LMFDB - Embedded newform 3969.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, 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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3969 = 3^{4} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3969.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.574857.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 3x - 3$ x^5 - 2x^4 - 5x^3 + 9x^2 + 3x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.84124$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84124 q^{2} +1.39017 q^{4} +1.33475 q^{5} +1.12285 q^{8} -2.45760 q^{10} +1.51302 q^{11} +5.17599 q^{13} -4.84777 q^{16} -1.54893 q^{17} +2.50422 q^{19} +1.85553 q^{20} -2.78583 q^{22} -7.36079 q^{23} -3.21843 q^{25} -9.53025 q^{26} +0.0619427 q^{29} -3.84777 q^{31} +6.68021 q^{32} +2.85195 q^{34} +0.563216 q^{37} -4.61087 q^{38} +1.49873 q^{40} -9.02376 q^{41} -10.1998 q^{43} +2.10335 q^{44} +13.5530 q^{46} -9.51851 q^{47} +5.92591 q^{50} +7.19550 q^{52} -1.51075 q^{53} +2.01950 q^{55} -0.114051 q^{58} -8.44331 q^{59} +3.23917 q^{61} +7.08467 q^{62} -2.60434 q^{64} +6.90868 q^{65} +6.93339 q^{67} -2.15327 q^{68} -12.3304 q^{71} +2.75871 q^{73} -1.03702 q^{74} +3.48128 q^{76} -5.91938 q^{79} -6.47058 q^{80} +16.6149 q^{82} -5.60222 q^{83} -2.06744 q^{85} +18.7802 q^{86} +1.69889 q^{88} -1.40657 q^{89} -10.2327 q^{92} +17.5259 q^{94} +3.34251 q^{95} +12.1943 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$5 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{8} + 7 q^{10} - 4 q^{11} + 8 q^{13} - 2 q^{16} - 12 q^{17} - q^{19} - 5 q^{20} + q^{22} - 3 q^{23} + q^{25} - 11 q^{26} - 7 q^{29} + 3 q^{31} + 2 q^{32} - 3 q^{34}+ \cdots + 12 q^{97}+O(q^{100})$ 5 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^8 + 7 * q^10 - 4 * q^11 + 8 * q^13 - 2 * q^16 - 12 * q^17 - q^19 - 5 * q^20 + q^22 - 3 * q^23 + q^25 - 11 * q^26 - 7 * q^29 + 3 * q^31 + 2 * q^32 - 3 * q^34 - 20 * q^38 + 3 * q^40 - 5 * q^41 + 7 * q^43 + 10 * q^44 - 3 * q^46 - 27 * q^47 - 19 * q^50 + 10 * q^52 + 21 * q^53 + 2 * q^55 + 10 * q^58 - 30 * q^59 + 14 * q^61 - 6 * q^62 - 25 * q^64 + 11 * q^65 + 2 * q^67 - 27 * q^68 - 3 * q^71 - 15 * q^73 + 36 * q^74 - 5 * q^76 + 4 * q^79 - 20 * q^80 + 5 * q^82 - 9 * q^83 + 6 * q^85 + 8 * q^86 + 18 * q^88 - 28 * q^89 - 27 * q^92 + 3 * q^94 + 14 * q^95 + 12 * q^97

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

)

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

$2$	−1.84124	−1.30195	−0.650977	−	0.759098i	$-0.725641\pi$
$2$	−1.84124	−1.30195	−0.650977	+	0.759098i	$0.725641\pi$
$3$	0	0
$3$	0	0
$4$	1.39017	0.695084
$5$	1.33475	0.596920	0.298460	−	0.954422i	$-0.403527\pi$
$5$	1.33475	0.596920	0.298460	+	0.954422i	$0.403527\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.12285	0.396987
$9$	0	0
$10$	−2.45760	−0.777162
$11$	1.51302	0.456192	0.228096	−	0.973639i	$-0.426750\pi$
$11$	1.51302	0.456192	0.228096	+	0.973639i	$0.426750\pi$
$12$	0	0
$13$	5.17599	1.43556	0.717781	−	0.696269i	$-0.245158\pi$
$13$	5.17599	1.43556	0.717781	+	0.696269i	$0.245158\pi$
$14$	0	0
$15$	0	0
$16$	−4.84777	−1.21194
$17$	−1.54893	−0.375670	−0.187835	−	0.982201i	$-0.560147\pi$
$17$	−1.54893	−0.375670	−0.187835	+	0.982201i	$0.560147\pi$
$18$	0	0
$19$	2.50422	0.574507	0.287254	−	0.957855i	$-0.407258\pi$
$19$	2.50422	0.574507	0.287254	+	0.957855i	$0.407258\pi$
$20$	1.85553	0.414909
$21$	0	0
$22$	−2.78583	−0.593940
$23$	−7.36079	−1.53483	−0.767415	−	0.641151i	$-0.778457\pi$
$23$	−7.36079	−1.53483	−0.767415	+	0.641151i	$0.778457\pi$
$24$	0	0
$25$	−3.21843	−0.643687
$26$	−9.53025	−1.86904
$27$	0	0
$28$	0	0
$29$	0.0619427	0.0115025	0.00575123	−	0.999983i	$-0.498169\pi$
$29$	0.0619427	0.0115025	0.00575123	+	0.999983i	$0.498169\pi$
$30$	0	0
$31$	−3.84777	−0.691080	−0.345540	−	0.938404i	$-0.612304\pi$
$31$	−3.84777	−0.691080	−0.345540	+	0.938404i	$0.612304\pi$
$32$	6.68021	1.18091
$33$	0	0
$34$	2.85195	0.489105
$35$	0	0
$36$	0	0
$37$	0.563216	0.0925922	0.0462961	−	0.998928i	$-0.485258\pi$
$37$	0.563216	0.0925922	0.0462961	+	0.998928i	$0.485258\pi$
$38$	−4.61087	−0.747982
$39$	0	0
$40$	1.49873	0.236969
$41$	−9.02376	−1.40928	−0.704638	−	0.709567i	$-0.748890\pi$
$41$	−9.02376	−1.40928	−0.704638	+	0.709567i	$0.748890\pi$
$42$	0	0
$43$	−10.1998	−1.55545	−0.777724	−	0.628606i	$-0.783626\pi$
$43$	−10.1998	−1.55545	−0.777724	+	0.628606i	$0.783626\pi$
$44$	2.10335	0.317091
$45$	0	0
$46$	13.5530	1.99828
$47$	−9.51851	−1.38842	−0.694209	−	0.719774i	$-0.744245\pi$
$47$	−9.51851	−1.38842	−0.694209	+	0.719774i	$0.744245\pi$
$48$	0	0
$49$	0	0
$50$	5.92591	0.838050
$51$	0	0
$52$	7.19550	0.997836
$53$	−1.51075	−0.207517	−0.103759	−	0.994603i	$-0.533087\pi$
$53$	−1.51075	−0.207517	−0.103759	+	0.994603i	$0.533087\pi$
$54$	0	0
$55$	2.01950	0.272310
$56$	0	0
$57$	0	0
$58$	−0.114051	−0.0149757
$59$	−8.44331	−1.09923	−0.549613	−	0.835419i	$-0.685225\pi$
$59$	−8.44331	−1.09923	−0.549613	+	0.835419i	$0.685225\pi$
$60$	0	0
$61$	3.23917	0.414733	0.207367	−	0.978263i	$-0.433511\pi$
$61$	3.23917	0.414733	0.207367	+	0.978263i	$0.433511\pi$
$62$	7.08467	0.899754
$63$	0	0
$64$	−2.60434	−0.325543
$65$	6.90868	0.856916
$66$	0	0
$67$	6.93339	0.847049	0.423524	−	0.905885i	$-0.360793\pi$
$67$	6.93339	0.847049	0.423524	+	0.905885i	$0.360793\pi$
$68$	−2.15327	−0.261122
$69$	0	0
$70$	0	0
$71$	−12.3304	−1.46335	−0.731673	−	0.681656i	$-0.761260\pi$
$71$	−12.3304	−1.46335	−0.731673	+	0.681656i	$0.761260\pi$
$72$	0	0
$73$	2.75871	0.322883	0.161442	−	0.986882i	$-0.448386\pi$
$73$	2.75871	0.322883	0.161442	+	0.986882i	$0.448386\pi$
$74$	−1.03702	−0.120551
$75$	0	0
$76$	3.48128	0.399331
$77$	0	0
$78$	0	0
$79$	−5.91938	−0.665982	−0.332991	−	0.942930i	$-0.608058\pi$
$79$	−5.91938	−0.665982	−0.332991	+	0.942930i	$0.608058\pi$
$80$	−6.47058	−0.723432
$81$	0	0
$82$	16.6149	1.83481
$83$	−5.60222	−0.614924	−0.307462	−	0.951560i	$-0.599480\pi$
$83$	−5.60222	−0.614924	−0.307462	+	0.951560i	$0.599480\pi$
$84$	0	0
$85$	−2.06744	−0.224245
$86$	18.7802	2.02512
$87$	0	0
$88$	1.69889	0.181102
$89$	−1.40657	−0.149097	−0.0745483	−	0.997217i	$-0.523751\pi$
$89$	−1.40657	−0.149097	−0.0745483	+	0.997217i	$0.523751\pi$
$90$	0	0
$91$	0	0
$92$	−10.2327	−1.06684
$93$	0	0
$94$	17.5259	1.80765
$95$	3.34251	0.342935
$96$	0	0
$97$	12.1943	1.23814	0.619070	−	0.785336i	$-0.287510\pi$
$97$	12.1943	1.23814	0.619070	+	0.785336i	$0.287510\pi$
$98$	0	0
$99$	0	0
$100$	−4.47416	−0.447416
$101$	1.11867	0.111312	0.0556560	−	0.998450i	$-0.482275\pi$
$101$	1.11867	0.111312	0.0556560	+	0.998450i	$0.482275\pi$
$102$	0	0
$103$	1.93045	0.190213	0.0951063	−	0.995467i	$-0.469681\pi$
$103$	1.93045	0.190213	0.0951063	+	0.995467i	$0.469681\pi$
$104$	5.81186	0.569900
$105$	0	0
$106$	2.78165	0.270178
$107$	−5.77938	−0.558714	−0.279357	−	0.960187i	$-0.590121\pi$
$107$	−5.77938	−0.558714	−0.279357	+	0.960187i	$0.590121\pi$
$108$	0	0
$109$	8.24211	0.789451	0.394726	−	0.918799i	$-0.370840\pi$
$109$	8.24211	0.789451	0.394726	+	0.918799i	$0.370840\pi$
$110$	−3.71839	−0.354535
$111$	0	0
$112$	0	0
$113$	−14.5021	−1.36424	−0.682121	−	0.731239i	$-0.738942\pi$
$113$	−14.5021	−1.36424	−0.682121	+	0.731239i	$0.738942\pi$
$114$	0	0
$115$	−9.82483	−0.916170
$116$	0.0861107	0.00799518
$117$	0	0
$118$	15.5462	1.43114
$119$	0	0
$120$	0	0
$121$	−8.71078	−0.791889
$122$	−5.96409	−0.539963
$123$	0	0
$124$	−5.34904	−0.480358
$125$	−10.9696	−0.981149
$126$	0	0
$127$	8.50004	0.754257	0.377128	−	0.926161i	$-0.376912\pi$
$127$	8.50004	0.754257	0.377128	+	0.926161i	$0.376912\pi$
$128$	−8.56521	−0.757065
$129$	0	0
$130$	−12.7205	−1.11566
$131$	−2.01346	−0.175917	−0.0879585	−	0.996124i	$-0.528034\pi$
$131$	−2.01346	−0.175917	−0.0879585	+	0.996124i	$0.528034\pi$
$132$	0	0
$133$	0	0
$134$	−12.7660	−1.10282
$135$	0	0
$136$	−1.73921	−0.149136
$137$	2.21740	0.189445	0.0947225	−	0.995504i	$-0.469804\pi$
$137$	2.21740	0.189445	0.0947225	+	0.995504i	$0.469804\pi$
$138$	0	0
$139$	−0.755339	−0.0640670	−0.0320335	−	0.999487i	$-0.510198\pi$
$139$	−0.755339	−0.0640670	−0.0320335	+	0.999487i	$0.510198\pi$
$140$	0	0
$141$	0	0
$142$	22.7032	1.90521
$143$	7.83136	0.654891
$144$	0	0
$145$	0.0826782	0.00686605
$146$	−5.07946	−0.420379
$147$	0	0
$148$	0.782965	0.0643593
$149$	6.58499	0.539463	0.269732	−	0.962936i	$-0.413065\pi$
$149$	6.58499	0.539463	0.269732	+	0.962936i	$0.413065\pi$
$150$	0	0
$151$	12.6671	1.03083	0.515417	−	0.856939i	$-0.327637\pi$
$151$	12.6671	1.03083	0.515417	+	0.856939i	$0.327637\pi$
$152$	2.81186	0.228072
$153$	0	0
$154$	0	0
$155$	−5.13582	−0.412519
$156$	0	0
$157$	−17.3074	−1.38128	−0.690642	−	0.723197i	$-0.742672\pi$
$157$	−17.3074	−1.38128	−0.690642	+	0.723197i	$0.742672\pi$
$158$	10.8990	0.867078
$159$	0	0
$160$	8.91644	0.704906
$161$	0	0
$162$	0	0
$163$	−12.2193	−0.957086	−0.478543	−	0.878064i	$-0.658835\pi$
$163$	−12.2193	−0.957086	−0.478543	+	0.878064i	$0.658835\pi$
$164$	−12.5445	−0.979564
$165$	0	0
$166$	10.3150	0.800602
$167$	−3.52495	−0.272769	−0.136385	−	0.990656i	$-0.543548\pi$
$167$	−3.52495	−0.272769	−0.136385	+	0.990656i	$0.543548\pi$
$168$	0	0
$169$	13.7909	1.06084
$170$	3.80665	0.291956
$171$	0	0
$172$	−14.1794	−1.08117
$173$	10.1409	0.770999	0.385500	−	0.922708i	$-0.374029\pi$
$173$	10.1409	0.770999	0.385500	+	0.922708i	$0.374029\pi$
$174$	0	0
$175$	0	0
$176$	−7.33475	−0.552878
$177$	0	0
$178$	2.58984	0.194117
$179$	−1.70116	−0.127150	−0.0635752	−	0.997977i	$-0.520250\pi$
$179$	−1.70116	−0.127150	−0.0635752	+	0.997977i	$0.520250\pi$
$180$	0	0
$181$	−16.9941	−1.26316	−0.631581	−	0.775310i	$-0.717594\pi$
$181$	−16.9941	−1.26316	−0.631581	+	0.775310i	$0.717594\pi$
$182$	0	0
$183$	0	0
$184$	−8.26505	−0.609308
$185$	0.751755	0.0552701
$186$	0	0
$187$	−2.34355	−0.171377
$188$	−13.2323	−0.965066
$189$	0	0
$190$	−6.15437	−0.446485
$191$	22.6939	1.64208	0.821038	−	0.570873i	$-0.193395\pi$
$191$	22.6939	1.64208	0.821038	+	0.570873i	$0.193395\pi$
$192$	0	0
$193$	6.18698	0.445348	0.222674	−	0.974893i	$-0.428521\pi$
$193$	6.18698	0.445348	0.222674	+	0.974893i	$0.428521\pi$
$194$	−22.4526	−1.61200
$195$	0	0
$196$	0	0
$197$	9.77010	0.696091	0.348045	−	0.937478i	$-0.386846\pi$
$197$	9.77010	0.696091	0.348045	+	0.937478i	$0.386846\pi$
$198$	0	0
$199$	8.67947	0.615271	0.307636	−	0.951504i	$-0.400462\pi$
$199$	8.67947	0.615271	0.307636	+	0.951504i	$0.400462\pi$
$200$	−3.61381	−0.255535
$201$	0	0
$202$	−2.05974	−0.144923
$203$	0	0
$204$	0	0
$205$	−12.0445	−0.841224
$206$	−3.55442	−0.247648
$207$	0	0
$208$	−25.0920	−1.73982
$209$	3.78892	0.262085
$210$	0	0
$211$	5.68439	0.391330	0.195665	−	0.980671i	$-0.437314\pi$
$211$	5.68439	0.391330	0.195665	+	0.980671i	$0.437314\pi$
$212$	−2.10019	−0.144242
$213$	0	0
$214$	10.6412	0.727420
$215$	−13.6142	−0.928478
$216$	0	0
$217$	0	0
$218$	−15.1757	−1.02783
$219$	0	0
$220$	2.80745	0.189278
$221$	−8.01723	−0.539298
$222$	0	0
$223$	−11.7227	−0.785007	−0.392503	−	0.919751i	$-0.628391\pi$
$223$	−11.7227	−0.785007	−0.392503	+	0.919751i	$0.628391\pi$
$224$	0	0
$225$	0	0
$226$	26.7019	1.77618
$227$	11.1831	0.742247	0.371123	−	0.928584i	$-0.378973\pi$
$227$	11.1831	0.742247	0.371123	+	0.928584i	$0.378973\pi$
$228$	0	0
$229$	−9.65647	−0.638118	−0.319059	−	0.947735i	$-0.603367\pi$
$229$	−9.65647	−0.638118	−0.319059	+	0.947735i	$0.603367\pi$
$230$	18.0899	1.19281
$231$	0	0
$232$	0.0695523	0.00456633
$233$	19.2898	1.26372	0.631860	−	0.775083i	$-0.282292\pi$
$233$	19.2898	1.26372	0.631860	+	0.775083i	$0.282292\pi$
$234$	0	0
$235$	−12.7049	−0.828774
$236$	−11.7376	−0.764054
$237$	0	0
$238$	0	0
$239$	0.389282	0.0251806	0.0125903	−	0.999921i	$-0.495992\pi$
$239$	0.389282	0.0251806	0.0125903	+	0.999921i	$0.495992\pi$
$240$	0	0
$241$	10.6361	0.685134	0.342567	−	0.939493i	$-0.388704\pi$
$241$	10.6361	0.685134	0.342567	+	0.939493i	$0.388704\pi$
$242$	16.0386	1.03100
$243$	0	0
$244$	4.50299	0.288274
$245$	0	0
$246$	0	0
$247$	12.9618	0.824741
$248$	−4.32046	−0.274350
$249$	0	0
$250$	20.1976	1.27741
$251$	−3.26628	−0.206166	−0.103083	−	0.994673i	$-0.532871\pi$
$251$	−3.26628	−0.206166	−0.103083	+	0.994673i	$0.532871\pi$
$252$	0	0
$253$	−11.1370	−0.700176
$254$	−15.6506	−0.982007
$255$	0	0
$256$	20.9793	1.31121
$257$	−4.69573	−0.292912	−0.146456	−	0.989217i	$-0.546787\pi$
$257$	−4.69573	−0.292912	−0.146456	+	0.989217i	$0.546787\pi$
$258$	0	0
$259$	0	0
$260$	9.60421	0.595628
$261$	0	0
$262$	3.70727	0.229036
$263$	19.5498	1.20549	0.602747	−	0.797932i	$-0.294073\pi$
$263$	19.5498	1.20549	0.602747	+	0.797932i	$0.294073\pi$
$264$	0	0
$265$	−2.01648	−0.123871
$266$	0	0
$267$	0	0
$268$	9.63858	0.588770
$269$	−15.7673	−0.961349	−0.480675	−	0.876899i	$-0.659608\pi$
$269$	−15.7673	−0.961349	−0.480675	+	0.876899i	$0.659608\pi$
$270$	0	0
$271$	−14.7976	−0.898893	−0.449446	−	0.893307i	$-0.648379\pi$
$271$	−14.7976	−0.898893	−0.449446	+	0.893307i	$0.648379\pi$
$272$	7.50884	0.455290
$273$	0	0
$274$	−4.08276	−0.246649
$275$	−4.86954	−0.293644
$276$	0	0
$277$	−7.45122	−0.447701	−0.223850	−	0.974624i	$-0.571863\pi$
$277$	−7.45122	−0.447701	−0.223850	+	0.974624i	$0.571863\pi$
$278$	1.39076	0.0834123
$279$	0	0
$280$	0	0
$281$	25.9876	1.55029	0.775146	−	0.631782i	$-0.217676\pi$
$281$	25.9876	1.55029	0.775146	+	0.631782i	$0.217676\pi$
$282$	0	0
$283$	18.7554	1.11489	0.557445	−	0.830214i	$-0.311782\pi$
$283$	18.7554	1.11489	0.557445	+	0.830214i	$0.311782\pi$
$284$	−17.1413	−1.01715
$285$	0	0
$286$	−14.4194	−0.852638
$287$	0	0
$288$	0	0
$289$	−14.6008	−0.858872
$290$	−0.152230	−0.00893928
$291$	0	0
$292$	3.83507	0.224431
$293$	2.46178	0.143819	0.0719093	−	0.997411i	$-0.477091\pi$
$293$	2.46178	0.143819	0.0719093	+	0.997411i	$0.477091\pi$
$294$	0	0
$295$	−11.2697	−0.656150
$296$	0.632407	0.0367579
$297$	0	0
$298$	−12.1245	−0.702356
$299$	−38.0994	−2.20334
$300$	0	0
$301$	0	0
$302$	−23.3232	−1.34210
$303$	0	0
$304$	−12.1399	−0.696270
$305$	4.32349	0.247562
$306$	0	0
$307$	−4.66277	−0.266118	−0.133059	−	0.991108i	$-0.542480\pi$
$307$	−4.66277	−0.266118	−0.133059	+	0.991108i	$0.542480\pi$
$308$	0	0
$309$	0	0
$310$	9.45629	0.537081
$311$	27.4821	1.55837	0.779183	−	0.626797i	$-0.215634\pi$
$311$	27.4821	1.55837	0.779183	+	0.626797i	$0.215634\pi$
$312$	0	0
$313$	5.49332	0.310501	0.155250	−	0.987875i	$-0.450382\pi$
$313$	5.49332	0.310501	0.155250	+	0.987875i	$0.450382\pi$
$314$	31.8671	1.79837
$315$	0	0
$316$	−8.22893	−0.462914
$317$	9.87758	0.554780	0.277390	−	0.960757i	$-0.410531\pi$
$317$	9.87758	0.554780	0.277390	+	0.960757i	$0.410531\pi$
$318$	0	0
$319$	0.0937203	0.00524733
$320$	−3.47615	−0.194323
$321$	0	0
$322$	0	0
$323$	−3.87885	−0.215825
$324$	0	0
$325$	−16.6586	−0.924052
$326$	22.4986	1.24608
$327$	0	0
$328$	−10.1323	−0.559464
$329$	0	0
$330$	0	0
$331$	−20.6942	−1.13746	−0.568729	−	0.822525i	$-0.692565\pi$
$331$	−20.6942	−1.13746	−0.568729	+	0.822525i	$0.692565\pi$
$332$	−7.78803	−0.427424
$333$	0	0
$334$	6.49029	0.355133
$335$	9.25437	0.505620
$336$	0	0
$337$	−1.49749	−0.0815737	−0.0407869	−	0.999168i	$-0.512986\pi$
$337$	−1.49749	−0.0815737	−0.0407869	+	0.999168i	$0.512986\pi$
$338$	−25.3924	−1.38116
$339$	0	0
$340$	−2.87408	−0.155869
$341$	−5.82174	−0.315265
$342$	0	0
$343$	0	0
$344$	−11.4528	−0.617493
$345$	0	0
$346$	−18.6719	−1.00381
$347$	−29.5388	−1.58572	−0.792862	−	0.609401i	$-0.791410\pi$
$347$	−29.5388	−1.58572	−0.792862	+	0.609401i	$0.791410\pi$
$348$	0	0
$349$	−36.0013	−1.92710	−0.963551	−	0.267523i	$-0.913795\pi$
$349$	−36.0013	−1.92710	−0.963551	+	0.267523i	$0.913795\pi$
$350$	0	0
$351$	0	0
$352$	10.1073	0.538719
$353$	−29.4930	−1.56975	−0.784877	−	0.619652i	$-0.787274\pi$
$353$	−29.4930	−1.56975	−0.784877	+	0.619652i	$0.787274\pi$
$354$	0	0
$355$	−16.4580	−0.873500
$356$	−1.95537	−0.103635
$357$	0	0
$358$	3.13224	0.165544
$359$	−5.41069	−0.285566	−0.142783	−	0.989754i	$-0.545605\pi$
$359$	−5.41069	−0.285566	−0.142783	+	0.989754i	$0.545605\pi$
$360$	0	0
$361$	−12.7289	−0.669942
$362$	31.2902	1.64458
$363$	0	0
$364$	0	0
$365$	3.68220	0.192735
$366$	0	0
$367$	−23.0843	−1.20499	−0.602496	−	0.798122i	$-0.705827\pi$
$367$	−23.0843	−1.20499	−0.602496	+	0.798122i	$0.705827\pi$
$368$	35.6834	1.86013
$369$	0	0
$370$	−1.38416	−0.0719591
$371$	0	0
$372$	0	0
$373$	21.5030	1.11338	0.556692	−	0.830719i	$-0.312070\pi$
$373$	21.5030	1.11338	0.556692	+	0.830719i	$0.312070\pi$
$374$	4.31504	0.223125
$375$	0	0
$376$	−10.6878	−0.551184
$377$	0.320615	0.0165125
$378$	0	0
$379$	5.72168	0.293903	0.146952	−	0.989144i	$-0.453054\pi$
$379$	5.72168	0.293903	0.146952	+	0.989144i	$0.453054\pi$
$380$	4.64665	0.238368
$381$	0	0
$382$	−41.7850	−2.13791
$383$	−34.9209	−1.78437	−0.892187	−	0.451666i	$-0.850830\pi$
$383$	−34.9209	−1.78437	−0.892187	+	0.451666i	$0.850830\pi$
$384$	0	0
$385$	0	0
$386$	−11.3917	−0.579823
$387$	0	0
$388$	16.9521	0.860611
$389$	−28.8822	−1.46438	−0.732192	−	0.681098i	$-0.761503\pi$
$389$	−28.8822	−1.46438	−0.732192	+	0.681098i	$0.761503\pi$
$390$	0	0
$391$	11.4013	0.576589
$392$	0	0
$393$	0	0
$394$	−17.9891	−0.906278
$395$	−7.90091	−0.397538
$396$	0	0
$397$	−11.1845	−0.561335	−0.280667	−	0.959805i	$-0.590556\pi$
$397$	−11.1845	−0.561335	−0.280667	+	0.959805i	$0.590556\pi$
$398$	−15.9810	−0.801055
$399$	0	0
$400$	15.6022	0.780111
$401$	−1.08212	−0.0540386	−0.0270193	−	0.999635i	$-0.508602\pi$
$401$	−1.08212	−0.0540386	−0.0270193	+	0.999635i	$0.508602\pi$
$402$	0	0
$403$	−19.9160	−0.992088
$404$	1.55514	0.0773711
$405$	0	0
$406$	0	0
$407$	0.852155	0.0422398
$408$	0	0
$409$	−21.7349	−1.07472	−0.537360	−	0.843353i	$-0.680578\pi$
$409$	−21.7349	−1.07472	−0.537360	+	0.843353i	$0.680578\pi$
$410$	22.1768	1.09524
$411$	0	0
$412$	2.68365	0.132214
$413$	0	0
$414$	0	0
$415$	−7.47759	−0.367060
$416$	34.5767	1.69526
$417$	0	0
$418$	−6.97632	−0.341223
$419$	−25.1811	−1.23018	−0.615090	−	0.788457i	$-0.710880\pi$
$419$	−25.1811	−1.23018	−0.615090	+	0.788457i	$0.710880\pi$
$420$	0	0
$421$	29.6607	1.44558	0.722788	−	0.691070i	$-0.242860\pi$
$421$	29.6607	1.44558	0.722788	+	0.691070i	$0.242860\pi$
$422$	−10.4663	−0.509493
$423$	0	0
$424$	−1.69634	−0.0823816
$425$	4.98512	0.241814
$426$	0	0
$427$	0	0
$428$	−8.03431	−0.388353
$429$	0	0
$430$	25.0669	1.20884
$431$	−4.89034	−0.235559	−0.117780	−	0.993040i	$-0.537578\pi$
$431$	−4.89034	−0.235559	−0.117780	+	0.993040i	$0.537578\pi$
$432$	0	0
$433$	9.71430	0.466839	0.233420	−	0.972376i	$-0.425008\pi$
$433$	9.71430	0.466839	0.233420	+	0.972376i	$0.425008\pi$
$434$	0	0
$435$	0	0
$436$	11.4579	0.548735
$437$	−18.4330	−0.881771
$438$	0	0
$439$	−14.8235	−0.707488	−0.353744	−	0.935342i	$-0.615092\pi$
$439$	−14.8235	−0.707488	−0.353744	+	0.935342i	$0.615092\pi$
$440$	2.26760	0.108103
$441$	0	0
$442$	14.7617	0.702141
$443$	−21.9020	−1.04059	−0.520297	−	0.853986i	$-0.674179\pi$
$443$	−21.9020	−1.04059	−0.520297	+	0.853986i	$0.674179\pi$
$444$	0	0
$445$	−1.87743	−0.0889987
$446$	21.5842	1.02204
$447$	0	0
$448$	0	0
$449$	21.4952	1.01442	0.507212	−	0.861822i	$-0.330676\pi$
$449$	21.4952	1.01442	0.507212	+	0.861822i	$0.330676\pi$
$450$	0	0
$451$	−13.6531	−0.642899
$452$	−20.1603	−0.948263
$453$	0	0
$454$	−20.5907	−0.966371
$455$	0	0
$456$	0	0
$457$	40.6255	1.90038	0.950190	−	0.311670i	$-0.100888\pi$
$457$	40.6255	1.90038	0.950190	+	0.311670i	$0.100888\pi$
$458$	17.7799	0.830800
$459$	0	0
$460$	−13.6582	−0.636815
$461$	−2.83081	−0.131844	−0.0659220	−	0.997825i	$-0.520999\pi$
$461$	−2.83081	−0.131844	−0.0659220	+	0.997825i	$0.520999\pi$
$462$	0	0
$463$	27.8648	1.29499	0.647494	−	0.762070i	$-0.275817\pi$
$463$	27.8648	1.29499	0.647494	+	0.762070i	$0.275817\pi$
$464$	−0.300284	−0.0139403
$465$	0	0
$466$	−35.5173	−1.64531
$467$	26.6438	1.23293	0.616464	−	0.787383i	$-0.288564\pi$
$467$	26.6438	1.23293	0.616464	+	0.787383i	$0.288564\pi$
$468$	0	0
$469$	0	0
$470$	23.3927	1.07903
$471$	0	0
$472$	−9.48056	−0.436378
$473$	−15.4324	−0.709582
$474$	0	0
$475$	−8.05966	−0.369803
$476$	0	0
$477$	0	0
$478$	−0.716762	−0.0327839
$479$	−31.5791	−1.44289	−0.721443	−	0.692474i	$-0.756521\pi$
$479$	−31.5791	−1.44289	−0.721443	+	0.692474i	$0.756521\pi$
$480$	0	0
$481$	2.91520	0.132922
$482$	−19.5837	−0.892013
$483$	0	0
$484$	−12.1094	−0.550429
$485$	16.2763	0.739070
$486$	0	0
$487$	0.306174	0.0138741	0.00693703	−	0.999976i	$-0.497792\pi$
$487$	0.306174	0.0138741	0.00693703	+	0.999976i	$0.497792\pi$
$488$	3.63710	0.164644
$489$	0	0
$490$	0	0
$491$	18.1396	0.818629	0.409315	−	0.912393i	$-0.365768\pi$
$491$	18.1396	0.818629	0.409315	+	0.912393i	$0.365768\pi$
$492$	0	0
$493$	−0.0959447	−0.00432113
$494$	−23.8658	−1.07377
$495$	0	0
$496$	18.6531	0.837549
$497$	0	0
$498$	0	0
$499$	−21.3091	−0.953928	−0.476964	−	0.878923i	$-0.658263\pi$
$499$	−21.3091	−0.953928	−0.476964	+	0.878923i	$0.658263\pi$
$500$	−15.2496	−0.681981
$501$	0	0
$502$	6.01401	0.268418
$503$	−17.0738	−0.761285	−0.380642	−	0.924722i	$-0.624297\pi$
$503$	−17.0738	−0.761285	−0.380642	+	0.924722i	$0.624297\pi$
$504$	0	0
$505$	1.49315	0.0664443
$506$	20.5059	0.911597
$507$	0	0
$508$	11.8165	0.524271
$509$	36.7735	1.62996	0.814979	−	0.579490i	$-0.196748\pi$
$509$	36.7735	1.62996	0.814979	+	0.579490i	$0.196748\pi$
$510$	0	0
$511$	0	0
$512$	−21.4975	−0.950065
$513$	0	0
$514$	8.64598	0.381358
$515$	2.57667	0.113542
$516$	0	0
$517$	−14.4017	−0.633384
$518$	0	0
$519$	0	0
$520$	7.75740	0.340184
$521$	19.1507	0.839008	0.419504	−	0.907754i	$-0.362204\pi$
$521$	19.1507	0.839008	0.419504	+	0.907754i	$0.362204\pi$
$522$	0	0
$523$	41.9429	1.83404	0.917018	−	0.398847i	$-0.130589\pi$
$523$	41.9429	1.83404	0.917018	+	0.398847i	$0.130589\pi$
$524$	−2.79905	−0.122277
$525$	0	0
$526$	−35.9959	−1.56950
$527$	5.95991	0.259618
$528$	0	0
$529$	31.1812	1.35570
$530$	3.71282	0.161274
$531$	0	0
$532$	0	0
$533$	−46.7069	−2.02310
$534$	0	0
$535$	−7.71405	−0.333507
$536$	7.78515	0.336267
$537$	0	0
$538$	29.0314	1.25163
$539$	0	0
$540$	0	0
$541$	2.88544	0.124055	0.0620273	−	0.998074i	$-0.480243\pi$
$541$	2.88544	0.124055	0.0620273	+	0.998074i	$0.480243\pi$
$542$	27.2460	1.17032
$543$	0	0
$544$	−10.3472	−0.443631
$545$	11.0012	0.471239
$546$	0	0
$547$	−2.77476	−0.118640	−0.0593201	−	0.998239i	$-0.518893\pi$
$547$	−2.77476	−0.118640	−0.0593201	+	0.998239i	$0.518893\pi$
$548$	3.08255	0.131680
$549$	0	0
$550$	8.96600	0.382311
$551$	0.155118	0.00660825
$552$	0	0
$553$	0	0
$554$	13.7195	0.582886
$555$	0	0
$556$	−1.05005	−0.0445319
$557$	−31.0688	−1.31643	−0.658214	−	0.752831i	$-0.728688\pi$
$557$	−31.0688	−1.31643	−0.658214	+	0.752831i	$0.728688\pi$
$558$	0	0
$559$	−52.7939	−2.23294
$560$	0	0
$561$	0	0
$562$	−47.8495	−2.01841
$563$	0.288041	0.0121395	0.00606973	−	0.999982i	$-0.498068\pi$
$563$	0.288041	0.0121395	0.00606973	+	0.999982i	$0.498068\pi$
$564$	0	0
$565$	−19.3567	−0.814344
$566$	−34.5331	−1.45154
$567$	0	0
$568$	−13.8451	−0.580929
$569$	−16.0801	−0.674112	−0.337056	−	0.941485i	$-0.609431\pi$
$569$	−16.0801	−0.674112	−0.337056	+	0.941485i	$0.609431\pi$
$570$	0	0
$571$	−15.2858	−0.639690	−0.319845	−	0.947470i	$-0.603631\pi$
$571$	−15.2858	−0.639690	−0.319845	+	0.947470i	$0.603631\pi$
$572$	10.8869	0.455204
$573$	0	0
$574$	0	0
$575$	23.6902	0.987950
$576$	0	0
$577$	−24.1625	−1.00590	−0.502949	−	0.864316i	$-0.667752\pi$
$577$	−24.1625	−1.00590	−0.502949	+	0.864316i	$0.667752\pi$
$578$	26.8836	1.11821
$579$	0	0
$580$	0.114937	0.00477248
$581$	0	0
$582$	0	0
$583$	−2.28579	−0.0946676
$584$	3.09762	0.128180
$585$	0	0
$586$	−4.53273	−0.187245
$587$	−36.0289	−1.48707	−0.743537	−	0.668695i	$-0.766853\pi$
$587$	−36.0289	−1.48707	−0.743537	+	0.668695i	$0.766853\pi$
$588$	0	0
$589$	−9.63566	−0.397030
$590$	20.7503	0.854276
$591$	0	0
$592$	−2.73034	−0.112216
$593$	−24.9337	−1.02390	−0.511951	−	0.859014i	$-0.671077\pi$
$593$	−24.9337	−1.02390	−0.511951	+	0.859014i	$0.671077\pi$
$594$	0	0
$595$	0	0
$596$	9.15424	0.374972
$597$	0	0
$598$	70.1501	2.86865
$599$	39.5283	1.61508	0.807542	−	0.589810i	$-0.200797\pi$
$599$	39.5283	1.61508	0.807542	+	0.589810i	$0.200797\pi$
$600$	0	0
$601$	−3.72895	−0.152107	−0.0760534	−	0.997104i	$-0.524232\pi$
$601$	−3.72895	−0.152107	−0.0760534	+	0.997104i	$0.524232\pi$
$602$	0	0
$603$	0	0
$604$	17.6094	0.716516
$605$	−11.6267	−0.472694
$606$	0	0
$607$	23.6528	0.960036	0.480018	−	0.877259i	$-0.340630\pi$
$607$	23.6528	0.960036	0.480018	+	0.877259i	$0.340630\pi$
$608$	16.7287	0.678439
$609$	0	0
$610$	−7.96059	−0.322315
$611$	−49.2677	−1.99316
$612$	0	0
$613$	−3.79903	−0.153442	−0.0767208	−	0.997053i	$-0.524445\pi$
$613$	−3.79903	−0.153442	−0.0767208	+	0.997053i	$0.524445\pi$
$614$	8.58528	0.346474
$615$	0	0
$616$	0	0
$617$	35.1230	1.41400	0.706999	−	0.707214i	$-0.250048\pi$
$617$	35.1230	1.41400	0.706999	+	0.707214i	$0.250048\pi$
$618$	0	0
$619$	−21.1632	−0.850622	−0.425311	−	0.905047i	$-0.639835\pi$
$619$	−21.1632	−0.850622	−0.425311	+	0.905047i	$0.639835\pi$
$620$	−7.13965	−0.286735
$621$	0	0
$622$	−50.6011	−2.02892
$623$	0	0
$624$	0	0
$625$	1.45048	0.0580192
$626$	−10.1145	−0.404257
$627$	0	0
$628$	−24.0602	−0.960107
$629$	−0.872381	−0.0347841
$630$	0	0
$631$	4.74845	0.189033	0.0945164	−	0.995523i	$-0.469870\pi$
$631$	4.74845	0.189033	0.0945164	+	0.995523i	$0.469870\pi$
$632$	−6.64657	−0.264386
$633$	0	0
$634$	−18.1870	−0.722298
$635$	11.3455	0.450231
$636$	0	0
$637$	0	0
$638$	−0.172562	−0.00683178
$639$	0	0
$640$	−11.4324	−0.451907
$641$	−9.87469	−0.390027	−0.195013	−	0.980801i	$-0.562475\pi$
$641$	−9.87469	−0.390027	−0.195013	+	0.980801i	$0.562475\pi$
$642$	0	0
$643$	−43.9496	−1.73320	−0.866602	−	0.499000i	$-0.833701\pi$
$643$	−43.9496	−1.73320	−0.866602	+	0.499000i	$0.833701\pi$
$644$	0	0
$645$	0	0
$646$	7.14190	0.280994
$647$	−44.3872	−1.74504	−0.872521	−	0.488577i	$-0.837516\pi$
$647$	−44.3872	−1.74504	−0.872521	+	0.488577i	$0.837516\pi$
$648$	0	0
$649$	−12.7749	−0.501457
$650$	30.6725	1.20307
$651$	0	0
$652$	−16.9868	−0.665255
$653$	41.9912	1.64324	0.821622	−	0.570033i	$-0.193070\pi$
$653$	41.9912	1.64324	0.821622	+	0.570033i	$0.193070\pi$
$654$	0	0
$655$	−2.68748	−0.105008
$656$	43.7451	1.70796
$657$	0	0
$658$	0	0
$659$	39.2729	1.52986	0.764928	−	0.644115i	$-0.222774\pi$
$659$	39.2729	1.52986	0.764928	+	0.644115i	$0.222774\pi$
$660$	0	0
$661$	−0.186739	−0.00726330	−0.00363165	−	0.999993i	$-0.501156\pi$
$661$	−0.186739	−0.00726330	−0.00363165	+	0.999993i	$0.501156\pi$
$662$	38.1030	1.48092
$663$	0	0
$664$	−6.29045	−0.244117
$665$	0	0
$666$	0	0
$667$	−0.455947	−0.0176543
$668$	−4.90028	−0.189597
$669$	0	0
$670$	−17.0395	−0.658294
$671$	4.90091	0.189198
$672$	0	0
$673$	10.8676	0.418917	0.209458	−	0.977818i	$-0.432830\pi$
$673$	10.8676	0.418917	0.209458	+	0.977818i	$0.432830\pi$
$674$	2.75725	0.106205
$675$	0	0
$676$	19.1717	0.737372
$677$	28.3901	1.09112	0.545560	−	0.838072i	$-0.316317\pi$
$677$	28.3901	1.09112	0.545560	+	0.838072i	$0.316317\pi$
$678$	0	0
$679$	0	0
$680$	−2.32142	−0.0890223
$681$	0	0
$682$	10.7192	0.410460
$683$	−11.8407	−0.453071	−0.226536	−	0.974003i	$-0.572740\pi$
$683$	−11.8407	−0.453071	−0.226536	+	0.974003i	$0.572740\pi$
$684$	0	0
$685$	2.95968	0.113083
$686$	0	0
$687$	0	0
$688$	49.4461	1.88511
$689$	−7.81962	−0.297904
$690$	0	0
$691$	11.9083	0.453014	0.226507	−	0.974010i	$-0.427269\pi$
$691$	11.9083	0.453014	0.226507	+	0.974010i	$0.427269\pi$
$692$	14.0976	0.535909
$693$	0	0
$694$	54.3880	2.06454
$695$	−1.00819	−0.0382429
$696$	0	0
$697$	13.9771	0.529422
$698$	66.2870	2.50900
$699$	0	0
$700$	0	0
$701$	−31.3902	−1.18559	−0.592795	−	0.805353i	$-0.701976\pi$
$701$	−31.3902	−1.18559	−0.592795	+	0.805353i	$0.701976\pi$
$702$	0	0
$703$	1.41042	0.0531949
$704$	−3.94041	−0.148510
$705$	0	0
$706$	54.3037	2.04375
$707$	0	0
$708$	0	0
$709$	0.625218	0.0234806	0.0117403	−	0.999931i	$-0.496263\pi$
$709$	0.625218	0.0234806	0.0117403	+	0.999931i	$0.496263\pi$
$710$	30.3031	1.13726
$711$	0	0
$712$	−1.57937	−0.0591894
$713$	28.3226	1.06069
$714$	0	0
$715$	10.4529	0.390918
$716$	−2.36489	−0.0883802
$717$	0	0
$718$	9.96239	0.371793
$719$	−24.3939	−0.909739	−0.454869	−	0.890558i	$-0.650314\pi$
$719$	−24.3939	−0.909739	−0.454869	+	0.890558i	$0.650314\pi$
$720$	0	0
$721$	0	0
$722$	23.4369	0.872233
$723$	0	0
$724$	−23.6246	−0.878003
$725$	−0.199358	−0.00740399
$726$	0	0
$727$	37.8506	1.40380	0.701900	−	0.712275i	$-0.252335\pi$
$727$	37.8506	1.40380	0.701900	+	0.712275i	$0.252335\pi$
$728$	0	0
$729$	0	0
$730$	−6.77982	−0.250932
$731$	15.7987	0.584335
$732$	0	0
$733$	2.40155	0.0887033	0.0443516	−	0.999016i	$-0.485878\pi$
$733$	2.40155	0.0887033	0.0443516	+	0.999016i	$0.485878\pi$
$734$	42.5038	1.56884
$735$	0	0
$736$	−49.1716	−1.81249
$737$	10.4903	0.386416
$738$	0	0
$739$	30.3880	1.11784	0.558920	−	0.829222i	$-0.311216\pi$
$739$	30.3880	1.11784	0.558920	+	0.829222i	$0.311216\pi$
$740$	1.04507	0.0384174
$741$	0	0
$742$	0	0
$743$	5.09570	0.186943	0.0934715	−	0.995622i	$-0.470204\pi$
$743$	5.09570	0.186943	0.0934715	+	0.995622i	$0.470204\pi$
$744$	0	0
$745$	8.78934	0.322016
$746$	−39.5922	−1.44957
$747$	0	0
$748$	−3.25793	−0.119122
$749$	0	0
$750$	0	0
$751$	−0.975011	−0.0355787	−0.0177893	−	0.999842i	$-0.505663\pi$
$751$	−0.975011	−0.0355787	−0.0177893	+	0.999842i	$0.505663\pi$
$752$	46.1435	1.68268
$753$	0	0
$754$	−0.590329	−0.0214985
$755$	16.9075	0.615326
$756$	0	0
$757$	11.6346	0.422865	0.211433	−	0.977393i	$-0.432187\pi$
$757$	11.6346	0.422865	0.211433	+	0.977393i	$0.432187\pi$
$758$	−10.5350	−0.382648
$759$	0	0
$760$	3.75314	0.136141
$761$	−54.1749	−1.96384	−0.981920	−	0.189298i	$-0.939379\pi$
$761$	−54.1749	−1.96384	−0.981920	+	0.189298i	$0.939379\pi$
$762$	0	0
$763$	0	0
$764$	31.5484	1.14138
$765$	0	0
$766$	64.2978	2.32317
$767$	−43.7025	−1.57801
$768$	0	0
$769$	20.8652	0.752417	0.376208	−	0.926535i	$-0.377228\pi$
$769$	20.8652	0.752417	0.376208	+	0.926535i	$0.377228\pi$
$770$	0	0
$771$	0	0
$772$	8.60094	0.309554
$773$	54.9945	1.97801	0.989007	−	0.147868i	$-0.0472409\pi$
$773$	54.9945	1.97801	0.989007	+	0.147868i	$0.0472409\pi$
$774$	0	0
$775$	12.3838	0.444839
$776$	13.6923	0.491526
$777$	0	0
$778$	53.1790	1.90656
$779$	−22.5975	−0.809639
$780$	0	0
$781$	−18.6560	−0.667566
$782$	−20.9926	−0.750693
$783$	0	0
$784$	0	0
$785$	−23.1012	−0.824516
$786$	0	0
$787$	9.18949	0.327570	0.163785	−	0.986496i	$-0.447630\pi$
$787$	9.18949	0.327570	0.163785	+	0.986496i	$0.447630\pi$
$788$	13.5821	0.483841
$789$	0	0
$790$	14.5475	0.517576
$791$	0	0
$792$	0	0
$793$	16.7659	0.595375
$794$	20.5934	0.730832
$795$	0	0
$796$	12.0659	0.427665
$797$	−7.07547	−0.250626	−0.125313	−	0.992117i	$-0.539994\pi$
$797$	−7.07547	−0.250626	−0.125313	+	0.992117i	$0.539994\pi$
$798$	0	0
$799$	14.7435	0.521586
$800$	−21.4998	−0.760133
$801$	0	0
$802$	1.99245	0.0703558
$803$	4.17398	0.147297
$804$	0	0
$805$	0	0
$806$	36.6702	1.29165
$807$	0	0
$808$	1.25610	0.0441894
$809$	5.94119	0.208881	0.104441	−	0.994531i	$-0.466695\pi$
$809$	5.94119	0.208881	0.104441	+	0.994531i	$0.466695\pi$
$810$	0	0
$811$	44.4139	1.55958	0.779791	−	0.626039i	$-0.215325\pi$
$811$	44.4139	1.55958	0.779791	+	0.626039i	$0.215325\pi$
$812$	0	0
$813$	0	0
$814$	−1.56902	−0.0549942
$815$	−16.3097	−0.571304
$816$	0	0
$817$	−25.5424	−0.893616
$818$	40.0191	1.39924
$819$	0	0
$820$	−16.7439	−0.584721
$821$	6.35522	0.221799	0.110899	−	0.993832i	$-0.464627\pi$
$821$	6.35522	0.221799	0.110899	+	0.993832i	$0.464627\pi$
$822$	0	0
$823$	−9.46433	−0.329906	−0.164953	−	0.986301i	$-0.552747\pi$
$823$	−9.46433	−0.329906	−0.164953	+	0.986301i	$0.552747\pi$
$824$	2.16760	0.0755120
$825$	0	0
$826$	0	0
$827$	−4.86261	−0.169090	−0.0845448	−	0.996420i	$-0.526944\pi$
$827$	−4.86261	−0.169090	−0.0845448	+	0.996420i	$0.526944\pi$
$828$	0	0
$829$	−40.7853	−1.41653	−0.708266	−	0.705946i	$-0.750522\pi$
$829$	−40.7853	−1.41653	−0.708266	+	0.705946i	$0.750522\pi$
$830$	13.7680	0.477896
$831$	0	0
$832$	−13.4801	−0.467337
$833$	0	0
$834$	0	0
$835$	−4.70494	−0.162821
$836$	5.26724	0.182171
$837$	0	0
$838$	46.3645	1.60164
$839$	−19.2034	−0.662976	−0.331488	−	0.943459i	$-0.607551\pi$
$839$	−19.2034	−0.662976	−0.331488	+	0.943459i	$0.607551\pi$
$840$	0	0
$841$	−28.9962	−0.999868
$842$	−54.6126	−1.88207
$843$	0	0
$844$	7.90225	0.272007
$845$	18.4075	0.633236
$846$	0	0
$847$	0	0
$848$	7.32376	0.251499
$849$	0	0
$850$	−9.17880	−0.314830
$851$	−4.14571	−0.142113
$852$	0	0
$853$	13.9011	0.475965	0.237982	−	0.971269i	$-0.423514\pi$
$853$	13.9011	0.475965	0.237982	+	0.971269i	$0.423514\pi$
$854$	0	0
$855$	0	0
$856$	−6.48937	−0.221802
$857$	56.9838	1.94653	0.973265	−	0.229686i	$-0.0737700\pi$
$857$	56.9838	1.94653	0.973265	+	0.229686i	$0.0737700\pi$
$858$	0	0
$859$	−20.1002	−0.685810	−0.342905	−	0.939370i	$-0.611411\pi$
$859$	−20.1002	−0.685810	−0.342905	+	0.939370i	$0.611411\pi$
$860$	−18.9260	−0.645370
$861$	0	0
$862$	9.00429	0.306687
$863$	6.17786	0.210297	0.105148	−	0.994457i	$-0.466468\pi$
$863$	6.17786	0.210297	0.105148	+	0.994457i	$0.466468\pi$
$864$	0	0
$865$	13.5356	0.460225
$866$	−17.8864	−0.607803
$867$	0	0
$868$	0	0
$869$	−8.95612	−0.303816
$870$	0	0
$871$	35.8872	1.21599
$872$	9.25465	0.313402
$873$	0	0
$874$	33.9396	1.14802
$875$	0	0
$876$	0	0
$877$	−37.2574	−1.25809	−0.629046	−	0.777368i	$-0.716554\pi$
$877$	−37.2574	−1.25809	−0.629046	+	0.777368i	$0.716554\pi$
$878$	27.2937	0.921117
$879$	0	0
$880$	−9.79009	−0.330024
$881$	−11.7848	−0.397041	−0.198520	−	0.980097i	$-0.563614\pi$
$881$	−11.7848	−0.397041	−0.198520	+	0.980097i	$0.563614\pi$
$882$	0	0
$883$	−29.2308	−0.983693	−0.491847	−	0.870682i	$-0.663678\pi$
$883$	−29.2308	−0.983693	−0.491847	+	0.870682i	$0.663678\pi$
$884$	−11.1453	−0.374857
$885$	0	0
$886$	40.3268	1.35480
$887$	28.5161	0.957479	0.478739	−	0.877957i	$-0.341094\pi$
$887$	28.5161	0.957479	0.478739	+	0.877957i	$0.341094\pi$
$888$	0	0
$889$	0	0
$890$	3.45680	0.115872
$891$	0	0
$892$	−16.2964	−0.545645
$893$	−23.8364	−0.797656
$894$	0	0
$895$	−2.27063	−0.0758986
$896$	0	0
$897$	0	0
$898$	−39.5779	−1.32073
$899$	−0.238341	−0.00794912
$900$	0	0
$901$	2.34004	0.0779579
$902$	25.1386	0.837025
$903$	0	0
$904$	−16.2837	−0.541587
$905$	−22.6829	−0.754006
$906$	0	0
$907$	−7.89155	−0.262035	−0.131017	−	0.991380i	$-0.541824\pi$
$907$	−7.89155	−0.262035	−0.131017	+	0.991380i	$0.541824\pi$
$908$	15.5463	0.515923
$909$	0	0
$910$	0	0
$911$	28.4412	0.942299	0.471150	−	0.882053i	$-0.343839\pi$
$911$	28.4412	0.942299	0.471150	+	0.882053i	$0.343839\pi$
$912$	0	0
$913$	−8.47625	−0.280523
$914$	−74.8013	−2.47421
$915$	0	0
$916$	−13.4241	−0.443545
$917$	0	0
$918$	0	0
$919$	−7.98542	−0.263415	−0.131707	−	0.991289i	$-0.542046\pi$
$919$	−7.98542	−0.263415	−0.131707	+	0.991289i	$0.542046\pi$
$920$	−11.0318	−0.363708
$921$	0	0
$922$	5.21221	0.171655
$923$	−63.8219	−2.10072
$924$	0	0
$925$	−1.81267	−0.0596004
$926$	−51.3059	−1.68602
$927$	0	0
$928$	0.413790	0.0135833
$929$	18.8006	0.616829	0.308414	−	0.951252i	$-0.400202\pi$
$929$	18.8006	0.616829	0.308414	+	0.951252i	$0.400202\pi$
$930$	0	0
$931$	0	0
$932$	26.8161	0.878391
$933$	0	0
$934$	−49.0577	−1.60522
$935$	−3.12806	−0.102299
$936$	0	0
$937$	48.5788	1.58700	0.793500	−	0.608570i	$-0.208256\pi$
$937$	48.5788	1.58700	0.793500	+	0.608570i	$0.208256\pi$
$938$	0	0
$939$	0	0
$940$	−17.6619	−0.576067
$941$	20.4851	0.667795	0.333898	−	0.942609i	$-0.391636\pi$
$941$	20.4851	0.667795	0.333898	+	0.942609i	$0.391636\pi$
$942$	0	0
$943$	66.4220	2.16300
$944$	40.9312	1.33220
$945$	0	0
$946$	28.4148	0.923843
$947$	−14.8505	−0.482576	−0.241288	−	0.970454i	$-0.577570\pi$
$947$	−14.8505	−0.482576	−0.241288	+	0.970454i	$0.577570\pi$
$948$	0	0
$949$	14.2791	0.463519
$950$	14.8398	0.481466
$951$	0	0
$952$	0	0
$953$	46.4678	1.50524	0.752620	−	0.658456i	$-0.228790\pi$
$953$	46.4678	1.50524	0.752620	+	0.658456i	$0.228790\pi$
$954$	0	0
$955$	30.2908	0.980188
$956$	0.541167	0.0175026
$957$	0	0
$958$	58.1447	1.87857
$959$	0	0
$960$	0	0
$961$	−16.1947	−0.522409
$962$	−5.36759	−0.173058
$963$	0	0
$964$	14.7860	0.476226
$965$	8.25809	0.265837
$966$	0	0
$967$	−1.72734	−0.0555475	−0.0277738	−	0.999614i	$-0.508842\pi$
$967$	−1.72734	−0.0555475	−0.0277738	+	0.999614i	$0.508842\pi$
$968$	−9.78089	−0.314370
$969$	0	0
$970$	−29.9687	−0.962235
$971$	7.56171	0.242667	0.121333	−	0.992612i	$-0.461283\pi$
$971$	7.56171	0.242667	0.121333	+	0.992612i	$0.461283\pi$
$972$	0	0
$973$	0	0
$974$	−0.563740	−0.0180634
$975$	0	0
$976$	−15.7027	−0.502633
$977$	−56.6202	−1.81144	−0.905721	−	0.423875i	$-0.860670\pi$
$977$	−56.6202	−1.81144	−0.905721	+	0.423875i	$0.860670\pi$
$978$	0	0
$979$	−2.12817	−0.0680166
$980$	0	0
$981$	0	0
$982$	−33.3994	−1.06582
$983$	32.2972	1.03012	0.515061	−	0.857154i	$-0.327769\pi$
$983$	32.2972	1.03012	0.515061	+	0.857154i	$0.327769\pi$
$984$	0	0
$985$	13.0407	0.415510
$986$	0.176657	0.00562591
$987$	0	0
$988$	18.0191	0.573264
$989$	75.0782	2.38735
$990$	0	0
$991$	14.3100	0.454573	0.227287	−	0.973828i	$-0.427015\pi$
$991$	14.3100	0.454573	0.227287	+	0.973828i	$0.427015\pi$
$992$	−25.7039	−0.816100
$993$	0	0
$994$	0	0
$995$	11.5849	0.367268
$996$	0	0
$997$	56.2524	1.78153	0.890765	−	0.454463i	$-0.150169\pi$
$997$	56.2524	1.78153	0.890765	+	0.454463i	$0.150169\pi$
$998$	39.2353	1.24197
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.z.1.2		5
3.2	odd	2		3969.2.a.bc.1.4		5
7.2	even	3		567.2.e.f.487.4		10
7.4	even	3		567.2.e.f.163.4		10
7.6	odd	2		3969.2.a.ba.1.2		5
9.2	odd	6		1323.2.f.e.442.2		10
9.4	even	3		441.2.f.e.295.4		10
9.5	odd	6		1323.2.f.e.883.2		10
9.7	even	3		441.2.f.e.148.4		10
21.2	odd	6		567.2.e.e.487.2		10
21.11	odd	6		567.2.e.e.163.2		10
21.20	even	2		3969.2.a.bb.1.4		5
63.2	odd	6		189.2.g.b.172.2		10
63.4	even	3		63.2.g.b.16.4	yes	10
63.5	even	6		1323.2.h.f.802.4		10
63.11	odd	6		189.2.h.b.37.4		10
63.13	odd	6		441.2.f.f.295.4		10
63.16	even	3		63.2.g.b.4.4	✓	10
63.20	even	6		1323.2.f.f.442.2		10
63.23	odd	6		189.2.h.b.46.4		10
63.25	even	3		63.2.h.b.58.2	yes	10
63.31	odd	6		441.2.g.f.79.4		10
63.32	odd	6		189.2.g.b.100.2		10
63.34	odd	6		441.2.f.f.148.4		10
63.38	even	6		1323.2.h.f.226.4		10
63.40	odd	6		441.2.h.f.214.2		10
63.41	even	6		1323.2.f.f.883.2		10
63.47	even	6		1323.2.g.f.361.2		10
63.52	odd	6		441.2.h.f.373.2		10
63.58	even	3		63.2.h.b.25.2	yes	10
63.59	even	6		1323.2.g.f.667.2		10
63.61	odd	6		441.2.g.f.67.4		10
252.11	even	6		3024.2.q.i.2305.4		10
252.23	even	6		3024.2.q.i.2881.4		10
252.67	odd	6		1008.2.t.i.961.5		10
252.79	odd	6		1008.2.t.i.193.5		10
252.95	even	6		3024.2.t.i.289.2		10
252.151	odd	6		1008.2.q.i.625.2		10
252.191	even	6		3024.2.t.i.1873.2		10
252.247	odd	6		1008.2.q.i.529.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.g.b.4.4	✓	10	63.16	even	3
63.2.g.b.16.4	yes	10	63.4	even	3
63.2.h.b.25.2	yes	10	63.58	even	3
63.2.h.b.58.2	yes	10	63.25	even	3
189.2.g.b.100.2		10	63.32	odd	6
189.2.g.b.172.2		10	63.2	odd	6
189.2.h.b.37.4		10	63.11	odd	6
189.2.h.b.46.4		10	63.23	odd	6
441.2.f.e.148.4		10	9.7	even	3
441.2.f.e.295.4		10	9.4	even	3
441.2.f.f.148.4		10	63.34	odd	6
441.2.f.f.295.4		10	63.13	odd	6
441.2.g.f.67.4		10	63.61	odd	6
441.2.g.f.79.4		10	63.31	odd	6
441.2.h.f.214.2		10	63.40	odd	6
441.2.h.f.373.2		10	63.52	odd	6
567.2.e.e.163.2		10	21.11	odd	6
567.2.e.e.487.2		10	21.2	odd	6
567.2.e.f.163.4		10	7.4	even	3
567.2.e.f.487.4		10	7.2	even	3
1008.2.q.i.529.2		10	252.247	odd	6
1008.2.q.i.625.2		10	252.151	odd	6
1008.2.t.i.193.5		10	252.79	odd	6
1008.2.t.i.961.5		10	252.67	odd	6
1323.2.f.e.442.2		10	9.2	odd	6
1323.2.f.e.883.2		10	9.5	odd	6
1323.2.f.f.442.2		10	63.20	even	6
1323.2.f.f.883.2		10	63.41	even	6
1323.2.g.f.361.2		10	63.47	even	6
1323.2.g.f.667.2		10	63.59	even	6
1323.2.h.f.226.4		10	63.38	even	6
1323.2.h.f.802.4		10	63.5	even	6
3024.2.q.i.2305.4		10	252.11	even	6
3024.2.q.i.2881.4		10	252.23	even	6
3024.2.t.i.289.2		10	252.95	even	6
3024.2.t.i.1873.2		10	252.191	even	6
3969.2.a.z.1.2		5	1.1	even	1	trivial
3969.2.a.ba.1.2		5	7.6	odd	2
3969.2.a.bb.1.4		5	21.20	even	2
3969.2.a.bc.1.4		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3969.2.a.z.1.2

Level	$3969$
Weight	$2$
Character	3969.1
Self dual	yes
Analytic conductor	$31.693$
Analytic rank	$1$
Dimension	$5$
CM	no
Inner twists	$1$