706	1.41573	0.707863	+	0.706350i	$0.249659\pi$
$138$	0	0
$139$	10.6385	0.902347	0.451174	−	0.892436i	$-0.351006\pi$
$139$	10.6385	0.902347	0.451174	+	0.892436i	$0.351006\pi$
$140$	4.49571	0.379956
$141$	0	0
$142$	−3.26677	−0.274141
$143$	23.3480	1.95246
$144$	0	0
$145$	1.36942	0.113724
$146$	−3.64845	−0.301948
$147$	0	0
$148$	−1.91150	−0.157125
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−11.4508	−0.931854	−0.465927	−	0.884823i	$-0.654279\pi$
$151$	−11.4508	−0.931854	−0.465927	+	0.884823i	$0.654279\pi$
$152$	0.972350	0.0788680
$153$	0	0
$154$	15.3017	1.23304
$155$	10.1950	0.818883
$156$	0	0
$157$	−2.49203	−0.198886	−0.0994428	−	0.995043i	$-0.531706\pi$
$157$	−2.49203	−0.198886	−0.0994428	+	0.995043i	$0.531706\pi$
$158$	6.18956	0.492415
$159$	0	0
$160$	1.72525	0.136393
$161$	−3.78503	−0.298302
$162$	0	0
$163$	−23.2212	−1.81882	−0.909411	−	0.415899i	$-0.863467\pi$
$163$	−23.2212	−1.81882	−0.909411	+	0.415899i	$0.863467\pi$
$164$	−2.77935	−0.217030
$165$	0	0
$166$	14.2544	1.10635
$167$	18.5579	1.43606	0.718028	−	0.696014i	$-0.245045\pi$
$167$	18.5579	1.43606	0.718028	+	0.696014i	$0.245045\pi$
$168$	0	0
$169$	2.80941	0.216108
$170$	−6.36619	−0.488265
$171$	0	0
$172$	0.964786	0.0735642
$173$	17.3894	1.32209	0.661046	−	0.750345i	$-0.270113\pi$
$173$	17.3894	1.32209	0.661046	+	0.750345i	$0.270113\pi$
$174$	0	0
$175$	5.27296	0.398599
$176$	5.87208	0.442625
$177$	0	0
$178$	−2.08402	−0.156204
$179$	−0.962244	−0.0719215	−0.0359607	−	0.999353i	$-0.511449\pi$
$179$	−0.962244	−0.0719215	−0.0359607	+	0.999353i	$0.511449\pi$
$180$	0	0
$181$	−15.2649	−1.13463	−0.567314	−	0.823501i	$-0.692017\pi$
$181$	−15.2649	−1.13463	−0.567314	+	0.823501i	$0.692017\pi$
$182$	10.3611	0.768013
$183$	0	0
$184$	−1.45252	−0.107081
$185$	3.29781	0.242460
$186$	0	0
$187$	−21.6681	−1.58453
$188$	5.72077	0.417230
$189$	0	0
$190$	−1.67754	−0.121702
$191$	−13.3385	−0.965140	−0.482570	−	0.875858i	$-0.660297\pi$
$191$	−13.3385	−0.965140	−0.482570	+	0.875858i	$0.660297\pi$
$192$	0	0
$193$	−0.978390	−0.0704260	−0.0352130	−	0.999380i	$-0.511211\pi$
$193$	−0.978390	−0.0704260	−0.0352130	+	0.999380i	$0.511211\pi$
$194$	−4.60522	−0.330636
$195$	0	0
$196$	−0.209634	−0.0149739
$197$	−15.9854	−1.13891	−0.569455	−	0.822023i	$-0.692846\pi$
$197$	−15.9854	−1.13891	−0.569455	+	0.822023i	$0.692846\pi$
$198$	0	0
$199$	2.31903	0.164392	0.0821959	−	0.996616i	$-0.473807\pi$
$199$	2.31903	0.164392	0.0821959	+	0.996616i	$0.473807\pi$
$200$	2.02352	0.143085
$201$	0	0
$202$	−6.36787	−0.448042
$203$	2.06838	0.145172
$204$	0	0
$205$	4.79506	0.334901
$206$	−0.442994	−0.0308649
$207$	0	0
$208$	3.97610	0.275693
$209$	−5.70972	−0.394949
$210$	0	0
$211$	12.7331	0.876581	0.438291	−	0.898833i	$-0.355584\pi$
$211$	12.7331	0.876581	0.438291	+	0.898833i	$0.355584\pi$
$212$	12.1545	0.834775
$213$	0	0
$214$	−1.20231	−0.0821882
$215$	−1.66449	−0.113517
$216$	0	0
$217$	15.3987	1.04533
$218$	−0.528131	−0.0357696
$219$	0	0
$220$	−10.1308	−0.683017
$221$	−14.6719	−0.986939
$222$	0	0
$223$	−13.1207	−0.878625	−0.439313	−	0.898334i	$-0.644778\pi$
$223$	−13.1207	−0.878625	−0.439313	+	0.898334i	$0.644778\pi$
$224$	2.60583	0.174110
$225$	0	0
$226$	3.78400	0.251708
$227$	5.44603	0.361466	0.180733	−	0.983532i	$-0.442153\pi$
$227$	5.44603	0.361466	0.180733	+	0.983532i	$0.442153\pi$
$228$	0	0
$229$	13.9568	0.922294	0.461147	−	0.887324i	$-0.347438\pi$
$229$	13.9568	0.922294	0.461147	+	0.887324i	$0.347438\pi$
$230$	2.50596	0.165238
$231$	0	0
$232$	0.793752	0.0521124
$233$	−26.4164	−1.73059	−0.865297	−	0.501259i	$-0.832870\pi$
$233$	−26.4164	−1.73059	−0.865297	+	0.501259i	$0.832870\pi$
$234$	0	0
$235$	−9.86975	−0.643831
$236$	−1.69551	−0.110368
$237$	0	0
$238$	−9.61557	−0.623285
$239$	3.39802	0.219800	0.109900	−	0.993943i	$-0.464947\pi$
$239$	3.39802	0.219800	0.109900	+	0.993943i	$0.464947\pi$
$240$	0	0
$241$	−9.84720	−0.634314	−0.317157	−	0.948373i	$-0.602728\pi$
$241$	−9.84720	−0.634314	−0.317157	+	0.948373i	$0.602728\pi$
$242$	−23.4813	−1.50944
$243$	0	0
$244$	−8.91875	−0.570965
$245$	0.361671	0.0231063
$246$	0	0
$247$	−3.86617	−0.245998
$248$	5.90931	0.375241
$249$	0	0
$250$	−12.1173	−0.766366
$251$	−1.91010	−0.120565	−0.0602823	−	0.998181i	$-0.519200\pi$
$251$	−1.91010	−0.120565	−0.0602823	+	0.998181i	$0.519200\pi$
$252$	0	0
$253$	8.52931	0.536233
$254$	7.25200	0.455031
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.3629	0.833553	0.416777	−	0.909009i	$-0.363160\pi$
$257$	13.3629	0.833553	0.416777	+	0.909009i	$0.363160\pi$
$258$	0	0
$259$	4.98106	0.309508
$260$	−6.85976	−0.425424
$261$	0	0
$262$	−3.56564	−0.220286
$263$	14.1629	0.873320	0.436660	−	0.899627i	$-0.356161\pi$
$263$	14.1629	0.873320	0.436660	+	0.899627i	$0.356161\pi$
$264$	0	0
$265$	−20.9695	−1.28815
$266$	−2.53378	−0.155356
$267$	0	0
$268$	1.13575	0.0693770
$269$	−11.7518	−0.716522	−0.358261	−	0.933622i	$-0.616630\pi$
$269$	−11.7518	−0.716522	−0.358261	+	0.933622i	$0.616630\pi$
$270$	0	0
$271$	−23.2669	−1.41336	−0.706680	−	0.707533i	$-0.749808\pi$
$271$	−23.2669	−1.41336	−0.706680	+	0.707533i	$0.749808\pi$
$272$	−3.69002	−0.223740
$273$	0	0
$274$	−16.5706	−1.00107
$275$	−11.8823	−0.716529
$276$	0	0
$277$	−22.5303	−1.35371	−0.676857	−	0.736115i	$-0.736658\pi$
$277$	−22.5303	−1.35371	−0.676857	+	0.736115i	$0.736658\pi$
$278$	−10.6385	−0.638056
$279$	0	0
$280$	−4.49571	−0.268670
$281$	−26.5178	−1.58192	−0.790960	−	0.611868i	$-0.790418\pi$
$281$	−26.5178	−1.58192	−0.790960	+	0.611868i	$0.790418\pi$
$282$	0	0
$283$	3.43156	0.203985	0.101993	−	0.994785i	$-0.467478\pi$
$283$	3.43156	0.203985	0.101993	+	0.994785i	$0.467478\pi$
$284$	3.26677	0.193847
$285$	0	0
$286$	−23.3480	−1.38060
$287$	7.24251	0.427512
$288$	0	0
$289$	−3.38377	−0.199045
$290$	−1.36942	−0.0804150
$291$	0	0
$292$	3.64845	0.213509
$293$	−6.15832	−0.359773	−0.179886	−	0.983687i	$-0.557573\pi$
$293$	−6.15832	−0.359773	−0.179886	+	0.983687i	$0.557573\pi$
$294$	0	0
$295$	2.92518	0.170310
$296$	1.91150	0.111104
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	5.77537	0.333998
$300$	0	0
$301$	−2.51407	−0.144909
$302$	11.4508	0.658920
$303$	0	0
$304$	−0.972350	−0.0557681
$305$	15.3870	0.881060
$306$	0	0
$307$	−31.5787	−1.80229	−0.901145	−	0.433518i	$-0.857272\pi$
$307$	−31.5787	−1.80229	−0.901145	+	0.433518i	$0.857272\pi$
$308$	−15.3017	−0.871893
$309$	0	0
$310$	−10.1950	−0.579038
$311$	10.9006	0.618114	0.309057	−	0.951044i	$-0.399987\pi$
$311$	10.9006	0.618114	0.309057	+	0.951044i	$0.399987\pi$
$312$	0	0
$313$	2.80950	0.158803	0.0794013	−	0.996843i	$-0.474699\pi$
$313$	2.80950	0.158803	0.0794013	+	0.996843i	$0.474699\pi$
$314$	2.49203	0.140633
$315$	0	0
$316$	−6.18956	−0.348190
$317$	1.65172	0.0927696	0.0463848	−	0.998924i	$-0.485230\pi$
$317$	1.65172	0.0927696	0.0463848	+	0.998924i	$0.485230\pi$
$318$	0	0
$319$	−4.66097	−0.260964
$320$	−1.72525	−0.0964442
$321$	0	0
$322$	3.78503	0.210931
$323$	3.58799	0.199641
$324$	0	0
$325$	−8.04574	−0.446297
$326$	23.2212	1.28610
$327$	0	0
$328$	2.77935	0.153464
$329$	−14.9074	−0.821871
$330$	0	0
$331$	−24.8063	−1.36348	−0.681740	−	0.731595i	$-0.738776\pi$
$331$	−24.8063	−1.36348	−0.681740	+	0.731595i	$0.738776\pi$
$332$	−14.2544	−0.782309
$333$	0	0
$334$	−18.5579	−1.01544
$335$	−1.95945	−0.107056
$336$	0	0
$337$	6.76816	0.368685	0.184343	−	0.982862i	$-0.440984\pi$
$337$	6.76816	0.368685	0.184343	+	0.982862i	$0.440984\pi$
$338$	−2.80941	−0.152812
$339$	0	0
$340$	6.36619	0.345255
$341$	−34.6999	−1.87911
$342$	0	0
$343$	18.7871	1.01441
$344$	−0.964786	−0.0520177
$345$	0	0
$346$	−17.3894	−0.934860
$347$	−11.1010	−0.595932	−0.297966	−	0.954577i	$-0.596308\pi$
$347$	−11.1010	−0.595932	−0.297966	+	0.954577i	$0.596308\pi$
$348$	0	0
$349$	27.2718	1.45982	0.729912	−	0.683541i	$-0.239561\pi$
$349$	27.2718	1.45982	0.729912	+	0.683541i	$0.239561\pi$
$350$	−5.27296	−0.281852
$351$	0	0
$352$	−5.87208	−0.312983
$353$	−0.471858	−0.0251145	−0.0125572	−	0.999921i	$-0.503997\pi$
$353$	−0.471858	−0.0251145	−0.0125572	+	0.999921i	$0.503997\pi$
$354$	0	0
$355$	−5.63598	−0.299127
$356$	2.08402	0.110453
$357$	0	0
$358$	0.962244	0.0508562
$359$	−13.0395	−0.688197	−0.344099	−	0.938934i	$-0.611815\pi$
$359$	−13.0395	−0.688197	−0.344099	+	0.938934i	$0.611815\pi$
$360$	0	0
$361$	−18.0545	−0.950239
$362$	15.2649	0.802303
$363$	0	0
$364$	−10.3611	−0.543067
$365$	−6.29447	−0.329468
$366$	0	0
$367$	−3.53216	−0.184377	−0.0921886	−	0.995742i	$-0.529386\pi$
$367$	−3.53216	−0.184377	−0.0921886	+	0.995742i	$0.529386\pi$
$368$	1.45252	0.0757179
$369$	0	0
$370$	−3.29781	−0.171445
$371$	−31.6726	−1.64436
$372$	0	0
$373$	−30.5929	−1.58404	−0.792021	−	0.610494i	$-0.790971\pi$
$373$	−30.5929	−1.58404	−0.792021	+	0.610494i	$0.790971\pi$
$374$	21.6681	1.12043
$375$	0	0
$376$	−5.72077	−0.295026
$377$	−3.15604	−0.162544
$378$	0	0
$379$	−0.756326	−0.0388499	−0.0194249	−	0.999811i	$-0.506184\pi$
$379$	−0.756326	−0.0388499	−0.0194249	+	0.999811i	$0.506184\pi$
$380$	1.67754	0.0860562
$381$	0	0
$382$	13.3385	0.682457
$383$	15.5219	0.793130	0.396565	−	0.918007i	$-0.370202\pi$
$383$	15.5219	0.793130	0.396565	+	0.918007i	$0.370202\pi$
$384$	0	0
$385$	26.3991	1.34542
$386$	0.978390	0.0497987
$387$	0	0
$388$	4.60522	0.233795
$389$	−34.2558	−1.73684	−0.868420	−	0.495830i	$-0.834864\pi$
$389$	−34.2558	−1.73684	−0.868420	+	0.495830i	$0.834864\pi$
$390$	0	0
$391$	−5.35983	−0.271058
$392$	0.209634	0.0105881
$393$	0	0
$394$	15.9854	0.805331
$395$	10.6785	0.537295
$396$	0	0
$397$	−4.32270	−0.216950	−0.108475	−	0.994099i	$-0.534597\pi$
$397$	−4.32270	−0.216950	−0.108475	+	0.994099i	$0.534597\pi$
$398$	−2.31903	−0.116243
$399$	0	0
$400$	−2.02352	−0.101176
$401$	−21.3467	−1.06600	−0.533002	−	0.846114i	$-0.678936\pi$
$401$	−21.3467	−1.06600	−0.533002	+	0.846114i	$0.678936\pi$
$402$	0	0
$403$	−23.4960	−1.17042
$404$	6.36787	0.316814
$405$	0	0
$406$	−2.06838	−0.102652
$407$	−11.2245	−0.556377
$408$	0	0
$409$	33.9729	1.67985	0.839925	−	0.542702i	$-0.182599\pi$
$409$	33.9729	1.67985	0.839925	+	0.542702i	$0.182599\pi$
$410$	−4.79506	−0.236811
$411$	0	0
$412$	0.442994	0.0218248
$413$	4.41822	0.217406
$414$	0	0
$415$	24.5923	1.20719
$416$	−3.97610	−0.194945
$417$	0	0
$418$	5.70972	0.279271
$419$	18.8895	0.922813	0.461406	−	0.887189i	$-0.347345\pi$
$419$	18.8895	0.922813	0.461406	+	0.887189i	$0.347345\pi$
$420$	0	0
$421$	11.7138	0.570896	0.285448	−	0.958394i	$-0.407858\pi$
$421$	11.7138	0.570896	0.285448	+	0.958394i	$0.407858\pi$
$422$	−12.7331	−0.619837
$423$	0	0
$424$	−12.1545	−0.590275
$425$	7.46684	0.362195
$426$	0	0
$427$	23.2408	1.12470
$428$	1.20231	0.0581158
$429$	0	0
$430$	1.66449	0.0802690
$431$	−17.8470	−0.859661	−0.429830	−	0.902910i	$-0.641427\pi$
$431$	−17.8470	−0.859661	−0.429830	+	0.902910i	$0.641427\pi$
$432$	0	0
$433$	−34.1206	−1.63973	−0.819866	−	0.572555i	$-0.805952\pi$
$433$	−34.1206	−1.63973	−0.819866	+	0.572555i	$0.805952\pi$
$434$	−15.3987	−0.739160
$435$	0	0
$436$	0.528131	0.0252929
$437$	−1.41236	−0.0675623
$438$	0	0
$439$	−2.87290	−0.137116	−0.0685579	−	0.997647i	$-0.521840\pi$
$439$	−2.87290	−0.137116	−0.0685579	+	0.997647i	$0.521840\pi$
$440$	10.1308	0.482966
$441$	0	0
$442$	14.6719	0.697871
$443$	−29.2501	−1.38971	−0.694856	−	0.719149i	$-0.744532\pi$
$443$	−29.2501	−1.38971	−0.694856	+	0.719149i	$0.744532\pi$
$444$	0	0
$445$	−3.59545	−0.170441
$446$	13.1207	0.621282
$447$	0	0
$448$	−2.60583	−0.123114
$449$	−11.4354	−0.539670	−0.269835	−	0.962907i	$-0.586969\pi$
$449$	−11.4354	−0.539670	−0.269835	+	0.962907i	$0.586969\pi$
$450$	0	0
$451$	−16.3205	−0.768504
$452$	−3.78400	−0.177984
$453$	0	0
$454$	−5.44603	−0.255595
$455$	17.8754	0.838011
$456$	0	0
$457$	13.2066	0.617781	0.308890	−	0.951098i	$-0.400042\pi$
$457$	13.2066	0.617781	0.308890	+	0.951098i	$0.400042\pi$
$458$	−13.9568	−0.652160
$459$	0	0
$460$	−2.50596	−0.116841
$461$	−23.7293	−1.10518	−0.552592	−	0.833452i	$-0.686361\pi$
$461$	−23.7293	−1.10518	−0.552592	+	0.833452i	$0.686361\pi$
$462$	0	0
$463$	−6.33611	−0.294464	−0.147232	−	0.989102i	$-0.547036\pi$
$463$	−6.33611	−0.294464	−0.147232	+	0.989102i	$0.547036\pi$
$464$	−0.793752	−0.0368490
$465$	0	0
$466$	26.4164	1.22371
$467$	−10.5195	−0.486782	−0.243391	−	0.969928i	$-0.578260\pi$
$467$	−10.5195	−0.486782	−0.243391	+	0.969928i	$0.578260\pi$
$468$	0	0
$469$	−2.95958	−0.136661
$470$	9.86975	0.455258
$471$	0	0
$472$	1.69551	0.0780422
$473$	5.66530	0.260491
$474$	0	0
$475$	1.96757	0.0902784
$476$	9.61557	0.440729
$477$	0	0
$478$	−3.39802	−0.155422
$479$	−0.598309	−0.0273374	−0.0136687	−	0.999907i	$-0.504351\pi$
$479$	−0.598309	−0.0273374	−0.0136687	+	0.999907i	$0.504351\pi$
$480$	0	0
$481$	−7.60033	−0.346545
$482$	9.84720	0.448528
$483$	0	0
$484$	23.4813	1.06733
$485$	−7.94514	−0.360770
$486$	0	0
$487$	3.97672	0.180203	0.0901013	−	0.995933i	$-0.471281\pi$
$487$	3.97672	0.180203	0.0901013	+	0.995933i	$0.471281\pi$
$488$	8.91875	0.403733
$489$	0	0
$490$	−0.361671	−0.0163386
$491$	−12.4900	−0.563664	−0.281832	−	0.959464i	$-0.590942\pi$
$491$	−12.4900	−0.563664	−0.281832	+	0.959464i	$0.590942\pi$
$492$	0	0
$493$	2.92896	0.131914
$494$	3.86617	0.173947
$495$	0	0
$496$	−5.90931	−0.265336
$497$	−8.51264	−0.381844
$498$	0	0
$499$	30.1380	1.34916	0.674580	−	0.738202i	$-0.264325\pi$
$499$	30.1380	1.34916	0.674580	+	0.738202i	$0.264325\pi$
$500$	12.1173	0.541903
$501$	0	0
$502$	1.91010	0.0852520
$503$	4.87062	0.217170	0.108585	−	0.994087i	$-0.465368\pi$
$503$	4.87062	0.217170	0.108585	+	0.994087i	$0.465368\pi$
$504$	0	0
$505$	−10.9862	−0.488877
$506$	−8.52931	−0.379174
$507$	0	0
$508$	−7.25200	−0.321755
$509$	0.234024	0.0103729	0.00518647	−	0.999987i	$-0.498349\pi$
$509$	0.234024	0.0103729	0.00518647	+	0.999987i	$0.498349\pi$
$510$	0	0
$511$	−9.50724	−0.420576
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−13.3629	−0.589411
$515$	−0.764274	−0.0336780
$516$	0	0
$517$	33.5928	1.47741
$518$	−4.98106	−0.218855
$519$	0	0
$520$	6.85976	0.300820
$521$	37.9004	1.66045	0.830223	−	0.557432i	$-0.188213\pi$
$521$	37.9004	1.66045	0.830223	+	0.557432i	$0.188213\pi$
$522$	0	0
$523$	−11.1086	−0.485744	−0.242872	−	0.970058i	$-0.578090\pi$
$523$	−11.1086	−0.485744	−0.242872	+	0.970058i	$0.578090\pi$
$524$	3.56564	0.155765
$525$	0	0
$526$	−14.1629	−0.617530
$527$	21.8055	0.949861
$528$	0	0
$529$	−20.8902	−0.908269
$530$	20.9695	0.910858
$531$	0	0
$532$	2.53378	0.109853
$533$	−11.0510	−0.478671
$534$	0	0
$535$	−2.07428	−0.0896790
$536$	−1.13575	−0.0490570
$537$	0	0
$538$	11.7518	0.506657
$539$	−1.23099	−0.0530225
$540$	0	0
$541$	−32.4740	−1.39617	−0.698084	−	0.716016i	$-0.745964\pi$
$541$	−32.4740	−1.39617	−0.698084	+	0.716016i	$0.745964\pi$
$542$	23.2669	0.999397
$543$	0	0
$544$	3.69002	0.158208
$545$	−0.911157	−0.0390297
$546$	0	0
$547$	−26.9889	−1.15396	−0.576981	−	0.816757i	$-0.695769\pi$
$547$	−26.9889	−1.15396	−0.576981	+	0.816757i	$0.695769\pi$
$548$	16.5706	0.707863
$549$	0	0
$550$	11.8823	0.506662
$551$	0.771805	0.0328800
$552$	0	0
$553$	16.1290	0.685873
$554$	22.5303	0.957220
$555$	0	0
$556$	10.6385	0.451174
$557$	−37.9551	−1.60821	−0.804105	−	0.594487i	$-0.797355\pi$
$557$	−37.9551	−1.60821	−0.804105	+	0.594487i	$0.797355\pi$
$558$	0	0
$559$	3.83609	0.162249
$560$	4.49571	0.189978
$561$	0	0
$562$	26.5178	1.11859
$563$	−10.7327	−0.452330	−0.226165	−	0.974089i	$-0.572619\pi$
$563$	−10.7327	−0.452330	−0.226165	+	0.974089i	$0.572619\pi$
$564$	0	0
$565$	6.52833	0.274649
$566$	−3.43156	−0.144239
$567$	0	0
$568$	−3.26677	−0.137070
$569$	−26.7458	−1.12124	−0.560621	−	0.828072i	$-0.689438\pi$
$569$	−26.7458	−1.12124	−0.560621	+	0.828072i	$0.689438\pi$
$570$	0	0
$571$	−32.2845	−1.35106	−0.675532	−	0.737331i	$-0.736086\pi$
$571$	−32.2845	−1.35106	−0.675532	+	0.737331i	$0.736086\pi$
$572$	23.3480	0.976229
$573$	0	0
$574$	−7.24251	−0.302297
$575$	−2.93921	−0.122573
$576$	0	0
$577$	43.5328	1.81230	0.906148	−	0.422961i	$-0.139009\pi$
$577$	43.5328	1.81230	0.906148	+	0.422961i	$0.139009\pi$
$578$	3.38377	0.140746
$579$	0	0
$580$	1.36942	0.0568620
$581$	37.1445	1.54101
$582$	0	0
$583$	71.3722	2.95593
$584$	−3.64845	−0.150974
$585$	0	0
$586$	6.15832	0.254398
$587$	−9.91387	−0.409189	−0.204595	−	0.978847i	$-0.565588\pi$
$587$	−9.91387	−0.409189	−0.204595	+	0.978847i	$0.565588\pi$
$588$	0	0
$589$	5.74592	0.236756
$590$	−2.92518	−0.120428
$591$	0	0
$592$	−1.91150	−0.0785623
$593$	−21.5675	−0.885673	−0.442836	−	0.896602i	$-0.646028\pi$
$593$	−21.5675	−0.885673	−0.442836	+	0.896602i	$0.646028\pi$
$594$	0	0
$595$	−16.5892	−0.680092
$596$	1.00000	0.0409616
$597$	0	0
$598$	−5.77537	−0.236173
$599$	−4.47812	−0.182971	−0.0914854	−	0.995806i	$-0.529161\pi$
$599$	−4.47812	−0.182971	−0.0914854	+	0.995806i	$0.529161\pi$
$600$	0	0
$601$	−9.29537	−0.379166	−0.189583	−	0.981865i	$-0.560714\pi$
$601$	−9.29537	−0.379166	−0.189583	+	0.981865i	$0.560714\pi$
$602$	2.51407	0.102466
$603$	0	0
$604$	−11.4508	−0.465927
$605$	−40.5110	−1.64701
$606$	0	0
$607$	35.9673	1.45987	0.729933	−	0.683518i	$-0.239551\pi$
$607$	35.9673	1.45987	0.729933	+	0.683518i	$0.239551\pi$
$608$	0.972350	0.0394340
$609$	0	0
$610$	−15.3870	−0.623003
$611$	22.7464	0.920221
$612$	0	0
$613$	32.3275	1.30570	0.652849	−	0.757488i	$-0.273574\pi$
$613$	32.3275	1.30570	0.652849	+	0.757488i	$0.273574\pi$
$614$	31.5787	1.27441
$615$	0	0
$616$	15.3017	0.616521
$617$	33.4194	1.34541	0.672707	−	0.739909i	$-0.265131\pi$
$617$	33.4194	1.34541	0.672707	+	0.739909i	$0.265131\pi$
$618$	0	0
$619$	−26.3165	−1.05775	−0.528875	−	0.848700i	$-0.677386\pi$
$619$	−26.3165	−1.05775	−0.528875	+	0.848700i	$0.677386\pi$
$620$	10.1950	0.409442
$621$	0	0
$622$	−10.9006	−0.437072
$623$	−5.43061	−0.217573
$624$	0	0
$625$	−10.7877	−0.431509
$626$	−2.80950	−0.112290
$627$	0	0
$628$	−2.49203	−0.0994428
$629$	7.05348	0.281241
$630$	0	0
$631$	0.902840	0.0359415	0.0179707	−	0.999839i	$-0.494279\pi$
$631$	0.902840	0.0359415	0.0179707	+	0.999839i	$0.494279\pi$
$632$	6.18956	0.246208
$633$	0	0
$634$	−1.65172	−0.0655980
$635$	12.5115	0.496503
$636$	0	0
$637$	−0.833528	−0.0330256
$638$	4.66097	0.184530
$639$	0	0
$640$	1.72525	0.0681964
$641$	−5.47310	−0.216174	−0.108087	−	0.994141i	$-0.534473\pi$
$641$	−5.47310	−0.216174	−0.108087	+	0.994141i	$0.534473\pi$
$642$	0	0
$643$	14.4305	0.569082	0.284541	−	0.958664i	$-0.408159\pi$
$643$	14.4305	0.569082	0.284541	+	0.958664i	$0.408159\pi$
$644$	−3.78503	−0.149151
$645$	0	0
$646$	−3.58799	−0.141168
$647$	14.4809	0.569303	0.284652	−	0.958631i	$-0.408122\pi$
$647$	14.4809	0.569303	0.284652	+	0.958631i	$0.408122\pi$
$648$	0	0
$649$	−9.95618	−0.390814
$650$	8.04574	0.315580
$651$	0	0
$652$	−23.2212	−0.909411
$653$	−2.85456	−0.111708	−0.0558538	−	0.998439i	$-0.517788\pi$
$653$	−2.85456	−0.111708	−0.0558538	+	0.998439i	$0.517788\pi$
$654$	0	0
$655$	−6.15160	−0.240363
$656$	−2.77935	−0.108515
$657$	0	0
$658$	14.9074	0.581150
$659$	−11.4305	−0.445268	−0.222634	−	0.974902i	$-0.571466\pi$
$659$	−11.4305	−0.445268	−0.222634	+	0.974902i	$0.571466\pi$
$660$	0	0
$661$	−25.4333	−0.989243	−0.494621	−	0.869109i	$-0.664693\pi$
$661$	−25.4333	−0.989243	−0.494621	+	0.869109i	$0.664693\pi$
$662$	24.8063	0.964126
$663$	0	0
$664$	14.2544	0.553176
$665$	−4.37140	−0.169516
$666$	0	0
$667$	−1.15294	−0.0446420
$668$	18.5579	0.718028
$669$	0	0
$670$	1.95945	0.0757002
$671$	−52.3716	−2.02178
$672$	0	0
$673$	14.5973	0.562686	0.281343	−	0.959607i	$-0.409220\pi$
$673$	14.5973	0.562686	0.281343	+	0.959607i	$0.409220\pi$
$674$	−6.76816	−0.260700
$675$	0	0
$676$	2.80941	0.108054
$677$	−5.17809	−0.199010	−0.0995051	−	0.995037i	$-0.531726\pi$
$677$	−5.17809	−0.199010	−0.0995051	+	0.995037i	$0.531726\pi$
$678$	0	0
$679$	−12.0004	−0.460534
$680$	−6.36619	−0.244132
$681$	0	0
$682$	34.6999	1.32873
$683$	19.5785	0.749151	0.374575	−	0.927196i	$-0.377789\pi$
$683$	19.5785	0.749151	0.374575	+	0.927196i	$0.377789\pi$
$684$	0	0
$685$	−28.5885	−1.09231
$686$	−18.7871	−0.717295
$687$	0	0
$688$	0.964786	0.0367821
$689$	48.3276	1.84113
$690$	0	0
$691$	17.8111	0.677567	0.338784	−	0.940864i	$-0.389985\pi$
$691$	17.8111	0.677567	0.338784	+	0.940864i	$0.389985\pi$
$692$	17.3894	0.661046
$693$	0	0
$694$	11.1010	0.421387
$695$	−18.3541	−0.696209
$696$	0	0
$697$	10.2558	0.388468
$698$	−27.2718	−1.03225
$699$	0	0
$700$	5.27296	0.199299
$701$	−0.257621	−0.00973021	−0.00486510	−	0.999988i	$-0.501549\pi$
$701$	−0.257621	−0.00973021	−0.00486510	+	0.999988i	$0.501549\pi$
$702$	0	0
$703$	1.85865	0.0701003
$704$	5.87208	0.221312
$705$	0	0
$706$	0.471858	0.0177586
$707$	−16.5936	−0.624067
$708$	0	0
$709$	−50.2989	−1.88902	−0.944508	−	0.328488i	$-0.893461\pi$
$709$	−50.2989	−1.88902	−0.944508	+	0.328488i	$0.893461\pi$
$710$	5.63598	0.211514
$711$	0	0
$712$	−2.08402	−0.0781019
$713$	−8.58339	−0.321451
$714$	0	0
$715$	−40.2811	−1.50643
$716$	−0.962244	−0.0359607
$717$	0	0
$718$	13.0395	0.486629
$719$	28.5044	1.06304	0.531518	−	0.847047i	$-0.321622\pi$
$719$	28.5044	1.06304	0.531518	+	0.847047i	$0.321622\pi$
$720$	0	0
$721$	−1.15437	−0.0429909
$722$	18.0545	0.671920
$723$	0	0
$724$	−15.2649	−0.567314
$725$	1.60618	0.0596519
$726$	0	0
$727$	45.6430	1.69281	0.846403	−	0.532543i	$-0.178763\pi$
$727$	45.6430	1.69281	0.846403	+	0.532543i	$0.178763\pi$
$728$	10.3611	0.384007
$729$	0	0
$730$	6.29447	0.232969
$731$	−3.56008	−0.131674
$732$	0	0
$733$	15.5822	0.575540	0.287770	−	0.957700i	$-0.407086\pi$
$733$	15.5822	0.575540	0.287770	+	0.957700i	$0.407086\pi$
$734$	3.53216	0.130374
$735$	0	0
$736$	−1.45252	−0.0535406
$737$	6.66922	0.245664
$738$	0	0
$739$	−0.739981	−0.0272207	−0.0136103	−	0.999907i	$-0.504332\pi$
$739$	−0.739981	−0.0272207	−0.0136103	+	0.999907i	$0.504332\pi$
$740$	3.29781	0.121230
$741$	0	0
$742$	31.6726	1.16274
$743$	−2.42744	−0.0890540	−0.0445270	−	0.999008i	$-0.514178\pi$
$743$	−2.42744	−0.0890540	−0.0445270	+	0.999008i	$0.514178\pi$
$744$	0	0
$745$	−1.72525	−0.0632082
$746$	30.5929	1.12009
$747$	0	0
$748$	−21.6681	−0.792263
$749$	−3.13302	−0.114478
$750$	0	0
$751$	−43.3867	−1.58320	−0.791601	−	0.611038i	$-0.790752\pi$
$751$	−43.3867	−1.58320	−0.791601	+	0.611038i	$0.790752\pi$
$752$	5.72077	0.208615
$753$	0	0
$754$	3.15604	0.114936
$755$	19.7555	0.718975
$756$	0	0
$757$	16.9892	0.617481	0.308741	−	0.951146i	$-0.400092\pi$
$757$	16.9892	0.617481	0.308741	+	0.951146i	$0.400092\pi$
$758$	0.756326	0.0274710
$759$	0	0
$760$	−1.67754	−0.0608509
$761$	−16.1343	−0.584866	−0.292433	−	0.956286i	$-0.594465\pi$
$761$	−16.1343	−0.584866	−0.292433	+	0.956286i	$0.594465\pi$
$762$	0	0
$763$	−1.37622	−0.0498226
$764$	−13.3385	−0.482570
$765$	0	0
$766$	−15.5219	−0.560828
$767$	−6.74153	−0.243423
$768$	0	0
$769$	−22.3888	−0.807360	−0.403680	−	0.914900i	$-0.632269\pi$
$769$	−22.3888	−0.807360	−0.403680	+	0.914900i	$0.632269\pi$
$770$	−26.3991	−0.951359
$771$	0	0
$772$	−0.978390	−0.0352130
$773$	20.7715	0.747100	0.373550	−	0.927610i	$-0.378141\pi$
$773$	20.7715	0.747100	0.373550	+	0.927610i	$0.378141\pi$
$774$	0	0
$775$	11.9576	0.429531
$776$	−4.60522	−0.165318
$777$	0	0
$778$	34.2558	1.22813
$779$	2.70250	0.0968270
$780$	0	0
$781$	19.1827	0.686411
$782$	5.35983	0.191667
$783$	0	0
$784$	−0.209634	−0.00748694
$785$	4.29937	0.153451
$786$	0	0
$787$	−3.27862	−0.116870	−0.0584350	−	0.998291i	$-0.518611\pi$
$787$	−3.27862	−0.116870	−0.0584350	+	0.998291i	$0.518611\pi$
$788$	−15.9854	−0.569455
$789$	0	0
$790$	−10.6785	−0.379925
$791$	9.86047	0.350598
$792$	0	0
$793$	−35.4619	−1.25929
$794$	4.32270	0.153407
$795$	0	0
$796$	2.31903	0.0821959
$797$	31.0764	1.10078	0.550392	−	0.834906i	$-0.314478\pi$
$797$	31.0764	1.10078	0.550392	+	0.834906i	$0.314478\pi$
$798$	0	0
$799$	−21.1098	−0.746810
$800$	2.02352	0.0715424
$801$	0	0
$802$	21.3467	0.753778
$803$	21.4240	0.756035
$804$	0	0
$805$	6.53010	0.230156
$806$	23.4960	0.827612
$807$	0	0
$808$	−6.36787	−0.224021
$809$	−51.4149	−1.80765	−0.903826	−	0.427901i	$-0.859253\pi$
$809$	−51.4149	−1.80765	−0.903826	+	0.427901i	$0.859253\pi$
$810$	0	0
$811$	13.6647	0.479831	0.239915	−	0.970794i	$-0.422880\pi$
$811$	13.6647	0.479831	0.239915	+	0.970794i	$0.422880\pi$
$812$	2.06838	0.0725861
$813$	0	0
$814$	11.2245	0.393418
$815$	40.0622	1.40332
$816$	0	0
$817$	−0.938109	−0.0328203
$818$	−33.9729	−1.18783
$819$	0	0
$820$	4.79506	0.167451
$821$	−37.8782	−1.32196	−0.660979	−	0.750404i	$-0.729859\pi$
$821$	−37.8782	−1.32196	−0.660979	+	0.750404i	$0.729859\pi$
$822$	0	0
$823$	2.26224	0.0788566	0.0394283	−	0.999222i	$-0.487446\pi$
$823$	2.26224	0.0788566	0.0394283	+	0.999222i	$0.487446\pi$
$824$	−0.442994	−0.0154324
$825$	0	0
$826$	−4.41822	−0.153730
$827$	−31.6475	−1.10049	−0.550246	−	0.835003i	$-0.685466\pi$
$827$	−31.6475	−1.10049	−0.550246	+	0.835003i	$0.685466\pi$
$828$	0	0
$829$	36.3626	1.26293	0.631463	−	0.775406i	$-0.282455\pi$
$829$	36.3626	1.26293	0.631463	+	0.775406i	$0.282455\pi$
$830$	−24.5923	−0.853611
$831$	0	0
$832$	3.97610	0.137847
$833$	0.773554	0.0268021
$834$	0	0
$835$	−32.0170	−1.10799
$836$	−5.70972	−0.197475
$837$	0	0
$838$	−18.8895	−0.652527
$839$	−10.2052	−0.352324	−0.176162	−	0.984361i	$-0.556368\pi$
$839$	−10.2052	−0.352324	−0.176162	+	0.984361i	$0.556368\pi$
$840$	0	0
$841$	−28.3700	−0.978274
$842$	−11.7138	−0.403684
$843$	0	0
$844$	12.7331	0.438291
$845$	−4.84692	−0.166739
$846$	0	0
$847$	−61.1884	−2.10246
$848$	12.1545	0.417387
$849$	0	0
$850$	−7.46684	−0.256110
$851$	−2.77650	−0.0951771
$852$	0	0
$853$	−45.4657	−1.55672	−0.778358	−	0.627821i	$-0.783947\pi$
$853$	−45.4657	−1.55672	−0.778358	+	0.627821i	$0.783947\pi$
$854$	−23.2408	−0.795283
$855$	0	0
$856$	−1.20231	−0.0410941
$857$	−24.5848	−0.839800	−0.419900	−	0.907570i	$-0.637935\pi$
$857$	−24.5848	−0.839800	−0.419900	+	0.907570i	$0.637935\pi$
$858$	0	0
$859$	−2.10387	−0.0717832	−0.0358916	−	0.999356i	$-0.511427\pi$
$859$	−2.10387	−0.0717832	−0.0358916	+	0.999356i	$0.511427\pi$
$860$	−1.66449	−0.0567587
$861$	0	0
$862$	17.8470	0.607872
$863$	−21.5707	−0.734276	−0.367138	−	0.930166i	$-0.619662\pi$
$863$	−21.5707	−0.734276	−0.367138	+	0.930166i	$0.619662\pi$
$864$	0	0
$865$	−30.0010	−1.02007
$866$	34.1206	1.15947
$867$	0	0
$868$	15.3987	0.522665
$869$	−36.3456	−1.23294
$870$	0	0
$871$	4.51587	0.153014
$872$	−0.528131	−0.0178848
$873$	0	0
$874$	1.41236	0.0477737
$875$	−31.5757	−1.06745
$876$	0	0
$877$	16.2268	0.547940	0.273970	−	0.961738i	$-0.411663\pi$
$877$	16.2268	0.547940	0.273970	+	0.961738i	$0.411663\pi$
$878$	2.87290	0.0969556
$879$	0	0
$880$	−10.1308	−0.341509
$881$	−14.5370	−0.489763	−0.244882	−	0.969553i	$-0.578749\pi$
$881$	−14.5370	−0.489763	−0.244882	+	0.969553i	$0.578749\pi$
$882$	0	0
$883$	−17.0988	−0.575421	−0.287711	−	0.957717i	$-0.592894\pi$
$883$	−17.0988	−0.575421	−0.287711	+	0.957717i	$0.592894\pi$
$884$	−14.6719	−0.493469
$885$	0	0
$886$	29.2501	0.982675
$887$	52.0091	1.74629	0.873147	−	0.487456i	$-0.162075\pi$
$887$	52.0091	1.74629	0.873147	+	0.487456i	$0.162075\pi$
$888$	0	0
$889$	18.8975	0.633802
$890$	3.59545	0.120520
$891$	0	0
$892$	−13.1207	−0.439313
$893$	−5.56259	−0.186145
$894$	0	0
$895$	1.66011	0.0554913
$896$	2.60583	0.0870548
$897$	0	0
$898$	11.4354	0.381604
$899$	4.69053	0.156438
$900$	0	0
$901$	−44.8503	−1.49418
$902$	16.3205	0.543415
$903$	0	0
$904$	3.78400	0.125854
$905$	26.3357	0.875427
$906$	0	0
$907$	−25.0791	−0.832737	−0.416368	−	0.909196i	$-0.636697\pi$
$907$	−25.0791	−0.832737	−0.416368	+	0.909196i	$0.636697\pi$
$908$	5.44603	0.180733
$909$	0	0
$910$	−17.8754	−0.592564
$911$	−5.94833	−0.197077	−0.0985386	−	0.995133i	$-0.531417\pi$
$911$	−5.94833	−0.197077	−0.0985386	+	0.995133i	$0.531417\pi$
$912$	0	0
$913$	−83.7027	−2.77015
$914$	−13.2066	−0.436837
$915$	0	0
$916$	13.9568	0.461147
$917$	−9.29145	−0.306831
$918$	0	0
$919$	58.6908	1.93603	0.968015	−	0.250891i	$-0.0807235\pi$
$919$	58.6908	1.93603	0.968015	+	0.250891i	$0.0807235\pi$
$920$	2.50596	0.0826189
$921$	0	0
$922$	23.7293	0.781483
$923$	12.9890	0.427538
$924$	0	0
$925$	3.86797	0.127178
$926$	6.33611	0.208218
$927$	0	0
$928$	0.793752	0.0260562
$929$	9.12012	0.299222	0.149611	−	0.988745i	$-0.452198\pi$
$929$	9.12012	0.299222	0.149611	+	0.988745i	$0.452198\pi$
$930$	0	0
$931$	0.203838	0.00668052
$932$	−26.4164	−0.865297
$933$	0	0
$934$	10.5195	0.344207
$935$	37.3828	1.22255
$936$	0	0
$937$	7.19910	0.235184	0.117592	−	0.993062i	$-0.462482\pi$
$937$	7.19910	0.235184	0.117592	+	0.993062i	$0.462482\pi$
$938$	2.95958	0.0966336
$939$	0	0
$940$	−9.86975	−0.321916
$941$	−4.41171	−0.143818	−0.0719088	−	0.997411i	$-0.522909\pi$
$941$	−4.41171	−0.143818	−0.0719088	+	0.997411i	$0.522909\pi$
$942$	0	0
$943$	−4.03706	−0.131465
$944$	−1.69551	−0.0551842
$945$	0	0
$946$	−5.66530	−0.184195
$947$	2.69955	0.0877237	0.0438618	−	0.999038i	$-0.486034\pi$
$947$	2.69955	0.0877237	0.0438618	+	0.999038i	$0.486034\pi$
$948$	0	0
$949$	14.5066	0.470904
$950$	−1.96757	−0.0638365
$951$	0	0
$952$	−9.61557	−0.311642
$953$	55.7558	1.80611	0.903054	−	0.429528i	$-0.141320\pi$
$953$	55.7558	1.80611	0.903054	+	0.429528i	$0.141320\pi$
$954$	0	0
$955$	23.0122	0.744657
$956$	3.39802	0.109900
$957$	0	0
$958$	0.598309	0.0193305
$959$	−43.1803	−1.39437
$960$	0	0
$961$	3.91993	0.126449
$962$	7.60033	0.245045
$963$	0	0
$964$	−9.84720	−0.317157
$965$	1.68796	0.0543375
$966$	0	0
$967$	42.8115	1.37672	0.688362	−	0.725367i	$-0.258330\pi$
$967$	42.8115	1.37672	0.688362	+	0.725367i	$0.258330\pi$
$968$	−23.4813	−0.754718
$969$	0	0
$970$	7.94514	0.255103
$971$	−12.2179	−0.392092	−0.196046	−	0.980595i	$-0.562810\pi$
$971$	−12.2179	−0.392092	−0.196046	+	0.980595i	$0.562810\pi$
$972$	0	0
$973$	−27.7222	−0.888733
$974$	−3.97672	−0.127422
$975$	0	0
$976$	−8.91875	−0.285482
$977$	−23.1547	−0.740786	−0.370393	−	0.928875i	$-0.620777\pi$
$977$	−23.1547	−0.740786	−0.370393	+	0.928875i	$0.620777\pi$
$978$	0	0
$979$	12.2375	0.391113
$980$	0.361671	0.0115532
$981$	0	0
$982$	12.4900	0.398571
$983$	4.54691	0.145024	0.0725120	−	0.997368i	$-0.476898\pi$
$983$	4.54691	0.145024	0.0725120	+	0.997368i	$0.476898\pi$
$984$	0	0
$985$	27.5787	0.878730
$986$	−2.92896	−0.0932771
$987$	0	0
$988$	−3.86617	−0.122999
$989$	1.40137	0.0445610
$990$	0	0
$991$	−35.1004	−1.11500	−0.557501	−	0.830176i	$-0.688240\pi$
$991$	−35.1004	−1.11500	−0.557501	+	0.830176i	$0.688240\pi$
$992$	5.90931	0.187621
$993$	0	0
$994$	8.51264	0.270005
$995$	−4.00090	−0.126837
$996$	0	0
$997$	−1.65780	−0.0525031	−0.0262515	−	0.999655i	$-0.508357\pi$
$997$	−1.65780	−0.0525031	−0.0262515	+	0.999655i	$0.508357\pi$
$998$	−30.1380	−0.954000
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.g.1.3	✓	9
3.2	odd	2		8046.2.a.h.1.7	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.g.1.3	✓	9	1.1	even	1	trivial
8046.2.a.h.1.7	yes	9	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.72525 q^{5} -2.60583 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.72525 q^{5} -2.60583 q^{7} -1.00000 q^{8} +1.72525 q^{10} +5.87208 q^{11} +3.97610 q^{13} +2.60583 q^{14} +1.00000 q^{16} -3.69002 q^{17} -0.972350 q^{19} -1.72525 q^{20} -5.87208 q^{22} +1.45252 q^{23} -2.02352 q^{25} -3.97610 q^{26} -2.60583 q^{28} -0.793752 q^{29} -5.90931 q^{31} -1.00000 q^{32} +3.69002 q^{34} +4.49571 q^{35} -1.91150 q^{37} +0.972350 q^{38} +1.72525 q^{40} -2.77935 q^{41} +0.964786 q^{43} +5.87208 q^{44} -1.45252 q^{46} +5.72077 q^{47} -0.209634 q^{49} +2.02352 q^{50} +3.97610 q^{52} +12.1545 q^{53} -10.1308 q^{55} +2.60583 q^{56} +0.793752 q^{58} -1.69551 q^{59} -8.91875 q^{61} +5.90931 q^{62} +1.00000 q^{64} -6.85976 q^{65} +1.13575 q^{67} -3.69002 q^{68} -4.49571 q^{70} +3.26677 q^{71} +3.64845 q^{73} +1.91150 q^{74} -0.972350 q^{76} -15.3017 q^{77} -6.18956 q^{79} -1.72525 q^{80} +2.77935 q^{82} -14.2544 q^{83} +6.36619 q^{85} -0.964786 q^{86} -5.87208 q^{88} +2.08402 q^{89} -10.3611 q^{91} +1.45252 q^{92} -5.72077 q^{94} +1.67754 q^{95} +4.60522 q^{97} +0.209634 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) \) 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8	\( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8} - 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 9 q^{16} + q^{17} - 10 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 3 q^{25} + 8 q^{26} - 4 q^{28} + 4 q^{29} - 17 q^{31} - 9 q^{32} - q^{34} + 10 q^{35} - 11 q^{37} + 10 q^{38} - 4 q^{40} - 16 q^{43} + 4 q^{44} - 8 q^{46} + 7 q^{47} - 5 q^{49} + 3 q^{50} - 8 q^{52} + 12 q^{53} - 23 q^{55} + 4 q^{56} - 4 q^{58} + 6 q^{59} - 13 q^{61} + 17 q^{62} + 9 q^{64} - 24 q^{65} - 14 q^{67} + q^{68} - 10 q^{70} + 30 q^{71} - 12 q^{73} + 11 q^{74} - 10 q^{76} + 12 q^{77} - 35 q^{79} + 4 q^{80} + 5 q^{83} - 27 q^{85} + 16 q^{86} - 4 q^{88} + 23 q^{89} - 28 q^{91} + 8 q^{92} - 7 q^{94} + 32 q^{95} - 21 q^{97} + 5 q^{98}+O(q^{100}) \) 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8 - 4 * q^10 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 9 * q^16 + q^17 - 10 * q^19 + 4 * q^20 - 4 * q^22 + 8 * q^23 - 3 * q^25 + 8 * q^26 - 4 * q^28 + 4 * q^29 - 17 * q^31 - 9 * q^32 - q^34 + 10 * q^35 - 11 * q^37 + 10 * q^38 - 4 * q^40 - 16 * q^43 + 4 * q^44 - 8 * q^46 + 7 * q^47 - 5 * q^49 + 3 * q^50 - 8 * q^52 + 12 * q^53 - 23 * q^55 + 4 * q^56 - 4 * q^58 + 6 * q^59 - 13 * q^61 + 17 * q^62 + 9 * q^64 - 24 * q^65 - 14 * q^67 + q^68 - 10 * q^70 + 30 * q^71 - 12 * q^73 + 11 * q^74 - 10 * q^76 + 12 * q^77 - 35 * q^79 + 4 * q^80 + 5 * q^83 - 27 * q^85 + 16 * q^86 - 4 * q^88 + 23 * q^89 - 28 * q^91 + 8 * q^92 - 7 * q^94 + 32 * q^95 - 21 * q^97 + 5 * q^98

Introduction

L-functions

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