LMFDB - Embedded newform 3969.2.a.bb.1.4

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			1.4
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)

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

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 $

$2$	1.84124	1.30195	0.650977	−	0.759098i	$-0.274359\pi$
$2$	1.84124	1.30195	0.650977	+	0.759098i	$0.274359\pi$
$3$	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	$-0.573250\pi$
$11$	−1.51302	−0.456192	−0.228096	+	0.973639i	$0.573250\pi$
$12$	0	0
$13$	−5.17599	−1.43556	−0.717781	−	0.696269i	$-0.754842\pi$
$13$	−5.17599	−1.43556	−0.717781	+	0.696269i	$0.754842\pi$
$14$	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	$-0.592742\pi$
$19$	−2.50422	−0.574507	−0.287254	+	0.957855i	$0.592742\pi$
$20$	1.85553	0.414909
$21$	0	0
$22$	−2.78583	−0.593940
$23$	7.36079	1.53483	0.767415	−	0.641151i	$-0.221543\pi$
$23$	7.36079	1.53483	0.767415	+	0.641151i	$0.221543\pi$
$24$	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	$-0.501831\pi$
$29$	−0.0619427	−0.0115025	−0.00575123	+	0.999983i	$0.501831\pi$
$30$	0	0
$31$	3.84777	0.691080	0.345540	−	0.938404i	$-0.387696\pi$
$31$	3.84777	0.691080	0.345540	+	0.938404i	$0.387696\pi$
$32$	−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	$-0.466913\pi$
$53$	1.51075	0.207517	0.103759	+	0.994603i	$0.466913\pi$
$54$	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	$-0.566489\pi$
$61$	−3.23917	−0.414733	−0.207367	+	0.978263i	$0.566489\pi$
$62$	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	$-0.238740\pi$
$71$	12.3304	1.46335	0.731673	+	0.681656i	$0.238740\pi$
$72$	0	0
$73$	−2.75871	−0.322883	−0.161442	−	0.986882i	$-0.551614\pi$
$73$	−2.75871	−0.322883	−0.161442	+	0.986882i	$0.551614\pi$
$74$	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	$-0.712490\pi$
$97$	−12.1943	−1.23814	−0.619070	+	0.785336i	$0.712490\pi$
$98$	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	$-0.530319\pi$
$103$	−1.93045	−0.190213	−0.0951063	+	0.995467i	$0.530319\pi$
$104$	5.81186	0.569900
$105$	0	0
$106$	2.78165	0.270178
$107$	5.77938	0.558714	0.279357	−	0.960187i	$-0.409879\pi$
$107$	5.77938	0.558714	0.279357	+	0.960187i	$0.409879\pi$
$108$	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	$-0.261058\pi$
$113$	14.5021	1.36424	0.682121	+	0.731239i	$0.261058\pi$
$114$	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	$-0.530196\pi$
$137$	−2.21740	−0.189445	−0.0947225	+	0.995504i	$0.530196\pi$
$138$	0	0
$139$	0.755339	0.0640670	0.0320335	−	0.999487i	$-0.489802\pi$
$139$	0.755339	0.0640670	0.0320335	+	0.999487i	$0.489802\pi$
$140$	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	$-0.586935\pi$
$149$	−6.58499	−0.539463	−0.269732	+	0.962936i	$0.586935\pi$
$150$	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	$-0.257328\pi$
$157$	17.3074	1.38128	0.690642	+	0.723197i	$0.257328\pi$
$158$	−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	$-0.479750\pi$
$179$	1.70116	0.127150	0.0635752	+	0.997977i	$0.479750\pi$
$180$	0	0
$181$	16.9941	1.26316	0.631581	−	0.775310i	$-0.282406\pi$
$181$	16.9941	1.26316	0.631581	+	0.775310i	$0.282406\pi$
$182$	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	$-0.806605\pi$
$191$	−22.6939	−1.64208	−0.821038	+	0.570873i	$0.806605\pi$
$192$	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	$-0.613154\pi$
$197$	−9.77010	−0.696091	−0.348045	+	0.937478i	$0.613154\pi$
$198$	0	0
$199$	−8.67947	−0.615271	−0.307636	−	0.951504i	$-0.599538\pi$
$199$	−8.67947	−0.615271	−0.307636	+	0.951504i	$0.599538\pi$
$200$	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	$-0.371609\pi$
$223$	11.7227	0.785007	0.392503	+	0.919751i	$0.371609\pi$
$224$	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	$-0.396633\pi$
$229$	9.65647	0.638118	0.319059	+	0.947735i	$0.396633\pi$
$230$	18.0899	1.19281
$231$	0	0
$232$	0.0695523	0.00456633
$233$	−19.2898	−1.26372	−0.631860	−	0.775083i	$-0.717708\pi$
$233$	−19.2898	−1.26372	−0.631860	+	0.775083i	$0.717708\pi$
$234$	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	$-0.504008\pi$
$239$	−0.389282	−0.0251806	−0.0125903	+	0.999921i	$0.504008\pi$
$240$	0	0
$241$	−10.6361	−0.685134	−0.342567	−	0.939493i	$-0.611296\pi$
$241$	−10.6361	−0.685134	−0.342567	+	0.939493i	$0.611296\pi$
$242$	−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	$-0.705927\pi$
$263$	−19.5498	−1.20549	−0.602747	+	0.797932i	$0.705927\pi$
$264$	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	$-0.351621\pi$
$271$	14.7976	0.898893	0.449446	+	0.893307i	$0.351621\pi$
$272$	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	$-0.782324\pi$
$281$	−25.9876	−1.55029	−0.775146	+	0.631782i	$0.782324\pi$
$282$	0	0
$283$	−18.7554	−1.11489	−0.557445	−	0.830214i	$-0.688218\pi$
$283$	−18.7554	−1.11489	−0.557445	+	0.830214i	$0.688218\pi$
$284$	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	$-0.457520\pi$
$307$	4.66277	0.266118	0.133059	+	0.991108i	$0.457520\pi$
$308$	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	$-0.549618\pi$
$313$	−5.49332	−0.310501	−0.155250	+	0.987875i	$0.549618\pi$
$314$	31.8671	1.79837
$315$	0	0
$316$	−8.22893	−0.462914
$317$	−9.87758	−0.554780	−0.277390	−	0.960757i	$-0.589469\pi$
$317$	−9.87758	−0.554780	−0.277390	+	0.960757i	$0.589469\pi$
$318$	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	$-0.208590\pi$
$347$	29.5388	1.58572	0.792862	+	0.609401i	$0.208590\pi$
$348$	0	0
$349$	36.0013	1.92710	0.963551	−	0.267523i	$-0.0862052\pi$
$349$	36.0013	1.92710	0.963551	+	0.267523i	$0.0862052\pi$
$350$	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	$-0.454395\pi$
$359$	5.41069	0.285566	0.142783	+	0.989754i	$0.454395\pi$
$360$	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	$-0.294173\pi$
$367$	23.0843	1.20499	0.602496	+	0.798122i	$0.294173\pi$
$368$	−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	$-0.238497\pi$
$389$	28.8822	1.46438	0.732192	+	0.681098i	$0.238497\pi$
$390$	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	$-0.409444\pi$
$397$	11.1845	0.561335	0.280667	+	0.959805i	$0.409444\pi$
$398$	−15.9810	−0.801055
$399$	0	0
$400$	15.6022	0.780111
$401$	1.08212	0.0540386	0.0270193	−	0.999635i	$-0.491398\pi$
$401$	1.08212	0.0540386	0.0270193	+	0.999635i	$0.491398\pi$
$402$	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	$-0.319422\pi$
$409$	21.7349	1.07472	0.537360	+	0.843353i	$0.319422\pi$
$410$	−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	$-0.462422\pi$
$431$	4.89034	0.235559	0.117780	+	0.993040i	$0.462422\pi$
$432$	0	0
$433$	−9.71430	−0.466839	−0.233420	−	0.972376i	$-0.574992\pi$
$433$	−9.71430	−0.466839	−0.233420	+	0.972376i	$0.574992\pi$
$434$	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	$-0.384908\pi$
$439$	14.8235	0.707488	0.353744	+	0.935342i	$0.384908\pi$
$440$	2.26760	0.108103
$441$	0	0
$442$	14.7617	0.702141
$443$	21.9020	1.04059	0.520297	−	0.853986i	$-0.325821\pi$
$443$	21.9020	1.04059	0.520297	+	0.853986i	$0.325821\pi$
$444$	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	$-0.669324\pi$
$449$	−21.4952	−1.01442	−0.507212	+	0.861822i	$0.669324\pi$
$450$	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	$-0.634232\pi$
$491$	−18.1396	−0.818629	−0.409315	+	0.912393i	$0.634232\pi$
$492$	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	$-0.869411\pi$
$523$	−41.9429	−1.83404	−0.917018	+	0.398847i	$0.869411\pi$
$524$	−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	$-0.271312\pi$
$557$	31.0688	1.31643	0.658214	+	0.752831i	$0.271312\pi$
$558$	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	$-0.390569\pi$
$569$	16.0801	0.674112	0.337056	+	0.941485i	$0.390569\pi$
$570$	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	$-0.332248\pi$
$577$	24.1625	1.00590	0.502949	+	0.864316i	$0.332248\pi$
$578$	−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	$-0.799203\pi$
$599$	−39.5283	−1.61508	−0.807542	+	0.589810i	$0.799203\pi$
$600$	0	0
$601$	3.72895	0.152107	0.0760534	−	0.997104i	$-0.475768\pi$
$601$	3.72895	0.152107	0.0760534	+	0.997104i	$0.475768\pi$
$602$	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	$-0.659370\pi$
$607$	−23.6528	−0.960036	−0.480018	+	0.877259i	$0.659370\pi$
$608$	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	$-0.749952\pi$
$617$	−35.1230	−1.41400	−0.706999	+	0.707214i	$0.749952\pi$
$618$	0	0
$619$	21.1632	0.850622	0.425311	−	0.905047i	$-0.360165\pi$
$619$	21.1632	0.850622	0.425311	+	0.905047i	$0.360165\pi$
$620$	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	$-0.437525\pi$
$641$	9.87469	0.390027	0.195013	+	0.980801i	$0.437525\pi$
$642$	0	0
$643$	43.9496	1.73320	0.866602	−	0.499000i	$-0.166299\pi$
$643$	43.9496	1.73320	0.866602	+	0.499000i	$0.166299\pi$
$644$	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	$-0.806930\pi$
$653$	−41.9912	−1.64324	−0.821622	+	0.570033i	$0.806930\pi$
$654$	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	$-0.777226\pi$
$659$	−39.2729	−1.52986	−0.764928	+	0.644115i	$0.777226\pi$
$660$	0	0
$661$	0.186739	0.00726330	0.00363165	−	0.999993i	$-0.498844\pi$
$661$	0.186739	0.00726330	0.00363165	+	0.999993i	$0.498844\pi$
$662$	−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	$-0.427260\pi$
$683$	11.8407	0.453071	0.226536	+	0.974003i	$0.427260\pi$
$684$	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	$-0.572731\pi$
$691$	−11.9083	−0.453014	−0.226507	+	0.974010i	$0.572731\pi$
$692$	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	$-0.298024\pi$
$701$	31.3902	1.18559	0.592795	+	0.805353i	$0.298024\pi$
$702$	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	$-0.747665\pi$
$727$	−37.8506	−1.40380	−0.701900	+	0.712275i	$0.747665\pi$
$728$	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	$-0.514122\pi$
$733$	−2.40155	−0.0887033	−0.0443516	+	0.999016i	$0.514122\pi$
$734$	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	$-0.529796\pi$
$743$	−5.09570	−0.186943	−0.0934715	+	0.995622i	$0.529796\pi$
$744$	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	$-0.622772\pi$
$769$	−20.8652	−0.752417	−0.376208	+	0.926535i	$0.622772\pi$
$770$	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	$-0.552370\pi$
$787$	−9.18949	−0.327570	−0.163785	+	0.986496i	$0.552370\pi$
$788$	−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	$-0.533305\pi$
$809$	−5.94119	−0.208881	−0.104441	+	0.994531i	$0.533305\pi$
$810$	0	0
$811$	−44.4139	−1.55958	−0.779791	−	0.626039i	$-0.784675\pi$
$811$	−44.4139	−1.55958	−0.779791	+	0.626039i	$0.784675\pi$
$812$	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	$-0.535373\pi$
$821$	−6.35522	−0.221799	−0.110899	+	0.993832i	$0.535373\pi$
$822$	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	$-0.473056\pi$
$827$	4.86261	0.169090	0.0845448	+	0.996420i	$0.473056\pi$
$828$	0	0
$829$	40.7853	1.41653	0.708266	−	0.705946i	$-0.249478\pi$
$829$	40.7853	1.41653	0.708266	+	0.705946i	$0.249478\pi$
$830$	−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	$-0.576486\pi$
$853$	−13.9011	−0.475965	−0.237982	+	0.971269i	$0.576486\pi$
$854$	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	$-0.388589\pi$
$859$	20.1002	0.685810	0.342905	+	0.939370i	$0.388589\pi$
$860$	−18.9260	−0.645370
$861$	0	0
$862$	9.00429	0.306687
$863$	−6.17786	−0.210297	−0.105148	−	0.994457i	$-0.533532\pi$
$863$	−6.17786	−0.210297	−0.105148	+	0.994457i	$0.533532\pi$
$864$	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	$-0.656161\pi$
$911$	−28.4412	−0.942299	−0.471150	+	0.882053i	$0.656161\pi$
$912$	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	$-0.791744\pi$
$937$	−48.5788	−1.58700	−0.793500	+	0.608570i	$0.791744\pi$
$938$	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	$-0.422430\pi$
$947$	14.8505	0.482576	0.241288	+	0.970454i	$0.422430\pi$
$948$	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	$-0.771210\pi$
$953$	−46.4678	−1.50524	−0.752620	+	0.658456i	$0.771210\pi$
$954$	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	$-0.139330\pi$
$977$	56.6202	1.81144	0.905721	+	0.423875i	$0.139330\pi$
$978$	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	$-0.849831\pi$
$997$	−56.2524	−1.78153	−0.890765	+	0.454463i	$0.849831\pi$
$998$	−39.2353	−1.24197
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.g.b.4.4	✓	10	63.47	even	6
63.2.g.b.16.4	yes	10	63.59	even	6
63.2.h.b.25.2	yes	10	63.5	even	6
63.2.h.b.58.2	yes	10	63.38	even	6
189.2.g.b.100.2		10	63.31	odd	6
189.2.g.b.172.2		10	63.61	odd	6
189.2.h.b.37.4		10	63.52	odd	6
189.2.h.b.46.4		10	63.40	odd	6
441.2.f.e.148.4		10	63.20	even	6
441.2.f.e.295.4		10	63.41	even	6
441.2.f.f.148.4		10	9.2	odd	6
441.2.f.f.295.4		10	9.5	odd	6
441.2.g.f.67.4		10	63.2	odd	6
441.2.g.f.79.4		10	63.32	odd	6
441.2.h.f.214.2		10	63.23	odd	6
441.2.h.f.373.2		10	63.11	odd	6
567.2.e.e.163.2		10	7.3	odd	6
567.2.e.e.487.2		10	7.5	odd	6
567.2.e.f.163.4		10	21.17	even	6
567.2.e.f.487.4		10	21.5	even	6
1008.2.q.i.529.2		10	252.131	odd	6
1008.2.q.i.625.2		10	252.227	odd	6
1008.2.t.i.193.5		10	252.47	odd	6
1008.2.t.i.961.5		10	252.59	odd	6
1323.2.f.e.442.2		10	63.34	odd	6
1323.2.f.e.883.2		10	63.13	odd	6
1323.2.f.f.442.2		10	9.7	even	3
1323.2.f.f.883.2		10	9.4	even	3
1323.2.g.f.361.2		10	63.16	even	3
1323.2.g.f.667.2		10	63.4	even	3
1323.2.h.f.226.4		10	63.25	even	3
1323.2.h.f.802.4		10	63.58	even	3
3024.2.q.i.2305.4		10	252.115	even	6
3024.2.q.i.2881.4		10	252.103	even	6
3024.2.t.i.289.2		10	252.31	even	6
3024.2.t.i.1873.2		10	252.187	even	6
3969.2.a.z.1.2		5	21.20	even	2
3969.2.a.ba.1.2		5	3.2	odd	2
3969.2.a.bb.1.4		5	1.1	even	1	trivial
3969.2.a.bc.1.4		5	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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

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