LMFDB - Embedded newform 8001.2.a.l.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	7.7.118870813.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 9x^{5} - 3x^{4} + 20x^{3} + 7x^{2} - 13x - 4 $ x^7 - 9x^5 - 3x^4 + 20x^3 + 7x^2 - 13*x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.20244$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.24280 q^{2} -0.455452 q^{4} +3.33400 q^{5} +1.00000 q^{7} -3.05163 q^{8} +O(q^{10})$	\(q+1.24280 q^{2} -0.455452 q^{4} +3.33400 q^{5} +1.00000 q^{7} -3.05163 q^{8} +4.14349 q^{10} -2.90717 q^{11} -4.59254 q^{13} +1.24280 q^{14} -2.88166 q^{16} +5.29249 q^{17} +1.86822 q^{19} -1.51848 q^{20} -3.61302 q^{22} -5.03184 q^{23} +6.11557 q^{25} -5.70761 q^{26} -0.455452 q^{28} -3.69938 q^{29} +0.968597 q^{31} +2.52194 q^{32} +6.57750 q^{34} +3.33400 q^{35} -10.5381 q^{37} +2.32183 q^{38} -10.1741 q^{40} -11.5606 q^{41} +2.54868 q^{43} +1.32408 q^{44} -6.25356 q^{46} -7.34702 q^{47} +1.00000 q^{49} +7.60043 q^{50} +2.09168 q^{52} +11.2837 q^{53} -9.69250 q^{55} -3.05163 q^{56} -4.59758 q^{58} +13.3410 q^{59} +0.666722 q^{61} +1.20377 q^{62} +8.89758 q^{64} -15.3116 q^{65} -14.0335 q^{67} -2.41048 q^{68} +4.14349 q^{70} -9.88541 q^{71} -2.57222 q^{73} -13.0968 q^{74} -0.850887 q^{76} -2.90717 q^{77} +14.5706 q^{79} -9.60746 q^{80} -14.3675 q^{82} -0.493442 q^{83} +17.6452 q^{85} +3.16749 q^{86} +8.87160 q^{88} -12.1652 q^{89} -4.59254 q^{91} +2.29176 q^{92} -9.13087 q^{94} +6.22866 q^{95} -14.7006 q^{97} +1.24280 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8}+O(q^{10}) $ 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8	$ 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8} + 3 q^{11} - 23 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} - 9 q^{19} + 9 q^{20} - 19 q^{22} - 12 q^{23} + 3 q^{25} - 18 q^{26} + 4 q^{28} + 9 q^{29} - 33 q^{31} - 10 q^{32} - 2 q^{34} + 8 q^{35} - 33 q^{37} + 3 q^{38} - 9 q^{40} + 3 q^{41} - 9 q^{43} - 2 q^{44} - 32 q^{46} - 11 q^{47} + 7 q^{49} - 29 q^{50} - 21 q^{52} - q^{53} - 16 q^{55} + 9 q^{56} - 5 q^{58} + 30 q^{59} - 19 q^{61} - 3 q^{62} - 21 q^{64} - 14 q^{65} - 30 q^{67} - 24 q^{68} - 8 q^{71} - 20 q^{73} + 9 q^{74} - 42 q^{76} + 3 q^{77} + 8 q^{79} - 12 q^{80} + 10 q^{82} + 34 q^{83} - 28 q^{85} - 24 q^{86} - q^{88} + 12 q^{89} - 23 q^{91} - 60 q^{92} - 3 q^{94} - 12 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8 + 3 * q^11 - 23 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 - 9 * q^19 + 9 * q^20 - 19 * q^22 - 12 * q^23 + 3 * q^25 - 18 * q^26 + 4 * q^28 + 9 * q^29 - 33 * q^31 - 10 * q^32 - 2 * q^34 + 8 * q^35 - 33 * q^37 + 3 * q^38 - 9 * q^40 + 3 * q^41 - 9 * q^43 - 2 * q^44 - 32 * q^46 - 11 * q^47 + 7 * q^49 - 29 * q^50 - 21 * q^52 - q^53 - 16 * q^55 + 9 * q^56 - 5 * q^58 + 30 * q^59 - 19 * q^61 - 3 * q^62 - 21 * q^64 - 14 * q^65 - 30 * q^67 - 24 * q^68 - 8 * q^71 - 20 * q^73 + 9 * q^74 - 42 * q^76 + 3 * q^77 + 8 * q^79 - 12 * q^80 + 10 * q^82 + 34 * q^83 - 28 * q^85 - 24 * q^86 - q^88 + 12 * q^89 - 23 * q^91 - 60 * q^92 - 3 * q^94 - 12 * q^95 + 7 * q^97 + 2 * 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 $

$2$	1.24280	0.878791	0.439396	−	0.898294i	$-0.355193\pi$
$2$	1.24280	0.878791	0.439396	+	0.898294i	$0.355193\pi$
$3$	0	0
$3$	0	0
$4$	−0.455452	−0.227726
$5$	3.33400	1.49101	0.745506	−	0.666499i	$-0.232208\pi$
$5$	3.33400	1.49101	0.745506	+	0.666499i	$0.232208\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.05163	−1.07891
$9$	0	0
$10$	4.14349	1.31029
$11$	−2.90717	−0.876544	−0.438272	−	0.898842i	$-0.644409\pi$
$11$	−2.90717	−0.876544	−0.438272	+	0.898842i	$0.644409\pi$
$12$	0	0
$13$	−4.59254	−1.27374	−0.636871	−	0.770970i	$-0.719772\pi$
$13$	−4.59254	−1.27374	−0.636871	+	0.770970i	$0.719772\pi$
$14$	1.24280	0.332152
$15$	0	0
$16$	−2.88166	−0.720415
$17$	5.29249	1.28362	0.641809	−	0.766865i	$-0.278184\pi$
$17$	5.29249	1.28362	0.641809	+	0.766865i	$0.278184\pi$
$18$	0	0
$19$	1.86822	0.428600	0.214300	−	0.976768i	$-0.431253\pi$
$19$	1.86822	0.428600	0.214300	+	0.976768i	$0.431253\pi$
$20$	−1.51848	−0.339542
$21$	0	0
$22$	−3.61302	−0.770299
$23$	−5.03184	−1.04921	−0.524606	−	0.851345i	$-0.675787\pi$
$23$	−5.03184	−1.04921	−0.524606	+	0.851345i	$0.675787\pi$
$24$	0	0
$25$	6.11557	1.22311
$26$	−5.70761	−1.11935
$27$	0	0
$28$	−0.455452	−0.0860724
$29$	−3.69938	−0.686957	−0.343478	−	0.939161i	$-0.611605\pi$
$29$	−3.69938	−0.686957	−0.343478	+	0.939161i	$0.611605\pi$
$30$	0	0
$31$	0.968597	0.173965	0.0869826	−	0.996210i	$-0.472278\pi$
$31$	0.968597	0.173965	0.0869826	+	0.996210i	$0.472278\pi$
$32$	2.52194	0.445821
$33$	0	0
$34$	6.57750	1.12803
$35$	3.33400	0.563549
$36$	0	0
$37$	−10.5381	−1.73246	−0.866230	−	0.499645i	$-0.833464\pi$
$37$	−10.5381	−1.73246	−0.866230	+	0.499645i	$0.833464\pi$
$38$	2.32183	0.376650
$39$	0	0
$40$	−10.1741	−1.60867
$41$	−11.5606	−1.80546	−0.902731	−	0.430205i	$-0.858441\pi$
$41$	−11.5606	−1.80546	−0.902731	+	0.430205i	$0.858441\pi$
$42$	0	0
$43$	2.54868	0.388670	0.194335	−	0.980935i	$-0.437745\pi$
$43$	2.54868	0.388670	0.194335	+	0.980935i	$0.437745\pi$
$44$	1.32408	0.199612
$45$	0	0
$46$	−6.25356	−0.922038
$47$	−7.34702	−1.07167	−0.535837	−	0.844322i	$-0.680004\pi$
$47$	−7.34702	−1.07167	−0.535837	+	0.844322i	$0.680004\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.60043	1.07486
$51$	0	0
$52$	2.09168	0.290064
$53$	11.2837	1.54993	0.774966	−	0.632003i	$-0.217767\pi$
$53$	11.2837	1.54993	0.774966	+	0.632003i	$0.217767\pi$
$54$	0	0
$55$	−9.69250	−1.30694
$56$	−3.05163	−0.407791
$57$	0	0
$58$	−4.59758	−0.603691
$59$	13.3410	1.73685	0.868425	−	0.495820i	$-0.165132\pi$
$59$	13.3410	1.73685	0.868425	+	0.495820i	$0.165132\pi$
$60$	0	0
$61$	0.666722	0.0853651	0.0426825	−	0.999089i	$-0.486410\pi$
$61$	0.666722	0.0853651	0.0426825	+	0.999089i	$0.486410\pi$
$62$	1.20377	0.152879
$63$	0	0
$64$	8.89758	1.11220
$65$	−15.3116	−1.89916
$66$	0	0
$67$	−14.0335	−1.71447	−0.857233	−	0.514929i	$-0.827818\pi$
$67$	−14.0335	−1.71447	−0.857233	+	0.514929i	$0.827818\pi$
$68$	−2.41048	−0.292313
$69$	0	0
$70$	4.14349	0.495242
$71$	−9.88541	−1.17318	−0.586591	−	0.809883i	$-0.699530\pi$
$71$	−9.88541	−1.17318	−0.586591	+	0.809883i	$0.699530\pi$
$72$	0	0
$73$	−2.57222	−0.301056	−0.150528	−	0.988606i	$-0.548097\pi$
$73$	−2.57222	−0.301056	−0.150528	+	0.988606i	$0.548097\pi$
$74$	−13.0968	−1.52247
$75$	0	0
$76$	−0.850887	−0.0976034
$77$	−2.90717	−0.331302
$78$	0	0
$79$	14.5706	1.63932	0.819661	−	0.572849i	$-0.194162\pi$
$79$	14.5706	1.63932	0.819661	+	0.572849i	$0.194162\pi$
$80$	−9.60746	−1.07415
$81$	0	0
$82$	−14.3675	−1.58662
$83$	−0.493442	−0.0541623	−0.0270811	−	0.999633i	$-0.508621\pi$
$83$	−0.493442	−0.0541623	−0.0270811	+	0.999633i	$0.508621\pi$
$84$	0	0
$85$	17.6452	1.91389
$86$	3.16749	0.341560
$87$	0	0
$88$	8.87160	0.945716
$89$	−12.1652	−1.28951	−0.644754	−	0.764390i	$-0.723040\pi$
$89$	−12.1652	−1.28951	−0.644754	+	0.764390i	$0.723040\pi$
$90$	0	0
$91$	−4.59254	−0.481429
$92$	2.29176	0.238933
$93$	0	0
$94$	−9.13087	−0.941777
$95$	6.22866	0.639047
$96$	0	0
$97$	−14.7006	−1.49262	−0.746312	−	0.665596i	$-0.768177\pi$
$97$	−14.7006	−1.49262	−0.746312	+	0.665596i	$0.768177\pi$
$98$	1.24280	0.125542
$99$	0	0
$100$	−2.78535	−0.278535
$101$	7.69701	0.765881	0.382941	−	0.923773i	$-0.374911\pi$
$101$	7.69701	0.765881	0.382941	+	0.923773i	$0.374911\pi$
$102$	0	0
$103$	−18.0952	−1.78297	−0.891486	−	0.453048i	$-0.850337\pi$
$103$	−18.0952	−1.78297	−0.891486	+	0.453048i	$0.850337\pi$
$104$	14.0148	1.37426
$105$	0	0
$106$	14.0233	1.36207
$107$	2.30658	0.222986	0.111493	−	0.993765i	$-0.464437\pi$
$107$	2.30658	0.222986	0.111493	+	0.993765i	$0.464437\pi$
$108$	0	0
$109$	−0.892079	−0.0854457	−0.0427228	−	0.999087i	$-0.513603\pi$
$109$	−0.892079	−0.0854457	−0.0427228	+	0.999087i	$0.513603\pi$
$110$	−12.0458	−1.14852
$111$	0	0
$112$	−2.88166	−0.272291
$113$	−4.35699	−0.409872	−0.204936	−	0.978775i	$-0.565699\pi$
$113$	−4.35699	−0.409872	−0.204936	+	0.978775i	$0.565699\pi$
$114$	0	0
$115$	−16.7762	−1.56439
$116$	1.68489	0.156438
$117$	0	0
$118$	16.5802	1.52633
$119$	5.29249	0.485162
$120$	0	0
$121$	−2.54838	−0.231671
$122$	0.828601	0.0750181
$123$	0	0
$124$	−0.441150	−0.0396164
$125$	3.71933	0.332667
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	6.01402	0.531569
$129$	0	0
$130$	−19.0292	−1.66897
$131$	−8.71812	−0.761706	−0.380853	−	0.924636i	$-0.624370\pi$
$131$	−8.71812	−0.761706	−0.380853	+	0.924636i	$0.624370\pi$
$132$	0	0
$133$	1.86822	0.161996
$134$	−17.4408	−1.50666
$135$	0	0
$136$	−16.1507	−1.38491
$137$	4.66285	0.398374	0.199187	−	0.979962i	$-0.436170\pi$
$137$	4.66285	0.398374	0.199187	+	0.979962i	$0.436170\pi$
$138$	0	0
$139$	−19.7441	−1.67467	−0.837334	−	0.546691i	$-0.815887\pi$
$139$	−19.7441	−1.67467	−0.837334	+	0.546691i	$0.815887\pi$
$140$	−1.51848	−0.128335
$141$	0	0
$142$	−12.2856	−1.03098
$143$	13.3513	1.11649
$144$	0	0
$145$	−12.3337	−1.02426
$146$	−3.19675	−0.264565
$147$	0	0
$148$	4.79962	0.394526
$149$	5.49864	0.450467	0.225233	−	0.974305i	$-0.427686\pi$
$149$	5.49864	0.450467	0.225233	+	0.974305i	$0.427686\pi$
$150$	0	0
$151$	−11.8169	−0.961645	−0.480822	−	0.876818i	$-0.659662\pi$
$151$	−11.8169	−0.961645	−0.480822	+	0.876818i	$0.659662\pi$
$152$	−5.70113	−0.462423
$153$	0	0
$154$	−3.61302	−0.291146
$155$	3.22931	0.259384
$156$	0	0
$157$	3.77762	0.301487	0.150744	−	0.988573i	$-0.451833\pi$
$157$	3.77762	0.301487	0.150744	+	0.988573i	$0.451833\pi$
$158$	18.1083	1.44062
$159$	0	0
$160$	8.40817	0.664724
$161$	−5.03184	−0.396565
$162$	0	0
$163$	−20.0111	−1.56739	−0.783694	−	0.621147i	$-0.786667\pi$
$163$	−20.0111	−1.56739	−0.783694	+	0.621147i	$0.786667\pi$
$164$	5.26530	0.411151
$165$	0	0
$166$	−0.613249	−0.0475973
$167$	16.4295	1.27135	0.635675	−	0.771957i	$-0.280722\pi$
$167$	16.4295	1.27135	0.635675	+	0.771957i	$0.280722\pi$
$168$	0	0
$169$	8.09145	0.622419
$170$	21.9294	1.68191
$171$	0	0
$172$	−1.16080	−0.0885103
$173$	16.2436	1.23498	0.617489	−	0.786580i	$-0.288150\pi$
$173$	16.2436	1.23498	0.617489	+	0.786580i	$0.288150\pi$
$174$	0	0
$175$	6.11557	0.462294
$176$	8.37746	0.631475
$177$	0	0
$178$	−15.1189	−1.13321
$179$	4.00381	0.299259	0.149630	−	0.988742i	$-0.452192\pi$
$179$	4.00381	0.299259	0.149630	+	0.988742i	$0.452192\pi$
$180$	0	0
$181$	−24.4061	−1.81409	−0.907044	−	0.421035i	$-0.861667\pi$
$181$	−24.4061	−1.81409	−0.907044	+	0.421035i	$0.861667\pi$
$182$	−5.70761	−0.423076
$183$	0	0
$184$	15.3553	1.13201
$185$	−35.1342	−2.58312
$186$	0	0
$187$	−15.3862	−1.12515
$188$	3.34622	0.244048
$189$	0	0
$190$	7.74097	0.561589
$191$	22.5671	1.63290	0.816450	−	0.577416i	$-0.195939\pi$
$191$	22.5671	1.63290	0.816450	+	0.577416i	$0.195939\pi$
$192$	0	0
$193$	20.7333	1.49242	0.746208	−	0.665713i	$-0.231873\pi$
$193$	20.7333	1.49242	0.746208	+	0.665713i	$0.231873\pi$
$194$	−18.2699	−1.31170
$195$	0	0
$196$	−0.455452	−0.0325323
$197$	3.77768	0.269148	0.134574	−	0.990904i	$-0.457033\pi$
$197$	3.77768	0.269148	0.134574	+	0.990904i	$0.457033\pi$
$198$	0	0
$199$	−9.73569	−0.690145	−0.345072	−	0.938576i	$-0.612146\pi$
$199$	−9.73569	−0.690145	−0.345072	+	0.938576i	$0.612146\pi$
$200$	−18.6625	−1.31964
$201$	0	0
$202$	9.56583	0.673050
$203$	−3.69938	−0.259645
$204$	0	0
$205$	−38.5431	−2.69196
$206$	−22.4887	−1.56686
$207$	0	0
$208$	13.2341	0.917623
$209$	−5.43124	−0.375687
$210$	0	0
$211$	4.66669	0.321268	0.160634	−	0.987014i	$-0.448646\pi$
$211$	4.66669	0.321268	0.160634	+	0.987014i	$0.448646\pi$
$212$	−5.13918	−0.352960
$213$	0	0
$214$	2.86662	0.195958
$215$	8.49730	0.579511
$216$	0	0
$217$	0.968597	0.0657527
$218$	−1.10867	−0.0750889
$219$	0	0
$220$	4.41447	0.297624
$221$	−24.3060	−1.63500
$222$	0	0
$223$	5.37863	0.360180	0.180090	−	0.983650i	$-0.442361\pi$
$223$	5.37863	0.360180	0.180090	+	0.983650i	$0.442361\pi$
$224$	2.52194	0.168504
$225$	0	0
$226$	−5.41486	−0.360191
$227$	11.5084	0.763837	0.381918	−	0.924196i	$-0.375264\pi$
$227$	11.5084	0.763837	0.381918	+	0.924196i	$0.375264\pi$
$228$	0	0
$229$	−1.87862	−0.124142	−0.0620712	−	0.998072i	$-0.519771\pi$
$229$	−1.87862	−0.124142	−0.0620712	+	0.998072i	$0.519771\pi$
$230$	−20.8494	−1.37477
$231$	0	0
$232$	11.2891	0.741168
$233$	−13.5663	−0.888755	−0.444378	−	0.895840i	$-0.646575\pi$
$233$	−13.5663	−0.888755	−0.444378	+	0.895840i	$0.646575\pi$
$234$	0	0
$235$	−24.4950	−1.59788
$236$	−6.07619	−0.395526
$237$	0	0
$238$	6.57750	0.426356
$239$	−22.4619	−1.45294	−0.726468	−	0.687200i	$-0.758840\pi$
$239$	−22.4619	−1.45294	−0.726468	+	0.687200i	$0.758840\pi$
$240$	0	0
$241$	12.0289	0.774848	0.387424	−	0.921902i	$-0.373365\pi$
$241$	12.0289	0.774848	0.387424	+	0.921902i	$0.373365\pi$
$242$	−3.16712	−0.203590
$243$	0	0
$244$	−0.303660	−0.0194399
$245$	3.33400	0.213002
$246$	0	0
$247$	−8.57990	−0.545926
$248$	−2.95580	−0.187694
$249$	0	0
$250$	4.62237	0.292345
$251$	−5.27452	−0.332924	−0.166462	−	0.986048i	$-0.553234\pi$
$251$	−5.27452	−0.332924	−0.166462	+	0.986048i	$0.553234\pi$
$252$	0	0
$253$	14.6284	0.919680
$254$	−1.24280	−0.0779801
$255$	0	0
$256$	−10.3210	−0.645060
$257$	28.6479	1.78701	0.893504	−	0.449055i	$-0.148239\pi$
$257$	28.6479	1.78701	0.893504	+	0.449055i	$0.148239\pi$
$258$	0	0
$259$	−10.5381	−0.654808
$260$	6.97368	0.432489
$261$	0	0
$262$	−10.8349	−0.669380
$263$	6.33902	0.390881	0.195440	−	0.980716i	$-0.437386\pi$
$263$	6.33902	0.390881	0.195440	+	0.980716i	$0.437386\pi$
$264$	0	0
$265$	37.6198	2.31097
$266$	2.32183	0.142360
$267$	0	0
$268$	6.39159	0.390429
$269$	−1.26346	−0.0770348	−0.0385174	−	0.999258i	$-0.512264\pi$
$269$	−1.26346	−0.0770348	−0.0385174	+	0.999258i	$0.512264\pi$
$270$	0	0
$271$	−5.15641	−0.313230	−0.156615	−	0.987660i	$-0.550058\pi$
$271$	−5.15641	−0.313230	−0.156615	+	0.987660i	$0.550058\pi$
$272$	−15.2512	−0.924737
$273$	0	0
$274$	5.79498	0.350087
$275$	−17.7790	−1.07211
$276$	0	0
$277$	2.77471	0.166716	0.0833580	−	0.996520i	$-0.473435\pi$
$277$	2.77471	0.166716	0.0833580	+	0.996520i	$0.473435\pi$
$278$	−24.5379	−1.47168
$279$	0	0
$280$	−10.1741	−0.608022
$281$	−5.01385	−0.299101	−0.149551	−	0.988754i	$-0.547783\pi$
$281$	−5.01385	−0.299101	−0.149551	+	0.988754i	$0.547783\pi$
$282$	0	0
$283$	−8.84064	−0.525521	−0.262761	−	0.964861i	$-0.584633\pi$
$283$	−8.84064	−0.525521	−0.262761	+	0.964861i	$0.584633\pi$
$284$	4.50233	0.267164
$285$	0	0
$286$	16.5930	0.981162
$287$	−11.5606	−0.682401
$288$	0	0
$289$	11.0105	0.647674
$290$	−15.3283	−0.900111
$291$	0	0
$292$	1.17152	0.0685583
$293$	16.5617	0.967545	0.483772	−	0.875194i	$-0.339266\pi$
$293$	16.5617	0.967545	0.483772	+	0.875194i	$0.339266\pi$
$294$	0	0
$295$	44.4789	2.58966
$296$	32.1585	1.86918
$297$	0	0
$298$	6.83371	0.395866
$299$	23.1089	1.33643
$300$	0	0
$301$	2.54868	0.146903
$302$	−14.6860	−0.845085
$303$	0	0
$304$	−5.38358	−0.308770
$305$	2.22285	0.127280
$306$	0	0
$307$	−23.4209	−1.33670	−0.668352	−	0.743845i	$-0.733000\pi$
$307$	−23.4209	−1.33670	−0.668352	+	0.743845i	$0.733000\pi$
$308$	1.32408	0.0754462
$309$	0	0
$310$	4.01338	0.227944
$311$	19.5726	1.10986	0.554929	−	0.831898i	$-0.312745\pi$
$311$	19.5726	1.10986	0.554929	+	0.831898i	$0.312745\pi$
$312$	0	0
$313$	5.10217	0.288392	0.144196	−	0.989549i	$-0.453940\pi$
$313$	5.10217	0.288392	0.144196	+	0.989549i	$0.453940\pi$
$314$	4.69483	0.264944
$315$	0	0
$316$	−6.63622	−0.373316
$317$	19.6691	1.10472	0.552362	−	0.833604i	$-0.313727\pi$
$317$	19.6691	1.10472	0.552362	+	0.833604i	$0.313727\pi$
$318$	0	0
$319$	10.7547	0.602148
$320$	29.6646	1.65830
$321$	0	0
$322$	−6.25356	−0.348497
$323$	9.88756	0.550158
$324$	0	0
$325$	−28.0860	−1.55793
$326$	−24.8697	−1.37741
$327$	0	0
$328$	35.2787	1.94794
$329$	−7.34702	−0.405055
$330$	0	0
$331$	15.1305	0.831648	0.415824	−	0.909445i	$-0.363493\pi$
$331$	15.1305	0.831648	0.415824	+	0.909445i	$0.363493\pi$
$332$	0.224739	0.0123342
$333$	0	0
$334$	20.4185	1.11725
$335$	−46.7877	−2.55629
$336$	0	0
$337$	−7.21235	−0.392882	−0.196441	−	0.980516i	$-0.562938\pi$
$337$	−7.21235	−0.392882	−0.196441	+	0.980516i	$0.562938\pi$
$338$	10.0560	0.546977
$339$	0	0
$340$	−8.03654	−0.435842
$341$	−2.81587	−0.152488
$342$	0	0
$343$	1.00000	0.0539949
$344$	−7.77763	−0.419342
$345$	0	0
$346$	20.1875	1.08529
$347$	−28.7813	−1.54506	−0.772531	−	0.634977i	$-0.781010\pi$
$347$	−28.7813	−1.54506	−0.772531	+	0.634977i	$0.781010\pi$
$348$	0	0
$349$	−28.8201	−1.54270	−0.771352	−	0.636409i	$-0.780419\pi$
$349$	−28.8201	−1.54270	−0.771352	+	0.636409i	$0.780419\pi$
$350$	7.60043	0.406260
$351$	0	0
$352$	−7.33171	−0.390782
$353$	−26.6489	−1.41838	−0.709188	−	0.705019i	$-0.750938\pi$
$353$	−26.6489	−1.41838	−0.709188	+	0.705019i	$0.750938\pi$
$354$	0	0
$355$	−32.9580	−1.74923
$356$	5.54066	0.293655
$357$	0	0
$358$	4.97593	0.262986
$359$	14.4278	0.761469	0.380734	−	0.924684i	$-0.375671\pi$
$359$	14.4278	0.761469	0.380734	+	0.924684i	$0.375671\pi$
$360$	0	0
$361$	−15.5097	−0.816302
$362$	−30.3318	−1.59421
$363$	0	0
$364$	2.09168	0.109634
$365$	−8.57580	−0.448878
$366$	0	0
$367$	−8.35795	−0.436282	−0.218141	−	0.975917i	$-0.569999\pi$
$367$	−8.35795	−0.436282	−0.218141	+	0.975917i	$0.569999\pi$
$368$	14.5000	0.755867
$369$	0	0
$370$	−43.6647	−2.27002
$371$	11.2837	0.585819
$372$	0	0
$373$	30.3153	1.56966	0.784832	−	0.619708i	$-0.212749\pi$
$373$	30.3153	1.56966	0.784832	+	0.619708i	$0.212749\pi$
$374$	−19.1219	−0.988769
$375$	0	0
$376$	22.4204	1.15624
$377$	16.9895	0.875006
$378$	0	0
$379$	2.57847	0.132447	0.0662235	−	0.997805i	$-0.478905\pi$
$379$	2.57847	0.132447	0.0662235	+	0.997805i	$0.478905\pi$
$380$	−2.83686	−0.145528
$381$	0	0
$382$	28.0464	1.43498
$383$	14.6277	0.747438	0.373719	−	0.927542i	$-0.378082\pi$
$383$	14.6277	0.747438	0.373719	+	0.927542i	$0.378082\pi$
$384$	0	0
$385$	−9.69250	−0.493976
$386$	25.7673	1.31152
$387$	0	0
$388$	6.69544	0.339909
$389$	−19.5586	−0.991663	−0.495831	−	0.868419i	$-0.665137\pi$
$389$	−19.5586	−0.991663	−0.495831	+	0.868419i	$0.665137\pi$
$390$	0	0
$391$	−26.6310	−1.34679
$392$	−3.05163	−0.154131
$393$	0	0
$394$	4.69489	0.236525
$395$	48.5784	2.44425
$396$	0	0
$397$	−8.35455	−0.419303	−0.209651	−	0.977776i	$-0.567233\pi$
$397$	−8.35455	−0.419303	−0.209651	+	0.977776i	$0.567233\pi$
$398$	−12.0995	−0.606493
$399$	0	0
$400$	−17.6230	−0.881150
$401$	−7.25894	−0.362494	−0.181247	−	0.983438i	$-0.558013\pi$
$401$	−7.25894	−0.362494	−0.181247	+	0.983438i	$0.558013\pi$
$402$	0	0
$403$	−4.44832	−0.221587
$404$	−3.50562	−0.174411
$405$	0	0
$406$	−4.59758	−0.228174
$407$	30.6361	1.51858
$408$	0	0
$409$	8.46320	0.418478	0.209239	−	0.977864i	$-0.432901\pi$
$409$	8.46320	0.418478	0.209239	+	0.977864i	$0.432901\pi$
$410$	−47.9013	−2.36567
$411$	0	0
$412$	8.24150	0.406029
$413$	13.3410	0.656468
$414$	0	0
$415$	−1.64514	−0.0807566
$416$	−11.5821	−0.567861
$417$	0	0
$418$	−6.74994	−0.330150
$419$	−26.7971	−1.30912	−0.654561	−	0.756009i	$-0.727146\pi$
$419$	−26.7971	−1.30912	−0.654561	+	0.756009i	$0.727146\pi$
$420$	0	0
$421$	14.1261	0.688463	0.344231	−	0.938885i	$-0.388140\pi$
$421$	14.1261	0.688463	0.344231	+	0.938885i	$0.388140\pi$
$422$	5.79976	0.282328
$423$	0	0
$424$	−34.4336	−1.67225
$425$	32.3666	1.57001
$426$	0	0
$427$	0.666722	0.0322650
$428$	−1.05054	−0.0507797
$429$	0	0
$430$	10.5604	0.509269
$431$	16.6128	0.800211	0.400105	−	0.916469i	$-0.368974\pi$
$431$	16.6128	0.800211	0.400105	+	0.916469i	$0.368974\pi$
$432$	0	0
$433$	24.5878	1.18161	0.590807	−	0.806813i	$-0.298810\pi$
$433$	24.5878	1.18161	0.590807	+	0.806813i	$0.298810\pi$
$434$	1.20377	0.0577829
$435$	0	0
$436$	0.406300	0.0194582
$437$	−9.40061	−0.449692
$438$	0	0
$439$	−38.1271	−1.81971	−0.909853	−	0.414930i	$-0.863806\pi$
$439$	−38.1271	−1.81971	−0.909853	+	0.414930i	$0.863806\pi$
$440$	29.5780	1.41007
$441$	0	0
$442$	−30.2074	−1.43682
$443$	−22.7120	−1.07908	−0.539541	−	0.841959i	$-0.681402\pi$
$443$	−22.7120	−1.07908	−0.539541	+	0.841959i	$0.681402\pi$
$444$	0	0
$445$	−40.5588	−1.92267
$446$	6.68455	0.316523
$447$	0	0
$448$	8.89758	0.420371
$449$	29.0754	1.37215	0.686075	−	0.727531i	$-0.259332\pi$
$449$	29.0754	1.37215	0.686075	+	0.727531i	$0.259332\pi$
$450$	0	0
$451$	33.6086	1.58257
$452$	1.98440	0.0933385
$453$	0	0
$454$	14.3026	0.671253
$455$	−15.3116	−0.717817
$456$	0	0
$457$	25.5196	1.19376	0.596879	−	0.802332i	$-0.296407\pi$
$457$	25.5196	1.19376	0.596879	+	0.802332i	$0.296407\pi$
$458$	−2.33474	−0.109095
$459$	0	0
$460$	7.64075	0.356252
$461$	10.8134	0.503631	0.251815	−	0.967775i	$-0.418972\pi$
$461$	10.8134	0.503631	0.251815	+	0.967775i	$0.418972\pi$
$462$	0	0
$463$	5.98793	0.278283	0.139141	−	0.990273i	$-0.455566\pi$
$463$	5.98793	0.278283	0.139141	+	0.990273i	$0.455566\pi$
$464$	10.6603	0.494894
$465$	0	0
$466$	−16.8601	−0.781030
$467$	−41.9564	−1.94151	−0.970755	−	0.240074i	$-0.922828\pi$
$467$	−41.9564	−1.94151	−0.970755	+	0.240074i	$0.922828\pi$
$468$	0	0
$469$	−14.0335	−0.648007
$470$	−30.4423	−1.40420
$471$	0	0
$472$	−40.7118	−1.87391
$473$	−7.40943	−0.340686
$474$	0	0
$475$	11.4253	0.524227
$476$	−2.41048	−0.110484
$477$	0	0
$478$	−27.9156	−1.27683
$479$	9.65088	0.440960	0.220480	−	0.975392i	$-0.429238\pi$
$479$	9.65088	0.440960	0.220480	+	0.975392i	$0.429238\pi$
$480$	0	0
$481$	48.3969	2.20671
$482$	14.9495	0.680930
$483$	0	0
$484$	1.16067	0.0527575
$485$	−49.0120	−2.22552
$486$	0	0
$487$	−17.7189	−0.802920	−0.401460	−	0.915877i	$-0.631497\pi$
$487$	−17.7189	−0.802920	−0.401460	+	0.915877i	$0.631497\pi$
$488$	−2.03459	−0.0921016
$489$	0	0
$490$	4.14349	0.187184
$491$	4.97599	0.224563	0.112282	−	0.993676i	$-0.464184\pi$
$491$	4.97599	0.224563	0.112282	+	0.993676i	$0.464184\pi$
$492$	0	0
$493$	−19.5789	−0.881790
$494$	−10.6631	−0.479755
$495$	0	0
$496$	−2.79117	−0.125327
$497$	−9.88541	−0.443421
$498$	0	0
$499$	44.0654	1.97264	0.986318	−	0.164851i	$-0.0527143\pi$
$499$	44.0654	1.97264	0.986318	+	0.164851i	$0.0527143\pi$
$500$	−1.69398	−0.0757569
$501$	0	0
$502$	−6.55516	−0.292571
$503$	−5.31840	−0.237136	−0.118568	−	0.992946i	$-0.537830\pi$
$503$	−5.31840	−0.237136	−0.118568	+	0.992946i	$0.537830\pi$
$504$	0	0
$505$	25.6619	1.14194
$506$	18.1802	0.808207
$507$	0	0
$508$	0.455452	0.0202074
$509$	−10.4607	−0.463661	−0.231830	−	0.972756i	$-0.574471\pi$
$509$	−10.4607	−0.463661	−0.231830	+	0.972756i	$0.574471\pi$
$510$	0	0
$511$	−2.57222	−0.113788
$512$	−24.8549	−1.09844
$513$	0	0
$514$	35.6036	1.57041
$515$	−60.3294	−2.65843
$516$	0	0
$517$	21.3590	0.939369
$518$	−13.0968	−0.575440
$519$	0	0
$520$	46.7252	2.04904
$521$	−35.6052	−1.55989	−0.779947	−	0.625846i	$-0.784754\pi$
$521$	−35.6052	−1.55989	−0.779947	+	0.625846i	$0.784754\pi$
$522$	0	0
$523$	23.5608	1.03024	0.515120	−	0.857118i	$-0.327747\pi$
$523$	23.5608	1.03024	0.515120	+	0.857118i	$0.327747\pi$
$524$	3.97069	0.173460
$525$	0	0
$526$	7.87813	0.343503
$527$	5.12629	0.223305
$528$	0	0
$529$	2.31943	0.100845
$530$	46.7538	2.03086
$531$	0	0
$532$	−0.850887	−0.0368906
$533$	53.0926	2.29969
$534$	0	0
$535$	7.69015	0.332474
$536$	42.8251	1.84976
$537$	0	0
$538$	−1.57023	−0.0676975
$539$	−2.90717	−0.125221
$540$	0	0
$541$	7.71667	0.331765	0.165883	−	0.986145i	$-0.446953\pi$
$541$	7.71667	0.331765	0.165883	+	0.986145i	$0.446953\pi$
$542$	−6.40838	−0.275264
$543$	0	0
$544$	13.3474	0.572263
$545$	−2.97419	−0.127400
$546$	0	0
$547$	−15.9737	−0.682987	−0.341494	−	0.939884i	$-0.610933\pi$
$547$	−15.9737	−0.682987	−0.341494	+	0.939884i	$0.610933\pi$
$548$	−2.12370	−0.0907201
$549$	0	0
$550$	−22.0957	−0.942164
$551$	−6.91126	−0.294430
$552$	0	0
$553$	14.5706	0.619605
$554$	3.44840	0.146509
$555$	0	0
$556$	8.99247	0.381366
$557$	−0.464136	−0.0196661	−0.00983304	−	0.999952i	$-0.503130\pi$
$557$	−0.464136	−0.0196661	−0.00983304	+	0.999952i	$0.503130\pi$
$558$	0	0
$559$	−11.7049	−0.495065
$560$	−9.60746	−0.405989
$561$	0	0
$562$	−6.23121	−0.262848
$563$	−35.9885	−1.51673	−0.758367	−	0.651828i	$-0.774002\pi$
$563$	−35.9885	−1.51673	−0.758367	+	0.651828i	$0.774002\pi$
$564$	0	0
$565$	−14.5262	−0.611123
$566$	−10.9871	−0.461824
$567$	0	0
$568$	30.1666	1.26576
$569$	11.1032	0.465469	0.232735	−	0.972540i	$-0.425233\pi$
$569$	11.1032	0.465469	0.232735	+	0.972540i	$0.425233\pi$
$570$	0	0
$571$	10.2214	0.427753	0.213877	−	0.976861i	$-0.431391\pi$
$571$	10.2214	0.427753	0.213877	+	0.976861i	$0.431391\pi$
$572$	−6.08088	−0.254254
$573$	0	0
$574$	−14.3675	−0.599688
$575$	−30.7726	−1.28331
$576$	0	0
$577$	10.4490	0.434999	0.217500	−	0.976060i	$-0.430210\pi$
$577$	10.4490	0.434999	0.217500	+	0.976060i	$0.430210\pi$
$578$	13.6838	0.569170
$579$	0	0
$580$	5.61742	0.233251
$581$	−0.493442	−0.0204714
$582$	0	0
$583$	−32.8035	−1.35858
$584$	7.84948	0.324814
$585$	0	0
$586$	20.5829	0.850270
$587$	−37.8825	−1.56358	−0.781789	−	0.623543i	$-0.785693\pi$
$587$	−37.8825	−1.56358	−0.781789	+	0.623543i	$0.785693\pi$
$588$	0	0
$589$	1.80956	0.0745615
$590$	55.2783	2.27577
$591$	0	0
$592$	30.3673	1.24809
$593$	43.2711	1.77693	0.888466	−	0.458942i	$-0.151772\pi$
$593$	43.2711	1.77693	0.888466	+	0.458942i	$0.151772\pi$
$594$	0	0
$595$	17.6452	0.723382
$596$	−2.50437	−0.102583
$597$	0	0
$598$	28.7198	1.17444
$599$	28.8279	1.17787	0.588937	−	0.808179i	$-0.299547\pi$
$599$	28.8279	1.17787	0.588937	+	0.808179i	$0.299547\pi$
$600$	0	0
$601$	29.8137	1.21612	0.608062	−	0.793889i	$-0.291947\pi$
$601$	29.8137	1.21612	0.608062	+	0.793889i	$0.291947\pi$
$602$	3.16749	0.129097
$603$	0	0
$604$	5.38203	0.218992
$605$	−8.49630	−0.345424
$606$	0	0
$607$	−15.0035	−0.608974	−0.304487	−	0.952516i	$-0.598485\pi$
$607$	−15.0035	−0.608974	−0.304487	+	0.952516i	$0.598485\pi$
$608$	4.71156	0.191079
$609$	0	0
$610$	2.76256	0.111853
$611$	33.7415	1.36504
$612$	0	0
$613$	−5.68365	−0.229561	−0.114780	−	0.993391i	$-0.536616\pi$
$613$	−5.68365	−0.229561	−0.114780	+	0.993391i	$0.536616\pi$
$614$	−29.1075	−1.17468
$615$	0	0
$616$	8.87160	0.357447
$617$	−13.5363	−0.544952	−0.272476	−	0.962163i	$-0.587842\pi$
$617$	−13.5363	−0.544952	−0.272476	+	0.962163i	$0.587842\pi$
$618$	0	0
$619$	−27.0321	−1.08651	−0.543256	−	0.839567i	$-0.682809\pi$
$619$	−27.0321	−1.08651	−0.543256	+	0.839567i	$0.682809\pi$
$620$	−1.47079	−0.0590685
$621$	0	0
$622$	24.3247	0.975334
$623$	−12.1652	−0.487388
$624$	0	0
$625$	−18.1776	−0.727105
$626$	6.34097	0.253436
$627$	0	0
$628$	−1.72053	−0.0686565
$629$	−55.7730	−2.22382
$630$	0	0
$631$	2.75817	0.109801	0.0549004	−	0.998492i	$-0.482516\pi$
$631$	2.75817	0.109801	0.0549004	+	0.998492i	$0.482516\pi$
$632$	−44.4641	−1.76869
$633$	0	0
$634$	24.4447	0.970822
$635$	−3.33400	−0.132306
$636$	0	0
$637$	−4.59254	−0.181963
$638$	13.3659	0.529162
$639$	0	0
$640$	20.0507	0.792575
$641$	10.2155	0.403489	0.201744	−	0.979438i	$-0.435339\pi$
$641$	10.2155	0.403489	0.201744	+	0.979438i	$0.435339\pi$
$642$	0	0
$643$	−25.3762	−1.00074	−0.500370	−	0.865812i	$-0.666803\pi$
$643$	−25.3762	−1.00074	−0.500370	+	0.865812i	$0.666803\pi$
$644$	2.29176	0.0903081
$645$	0	0
$646$	12.2882	0.483474
$647$	22.1567	0.871068	0.435534	−	0.900172i	$-0.356560\pi$
$647$	22.1567	0.871068	0.435534	+	0.900172i	$0.356560\pi$
$648$	0	0
$649$	−38.7845	−1.52243
$650$	−34.9053	−1.36910
$651$	0	0
$652$	9.11409	0.356935
$653$	9.65817	0.377954	0.188977	−	0.981982i	$-0.439483\pi$
$653$	9.65817	0.377954	0.188977	+	0.981982i	$0.439483\pi$
$654$	0	0
$655$	−29.0662	−1.13571
$656$	33.3137	1.30068
$657$	0	0
$658$	−9.13087	−0.355958
$659$	14.6717	0.571527	0.285764	−	0.958300i	$-0.407753\pi$
$659$	14.6717	0.571527	0.285764	+	0.958300i	$0.407753\pi$
$660$	0	0
$661$	33.3172	1.29589	0.647945	−	0.761687i	$-0.275629\pi$
$661$	33.3172	1.29589	0.647945	+	0.761687i	$0.275629\pi$
$662$	18.8042	0.730845
$663$	0	0
$664$	1.50580	0.0584365
$665$	6.22866	0.241537
$666$	0	0
$667$	18.6147	0.720763
$668$	−7.48283	−0.289519
$669$	0	0
$670$	−58.1477	−2.24644
$671$	−1.93827	−0.0748262
$672$	0	0
$673$	1.32670	0.0511404	0.0255702	−	0.999673i	$-0.491860\pi$
$673$	1.32670	0.0511404	0.0255702	+	0.999673i	$0.491860\pi$
$674$	−8.96350	−0.345261
$675$	0	0
$676$	−3.68527	−0.141741
$677$	−11.9106	−0.457761	−0.228881	−	0.973454i	$-0.573507\pi$
$677$	−11.9106	−0.457761	−0.228881	+	0.973454i	$0.573507\pi$
$678$	0	0
$679$	−14.7006	−0.564159
$680$	−53.8466	−2.06492
$681$	0	0
$682$	−3.49956	−0.134005
$683$	6.04534	0.231318	0.115659	−	0.993289i	$-0.463102\pi$
$683$	6.04534	0.231318	0.115659	+	0.993289i	$0.463102\pi$
$684$	0	0
$685$	15.5459	0.593980
$686$	1.24280	0.0474503
$687$	0	0
$688$	−7.34442	−0.280003
$689$	−51.8208	−1.97421
$690$	0	0
$691$	−26.4623	−1.00667	−0.503336	−	0.864090i	$-0.667894\pi$
$691$	−26.4623	−1.00667	−0.503336	+	0.864090i	$0.667894\pi$
$692$	−7.39818	−0.281237
$693$	0	0
$694$	−35.7694	−1.35779
$695$	−65.8267	−2.49695
$696$	0	0
$697$	−61.1844	−2.31752
$698$	−35.8176	−1.35571
$699$	0	0
$700$	−2.78535	−0.105276
$701$	−27.8873	−1.05329	−0.526645	−	0.850085i	$-0.676550\pi$
$701$	−27.8873	−1.05329	−0.526645	+	0.850085i	$0.676550\pi$
$702$	0	0
$703$	−19.6876	−0.742532
$704$	−25.8668	−0.974890
$705$	0	0
$706$	−33.1192	−1.24646
$707$	7.69701	0.289476
$708$	0	0
$709$	−21.7892	−0.818312	−0.409156	−	0.912465i	$-0.634177\pi$
$709$	−21.7892	−0.818312	−0.409156	+	0.912465i	$0.634177\pi$
$710$	−40.9601	−1.53721
$711$	0	0
$712$	37.1237	1.39127
$713$	−4.87383	−0.182526
$714$	0	0
$715$	44.5132	1.66470
$716$	−1.82355	−0.0681491
$717$	0	0
$718$	17.9308	0.669172
$719$	−0.503768	−0.0187874	−0.00939368	−	0.999956i	$-0.502990\pi$
$719$	−0.503768	−0.0187874	−0.00939368	+	0.999956i	$0.502990\pi$
$720$	0	0
$721$	−18.0952	−0.673900
$722$	−19.2755	−0.717359
$723$	0	0
$724$	11.1158	0.413115
$725$	−22.6238	−0.840227
$726$	0	0
$727$	23.4153	0.868427	0.434213	−	0.900810i	$-0.357026\pi$
$727$	23.4153	0.868427	0.434213	+	0.900810i	$0.357026\pi$
$728$	14.0148	0.519421
$729$	0	0
$730$	−10.6580	−0.394470
$731$	13.4889	0.498903
$732$	0	0
$733$	52.3311	1.93289	0.966446	−	0.256871i	$-0.0826915\pi$
$733$	52.3311	1.93289	0.966446	+	0.256871i	$0.0826915\pi$
$734$	−10.3872	−0.383400
$735$	0	0
$736$	−12.6900	−0.467760
$737$	40.7978	1.50280
$738$	0	0
$739$	−11.4312	−0.420504	−0.210252	−	0.977647i	$-0.567429\pi$
$739$	−11.4312	−0.420504	−0.210252	+	0.977647i	$0.567429\pi$
$740$	16.0020	0.588243
$741$	0	0
$742$	14.0233	0.514813
$743$	−33.8238	−1.24087	−0.620437	−	0.784256i	$-0.713045\pi$
$743$	−33.8238	−1.24087	−0.620437	+	0.784256i	$0.713045\pi$
$744$	0	0
$745$	18.3325	0.671651
$746$	37.6758	1.37941
$747$	0	0
$748$	7.00766	0.256225
$749$	2.30658	0.0842807
$750$	0	0
$751$	−50.0895	−1.82779	−0.913896	−	0.405949i	$-0.866941\pi$
$751$	−50.0895	−1.82779	−0.913896	+	0.405949i	$0.866941\pi$
$752$	21.1716	0.772049
$753$	0	0
$754$	21.1146	0.768947
$755$	−39.3975	−1.43382
$756$	0	0
$757$	16.9391	0.615663	0.307831	−	0.951441i	$-0.400397\pi$
$757$	16.9391	0.615663	0.307831	+	0.951441i	$0.400397\pi$
$758$	3.20451	0.116393
$759$	0	0
$760$	−19.0076	−0.689478
$761$	32.7577	1.18747	0.593734	−	0.804662i	$-0.297653\pi$
$761$	32.7577	1.18747	0.593734	+	0.804662i	$0.297653\pi$
$762$	0	0
$763$	−0.892079	−0.0322954
$764$	−10.2783	−0.371854
$765$	0	0
$766$	18.1792	0.656842
$767$	−61.2691	−2.21230
$768$	0	0
$769$	47.6458	1.71815	0.859075	−	0.511850i	$-0.171040\pi$
$769$	47.6458	1.71815	0.859075	+	0.511850i	$0.171040\pi$
$770$	−12.0458	−0.434101
$771$	0	0
$772$	−9.44303	−0.339862
$773$	47.1769	1.69683	0.848417	−	0.529328i	$-0.177556\pi$
$773$	47.1769	1.69683	0.848417	+	0.529328i	$0.177556\pi$
$774$	0	0
$775$	5.92353	0.212779
$776$	44.8609	1.61041
$777$	0	0
$778$	−24.3075	−0.871464
$779$	−21.5978	−0.773821
$780$	0	0
$781$	28.7385	1.02835
$782$	−33.0969	−1.18354
$783$	0	0
$784$	−2.88166	−0.102916
$785$	12.5946	0.449521
$786$	0	0
$787$	−15.6611	−0.558257	−0.279129	−	0.960254i	$-0.590046\pi$
$787$	−15.6611	−0.558257	−0.279129	+	0.960254i	$0.590046\pi$
$788$	−1.72055	−0.0612921
$789$	0	0
$790$	60.3732	2.14798
$791$	−4.35699	−0.154917
$792$	0	0
$793$	−3.06195	−0.108733
$794$	−10.3830	−0.368479
$795$	0	0
$796$	4.43414	0.157164
$797$	2.11085	0.0747702	0.0373851	−	0.999301i	$-0.488097\pi$
$797$	2.11085	0.0747702	0.0373851	+	0.999301i	$0.488097\pi$
$798$	0	0
$799$	−38.8841	−1.37562
$800$	15.4231	0.545290
$801$	0	0
$802$	−9.02139	−0.318556
$803$	7.47788	0.263889
$804$	0	0
$805$	−16.7762	−0.591282
$806$	−5.52837	−0.194729
$807$	0	0
$808$	−23.4884	−0.826321
$809$	10.8674	0.382079	0.191039	−	0.981582i	$-0.438814\pi$
$809$	10.8674	0.382079	0.191039	+	0.981582i	$0.438814\pi$
$810$	0	0
$811$	−46.4996	−1.63282	−0.816411	−	0.577472i	$-0.804039\pi$
$811$	−46.4996	−1.63282	−0.816411	+	0.577472i	$0.804039\pi$
$812$	1.68489	0.0591280
$813$	0	0
$814$	38.0746	1.33451
$815$	−66.7170	−2.33699
$816$	0	0
$817$	4.76150	0.166584
$818$	10.5181	0.367755
$819$	0	0
$820$	17.5545	0.613031
$821$	26.4225	0.922152	0.461076	−	0.887361i	$-0.347464\pi$
$821$	26.4225	0.922152	0.461076	+	0.887361i	$0.347464\pi$
$822$	0	0
$823$	−26.4500	−0.921989	−0.460995	−	0.887403i	$-0.652507\pi$
$823$	−26.4500	−0.921989	−0.460995	+	0.887403i	$0.652507\pi$
$824$	55.2199	1.92368
$825$	0	0
$826$	16.5802	0.576898
$827$	−31.2695	−1.08735	−0.543673	−	0.839297i	$-0.682967\pi$
$827$	−31.2695	−1.08735	−0.543673	+	0.839297i	$0.682967\pi$
$828$	0	0
$829$	−52.6203	−1.82758	−0.913790	−	0.406187i	$-0.866858\pi$
$829$	−52.6203	−1.82758	−0.913790	+	0.406187i	$0.866858\pi$
$830$	−2.04457	−0.0709682
$831$	0	0
$832$	−40.8625	−1.41665
$833$	5.29249	0.183374
$834$	0	0
$835$	54.7758	1.89560
$836$	2.47367	0.0855537
$837$	0	0
$838$	−33.3033	−1.15044
$839$	29.2755	1.01070	0.505352	−	0.862913i	$-0.331363\pi$
$839$	29.2755	1.01070	0.505352	+	0.862913i	$0.331363\pi$
$840$	0	0
$841$	−15.3146	−0.528090
$842$	17.5559	0.605015
$843$	0	0
$844$	−2.12546	−0.0731612
$845$	26.9769	0.928034
$846$	0	0
$847$	−2.54838	−0.0875633
$848$	−32.5157	−1.11659
$849$	0	0
$850$	40.2252	1.37971
$851$	53.0263	1.81772
$852$	0	0
$853$	−22.4780	−0.769631	−0.384816	−	0.922994i	$-0.625735\pi$
$853$	−22.4780	−0.769631	−0.384816	+	0.922994i	$0.625735\pi$
$854$	0.828601	0.0283542
$855$	0	0
$856$	−7.03884	−0.240583
$857$	35.5634	1.21482	0.607412	−	0.794387i	$-0.292208\pi$
$857$	35.5634	1.21482	0.607412	+	0.794387i	$0.292208\pi$
$858$	0	0
$859$	38.2979	1.30671	0.653354	−	0.757053i	$-0.273361\pi$
$859$	38.2979	1.30671	0.653354	+	0.757053i	$0.273361\pi$
$860$	−3.87011	−0.131970
$861$	0	0
$862$	20.6464	0.703218
$863$	−47.7198	−1.62440	−0.812201	−	0.583378i	$-0.801730\pi$
$863$	−47.7198	−1.62440	−0.812201	+	0.583378i	$0.801730\pi$
$864$	0	0
$865$	54.1562	1.84137
$866$	30.5577	1.03839
$867$	0	0
$868$	−0.441150	−0.0149736
$869$	−42.3592	−1.43694
$870$	0	0
$871$	64.4495	2.18379
$872$	2.72230	0.0921886
$873$	0	0
$874$	−11.6831	−0.395185
$875$	3.71933	0.125736
$876$	0	0
$877$	−20.9916	−0.708837	−0.354418	−	0.935087i	$-0.615321\pi$
$877$	−20.9916	−0.708837	−0.354418	+	0.935087i	$0.615321\pi$
$878$	−47.3843	−1.59914
$879$	0	0
$880$	27.9305	0.941536
$881$	47.8891	1.61343	0.806713	−	0.590943i	$-0.201244\pi$
$881$	47.8891	1.61343	0.806713	+	0.590943i	$0.201244\pi$
$882$	0	0
$883$	10.1634	0.342026	0.171013	−	0.985269i	$-0.445296\pi$
$883$	10.1634	0.342026	0.171013	+	0.985269i	$0.445296\pi$
$884$	11.0702	0.372332
$885$	0	0
$886$	−28.2265	−0.948287
$887$	49.2135	1.65243	0.826213	−	0.563357i	$-0.190490\pi$
$887$	49.2135	1.65243	0.826213	+	0.563357i	$0.190490\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−50.4064	−1.68963
$891$	0	0
$892$	−2.44971	−0.0820223
$893$	−13.7259	−0.459319
$894$	0	0
$895$	13.3487	0.446199
$896$	6.01402	0.200914
$897$	0	0
$898$	36.1348	1.20583
$899$	−3.58320	−0.119507
$900$	0	0
$901$	59.7188	1.98952
$902$	41.7687	1.39075
$903$	0	0
$904$	13.2959	0.442216
$905$	−81.3699	−2.70483
$906$	0	0
$907$	50.4233	1.67428	0.837139	−	0.546991i	$-0.184227\pi$
$907$	50.4233	1.67428	0.837139	+	0.546991i	$0.184227\pi$
$908$	−5.24151	−0.173946
$909$	0	0
$910$	−19.0292	−0.630811
$911$	−19.4045	−0.642899	−0.321449	−	0.946927i	$-0.604170\pi$
$911$	−19.4045	−0.642899	−0.321449	+	0.946927i	$0.604170\pi$
$912$	0	0
$913$	1.43452	0.0474756
$914$	31.7157	1.04906
$915$	0	0
$916$	0.855620	0.0282705
$917$	−8.71812	−0.287898
$918$	0	0
$919$	−38.9767	−1.28572	−0.642861	−	0.765983i	$-0.722253\pi$
$919$	−38.9767	−1.28572	−0.642861	+	0.765983i	$0.722253\pi$
$920$	51.1947	1.68784
$921$	0	0
$922$	13.4389	0.442586
$923$	45.3992	1.49433
$924$	0	0
$925$	−64.4468	−2.11900
$926$	7.44179	0.244552
$927$	0	0
$928$	−9.32962	−0.306260
$929$	2.48794	0.0816266	0.0408133	−	0.999167i	$-0.487005\pi$
$929$	2.48794	0.0816266	0.0408133	+	0.999167i	$0.487005\pi$
$930$	0	0
$931$	1.86822	0.0612286
$932$	6.17878	0.202393
$933$	0	0
$934$	−52.1433	−1.70618
$935$	−51.2975	−1.67761
$936$	0	0
$937$	42.9656	1.40362	0.701812	−	0.712362i	$-0.252375\pi$
$937$	42.9656	1.40362	0.701812	+	0.712362i	$0.252375\pi$
$938$	−17.4408	−0.569463
$939$	0	0
$940$	11.1563	0.363878
$941$	−49.3601	−1.60909	−0.804547	−	0.593889i	$-0.797592\pi$
$941$	−49.3601	−1.60909	−0.804547	+	0.593889i	$0.797592\pi$
$942$	0	0
$943$	58.1711	1.89431
$944$	−38.4442	−1.25125
$945$	0	0
$946$	−9.20843	−0.299392
$947$	7.03435	0.228586	0.114293	−	0.993447i	$-0.463540\pi$
$947$	7.03435	0.228586	0.114293	+	0.993447i	$0.463540\pi$
$948$	0	0
$949$	11.8130	0.383468
$950$	14.1993	0.460686
$951$	0	0
$952$	−16.1507	−0.523448
$953$	−12.7667	−0.413553	−0.206777	−	0.978388i	$-0.566297\pi$
$953$	−12.7667	−0.413553	−0.206777	+	0.978388i	$0.566297\pi$
$954$	0	0
$955$	75.2389	2.43467
$956$	10.2303	0.330872
$957$	0	0
$958$	11.9941	0.387512
$959$	4.66285	0.150571
$960$	0	0
$961$	−30.0618	−0.969736
$962$	60.1476	1.93924
$963$	0	0
$964$	−5.47858	−0.176453
$965$	69.1249	2.22521
$966$	0	0
$967$	53.8825	1.73274	0.866372	−	0.499400i	$-0.166446\pi$
$967$	53.8825	1.73274	0.866372	+	0.499400i	$0.166446\pi$
$968$	7.77671	0.249953
$969$	0	0
$970$	−60.9120	−1.95577
$971$	55.2100	1.77177	0.885887	−	0.463901i	$-0.153551\pi$
$971$	55.2100	1.77177	0.885887	+	0.463901i	$0.153551\pi$
$972$	0	0
$973$	−19.7441	−0.632965
$974$	−22.0210	−0.705599
$975$	0	0
$976$	−1.92127	−0.0614982
$977$	38.3352	1.22645	0.613226	−	0.789908i	$-0.289872\pi$
$977$	38.3352	1.22645	0.613226	+	0.789908i	$0.289872\pi$
$978$	0	0
$979$	35.3662	1.13031
$980$	−1.51848	−0.0485060
$981$	0	0
$982$	6.18415	0.197344
$983$	19.8976	0.634635	0.317317	−	0.948319i	$-0.397218\pi$
$983$	19.8976	0.634635	0.317317	+	0.948319i	$0.397218\pi$
$984$	0	0
$985$	12.5948	0.401303
$986$	−24.3326	−0.774909
$987$	0	0
$988$	3.90774	0.124322
$989$	−12.8245	−0.407797
$990$	0	0
$991$	5.55649	0.176508	0.0882538	−	0.996098i	$-0.471871\pi$
$991$	5.55649	0.176508	0.0882538	+	0.996098i	$0.471871\pi$
$992$	2.44275	0.0775573
$993$	0	0
$994$	−12.2856	−0.389675
$995$	−32.4588	−1.02901
$996$	0	0
$997$	4.32640	0.137019	0.0685093	−	0.997650i	$-0.478176\pi$
$997$	4.32640	0.137019	0.0685093	+	0.997650i	$0.478176\pi$
$998$	54.7644	1.73354
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.l.1.5		7
3.2	odd	2		2667.2.a.j.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.j.1.3	✓	7	3.2	odd	2
8001.2.a.l.1.5		7	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.24280 q^{2} -0.455452 q^{4} +3.33400 q^{5} +1.00000 q^{7} -3.05163 q^{8} +O(q^{10})\)	\(q+1.24280 q^{2} -0.455452 q^{4} +3.33400 q^{5} +1.00000 q^{7} -3.05163 q^{8} +4.14349 q^{10} -2.90717 q^{11} -4.59254 q^{13} +1.24280 q^{14} -2.88166 q^{16} +5.29249 q^{17} +1.86822 q^{19} -1.51848 q^{20} -3.61302 q^{22} -5.03184 q^{23} +6.11557 q^{25} -5.70761 q^{26} -0.455452 q^{28} -3.69938 q^{29} +0.968597 q^{31} +2.52194 q^{32} +6.57750 q^{34} +3.33400 q^{35} -10.5381 q^{37} +2.32183 q^{38} -10.1741 q^{40} -11.5606 q^{41} +2.54868 q^{43} +1.32408 q^{44} -6.25356 q^{46} -7.34702 q^{47} +1.00000 q^{49} +7.60043 q^{50} +2.09168 q^{52} +11.2837 q^{53} -9.69250 q^{55} -3.05163 q^{56} -4.59758 q^{58} +13.3410 q^{59} +0.666722 q^{61} +1.20377 q^{62} +8.89758 q^{64} -15.3116 q^{65} -14.0335 q^{67} -2.41048 q^{68} +4.14349 q^{70} -9.88541 q^{71} -2.57222 q^{73} -13.0968 q^{74} -0.850887 q^{76} -2.90717 q^{77} +14.5706 q^{79} -9.60746 q^{80} -14.3675 q^{82} -0.493442 q^{83} +17.6452 q^{85} +3.16749 q^{86} +8.87160 q^{88} -12.1652 q^{89} -4.59254 q^{91} +2.29176 q^{92} -9.13087 q^{94} +6.22866 q^{95} -14.7006 q^{97} +1.24280 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8}+O(q^{10}) \) 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8	\( 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8} + 3 q^{11} - 23 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} - 9 q^{19} + 9 q^{20} - 19 q^{22} - 12 q^{23} + 3 q^{25} - 18 q^{26} + 4 q^{28} + 9 q^{29} - 33 q^{31} - 10 q^{32} - 2 q^{34} + 8 q^{35} - 33 q^{37} + 3 q^{38} - 9 q^{40} + 3 q^{41} - 9 q^{43} - 2 q^{44} - 32 q^{46} - 11 q^{47} + 7 q^{49} - 29 q^{50} - 21 q^{52} - q^{53} - 16 q^{55} + 9 q^{56} - 5 q^{58} + 30 q^{59} - 19 q^{61} - 3 q^{62} - 21 q^{64} - 14 q^{65} - 30 q^{67} - 24 q^{68} - 8 q^{71} - 20 q^{73} + 9 q^{74} - 42 q^{76} + 3 q^{77} + 8 q^{79} - 12 q^{80} + 10 q^{82} + 34 q^{83} - 28 q^{85} - 24 q^{86} - q^{88} + 12 q^{89} - 23 q^{91} - 60 q^{92} - 3 q^{94} - 12 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8 + 3 * q^11 - 23 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 - 9 * q^19 + 9 * q^20 - 19 * q^22 - 12 * q^23 + 3 * q^25 - 18 * q^26 + 4 * q^28 + 9 * q^29 - 33 * q^31 - 10 * q^32 - 2 * q^34 + 8 * q^35 - 33 * q^37 + 3 * q^38 - 9 * q^40 + 3 * q^41 - 9 * q^43 - 2 * q^44 - 32 * q^46 - 11 * q^47 + 7 * q^49 - 29 * q^50 - 21 * q^52 - q^53 - 16 * q^55 + 9 * q^56 - 5 * q^58 + 30 * q^59 - 19 * q^61 - 3 * q^62 - 21 * q^64 - 14 * q^65 - 30 * q^67 - 24 * q^68 - 8 * q^71 - 20 * q^73 + 9 * q^74 - 42 * q^76 + 3 * q^77 + 8 * q^79 - 12 * q^80 + 10 * q^82 + 34 * q^83 - 28 * q^85 - 24 * q^86 - q^88 + 12 * q^89 - 23 * q^91 - 60 * q^92 - 3 * q^94 - 12 * q^95 + 7 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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