LMFDB - Embedded newform 8046.2.a.r.1.12

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:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + \cdots - 7083 $ x^14 - 2x^13 - 42x^12 + 70x^11 + 680x^10 - 882x^9 - 5559x^8 + 5066x^7 + 24455x^6 - 12990x^5 - 55580x^4 + 9808x^3 + 53551x^2 + 6282*x - 7083
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$2.95487$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.95487 q^{5} +2.46629 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.95487 q^{5} +2.46629 q^{7} +1.00000 q^{8} +2.95487 q^{10} -0.814872 q^{11} +5.11356 q^{13} +2.46629 q^{14} +1.00000 q^{16} +3.17081 q^{17} -4.28273 q^{19} +2.95487 q^{20} -0.814872 q^{22} +3.02875 q^{23} +3.73123 q^{25} +5.11356 q^{26} +2.46629 q^{28} +0.136682 q^{29} +4.98954 q^{31} +1.00000 q^{32} +3.17081 q^{34} +7.28755 q^{35} +8.76535 q^{37} -4.28273 q^{38} +2.95487 q^{40} -10.5302 q^{41} -11.8541 q^{43} -0.814872 q^{44} +3.02875 q^{46} +6.64266 q^{47} -0.917427 q^{49} +3.73123 q^{50} +5.11356 q^{52} -10.5309 q^{53} -2.40784 q^{55} +2.46629 q^{56} +0.136682 q^{58} +1.90444 q^{59} +13.2595 q^{61} +4.98954 q^{62} +1.00000 q^{64} +15.1099 q^{65} -7.86487 q^{67} +3.17081 q^{68} +7.28755 q^{70} +5.04717 q^{71} -2.80629 q^{73} +8.76535 q^{74} -4.28273 q^{76} -2.00971 q^{77} +12.5831 q^{79} +2.95487 q^{80} -10.5302 q^{82} +11.6904 q^{83} +9.36930 q^{85} -11.8541 q^{86} -0.814872 q^{88} -10.7261 q^{89} +12.6115 q^{91} +3.02875 q^{92} +6.64266 q^{94} -12.6549 q^{95} -4.75485 q^{97} -0.917427 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8}+O(q^{10}) $ 14 * q + 14 * q^2 + 14 * q^4 + 2 * q^5 + 4 * q^7 + 14 * q^8	$ 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{13} + 4 q^{14} + 14 q^{16} + 9 q^{17} + 14 q^{19} + 2 q^{20} + 2 q^{22} + 30 q^{23} + 18 q^{25} + 4 q^{26} + 4 q^{28} + 6 q^{29} + 11 q^{31} + 14 q^{32} + 9 q^{34} - 18 q^{35} + 13 q^{37} + 14 q^{38} + 2 q^{40} - 2 q^{41} + 12 q^{43} + 2 q^{44} + 30 q^{46} + 21 q^{47} + 32 q^{49} + 18 q^{50} + 4 q^{52} + 22 q^{53} - 7 q^{55} + 4 q^{56} + 6 q^{58} + 14 q^{59} + 31 q^{61} + 11 q^{62} + 14 q^{64} + 24 q^{67} + 9 q^{68} - 18 q^{70} + 28 q^{71} + 24 q^{73} + 13 q^{74} + 14 q^{76} + 16 q^{77} + 65 q^{79} + 2 q^{80} - 2 q^{82} - 15 q^{83} - 19 q^{85} + 12 q^{86} + 2 q^{88} - 11 q^{89} + 68 q^{91} + 30 q^{92} + 21 q^{94} + 8 q^{95} + 23 q^{97} + 32 q^{98}+O(q^{100}) $ 14 * q + 14 * q^2 + 14 * q^4 + 2 * q^5 + 4 * q^7 + 14 * q^8 + 2 * q^10 + 2 * q^11 + 4 * q^13 + 4 * q^14 + 14 * q^16 + 9 * q^17 + 14 * q^19 + 2 * q^20 + 2 * q^22 + 30 * q^23 + 18 * q^25 + 4 * q^26 + 4 * q^28 + 6 * q^29 + 11 * q^31 + 14 * q^32 + 9 * q^34 - 18 * q^35 + 13 * q^37 + 14 * q^38 + 2 * q^40 - 2 * q^41 + 12 * q^43 + 2 * q^44 + 30 * q^46 + 21 * q^47 + 32 * q^49 + 18 * q^50 + 4 * q^52 + 22 * q^53 - 7 * q^55 + 4 * q^56 + 6 * q^58 + 14 * q^59 + 31 * q^61 + 11 * q^62 + 14 * q^64 + 24 * q^67 + 9 * q^68 - 18 * q^70 + 28 * q^71 + 24 * q^73 + 13 * q^74 + 14 * q^76 + 16 * q^77 + 65 * q^79 + 2 * q^80 - 2 * q^82 - 15 * q^83 - 19 * q^85 + 12 * q^86 + 2 * q^88 - 11 * q^89 + 68 * q^91 + 30 * q^92 + 21 * q^94 + 8 * q^95 + 23 * q^97 + 32 * 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
$5$	2.95487	1.32146	0.660728	−	0.750625i	$-0.270248\pi$
$5$	2.95487	1.32146	0.660728	+	0.750625i	$0.270248\pi$
$6$	0	0
$7$	2.46629	0.932169	0.466085	−	0.884740i	$-0.345664\pi$
$7$	2.46629	0.932169	0.466085	+	0.884740i	$0.345664\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.95487	0.934410
$11$	−0.814872	−0.245693	−0.122847	−	0.992426i	$-0.539202\pi$
$11$	−0.814872	−0.245693	−0.122847	+	0.992426i	$0.539202\pi$
$12$	0	0
$13$	5.11356	1.41825	0.709123	−	0.705085i	$-0.249091\pi$
$13$	5.11356	1.41825	0.709123	+	0.705085i	$0.249091\pi$
$14$	2.46629	0.659143
$15$	0	0
$16$	1.00000	0.250000
$17$	3.17081	0.769033	0.384517	−	0.923118i	$-0.374368\pi$
$17$	3.17081	0.769033	0.384517	+	0.923118i	$0.374368\pi$
$18$	0	0
$19$	−4.28273	−0.982526	−0.491263	−	0.871011i	$-0.663465\pi$
$19$	−4.28273	−0.982526	−0.491263	+	0.871011i	$0.663465\pi$
$20$	2.95487	0.660728
$21$	0	0
$22$	−0.814872	−0.173731
$23$	3.02875	0.631537	0.315769	−	0.948836i	$-0.397738\pi$
$23$	3.02875	0.631537	0.315769	+	0.948836i	$0.397738\pi$
$24$	0	0
$25$	3.73123	0.746246
$26$	5.11356	1.00285
$27$	0	0
$28$	2.46629	0.466085
$29$	0.136682	0.0253812	0.0126906	−	0.999919i	$-0.495960\pi$
$29$	0.136682	0.0253812	0.0126906	+	0.999919i	$0.495960\pi$
$30$	0	0
$31$	4.98954	0.896148	0.448074	−	0.893997i	$-0.352110\pi$
$31$	4.98954	0.896148	0.448074	+	0.893997i	$0.352110\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.17081	0.543789
$35$	7.28755	1.23182
$36$	0	0
$37$	8.76535	1.44101	0.720507	−	0.693447i	$-0.243909\pi$
$37$	8.76535	1.44101	0.720507	+	0.693447i	$0.243909\pi$
$38$	−4.28273	−0.694751
$39$	0	0
$40$	2.95487	0.467205
$41$	−10.5302	−1.64454	−0.822270	−	0.569097i	$-0.807293\pi$
$41$	−10.5302	−1.64454	−0.822270	+	0.569097i	$0.807293\pi$
$42$	0	0
$43$	−11.8541	−1.80774	−0.903869	−	0.427809i	$-0.859285\pi$
$43$	−11.8541	−1.80774	−0.903869	+	0.427809i	$0.859285\pi$
$44$	−0.814872	−0.122847
$45$	0	0
$46$	3.02875	0.446564
$47$	6.64266	0.968932	0.484466	−	0.874810i	$-0.339014\pi$
$47$	6.64266	0.968932	0.484466	+	0.874810i	$0.339014\pi$
$48$	0	0
$49$	−0.917427	−0.131061
$50$	3.73123	0.527675
$51$	0	0
$52$	5.11356	0.709123
$53$	−10.5309	−1.44653	−0.723263	−	0.690572i	$-0.757359\pi$
$53$	−10.5309	−1.44653	−0.723263	+	0.690572i	$0.757359\pi$
$54$	0	0
$55$	−2.40784	−0.324672
$56$	2.46629	0.329572
$57$	0	0
$58$	0.136682	0.0179472
$59$	1.90444	0.247937	0.123969	−	0.992286i	$-0.460438\pi$
$59$	1.90444	0.247937	0.123969	+	0.992286i	$0.460438\pi$
$60$	0	0
$61$	13.2595	1.69771	0.848855	−	0.528626i	$-0.177293\pi$
$61$	13.2595	1.69771	0.848855	+	0.528626i	$0.177293\pi$
$62$	4.98954	0.633672
$63$	0	0
$64$	1.00000	0.125000
$65$	15.1099	1.87415
$66$	0	0
$67$	−7.86487	−0.960847	−0.480423	−	0.877037i	$-0.659517\pi$
$67$	−7.86487	−0.960847	−0.480423	+	0.877037i	$0.659517\pi$
$68$	3.17081	0.384517
$69$	0	0
$70$	7.28755	0.871028
$71$	5.04717	0.598989	0.299495	−	0.954098i	$-0.403182\pi$
$71$	5.04717	0.598989	0.299495	+	0.954098i	$0.403182\pi$
$72$	0	0
$73$	−2.80629	−0.328452	−0.164226	−	0.986423i	$-0.552513\pi$
$73$	−2.80629	−0.328452	−0.164226	+	0.986423i	$0.552513\pi$
$74$	8.76535	1.01895
$75$	0	0
$76$	−4.28273	−0.491263
$77$	−2.00971	−0.229027
$78$	0	0
$79$	12.5831	1.41571	0.707857	−	0.706356i	$-0.249662\pi$
$79$	12.5831	1.41571	0.707857	+	0.706356i	$0.249662\pi$
$80$	2.95487	0.330364
$81$	0	0
$82$	−10.5302	−1.16287
$83$	11.6904	1.28319	0.641594	−	0.767045i	$-0.278274\pi$
$83$	11.6904	1.28319	0.641594	+	0.767045i	$0.278274\pi$
$84$	0	0
$85$	9.36930	1.01624
$86$	−11.8541	−1.27826
$87$	0	0
$88$	−0.814872	−0.0868656
$89$	−10.7261	−1.13696	−0.568480	−	0.822697i	$-0.692468\pi$
$89$	−10.7261	−1.13696	−0.568480	+	0.822697i	$0.692468\pi$
$90$	0	0
$91$	12.6115	1.32204
$92$	3.02875	0.315769
$93$	0	0
$94$	6.64266	0.685138
$95$	−12.6549	−1.29836
$96$	0	0
$97$	−4.75485	−0.482782	−0.241391	−	0.970428i	$-0.577604\pi$
$97$	−4.75485	−0.482782	−0.241391	+	0.970428i	$0.577604\pi$
$98$	−0.917427	−0.0926741
$99$	0	0
$100$	3.73123	0.373123
$101$	−19.1880	−1.90928	−0.954640	−	0.297763i	$-0.903759\pi$
$101$	−19.1880	−1.90928	−0.954640	+	0.297763i	$0.903759\pi$
$102$	0	0
$103$	1.55792	0.153507	0.0767533	−	0.997050i	$-0.475545\pi$
$103$	1.55792	0.153507	0.0767533	+	0.997050i	$0.475545\pi$
$104$	5.11356	0.501425
$105$	0	0
$106$	−10.5309	−1.02285
$107$	−2.86752	−0.277214	−0.138607	−	0.990347i	$-0.544262\pi$
$107$	−2.86752	−0.277214	−0.138607	+	0.990347i	$0.544262\pi$
$108$	0	0
$109$	−9.78165	−0.936913	−0.468456	−	0.883487i	$-0.655190\pi$
$109$	−9.78165	−0.936913	−0.468456	+	0.883487i	$0.655190\pi$
$110$	−2.40784	−0.229578
$111$	0	0
$112$	2.46629	0.233042
$113$	−14.9223	−1.40377	−0.701886	−	0.712289i	$-0.747658\pi$
$113$	−14.9223	−1.40377	−0.701886	+	0.712289i	$0.747658\pi$
$114$	0	0
$115$	8.94954	0.834549
$116$	0.136682	0.0126906
$117$	0	0
$118$	1.90444	0.175318
$119$	7.82012	0.716869
$120$	0	0
$121$	−10.3360	−0.939635
$122$	13.2595	1.20046
$123$	0	0
$124$	4.98954	0.448074
$125$	−3.74905	−0.335325
$126$	0	0
$127$	5.75872	0.511004	0.255502	−	0.966809i	$-0.417759\pi$
$127$	5.75872	0.511004	0.255502	+	0.966809i	$0.417759\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	15.1099	1.32522
$131$	16.6424	1.45405	0.727026	−	0.686609i	$-0.240902\pi$
$131$	16.6424	1.45405	0.727026	+	0.686609i	$0.240902\pi$
$132$	0	0
$133$	−10.5624	−0.915880
$134$	−7.86487	−0.679421
$135$	0	0
$136$	3.17081	0.271894
$137$	2.95327	0.252315	0.126157	−	0.992010i	$-0.459736\pi$
$137$	2.95327	0.252315	0.126157	+	0.992010i	$0.459736\pi$
$138$	0	0
$139$	1.33001	0.112810	0.0564048	−	0.998408i	$-0.482036\pi$
$139$	1.33001	0.112810	0.0564048	+	0.998408i	$0.482036\pi$
$140$	7.28755	0.615910
$141$	0	0
$142$	5.04717	0.423549
$143$	−4.16689	−0.348453
$144$	0	0
$145$	0.403876	0.0335401
$146$	−2.80629	−0.232250
$147$	0	0
$148$	8.76535	0.720507
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−7.77058	−0.632361	−0.316180	−	0.948699i	$-0.602401\pi$
$151$	−7.77058	−0.632361	−0.316180	+	0.948699i	$0.602401\pi$
$152$	−4.28273	−0.347375
$153$	0	0
$154$	−2.00971	−0.161947
$155$	14.7434	1.18422
$156$	0	0
$157$	−22.0189	−1.75730	−0.878648	−	0.477470i	$-0.841554\pi$
$157$	−22.0189	−1.75730	−0.878648	+	0.477470i	$0.841554\pi$
$158$	12.5831	1.00106
$159$	0	0
$160$	2.95487	0.233603
$161$	7.46976	0.588700
$162$	0	0
$163$	18.5899	1.45607	0.728037	−	0.685538i	$-0.240433\pi$
$163$	18.5899	1.45607	0.728037	+	0.685538i	$0.240433\pi$
$164$	−10.5302	−0.822270
$165$	0	0
$166$	11.6904	0.907351
$167$	14.3337	1.10918	0.554589	−	0.832125i	$-0.312876\pi$
$167$	14.3337	1.10918	0.554589	+	0.832125i	$0.312876\pi$
$168$	0	0
$169$	13.1485	1.01142
$170$	9.36930	0.718593
$171$	0	0
$172$	−11.8541	−0.903869
$173$	0.492601	0.0374517	0.0187259	−	0.999825i	$-0.494039\pi$
$173$	0.492601	0.0374517	0.0187259	+	0.999825i	$0.494039\pi$
$174$	0	0
$175$	9.20228	0.695627
$176$	−0.814872	−0.0614233
$177$	0	0
$178$	−10.7261	−0.803952
$179$	20.2737	1.51533	0.757664	−	0.652645i	$-0.226341\pi$
$179$	20.2737	1.51533	0.757664	+	0.652645i	$0.226341\pi$
$180$	0	0
$181$	−2.10348	−0.156351	−0.0781753	−	0.996940i	$-0.524909\pi$
$181$	−2.10348	−0.156351	−0.0781753	+	0.996940i	$0.524909\pi$
$182$	12.6115	0.934826
$183$	0	0
$184$	3.02875	0.223282
$185$	25.9004	1.90424
$186$	0	0
$187$	−2.58380	−0.188946
$188$	6.64266	0.484466
$189$	0	0
$190$	−12.6549	−0.918082
$191$	−2.56166	−0.185355	−0.0926777	−	0.995696i	$-0.529543\pi$
$191$	−2.56166	−0.185355	−0.0926777	+	0.995696i	$0.529543\pi$
$192$	0	0
$193$	18.7674	1.35091	0.675453	−	0.737403i	$-0.263948\pi$
$193$	18.7674	1.35091	0.675453	+	0.737403i	$0.263948\pi$
$194$	−4.75485	−0.341379
$195$	0	0
$196$	−0.917427	−0.0655305
$197$	−4.83671	−0.344601	−0.172301	−	0.985044i	$-0.555120\pi$
$197$	−4.83671	−0.344601	−0.172301	+	0.985044i	$0.555120\pi$
$198$	0	0
$199$	−10.3657	−0.734805	−0.367403	−	0.930062i	$-0.619753\pi$
$199$	−10.3657	−0.734805	−0.367403	+	0.930062i	$0.619753\pi$
$200$	3.73123	0.263838
$201$	0	0
$202$	−19.1880	−1.35006
$203$	0.337097	0.0236595
$204$	0	0
$205$	−31.1153	−2.17319
$206$	1.55792	0.108546
$207$	0	0
$208$	5.11356	0.354561
$209$	3.48988	0.241400
$210$	0	0
$211$	−23.6423	−1.62760	−0.813800	−	0.581145i	$-0.802605\pi$
$211$	−23.6423	−1.62760	−0.813800	+	0.581145i	$0.802605\pi$
$212$	−10.5309	−0.723263
$213$	0	0
$214$	−2.86752	−0.196020
$215$	−35.0274	−2.38885
$216$	0	0
$217$	12.3056	0.835361
$218$	−9.78165	−0.662497
$219$	0	0
$220$	−2.40784	−0.162336
$221$	16.2141	1.09068
$222$	0	0
$223$	16.1009	1.07820	0.539098	−	0.842243i	$-0.318765\pi$
$223$	16.1009	1.07820	0.539098	+	0.842243i	$0.318765\pi$
$224$	2.46629	0.164786
$225$	0	0
$226$	−14.9223	−0.992617
$227$	16.9613	1.12576	0.562880	−	0.826539i	$-0.309693\pi$
$227$	16.9613	1.12576	0.562880	+	0.826539i	$0.309693\pi$
$228$	0	0
$229$	25.9949	1.71779	0.858897	−	0.512149i	$-0.171150\pi$
$229$	25.9949	1.71779	0.858897	+	0.512149i	$0.171150\pi$
$230$	8.94954	0.590115
$231$	0	0
$232$	0.136682	0.00897360
$233$	−9.06092	−0.593601	−0.296800	−	0.954940i	$-0.595920\pi$
$233$	−9.06092	−0.593601	−0.296800	+	0.954940i	$0.595920\pi$
$234$	0	0
$235$	19.6282	1.28040
$236$	1.90444	0.123969
$237$	0	0
$238$	7.82012	0.506903
$239$	−20.1593	−1.30399	−0.651997	−	0.758222i	$-0.726069\pi$
$239$	−20.1593	−1.30399	−0.651997	+	0.758222i	$0.726069\pi$
$240$	0	0
$241$	−25.7413	−1.65815	−0.829073	−	0.559141i	$-0.811131\pi$
$241$	−25.7413	−1.65815	−0.829073	+	0.559141i	$0.811131\pi$
$242$	−10.3360	−0.664422
$243$	0	0
$244$	13.2595	0.848855
$245$	−2.71087	−0.173191
$246$	0	0
$247$	−21.9000	−1.39346
$248$	4.98954	0.316836
$249$	0	0
$250$	−3.74905	−0.237111
$251$	−26.9096	−1.69852	−0.849259	−	0.527976i	$-0.822951\pi$
$251$	−26.9096	−1.69852	−0.849259	+	0.527976i	$0.822951\pi$
$252$	0	0
$253$	−2.46804	−0.155164
$254$	5.75872	0.361335
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.54377	0.158676	0.0793379	−	0.996848i	$-0.474719\pi$
$257$	2.54377	0.158676	0.0793379	+	0.996848i	$0.474719\pi$
$258$	0	0
$259$	21.6179	1.34327
$260$	15.1099	0.937074
$261$	0	0
$262$	16.6424	1.02817
$263$	−20.3203	−1.25301	−0.626503	−	0.779419i	$-0.715514\pi$
$263$	−20.3203	−1.25301	−0.626503	+	0.779419i	$0.715514\pi$
$264$	0	0
$265$	−31.1173	−1.91152
$266$	−10.5624	−0.647625
$267$	0	0
$268$	−7.86487	−0.480423
$269$	−14.3982	−0.877874	−0.438937	−	0.898518i	$-0.644645\pi$
$269$	−14.3982	−0.877874	−0.438937	+	0.898518i	$0.644645\pi$
$270$	0	0
$271$	14.9393	0.907497	0.453749	−	0.891130i	$-0.350086\pi$
$271$	14.9393	0.907497	0.453749	+	0.891130i	$0.350086\pi$
$272$	3.17081	0.192258
$273$	0	0
$274$	2.95327	0.178414
$275$	−3.04047	−0.183347
$276$	0	0
$277$	19.0226	1.14296	0.571479	−	0.820616i	$-0.306370\pi$
$277$	19.0226	1.14296	0.571479	+	0.820616i	$0.306370\pi$
$278$	1.33001	0.0797685
$279$	0	0
$280$	7.28755	0.435514
$281$	−27.7138	−1.65327	−0.826633	−	0.562742i	$-0.809746\pi$
$281$	−27.7138	−1.65327	−0.826633	+	0.562742i	$0.809746\pi$
$282$	0	0
$283$	−10.5380	−0.626416	−0.313208	−	0.949684i	$-0.601404\pi$
$283$	−10.5380	−0.626416	−0.313208	+	0.949684i	$0.601404\pi$
$284$	5.04717	0.299495
$285$	0	0
$286$	−4.16689	−0.246393
$287$	−25.9705	−1.53299
$288$	0	0
$289$	−6.94599	−0.408588
$290$	0.403876	0.0237164
$291$	0	0
$292$	−2.80629	−0.164226
$293$	−24.8169	−1.44982	−0.724910	−	0.688844i	$-0.758119\pi$
$293$	−24.8169	−1.44982	−0.724910	+	0.688844i	$0.758119\pi$
$294$	0	0
$295$	5.62737	0.327638
$296$	8.76535	0.509476
$297$	0	0
$298$	1.00000	0.0579284
$299$	15.4877	0.895675
$300$	0	0
$301$	−29.2357	−1.68512
$302$	−7.77058	−0.447147
$303$	0	0
$304$	−4.28273	−0.245631
$305$	39.1801	2.24345
$306$	0	0
$307$	20.7304	1.18315	0.591573	−	0.806251i	$-0.298507\pi$
$307$	20.7304	1.18315	0.591573	+	0.806251i	$0.298507\pi$
$308$	−2.00971	−0.114514
$309$	0	0
$310$	14.7434	0.837370
$311$	12.2882	0.696802	0.348401	−	0.937346i	$-0.386725\pi$
$311$	12.2882	0.696802	0.348401	+	0.937346i	$0.386725\pi$
$312$	0	0
$313$	3.12802	0.176806	0.0884031	−	0.996085i	$-0.471824\pi$
$313$	3.12802	0.176806	0.0884031	+	0.996085i	$0.471824\pi$
$314$	−22.0189	−1.24260
$315$	0	0
$316$	12.5831	0.707857
$317$	11.5404	0.648175	0.324088	−	0.946027i	$-0.394943\pi$
$317$	11.5404	0.648175	0.324088	+	0.946027i	$0.394943\pi$
$318$	0	0
$319$	−0.111378	−0.00623598
$320$	2.95487	0.165182
$321$	0	0
$322$	7.46976	0.416273
$323$	−13.5797	−0.755595
$324$	0	0
$325$	19.0798	1.05836
$326$	18.5899	1.02960
$327$	0	0
$328$	−10.5302	−0.581433
$329$	16.3827	0.903208
$330$	0	0
$331$	25.9638	1.42710	0.713550	−	0.700605i	$-0.247086\pi$
$331$	25.9638	1.42710	0.713550	+	0.700605i	$0.247086\pi$
$332$	11.6904	0.641594
$333$	0	0
$334$	14.3337	0.784307
$335$	−23.2396	−1.26972
$336$	0	0
$337$	−7.37140	−0.401546	−0.200773	−	0.979638i	$-0.564345\pi$
$337$	−7.37140	−0.401546	−0.200773	+	0.979638i	$0.564345\pi$
$338$	13.1485	0.715182
$339$	0	0
$340$	9.36930	0.508122
$341$	−4.06583	−0.220177
$342$	0	0
$343$	−19.5266	−1.05434
$344$	−11.8541	−0.639132
$345$	0	0
$346$	0.492601	0.0264824
$347$	5.74761	0.308548	0.154274	−	0.988028i	$-0.450696\pi$
$347$	5.74761	0.308548	0.154274	+	0.988028i	$0.450696\pi$
$348$	0	0
$349$	27.1115	1.45125	0.725624	−	0.688092i	$-0.241551\pi$
$349$	27.1115	1.45125	0.725624	+	0.688092i	$0.241551\pi$
$350$	9.20228	0.491883
$351$	0	0
$352$	−0.814872	−0.0434328
$353$	34.0774	1.81376	0.906878	−	0.421394i	$-0.138459\pi$
$353$	34.0774	1.81376	0.906878	+	0.421394i	$0.138459\pi$
$354$	0	0
$355$	14.9137	0.791538
$356$	−10.7261	−0.568480
$357$	0	0
$358$	20.2737	1.07150
$359$	−1.67544	−0.0884265	−0.0442133	−	0.999022i	$-0.514078\pi$
$359$	−1.67544	−0.0884265	−0.0442133	+	0.999022i	$0.514078\pi$
$360$	0	0
$361$	−0.658214	−0.0346429
$362$	−2.10348	−0.110557
$363$	0	0
$364$	12.6115	0.661022
$365$	−8.29221	−0.434034
$366$	0	0
$367$	−2.42612	−0.126642	−0.0633212	−	0.997993i	$-0.520169\pi$
$367$	−2.42612	−0.126642	−0.0633212	+	0.997993i	$0.520169\pi$
$368$	3.02875	0.157884
$369$	0	0
$370$	25.9004	1.34650
$371$	−25.9722	−1.34841
$372$	0	0
$373$	−2.93436	−0.151935	−0.0759676	−	0.997110i	$-0.524205\pi$
$373$	−2.93436	−0.151935	−0.0759676	+	0.997110i	$0.524205\pi$
$374$	−2.58380	−0.133605
$375$	0	0
$376$	6.64266	0.342569
$377$	0.698930	0.0359967
$378$	0	0
$379$	−2.60784	−0.133956	−0.0669778	−	0.997754i	$-0.521336\pi$
$379$	−2.60784	−0.133956	−0.0669778	+	0.997754i	$0.521336\pi$
$380$	−12.6549	−0.649182
$381$	0	0
$382$	−2.56166	−0.131066
$383$	−7.68419	−0.392644	−0.196322	−	0.980540i	$-0.562900\pi$
$383$	−7.68419	−0.392644	−0.196322	+	0.980540i	$0.562900\pi$
$384$	0	0
$385$	−5.93841	−0.302650
$386$	18.7674	0.955235
$387$	0	0
$388$	−4.75485	−0.241391
$389$	−23.8283	−1.20814	−0.604072	−	0.796930i	$-0.706456\pi$
$389$	−23.8283	−1.20814	−0.604072	+	0.796930i	$0.706456\pi$
$390$	0	0
$391$	9.60357	0.485673
$392$	−0.917427	−0.0463370
$393$	0	0
$394$	−4.83671	−0.243670
$395$	37.1815	1.87080
$396$	0	0
$397$	9.60761	0.482192	0.241096	−	0.970501i	$-0.422493\pi$
$397$	9.60761	0.482192	0.241096	+	0.970501i	$0.422493\pi$
$398$	−10.3657	−0.519586
$399$	0	0
$400$	3.73123	0.186561
$401$	−30.3133	−1.51377	−0.756887	−	0.653546i	$-0.773281\pi$
$401$	−30.3133	−1.51377	−0.756887	+	0.653546i	$0.773281\pi$
$402$	0	0
$403$	25.5143	1.27096
$404$	−19.1880	−0.954640
$405$	0	0
$406$	0.337097	0.0167298
$407$	−7.14263	−0.354047
$408$	0	0
$409$	−15.3506	−0.759039	−0.379520	−	0.925184i	$-0.623911\pi$
$409$	−15.3506	−0.759039	−0.379520	+	0.925184i	$0.623911\pi$
$410$	−31.1153	−1.53668
$411$	0	0
$412$	1.55792	0.0767533
$413$	4.69690	0.231120
$414$	0	0
$415$	34.5435	1.69568
$416$	5.11356	0.250713
$417$	0	0
$418$	3.48988	0.170695
$419$	1.08706	0.0531064	0.0265532	−	0.999647i	$-0.491547\pi$
$419$	1.08706	0.0531064	0.0265532	+	0.999647i	$0.491547\pi$
$420$	0	0
$421$	−3.91425	−0.190769	−0.0953845	−	0.995441i	$-0.530408\pi$
$421$	−3.91425	−0.190769	−0.0953845	+	0.995441i	$0.530408\pi$
$422$	−23.6423	−1.15089
$423$	0	0
$424$	−10.5309	−0.511424
$425$	11.8310	0.573888
$426$	0	0
$427$	32.7018	1.58255
$428$	−2.86752	−0.138607
$429$	0	0
$430$	−35.0274	−1.68917
$431$	−11.9555	−0.575877	−0.287938	−	0.957649i	$-0.592970\pi$
$431$	−11.9555	−0.575877	−0.287938	+	0.957649i	$0.592970\pi$
$432$	0	0
$433$	27.0459	1.29974	0.649871	−	0.760045i	$-0.274823\pi$
$433$	27.0459	1.29974	0.649871	+	0.760045i	$0.274823\pi$
$434$	12.3056	0.590689
$435$	0	0
$436$	−9.78165	−0.468456
$437$	−12.9713	−0.620502
$438$	0	0
$439$	19.2418	0.918363	0.459181	−	0.888342i	$-0.348143\pi$
$439$	19.2418	0.918363	0.459181	+	0.888342i	$0.348143\pi$
$440$	−2.40784	−0.114789
$441$	0	0
$442$	16.2141	0.771226
$443$	−35.5408	−1.68859	−0.844297	−	0.535876i	$-0.819981\pi$
$443$	−35.5408	−1.68859	−0.844297	+	0.535876i	$0.819981\pi$
$444$	0	0
$445$	−31.6940	−1.50244
$446$	16.1009	0.762400
$447$	0	0
$448$	2.46629	0.116521
$449$	27.8674	1.31514	0.657572	−	0.753391i	$-0.271583\pi$
$449$	27.8674	1.31514	0.657572	+	0.753391i	$0.271583\pi$
$450$	0	0
$451$	8.58076	0.404052
$452$	−14.9223	−0.701886
$453$	0	0
$454$	16.9613	0.796032
$455$	37.2653	1.74702
$456$	0	0
$457$	22.2773	1.04209	0.521043	−	0.853530i	$-0.325543\pi$
$457$	22.2773	1.04209	0.521043	+	0.853530i	$0.325543\pi$
$458$	25.9949	1.21466
$459$	0	0
$460$	8.94954	0.417274
$461$	−28.6522	−1.33447	−0.667233	−	0.744849i	$-0.732521\pi$
$461$	−28.6522	−1.33447	−0.667233	+	0.744849i	$0.732521\pi$
$462$	0	0
$463$	30.9993	1.44066	0.720331	−	0.693631i	$-0.243990\pi$
$463$	30.9993	1.44066	0.720331	+	0.693631i	$0.243990\pi$
$464$	0.136682	0.00634529
$465$	0	0
$466$	−9.06092	−0.419739
$467$	9.64691	0.446406	0.223203	−	0.974772i	$-0.428349\pi$
$467$	9.64691	0.446406	0.223203	+	0.974772i	$0.428349\pi$
$468$	0	0
$469$	−19.3970	−0.895672
$470$	19.6282	0.905380
$471$	0	0
$472$	1.90444	0.0876591
$473$	9.65959	0.444149
$474$	0	0
$475$	−15.9798	−0.733206
$476$	7.82012	0.358434
$477$	0	0
$478$	−20.1593	−0.922063
$479$	−3.24827	−0.148417	−0.0742086	−	0.997243i	$-0.523643\pi$
$479$	−3.24827	−0.148417	−0.0742086	+	0.997243i	$0.523643\pi$
$480$	0	0
$481$	44.8221	2.04371
$482$	−25.7413	−1.17249
$483$	0	0
$484$	−10.3360	−0.469817
$485$	−14.0500	−0.637975
$486$	0	0
$487$	26.7297	1.21124	0.605618	−	0.795755i	$-0.292926\pi$
$487$	26.7297	1.21124	0.605618	+	0.795755i	$0.292926\pi$
$488$	13.2595	0.600231
$489$	0	0
$490$	−2.71087	−0.122465
$491$	26.5883	1.19991	0.599956	−	0.800033i	$-0.295185\pi$
$491$	26.5883	1.19991	0.599956	+	0.800033i	$0.295185\pi$
$492$	0	0
$493$	0.433391	0.0195190
$494$	−21.9000	−0.985327
$495$	0	0
$496$	4.98954	0.224037
$497$	12.4478	0.558359
$498$	0	0
$499$	0.886424	0.0396818	0.0198409	−	0.999803i	$-0.493684\pi$
$499$	0.886424	0.0396818	0.0198409	+	0.999803i	$0.493684\pi$
$500$	−3.74905	−0.167663
$501$	0	0
$502$	−26.9096	−1.20103
$503$	−24.2698	−1.08214	−0.541068	−	0.840979i	$-0.681980\pi$
$503$	−24.2698	−1.08214	−0.541068	+	0.840979i	$0.681980\pi$
$504$	0	0
$505$	−56.6980	−2.52303
$506$	−2.46804	−0.109718
$507$	0	0
$508$	5.75872	0.255502
$509$	22.6022	1.00182	0.500912	−	0.865498i	$-0.332998\pi$
$509$	22.6022	1.00182	0.500912	+	0.865498i	$0.332998\pi$
$510$	0	0
$511$	−6.92112	−0.306172
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.54377	0.112201
$515$	4.60345	0.202852
$516$	0	0
$517$	−5.41292	−0.238060
$518$	21.6179	0.949835
$519$	0	0
$520$	15.1099	0.662612
$521$	−35.7817	−1.56762	−0.783812	−	0.620999i	$-0.786727\pi$
$521$	−35.7817	−1.56762	−0.783812	+	0.620999i	$0.786727\pi$
$522$	0	0
$523$	43.7790	1.91432	0.957160	−	0.289559i	$-0.0935088\pi$
$523$	43.7790	1.91432	0.957160	+	0.289559i	$0.0935088\pi$
$524$	16.6424	0.727026
$525$	0	0
$526$	−20.3203	−0.886008
$527$	15.8209	0.689167
$528$	0	0
$529$	−13.8267	−0.601161
$530$	−31.1173	−1.35165
$531$	0	0
$532$	−10.5624	−0.457940
$533$	−53.8467	−2.33236
$534$	0	0
$535$	−8.47314	−0.366326
$536$	−7.86487	−0.339711
$537$	0	0
$538$	−14.3982	−0.620751
$539$	0.747585	0.0322008
$540$	0	0
$541$	−9.54169	−0.410229	−0.205115	−	0.978738i	$-0.565757\pi$
$541$	−9.54169	−0.410229	−0.205115	+	0.978738i	$0.565757\pi$
$542$	14.9393	0.641697
$543$	0	0
$544$	3.17081	0.135947
$545$	−28.9035	−1.23809
$546$	0	0
$547$	−30.6787	−1.31172	−0.655862	−	0.754880i	$-0.727695\pi$
$547$	−30.6787	−1.31172	−0.655862	+	0.754880i	$0.727695\pi$
$548$	2.95327	0.126157
$549$	0	0
$550$	−3.04047	−0.129646
$551$	−0.585371	−0.0249377
$552$	0	0
$553$	31.0337	1.31969
$554$	19.0226	0.808194
$555$	0	0
$556$	1.33001	0.0564048
$557$	−1.79395	−0.0760122	−0.0380061	−	0.999278i	$-0.512101\pi$
$557$	−1.79395	−0.0760122	−0.0380061	+	0.999278i	$0.512101\pi$
$558$	0	0
$559$	−60.6168	−2.56382
$560$	7.28755	0.307955
$561$	0	0
$562$	−27.7138	−1.16904
$563$	33.3141	1.40402	0.702012	−	0.712165i	$-0.252285\pi$
$563$	33.3141	1.40402	0.702012	+	0.712165i	$0.252285\pi$
$564$	0	0
$565$	−44.0934	−1.85502
$566$	−10.5380	−0.442943
$567$	0	0
$568$	5.04717	0.211775
$569$	7.31103	0.306494	0.153247	−	0.988188i	$-0.451027\pi$
$569$	7.31103	0.306494	0.153247	+	0.988188i	$0.451027\pi$
$570$	0	0
$571$	−15.0441	−0.629578	−0.314789	−	0.949162i	$-0.601934\pi$
$571$	−15.0441	−0.629578	−0.314789	+	0.949162i	$0.601934\pi$
$572$	−4.16689	−0.174226
$573$	0	0
$574$	−25.9705	−1.08399
$575$	11.3009	0.471282
$576$	0	0
$577$	22.7885	0.948699	0.474349	−	0.880337i	$-0.342683\pi$
$577$	22.7885	0.948699	0.474349	+	0.880337i	$0.342683\pi$
$578$	−6.94599	−0.288915
$579$	0	0
$580$	0.403876	0.0167701
$581$	28.8319	1.19615
$582$	0	0
$583$	8.58131	0.355402
$584$	−2.80629	−0.116125
$585$	0	0
$586$	−24.8169	−1.02518
$587$	6.08972	0.251349	0.125675	−	0.992072i	$-0.459890\pi$
$587$	6.08972	0.251349	0.125675	+	0.992072i	$0.459890\pi$
$588$	0	0
$589$	−21.3689	−0.880488
$590$	5.62737	0.231675
$591$	0	0
$592$	8.76535	0.360254
$593$	−5.19414	−0.213298	−0.106649	−	0.994297i	$-0.534012\pi$
$593$	−5.19414	−0.213298	−0.106649	+	0.994297i	$0.534012\pi$
$594$	0	0
$595$	23.1074	0.947311
$596$	1.00000	0.0409616
$597$	0	0
$598$	15.4877	0.633338
$599$	−34.0879	−1.39279	−0.696397	−	0.717657i	$-0.745215\pi$
$599$	−34.0879	−1.39279	−0.696397	+	0.717657i	$0.745215\pi$
$600$	0	0
$601$	29.3161	1.19583	0.597915	−	0.801560i	$-0.295996\pi$
$601$	29.3161	1.19583	0.597915	+	0.801560i	$0.295996\pi$
$602$	−29.2357	−1.19156
$603$	0	0
$604$	−7.77058	−0.316180
$605$	−30.5414	−1.24169
$606$	0	0
$607$	43.1002	1.74939	0.874693	−	0.484678i	$-0.161063\pi$
$607$	43.1002	1.74939	0.874693	+	0.484678i	$0.161063\pi$
$608$	−4.28273	−0.173688
$609$	0	0
$610$	39.1801	1.58636
$611$	33.9676	1.37418
$612$	0	0
$613$	36.7723	1.48522	0.742610	−	0.669725i	$-0.233588\pi$
$613$	36.7723	1.48522	0.742610	+	0.669725i	$0.233588\pi$
$614$	20.7304	0.836611
$615$	0	0
$616$	−2.00971	−0.0809734
$617$	−9.72411	−0.391478	−0.195739	−	0.980656i	$-0.562711\pi$
$617$	−9.72411	−0.391478	−0.195739	+	0.980656i	$0.562711\pi$
$618$	0	0
$619$	−41.0184	−1.64867	−0.824335	−	0.566102i	$-0.808451\pi$
$619$	−41.0184	−1.64867	−0.824335	+	0.566102i	$0.808451\pi$
$620$	14.7434	0.592110
$621$	0	0
$622$	12.2882	0.492713
$623$	−26.4535	−1.05984
$624$	0	0
$625$	−29.7341	−1.18936
$626$	3.12802	0.125021
$627$	0	0
$628$	−22.0189	−0.878648
$629$	27.7932	1.10819
$630$	0	0
$631$	−24.9638	−0.993792	−0.496896	−	0.867810i	$-0.665527\pi$
$631$	−24.9638	−0.993792	−0.496896	+	0.867810i	$0.665527\pi$
$632$	12.5831	0.500531
$633$	0	0
$634$	11.5404	0.458329
$635$	17.0163	0.675269
$636$	0	0
$637$	−4.69131	−0.185877
$638$	−0.111378	−0.00440950
$639$	0	0
$640$	2.95487	0.116801
$641$	−8.64163	−0.341324	−0.170662	−	0.985330i	$-0.554591\pi$
$641$	−8.64163	−0.341324	−0.170662	+	0.985330i	$0.554591\pi$
$642$	0	0
$643$	12.8702	0.507552	0.253776	−	0.967263i	$-0.418327\pi$
$643$	12.8702	0.507552	0.253776	+	0.967263i	$0.418327\pi$
$644$	7.46976	0.294350
$645$	0	0
$646$	−13.5797	−0.534286
$647$	−6.91319	−0.271785	−0.135893	−	0.990724i	$-0.543390\pi$
$647$	−6.91319	−0.271785	−0.135893	+	0.990724i	$0.543390\pi$
$648$	0	0
$649$	−1.55188	−0.0609165
$650$	19.0798	0.748373
$651$	0	0
$652$	18.5899	0.728037
$653$	11.1149	0.434960	0.217480	−	0.976065i	$-0.430216\pi$
$653$	11.1149	0.434960	0.217480	+	0.976065i	$0.430216\pi$
$654$	0	0
$655$	49.1760	1.92147
$656$	−10.5302	−0.411135
$657$	0	0
$658$	16.3827	0.638665
$659$	−30.6387	−1.19351	−0.596756	−	0.802423i	$-0.703544\pi$
$659$	−30.6387	−1.19351	−0.596756	+	0.802423i	$0.703544\pi$
$660$	0	0
$661$	30.4804	1.18555	0.592776	−	0.805368i	$-0.298032\pi$
$661$	30.4804	1.18555	0.592776	+	0.805368i	$0.298032\pi$
$662$	25.9638	1.00911
$663$	0	0
$664$	11.6904	0.453675
$665$	−31.2106	−1.21030
$666$	0	0
$667$	0.413975	0.0160292
$668$	14.3337	0.554589
$669$	0	0
$670$	−23.2396	−0.897825
$671$	−10.8048	−0.417115
$672$	0	0
$673$	48.1223	1.85498	0.927490	−	0.373848i	$-0.121962\pi$
$673$	48.1223	1.85498	0.927490	+	0.373848i	$0.121962\pi$
$674$	−7.37140	−0.283936
$675$	0	0
$676$	13.1485	0.505710
$677$	34.6652	1.33229	0.666146	−	0.745821i	$-0.267943\pi$
$677$	34.6652	1.33229	0.666146	+	0.745821i	$0.267943\pi$
$678$	0	0
$679$	−11.7268	−0.450035
$680$	9.36930	0.359296
$681$	0	0
$682$	−4.06583	−0.155689
$683$	−0.608512	−0.0232841	−0.0116420	−	0.999932i	$-0.503706\pi$
$683$	−0.608512	−0.0232841	−0.0116420	+	0.999932i	$0.503706\pi$
$684$	0	0
$685$	8.72651	0.333423
$686$	−19.5266	−0.745531
$687$	0	0
$688$	−11.8541	−0.451934
$689$	−53.8502	−2.05153
$690$	0	0
$691$	−1.38991	−0.0528748	−0.0264374	−	0.999650i	$-0.508416\pi$
$691$	−1.38991	−0.0528748	−0.0264374	+	0.999650i	$0.508416\pi$
$692$	0.492601	0.0187259
$693$	0	0
$694$	5.74761	0.218176
$695$	3.92999	0.149073
$696$	0	0
$697$	−33.3892	−1.26471
$698$	27.1115	1.02619
$699$	0	0
$700$	9.20228	0.347813
$701$	38.2618	1.44513	0.722565	−	0.691303i	$-0.242963\pi$
$701$	38.2618	1.44513	0.722565	+	0.691303i	$0.242963\pi$
$702$	0	0
$703$	−37.5396	−1.41583
$704$	−0.814872	−0.0307116
$705$	0	0
$706$	34.0774	1.28252
$707$	−47.3232	−1.77977
$708$	0	0
$709$	−13.6003	−0.510772	−0.255386	−	0.966839i	$-0.582203\pi$
$709$	−13.6003	−0.510772	−0.255386	+	0.966839i	$0.582203\pi$
$710$	14.9137	0.559702
$711$	0	0
$712$	−10.7261	−0.401976
$713$	15.1121	0.565951
$714$	0	0
$715$	−12.3126	−0.460465
$716$	20.2737	0.757664
$717$	0	0
$718$	−1.67544	−0.0625270
$719$	−11.6490	−0.434436	−0.217218	−	0.976123i	$-0.569698\pi$
$719$	−11.6490	−0.434436	−0.217218	+	0.976123i	$0.569698\pi$
$720$	0	0
$721$	3.84228	0.143094
$722$	−0.658214	−0.0244962
$723$	0	0
$724$	−2.10348	−0.0781753
$725$	0.509991	0.0189406
$726$	0	0
$727$	30.1568	1.11846	0.559228	−	0.829014i	$-0.311098\pi$
$727$	30.1568	1.11846	0.559228	+	0.829014i	$0.311098\pi$
$728$	12.6115	0.467413
$729$	0	0
$730$	−8.29221	−0.306909
$731$	−37.5871	−1.39021
$732$	0	0
$733$	−16.4843	−0.608860	−0.304430	−	0.952535i	$-0.598466\pi$
$733$	−16.4843	−0.608860	−0.304430	+	0.952535i	$0.598466\pi$
$734$	−2.42612	−0.0895497
$735$	0	0
$736$	3.02875	0.111641
$737$	6.40886	0.236073
$738$	0	0
$739$	28.5601	1.05060	0.525300	−	0.850917i	$-0.323953\pi$
$739$	28.5601	1.05060	0.525300	+	0.850917i	$0.323953\pi$
$740$	25.9004	0.952119
$741$	0	0
$742$	−25.9722	−0.953468
$743$	7.46987	0.274043	0.137021	−	0.990568i	$-0.456247\pi$
$743$	7.46987	0.274043	0.137021	+	0.990568i	$0.456247\pi$
$744$	0	0
$745$	2.95487	0.108258
$746$	−2.93436	−0.107434
$747$	0	0
$748$	−2.58380	−0.0944731
$749$	−7.07213	−0.258410
$750$	0	0
$751$	−12.2998	−0.448827	−0.224413	−	0.974494i	$-0.572047\pi$
$751$	−12.2998	−0.448827	−0.224413	+	0.974494i	$0.572047\pi$
$752$	6.64266	0.242233
$753$	0	0
$754$	0.698930	0.0254535
$755$	−22.9610	−0.835637
$756$	0	0
$757$	10.1460	0.368763	0.184382	−	0.982855i	$-0.440972\pi$
$757$	10.1460	0.368763	0.184382	+	0.982855i	$0.440972\pi$
$758$	−2.60784	−0.0947210
$759$	0	0
$760$	−12.6549	−0.459041
$761$	36.0378	1.30637	0.653184	−	0.757199i	$-0.273433\pi$
$761$	36.0378	1.30637	0.653184	+	0.757199i	$0.273433\pi$
$762$	0	0
$763$	−24.1244	−0.873361
$764$	−2.56166	−0.0926777
$765$	0	0
$766$	−7.68419	−0.277641
$767$	9.73848	0.351636
$768$	0	0
$769$	−0.0930972	−0.00335717	−0.00167859	−	0.999999i	$-0.500534\pi$
$769$	−0.0930972	−0.00335717	−0.00167859	+	0.999999i	$0.500534\pi$
$770$	−5.93841	−0.214006
$771$	0	0
$772$	18.7674	0.675453
$773$	31.2361	1.12348	0.561742	−	0.827312i	$-0.310131\pi$
$773$	31.2361	1.12348	0.561742	+	0.827312i	$0.310131\pi$
$774$	0	0
$775$	18.6171	0.668746
$776$	−4.75485	−0.170689
$777$	0	0
$778$	−23.8283	−0.854287
$779$	45.0980	1.61580
$780$	0	0
$781$	−4.11280	−0.147167
$782$	9.60357	0.343423
$783$	0	0
$784$	−0.917427	−0.0327652
$785$	−65.0627	−2.32219
$786$	0	0
$787$	−48.8787	−1.74234	−0.871169	−	0.490983i	$-0.836638\pi$
$787$	−48.8787	−1.74234	−0.871169	+	0.490983i	$0.836638\pi$
$788$	−4.83671	−0.172301
$789$	0	0
$790$	37.1815	1.32286
$791$	−36.8027	−1.30855
$792$	0	0
$793$	67.8034	2.40777
$794$	9.60761	0.340961
$795$	0	0
$796$	−10.3657	−0.367403
$797$	6.53100	0.231340	0.115670	−	0.993288i	$-0.463099\pi$
$797$	6.53100	0.231340	0.115670	+	0.993288i	$0.463099\pi$
$798$	0	0
$799$	21.0626	0.745141
$800$	3.73123	0.131919
$801$	0	0
$802$	−30.3133	−1.07040
$803$	2.28677	0.0806983
$804$	0	0
$805$	22.0721	0.777940
$806$	25.5143	0.898702
$807$	0	0
$808$	−19.1880	−0.675032
$809$	−33.4029	−1.17438	−0.587191	−	0.809449i	$-0.699766\pi$
$809$	−33.4029	−1.17438	−0.587191	+	0.809449i	$0.699766\pi$
$810$	0	0
$811$	43.8883	1.54113	0.770564	−	0.637363i	$-0.219975\pi$
$811$	43.8883	1.54113	0.770564	+	0.637363i	$0.219975\pi$
$812$	0.337097	0.0118298
$813$	0	0
$814$	−7.14263	−0.250349
$815$	54.9307	1.92414
$816$	0	0
$817$	50.7681	1.77615
$818$	−15.3506	−0.536722
$819$	0	0
$820$	−31.1153	−1.08659
$821$	2.30719	0.0805216	0.0402608	−	0.999189i	$-0.487181\pi$
$821$	2.30719	0.0805216	0.0402608	+	0.999189i	$0.487181\pi$
$822$	0	0
$823$	−36.1175	−1.25898	−0.629488	−	0.777010i	$-0.716735\pi$
$823$	−36.1175	−1.25898	−0.629488	+	0.777010i	$0.716735\pi$
$824$	1.55792	0.0542728
$825$	0	0
$826$	4.69690	0.163426
$827$	5.96215	0.207324	0.103662	−	0.994613i	$-0.466944\pi$
$827$	5.96215	0.207324	0.103662	+	0.994613i	$0.466944\pi$
$828$	0	0
$829$	39.6099	1.37571	0.687854	−	0.725849i	$-0.258553\pi$
$829$	39.6099	1.37571	0.687854	+	0.725849i	$0.258553\pi$
$830$	34.5435	1.19902
$831$	0	0
$832$	5.11356	0.177281
$833$	−2.90898	−0.100790
$834$	0	0
$835$	42.3542	1.46573
$836$	3.48988	0.120700
$837$	0	0
$838$	1.08706	0.0375519
$839$	−35.6879	−1.23208	−0.616041	−	0.787714i	$-0.711264\pi$
$839$	−35.6879	−1.23208	−0.616041	+	0.787714i	$0.711264\pi$
$840$	0	0
$841$	−28.9813	−0.999356
$842$	−3.91425	−0.134894
$843$	0	0
$844$	−23.6423	−0.813800
$845$	38.8519	1.33655
$846$	0	0
$847$	−25.4915	−0.875899
$848$	−10.5309	−0.361632
$849$	0	0
$850$	11.8310	0.405800
$851$	26.5480	0.910055
$852$	0	0
$853$	10.3670	0.354959	0.177480	−	0.984124i	$-0.443206\pi$
$853$	10.3670	0.354959	0.177480	+	0.984124i	$0.443206\pi$
$854$	32.7018	1.11903
$855$	0	0
$856$	−2.86752	−0.0980098
$857$	−49.6954	−1.69756	−0.848781	−	0.528744i	$-0.822663\pi$
$857$	−49.6954	−1.69756	−0.848781	+	0.528744i	$0.822663\pi$
$858$	0	0
$859$	−45.5021	−1.55251	−0.776255	−	0.630419i	$-0.782883\pi$
$859$	−45.5021	−1.55251	−0.776255	+	0.630419i	$0.782883\pi$
$860$	−35.0274	−1.19442
$861$	0	0
$862$	−11.9555	−0.407207
$863$	44.5704	1.51719	0.758597	−	0.651560i	$-0.225885\pi$
$863$	44.5704	1.51719	0.758597	+	0.651560i	$0.225885\pi$
$864$	0	0
$865$	1.45557	0.0494908
$866$	27.0459	0.919056
$867$	0	0
$868$	12.3056	0.417681
$869$	−10.2536	−0.347831
$870$	0	0
$871$	−40.2175	−1.36272
$872$	−9.78165	−0.331249
$873$	0	0
$874$	−12.9713	−0.438761
$875$	−9.24624	−0.312580
$876$	0	0
$877$	−22.2425	−0.751077	−0.375538	−	0.926807i	$-0.622542\pi$
$877$	−22.2425	−0.751077	−0.375538	+	0.926807i	$0.622542\pi$
$878$	19.2418	0.649381
$879$	0	0
$880$	−2.40784	−0.0811681
$881$	−14.1929	−0.478172	−0.239086	−	0.970998i	$-0.576848\pi$
$881$	−14.1929	−0.478172	−0.239086	+	0.970998i	$0.576848\pi$
$882$	0	0
$883$	16.5105	0.555623	0.277811	−	0.960636i	$-0.410391\pi$
$883$	16.5105	0.555623	0.277811	+	0.960636i	$0.410391\pi$
$884$	16.2141	0.545339
$885$	0	0
$886$	−35.5408	−1.19402
$887$	−40.8925	−1.37303	−0.686517	−	0.727114i	$-0.740861\pi$
$887$	−40.8925	−1.37303	−0.686517	+	0.727114i	$0.740861\pi$
$888$	0	0
$889$	14.2027	0.476342
$890$	−31.6940	−1.06239
$891$	0	0
$892$	16.1009	0.539098
$893$	−28.4487	−0.952001
$894$	0	0
$895$	59.9060	2.00244
$896$	2.46629	0.0823929
$897$	0	0
$898$	27.8674	0.929948
$899$	0.681979	0.0227453
$900$	0	0
$901$	−33.3914	−1.11243
$902$	8.58076	0.285708
$903$	0	0
$904$	−14.9223	−0.496308
$905$	−6.21550	−0.206610
$906$	0	0
$907$	−1.27573	−0.0423598	−0.0211799	−	0.999776i	$-0.506742\pi$
$907$	−1.27573	−0.0423598	−0.0211799	+	0.999776i	$0.506742\pi$
$908$	16.9613	0.562880
$909$	0	0
$910$	37.2653	1.23533
$911$	−39.0453	−1.29363	−0.646814	−	0.762648i	$-0.723899\pi$
$911$	−39.0453	−1.29363	−0.646814	+	0.762648i	$0.723899\pi$
$912$	0	0
$913$	−9.52617	−0.315270
$914$	22.2773	0.736866
$915$	0	0
$916$	25.9949	0.858897
$917$	41.0449	1.35542
$918$	0	0
$919$	20.8159	0.686652	0.343326	−	0.939216i	$-0.388446\pi$
$919$	20.8159	0.686652	0.343326	+	0.939216i	$0.388446\pi$
$920$	8.94954	0.295058
$921$	0	0
$922$	−28.6522	−0.943609
$923$	25.8090	0.849513
$924$	0	0
$925$	32.7055	1.07535
$926$	30.9993	1.01870
$927$	0	0
$928$	0.136682	0.00448680
$929$	−4.14522	−0.136000	−0.0680001	−	0.997685i	$-0.521662\pi$
$929$	−4.14522	−0.136000	−0.0680001	+	0.997685i	$0.521662\pi$
$930$	0	0
$931$	3.92909	0.128771
$932$	−9.06092	−0.296800
$933$	0	0
$934$	9.64691	0.315657
$935$	−7.63478	−0.249684
$936$	0	0
$937$	−12.9454	−0.422907	−0.211453	−	0.977388i	$-0.567820\pi$
$937$	−12.9454	−0.422907	−0.211453	+	0.977388i	$0.567820\pi$
$938$	−19.3970	−0.633336
$939$	0	0
$940$	19.6282	0.640200
$941$	−58.9335	−1.92118	−0.960588	−	0.277975i	$-0.910337\pi$
$941$	−58.9335	−1.92118	−0.960588	+	0.277975i	$0.910337\pi$
$942$	0	0
$943$	−31.8933	−1.03859
$944$	1.90444	0.0619843
$945$	0	0
$946$	9.65959	0.314061
$947$	−15.8253	−0.514253	−0.257127	−	0.966378i	$-0.582776\pi$
$947$	−15.8253	−0.514253	−0.257127	+	0.966378i	$0.582776\pi$
$948$	0	0
$949$	−14.3501	−0.465825
$950$	−15.9798	−0.518455
$951$	0	0
$952$	7.82012	0.253451
$953$	3.22399	0.104435	0.0522177	−	0.998636i	$-0.483371\pi$
$953$	3.22399	0.104435	0.0522177	+	0.998636i	$0.483371\pi$
$954$	0	0
$955$	−7.56937	−0.244939
$956$	−20.1593	−0.651997
$957$	0	0
$958$	−3.24827	−0.104947
$959$	7.28361	0.235200
$960$	0	0
$961$	−6.10450	−0.196919
$962$	44.8221	1.44512
$963$	0	0
$964$	−25.7413	−0.829073
$965$	55.4551	1.78516
$966$	0	0
$967$	33.6919	1.08346	0.541729	−	0.840553i	$-0.317770\pi$
$967$	33.6919	1.08346	0.541729	+	0.840553i	$0.317770\pi$
$968$	−10.3360	−0.332211
$969$	0	0
$970$	−14.0500	−0.451117
$971$	−31.9670	−1.02587	−0.512935	−	0.858427i	$-0.671442\pi$
$971$	−31.9670	−1.02587	−0.512935	+	0.858427i	$0.671442\pi$
$972$	0	0
$973$	3.28018	0.105158
$974$	26.7297	0.856474
$975$	0	0
$976$	13.2595	0.424427
$977$	32.6349	1.04408	0.522042	−	0.852920i	$-0.325170\pi$
$977$	32.6349	1.04408	0.522042	+	0.852920i	$0.325170\pi$
$978$	0	0
$979$	8.74036	0.279343
$980$	−2.71087	−0.0865956
$981$	0	0
$982$	26.5883	0.848466
$983$	−16.0557	−0.512098	−0.256049	−	0.966664i	$-0.582421\pi$
$983$	−16.0557	−0.512098	−0.256049	+	0.966664i	$0.582421\pi$
$984$	0	0
$985$	−14.2918	−0.455375
$986$	0.433391	0.0138020
$987$	0	0
$988$	−21.9000	−0.696731
$989$	−35.9032	−1.14165
$990$	0	0
$991$	−6.89766	−0.219111	−0.109556	−	0.993981i	$-0.534943\pi$
$991$	−6.89766	−0.219111	−0.109556	+	0.993981i	$0.534943\pi$
$992$	4.98954	0.158418
$993$	0	0
$994$	12.4478	0.394820
$995$	−30.6293	−0.971013
$996$	0	0
$997$	14.3898	0.455729	0.227864	−	0.973693i	$-0.426826\pi$
$997$	14.3898	0.455729	0.227864	+	0.973693i	$0.426826\pi$
$998$	0.886424	0.0280593
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.r.1.12	yes	14
3.2	odd	2		8046.2.a.q.1.3	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.q.1.3	✓	14	3.2	odd	2
8046.2.a.r.1.12	yes	14	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.95487 q^{5} +2.46629 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.95487 q^{5} +2.46629 q^{7} +1.00000 q^{8} +2.95487 q^{10} -0.814872 q^{11} +5.11356 q^{13} +2.46629 q^{14} +1.00000 q^{16} +3.17081 q^{17} -4.28273 q^{19} +2.95487 q^{20} -0.814872 q^{22} +3.02875 q^{23} +3.73123 q^{25} +5.11356 q^{26} +2.46629 q^{28} +0.136682 q^{29} +4.98954 q^{31} +1.00000 q^{32} +3.17081 q^{34} +7.28755 q^{35} +8.76535 q^{37} -4.28273 q^{38} +2.95487 q^{40} -10.5302 q^{41} -11.8541 q^{43} -0.814872 q^{44} +3.02875 q^{46} +6.64266 q^{47} -0.917427 q^{49} +3.73123 q^{50} +5.11356 q^{52} -10.5309 q^{53} -2.40784 q^{55} +2.46629 q^{56} +0.136682 q^{58} +1.90444 q^{59} +13.2595 q^{61} +4.98954 q^{62} +1.00000 q^{64} +15.1099 q^{65} -7.86487 q^{67} +3.17081 q^{68} +7.28755 q^{70} +5.04717 q^{71} -2.80629 q^{73} +8.76535 q^{74} -4.28273 q^{76} -2.00971 q^{77} +12.5831 q^{79} +2.95487 q^{80} -10.5302 q^{82} +11.6904 q^{83} +9.36930 q^{85} -11.8541 q^{86} -0.814872 q^{88} -10.7261 q^{89} +12.6115 q^{91} +3.02875 q^{92} +6.64266 q^{94} -12.6549 q^{95} -4.75485 q^{97} -0.917427 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8}+O(q^{10}) \) 14 * q + 14 * q^2 + 14 * q^4 + 2 * q^5 + 4 * q^7 + 14 * q^8	\( 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{13} + 4 q^{14} + 14 q^{16} + 9 q^{17} + 14 q^{19} + 2 q^{20} + 2 q^{22} + 30 q^{23} + 18 q^{25} + 4 q^{26} + 4 q^{28} + 6 q^{29} + 11 q^{31} + 14 q^{32} + 9 q^{34} - 18 q^{35} + 13 q^{37} + 14 q^{38} + 2 q^{40} - 2 q^{41} + 12 q^{43} + 2 q^{44} + 30 q^{46} + 21 q^{47} + 32 q^{49} + 18 q^{50} + 4 q^{52} + 22 q^{53} - 7 q^{55} + 4 q^{56} + 6 q^{58} + 14 q^{59} + 31 q^{61} + 11 q^{62} + 14 q^{64} + 24 q^{67} + 9 q^{68} - 18 q^{70} + 28 q^{71} + 24 q^{73} + 13 q^{74} + 14 q^{76} + 16 q^{77} + 65 q^{79} + 2 q^{80} - 2 q^{82} - 15 q^{83} - 19 q^{85} + 12 q^{86} + 2 q^{88} - 11 q^{89} + 68 q^{91} + 30 q^{92} + 21 q^{94} + 8 q^{95} + 23 q^{97} + 32 q^{98}+O(q^{100}) \) 14 * q + 14 * q^2 + 14 * q^4 + 2 * q^5 + 4 * q^7 + 14 * q^8 + 2 * q^10 + 2 * q^11 + 4 * q^13 + 4 * q^14 + 14 * q^16 + 9 * q^17 + 14 * q^19 + 2 * q^20 + 2 * q^22 + 30 * q^23 + 18 * q^25 + 4 * q^26 + 4 * q^28 + 6 * q^29 + 11 * q^31 + 14 * q^32 + 9 * q^34 - 18 * q^35 + 13 * q^37 + 14 * q^38 + 2 * q^40 - 2 * q^41 + 12 * q^43 + 2 * q^44 + 30 * q^46 + 21 * q^47 + 32 * q^49 + 18 * q^50 + 4 * q^52 + 22 * q^53 - 7 * q^55 + 4 * q^56 + 6 * q^58 + 14 * q^59 + 31 * q^61 + 11 * q^62 + 14 * q^64 + 24 * q^67 + 9 * q^68 - 18 * q^70 + 28 * q^71 + 24 * q^73 + 13 * q^74 + 14 * q^76 + 16 * q^77 + 65 * q^79 + 2 * q^80 - 2 * q^82 - 15 * q^83 - 19 * q^85 + 12 * q^86 + 2 * q^88 - 11 * q^89 + 68 * q^91 + 30 * q^92 + 21 * q^94 + 8 * q^95 + 23 * q^97 + 32 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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