LMFDB - Embedded newform 4005.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
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} - 4x^{8} - 6x^{7} + 31x^{6} + 13x^{5} - 75x^{4} - 17x^{3} + 52x^{2} + 11x - 1 $ x^9 - 4x^8 - 6x^7 + 31x^6 + 13x^5 - 75x^4 - 17x^3 + 52x^2 + 11x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.70432$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70432 q^{2} +5.31333 q^{4} +1.00000 q^{5} +3.91511 q^{7} -8.96030 q^{8} +O(q^{10})$	\(q-2.70432 q^{2} +5.31333 q^{4} +1.00000 q^{5} +3.91511 q^{7} -8.96030 q^{8} -2.70432 q^{10} -0.290736 q^{11} -2.36069 q^{13} -10.5877 q^{14} +13.6048 q^{16} -4.13501 q^{17} -7.64191 q^{19} +5.31333 q^{20} +0.786242 q^{22} +6.01817 q^{23} +1.00000 q^{25} +6.38406 q^{26} +20.8023 q^{28} -8.87608 q^{29} +7.33038 q^{31} -18.8712 q^{32} +11.1824 q^{34} +3.91511 q^{35} +0.927874 q^{37} +20.6661 q^{38} -8.96030 q^{40} +10.6233 q^{41} -9.40345 q^{43} -1.54478 q^{44} -16.2751 q^{46} -11.2630 q^{47} +8.32807 q^{49} -2.70432 q^{50} -12.5431 q^{52} -4.74498 q^{53} -0.290736 q^{55} -35.0805 q^{56} +24.0037 q^{58} +8.49811 q^{59} -2.98630 q^{61} -19.8237 q^{62} +23.8239 q^{64} -2.36069 q^{65} -7.70384 q^{67} -21.9706 q^{68} -10.5877 q^{70} -4.65945 q^{71} -11.9200 q^{73} -2.50926 q^{74} -40.6040 q^{76} -1.13826 q^{77} +13.5279 q^{79} +13.6048 q^{80} -28.7288 q^{82} -3.43747 q^{83} -4.13501 q^{85} +25.4299 q^{86} +2.60508 q^{88} -1.00000 q^{89} -9.24237 q^{91} +31.9765 q^{92} +30.4587 q^{94} -7.64191 q^{95} -6.74861 q^{97} -22.5218 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8}+O(q^{10}) $ 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8	$ 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8} - 5 q^{10} - 4 q^{11} + 5 q^{13} + q^{14} + 15 q^{16} - 21 q^{17} - 18 q^{19} + 11 q^{20} + 6 q^{22} - 16 q^{23} + 9 q^{25} - 8 q^{26} + 6 q^{28} - 3 q^{29} - 6 q^{31} - 46 q^{32} + 12 q^{34} - 3 q^{35} + 11 q^{37} - 20 q^{38} - 18 q^{40} + q^{41} - 3 q^{43} - 38 q^{44} + 16 q^{46} - 27 q^{47} + 24 q^{49} - 5 q^{50} + 17 q^{52} - 43 q^{53} - 4 q^{55} - 5 q^{56} + 34 q^{58} - 3 q^{59} - 30 q^{61} - 36 q^{62} + 50 q^{64} + 5 q^{65} - 12 q^{67} - 64 q^{68} + q^{70} + 4 q^{71} + 26 q^{73} - 2 q^{74} - 12 q^{76} - 34 q^{77} + q^{79} + 15 q^{80} - 51 q^{82} - 24 q^{83} - 21 q^{85} - 18 q^{86} + 64 q^{88} - 9 q^{89} - 50 q^{91} - 10 q^{92} - 11 q^{94} - 18 q^{95} - 4 q^{97} - 75 q^{98}+O(q^{100}) $ 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8 - 5 * q^10 - 4 * q^11 + 5 * q^13 + q^14 + 15 * q^16 - 21 * q^17 - 18 * q^19 + 11 * q^20 + 6 * q^22 - 16 * q^23 + 9 * q^25 - 8 * q^26 + 6 * q^28 - 3 * q^29 - 6 * q^31 - 46 * q^32 + 12 * q^34 - 3 * q^35 + 11 * q^37 - 20 * q^38 - 18 * q^40 + q^41 - 3 * q^43 - 38 * q^44 + 16 * q^46 - 27 * q^47 + 24 * q^49 - 5 * q^50 + 17 * q^52 - 43 * q^53 - 4 * q^55 - 5 * q^56 + 34 * q^58 - 3 * q^59 - 30 * q^61 - 36 * q^62 + 50 * q^64 + 5 * q^65 - 12 * q^67 - 64 * q^68 + q^70 + 4 * q^71 + 26 * q^73 - 2 * q^74 - 12 * q^76 - 34 * q^77 + q^79 + 15 * q^80 - 51 * q^82 - 24 * q^83 - 21 * q^85 - 18 * q^86 + 64 * q^88 - 9 * q^89 - 50 * q^91 - 10 * q^92 - 11 * q^94 - 18 * q^95 - 4 * q^97 - 75 * 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 $

$2$	−2.70432	−1.91224	−0.956120	−	0.292974i	$-0.905355\pi$
$2$	−2.70432	−1.91224	−0.956120	+	0.292974i	$0.905355\pi$
$3$	0	0
$3$	0	0
$4$	5.31333	2.65667
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.91511	1.47977	0.739886	−	0.672732i	$-0.234879\pi$
$7$	3.91511	1.47977	0.739886	+	0.672732i	$0.234879\pi$
$8$	−8.96030	−3.16794
$9$	0	0
$10$	−2.70432	−0.855180
$11$	−0.290736	−0.0876602	−0.0438301	−	0.999039i	$-0.513956\pi$
$11$	−0.290736	−0.0876602	−0.0438301	+	0.999039i	$0.513956\pi$
$12$	0	0
$13$	−2.36069	−0.654738	−0.327369	−	0.944896i	$-0.606162\pi$
$13$	−2.36069	−0.654738	−0.327369	+	0.944896i	$0.606162\pi$
$14$	−10.5877	−2.82968
$15$	0	0
$16$	13.6048	3.40120
$17$	−4.13501	−1.00289	−0.501443	−	0.865191i	$-0.667197\pi$
$17$	−4.13501	−1.00289	−0.501443	+	0.865191i	$0.667197\pi$
$18$	0	0
$19$	−7.64191	−1.75317	−0.876587	−	0.481244i	$-0.840185\pi$
$19$	−7.64191	−1.75317	−0.876587	+	0.481244i	$0.840185\pi$
$20$	5.31333	1.18810
$21$	0	0
$22$	0.786242	0.167627
$23$	6.01817	1.25488	0.627438	−	0.778667i	$-0.284104\pi$
$23$	6.01817	1.25488	0.627438	+	0.778667i	$0.284104\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.38406	1.25202
$27$	0	0
$28$	20.8023	3.93126
$29$	−8.87608	−1.64825	−0.824124	−	0.566410i	$-0.808332\pi$
$29$	−8.87608	−1.64825	−0.824124	+	0.566410i	$0.808332\pi$
$30$	0	0
$31$	7.33038	1.31658	0.658288	−	0.752766i	$-0.271281\pi$
$31$	7.33038	1.31658	0.658288	+	0.752766i	$0.271281\pi$
$32$	−18.8712	−3.33598
$33$	0	0
$34$	11.1824	1.91776
$35$	3.91511	0.661774
$36$	0	0
$37$	0.927874	0.152541	0.0762707	−	0.997087i	$-0.475699\pi$
$37$	0.927874	0.152541	0.0762707	+	0.997087i	$0.475699\pi$
$38$	20.6661	3.35249
$39$	0	0
$40$	−8.96030	−1.41675
$41$	10.6233	1.65908	0.829541	−	0.558446i	$-0.188602\pi$
$41$	10.6233	1.65908	0.829541	+	0.558446i	$0.188602\pi$
$42$	0	0
$43$	−9.40345	−1.43401	−0.717007	−	0.697066i	$-0.754488\pi$
$43$	−9.40345	−1.43401	−0.717007	+	0.697066i	$0.754488\pi$
$44$	−1.54478	−0.232884
$45$	0	0
$46$	−16.2751	−2.39963
$47$	−11.2630	−1.64288	−0.821438	−	0.570298i	$-0.806828\pi$
$47$	−11.2630	−1.64288	−0.821438	+	0.570298i	$0.806828\pi$
$48$	0	0
$49$	8.32807	1.18972
$50$	−2.70432	−0.382448
$51$	0	0
$52$	−12.5431	−1.73942
$53$	−4.74498	−0.651773	−0.325887	−	0.945409i	$-0.605663\pi$
$53$	−4.74498	−0.651773	−0.325887	+	0.945409i	$0.605663\pi$
$54$	0	0
$55$	−0.290736	−0.0392028
$56$	−35.0805	−4.68783
$57$	0	0
$58$	24.0037	3.15185
$59$	8.49811	1.10636	0.553180	−	0.833062i	$-0.313414\pi$
$59$	8.49811	1.10636	0.553180	+	0.833062i	$0.313414\pi$
$60$	0	0
$61$	−2.98630	−0.382356	−0.191178	−	0.981555i	$-0.561231\pi$
$61$	−2.98630	−0.382356	−0.191178	+	0.981555i	$0.561231\pi$
$62$	−19.8237	−2.51761
$63$	0	0
$64$	23.8239	2.97799
$65$	−2.36069	−0.292808
$66$	0	0
$67$	−7.70384	−0.941174	−0.470587	−	0.882354i	$-0.655958\pi$
$67$	−7.70384	−0.941174	−0.470587	+	0.882354i	$0.655958\pi$
$68$	−21.9706	−2.66433
$69$	0	0
$70$	−10.5877	−1.26547
$71$	−4.65945	−0.552975	−0.276488	−	0.961017i	$-0.589170\pi$
$71$	−4.65945	−0.552975	−0.276488	+	0.961017i	$0.589170\pi$
$72$	0	0
$73$	−11.9200	−1.39513	−0.697566	−	0.716520i	$-0.745734\pi$
$73$	−11.9200	−1.39513	−0.697566	+	0.716520i	$0.745734\pi$
$74$	−2.50926	−0.291696
$75$	0	0
$76$	−40.6040	−4.65760
$77$	−1.13826	−0.129717
$78$	0	0
$79$	13.5279	1.52201	0.761004	−	0.648748i	$-0.224707\pi$
$79$	13.5279	1.52201	0.761004	+	0.648748i	$0.224707\pi$
$80$	13.6048	1.52107
$81$	0	0
$82$	−28.7288	−3.17257
$83$	−3.43747	−0.377311	−0.188655	−	0.982043i	$-0.560413\pi$
$83$	−3.43747	−0.377311	−0.188655	+	0.982043i	$0.560413\pi$
$84$	0	0
$85$	−4.13501	−0.448504
$86$	25.4299	2.74218
$87$	0	0
$88$	2.60508	0.277702
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−9.24237	−0.968864
$92$	31.9765	3.33379
$93$	0	0
$94$	30.4587	3.14157
$95$	−7.64191	−0.784043
$96$	0	0
$97$	−6.74861	−0.685218	−0.342609	−	0.939478i	$-0.611311\pi$
$97$	−6.74861	−0.685218	−0.342609	+	0.939478i	$0.611311\pi$
$98$	−22.5218	−2.27504
$99$	0	0
$100$	5.31333	0.531333
$101$	2.94829	0.293366	0.146683	−	0.989184i	$-0.453140\pi$
$101$	2.94829	0.293366	0.146683	+	0.989184i	$0.453140\pi$
$102$	0	0
$103$	−2.72989	−0.268984	−0.134492	−	0.990915i	$-0.542940\pi$
$103$	−2.72989	−0.268984	−0.134492	+	0.990915i	$0.542940\pi$
$104$	21.1525	2.07417
$105$	0	0
$106$	12.8319	1.24635
$107$	−5.39163	−0.521229	−0.260614	−	0.965443i	$-0.583925\pi$
$107$	−5.39163	−0.521229	−0.260614	+	0.965443i	$0.583925\pi$
$108$	0	0
$109$	−13.3308	−1.27686	−0.638430	−	0.769680i	$-0.720416\pi$
$109$	−13.3308	−1.27686	−0.638430	+	0.769680i	$0.720416\pi$
$110$	0.786242	0.0749652
$111$	0	0
$112$	53.2643	5.03301
$113$	5.75372	0.541264	0.270632	−	0.962683i	$-0.412767\pi$
$113$	5.75372	0.541264	0.270632	+	0.962683i	$0.412767\pi$
$114$	0	0
$115$	6.01817	0.561198
$116$	−47.1616	−4.37884
$117$	0	0
$118$	−22.9816	−2.11563
$119$	−16.1890	−1.48404
$120$	0	0
$121$	−10.9155	−0.992316
$122$	8.07590	0.731157
$123$	0	0
$124$	38.9487	3.49770
$125$	1.00000	0.0894427
$126$	0	0
$127$	−15.5337	−1.37839	−0.689197	−	0.724574i	$-0.742037\pi$
$127$	−15.5337	−1.37839	−0.689197	+	0.724574i	$0.742037\pi$
$128$	−26.6852	−2.35866
$129$	0	0
$130$	6.38406	0.559919
$131$	8.52018	0.744411	0.372206	−	0.928150i	$-0.378602\pi$
$131$	8.52018	0.744411	0.372206	+	0.928150i	$0.378602\pi$
$132$	0	0
$133$	−29.9189	−2.59430
$134$	20.8336	1.79975
$135$	0	0
$136$	37.0509	3.17709
$137$	−11.2125	−0.957949	−0.478975	−	0.877829i	$-0.658991\pi$
$137$	−11.2125	−0.957949	−0.478975	+	0.877829i	$0.658991\pi$
$138$	0	0
$139$	−20.0296	−1.69889	−0.849446	−	0.527676i	$-0.823064\pi$
$139$	−20.0296	−1.69889	−0.849446	+	0.527676i	$0.823064\pi$
$140$	20.8023	1.75811
$141$	0	0
$142$	12.6006	1.05742
$143$	0.686338	0.0573945
$144$	0	0
$145$	−8.87608	−0.737118
$146$	32.2355	2.66783
$147$	0	0
$148$	4.93010	0.405252
$149$	−16.2201	−1.32880	−0.664402	−	0.747375i	$-0.731314\pi$
$149$	−16.2201	−1.32880	−0.664402	+	0.747375i	$0.731314\pi$
$150$	0	0
$151$	12.8001	1.04166	0.520828	−	0.853662i	$-0.325623\pi$
$151$	12.8001	1.04166	0.520828	+	0.853662i	$0.325623\pi$
$152$	68.4737	5.55395
$153$	0	0
$154$	3.07822	0.248050
$155$	7.33038	0.588790
$156$	0	0
$157$	1.96635	0.156932	0.0784659	−	0.996917i	$-0.474998\pi$
$157$	1.96635	0.156932	0.0784659	+	0.996917i	$0.474998\pi$
$158$	−36.5837	−2.91044
$159$	0	0
$160$	−18.8712	−1.49190
$161$	23.5618	1.85693
$162$	0	0
$163$	0.152079	0.0119117	0.00595586	−	0.999982i	$-0.498104\pi$
$163$	0.152079	0.0119117	0.00595586	+	0.999982i	$0.498104\pi$
$164$	56.4452	4.40763
$165$	0	0
$166$	9.29600	0.721509
$167$	9.41133	0.728271	0.364135	−	0.931346i	$-0.381365\pi$
$167$	9.41133	0.728271	0.364135	+	0.931346i	$0.381365\pi$
$168$	0	0
$169$	−7.42713	−0.571318
$170$	11.1824	0.857648
$171$	0	0
$172$	−49.9637	−3.80969
$173$	4.96550	0.377520	0.188760	−	0.982023i	$-0.439553\pi$
$173$	4.96550	0.377520	0.188760	+	0.982023i	$0.439553\pi$
$174$	0	0
$175$	3.91511	0.295954
$176$	−3.95541	−0.298150
$177$	0	0
$178$	2.70432	0.202697
$179$	22.0675	1.64940	0.824701	−	0.565569i	$-0.191343\pi$
$179$	22.0675	1.64940	0.824701	+	0.565569i	$0.191343\pi$
$180$	0	0
$181$	14.3322	1.06530	0.532651	−	0.846335i	$-0.321196\pi$
$181$	14.3322	1.06530	0.532651	+	0.846335i	$0.321196\pi$
$182$	24.9943	1.85270
$183$	0	0
$184$	−53.9246	−3.97538
$185$	0.927874	0.0682186
$186$	0	0
$187$	1.20219	0.0879132
$188$	−59.8440	−4.36457
$189$	0	0
$190$	20.6661	1.49928
$191$	6.35834	0.460073	0.230037	−	0.973182i	$-0.426115\pi$
$191$	6.35834	0.460073	0.230037	+	0.973182i	$0.426115\pi$
$192$	0	0
$193$	7.73028	0.556438	0.278219	−	0.960518i	$-0.410256\pi$
$193$	7.73028	0.556438	0.278219	+	0.960518i	$0.410256\pi$
$194$	18.2504	1.31030
$195$	0	0
$196$	44.2498	3.16070
$197$	−16.1717	−1.15218	−0.576092	−	0.817385i	$-0.695423\pi$
$197$	−16.1717	−1.15218	−0.576092	+	0.817385i	$0.695423\pi$
$198$	0	0
$199$	6.87573	0.487408	0.243704	−	0.969850i	$-0.421637\pi$
$199$	6.87573	0.487408	0.243704	+	0.969850i	$0.421637\pi$
$200$	−8.96030	−0.633589
$201$	0	0
$202$	−7.97312	−0.560987
$203$	−34.7508	−2.43903
$204$	0	0
$205$	10.6233	0.741964
$206$	7.38250	0.514363
$207$	0	0
$208$	−32.1168	−2.22690
$209$	2.22178	0.153684
$210$	0	0
$211$	−10.8883	−0.749584	−0.374792	−	0.927109i	$-0.622286\pi$
$211$	−10.8883	−0.749584	−0.374792	+	0.927109i	$0.622286\pi$
$212$	−25.2117	−1.73154
$213$	0	0
$214$	14.5807	0.996715
$215$	−9.40345	−0.641310
$216$	0	0
$217$	28.6992	1.94823
$218$	36.0507	2.44166
$219$	0	0
$220$	−1.54478	−0.104149
$221$	9.76148	0.656628
$222$	0	0
$223$	17.5914	1.17801	0.589003	−	0.808131i	$-0.299521\pi$
$223$	17.5914	1.17801	0.589003	+	0.808131i	$0.299521\pi$
$224$	−73.8826	−4.93649
$225$	0	0
$226$	−15.5599	−1.03503
$227$	−0.611391	−0.0405794	−0.0202897	−	0.999794i	$-0.506459\pi$
$227$	−0.611391	−0.0405794	−0.0202897	+	0.999794i	$0.506459\pi$
$228$	0	0
$229$	−3.20075	−0.211512	−0.105756	−	0.994392i	$-0.533726\pi$
$229$	−3.20075	−0.211512	−0.105756	+	0.994392i	$0.533726\pi$
$230$	−16.2751	−1.07315
$231$	0	0
$232$	79.5323	5.22155
$233$	−7.75674	−0.508161	−0.254080	−	0.967183i	$-0.581773\pi$
$233$	−7.75674	−0.508161	−0.254080	+	0.967183i	$0.581773\pi$
$234$	0	0
$235$	−11.2630	−0.734717
$236$	45.1532	2.93923
$237$	0	0
$238$	43.7802	2.83785
$239$	28.1059	1.81802	0.909011	−	0.416773i	$-0.136839\pi$
$239$	28.1059	1.81802	0.909011	+	0.416773i	$0.136839\pi$
$240$	0	0
$241$	−18.8774	−1.21600	−0.608001	−	0.793937i	$-0.708028\pi$
$241$	−18.8774	−1.21600	−0.608001	+	0.793937i	$0.708028\pi$
$242$	29.5189	1.89755
$243$	0	0
$244$	−15.8672	−1.01579
$245$	8.32807	0.532061
$246$	0	0
$247$	18.0402	1.14787
$248$	−65.6824	−4.17084
$249$	0	0
$250$	−2.70432	−0.171036
$251$	−9.95645	−0.628446	−0.314223	−	0.949349i	$-0.601744\pi$
$251$	−9.95645	−0.628446	−0.314223	+	0.949349i	$0.601744\pi$
$252$	0	0
$253$	−1.74970	−0.110003
$254$	42.0081	2.63582
$255$	0	0
$256$	24.5173	1.53233
$257$	8.21867	0.512666	0.256333	−	0.966588i	$-0.417486\pi$
$257$	8.21867	0.512666	0.256333	+	0.966588i	$0.417486\pi$
$258$	0	0
$259$	3.63273	0.225727
$260$	−12.5431	−0.777893
$261$	0	0
$262$	−23.0413	−1.42349
$263$	−21.1125	−1.30186	−0.650928	−	0.759140i	$-0.725620\pi$
$263$	−21.1125	−1.30186	−0.650928	+	0.759140i	$0.725620\pi$
$264$	0	0
$265$	−4.74498	−0.291482
$266$	80.9102	4.96092
$267$	0	0
$268$	−40.9330	−2.50038
$269$	1.76519	0.107625	0.0538126	−	0.998551i	$-0.482863\pi$
$269$	1.76519	0.107625	0.0538126	+	0.998551i	$0.482863\pi$
$270$	0	0
$271$	4.26799	0.259262	0.129631	−	0.991562i	$-0.458621\pi$
$271$	4.26799	0.259262	0.129631	+	0.991562i	$0.458621\pi$
$272$	−56.2560	−3.41102
$273$	0	0
$274$	30.3222	1.83183
$275$	−0.290736	−0.0175320
$276$	0	0
$277$	8.15122	0.489760	0.244880	−	0.969553i	$-0.421252\pi$
$277$	8.15122	0.489760	0.244880	+	0.969553i	$0.421252\pi$
$278$	54.1665	3.24869
$279$	0	0
$280$	−35.0805	−2.09646
$281$	−23.1441	−1.38066	−0.690330	−	0.723494i	$-0.742535\pi$
$281$	−23.1441	−1.38066	−0.690330	+	0.723494i	$0.742535\pi$
$282$	0	0
$283$	0.495284	0.0294416	0.0147208	−	0.999892i	$-0.495314\pi$
$283$	0.495284	0.0294416	0.0147208	+	0.999892i	$0.495314\pi$
$284$	−24.7572	−1.46907
$285$	0	0
$286$	−1.85608	−0.109752
$287$	41.5914	2.45506
$288$	0	0
$289$	0.0982690	0.00578053
$290$	24.0037	1.40955
$291$	0	0
$292$	−63.3350	−3.70640
$293$	−20.8859	−1.22017	−0.610084	−	0.792337i	$-0.708864\pi$
$293$	−20.8859	−1.22017	−0.610084	+	0.792337i	$0.708864\pi$
$294$	0	0
$295$	8.49811	0.494779
$296$	−8.31402	−0.483243
$297$	0	0
$298$	43.8644	2.54099
$299$	−14.2071	−0.821616
$300$	0	0
$301$	−36.8155	−2.12201
$302$	−34.6155	−1.99190
$303$	0	0
$304$	−103.967	−5.96290
$305$	−2.98630	−0.170995
$306$	0	0
$307$	26.2842	1.50012	0.750060	−	0.661370i	$-0.230025\pi$
$307$	26.2842	1.50012	0.750060	+	0.661370i	$0.230025\pi$
$308$	−6.04797	−0.344615
$309$	0	0
$310$	−19.8237	−1.12591
$311$	−8.05572	−0.456798	−0.228399	−	0.973568i	$-0.573349\pi$
$311$	−8.05572	−0.456798	−0.228399	+	0.973568i	$0.573349\pi$
$312$	0	0
$313$	19.8942	1.12449	0.562245	−	0.826971i	$-0.309938\pi$
$313$	19.8942	1.12449	0.562245	+	0.826971i	$0.309938\pi$
$314$	−5.31763	−0.300091
$315$	0	0
$316$	71.8782	4.04346
$317$	−9.08595	−0.510318	−0.255159	−	0.966899i	$-0.582128\pi$
$317$	−9.08595	−0.510318	−0.255159	+	0.966899i	$0.582128\pi$
$318$	0	0
$319$	2.58060	0.144486
$320$	23.8239	1.33180
$321$	0	0
$322$	−63.7186	−3.55090
$323$	31.5993	1.75823
$324$	0	0
$325$	−2.36069	−0.130948
$326$	−0.411269	−0.0227781
$327$	0	0
$328$	−95.1880	−5.25588
$329$	−44.0958	−2.43108
$330$	0	0
$331$	31.7850	1.74706	0.873530	−	0.486771i	$-0.161825\pi$
$331$	31.7850	1.74706	0.873530	+	0.486771i	$0.161825\pi$
$332$	−18.2644	−1.00239
$333$	0	0
$334$	−25.4512	−1.39263
$335$	−7.70384	−0.420906
$336$	0	0
$337$	3.71554	0.202398	0.101199	−	0.994866i	$-0.467732\pi$
$337$	3.71554	0.202398	0.101199	+	0.994866i	$0.467732\pi$
$338$	20.0853	1.09250
$339$	0	0
$340$	−21.9706	−1.19153
$341$	−2.13121	−0.115411
$342$	0	0
$343$	5.19955	0.280750
$344$	84.2577	4.54287
$345$	0	0
$346$	−13.4283	−0.721909
$347$	−8.95640	−0.480805	−0.240402	−	0.970673i	$-0.577279\pi$
$347$	−8.95640	−0.480805	−0.240402	+	0.970673i	$0.577279\pi$
$348$	0	0
$349$	−6.54543	−0.350369	−0.175185	−	0.984536i	$-0.556052\pi$
$349$	−6.54543	−0.350369	−0.175185	+	0.984536i	$0.556052\pi$
$350$	−10.5877	−0.565936
$351$	0	0
$352$	5.48652	0.292433
$353$	−18.3177	−0.974953	−0.487476	−	0.873136i	$-0.662082\pi$
$353$	−18.3177	−0.974953	−0.487476	+	0.873136i	$0.662082\pi$
$354$	0	0
$355$	−4.65945	−0.247298
$356$	−5.31333	−0.281606
$357$	0	0
$358$	−59.6775	−3.15406
$359$	−11.6477	−0.614740	−0.307370	−	0.951590i	$-0.599449\pi$
$359$	−11.6477	−0.614740	−0.307370	+	0.951590i	$0.599449\pi$
$360$	0	0
$361$	39.3987	2.07362
$362$	−38.7587	−2.03711
$363$	0	0
$364$	−49.1078	−2.57395
$365$	−11.9200	−0.623922
$366$	0	0
$367$	−22.7882	−1.18954	−0.594768	−	0.803897i	$-0.702756\pi$
$367$	−22.7882	−1.18954	−0.594768	+	0.803897i	$0.702756\pi$
$368$	81.8762	4.26809
$369$	0	0
$370$	−2.50926	−0.130450
$371$	−18.5771	−0.964476
$372$	0	0
$373$	19.1659	0.992374	0.496187	−	0.868216i	$-0.334733\pi$
$373$	19.1659	0.992374	0.496187	+	0.868216i	$0.334733\pi$
$374$	−3.25112	−0.168111
$375$	0	0
$376$	100.920	5.20454
$377$	20.9537	1.07917
$378$	0	0
$379$	−36.4928	−1.87451	−0.937255	−	0.348646i	$-0.886642\pi$
$379$	−36.4928	−1.87451	−0.937255	+	0.348646i	$0.886642\pi$
$380$	−40.6040	−2.08294
$381$	0	0
$382$	−17.1950	−0.879770
$383$	23.1764	1.18426	0.592129	−	0.805843i	$-0.298287\pi$
$383$	23.1764	1.18426	0.592129	+	0.805843i	$0.298287\pi$
$384$	0	0
$385$	−1.13826	−0.0580112
$386$	−20.9051	−1.06404
$387$	0	0
$388$	−35.8576	−1.82039
$389$	−11.8330	−0.599955	−0.299978	−	0.953946i	$-0.596979\pi$
$389$	−11.8330	−0.599955	−0.299978	+	0.953946i	$0.596979\pi$
$390$	0	0
$391$	−24.8852	−1.25850
$392$	−74.6220	−3.76898
$393$	0	0
$394$	43.7334	2.20325
$395$	13.5279	0.680662
$396$	0	0
$397$	−6.87506	−0.345050	−0.172525	−	0.985005i	$-0.555192\pi$
$397$	−6.87506	−0.345050	−0.172525	+	0.985005i	$0.555192\pi$
$398$	−18.5942	−0.932041
$399$	0	0
$400$	13.6048	0.680241
$401$	−1.27432	−0.0636367	−0.0318184	−	0.999494i	$-0.510130\pi$
$401$	−1.27432	−0.0636367	−0.0318184	+	0.999494i	$0.510130\pi$
$402$	0	0
$403$	−17.3048	−0.862013
$404$	15.6653	0.779376
$405$	0	0
$406$	93.9772	4.66401
$407$	−0.269766	−0.0133718
$408$	0	0
$409$	28.2287	1.39582	0.697910	−	0.716185i	$-0.254113\pi$
$409$	28.2287	1.39582	0.697910	+	0.716185i	$0.254113\pi$
$410$	−28.7288	−1.41881
$411$	0	0
$412$	−14.5048	−0.714602
$413$	33.2710	1.63716
$414$	0	0
$415$	−3.43747	−0.168739
$416$	44.5490	2.18419
$417$	0	0
$418$	−6.00839	−0.293880
$419$	−12.3685	−0.604241	−0.302121	−	0.953270i	$-0.597695\pi$
$419$	−12.3685	−0.604241	−0.302121	+	0.953270i	$0.597695\pi$
$420$	0	0
$421$	−23.8509	−1.16242	−0.581210	−	0.813754i	$-0.697420\pi$
$421$	−23.8509	−1.16242	−0.581210	+	0.813754i	$0.697420\pi$
$422$	29.4455	1.43339
$423$	0	0
$424$	42.5164	2.06478
$425$	−4.13501	−0.200577
$426$	0	0
$427$	−11.6917	−0.565800
$428$	−28.6475	−1.38473
$429$	0	0
$430$	25.4299	1.22634
$431$	−2.87409	−0.138440	−0.0692200	−	0.997601i	$-0.522051\pi$
$431$	−2.87409	−0.138440	−0.0692200	+	0.997601i	$0.522051\pi$
$432$	0	0
$433$	23.0356	1.10702	0.553510	−	0.832843i	$-0.313288\pi$
$433$	23.0356	1.10702	0.553510	+	0.832843i	$0.313288\pi$
$434$	−77.6118	−3.72549
$435$	0	0
$436$	−70.8310	−3.39219
$437$	−45.9903	−2.20002
$438$	0	0
$439$	−7.98024	−0.380876	−0.190438	−	0.981699i	$-0.560991\pi$
$439$	−7.98024	−0.380876	−0.190438	+	0.981699i	$0.560991\pi$
$440$	2.60508	0.124192
$441$	0	0
$442$	−26.3981	−1.25563
$443$	−7.28163	−0.345961	−0.172980	−	0.984925i	$-0.555340\pi$
$443$	−7.28163	−0.345961	−0.172980	+	0.984925i	$0.555340\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−47.5727	−2.25263
$447$	0	0
$448$	93.2733	4.40675
$449$	−16.2637	−0.767531	−0.383765	−	0.923431i	$-0.625373\pi$
$449$	−16.2637	−0.767531	−0.383765	+	0.923431i	$0.625373\pi$
$450$	0	0
$451$	−3.08858	−0.145435
$452$	30.5714	1.43796
$453$	0	0
$454$	1.65339	0.0775977
$455$	−9.24237	−0.433289
$456$	0	0
$457$	29.1422	1.36321	0.681607	−	0.731719i	$-0.261282\pi$
$457$	29.1422	1.36321	0.681607	+	0.731719i	$0.261282\pi$
$458$	8.65585	0.404461
$459$	0	0
$460$	31.9765	1.49091
$461$	−7.11329	−0.331299	−0.165650	−	0.986185i	$-0.552972\pi$
$461$	−7.11329	−0.331299	−0.165650	+	0.986185i	$0.552972\pi$
$462$	0	0
$463$	−15.4906	−0.719911	−0.359955	−	0.932969i	$-0.617208\pi$
$463$	−15.4906	−0.719911	−0.359955	+	0.932969i	$0.617208\pi$
$464$	−120.757	−5.60603
$465$	0	0
$466$	20.9767	0.971726
$467$	−29.5394	−1.36692	−0.683461	−	0.729987i	$-0.739526\pi$
$467$	−29.5394	−1.36692	−0.683461	+	0.729987i	$0.739526\pi$
$468$	0	0
$469$	−30.1614	−1.39272
$470$	30.4587	1.40495
$471$	0	0
$472$	−76.1455	−3.50488
$473$	2.73392	0.125706
$474$	0	0
$475$	−7.64191	−0.350635
$476$	−86.0175	−3.94260
$477$	0	0
$478$	−76.0073	−3.47649
$479$	4.05176	0.185129	0.0925647	−	0.995707i	$-0.470493\pi$
$479$	4.05176	0.185129	0.0925647	+	0.995707i	$0.470493\pi$
$480$	0	0
$481$	−2.19042	−0.0998748
$482$	51.0505	2.32529
$483$	0	0
$484$	−57.9975	−2.63625
$485$	−6.74861	−0.306439
$486$	0	0
$487$	−18.7820	−0.851092	−0.425546	−	0.904937i	$-0.639918\pi$
$487$	−18.7820	−0.851092	−0.425546	+	0.904937i	$0.639918\pi$
$488$	26.7581	1.21128
$489$	0	0
$490$	−22.5218	−1.01743
$491$	5.77459	0.260604	0.130302	−	0.991474i	$-0.458405\pi$
$491$	5.77459	0.260604	0.130302	+	0.991474i	$0.458405\pi$
$492$	0	0
$493$	36.7026	1.65300
$494$	−48.7864	−2.19500
$495$	0	0
$496$	99.7285	4.47794
$497$	−18.2423	−0.818277
$498$	0	0
$499$	−25.9609	−1.16217	−0.581086	−	0.813842i	$-0.697372\pi$
$499$	−25.9609	−1.16217	−0.581086	+	0.813842i	$0.697372\pi$
$500$	5.31333	0.237619
$501$	0	0
$502$	26.9254	1.20174
$503$	−19.4539	−0.867407	−0.433704	−	0.901056i	$-0.642794\pi$
$503$	−19.4539	−0.867407	−0.433704	+	0.901056i	$0.642794\pi$
$504$	0	0
$505$	2.94829	0.131197
$506$	4.73174	0.210352
$507$	0	0
$508$	−82.5358	−3.66193
$509$	−6.14500	−0.272372	−0.136186	−	0.990683i	$-0.543485\pi$
$509$	−6.14500	−0.272372	−0.136186	+	0.990683i	$0.543485\pi$
$510$	0	0
$511$	−46.6682	−2.06448
$512$	−12.9322	−0.571529
$513$	0	0
$514$	−22.2259	−0.980342
$515$	−2.72989	−0.120294
$516$	0	0
$517$	3.27456	0.144015
$518$	−9.82404	−0.431644
$519$	0	0
$520$	21.1525	0.927599
$521$	10.9960	0.481742	0.240871	−	0.970557i	$-0.422567\pi$
$521$	10.9960	0.481742	0.240871	+	0.970557i	$0.422567\pi$
$522$	0	0
$523$	37.9162	1.65796	0.828979	−	0.559280i	$-0.188922\pi$
$523$	37.9162	1.65796	0.828979	+	0.559280i	$0.188922\pi$
$524$	45.2705	1.97765
$525$	0	0
$526$	57.0950	2.48946
$527$	−30.3112	−1.32038
$528$	0	0
$529$	13.2184	0.574714
$530$	12.8319	0.557384
$531$	0	0
$532$	−158.969	−6.89218
$533$	−25.0784	−1.08626
$534$	0	0
$535$	−5.39163	−0.233101
$536$	69.0287	2.98158
$537$	0	0
$538$	−4.77362	−0.205805
$539$	−2.42127	−0.104291
$540$	0	0
$541$	3.31521	0.142532	0.0712660	−	0.997457i	$-0.477296\pi$
$541$	3.31521	0.142532	0.0712660	+	0.997457i	$0.477296\pi$
$542$	−11.5420	−0.495771
$543$	0	0
$544$	78.0323	3.34561
$545$	−13.3308	−0.571029
$546$	0	0
$547$	−19.7730	−0.845432	−0.422716	−	0.906262i	$-0.638923\pi$
$547$	−19.7730	−0.845432	−0.422716	+	0.906262i	$0.638923\pi$
$548$	−59.5758	−2.54495
$549$	0	0
$550$	0.786242	0.0335255
$551$	67.8302	2.88966
$552$	0	0
$553$	52.9632	2.25222
$554$	−22.0435	−0.936538
$555$	0	0
$556$	−106.424	−4.51339
$557$	−17.3478	−0.735049	−0.367525	−	0.930014i	$-0.619795\pi$
$557$	−17.3478	−0.735049	−0.367525	+	0.930014i	$0.619795\pi$
$558$	0	0
$559$	22.1987	0.938904
$560$	53.2643	2.25083
$561$	0	0
$562$	62.5890	2.64016
$563$	24.6281	1.03795	0.518975	−	0.854789i	$-0.326314\pi$
$563$	24.6281	1.03795	0.518975	+	0.854789i	$0.326314\pi$
$564$	0	0
$565$	5.75372	0.242061
$566$	−1.33940	−0.0562993
$567$	0	0
$568$	41.7501	1.75179
$569$	30.3206	1.27111	0.635553	−	0.772057i	$-0.280772\pi$
$569$	30.3206	1.27111	0.635553	+	0.772057i	$0.280772\pi$
$570$	0	0
$571$	−9.48083	−0.396760	−0.198380	−	0.980125i	$-0.563568\pi$
$571$	−9.48083	−0.396760	−0.198380	+	0.980125i	$0.563568\pi$
$572$	3.64674	0.152478
$573$	0	0
$574$	−112.476	−4.69467
$575$	6.01817	0.250975
$576$	0	0
$577$	10.4548	0.435240	0.217620	−	0.976034i	$-0.430171\pi$
$577$	10.4548	0.435240	0.217620	+	0.976034i	$0.430171\pi$
$578$	−0.265750	−0.0110538
$579$	0	0
$580$	−47.1616	−1.95828
$581$	−13.4580	−0.558334
$582$	0	0
$583$	1.37954	0.0571346
$584$	106.807	4.41970
$585$	0	0
$586$	56.4821	2.33326
$587$	23.5836	0.973398	0.486699	−	0.873570i	$-0.338201\pi$
$587$	23.5836	0.973398	0.486699	+	0.873570i	$0.338201\pi$
$588$	0	0
$589$	−56.0181	−2.30819
$590$	−22.9816	−0.946136
$591$	0	0
$592$	12.6236	0.518825
$593$	−23.1956	−0.952530	−0.476265	−	0.879302i	$-0.658010\pi$
$593$	−23.1956	−0.952530	−0.476265	+	0.879302i	$0.658010\pi$
$594$	0	0
$595$	−16.1890	−0.663684
$596$	−86.1829	−3.53019
$597$	0	0
$598$	38.4204	1.57113
$599$	3.51234	0.143510	0.0717551	−	0.997422i	$-0.477140\pi$
$599$	3.51234	0.143510	0.0717551	+	0.997422i	$0.477140\pi$
$600$	0	0
$601$	5.13651	0.209522	0.104761	−	0.994497i	$-0.466592\pi$
$601$	5.13651	0.209522	0.104761	+	0.994497i	$0.466592\pi$
$602$	99.5609	4.05780
$603$	0	0
$604$	68.0110	2.76733
$605$	−10.9155	−0.443777
$606$	0	0
$607$	3.14756	0.127755	0.0638777	−	0.997958i	$-0.479653\pi$
$607$	3.14756	0.127755	0.0638777	+	0.997958i	$0.479653\pi$
$608$	144.212	5.84855
$609$	0	0
$610$	8.07590	0.326983
$611$	26.5885	1.07565
$612$	0	0
$613$	16.6025	0.670568	0.335284	−	0.942117i	$-0.391168\pi$
$613$	16.6025	0.670568	0.335284	+	0.942117i	$0.391168\pi$
$614$	−71.0809	−2.86859
$615$	0	0
$616$	10.1992	0.410936
$617$	41.0369	1.65208	0.826041	−	0.563610i	$-0.190588\pi$
$617$	41.0369	1.65208	0.826041	+	0.563610i	$0.190588\pi$
$618$	0	0
$619$	11.3158	0.454820	0.227410	−	0.973799i	$-0.426974\pi$
$619$	11.3158	0.454820	0.227410	+	0.973799i	$0.426974\pi$
$620$	38.9487	1.56422
$621$	0	0
$622$	21.7852	0.873507
$623$	−3.91511	−0.156856
$624$	0	0
$625$	1.00000	0.0400000
$626$	−53.8004	−2.15029
$627$	0	0
$628$	10.4479	0.416915
$629$	−3.83676	−0.152982
$630$	0	0
$631$	26.0023	1.03514	0.517568	−	0.855642i	$-0.326837\pi$
$631$	26.0023	1.03514	0.517568	+	0.855642i	$0.326837\pi$
$632$	−121.214	−4.82163
$633$	0	0
$634$	24.5713	0.975851
$635$	−15.5337	−0.616437
$636$	0	0
$637$	−19.6600	−0.778959
$638$	−6.97875	−0.276291
$639$	0	0
$640$	−26.6852	−1.05482
$641$	−34.0199	−1.34370	−0.671852	−	0.740685i	$-0.734501\pi$
$641$	−34.0199	−1.34370	−0.671852	+	0.740685i	$0.734501\pi$
$642$	0	0
$643$	−40.6945	−1.60483	−0.802417	−	0.596763i	$-0.796453\pi$
$643$	−40.6945	−1.60483	−0.802417	+	0.596763i	$0.796453\pi$
$644$	125.192	4.93324
$645$	0	0
$646$	−85.4546	−3.36217
$647$	−40.9166	−1.60860	−0.804300	−	0.594224i	$-0.797459\pi$
$647$	−40.9166	−1.60860	−0.804300	+	0.594224i	$0.797459\pi$
$648$	0	0
$649$	−2.47070	−0.0969836
$650$	6.38406	0.250404
$651$	0	0
$652$	0.808044	0.0316454
$653$	−24.3197	−0.951704	−0.475852	−	0.879525i	$-0.657860\pi$
$653$	−24.3197	−0.951704	−0.475852	+	0.879525i	$0.657860\pi$
$654$	0	0
$655$	8.52018	0.332911
$656$	144.528	5.64288
$657$	0	0
$658$	119.249	4.64881
$659$	−33.6065	−1.30912	−0.654561	−	0.756009i	$-0.727146\pi$
$659$	−33.6065	−1.30912	−0.654561	+	0.756009i	$0.727146\pi$
$660$	0	0
$661$	−23.4974	−0.913945	−0.456972	−	0.889481i	$-0.651066\pi$
$661$	−23.4974	−0.913945	−0.456972	+	0.889481i	$0.651066\pi$
$662$	−85.9566	−3.34080
$663$	0	0
$664$	30.8007	1.19530
$665$	−29.9189	−1.16020
$666$	0	0
$667$	−53.4178	−2.06835
$668$	50.0055	1.93477
$669$	0	0
$670$	20.8336	0.804873
$671$	0.868224	0.0335174
$672$	0	0
$673$	41.5321	1.60094	0.800472	−	0.599370i	$-0.204582\pi$
$673$	41.5321	1.60094	0.800472	+	0.599370i	$0.204582\pi$
$674$	−10.0480	−0.387034
$675$	0	0
$676$	−39.4628	−1.51780
$677$	−20.9719	−0.806017	−0.403008	−	0.915196i	$-0.632035\pi$
$677$	−20.9719	−0.806017	−0.403008	+	0.915196i	$0.632035\pi$
$678$	0	0
$679$	−26.4215	−1.01397
$680$	37.0509	1.42084
$681$	0	0
$682$	5.76345	0.220694
$683$	45.2377	1.73097	0.865486	−	0.500934i	$-0.167010\pi$
$683$	45.2377	1.73097	0.865486	+	0.500934i	$0.167010\pi$
$684$	0	0
$685$	−11.2125	−0.428408
$686$	−14.0612	−0.536861
$687$	0	0
$688$	−127.932	−4.87737
$689$	11.2014	0.426741
$690$	0	0
$691$	−13.5491	−0.515434	−0.257717	−	0.966220i	$-0.582970\pi$
$691$	−13.5491	−0.515434	−0.257717	+	0.966220i	$0.582970\pi$
$692$	26.3833	1.00294
$693$	0	0
$694$	24.2210	0.919415
$695$	−20.0296	−0.759767
$696$	0	0
$697$	−43.9274	−1.66387
$698$	17.7009	0.669990
$699$	0	0
$700$	20.8023	0.786252
$701$	39.1511	1.47871	0.739357	−	0.673313i	$-0.235129\pi$
$701$	39.1511	1.47871	0.739357	+	0.673313i	$0.235129\pi$
$702$	0	0
$703$	−7.09072	−0.267432
$704$	−6.92648	−0.261051
$705$	0	0
$706$	49.5369	1.86434
$707$	11.5429	0.434115
$708$	0	0
$709$	38.5042	1.44606	0.723028	−	0.690818i	$-0.242750\pi$
$709$	38.5042	1.44606	0.723028	+	0.690818i	$0.242750\pi$
$710$	12.6006	0.472894
$711$	0	0
$712$	8.96030	0.335801
$713$	44.1155	1.65214
$714$	0	0
$715$	0.686338	0.0256676
$716$	117.252	4.38191
$717$	0	0
$718$	31.4990	1.17553
$719$	10.7708	0.401682	0.200841	−	0.979624i	$-0.435633\pi$
$719$	10.7708	0.401682	0.200841	+	0.979624i	$0.435633\pi$
$720$	0	0
$721$	−10.6878	−0.398036
$722$	−106.547	−3.96526
$723$	0	0
$724$	76.1515	2.83015
$725$	−8.87608	−0.329649
$726$	0	0
$727$	−8.46822	−0.314069	−0.157034	−	0.987593i	$-0.550193\pi$
$727$	−8.46822	−0.314069	−0.157034	+	0.987593i	$0.550193\pi$
$728$	82.8144	3.06930
$729$	0	0
$730$	32.2355	1.19309
$731$	38.8833	1.43815
$732$	0	0
$733$	−19.7393	−0.729087	−0.364543	−	0.931186i	$-0.618775\pi$
$733$	−19.7393	−0.729087	−0.364543	+	0.931186i	$0.618775\pi$
$734$	61.6266	2.27468
$735$	0	0
$736$	−113.570	−4.18624
$737$	2.23978	0.0825035
$738$	0	0
$739$	−47.1294	−1.73368	−0.866842	−	0.498583i	$-0.833854\pi$
$739$	−47.1294	−1.73368	−0.866842	+	0.498583i	$0.833854\pi$
$740$	4.93010	0.181234
$741$	0	0
$742$	50.2384	1.84431
$743$	4.46755	0.163899	0.0819493	−	0.996637i	$-0.473885\pi$
$743$	4.46755	0.163899	0.0819493	+	0.996637i	$0.473885\pi$
$744$	0	0
$745$	−16.2201	−0.594259
$746$	−51.8307	−1.89766
$747$	0	0
$748$	6.38766	0.233556
$749$	−21.1088	−0.771300
$750$	0	0
$751$	−34.6927	−1.26595	−0.632977	−	0.774171i	$-0.718167\pi$
$751$	−34.6927	−1.26595	−0.632977	+	0.774171i	$0.718167\pi$
$752$	−153.231	−5.58776
$753$	0	0
$754$	−56.6655	−2.06363
$755$	12.8001	0.465842
$756$	0	0
$757$	−3.94366	−0.143335	−0.0716674	−	0.997429i	$-0.522832\pi$
$757$	−3.94366	−0.143335	−0.0716674	+	0.997429i	$0.522832\pi$
$758$	98.6881	3.58451
$759$	0	0
$760$	68.4737	2.48380
$761$	21.1330	0.766070	0.383035	−	0.923734i	$-0.374879\pi$
$761$	21.1330	0.766070	0.383035	+	0.923734i	$0.374879\pi$
$762$	0	0
$763$	−52.1916	−1.88946
$764$	33.7839	1.22226
$765$	0	0
$766$	−62.6763	−2.26459
$767$	−20.0614	−0.724376
$768$	0	0
$769$	18.5773	0.669914	0.334957	−	0.942233i	$-0.391278\pi$
$769$	18.5773	0.669914	0.334957	+	0.942233i	$0.391278\pi$
$770$	3.07822	0.110931
$771$	0	0
$772$	41.0735	1.47827
$773$	−8.54873	−0.307476	−0.153738	−	0.988112i	$-0.549131\pi$
$773$	−8.54873	−0.307476	−0.153738	+	0.988112i	$0.549131\pi$
$774$	0	0
$775$	7.33038	0.263315
$776$	60.4696	2.17073
$777$	0	0
$778$	32.0001	1.14726
$779$	−81.1823	−2.90866
$780$	0	0
$781$	1.35467	0.0484739
$782$	67.2974	2.40655
$783$	0	0
$784$	113.302	4.04650
$785$	1.96635	0.0701820
$786$	0	0
$787$	19.6235	0.699502	0.349751	−	0.936843i	$-0.386266\pi$
$787$	19.6235	0.699502	0.349751	+	0.936843i	$0.386266\pi$
$788$	−85.9255	−3.06097
$789$	0	0
$790$	−36.5837	−1.30159
$791$	22.5264	0.800948
$792$	0	0
$793$	7.04973	0.250343
$794$	18.5924	0.659818
$795$	0	0
$796$	36.5330	1.29488
$797$	−17.9984	−0.637534	−0.318767	−	0.947833i	$-0.603269\pi$
$797$	−17.9984	−0.637534	−0.318767	+	0.947833i	$0.603269\pi$
$798$	0	0
$799$	46.5725	1.64762
$800$	−18.8712	−0.667196
$801$	0	0
$802$	3.44618	0.121689
$803$	3.46558	0.122298
$804$	0	0
$805$	23.5618	0.830444
$806$	46.7976	1.64838
$807$	0	0
$808$	−26.4176	−0.929367
$809$	53.3900	1.87709	0.938547	−	0.345153i	$-0.112173\pi$
$809$	53.3900	1.87709	0.938547	+	0.345153i	$0.112173\pi$
$810$	0	0
$811$	41.3617	1.45241	0.726203	−	0.687480i	$-0.241283\pi$
$811$	41.3617	1.45241	0.726203	+	0.687480i	$0.241283\pi$
$812$	−184.643	−6.47969
$813$	0	0
$814$	0.729533	0.0255701
$815$	0.152079	0.00532708
$816$	0	0
$817$	71.8603	2.51407
$818$	−76.3394	−2.66915
$819$	0	0
$820$	56.4452	1.97115
$821$	51.9464	1.81294	0.906471	−	0.422268i	$-0.138766\pi$
$821$	51.9464	1.81294	0.906471	+	0.422268i	$0.138766\pi$
$822$	0	0
$823$	−24.7601	−0.863084	−0.431542	−	0.902093i	$-0.642030\pi$
$823$	−24.7601	−0.863084	−0.431542	+	0.902093i	$0.642030\pi$
$824$	24.4607	0.852127
$825$	0	0
$826$	−89.9753	−3.13064
$827$	5.69603	0.198070	0.0990352	−	0.995084i	$-0.468424\pi$
$827$	5.69603	0.198070	0.0990352	+	0.995084i	$0.468424\pi$
$828$	0	0
$829$	−30.6160	−1.06334	−0.531669	−	0.846952i	$-0.678435\pi$
$829$	−30.6160	−1.06334	−0.531669	+	0.846952i	$0.678435\pi$
$830$	9.29600	0.322669
$831$	0	0
$832$	−56.2410	−1.94981
$833$	−34.4366	−1.19316
$834$	0	0
$835$	9.41133	0.325692
$836$	11.8050	0.408286
$837$	0	0
$838$	33.4484	1.15546
$839$	−7.71064	−0.266201	−0.133100	−	0.991103i	$-0.542493\pi$
$839$	−7.71064	−0.266201	−0.133100	+	0.991103i	$0.542493\pi$
$840$	0	0
$841$	49.7848	1.71672
$842$	64.5003	2.22283
$843$	0	0
$844$	−57.8534	−1.99139
$845$	−7.42713	−0.255501
$846$	0	0
$847$	−42.7353	−1.46840
$848$	−64.5546	−2.21681
$849$	0	0
$850$	11.1824	0.383552
$851$	5.58410	0.191421
$852$	0	0
$853$	42.1577	1.44345	0.721727	−	0.692178i	$-0.243349\pi$
$853$	42.1577	1.44345	0.721727	+	0.692178i	$0.243349\pi$
$854$	31.6180	1.08195
$855$	0	0
$856$	48.3106	1.65122
$857$	11.5033	0.392946	0.196473	−	0.980509i	$-0.437051\pi$
$857$	11.5033	0.392946	0.196473	+	0.980509i	$0.437051\pi$
$858$	0	0
$859$	17.0400	0.581399	0.290699	−	0.956814i	$-0.406112\pi$
$859$	17.0400	0.581399	0.290699	+	0.956814i	$0.406112\pi$
$860$	−49.9637	−1.70375
$861$	0	0
$862$	7.77245	0.264731
$863$	−20.0375	−0.682083	−0.341042	−	0.940048i	$-0.610780\pi$
$863$	−20.0375	−0.682083	−0.341042	+	0.940048i	$0.610780\pi$
$864$	0	0
$865$	4.96550	0.168832
$866$	−62.2955	−2.11689
$867$	0	0
$868$	152.489	5.17580
$869$	−3.93305	−0.133419
$870$	0	0
$871$	18.1864	0.616223
$872$	119.448	4.04502
$873$	0	0
$874$	124.372	4.20696
$875$	3.91511	0.132355
$876$	0	0
$877$	31.0620	1.04889	0.524445	−	0.851445i	$-0.324273\pi$
$877$	31.0620	1.04889	0.524445	+	0.851445i	$0.324273\pi$
$878$	21.5811	0.728327
$879$	0	0
$880$	−3.95541	−0.133337
$881$	7.02503	0.236679	0.118340	−	0.992973i	$-0.462243\pi$
$881$	7.02503	0.236679	0.118340	+	0.992973i	$0.462243\pi$
$882$	0	0
$883$	−14.6289	−0.492303	−0.246151	−	0.969231i	$-0.579166\pi$
$883$	−14.6289	−0.492303	−0.246151	+	0.969231i	$0.579166\pi$
$884$	51.8660	1.74444
$885$	0	0
$886$	19.6918	0.661560
$887$	−45.2292	−1.51865	−0.759325	−	0.650712i	$-0.774471\pi$
$887$	−45.2292	−1.51865	−0.759325	+	0.650712i	$0.774471\pi$
$888$	0	0
$889$	−60.8162	−2.03971
$890$	2.70432	0.0906489
$891$	0	0
$892$	93.4688	3.12957
$893$	86.0707	2.88025
$894$	0	0
$895$	22.0675	0.737635
$896$	−104.475	−3.49028
$897$	0	0
$898$	43.9822	1.46770
$899$	−65.0651	−2.17004
$900$	0	0
$901$	19.6205	0.653654
$902$	8.35249	0.278108
$903$	0	0
$904$	−51.5550	−1.71469
$905$	14.3322	0.476417
$906$	0	0
$907$	13.3391	0.442917	0.221459	−	0.975170i	$-0.428918\pi$
$907$	13.3391	0.442917	0.221459	+	0.975170i	$0.428918\pi$
$908$	−3.24852	−0.107806
$909$	0	0
$910$	24.9943	0.828553
$911$	−2.51924	−0.0834663	−0.0417331	−	0.999129i	$-0.513288\pi$
$911$	−2.51924	−0.0834663	−0.0417331	+	0.999129i	$0.513288\pi$
$912$	0	0
$913$	0.999395	0.0330751
$914$	−78.8097	−2.60679
$915$	0	0
$916$	−17.0067	−0.561916
$917$	33.3574	1.10156
$918$	0	0
$919$	13.7219	0.452645	0.226322	−	0.974052i	$-0.427330\pi$
$919$	13.7219	0.452645	0.226322	+	0.974052i	$0.427330\pi$
$920$	−53.9246	−1.77784
$921$	0	0
$922$	19.2366	0.633524
$923$	10.9995	0.362054
$924$	0	0
$925$	0.927874	0.0305083
$926$	41.8916	1.37664
$927$	0	0
$928$	167.502	5.49852
$929$	21.1806	0.694912	0.347456	−	0.937696i	$-0.387046\pi$
$929$	21.1806	0.694912	0.347456	+	0.937696i	$0.387046\pi$
$930$	0	0
$931$	−63.6424	−2.08579
$932$	−41.2141	−1.35001
$933$	0	0
$934$	79.8840	2.61388
$935$	1.20219	0.0393160
$936$	0	0
$937$	0.680210	0.0222215	0.0111107	−	0.999938i	$-0.496463\pi$
$937$	0.680210	0.0222215	0.0111107	+	0.999938i	$0.496463\pi$
$938$	81.5659	2.66322
$939$	0	0
$940$	−59.8440	−1.95190
$941$	−33.5948	−1.09516	−0.547579	−	0.836754i	$-0.684451\pi$
$941$	−33.5948	−1.09516	−0.547579	+	0.836754i	$0.684451\pi$
$942$	0	0
$943$	63.9329	2.08194
$944$	115.615	3.76295
$945$	0	0
$946$	−7.39339	−0.240380
$947$	7.62796	0.247875	0.123938	−	0.992290i	$-0.460448\pi$
$947$	7.62796	0.247875	0.123938	+	0.992290i	$0.460448\pi$
$948$	0	0
$949$	28.1395	0.913447
$950$	20.6661	0.670498
$951$	0	0
$952$	145.058	4.70136
$953$	−47.5199	−1.53932	−0.769661	−	0.638453i	$-0.779575\pi$
$953$	−47.5199	−1.53932	−0.769661	+	0.638453i	$0.779575\pi$
$954$	0	0
$955$	6.35834	0.205751
$956$	149.336	4.82987
$957$	0	0
$958$	−10.9572	−0.354012
$959$	−43.8982	−1.41755
$960$	0	0
$961$	22.7345	0.733371
$962$	5.92360	0.190985
$963$	0	0
$964$	−100.302	−3.23051
$965$	7.73028	0.248847
$966$	0	0
$967$	0.611837	0.0196753	0.00983767	−	0.999952i	$-0.496869\pi$
$967$	0.611837	0.0196753	0.00983767	+	0.999952i	$0.496869\pi$
$968$	97.8059	3.14360
$969$	0	0
$970$	18.2504	0.585985
$971$	61.0779	1.96008	0.980041	−	0.198793i	$-0.0637022\pi$
$971$	61.0779	1.96008	0.980041	+	0.198793i	$0.0637022\pi$
$972$	0	0
$973$	−78.4182	−2.51397
$974$	50.7924	1.62749
$975$	0	0
$976$	−40.6280	−1.30047
$977$	−6.86858	−0.219745	−0.109873	−	0.993946i	$-0.535044\pi$
$977$	−6.86858	−0.219745	−0.109873	+	0.993946i	$0.535044\pi$
$978$	0	0
$979$	0.290736	0.00929196
$980$	44.2498	1.41351
$981$	0	0
$982$	−15.6163	−0.498337
$983$	−42.0513	−1.34123	−0.670615	−	0.741806i	$-0.733970\pi$
$983$	−42.0513	−1.34123	−0.670615	+	0.741806i	$0.733970\pi$
$984$	0	0
$985$	−16.1717	−0.515273
$986$	−99.2556	−3.16094
$987$	0	0
$988$	95.8535	3.04951
$989$	−56.5916	−1.79951
$990$	0	0
$991$	0.917498	0.0291453	0.0145726	−	0.999894i	$-0.495361\pi$
$991$	0.917498	0.0291453	0.0145726	+	0.999894i	$0.495361\pi$
$992$	−138.333	−4.39207
$993$	0	0
$994$	49.3329	1.56474
$995$	6.87573	0.217975
$996$	0	0
$997$	−0.150382	−0.00476264	−0.00238132	−	0.999997i	$-0.500758\pi$
$997$	−0.150382	−0.00476264	−0.00238132	+	0.999997i	$0.500758\pi$
$998$	70.2066	2.22235
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.q.1.2		9
3.2	odd	2		1335.2.a.h.1.8	✓	9
15.14	odd	2		6675.2.a.x.1.2		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.h.1.8	✓	9	3.2	odd	2
4005.2.a.q.1.2		9	1.1	even	1	trivial
6675.2.a.x.1.2		9	15.14	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70432 q^{2} +5.31333 q^{4} +1.00000 q^{5} +3.91511 q^{7} -8.96030 q^{8} +O(q^{10})\)	\(q-2.70432 q^{2} +5.31333 q^{4} +1.00000 q^{5} +3.91511 q^{7} -8.96030 q^{8} -2.70432 q^{10} -0.290736 q^{11} -2.36069 q^{13} -10.5877 q^{14} +13.6048 q^{16} -4.13501 q^{17} -7.64191 q^{19} +5.31333 q^{20} +0.786242 q^{22} +6.01817 q^{23} +1.00000 q^{25} +6.38406 q^{26} +20.8023 q^{28} -8.87608 q^{29} +7.33038 q^{31} -18.8712 q^{32} +11.1824 q^{34} +3.91511 q^{35} +0.927874 q^{37} +20.6661 q^{38} -8.96030 q^{40} +10.6233 q^{41} -9.40345 q^{43} -1.54478 q^{44} -16.2751 q^{46} -11.2630 q^{47} +8.32807 q^{49} -2.70432 q^{50} -12.5431 q^{52} -4.74498 q^{53} -0.290736 q^{55} -35.0805 q^{56} +24.0037 q^{58} +8.49811 q^{59} -2.98630 q^{61} -19.8237 q^{62} +23.8239 q^{64} -2.36069 q^{65} -7.70384 q^{67} -21.9706 q^{68} -10.5877 q^{70} -4.65945 q^{71} -11.9200 q^{73} -2.50926 q^{74} -40.6040 q^{76} -1.13826 q^{77} +13.5279 q^{79} +13.6048 q^{80} -28.7288 q^{82} -3.43747 q^{83} -4.13501 q^{85} +25.4299 q^{86} +2.60508 q^{88} -1.00000 q^{89} -9.24237 q^{91} +31.9765 q^{92} +30.4587 q^{94} -7.64191 q^{95} -6.74861 q^{97} -22.5218 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8}+O(q^{10}) \) 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8	\( 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8} - 5 q^{10} - 4 q^{11} + 5 q^{13} + q^{14} + 15 q^{16} - 21 q^{17} - 18 q^{19} + 11 q^{20} + 6 q^{22} - 16 q^{23} + 9 q^{25} - 8 q^{26} + 6 q^{28} - 3 q^{29} - 6 q^{31} - 46 q^{32} + 12 q^{34} - 3 q^{35} + 11 q^{37} - 20 q^{38} - 18 q^{40} + q^{41} - 3 q^{43} - 38 q^{44} + 16 q^{46} - 27 q^{47} + 24 q^{49} - 5 q^{50} + 17 q^{52} - 43 q^{53} - 4 q^{55} - 5 q^{56} + 34 q^{58} - 3 q^{59} - 30 q^{61} - 36 q^{62} + 50 q^{64} + 5 q^{65} - 12 q^{67} - 64 q^{68} + q^{70} + 4 q^{71} + 26 q^{73} - 2 q^{74} - 12 q^{76} - 34 q^{77} + q^{79} + 15 q^{80} - 51 q^{82} - 24 q^{83} - 21 q^{85} - 18 q^{86} + 64 q^{88} - 9 q^{89} - 50 q^{91} - 10 q^{92} - 11 q^{94} - 18 q^{95} - 4 q^{97} - 75 q^{98}+O(q^{100}) \) 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8 - 5 * q^10 - 4 * q^11 + 5 * q^13 + q^14 + 15 * q^16 - 21 * q^17 - 18 * q^19 + 11 * q^20 + 6 * q^22 - 16 * q^23 + 9 * q^25 - 8 * q^26 + 6 * q^28 - 3 * q^29 - 6 * q^31 - 46 * q^32 + 12 * q^34 - 3 * q^35 + 11 * q^37 - 20 * q^38 - 18 * q^40 + q^41 - 3 * q^43 - 38 * q^44 + 16 * q^46 - 27 * q^47 + 24 * q^49 - 5 * q^50 + 17 * q^52 - 43 * q^53 - 4 * q^55 - 5 * q^56 + 34 * q^58 - 3 * q^59 - 30 * q^61 - 36 * q^62 + 50 * q^64 + 5 * q^65 - 12 * q^67 - 64 * q^68 + q^70 + 4 * q^71 + 26 * q^73 - 2 * q^74 - 12 * q^76 - 34 * q^77 + q^79 + 15 * q^80 - 51 * q^82 - 24 * q^83 - 21 * q^85 - 18 * q^86 + 64 * q^88 - 9 * q^89 - 50 * q^91 - 10 * q^92 - 11 * q^94 - 18 * q^95 - 4 * q^97 - 75 * q^98

Introduction

L-functions

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