LMFDB - Embedded newform 4014.2.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, 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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.103354048.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 13x^{2} - 16x + 3 $ x^6 - 2x^5 - 8x^4 + 14x^3 + 13x^2 - 16*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.40037$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.03898 q^{5} -0.299718 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.03898 q^{5} -0.299718 q^{7} +1.00000 q^{8} -1.03898 q^{10} -0.525229 q^{11} +1.66111 q^{13} -0.299718 q^{14} +1.00000 q^{16} +4.52016 q^{17} +2.22045 q^{19} -1.03898 q^{20} -0.525229 q^{22} +5.40037 q^{23} -3.92053 q^{25} +1.66111 q^{26} -0.299718 q^{28} -4.11099 q^{29} -0.0896324 q^{31} +1.00000 q^{32} +4.52016 q^{34} +0.311399 q^{35} +1.75466 q^{37} +2.22045 q^{38} -1.03898 q^{40} +7.89429 q^{41} -7.27686 q^{43} -0.525229 q^{44} +5.40037 q^{46} -4.17641 q^{47} -6.91017 q^{49} -3.92053 q^{50} +1.66111 q^{52} +4.09356 q^{53} +0.545700 q^{55} -0.299718 q^{56} -4.11099 q^{58} -0.132135 q^{59} -1.00439 q^{61} -0.0896324 q^{62} +1.00000 q^{64} -1.72585 q^{65} +7.38541 q^{67} +4.52016 q^{68} +0.311399 q^{70} +12.2455 q^{71} +10.4015 q^{73} +1.75466 q^{74} +2.22045 q^{76} +0.157421 q^{77} +0.867469 q^{79} -1.03898 q^{80} +7.89429 q^{82} +14.7756 q^{83} -4.69634 q^{85} -7.27686 q^{86} -0.525229 q^{88} +4.84680 q^{89} -0.497864 q^{91} +5.40037 q^{92} -4.17641 q^{94} -2.30699 q^{95} +11.1559 q^{97} -6.91017 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 + 8 * q^7 + 6 * q^8	$ 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8} + 2 q^{10} + 2 q^{11} - 2 q^{13} + 8 q^{14} + 6 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{20} + 2 q^{22} + 22 q^{23} + 12 q^{25} - 2 q^{26} + 8 q^{28} + 8 q^{29} - 12 q^{31} + 6 q^{32} + 2 q^{34} + 20 q^{35} - 2 q^{38} + 2 q^{40} + 28 q^{41} + 14 q^{43} + 2 q^{44} + 22 q^{46} + 2 q^{47} + 12 q^{50} - 2 q^{52} + 26 q^{53} + 6 q^{55} + 8 q^{56} + 8 q^{58} + 4 q^{59} - 6 q^{61} - 12 q^{62} + 6 q^{64} + 16 q^{65} + 18 q^{67} + 2 q^{68} + 20 q^{70} + 12 q^{71} - 20 q^{73} - 2 q^{76} + 20 q^{77} + 12 q^{79} + 2 q^{80} + 28 q^{82} + 12 q^{83} - 10 q^{85} + 14 q^{86} + 2 q^{88} - 2 q^{89} - 22 q^{91} + 22 q^{92} + 2 q^{94} + 6 q^{95} - 6 q^{97}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 + 8 * q^7 + 6 * q^8 + 2 * q^10 + 2 * q^11 - 2 * q^13 + 8 * q^14 + 6 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^20 + 2 * q^22 + 22 * q^23 + 12 * q^25 - 2 * q^26 + 8 * q^28 + 8 * q^29 - 12 * q^31 + 6 * q^32 + 2 * q^34 + 20 * q^35 - 2 * q^38 + 2 * q^40 + 28 * q^41 + 14 * q^43 + 2 * q^44 + 22 * q^46 + 2 * q^47 + 12 * q^50 - 2 * q^52 + 26 * q^53 + 6 * q^55 + 8 * q^56 + 8 * q^58 + 4 * q^59 - 6 * q^61 - 12 * q^62 + 6 * q^64 + 16 * q^65 + 18 * q^67 + 2 * q^68 + 20 * q^70 + 12 * q^71 - 20 * q^73 - 2 * q^76 + 20 * q^77 + 12 * q^79 + 2 * q^80 + 28 * q^82 + 12 * q^83 - 10 * q^85 + 14 * q^86 + 2 * q^88 - 2 * q^89 - 22 * q^91 + 22 * q^92 + 2 * q^94 + 6 * q^95 - 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.03898	−0.464644	−0.232322	−	0.972639i	$-0.574632\pi$
$5$	−1.03898	−0.464644	−0.232322	+	0.972639i	$0.574632\pi$
$6$	0	0
$7$	−0.299718	−0.113283	−0.0566413	−	0.998395i	$-0.518039\pi$
$7$	−0.299718	−0.113283	−0.0566413	+	0.998395i	$0.518039\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.03898	−0.328553
$11$	−0.525229	−0.158363	−0.0791813	−	0.996860i	$-0.525231\pi$
$11$	−0.525229	−0.158363	−0.0791813	+	0.996860i	$0.525231\pi$
$12$	0	0
$13$	1.66111	0.460709	0.230354	−	0.973107i	$-0.426011\pi$
$13$	1.66111	0.460709	0.230354	+	0.973107i	$0.426011\pi$
$14$	−0.299718	−0.0801030
$15$	0	0
$16$	1.00000	0.250000
$17$	4.52016	1.09630	0.548150	−	0.836380i	$-0.315332\pi$
$17$	4.52016	1.09630	0.548150	+	0.836380i	$0.315332\pi$
$18$	0	0
$19$	2.22045	0.509405	0.254703	−	0.967019i	$-0.418022\pi$
$19$	2.22045	0.509405	0.254703	+	0.967019i	$0.418022\pi$
$20$	−1.03898	−0.232322
$21$	0	0
$22$	−0.525229	−0.111979
$23$	5.40037	1.12605	0.563027	−	0.826438i	$-0.309637\pi$
$23$	5.40037	1.12605	0.563027	+	0.826438i	$0.309637\pi$
$24$	0	0
$25$	−3.92053	−0.784106
$26$	1.66111	0.325770
$27$	0	0
$28$	−0.299718	−0.0566413
$29$	−4.11099	−0.763391	−0.381696	−	0.924288i	$-0.624660\pi$
$29$	−4.11099	−0.763391	−0.381696	+	0.924288i	$0.624660\pi$
$30$	0	0
$31$	−0.0896324	−0.0160984	−0.00804922	−	0.999968i	$-0.502562\pi$
$31$	−0.0896324	−0.0160984	−0.00804922	+	0.999968i	$0.502562\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.52016	0.775202
$35$	0.311399	0.0526361
$36$	0	0
$37$	1.75466	0.288465	0.144232	−	0.989544i	$-0.453929\pi$
$37$	1.75466	0.288465	0.144232	+	0.989544i	$0.453929\pi$
$38$	2.22045	0.360204
$39$	0	0
$40$	−1.03898	−0.164276
$41$	7.89429	1.23288	0.616440	−	0.787402i	$-0.288574\pi$
$41$	7.89429	1.23288	0.616440	+	0.787402i	$0.288574\pi$
$42$	0	0
$43$	−7.27686	−1.10971	−0.554855	−	0.831947i	$-0.687226\pi$
$43$	−7.27686	−1.10971	−0.554855	+	0.831947i	$0.687226\pi$
$44$	−0.525229	−0.0791813
$45$	0	0
$46$	5.40037	0.796241
$47$	−4.17641	−0.609191	−0.304596	−	0.952482i	$-0.598521\pi$
$47$	−4.17641	−0.609191	−0.304596	+	0.952482i	$0.598521\pi$
$48$	0	0
$49$	−6.91017	−0.987167
$50$	−3.92053	−0.554447
$51$	0	0
$52$	1.66111	0.230354
$53$	4.09356	0.562293	0.281147	−	0.959665i	$-0.409285\pi$
$53$	4.09356	0.562293	0.281147	+	0.959665i	$0.409285\pi$
$54$	0	0
$55$	0.545700	0.0735822
$56$	−0.299718	−0.0400515
$57$	0	0
$58$	−4.11099	−0.539799
$59$	−0.132135	−0.0172025	−0.00860124	−	0.999963i	$-0.502738\pi$
$59$	−0.132135	−0.0172025	−0.00860124	+	0.999963i	$0.502738\pi$
$60$	0	0
$61$	−1.00439	−0.128599	−0.0642995	−	0.997931i	$-0.520481\pi$
$61$	−1.00439	−0.128599	−0.0642995	+	0.997931i	$0.520481\pi$
$62$	−0.0896324	−0.0113833
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.72585	−0.214065
$66$	0	0
$67$	7.38541	0.902271	0.451136	−	0.892455i	$-0.351019\pi$
$67$	7.38541	0.902271	0.451136	+	0.892455i	$0.351019\pi$
$68$	4.52016	0.548150
$69$	0	0
$70$	0.311399	0.0372194
$71$	12.2455	1.45328	0.726640	−	0.687019i	$-0.241081\pi$
$71$	12.2455	1.45328	0.726640	+	0.687019i	$0.241081\pi$
$72$	0	0
$73$	10.4015	1.21741	0.608703	−	0.793398i	$-0.291690\pi$
$73$	10.4015	1.21741	0.608703	+	0.793398i	$0.291690\pi$
$74$	1.75466	0.203975
$75$	0	0
$76$	2.22045	0.254703
$77$	0.157421	0.0179397
$78$	0	0
$79$	0.867469	0.0975979	0.0487990	−	0.998809i	$-0.484461\pi$
$79$	0.867469	0.0975979	0.0487990	+	0.998809i	$0.484461\pi$
$80$	−1.03898	−0.116161
$81$	0	0
$82$	7.89429	0.871778
$83$	14.7756	1.62184	0.810919	−	0.585159i	$-0.198968\pi$
$83$	14.7756	1.62184	0.810919	+	0.585159i	$0.198968\pi$
$84$	0	0
$85$	−4.69634	−0.509390
$86$	−7.27686	−0.784684
$87$	0	0
$88$	−0.525229	−0.0559896
$89$	4.84680	0.513760	0.256880	−	0.966443i	$-0.417306\pi$
$89$	4.84680	0.513760	0.256880	+	0.966443i	$0.417306\pi$
$90$	0	0
$91$	−0.497864	−0.0521903
$92$	5.40037	0.563027
$93$	0	0
$94$	−4.17641	−0.430763
$95$	−2.30699	−0.236692
$96$	0	0
$97$	11.1559	1.13271	0.566356	−	0.824161i	$-0.308353\pi$
$97$	11.1559	1.13271	0.566356	+	0.824161i	$0.308353\pi$
$98$	−6.91017	−0.698033
$99$	0	0
$100$	−3.92053	−0.392053
$101$	19.5407	1.94437	0.972185	−	0.234214i	$-0.0752517\pi$
$101$	19.5407	1.94437	0.972185	+	0.234214i	$0.0752517\pi$
$102$	0	0
$103$	2.40563	0.237034	0.118517	−	0.992952i	$-0.462186\pi$
$103$	2.40563	0.237034	0.118517	+	0.992952i	$0.462186\pi$
$104$	1.66111	0.162885
$105$	0	0
$106$	4.09356	0.397601
$107$	1.99845	0.193197	0.0965987	−	0.995323i	$-0.469204\pi$
$107$	1.99845	0.193197	0.0965987	+	0.995323i	$0.469204\pi$
$108$	0	0
$109$	−0.860178	−0.0823901	−0.0411951	−	0.999151i	$-0.513117\pi$
$109$	−0.860178	−0.0823901	−0.0411951	+	0.999151i	$0.513117\pi$
$110$	0.545700	0.0520305
$111$	0	0
$112$	−0.299718	−0.0283207
$113$	−18.1891	−1.71108	−0.855541	−	0.517735i	$-0.826775\pi$
$113$	−18.1891	−1.71108	−0.855541	+	0.517735i	$0.826775\pi$
$114$	0	0
$115$	−5.61085	−0.523214
$116$	−4.11099	−0.381696
$117$	0	0
$118$	−0.132135	−0.0121640
$119$	−1.35477	−0.124192
$120$	0	0
$121$	−10.7241	−0.974921
$122$	−1.00439	−0.0909332
$123$	0	0
$124$	−0.0896324	−0.00804922
$125$	9.26821	0.828974
$126$	0	0
$127$	14.7556	1.30935	0.654676	−	0.755910i	$-0.272805\pi$
$127$	14.7556	1.30935	0.654676	+	0.755910i	$0.272805\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.72585	−0.151367
$131$	6.58779	0.575578	0.287789	−	0.957694i	$-0.407080\pi$
$131$	6.58779	0.575578	0.287789	+	0.957694i	$0.407080\pi$
$132$	0	0
$133$	−0.665507	−0.0577068
$134$	7.38541	0.638002
$135$	0	0
$136$	4.52016	0.387601
$137$	2.12723	0.181742	0.0908710	−	0.995863i	$-0.471035\pi$
$137$	2.12723	0.181742	0.0908710	+	0.995863i	$0.471035\pi$
$138$	0	0
$139$	18.7630	1.59146	0.795729	−	0.605653i	$-0.207088\pi$
$139$	18.7630	1.59146	0.795729	+	0.605653i	$0.207088\pi$
$140$	0.311399	0.0263181
$141$	0	0
$142$	12.2455	1.02762
$143$	−0.872463	−0.0729590
$144$	0	0
$145$	4.27122	0.354705
$146$	10.4015	0.860836
$147$	0	0
$148$	1.75466	0.144232
$149$	22.4018	1.83522	0.917611	−	0.397478i	$-0.130115\pi$
$149$	22.4018	1.83522	0.917611	+	0.397478i	$0.130115\pi$
$150$	0	0
$151$	−13.6293	−1.10913	−0.554567	−	0.832139i	$-0.687116\pi$
$151$	−13.6293	−1.10913	−0.554567	+	0.832139i	$0.687116\pi$
$152$	2.22045	0.180102
$153$	0	0
$154$	0.157421	0.0126853
$155$	0.0931258	0.00748005
$156$	0	0
$157$	5.34196	0.426335	0.213167	−	0.977016i	$-0.431622\pi$
$157$	5.34196	0.426335	0.213167	+	0.977016i	$0.431622\pi$
$158$	0.867469	0.0690122
$159$	0	0
$160$	−1.03898	−0.0821382
$161$	−1.61859	−0.127562
$162$	0	0
$163$	−10.0128	−0.784266	−0.392133	−	0.919909i	$-0.628263\pi$
$163$	−10.0128	−0.784266	−0.392133	+	0.919909i	$0.628263\pi$
$164$	7.89429	0.616440
$165$	0	0
$166$	14.7756	1.14681
$167$	4.18154	0.323577	0.161789	−	0.986825i	$-0.448274\pi$
$167$	4.18154	0.323577	0.161789	+	0.986825i	$0.448274\pi$
$168$	0	0
$169$	−10.2407	−0.787748
$170$	−4.69634	−0.360193
$171$	0	0
$172$	−7.27686	−0.554855
$173$	−6.93353	−0.527147	−0.263573	−	0.964639i	$-0.584901\pi$
$173$	−6.93353	−0.527147	−0.263573	+	0.964639i	$0.584901\pi$
$174$	0	0
$175$	1.17505	0.0888256
$176$	−0.525229	−0.0395906
$177$	0	0
$178$	4.84680	0.363283
$179$	6.64223	0.496463	0.248232	−	0.968701i	$-0.420151\pi$
$179$	6.64223	0.496463	0.248232	+	0.968701i	$0.420151\pi$
$180$	0	0
$181$	−22.6894	−1.68649	−0.843243	−	0.537532i	$-0.819357\pi$
$181$	−22.6894	−1.68649	−0.843243	+	0.537532i	$0.819357\pi$
$182$	−0.497864	−0.0369041
$183$	0	0
$184$	5.40037	0.398120
$185$	−1.82305	−0.134033
$186$	0	0
$187$	−2.37412	−0.173613
$188$	−4.17641	−0.304596
$189$	0	0
$190$	−2.30699	−0.167367
$191$	−13.9602	−1.01012	−0.505062	−	0.863083i	$-0.668530\pi$
$191$	−13.9602	−1.01012	−0.505062	+	0.863083i	$0.668530\pi$
$192$	0	0
$193$	22.3414	1.60817	0.804085	−	0.594515i	$-0.202656\pi$
$193$	22.3414	1.60817	0.804085	+	0.594515i	$0.202656\pi$
$194$	11.1559	0.800948
$195$	0	0
$196$	−6.91017	−0.493584
$197$	15.1604	1.08013	0.540066	−	0.841622i	$-0.318399\pi$
$197$	15.1604	1.08013	0.540066	+	0.841622i	$0.318399\pi$
$198$	0	0
$199$	−19.1639	−1.35849	−0.679246	−	0.733911i	$-0.737693\pi$
$199$	−19.1639	−1.35849	−0.679246	+	0.733911i	$0.737693\pi$
$200$	−3.92053	−0.277223
$201$	0	0
$202$	19.5407	1.37488
$203$	1.23214	0.0864790
$204$	0	0
$205$	−8.20197	−0.572850
$206$	2.40563	0.167608
$207$	0	0
$208$	1.66111	0.115177
$209$	−1.16624	−0.0806707
$210$	0	0
$211$	17.4035	1.19810	0.599052	−	0.800710i	$-0.295544\pi$
$211$	17.4035	1.19810	0.599052	+	0.800710i	$0.295544\pi$
$212$	4.09356	0.281147
$213$	0	0
$214$	1.99845	0.136611
$215$	7.56047	0.515620
$216$	0	0
$217$	0.0268644	0.00182368
$218$	−0.860178	−0.0582586
$219$	0	0
$220$	0.545700	0.0367911
$221$	7.50848	0.505075
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	−0.299718	−0.0200257
$225$	0	0
$226$	−18.1891	−1.20992
$227$	−18.7719	−1.24593	−0.622967	−	0.782248i	$-0.714073\pi$
$227$	−18.7719	−1.24593	−0.622967	+	0.782248i	$0.714073\pi$
$228$	0	0
$229$	−19.1687	−1.26671	−0.633353	−	0.773863i	$-0.718322\pi$
$229$	−19.1687	−1.26671	−0.633353	+	0.773863i	$0.718322\pi$
$230$	−5.61085	−0.369968
$231$	0	0
$232$	−4.11099	−0.269900
$233$	−16.5162	−1.08201	−0.541007	−	0.841018i	$-0.681957\pi$
$233$	−16.5162	−1.08201	−0.541007	+	0.841018i	$0.681957\pi$
$234$	0	0
$235$	4.33918	0.283057
$236$	−0.132135	−0.00860124
$237$	0	0
$238$	−1.35477	−0.0878169
$239$	−21.8402	−1.41272	−0.706362	−	0.707851i	$-0.749665\pi$
$239$	−21.8402	−1.41272	−0.706362	+	0.707851i	$0.749665\pi$
$240$	0	0
$241$	−17.2590	−1.11175	−0.555875	−	0.831266i	$-0.687617\pi$
$241$	−17.2590	−1.11175	−0.555875	+	0.831266i	$0.687617\pi$
$242$	−10.7241	−0.689373
$243$	0	0
$244$	−1.00439	−0.0642995
$245$	7.17950	0.458681
$246$	0	0
$247$	3.68840	0.234687
$248$	−0.0896324	−0.00569166
$249$	0	0
$250$	9.26821	0.586173
$251$	4.36375	0.275438	0.137719	−	0.990471i	$-0.456023\pi$
$251$	4.36375	0.275438	0.137719	+	0.990471i	$0.456023\pi$
$252$	0	0
$253$	−2.83643	−0.178325
$254$	14.7556	0.925851
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.5680	−1.28299	−0.641497	−	0.767125i	$-0.721686\pi$
$257$	−20.5680	−1.28299	−0.641497	+	0.767125i	$0.721686\pi$
$258$	0	0
$259$	−0.525904	−0.0326781
$260$	−1.72585	−0.107033
$261$	0	0
$262$	6.58779	0.406995
$263$	16.0190	0.987773	0.493886	−	0.869526i	$-0.335576\pi$
$263$	16.0190	0.987773	0.493886	+	0.869526i	$0.335576\pi$
$264$	0	0
$265$	−4.25310	−0.261266
$266$	−0.665507	−0.0408049
$267$	0	0
$268$	7.38541	0.451136
$269$	−12.4359	−0.758233	−0.379116	−	0.925349i	$-0.623772\pi$
$269$	−12.4359	−0.758233	−0.379116	+	0.925349i	$0.623772\pi$
$270$	0	0
$271$	27.9922	1.70040	0.850202	−	0.526457i	$-0.176480\pi$
$271$	27.9922	1.70040	0.850202	+	0.526457i	$0.176480\pi$
$272$	4.52016	0.274075
$273$	0	0
$274$	2.12723	0.128511
$275$	2.05918	0.124173
$276$	0	0
$277$	−1.75097	−0.105206	−0.0526028	−	0.998616i	$-0.516752\pi$
$277$	−1.75097	−0.105206	−0.0526028	+	0.998616i	$0.516752\pi$
$278$	18.7630	1.12533
$279$	0	0
$280$	0.311399	0.0186097
$281$	−0.628122	−0.0374706	−0.0187353	−	0.999824i	$-0.505964\pi$
$281$	−0.628122	−0.0374706	−0.0187353	+	0.999824i	$0.505964\pi$
$282$	0	0
$283$	−10.7408	−0.638476	−0.319238	−	0.947675i	$-0.603427\pi$
$283$	−10.7408	−0.638476	−0.319238	+	0.947675i	$0.603427\pi$
$284$	12.2455	0.726640
$285$	0	0
$286$	−0.872463	−0.0515898
$287$	−2.36606	−0.139664
$288$	0	0
$289$	3.43189	0.201876
$290$	4.27122	0.250814
$291$	0	0
$292$	10.4015	0.608703
$293$	6.48190	0.378676	0.189338	−	0.981912i	$-0.439366\pi$
$293$	6.48190	0.378676	0.189338	+	0.981912i	$0.439366\pi$
$294$	0	0
$295$	0.137285	0.00799303
$296$	1.75466	0.101988
$297$	0	0
$298$	22.4018	1.29770
$299$	8.97059	0.518783
$300$	0	0
$301$	2.18100	0.125711
$302$	−13.6293	−0.784276
$303$	0	0
$304$	2.22045	0.127351
$305$	1.04354	0.0597527
$306$	0	0
$307$	−25.9829	−1.48292	−0.741460	−	0.670997i	$-0.765866\pi$
$307$	−25.9829	−1.48292	−0.741460	+	0.670997i	$0.765866\pi$
$308$	0.157421	0.00896987
$309$	0	0
$310$	0.0931258	0.00528919
$311$	−17.0484	−0.966724	−0.483362	−	0.875421i	$-0.660584\pi$
$311$	−17.0484	−0.966724	−0.483362	+	0.875421i	$0.660584\pi$
$312$	0	0
$313$	−6.39442	−0.361434	−0.180717	−	0.983535i	$-0.557842\pi$
$313$	−6.39442	−0.361434	−0.180717	+	0.983535i	$0.557842\pi$
$314$	5.34196	0.301464
$315$	0	0
$316$	0.867469	0.0487990
$317$	24.4858	1.37526	0.687630	−	0.726061i	$-0.258651\pi$
$317$	24.4858	1.37526	0.687630	+	0.726061i	$0.258651\pi$
$318$	0	0
$319$	2.15921	0.120893
$320$	−1.03898	−0.0580805
$321$	0	0
$322$	−1.61859	−0.0902003
$323$	10.0368	0.558462
$324$	0	0
$325$	−6.51243	−0.361244
$326$	−10.0128	−0.554560
$327$	0	0
$328$	7.89429	0.435889
$329$	1.25174	0.0690108
$330$	0	0
$331$	20.1338	1.10665	0.553327	−	0.832964i	$-0.313358\pi$
$331$	20.1338	1.10665	0.553327	+	0.832964i	$0.313358\pi$
$332$	14.7756	0.810919
$333$	0	0
$334$	4.18154	0.228804
$335$	−7.67326	−0.419235
$336$	0	0
$337$	−13.9368	−0.759185	−0.379593	−	0.925154i	$-0.623936\pi$
$337$	−13.9368	−0.759185	−0.379593	+	0.925154i	$0.623936\pi$
$338$	−10.2407	−0.557022
$339$	0	0
$340$	−4.69634	−0.254695
$341$	0.0470775	0.00254939
$342$	0	0
$343$	4.16913	0.225112
$344$	−7.27686	−0.392342
$345$	0	0
$346$	−6.93353	−0.372749
$347$	28.5741	1.53394	0.766968	−	0.641685i	$-0.221764\pi$
$347$	28.5741	1.53394	0.766968	+	0.641685i	$0.221764\pi$
$348$	0	0
$349$	−17.9485	−0.960761	−0.480381	−	0.877060i	$-0.659501\pi$
$349$	−17.9485	−0.960761	−0.480381	+	0.877060i	$0.659501\pi$
$350$	1.17505	0.0628092
$351$	0	0
$352$	−0.525229	−0.0279948
$353$	−0.759446	−0.0404212	−0.0202106	−	0.999796i	$-0.506434\pi$
$353$	−0.759446	−0.0404212	−0.0202106	+	0.999796i	$0.506434\pi$
$354$	0	0
$355$	−12.7228	−0.675257
$356$	4.84680	0.256880
$357$	0	0
$358$	6.64223	0.351053
$359$	17.3328	0.914789	0.457394	−	0.889264i	$-0.348783\pi$
$359$	17.3328	0.914789	0.457394	+	0.889264i	$0.348783\pi$
$360$	0	0
$361$	−14.0696	−0.740506
$362$	−22.6894	−1.19253
$363$	0	0
$364$	−0.497864	−0.0260952
$365$	−10.8069	−0.565660
$366$	0	0
$367$	−25.8394	−1.34881	−0.674403	−	0.738363i	$-0.735599\pi$
$367$	−25.8394	−1.34881	−0.674403	+	0.738363i	$0.735599\pi$
$368$	5.40037	0.281514
$369$	0	0
$370$	−1.82305	−0.0947760
$371$	−1.22691	−0.0636981
$372$	0	0
$373$	18.4581	0.955726	0.477863	−	0.878434i	$-0.341412\pi$
$373$	18.4581	0.955726	0.477863	+	0.878434i	$0.341412\pi$
$374$	−2.37412	−0.122763
$375$	0	0
$376$	−4.17641	−0.215382
$377$	−6.82880	−0.351701
$378$	0	0
$379$	−37.9631	−1.95003	−0.975017	−	0.222132i	$-0.928699\pi$
$379$	−37.9631	−1.95003	−0.975017	+	0.222132i	$0.928699\pi$
$380$	−2.30699	−0.118346
$381$	0	0
$382$	−13.9602	−0.714266
$383$	18.5508	0.947903	0.473952	−	0.880551i	$-0.342827\pi$
$383$	18.5508	0.947903	0.473952	+	0.880551i	$0.342827\pi$
$384$	0	0
$385$	−0.163556	−0.00833559
$386$	22.3414	1.13715
$387$	0	0
$388$	11.1559	0.566356
$389$	13.7362	0.696451	0.348225	−	0.937411i	$-0.386784\pi$
$389$	13.7362	0.696451	0.348225	+	0.937411i	$0.386784\pi$
$390$	0	0
$391$	24.4105	1.23449
$392$	−6.91017	−0.349016
$393$	0	0
$394$	15.1604	0.763769
$395$	−0.901279	−0.0453483
$396$	0	0
$397$	−36.7502	−1.84444	−0.922220	−	0.386665i	$-0.873627\pi$
$397$	−36.7502	−1.84444	−0.922220	+	0.386665i	$0.873627\pi$
$398$	−19.1639	−0.960599
$399$	0	0
$400$	−3.92053	−0.196027
$401$	22.6156	1.12937	0.564685	−	0.825307i	$-0.308998\pi$
$401$	22.6156	1.12937	0.564685	+	0.825307i	$0.308998\pi$
$402$	0	0
$403$	−0.148889	−0.00741669
$404$	19.5407	0.972185
$405$	0	0
$406$	1.23214	0.0611499
$407$	−0.921601	−0.0456820
$408$	0	0
$409$	−18.1174	−0.895849	−0.447924	−	0.894071i	$-0.647837\pi$
$409$	−18.1174	−0.895849	−0.447924	+	0.894071i	$0.647837\pi$
$410$	−8.20197	−0.405066
$411$	0	0
$412$	2.40563	0.118517
$413$	0.0396031	0.00194874
$414$	0	0
$415$	−15.3515	−0.753577
$416$	1.66111	0.0814425
$417$	0	0
$418$	−1.16624	−0.0570428
$419$	23.4977	1.14794	0.573969	−	0.818877i	$-0.305403\pi$
$419$	23.4977	1.14794	0.573969	+	0.818877i	$0.305403\pi$
$420$	0	0
$421$	7.09188	0.345637	0.172819	−	0.984954i	$-0.444713\pi$
$421$	7.09188	0.345637	0.172819	+	0.984954i	$0.444713\pi$
$422$	17.4035	0.847188
$423$	0	0
$424$	4.09356	0.198801
$425$	−17.7214	−0.859616
$426$	0	0
$427$	0.301034	0.0145680
$428$	1.99845	0.0965987
$429$	0	0
$430$	7.56047	0.364598
$431$	−16.5315	−0.796295	−0.398148	−	0.917321i	$-0.630347\pi$
$431$	−16.5315	−0.796295	−0.398148	+	0.917321i	$0.630347\pi$
$432$	0	0
$433$	−8.98537	−0.431809	−0.215905	−	0.976414i	$-0.569270\pi$
$433$	−8.98537	−0.431809	−0.215905	+	0.976414i	$0.569270\pi$
$434$	0.0268644	0.00128953
$435$	0	0
$436$	−0.860178	−0.0411951
$437$	11.9912	0.573618
$438$	0	0
$439$	1.60272	0.0764935	0.0382468	−	0.999268i	$-0.487823\pi$
$439$	1.60272	0.0764935	0.0382468	+	0.999268i	$0.487823\pi$
$440$	0.545700	0.0260152
$441$	0	0
$442$	7.50848	0.357142
$443$	−14.8914	−0.707512	−0.353756	−	0.935338i	$-0.615096\pi$
$443$	−14.8914	−0.707512	−0.353756	+	0.935338i	$0.615096\pi$
$444$	0	0
$445$	−5.03571	−0.238715
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	−0.299718	−0.0141603
$449$	−0.0708914	−0.00334557	−0.00167279	−	0.999999i	$-0.500532\pi$
$449$	−0.0708914	−0.00334557	−0.00167279	+	0.999999i	$0.500532\pi$
$450$	0	0
$451$	−4.14631	−0.195242
$452$	−18.1891	−0.855541
$453$	0	0
$454$	−18.7719	−0.881009
$455$	0.517268	0.0242499
$456$	0	0
$457$	5.76447	0.269651	0.134825	−	0.990869i	$-0.456953\pi$
$457$	5.76447	0.269651	0.134825	+	0.990869i	$0.456953\pi$
$458$	−19.1687	−0.895696
$459$	0	0
$460$	−5.61085	−0.261607
$461$	−13.4012	−0.624155	−0.312077	−	0.950057i	$-0.601025\pi$
$461$	−13.4012	−0.624155	−0.312077	+	0.950057i	$0.601025\pi$
$462$	0	0
$463$	1.24896	0.0580439	0.0290220	−	0.999579i	$-0.490761\pi$
$463$	1.24896	0.0580439	0.0290220	+	0.999579i	$0.490761\pi$
$464$	−4.11099	−0.190848
$465$	0	0
$466$	−16.5162	−0.765100
$467$	−16.4228	−0.759958	−0.379979	−	0.924995i	$-0.624069\pi$
$467$	−16.4228	−0.759958	−0.379979	+	0.924995i	$0.624069\pi$
$468$	0	0
$469$	−2.21354	−0.102212
$470$	4.33918	0.200152
$471$	0	0
$472$	−0.132135	−0.00608199
$473$	3.82202	0.175737
$474$	0	0
$475$	−8.70533	−0.399428
$476$	−1.35477	−0.0620960
$477$	0	0
$478$	−21.8402	−0.998947
$479$	8.08026	0.369197	0.184598	−	0.982814i	$-0.440902\pi$
$479$	8.08026	0.369197	0.184598	+	0.982814i	$0.440902\pi$
$480$	0	0
$481$	2.91469	0.132898
$482$	−17.2590	−0.786125
$483$	0	0
$484$	−10.7241	−0.487461
$485$	−11.5907	−0.526308
$486$	0	0
$487$	24.2405	1.09844	0.549221	−	0.835677i	$-0.314925\pi$
$487$	24.2405	1.09844	0.549221	+	0.835677i	$0.314925\pi$
$488$	−1.00439	−0.0454666
$489$	0	0
$490$	7.17950	0.324337
$491$	−17.5667	−0.792775	−0.396388	−	0.918083i	$-0.629736\pi$
$491$	−17.5667	−0.792775	−0.396388	+	0.918083i	$0.629736\pi$
$492$	0	0
$493$	−18.5823	−0.836907
$494$	3.68840	0.165949
$495$	0	0
$496$	−0.0896324	−0.00402461
$497$	−3.67021	−0.164631
$498$	0	0
$499$	7.77382	0.348004	0.174002	−	0.984745i	$-0.444330\pi$
$499$	7.77382	0.348004	0.174002	+	0.984745i	$0.444330\pi$
$500$	9.26821	0.414487
$501$	0	0
$502$	4.36375	0.194764
$503$	1.63596	0.0729438	0.0364719	−	0.999335i	$-0.488388\pi$
$503$	1.63596	0.0729438	0.0364719	+	0.999335i	$0.488388\pi$
$504$	0	0
$505$	−20.3023	−0.903440
$506$	−2.83643	−0.126095
$507$	0	0
$508$	14.7556	0.654676
$509$	−10.7192	−0.475120	−0.237560	−	0.971373i	$-0.576348\pi$
$509$	−10.7192	−0.475120	−0.237560	+	0.971373i	$0.576348\pi$
$510$	0	0
$511$	−3.11752	−0.137911
$512$	1.00000	0.0441942
$513$	0	0
$514$	−20.5680	−0.907214
$515$	−2.49939	−0.110136
$516$	0	0
$517$	2.19357	0.0964731
$518$	−0.525904	−0.0231069
$519$	0	0
$520$	−1.72585	−0.0756836
$521$	14.1036	0.617889	0.308945	−	0.951080i	$-0.400024\pi$
$521$	14.1036	0.617889	0.308945	+	0.951080i	$0.400024\pi$
$522$	0	0
$523$	41.0860	1.79656	0.898282	−	0.439420i	$-0.144816\pi$
$523$	41.0860	1.79656	0.898282	+	0.439420i	$0.144816\pi$
$524$	6.58779	0.287789
$525$	0	0
$526$	16.0190	0.698461
$527$	−0.405153	−0.0176487
$528$	0	0
$529$	6.16395	0.267998
$530$	−4.25310	−0.184743
$531$	0	0
$532$	−0.665507	−0.0288534
$533$	13.1133	0.567999
$534$	0	0
$535$	−2.07634	−0.0897679
$536$	7.38541	0.319001
$537$	0	0
$538$	−12.4359	−0.536151
$539$	3.62942	0.156330
$540$	0	0
$541$	−43.2069	−1.85761	−0.928805	−	0.370568i	$-0.879163\pi$
$541$	−43.2069	−1.85761	−0.928805	+	0.370568i	$0.879163\pi$
$542$	27.9922	1.20237
$543$	0	0
$544$	4.52016	0.193800
$545$	0.893704	0.0382821
$546$	0	0
$547$	36.9636	1.58045	0.790224	−	0.612818i	$-0.209964\pi$
$547$	36.9636	1.58045	0.790224	+	0.612818i	$0.209964\pi$
$548$	2.12723	0.0908710
$549$	0	0
$550$	2.05918	0.0878036
$551$	−9.12823	−0.388876
$552$	0	0
$553$	−0.259996	−0.0110562
$554$	−1.75097	−0.0743917
$555$	0	0
$556$	18.7630	0.795729
$557$	11.9366	0.505771	0.252886	−	0.967496i	$-0.418620\pi$
$557$	11.9366	0.505771	0.252886	+	0.967496i	$0.418620\pi$
$558$	0	0
$559$	−12.0876	−0.511253
$560$	0.311399	0.0131590
$561$	0	0
$562$	−0.628122	−0.0264957
$563$	0.316299	0.0133304	0.00666520	−	0.999978i	$-0.497878\pi$
$563$	0.316299	0.0133304	0.00666520	+	0.999978i	$0.497878\pi$
$564$	0	0
$565$	18.8980	0.795044
$566$	−10.7408	−0.451470
$567$	0	0
$568$	12.2455	0.513812
$569$	−14.5044	−0.608056	−0.304028	−	0.952663i	$-0.598332\pi$
$569$	−14.5044	−0.608056	−0.304028	+	0.952663i	$0.598332\pi$
$570$	0	0
$571$	−31.2357	−1.30717	−0.653586	−	0.756852i	$-0.726736\pi$
$571$	−31.2357	−1.30717	−0.653586	+	0.756852i	$0.726736\pi$
$572$	−0.872463	−0.0364795
$573$	0	0
$574$	−2.36606	−0.0987574
$575$	−21.1723	−0.882946
$576$	0	0
$577$	−22.9555	−0.955650	−0.477825	−	0.878455i	$-0.658575\pi$
$577$	−22.9555	−0.955650	−0.477825	+	0.878455i	$0.658575\pi$
$578$	3.43189	0.142748
$579$	0	0
$580$	4.27122	0.177353
$581$	−4.42852	−0.183726
$582$	0	0
$583$	−2.15006	−0.0890462
$584$	10.4015	0.430418
$585$	0	0
$586$	6.48190	0.267765
$587$	−21.8544	−0.902029	−0.451015	−	0.892517i	$-0.648938\pi$
$587$	−21.8544	−0.902029	−0.451015	+	0.892517i	$0.648938\pi$
$588$	0	0
$589$	−0.199024	−0.00820064
$590$	0.137285	0.00565192
$591$	0	0
$592$	1.75466	0.0721162
$593$	13.1570	0.540294	0.270147	−	0.962819i	$-0.412928\pi$
$593$	13.1570	0.540294	0.270147	+	0.962819i	$0.412928\pi$
$594$	0	0
$595$	1.40758	0.0577050
$596$	22.4018	0.917611
$597$	0	0
$598$	8.97059	0.366835
$599$	−35.0228	−1.43099	−0.715496	−	0.698617i	$-0.753799\pi$
$599$	−35.0228	−1.43099	−0.715496	+	0.698617i	$0.753799\pi$
$600$	0	0
$601$	−16.7277	−0.682338	−0.341169	−	0.940002i	$-0.610823\pi$
$601$	−16.7277	−0.682338	−0.341169	+	0.940002i	$0.610823\pi$
$602$	2.18100	0.0888911
$603$	0	0
$604$	−13.6293	−0.554567
$605$	11.1421	0.452991
$606$	0	0
$607$	−7.57784	−0.307575	−0.153787	−	0.988104i	$-0.549147\pi$
$607$	−7.57784	−0.307575	−0.153787	+	0.988104i	$0.549147\pi$
$608$	2.22045	0.0900510
$609$	0	0
$610$	1.04354	0.0422516
$611$	−6.93746	−0.280660
$612$	0	0
$613$	−5.19833	−0.209959	−0.104979	−	0.994474i	$-0.533478\pi$
$613$	−5.19833	−0.209959	−0.104979	+	0.994474i	$0.533478\pi$
$614$	−25.9829	−1.04858
$615$	0	0
$616$	0.157421	0.00634266
$617$	46.2097	1.86033	0.930167	−	0.367136i	$-0.119662\pi$
$617$	46.2097	1.86033	0.930167	+	0.367136i	$0.119662\pi$
$618$	0	0
$619$	−2.06464	−0.0829848	−0.0414924	−	0.999139i	$-0.513211\pi$
$619$	−2.06464	−0.0829848	−0.0414924	+	0.999139i	$0.513211\pi$
$620$	0.0931258	0.00374002
$621$	0	0
$622$	−17.0484	−0.683577
$623$	−1.45267	−0.0582001
$624$	0	0
$625$	9.97321	0.398928
$626$	−6.39442	−0.255573
$627$	0	0
$628$	5.34196	0.213167
$629$	7.93137	0.316244
$630$	0	0
$631$	−11.7882	−0.469282	−0.234641	−	0.972082i	$-0.575391\pi$
$631$	−11.7882	−0.469282	−0.234641	+	0.972082i	$0.575391\pi$
$632$	0.867469	0.0345061
$633$	0	0
$634$	24.4858	0.972456
$635$	−15.3307	−0.608382
$636$	0	0
$637$	−11.4785	−0.454796
$638$	2.15921	0.0854840
$639$	0	0
$640$	−1.03898	−0.0410691
$641$	1.11471	0.0440284	0.0220142	−	0.999758i	$-0.492992\pi$
$641$	1.11471	0.0440284	0.0220142	+	0.999758i	$0.492992\pi$
$642$	0	0
$643$	−24.9371	−0.983423	−0.491712	−	0.870758i	$-0.663629\pi$
$643$	−24.9371	−0.983423	−0.491712	+	0.870758i	$0.663629\pi$
$644$	−1.61859	−0.0637812
$645$	0	0
$646$	10.0368	0.394892
$647$	−0.343948	−0.0135220	−0.00676099	−	0.999977i	$-0.502152\pi$
$647$	−0.343948	−0.0135220	−0.00676099	+	0.999977i	$0.502152\pi$
$648$	0	0
$649$	0.0694010	0.00272423
$650$	−6.51243	−0.255438
$651$	0	0
$652$	−10.0128	−0.392133
$653$	−28.7469	−1.12495	−0.562477	−	0.826813i	$-0.690152\pi$
$653$	−28.7469	−1.12495	−0.562477	+	0.826813i	$0.690152\pi$
$654$	0	0
$655$	−6.84455	−0.267439
$656$	7.89429	0.308220
$657$	0	0
$658$	1.25174	0.0487980
$659$	−34.3845	−1.33943	−0.669714	−	0.742619i	$-0.733583\pi$
$659$	−34.3845	−1.33943	−0.669714	+	0.742619i	$0.733583\pi$
$660$	0	0
$661$	17.7861	0.691801	0.345900	−	0.938271i	$-0.387574\pi$
$661$	17.7861	0.691801	0.345900	+	0.938271i	$0.387574\pi$
$662$	20.1338	0.782522
$663$	0	0
$664$	14.7756	0.573406
$665$	0.691446	0.0268131
$666$	0	0
$667$	−22.2008	−0.859620
$668$	4.18154	0.161789
$669$	0	0
$670$	−7.67326	−0.296444
$671$	0.527535	0.0203653
$672$	0	0
$673$	−21.4658	−0.827446	−0.413723	−	0.910403i	$-0.635772\pi$
$673$	−21.4658	−0.827446	−0.413723	+	0.910403i	$0.635772\pi$
$674$	−13.9368	−0.536825
$675$	0	0
$676$	−10.2407	−0.393874
$677$	−11.5286	−0.443080	−0.221540	−	0.975151i	$-0.571108\pi$
$677$	−11.5286	−0.443080	−0.221540	+	0.975151i	$0.571108\pi$
$678$	0	0
$679$	−3.34363	−0.128317
$680$	−4.69634	−0.180096
$681$	0	0
$682$	0.0470775	0.00180269
$683$	−10.5658	−0.404291	−0.202145	−	0.979356i	$-0.564791\pi$
$683$	−10.5658	−0.404291	−0.202145	+	0.979356i	$0.564791\pi$
$684$	0	0
$685$	−2.21014	−0.0844453
$686$	4.16913	0.159178
$687$	0	0
$688$	−7.27686	−0.277428
$689$	6.79984	0.259053
$690$	0	0
$691$	35.6000	1.35429	0.677144	−	0.735850i	$-0.263217\pi$
$691$	35.6000	1.35429	0.677144	+	0.735850i	$0.263217\pi$
$692$	−6.93353	−0.263573
$693$	0	0
$694$	28.5741	1.08466
$695$	−19.4943	−0.739461
$696$	0	0
$697$	35.6835	1.35161
$698$	−17.9485	−0.679361
$699$	0	0
$700$	1.17505	0.0444128
$701$	−13.8120	−0.521671	−0.260835	−	0.965383i	$-0.583998\pi$
$701$	−13.8120	−0.521671	−0.260835	+	0.965383i	$0.583998\pi$
$702$	0	0
$703$	3.89614	0.146946
$704$	−0.525229	−0.0197953
$705$	0	0
$706$	−0.759446	−0.0285821
$707$	−5.85669	−0.220263
$708$	0	0
$709$	34.7849	1.30637	0.653187	−	0.757197i	$-0.273432\pi$
$709$	34.7849	1.30637	0.653187	+	0.757197i	$0.273432\pi$
$710$	−12.7228	−0.477479
$711$	0	0
$712$	4.84680	0.181642
$713$	−0.484048	−0.0181277
$714$	0	0
$715$	0.906467	0.0339000
$716$	6.64223	0.248232
$717$	0	0
$718$	17.3328	0.646853
$719$	36.8984	1.37608	0.688039	−	0.725674i	$-0.258472\pi$
$719$	36.8984	1.37608	0.688039	+	0.725674i	$0.258472\pi$
$720$	0	0
$721$	−0.721010	−0.0268518
$722$	−14.0696	−0.523617
$723$	0	0
$724$	−22.6894	−0.843243
$725$	16.1173	0.598580
$726$	0	0
$727$	21.5881	0.800658	0.400329	−	0.916372i	$-0.368896\pi$
$727$	21.5881	0.800658	0.400329	+	0.916372i	$0.368896\pi$
$728$	−0.497864	−0.0184521
$729$	0	0
$730$	−10.8069	−0.399982
$731$	−32.8926	−1.21658
$732$	0	0
$733$	43.7293	1.61518	0.807588	−	0.589746i	$-0.200772\pi$
$733$	43.7293	1.61518	0.807588	+	0.589746i	$0.200772\pi$
$734$	−25.8394	−0.953750
$735$	0	0
$736$	5.40037	0.199060
$737$	−3.87903	−0.142886
$738$	0	0
$739$	6.03009	0.221821	0.110910	−	0.993830i	$-0.464623\pi$
$739$	6.03009	0.221821	0.110910	+	0.993830i	$0.464623\pi$
$740$	−1.82305	−0.0670167
$741$	0	0
$742$	−1.22691	−0.0450413
$743$	−26.4190	−0.969221	−0.484610	−	0.874730i	$-0.661039\pi$
$743$	−26.4190	−0.969221	−0.484610	+	0.874730i	$0.661039\pi$
$744$	0	0
$745$	−23.2749	−0.852725
$746$	18.4581	0.675800
$747$	0	0
$748$	−2.37412	−0.0868065
$749$	−0.598971	−0.0218859
$750$	0	0
$751$	21.0778	0.769139	0.384570	−	0.923096i	$-0.374350\pi$
$751$	21.0778	0.769139	0.384570	+	0.923096i	$0.374350\pi$
$752$	−4.17641	−0.152298
$753$	0	0
$754$	−6.82880	−0.248690
$755$	14.1605	0.515352
$756$	0	0
$757$	35.1247	1.27663	0.638315	−	0.769775i	$-0.279632\pi$
$757$	35.1247	1.27663	0.638315	+	0.769775i	$0.279632\pi$
$758$	−37.9631	−1.37888
$759$	0	0
$760$	−2.30699	−0.0836833
$761$	−49.8993	−1.80885	−0.904424	−	0.426636i	$-0.859699\pi$
$761$	−49.8993	−1.80885	−0.904424	+	0.426636i	$0.859699\pi$
$762$	0	0
$763$	0.257811	0.00933338
$764$	−13.9602	−0.505062
$765$	0	0
$766$	18.5508	0.670269
$767$	−0.219490	−0.00792533
$768$	0	0
$769$	−8.63920	−0.311538	−0.155769	−	0.987794i	$-0.549785\pi$
$769$	−8.63920	−0.311538	−0.155769	+	0.987794i	$0.549785\pi$
$770$	−0.163556	−0.00589415
$771$	0	0
$772$	22.3414	0.804085
$773$	−1.97552	−0.0710545	−0.0355273	−	0.999369i	$-0.511311\pi$
$773$	−1.97552	−0.0710545	−0.0355273	+	0.999369i	$0.511311\pi$
$774$	0	0
$775$	0.351406	0.0126229
$776$	11.1559	0.400474
$777$	0	0
$778$	13.7362	0.492465
$779$	17.5288	0.628036
$780$	0	0
$781$	−6.43172	−0.230145
$782$	24.4105	0.872919
$783$	0	0
$784$	−6.91017	−0.246792
$785$	−5.55017	−0.198094
$786$	0	0
$787$	−27.2295	−0.970627	−0.485314	−	0.874340i	$-0.661295\pi$
$787$	−27.2295	−0.970627	−0.485314	+	0.874340i	$0.661295\pi$
$788$	15.1604	0.540066
$789$	0	0
$790$	−0.901279	−0.0320661
$791$	5.45158	0.193836
$792$	0	0
$793$	−1.66840	−0.0592467
$794$	−36.7502	−1.30422
$795$	0	0
$796$	−19.1639	−0.679246
$797$	−18.1553	−0.643092	−0.321546	−	0.946894i	$-0.604202\pi$
$797$	−18.1553	−0.643092	−0.321546	+	0.946894i	$0.604202\pi$
$798$	0	0
$799$	−18.8780	−0.667857
$800$	−3.92053	−0.138612
$801$	0	0
$802$	22.6156	0.798585
$803$	−5.46318	−0.192792
$804$	0	0
$805$	1.68167	0.0592711
$806$	−0.148889	−0.00524439
$807$	0	0
$808$	19.5407	0.687439
$809$	48.7225	1.71299	0.856496	−	0.516154i	$-0.172637\pi$
$809$	48.7225	1.71299	0.856496	+	0.516154i	$0.172637\pi$
$810$	0	0
$811$	29.2960	1.02872	0.514361	−	0.857574i	$-0.328029\pi$
$811$	29.2960	1.02872	0.514361	+	0.857574i	$0.328029\pi$
$812$	1.23214	0.0432395
$813$	0	0
$814$	−0.921601	−0.0323021
$815$	10.4031	0.364404
$816$	0	0
$817$	−16.1579	−0.565292
$818$	−18.1174	−0.633461
$819$	0	0
$820$	−8.20197	−0.286425
$821$	−34.2849	−1.19655	−0.598275	−	0.801291i	$-0.704147\pi$
$821$	−34.2849	−1.19655	−0.598275	+	0.801291i	$0.704147\pi$
$822$	0	0
$823$	34.1489	1.19036	0.595178	−	0.803594i	$-0.297082\pi$
$823$	34.1489	1.19036	0.595178	+	0.803594i	$0.297082\pi$
$824$	2.40563	0.0838040
$825$	0	0
$826$	0.0396031	0.00137797
$827$	12.5893	0.437773	0.218886	−	0.975750i	$-0.429758\pi$
$827$	12.5893	0.437773	0.218886	+	0.975750i	$0.429758\pi$
$828$	0	0
$829$	29.1743	1.01327	0.506633	−	0.862162i	$-0.330890\pi$
$829$	29.1743	1.01327	0.506633	+	0.862162i	$0.330890\pi$
$830$	−15.3515	−0.532859
$831$	0	0
$832$	1.66111	0.0575886
$833$	−31.2351	−1.08223
$834$	0	0
$835$	−4.34452	−0.150348
$836$	−1.16624	−0.0403354
$837$	0	0
$838$	23.4977	0.811714
$839$	−22.1132	−0.763432	−0.381716	−	0.924280i	$-0.624667\pi$
$839$	−22.1132	−0.763432	−0.381716	+	0.924280i	$0.624667\pi$
$840$	0	0
$841$	−12.0998	−0.417233
$842$	7.09188	0.244402
$843$	0	0
$844$	17.4035	0.599052
$845$	10.6399	0.366022
$846$	0	0
$847$	3.21421	0.110442
$848$	4.09356	0.140573
$849$	0	0
$850$	−17.7214	−0.607840
$851$	9.47583	0.324827
$852$	0	0
$853$	−4.11611	−0.140933	−0.0704665	−	0.997514i	$-0.522449\pi$
$853$	−4.11611	−0.140933	−0.0704665	+	0.997514i	$0.522449\pi$
$854$	0.301034	0.0103012
$855$	0	0
$856$	1.99845	0.0683056
$857$	29.6170	1.01170	0.505848	−	0.862622i	$-0.331180\pi$
$857$	29.6170	1.01170	0.505848	+	0.862622i	$0.331180\pi$
$858$	0	0
$859$	27.5026	0.938375	0.469188	−	0.883099i	$-0.344547\pi$
$859$	27.5026	0.938375	0.469188	+	0.883099i	$0.344547\pi$
$860$	7.56047	0.257810
$861$	0	0
$862$	−16.5315	−0.563066
$863$	−19.0973	−0.650081	−0.325040	−	0.945700i	$-0.605378\pi$
$863$	−19.0973	−0.650081	−0.325040	+	0.945700i	$0.605378\pi$
$864$	0	0
$865$	7.20377	0.244936
$866$	−8.98537	−0.305335
$867$	0	0
$868$	0.0268644	0.000911838 0
$869$	−0.455620	−0.0154559
$870$	0	0
$871$	12.2680	0.415684
$872$	−0.860178	−0.0291293
$873$	0	0
$874$	11.9912	0.405609
$875$	−2.77785	−0.0939084
$876$	0	0
$877$	4.72369	0.159508	0.0797539	−	0.996815i	$-0.474587\pi$
$877$	4.72369	0.159508	0.0797539	+	0.996815i	$0.474587\pi$
$878$	1.60272	0.0540891
$879$	0	0
$880$	0.545700	0.0183956
$881$	−36.6557	−1.23496	−0.617481	−	0.786586i	$-0.711847\pi$
$881$	−36.6557	−1.23496	−0.617481	+	0.786586i	$0.711847\pi$
$882$	0	0
$883$	0.492159	0.0165625	0.00828124	−	0.999966i	$-0.497364\pi$
$883$	0.492159	0.0165625	0.00828124	+	0.999966i	$0.497364\pi$
$884$	7.50848	0.252538
$885$	0	0
$886$	−14.8914	−0.500286
$887$	−38.6278	−1.29699	−0.648496	−	0.761218i	$-0.724602\pi$
$887$	−38.6278	−1.29699	−0.648496	+	0.761218i	$0.724602\pi$
$888$	0	0
$889$	−4.42253	−0.148327
$890$	−5.03571	−0.168797
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	−9.27349	−0.310325
$894$	0	0
$895$	−6.90111	−0.230679
$896$	−0.299718	−0.0100129
$897$	0	0
$898$	−0.0708914	−0.00236568
$899$	0.368478	0.0122894
$900$	0	0
$901$	18.5035	0.616442
$902$	−4.14631	−0.138057
$903$	0	0
$904$	−18.1891	−0.604959
$905$	23.5737	0.783616
$906$	0	0
$907$	−12.8858	−0.427865	−0.213933	−	0.976848i	$-0.568627\pi$
$907$	−12.8858	−0.427865	−0.213933	+	0.976848i	$0.568627\pi$
$908$	−18.7719	−0.622967
$909$	0	0
$910$	0.517268	0.0171473
$911$	13.9797	0.463168	0.231584	−	0.972815i	$-0.425609\pi$
$911$	13.9797	0.463168	0.231584	+	0.972815i	$0.425609\pi$
$912$	0	0
$913$	−7.76060	−0.256838
$914$	5.76447	0.190672
$915$	0	0
$916$	−19.1687	−0.633353
$917$	−1.97448	−0.0652030
$918$	0	0
$919$	−21.4656	−0.708083	−0.354042	−	0.935230i	$-0.615193\pi$
$919$	−21.4656	−0.708083	−0.354042	+	0.935230i	$0.615193\pi$
$920$	−5.61085	−0.184984
$921$	0	0
$922$	−13.4012	−0.441344
$923$	20.3412	0.669538
$924$	0	0
$925$	−6.87921	−0.226187
$926$	1.24896	0.0410433
$927$	0	0
$928$	−4.11099	−0.134950
$929$	−11.3647	−0.372863	−0.186431	−	0.982468i	$-0.559692\pi$
$929$	−11.3647	−0.372863	−0.186431	+	0.982468i	$0.559692\pi$
$930$	0	0
$931$	−15.3437	−0.502868
$932$	−16.5162	−0.541007
$933$	0	0
$934$	−16.4228	−0.537371
$935$	2.46665	0.0806682
$936$	0	0
$937$	−34.6117	−1.13071	−0.565357	−	0.824846i	$-0.691262\pi$
$937$	−34.6117	−1.13071	−0.565357	+	0.824846i	$0.691262\pi$
$938$	−2.21354	−0.0722746
$939$	0	0
$940$	4.33918	0.141529
$941$	25.8236	0.841825	0.420912	−	0.907101i	$-0.361710\pi$
$941$	25.8236	0.841825	0.420912	+	0.907101i	$0.361710\pi$
$942$	0	0
$943$	42.6320	1.38829
$944$	−0.132135	−0.00430062
$945$	0	0
$946$	3.82202	0.124265
$947$	−15.2505	−0.495576	−0.247788	−	0.968814i	$-0.579704\pi$
$947$	−15.2505	−0.495576	−0.247788	+	0.968814i	$0.579704\pi$
$948$	0	0
$949$	17.2781	0.560869
$950$	−8.70533	−0.282438
$951$	0	0
$952$	−1.35477	−0.0439085
$953$	−20.4875	−0.663656	−0.331828	−	0.943340i	$-0.607665\pi$
$953$	−20.4875	−0.663656	−0.331828	+	0.943340i	$0.607665\pi$
$954$	0	0
$955$	14.5043	0.469348
$956$	−21.8402	−0.706362
$957$	0	0
$958$	8.08026	0.261061
$959$	−0.637570	−0.0205882
$960$	0	0
$961$	−30.9920	−0.999741
$962$	2.91469	0.0939733
$963$	0	0
$964$	−17.2590	−0.555875
$965$	−23.2122	−0.747226
$966$	0	0
$967$	46.0193	1.47988	0.739940	−	0.672673i	$-0.234854\pi$
$967$	46.0193	1.47988	0.739940	+	0.672673i	$0.234854\pi$
$968$	−10.7241	−0.344687
$969$	0	0
$970$	−11.5907	−0.372156
$971$	49.0166	1.57302	0.786509	−	0.617579i	$-0.211886\pi$
$971$	49.0166	1.57302	0.786509	+	0.617579i	$0.211886\pi$
$972$	0	0
$973$	−5.62361	−0.180285
$974$	24.2405	0.776716
$975$	0	0
$976$	−1.00439	−0.0321498
$977$	−47.7323	−1.52709	−0.763545	−	0.645755i	$-0.776543\pi$
$977$	−47.7323	−1.52709	−0.763545	+	0.645755i	$0.776543\pi$
$978$	0	0
$979$	−2.54568	−0.0813603
$980$	7.17950	0.229341
$981$	0	0
$982$	−17.5667	−0.560577
$983$	33.5372	1.06967	0.534835	−	0.844956i	$-0.320374\pi$
$983$	33.5372	1.06967	0.534835	+	0.844956i	$0.320374\pi$
$984$	0	0
$985$	−15.7513	−0.501877
$986$	−18.5823	−0.591782
$987$	0	0
$988$	3.68840	0.117344
$989$	−39.2977	−1.24959
$990$	0	0
$991$	10.9117	0.346621	0.173310	−	0.984867i	$-0.444554\pi$
$991$	10.9117	0.346621	0.173310	+	0.984867i	$0.444554\pi$
$992$	−0.0896324	−0.00284583
$993$	0	0
$994$	−3.67021	−0.116412
$995$	19.9108	0.631215
$996$	0	0
$997$	−1.66662	−0.0527822	−0.0263911	−	0.999652i	$-0.508402\pi$
$997$	−1.66662	−0.0527822	−0.0263911	+	0.999652i	$0.508402\pi$
$998$	7.77382	0.246076
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.u.1.3	yes	6
3.2	odd	2		4014.2.a.t.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.a.t.1.4	✓	6	3.2	odd	2
4014.2.a.u.1.3	yes	6	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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.03898 q^{5} -0.299718 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.03898 q^{5} -0.299718 q^{7} +1.00000 q^{8} -1.03898 q^{10} -0.525229 q^{11} +1.66111 q^{13} -0.299718 q^{14} +1.00000 q^{16} +4.52016 q^{17} +2.22045 q^{19} -1.03898 q^{20} -0.525229 q^{22} +5.40037 q^{23} -3.92053 q^{25} +1.66111 q^{26} -0.299718 q^{28} -4.11099 q^{29} -0.0896324 q^{31} +1.00000 q^{32} +4.52016 q^{34} +0.311399 q^{35} +1.75466 q^{37} +2.22045 q^{38} -1.03898 q^{40} +7.89429 q^{41} -7.27686 q^{43} -0.525229 q^{44} +5.40037 q^{46} -4.17641 q^{47} -6.91017 q^{49} -3.92053 q^{50} +1.66111 q^{52} +4.09356 q^{53} +0.545700 q^{55} -0.299718 q^{56} -4.11099 q^{58} -0.132135 q^{59} -1.00439 q^{61} -0.0896324 q^{62} +1.00000 q^{64} -1.72585 q^{65} +7.38541 q^{67} +4.52016 q^{68} +0.311399 q^{70} +12.2455 q^{71} +10.4015 q^{73} +1.75466 q^{74} +2.22045 q^{76} +0.157421 q^{77} +0.867469 q^{79} -1.03898 q^{80} +7.89429 q^{82} +14.7756 q^{83} -4.69634 q^{85} -7.27686 q^{86} -0.525229 q^{88} +4.84680 q^{89} -0.497864 q^{91} +5.40037 q^{92} -4.17641 q^{94} -2.30699 q^{95} +11.1559 q^{97} -6.91017 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 + 8 * q^7 + 6 * q^8	\( 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8} + 2 q^{10} + 2 q^{11} - 2 q^{13} + 8 q^{14} + 6 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{20} + 2 q^{22} + 22 q^{23} + 12 q^{25} - 2 q^{26} + 8 q^{28} + 8 q^{29} - 12 q^{31} + 6 q^{32} + 2 q^{34} + 20 q^{35} - 2 q^{38} + 2 q^{40} + 28 q^{41} + 14 q^{43} + 2 q^{44} + 22 q^{46} + 2 q^{47} + 12 q^{50} - 2 q^{52} + 26 q^{53} + 6 q^{55} + 8 q^{56} + 8 q^{58} + 4 q^{59} - 6 q^{61} - 12 q^{62} + 6 q^{64} + 16 q^{65} + 18 q^{67} + 2 q^{68} + 20 q^{70} + 12 q^{71} - 20 q^{73} - 2 q^{76} + 20 q^{77} + 12 q^{79} + 2 q^{80} + 28 q^{82} + 12 q^{83} - 10 q^{85} + 14 q^{86} + 2 q^{88} - 2 q^{89} - 22 q^{91} + 22 q^{92} + 2 q^{94} + 6 q^{95} - 6 q^{97}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 + 8 * q^7 + 6 * q^8 + 2 * q^10 + 2 * q^11 - 2 * q^13 + 8 * q^14 + 6 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^20 + 2 * q^22 + 22 * q^23 + 12 * q^25 - 2 * q^26 + 8 * q^28 + 8 * q^29 - 12 * q^31 + 6 * q^32 + 2 * q^34 + 20 * q^35 - 2 * q^38 + 2 * q^40 + 28 * q^41 + 14 * q^43 + 2 * q^44 + 22 * q^46 + 2 * q^47 + 12 * q^50 - 2 * q^52 + 26 * q^53 + 6 * q^55 + 8 * q^56 + 8 * q^58 + 4 * q^59 - 6 * q^61 - 12 * q^62 + 6 * q^64 + 16 * q^65 + 18 * q^67 + 2 * q^68 + 20 * q^70 + 12 * q^71 - 20 * q^73 - 2 * q^76 + 20 * q^77 + 12 * q^79 + 2 * q^80 + 28 * q^82 + 12 * q^83 - 10 * q^85 + 14 * q^86 + 2 * q^88 - 2 * q^89 - 22 * q^91 + 22 * q^92 + 2 * q^94 + 6 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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