LMFDB - Embedded newform 3969.2.a.be.1.3

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.59351616.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 21x^{2} - 9 $ x^6 - 12x^4 + 21x^2 - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 441)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.15750$ 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)

$f(q)$	$=$	$q+0.239123 q^{2} -1.94282 q^{4} -2.59179 q^{5} -0.942820 q^{8} +O(q^{10})$	$q+0.239123 q^{2} -1.94282 q^{4} -2.59179 q^{5} -0.942820 q^{8} -0.619757 q^{10} +4.18194 q^{11} +3.68310 q^{13} +3.66019 q^{16} +1.71107 q^{17} -7.15561 q^{19} +5.03538 q^{20} +1.00000 q^{22} -5.12476 q^{23} +1.71737 q^{25} +0.880716 q^{26} -2.12476 q^{29} -6.53585 q^{31} +2.76088 q^{32} +0.409157 q^{34} +1.66019 q^{37} -1.71107 q^{38} +2.44359 q^{40} -10.2190 q^{41} -1.66019 q^{43} -8.12476 q^{44} -1.22545 q^{46} +9.33824 q^{47} +0.410663 q^{50} -7.15561 q^{52} +10.6465 q^{53} -10.8387 q^{55} -0.508080 q^{58} +6.06429 q^{59} -7.98597 q^{61} -1.56287 q^{62} -6.66019 q^{64} -9.54583 q^{65} +8.26320 q^{67} -3.32431 q^{68} +6.23912 q^{71} +7.15561 q^{73} +0.396990 q^{74} +13.9021 q^{76} -9.82846 q^{79} -9.48644 q^{80} -2.44359 q^{82} +6.89465 q^{83} -4.43474 q^{85} -0.396990 q^{86} -3.94282 q^{88} -5.03538 q^{89} +9.95649 q^{92} +2.23299 q^{94} +18.5458 q^{95} +3.06335 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 6 q^{4} + 12 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 6 * q^4 + 12 * q^8	$ 6 q + 2 q^{2} + 6 q^{4} + 12 q^{8} + 8 q^{11} + 6 q^{16} + 6 q^{22} + 4 q^{23} + 12 q^{25} + 22 q^{29} + 16 q^{32} - 6 q^{37} + 6 q^{43} - 14 q^{44} + 12 q^{46} + 56 q^{50} + 28 q^{53} + 18 q^{58} - 24 q^{64} - 6 q^{65} + 38 q^{71} + 36 q^{74} - 6 q^{79} - 30 q^{85} - 36 q^{86} - 6 q^{88} + 62 q^{92} + 60 q^{95}+O(q^{100}) $ 6 * q + 2 * q^2 + 6 * q^4 + 12 * q^8 + 8 * q^11 + 6 * q^16 + 6 * q^22 + 4 * q^23 + 12 * q^25 + 22 * q^29 + 16 * q^32 - 6 * q^37 + 6 * q^43 - 14 * q^44 + 12 * q^46 + 56 * q^50 + 28 * q^53 + 18 * q^58 - 24 * q^64 - 6 * q^65 + 38 * q^71 + 36 * q^74 - 6 * q^79 - 30 * q^85 - 36 * q^86 - 6 * q^88 + 62 * q^92 + 60 * q^95

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 $

$2$	0.239123	0.169086	0.0845428	−	0.996420i	$-0.473057\pi$
$2$	0.239123	0.169086	0.0845428	+	0.996420i	$0.473057\pi$
$3$	0	0
$3$	0	0
$4$	−1.94282	−0.971410
$5$	−2.59179	−1.15908	−0.579542	−	0.814943i	$-0.696768\pi$
$5$	−2.59179	−1.15908	−0.579542	+	0.814943i	$0.696768\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−0.942820	−0.333337
$9$	0	0
$10$	−0.619757	−0.195984
$11$	4.18194	1.26090	0.630452	−	0.776228i	$-0.282870\pi$
$11$	4.18194	1.26090	0.630452	+	0.776228i	$0.282870\pi$
$12$	0	0
$13$	3.68310	1.02151	0.510755	−	0.859726i	$-0.329366\pi$
$13$	3.68310	1.02151	0.510755	+	0.859726i	$0.329366\pi$
$14$	0	0
$15$	0	0
$16$	3.66019	0.915047
$17$	1.71107	0.414996	0.207498	−	0.978235i	$-0.433468\pi$
$17$	1.71107	0.414996	0.207498	+	0.978235i	$0.433468\pi$
$18$	0	0
$19$	−7.15561	−1.64161	−0.820805	−	0.571209i	$-0.806475\pi$
$19$	−7.15561	−1.64161	−0.820805	+	0.571209i	$0.806475\pi$
$20$	5.03538	1.12595
$21$	0	0
$22$	1.00000	0.213201
$23$	−5.12476	−1.06859	−0.534294	−	0.845299i	$-0.679422\pi$
$23$	−5.12476	−1.06859	−0.534294	+	0.845299i	$0.679422\pi$
$24$	0	0
$25$	1.71737	0.343474
$26$	0.880716	0.172723
$27$	0	0
$28$	0	0
$29$	−2.12476	−0.394559	−0.197279	−	0.980347i	$-0.563211\pi$
$29$	−2.12476	−0.394559	−0.197279	+	0.980347i	$0.563211\pi$
$30$	0	0
$31$	−6.53585	−1.17387	−0.586937	−	0.809633i	$-0.699666\pi$
$31$	−6.53585	−1.17387	−0.586937	+	0.809633i	$0.699666\pi$
$32$	2.76088	0.488059
$33$	0	0
$34$	0.409157	0.0701699
$35$	0	0
$36$	0	0
$37$	1.66019	0.272934	0.136467	−	0.990645i	$-0.456425\pi$
$37$	1.66019	0.272934	0.136467	+	0.990645i	$0.456425\pi$
$38$	−1.71107	−0.277573
$39$	0	0
$40$	2.44359	0.386366
$41$	−10.2190	−1.59593	−0.797967	−	0.602702i	$-0.794091\pi$
$41$	−10.2190	−1.59593	−0.797967	+	0.602702i	$0.794091\pi$
$42$	0	0
$43$	−1.66019	−0.253177	−0.126588	−	0.991955i	$-0.540403\pi$
$43$	−1.66019	−0.253177	−0.126588	+	0.991955i	$0.540403\pi$
$44$	−8.12476	−1.22485
$45$	0	0
$46$	−1.22545	−0.180683
$47$	9.33824	1.36212	0.681061	−	0.732226i	$-0.261519\pi$
$47$	9.33824	1.36212	0.681061	+	0.732226i	$0.261519\pi$
$48$	0	0
$49$	0	0
$50$	0.410663	0.0580765
$51$	0	0
$52$	−7.15561	−0.992305
$53$	10.6465	1.46241	0.731206	−	0.682157i	$-0.238958\pi$
$53$	10.6465	1.46241	0.731206	+	0.682157i	$0.238958\pi$
$54$	0	0
$55$	−10.8387	−1.46149
$56$	0	0
$57$	0	0
$58$	−0.508080	−0.0667142
$59$	6.06429	0.789504	0.394752	−	0.918788i	$-0.370831\pi$
$59$	6.06429	0.789504	0.394752	+	0.918788i	$0.370831\pi$
$60$	0	0
$61$	−7.98597	−1.02250	−0.511249	−	0.859433i	$-0.670817\pi$
$61$	−7.98597	−1.02250	−0.511249	+	0.859433i	$0.670817\pi$
$62$	−1.56287	−0.198485
$63$	0	0
$64$	−6.66019	−0.832524
$65$	−9.54583	−1.18401
$66$	0	0
$67$	8.26320	1.00951	0.504755	−	0.863262i	$-0.331583\pi$
$67$	8.26320	1.00951	0.504755	+	0.863262i	$0.331583\pi$
$68$	−3.32431	−0.403131
$69$	0	0
$70$	0	0
$71$	6.23912	0.740448	0.370224	−	0.928943i	$-0.379281\pi$
$71$	6.23912	0.740448	0.370224	+	0.928943i	$0.379281\pi$
$72$	0	0
$73$	7.15561	0.837501	0.418750	−	0.908101i	$-0.362468\pi$
$73$	7.15561	0.837501	0.418750	+	0.908101i	$0.362468\pi$
$74$	0.396990	0.0461492
$75$	0	0
$76$	13.9021	1.59468
$77$	0	0
$78$	0	0
$79$	−9.82846	−1.10579	−0.552894	−	0.833252i	$-0.686477\pi$
$79$	−9.82846	−1.10579	−0.552894	+	0.833252i	$0.686477\pi$
$80$	−9.48644	−1.06062
$81$	0	0
$82$	−2.44359	−0.269849
$83$	6.89465	0.756786	0.378393	−	0.925645i	$-0.376477\pi$
$83$	6.89465	0.756786	0.378393	+	0.925645i	$0.376477\pi$
$84$	0	0
$85$	−4.43474	−0.481015
$86$	−0.396990	−0.0428085
$87$	0	0
$88$	−3.94282	−0.420306
$89$	−5.03538	−0.533749	−0.266875	−	0.963731i	$-0.585991\pi$
$89$	−5.03538	−0.533749	−0.266875	+	0.963731i	$0.585991\pi$
$90$	0	0
$91$	0	0
$92$	9.95649	1.03804
$93$	0	0
$94$	2.23299	0.230315
$95$	18.5458	1.90276
$96$	0	0
$97$	3.06335	0.311036	0.155518	−	0.987833i	$-0.450295\pi$
$97$	3.06335	0.311036	0.155518	+	0.987833i	$0.450295\pi$
$98$	0	0
$99$	0	0
$100$	−3.33654	−0.333654
$101$	−11.0997	−1.10446	−0.552229	−	0.833692i	$-0.686223\pi$
$101$	−11.0997	−1.10446	−0.552229	+	0.833692i	$0.686223\pi$
$102$	0	0
$103$	7.98597	0.786881	0.393440	−	0.919350i	$-0.371285\pi$
$103$	7.98597	0.786881	0.393440	+	0.919350i	$0.371285\pi$
$104$	−3.47250	−0.340507
$105$	0	0
$106$	2.54583	0.247273
$107$	3.95649	0.382489	0.191244	−	0.981542i	$-0.438748\pi$
$107$	3.95649	0.382489	0.191244	+	0.981542i	$0.438748\pi$
$108$	0	0
$109$	7.26320	0.695688	0.347844	−	0.937552i	$-0.386914\pi$
$109$	7.26320	0.695688	0.347844	+	0.937552i	$0.386914\pi$
$110$	−2.59179	−0.247117
$111$	0	0
$112$	0	0
$113$	6.92915	0.651839	0.325920	−	0.945397i	$-0.394326\pi$
$113$	6.92915	0.651839	0.325920	+	0.945397i	$0.394326\pi$
$114$	0	0
$115$	13.2823	1.23858
$116$	4.12803	0.383278
$117$	0	0
$118$	1.45011	0.133494
$119$	0	0
$120$	0	0
$121$	6.48865	0.589877
$122$	−1.90963	−0.172890
$123$	0	0
$124$	12.6980	1.14031
$125$	8.50788	0.760968
$126$	0	0
$127$	9.11109	0.808479	0.404239	−	0.914653i	$-0.367536\pi$
$127$	9.11109	0.808479	0.404239	+	0.914653i	$0.367536\pi$
$128$	−7.11436	−0.628827
$129$	0	0
$130$	−2.28263	−0.200200
$131$	−4.30286	−0.375943	−0.187971	−	0.982175i	$-0.560191\pi$
$131$	−4.30286	−0.375943	−0.187971	+	0.982175i	$0.560191\pi$
$132$	0	0
$133$	0	0
$134$	1.97592	0.170694
$135$	0	0
$136$	−1.61323	−0.138334
$137$	20.5893	1.75907	0.879533	−	0.475838i	$-0.157855\pi$
$137$	20.5893	1.75907	0.879533	+	0.475838i	$0.157855\pi$
$138$	0	0
$139$	−15.7613	−1.33686	−0.668429	−	0.743776i	$-0.733033\pi$
$139$	−15.7613	−1.33686	−0.668429	+	0.743776i	$0.733033\pi$
$140$	0	0
$141$	0	0
$142$	1.49192	0.125199
$143$	15.4025	1.28802
$144$	0	0
$145$	5.50694	0.457326
$146$	1.71107	0.141609
$147$	0	0
$148$	−3.22545	−0.265130
$149$	6.06758	0.497076	0.248538	−	0.968622i	$-0.420050\pi$
$149$	6.06758	0.497076	0.248538	+	0.968622i	$0.420050\pi$
$150$	0	0
$151$	4.48865	0.365281	0.182641	−	0.983180i	$-0.441536\pi$
$151$	4.48865	0.365281	0.182641	+	0.983180i	$0.441536\pi$
$152$	6.74645	0.547210
$153$	0	0
$154$	0	0
$155$	16.9396	1.36062
$156$	0	0
$157$	1.02891	0.0821163	0.0410582	−	0.999157i	$-0.486927\pi$
$157$	1.02891	0.0821163	0.0410582	+	0.999157i	$0.486927\pi$
$158$	−2.35021	−0.186973
$159$	0	0
$160$	−7.15561	−0.565701
$161$	0	0
$162$	0	0
$163$	6.82846	0.534846	0.267423	−	0.963579i	$-0.413828\pi$
$163$	6.82846	0.534846	0.267423	+	0.963579i	$0.413828\pi$
$164$	19.8536	1.55031
$165$	0	0
$166$	1.64867	0.127962
$167$	17.9943	1.39244	0.696221	−	0.717827i	$-0.254863\pi$
$167$	17.9943	1.39244	0.696221	+	0.717827i	$0.254863\pi$
$168$	0	0
$169$	0.565260	0.0434816
$170$	−1.06045	−0.0813328
$171$	0	0
$172$	3.22545	0.245938
$173$	−0.830357	−0.0631309	−0.0315654	−	0.999502i	$-0.510049\pi$
$173$	−0.830357	−0.0631309	−0.0315654	+	0.999502i	$0.510049\pi$
$174$	0	0
$175$	0	0
$176$	15.3067	1.15379
$177$	0	0
$178$	−1.20408	−0.0902493
$179$	7.57893	0.566476	0.283238	−	0.959050i	$-0.408591\pi$
$179$	7.57893	0.566476	0.283238	+	0.959050i	$0.408591\pi$
$180$	0	0
$181$	0.409157	0.0304124	0.0152062	−	0.999884i	$-0.495160\pi$
$181$	0.409157	0.0304124	0.0152062	+	0.999884i	$0.495160\pi$
$182$	0	0
$183$	0	0
$184$	4.83173	0.356200
$185$	−4.30286	−0.316353
$186$	0	0
$187$	7.15561	0.523270
$188$	−18.1425	−1.32318
$189$	0	0
$190$	4.43474	0.321730
$191$	16.0241	1.15946	0.579731	−	0.814808i	$-0.303158\pi$
$191$	16.0241	1.15946	0.579731	+	0.814808i	$0.303158\pi$
$192$	0	0
$193$	−12.3743	−0.890721	−0.445360	−	0.895351i	$-0.646924\pi$
$193$	−12.3743	−0.890721	−0.445360	+	0.895351i	$0.646924\pi$
$194$	0.732518	0.0525917
$195$	0	0
$196$	0	0
$197$	23.1021	1.64595	0.822977	−	0.568075i	$-0.192312\pi$
$197$	23.1021	1.64595	0.822977	+	0.568075i	$0.192312\pi$
$198$	0	0
$199$	−6.74645	−0.478243	−0.239122	−	0.970990i	$-0.576859\pi$
$199$	−6.74645	−0.478243	−0.239122	+	0.970990i	$0.576859\pi$
$200$	−1.61917	−0.114493
$201$	0	0
$202$	−2.65419	−0.186748
$203$	0	0
$204$	0	0
$205$	26.4854	1.84982
$206$	1.90963	0.133050
$207$	0	0
$208$	13.4809	0.934730
$209$	−29.9244	−2.06991
$210$	0	0
$211$	16.8856	1.16246	0.581228	−	0.813741i	$-0.302573\pi$
$211$	16.8856	1.16246	0.581228	+	0.813741i	$0.302573\pi$
$212$	−20.6843	−1.42060
$213$	0	0
$214$	0.946090	0.0646734
$215$	4.30286	0.293453
$216$	0	0
$217$	0	0
$218$	1.73680	0.117631
$219$	0	0
$220$	21.0577	1.41971
$221$	6.30206	0.423922
$222$	0	0
$223$	−4.50142	−0.301437	−0.150719	−	0.988577i	$-0.548159\pi$
$223$	−4.50142	−0.301437	−0.150719	+	0.988577i	$0.548159\pi$
$224$	0	0
$225$	0	0
$226$	1.65692	0.110217
$227$	−6.06429	−0.402501	−0.201251	−	0.979540i	$-0.564501\pi$
$227$	−6.06429	−0.402501	−0.201251	+	0.979540i	$0.564501\pi$
$228$	0	0
$229$	−11.0493	−0.730159	−0.365080	−	0.930976i	$-0.618958\pi$
$229$	−11.0493	−0.730159	−0.365080	+	0.930976i	$0.618958\pi$
$230$	3.17611	0.209426
$231$	0	0
$232$	2.00327	0.131521
$233$	−8.13844	−0.533167	−0.266583	−	0.963812i	$-0.585895\pi$
$233$	−8.13844	−0.533167	−0.266583	+	0.963812i	$0.585895\pi$
$234$	0	0
$235$	−24.2028	−1.57881
$236$	−11.7818	−0.766932
$237$	0	0
$238$	0	0
$239$	21.1625	1.36889	0.684445	−	0.729065i	$-0.260045\pi$
$239$	21.1625	1.36889	0.684445	+	0.729065i	$0.260045\pi$
$240$	0	0
$241$	13.6915	0.881945	0.440972	−	0.897521i	$-0.354634\pi$
$241$	13.6915	0.881945	0.440972	+	0.897521i	$0.354634\pi$
$242$	1.55159	0.0997398
$243$	0	0
$244$	15.5153	0.993265
$245$	0	0
$246$	0	0
$247$	−26.3549	−1.67692
$248$	6.16213	0.391296
$249$	0	0
$250$	2.03443	0.128669
$251$	15.2040	0.959667	0.479833	−	0.877360i	$-0.340697\pi$
$251$	15.2040	0.959667	0.479833	+	0.877360i	$0.340697\pi$
$252$	0	0
$253$	−21.4315	−1.34738
$254$	2.17867	0.136702
$255$	0	0
$256$	11.6192	0.726198
$257$	25.6215	1.59822	0.799112	−	0.601182i	$-0.205303\pi$
$257$	25.6215	1.59822	0.799112	+	0.601182i	$0.205303\pi$
$258$	0	0
$259$	0	0
$260$	18.5458	1.15016
$261$	0	0
$262$	−1.02891	−0.0635665
$263$	7.10069	0.437847	0.218924	−	0.975742i	$-0.429745\pi$
$263$	7.10069	0.437847	0.218924	+	0.975742i	$0.429745\pi$
$264$	0	0
$265$	−27.5935	−1.69506
$266$	0	0
$267$	0	0
$268$	−16.0539	−0.980649
$269$	−16.4314	−1.00184	−0.500922	−	0.865493i	$-0.667006\pi$
$269$	−16.4314	−1.00184	−0.500922	+	0.865493i	$0.667006\pi$
$270$	0	0
$271$	12.6980	0.771348	0.385674	−	0.922635i	$-0.373969\pi$
$271$	12.6980	0.771348	0.385674	+	0.922635i	$0.373969\pi$
$272$	6.26285	0.379741
$273$	0	0
$274$	4.92339	0.297433
$275$	7.18194	0.433087
$276$	0	0
$277$	−0.828460	−0.0497773	−0.0248887	−	0.999690i	$-0.507923\pi$
$277$	−0.828460	−0.0497773	−0.0248887	+	0.999690i	$0.507923\pi$
$278$	−3.76890	−0.226044
$279$	0	0
$280$	0	0
$281$	5.21969	0.311381	0.155690	−	0.987806i	$-0.450240\pi$
$281$	5.21969	0.311381	0.155690	+	0.987806i	$0.450240\pi$
$282$	0	0
$283$	−7.35417	−0.437160	−0.218580	−	0.975819i	$-0.570142\pi$
$283$	−7.35417	−0.437160	−0.218580	+	0.975819i	$0.570142\pi$
$284$	−12.1215	−0.719278
$285$	0	0
$286$	3.68310	0.217787
$287$	0	0
$288$	0	0
$289$	−14.0722	−0.827778
$290$	1.31684	0.0773273
$291$	0	0
$292$	−13.9021	−0.813557
$293$	7.82573	0.457184	0.228592	−	0.973522i	$-0.426588\pi$
$293$	7.82573	0.457184	0.228592	+	0.973522i	$0.426588\pi$
$294$	0	0
$295$	−15.7174	−0.915101
$296$	−1.56526	−0.0909789
$297$	0	0
$298$	1.45090	0.0840484
$299$	−18.8750	−1.09157
$300$	0	0
$301$	0	0
$302$	1.07334	0.0617638
$303$	0	0
$304$	−26.1909	−1.50215
$305$	20.6979	1.18516
$306$	0	0
$307$	22.6709	1.29390	0.646948	−	0.762534i	$-0.276045\pi$
$307$	22.6709	1.29390	0.646948	+	0.762534i	$0.276045\pi$
$308$	0	0
$309$	0	0
$310$	4.05064	0.230061
$311$	−32.3176	−1.83256	−0.916281	−	0.400536i	$-0.868824\pi$
$311$	−32.3176	−1.83256	−0.916281	+	0.400536i	$0.868824\pi$
$312$	0	0
$313$	−24.3196	−1.37462	−0.687312	−	0.726362i	$-0.741209\pi$
$313$	−24.3196	−1.37462	−0.687312	+	0.726362i	$0.741209\pi$
$314$	0.246037	0.0138847
$315$	0	0
$316$	19.0949	1.07417
$317$	5.13844	0.288603	0.144302	−	0.989534i	$-0.453906\pi$
$317$	5.13844	0.288603	0.144302	+	0.989534i	$0.453906\pi$
$318$	0	0
$319$	−8.88564	−0.497500
$320$	17.2618	0.964964
$321$	0	0
$322$	0	0
$323$	−12.2438	−0.681262
$324$	0	0
$325$	6.32525	0.350862
$326$	1.63284	0.0904349
$327$	0	0
$328$	9.63464	0.531984
$329$	0	0
$330$	0	0
$331$	−11.6979	−0.642977	−0.321488	−	0.946913i	$-0.604183\pi$
$331$	−11.6979	−0.642977	−0.321488	+	0.946913i	$0.604183\pi$
$332$	−13.3951	−0.735150
$333$	0	0
$334$	4.30286	0.235442
$335$	−21.4165	−1.17011
$336$	0	0
$337$	−33.6947	−1.83547	−0.917733	−	0.397198i	$-0.869983\pi$
$337$	−33.6947	−1.83547	−0.917733	+	0.397198i	$0.869983\pi$
$338$	0.135167	0.00735211
$339$	0	0
$340$	8.61590	0.467263
$341$	−27.3326	−1.48014
$342$	0	0
$343$	0	0
$344$	1.56526	0.0843932
$345$	0	0
$346$	−0.198558	−0.0106745
$347$	27.3114	1.46615	0.733075	−	0.680148i	$-0.238084\pi$
$347$	27.3114	1.46615	0.733075	+	0.680148i	$0.238084\pi$
$348$	0	0
$349$	22.9169	1.22672	0.613358	−	0.789805i	$-0.289818\pi$
$349$	22.9169	1.22672	0.613358	+	0.789805i	$0.289818\pi$
$350$	0	0
$351$	0	0
$352$	11.5458	0.615395
$353$	10.2693	0.546581	0.273290	−	0.961932i	$-0.411888\pi$
$353$	10.2693	0.546581	0.273290	+	0.961932i	$0.411888\pi$
$354$	0	0
$355$	−16.1705	−0.858241
$356$	9.78284	0.518489
$357$	0	0
$358$	1.81230	0.0957830
$359$	10.1007	0.533094	0.266547	−	0.963822i	$-0.414117\pi$
$359$	10.1007	0.533094	0.266547	+	0.963822i	$0.414117\pi$
$360$	0	0
$361$	32.2028	1.69488
$362$	0.0978390	0.00514231
$363$	0	0
$364$	0	0
$365$	−18.5458	−0.970733
$366$	0	0
$367$	−7.77537	−0.405871	−0.202935	−	0.979192i	$-0.565048\pi$
$367$	−7.77537	−0.405871	−0.202935	+	0.979192i	$0.565048\pi$
$368$	−18.7576	−0.977808
$369$	0	0
$370$	−1.02891	−0.0534907
$371$	0	0
$372$	0	0
$373$	24.1111	1.24842	0.624212	−	0.781255i	$-0.285420\pi$
$373$	24.1111	1.24842	0.624212	+	0.781255i	$0.285420\pi$
$374$	1.71107	0.0884775
$375$	0	0
$376$	−8.80428	−0.454046
$377$	−7.82573	−0.403045
$378$	0	0
$379$	−13.3581	−0.686161	−0.343081	−	0.939306i	$-0.611470\pi$
$379$	−13.3581	−0.686161	−0.343081	+	0.939306i	$0.611470\pi$
$380$	−36.0312	−1.84836
$381$	0	0
$382$	3.83173	0.196048
$383$	−9.24040	−0.472162	−0.236081	−	0.971733i	$-0.575863\pi$
$383$	−9.24040	−0.472162	−0.236081	+	0.971733i	$0.575863\pi$
$384$	0	0
$385$	0	0
$386$	−2.95898	−0.150608
$387$	0	0
$388$	−5.95153	−0.302143
$389$	10.4484	0.529756	0.264878	−	0.964282i	$-0.414668\pi$
$389$	10.4484	0.529756	0.264878	+	0.964282i	$0.414668\pi$
$390$	0	0
$391$	−8.76884	−0.443459
$392$	0	0
$393$	0	0
$394$	5.52424	0.278307
$395$	25.4733	1.28170
$396$	0	0
$397$	−0.409157	−0.0205350	−0.0102675	−	0.999947i	$-0.503268\pi$
$397$	−0.409157	−0.0205350	−0.0102675	+	0.999947i	$0.503268\pi$
$398$	−1.61323	−0.0808641
$399$	0	0
$400$	6.28590	0.314295
$401$	15.2528	0.761688	0.380844	−	0.924639i	$-0.375633\pi$
$401$	15.2528	0.761688	0.380844	+	0.924639i	$0.375633\pi$
$402$	0	0
$403$	−24.0722	−1.19912
$404$	21.5647	1.07288
$405$	0	0
$406$	0	0
$407$	6.94282	0.344143
$408$	0	0
$409$	−6.12670	−0.302946	−0.151473	−	0.988461i	$-0.548402\pi$
$409$	−6.12670	−0.302946	−0.151473	+	0.988461i	$0.548402\pi$
$410$	6.33327	0.312778
$411$	0	0
$412$	−15.5153	−0.764384
$413$	0	0
$414$	0	0
$415$	−17.8695	−0.877178
$416$	10.1686	0.498557
$417$	0	0
$418$	−7.15561	−0.349992
$419$	1.56287	0.0763514	0.0381757	−	0.999271i	$-0.487845\pi$
$419$	1.56287	0.0763514	0.0381757	+	0.999271i	$0.487845\pi$
$420$	0	0
$421$	23.2632	1.13378	0.566889	−	0.823794i	$-0.308147\pi$
$421$	23.2632	1.13378	0.566889	+	0.823794i	$0.308147\pi$
$422$	4.03775	0.196555
$423$	0	0
$424$	−10.0377	−0.487476
$425$	2.93854	0.142540
$426$	0	0
$427$	0	0
$428$	−7.68675	−0.371553
$429$	0	0
$430$	1.02891	0.0496187
$431$	1.00576	0.0484456	0.0242228	−	0.999707i	$-0.492289\pi$
$431$	1.00576	0.0484456	0.0242228	+	0.999707i	$0.492289\pi$
$432$	0	0
$433$	13.1071	0.629889	0.314945	−	0.949110i	$-0.398014\pi$
$433$	13.1071	0.629889	0.314945	+	0.949110i	$0.398014\pi$
$434$	0	0
$435$	0	0
$436$	−14.1111	−0.675799
$437$	36.6708	1.75420
$438$	0	0
$439$	18.6141	0.888402	0.444201	−	0.895927i	$-0.353488\pi$
$439$	18.6141	0.888402	0.444201	+	0.895927i	$0.353488\pi$
$440$	10.2190	0.487170
$441$	0	0
$442$	1.50697	0.0716792
$443$	−1.11901	−0.0531656	−0.0265828	−	0.999647i	$-0.508463\pi$
$443$	−1.11901	−0.0531656	−0.0265828	+	0.999647i	$0.508463\pi$
$444$	0	0
$445$	13.0506	0.618660
$446$	−1.07639	−0.0509687
$447$	0	0
$448$	0	0
$449$	−39.4419	−1.86138	−0.930689	−	0.365813i	$-0.880791\pi$
$449$	−39.4419	−1.86138	−0.930689	+	0.365813i	$0.880791\pi$
$450$	0	0
$451$	−42.7351	−2.01232
$452$	−13.4621	−0.633203
$453$	0	0
$454$	−1.45011	−0.0680572
$455$	0	0
$456$	0	0
$457$	−34.2405	−1.60170	−0.800852	−	0.598863i	$-0.795619\pi$
$457$	−34.2405	−1.60170	−0.800852	+	0.598863i	$0.795619\pi$
$458$	−2.64215	−0.123459
$459$	0	0
$460$	−25.8051	−1.20317
$461$	20.3876	0.949543	0.474772	−	0.880109i	$-0.342531\pi$
$461$	20.3876	0.949543	0.474772	+	0.880109i	$0.342531\pi$
$462$	0	0
$463$	6.80903	0.316442	0.158221	−	0.987404i	$-0.449424\pi$
$463$	6.80903	0.316442	0.158221	+	0.987404i	$0.449424\pi$
$464$	−7.77704	−0.361040
$465$	0	0
$466$	−1.94609	−0.0901509
$467$	24.7911	1.14720	0.573598	−	0.819137i	$-0.305547\pi$
$467$	24.7911	1.14720	0.573598	+	0.819137i	$0.305547\pi$
$468$	0	0
$469$	0	0
$470$	−5.78744	−0.266955
$471$	0	0
$472$	−5.71754	−0.263171
$473$	−6.94282	−0.319231
$474$	0	0
$475$	−12.2888	−0.563850
$476$	0	0
$477$	0	0
$478$	5.06045	0.231460
$479$	−11.0997	−0.507157	−0.253579	−	0.967315i	$-0.581608\pi$
$479$	−11.0997	−0.507157	−0.253579	+	0.967315i	$0.581608\pi$
$480$	0	0
$481$	6.11465	0.278804
$482$	3.27395	0.149124
$483$	0	0
$484$	−12.6063	−0.573013
$485$	−7.93955	−0.360516
$486$	0	0
$487$	−10.0377	−0.454854	−0.227427	−	0.973795i	$-0.573031\pi$
$487$	−10.0377	−0.454854	−0.227427	+	0.973795i	$0.573031\pi$
$488$	7.52933	0.340837
$489$	0	0
$490$	0	0
$491$	−12.3880	−0.559061	−0.279530	−	0.960137i	$-0.590179\pi$
$491$	−12.3880	−0.559061	−0.279530	+	0.960137i	$0.590179\pi$
$492$	0	0
$493$	−3.63562	−0.163740
$494$	−6.30206	−0.283543
$495$	0	0
$496$	−23.9225	−1.07415
$497$	0	0
$498$	0	0
$499$	10.2222	0.457608	0.228804	−	0.973473i	$-0.426519\pi$
$499$	10.2222	0.457608	0.228804	+	0.973473i	$0.426519\pi$
$500$	−16.5293	−0.739212
$501$	0	0
$502$	3.63562	0.162266
$503$	−8.45753	−0.377102	−0.188551	−	0.982063i	$-0.560379\pi$
$503$	−8.45753	−0.377102	−0.188551	+	0.982063i	$0.560379\pi$
$504$	0	0
$505$	28.7680	1.28016
$506$	−5.12476	−0.227824
$507$	0	0
$508$	−17.7012	−0.785364
$509$	−10.5657	−0.468317	−0.234159	−	0.972198i	$-0.575233\pi$
$509$	−10.5657	−0.468317	−0.234159	+	0.972198i	$0.575233\pi$
$510$	0	0
$511$	0	0
$512$	17.0071	0.751616
$513$	0	0
$514$	6.12670	0.270237
$515$	−20.6979	−0.912060
$516$	0	0
$517$	39.0520	1.71750
$518$	0	0
$519$	0	0
$520$	9.00000	0.394676
$521$	19.7558	0.865515	0.432758	−	0.901510i	$-0.357541\pi$
$521$	19.7558	0.865515	0.432758	+	0.901510i	$0.357541\pi$
$522$	0	0
$523$	−32.5282	−1.42236	−0.711179	−	0.703011i	$-0.751839\pi$
$523$	−32.5282	−1.42236	−0.711179	+	0.703011i	$0.751839\pi$
$524$	8.35969	0.365195
$525$	0	0
$526$	1.69794	0.0740337
$527$	−11.1833	−0.487153
$528$	0	0
$529$	3.26320	0.141878
$530$	−6.59825	−0.286610
$531$	0	0
$532$	0	0
$533$	−37.6375	−1.63026
$534$	0	0
$535$	−10.2544	−0.443336
$536$	−7.79071	−0.336507
$537$	0	0
$538$	−3.92914	−0.169397
$539$	0	0
$540$	0	0
$541$	15.2222	0.654453	0.327226	−	0.944946i	$-0.393886\pi$
$541$	15.2222	0.654453	0.327226	+	0.944946i	$0.393886\pi$
$542$	3.03638	0.130424
$543$	0	0
$544$	4.72406	0.202542
$545$	−18.8247	−0.806361
$546$	0	0
$547$	23.3743	0.999412	0.499706	−	0.866195i	$-0.333441\pi$
$547$	23.3743	0.999412	0.499706	+	0.866195i	$0.333441\pi$
$548$	−40.0014	−1.70877
$549$	0	0
$550$	1.71737	0.0732289
$551$	15.2040	0.647711
$552$	0	0
$553$	0	0
$554$	−0.198104	−0.00841664
$555$	0	0
$556$	30.6214	1.29864
$557$	27.6673	1.17230	0.586151	−	0.810202i	$-0.300643\pi$
$557$	27.6673	1.17230	0.586151	+	0.810202i	$0.300643\pi$
$558$	0	0
$559$	−6.11465	−0.258622
$560$	0	0
$561$	0	0
$562$	1.24815	0.0526500
$563$	−8.55824	−0.360687	−0.180343	−	0.983604i	$-0.557721\pi$
$563$	−8.55824	−0.360687	−0.180343	+	0.983604i	$0.557721\pi$
$564$	0	0
$565$	−17.9589	−0.755536
$566$	−1.75855	−0.0739175
$567$	0	0
$568$	−5.88237	−0.246819
$569$	13.7278	0.575498	0.287749	−	0.957706i	$-0.407093\pi$
$569$	13.7278	0.575498	0.287749	+	0.957706i	$0.407093\pi$
$570$	0	0
$571$	10.7174	0.448508	0.224254	−	0.974531i	$-0.428005\pi$
$571$	10.7174	0.448508	0.224254	+	0.974531i	$0.428005\pi$
$572$	−29.9244	−1.25120
$573$	0	0
$574$	0	0
$575$	−8.80111	−0.367032
$576$	0	0
$577$	45.6353	1.89982	0.949912	−	0.312518i	$-0.101173\pi$
$577$	45.6353	1.89982	0.949912	+	0.312518i	$0.101173\pi$
$578$	−3.36500	−0.139965
$579$	0	0
$580$	−10.6990	−0.444251
$581$	0	0
$582$	0	0
$583$	44.5231	1.84396
$584$	−6.74645	−0.279170
$585$	0	0
$586$	1.87131	0.0773032
$587$	10.2190	0.421782	0.210891	−	0.977510i	$-0.432364\pi$
$587$	10.2190	0.421782	0.210891	+	0.977510i	$0.432364\pi$
$588$	0	0
$589$	46.7680	1.92704
$590$	−3.75839	−0.154730
$591$	0	0
$592$	6.07661	0.249747
$593$	−11.3961	−0.467981	−0.233990	−	0.972239i	$-0.575178\pi$
$593$	−11.3961	−0.467981	−0.233990	+	0.972239i	$0.575178\pi$
$594$	0	0
$595$	0	0
$596$	−11.7882	−0.482864
$597$	0	0
$598$	−4.51346	−0.184569
$599$	−34.5746	−1.41268	−0.706339	−	0.707874i	$-0.749655\pi$
$599$	−34.5746	−1.41268	−0.706339	+	0.707874i	$0.749655\pi$
$600$	0	0
$601$	38.8414	1.58437	0.792187	−	0.610279i	$-0.208943\pi$
$601$	38.8414	1.58437	0.792187	+	0.610279i	$0.208943\pi$
$602$	0	0
$603$	0	0
$604$	−8.72064	−0.354838
$605$	−16.8172	−0.683717
$606$	0	0
$607$	41.3325	1.67763	0.838817	−	0.544414i	$-0.183248\pi$
$607$	41.3325	1.67763	0.838817	+	0.544414i	$0.183248\pi$
$608$	−19.7558	−0.801202
$609$	0	0
$610$	4.94936	0.200394
$611$	34.3937	1.39142
$612$	0	0
$613$	−28.6569	−1.15744	−0.578721	−	0.815526i	$-0.696448\pi$
$613$	−28.6569	−1.15744	−0.578721	+	0.815526i	$0.696448\pi$
$614$	5.42114	0.218779
$615$	0	0
$616$	0	0
$617$	−33.7037	−1.35686	−0.678430	−	0.734665i	$-0.737339\pi$
$617$	−33.7037	−1.35686	−0.678430	+	0.734665i	$0.737339\pi$
$618$	0	0
$619$	−1.43807	−0.0578010	−0.0289005	−	0.999582i	$-0.509201\pi$
$619$	−1.43807	−0.0578010	−0.0289005	+	0.999582i	$0.509201\pi$
$620$	−32.9105	−1.32172
$621$	0	0
$622$	−7.72789	−0.309860
$623$	0	0
$624$	0	0
$625$	−30.6375	−1.22550
$626$	−5.81538	−0.232429
$627$	0	0
$628$	−1.99900	−0.0797686
$629$	2.84071	0.113266
$630$	0	0
$631$	−30.7680	−1.22486	−0.612428	−	0.790527i	$-0.709807\pi$
$631$	−30.7680	−1.22486	−0.612428	+	0.790527i	$0.709807\pi$
$632$	9.26647	0.368600
$633$	0	0
$634$	1.22872	0.0487987
$635$	−23.6140	−0.937094
$636$	0	0
$637$	0	0
$638$	−2.12476	−0.0841202
$639$	0	0
$640$	18.4389	0.728862
$641$	9.23912	0.364923	0.182462	−	0.983213i	$-0.441593\pi$
$641$	9.23912	0.364923	0.182462	+	0.983213i	$0.441593\pi$
$642$	0	0
$643$	25.5591	1.00795	0.503976	−	0.863718i	$-0.331870\pi$
$643$	25.5591	1.00795	0.503976	+	0.863718i	$0.331870\pi$
$644$	0	0
$645$	0	0
$646$	−2.92777	−0.115192
$647$	−28.3111	−1.11302	−0.556512	−	0.830839i	$-0.687861\pi$
$647$	−28.3111	−1.11302	−0.556512	+	0.830839i	$0.687861\pi$
$648$	0	0
$649$	25.3605	0.995488
$650$	1.51252	0.0593257
$651$	0	0
$652$	−13.2665	−0.519555
$653$	−8.35021	−0.326769	−0.163385	−	0.986562i	$-0.552241\pi$
$653$	−8.35021	−0.326769	−0.163385	+	0.986562i	$0.552241\pi$
$654$	0	0
$655$	11.1521	0.435749
$656$	−37.4033	−1.46035
$657$	0	0
$658$	0	0
$659$	−33.5724	−1.30779	−0.653897	−	0.756583i	$-0.726867\pi$
$659$	−33.5724	−1.30779	−0.653897	+	0.756583i	$0.726867\pi$
$660$	0	0
$661$	16.9534	0.659410	0.329705	−	0.944084i	$-0.393051\pi$
$661$	16.9534	0.659410	0.329705	+	0.944084i	$0.393051\pi$
$662$	−2.79725	−0.108718
$663$	0	0
$664$	−6.50041	−0.252265
$665$	0	0
$666$	0	0
$667$	10.8889	0.421620
$668$	−34.9597	−1.35263
$669$	0	0
$670$	−5.12118	−0.197848
$671$	−33.3969	−1.28927
$672$	0	0
$673$	−44.4315	−1.71271	−0.856354	−	0.516390i	$-0.827276\pi$
$673$	−44.4315	−1.71271	−0.856354	+	0.516390i	$0.827276\pi$
$674$	−8.05718	−0.310351
$675$	0	0
$676$	−1.09820	−0.0422384
$677$	14.3736	0.552423	0.276212	−	0.961097i	$-0.410921\pi$
$677$	14.3736	0.552423	0.276212	+	0.961097i	$0.410921\pi$
$678$	0	0
$679$	0	0
$680$	4.18116	0.160340
$681$	0	0
$682$	−6.53585	−0.250271
$683$	−32.3092	−1.23628	−0.618138	−	0.786069i	$-0.712113\pi$
$683$	−32.3092	−1.23628	−0.618138	+	0.786069i	$0.712113\pi$
$684$	0	0
$685$	−53.3632	−2.03890
$686$	0	0
$687$	0	0
$688$	−6.07661	−0.231669
$689$	39.2122	1.49387
$690$	0	0
$691$	28.9962	1.10307	0.551533	−	0.834153i	$-0.314043\pi$
$691$	28.9962	1.10307	0.551533	+	0.834153i	$0.314043\pi$
$692$	1.61323	0.0613259
$693$	0	0
$694$	6.53078	0.247905
$695$	40.8500	1.54953
$696$	0	0
$697$	−17.4854	−0.662306
$698$	5.47997	0.207420
$699$	0	0
$700$	0	0
$701$	−26.3912	−0.996783	−0.498392	−	0.866952i	$-0.666076\pi$
$701$	−26.3912	−0.996783	−0.498392	+	0.866952i	$0.666076\pi$
$702$	0	0
$703$	−11.8797	−0.448050
$704$	−27.8525	−1.04973
$705$	0	0
$706$	2.45563	0.0924190
$707$	0	0
$708$	0	0
$709$	−7.88564	−0.296151	−0.148076	−	0.988976i	$-0.547308\pi$
$709$	−7.88564	−0.296151	−0.148076	+	0.988976i	$0.547308\pi$
$710$	−3.86674	−0.145116
$711$	0	0
$712$	4.74746	0.177918
$713$	33.4947	1.25439
$714$	0	0
$715$	−39.9201	−1.49293
$716$	−14.7245	−0.550281
$717$	0	0
$718$	2.41531	0.0901385
$719$	33.1508	1.23632	0.618159	−	0.786053i	$-0.287879\pi$
$719$	33.1508	1.23632	0.618159	+	0.786053i	$0.287879\pi$
$720$	0	0
$721$	0	0
$722$	7.70043	0.286580
$723$	0	0
$724$	−0.794919	−0.0295429
$725$	−3.64900	−0.135521
$726$	0	0
$727$	−33.1005	−1.22763	−0.613814	−	0.789451i	$-0.710366\pi$
$727$	−33.1005	−1.22763	−0.613814	+	0.789451i	$0.710366\pi$
$728$	0	0
$729$	0	0
$730$	−4.43474	−0.164137
$731$	−2.84071	−0.105067
$732$	0	0
$733$	44.5589	1.64582	0.822911	−	0.568170i	$-0.192349\pi$
$733$	44.5589	1.64582	0.822911	+	0.568170i	$0.192349\pi$
$734$	−1.85927	−0.0686270
$735$	0	0
$736$	−14.1488	−0.521533
$737$	34.5562	1.27290
$738$	0	0
$739$	39.8090	1.46440	0.732199	−	0.681090i	$-0.238494\pi$
$739$	39.8090	1.46440	0.732199	+	0.681090i	$0.238494\pi$
$740$	8.35969	0.307308
$741$	0	0
$742$	0	0
$743$	10.7414	0.394065	0.197033	−	0.980397i	$-0.436869\pi$
$743$	10.7414	0.394065	0.197033	+	0.980397i	$0.436869\pi$
$744$	0	0
$745$	−15.7259	−0.576152
$746$	5.76552	0.211091
$747$	0	0
$748$	−13.9021	−0.508310
$749$	0	0
$750$	0	0
$751$	19.7141	0.719378	0.359689	−	0.933072i	$-0.382883\pi$
$751$	19.7141	0.719378	0.359689	+	0.933072i	$0.382883\pi$
$752$	34.1797	1.24641
$753$	0	0
$754$	−1.87131	−0.0681492
$755$	−11.6336	−0.423391
$756$	0	0
$757$	35.3549	1.28499	0.642497	−	0.766288i	$-0.277898\pi$
$757$	35.3549	1.28499	0.642497	+	0.766288i	$0.277898\pi$
$758$	−3.19424	−0.116020
$759$	0	0
$760$	−17.4854	−0.634261
$761$	39.1144	1.41790	0.708948	−	0.705261i	$-0.249170\pi$
$761$	39.1144	1.41790	0.708948	+	0.705261i	$0.249170\pi$
$762$	0	0
$763$	0	0
$764$	−31.1319	−1.12631
$765$	0	0
$766$	−2.20960	−0.0798359
$767$	22.3354	0.806486
$768$	0	0
$769$	37.8479	1.36483	0.682415	−	0.730965i	$-0.260930\pi$
$769$	37.8479	1.36483	0.682415	+	0.730965i	$0.260930\pi$
$770$	0	0
$771$	0	0
$772$	24.0410	0.865255
$773$	−29.8265	−1.07279	−0.536393	−	0.843969i	$-0.680213\pi$
$773$	−29.8265	−1.07279	−0.536393	+	0.843969i	$0.680213\pi$
$774$	0	0
$775$	−11.2245	−0.403195
$776$	−2.88819	−0.103680
$777$	0	0
$778$	2.49846	0.0895741
$779$	73.1229	2.61990
$780$	0	0
$781$	26.0917	0.933633
$782$	−2.09683	−0.0749827
$783$	0	0
$784$	0	0
$785$	−2.66673	−0.0951796
$786$	0	0
$787$	−17.6206	−0.628107	−0.314053	−	0.949405i	$-0.601687\pi$
$787$	−17.6206	−0.628107	−0.314053	+	0.949405i	$0.601687\pi$
$788$	−44.8832	−1.59890
$789$	0	0
$790$	6.09126	0.216717
$791$	0	0
$792$	0	0
$793$	−29.4132	−1.04449
$794$	−0.0978390	−0.00347218
$795$	0	0
$796$	13.1071	0.464570
$797$	−10.1211	−0.358508	−0.179254	−	0.983803i	$-0.557368\pi$
$797$	−10.1211	−0.358508	−0.179254	+	0.983803i	$0.557368\pi$
$798$	0	0
$799$	15.9784	0.565276
$800$	4.74145	0.167635
$801$	0	0
$802$	3.64730	0.128791
$803$	29.9244	1.05601
$804$	0	0
$805$	0	0
$806$	−5.75623	−0.202755
$807$	0	0
$808$	10.4650	0.368157
$809$	−47.1469	−1.65760	−0.828799	−	0.559546i	$-0.810975\pi$
$809$	−47.1469	−1.65760	−0.828799	+	0.559546i	$0.810975\pi$
$810$	0	0
$811$	−21.0577	−0.739435	−0.369717	−	0.929144i	$-0.620546\pi$
$811$	−21.0577	−0.739435	−0.369717	+	0.929144i	$0.620546\pi$
$812$	0	0
$813$	0	0
$814$	1.66019	0.0581896
$815$	−17.6979	−0.619931
$816$	0	0
$817$	11.8797	0.415617
$818$	−1.46504	−0.0512238
$819$	0	0
$820$	−51.4563	−1.79693
$821$	11.1604	0.389499	0.194750	−	0.980853i	$-0.437611\pi$
$821$	11.1604	0.389499	0.194750	+	0.980853i	$0.437611\pi$
$822$	0	0
$823$	9.43474	0.328874	0.164437	−	0.986388i	$-0.447419\pi$
$823$	9.43474	0.328874	0.164437	+	0.986388i	$0.447419\pi$
$824$	−7.52933	−0.262297
$825$	0	0
$826$	0	0
$827$	17.2646	0.600348	0.300174	−	0.953884i	$-0.402955\pi$
$827$	17.2646	0.600348	0.300174	+	0.953884i	$0.402955\pi$
$828$	0	0
$829$	48.4526	1.68283	0.841415	−	0.540390i	$-0.181723\pi$
$829$	48.4526	1.68283	0.841415	+	0.540390i	$0.181723\pi$
$830$	−4.27301	−0.148318
$831$	0	0
$832$	−24.5302	−0.850431
$833$	0	0
$834$	0	0
$835$	−46.6375	−1.61396
$836$	58.1376	2.01073
$837$	0	0
$838$	0.373720	0.0129099
$839$	14.8686	0.513320	0.256660	−	0.966502i	$-0.417378\pi$
$839$	14.8686	0.513320	0.256660	+	0.966502i	$0.417378\pi$
$840$	0	0
$841$	−24.4854	−0.844323
$842$	5.56277	0.191706
$843$	0	0
$844$	−32.8058	−1.12922
$845$	−1.46504	−0.0503988
$846$	0	0
$847$	0	0
$848$	38.9683	1.33818
$849$	0	0
$850$	0.702674	0.0241015
$851$	−8.50808	−0.291653
$852$	0	0
$853$	−7.99801	−0.273847	−0.136923	−	0.990582i	$-0.543721\pi$
$853$	−7.99801	−0.273847	−0.136923	+	0.990582i	$0.543721\pi$
$854$	0	0
$855$	0	0
$856$	−3.73026	−0.127498
$857$	43.1322	1.47337	0.736684	−	0.676237i	$-0.236390\pi$
$857$	43.1322	1.47337	0.736684	+	0.676237i	$0.236390\pi$
$858$	0	0
$859$	2.44359	0.0833742	0.0416871	−	0.999131i	$-0.486727\pi$
$859$	2.44359	0.0833742	0.0416871	+	0.999131i	$0.486727\pi$
$860$	−8.35969	−0.285063
$861$	0	0
$862$	0.240500	0.00819146
$863$	25.7187	0.875476	0.437738	−	0.899103i	$-0.355780\pi$
$863$	25.7187	0.875476	0.437738	+	0.899103i	$0.355780\pi$
$864$	0	0
$865$	2.15211	0.0731739
$866$	3.13422	0.106505
$867$	0	0
$868$	0	0
$869$	−41.1021	−1.39429
$870$	0	0
$871$	30.4342	1.03122
$872$	−6.84789	−0.231899
$873$	0	0
$874$	8.76884	0.296611
$875$	0	0
$876$	0	0
$877$	−21.9590	−0.741502	−0.370751	−	0.928732i	$-0.620900\pi$
$877$	−21.9590	−0.741502	−0.370751	+	0.928732i	$0.620900\pi$
$878$	4.45106	0.150216
$879$	0	0
$880$	−39.6718	−1.33733
$881$	35.0576	1.18112	0.590560	−	0.806994i	$-0.298907\pi$
$881$	35.0576	1.18112	0.590560	+	0.806994i	$0.298907\pi$
$882$	0	0
$883$	26.3009	0.885097	0.442549	−	0.896744i	$-0.354074\pi$
$883$	26.3009	0.885097	0.442549	+	0.896744i	$0.354074\pi$
$884$	−12.2438	−0.411803
$885$	0	0
$886$	−0.267580	−0.00898954
$887$	−47.8180	−1.60557	−0.802785	−	0.596269i	$-0.796649\pi$
$887$	−47.8180	−1.60557	−0.802785	+	0.596269i	$0.796649\pi$
$888$	0	0
$889$	0	0
$890$	3.12071	0.104607
$891$	0	0
$892$	8.74545	0.292819
$893$	−66.8208	−2.23607
$894$	0	0
$895$	−19.6430	−0.656593
$896$	0	0
$897$	0	0
$898$	−9.43147	−0.314732
$899$	13.8871	0.463162
$900$	0	0
$901$	18.2170	0.606895
$902$	−10.2190	−0.340254
$903$	0	0
$904$	−6.53294	−0.217282
$905$	−1.06045	−0.0352505
$906$	0	0
$907$	−19.1144	−0.634682	−0.317341	−	0.948312i	$-0.602790\pi$
$907$	−19.1144	−0.634682	−0.317341	+	0.948312i	$0.602790\pi$
$908$	11.7818	0.390994
$909$	0	0
$910$	0	0
$911$	−18.0586	−0.598307	−0.299153	−	0.954205i	$-0.596704\pi$
$911$	−18.0586	−0.598307	−0.299153	+	0.954205i	$0.596704\pi$
$912$	0	0
$913$	28.8330	0.954234
$914$	−8.18770	−0.270825
$915$	0	0
$916$	21.4668	0.709284
$917$	0	0
$918$	0	0
$919$	16.2093	0.534695	0.267348	−	0.963600i	$-0.413853\pi$
$919$	16.2093	0.534695	0.267348	+	0.963600i	$0.413853\pi$
$920$	−12.5228	−0.412865
$921$	0	0
$922$	4.87514	0.160554
$923$	22.9793	0.756374
$924$	0	0
$925$	2.85116	0.0937456
$926$	1.62820	0.0535059
$927$	0	0
$928$	−5.86621	−0.192568
$929$	−22.6829	−0.744203	−0.372102	−	0.928192i	$-0.621363\pi$
$929$	−22.6829	−0.744203	−0.372102	+	0.928192i	$0.621363\pi$
$930$	0	0
$931$	0	0
$932$	15.8115	0.517923
$933$	0	0
$934$	5.92814	0.193975
$935$	−18.5458	−0.606513
$936$	0	0
$937$	51.2933	1.67568	0.837840	−	0.545915i	$-0.183818\pi$
$937$	51.2933	1.67568	0.837840	+	0.545915i	$0.183818\pi$
$938$	0	0
$939$	0	0
$940$	47.0216	1.53368
$941$	−31.9318	−1.04095	−0.520474	−	0.853878i	$-0.674245\pi$
$941$	−31.9318	−1.04095	−0.520474	+	0.853878i	$0.674245\pi$
$942$	0	0
$943$	52.3697	1.70539
$944$	22.1965	0.722433
$945$	0	0
$946$	−1.66019	−0.0539774
$947$	−4.49330	−0.146013	−0.0730063	−	0.997331i	$-0.523259\pi$
$947$	−4.49330	−0.146013	−0.0730063	+	0.997331i	$0.523259\pi$
$948$	0	0
$949$	26.3549	0.855515
$950$	−2.93854	−0.0953390
$951$	0	0
$952$	0	0
$953$	1.14635	0.0371340	0.0185670	−	0.999828i	$-0.494090\pi$
$953$	1.14635	0.0371340	0.0185670	+	0.999828i	$0.494090\pi$
$954$	0	0
$955$	−41.5310	−1.34391
$956$	−41.1150	−1.32975
$957$	0	0
$958$	−2.65419	−0.0857530
$959$	0	0
$960$	0	0
$961$	11.7174	0.377980
$962$	1.46216	0.0471418
$963$	0	0
$964$	−26.6000	−0.856730
$965$	32.0715	1.03242
$966$	0	0
$967$	49.6159	1.59554	0.797770	−	0.602962i	$-0.206013\pi$
$967$	49.6159	1.59554	0.797770	+	0.602962i	$0.206013\pi$
$968$	−6.11763	−0.196628
$969$	0	0
$970$	−1.89853	−0.0609582
$971$	−5.13322	−0.164733	−0.0823664	−	0.996602i	$-0.526248\pi$
$971$	−5.13322	−0.164733	−0.0823664	+	0.996602i	$0.526248\pi$
$972$	0	0
$973$	0	0
$974$	−2.40026	−0.0769093
$975$	0	0
$976$	−29.2302	−0.935634
$977$	31.1948	0.998011	0.499006	−	0.866599i	$-0.333699\pi$
$977$	31.1948	0.998011	0.499006	+	0.866599i	$0.333699\pi$
$978$	0	0
$979$	−21.0577	−0.673006
$980$	0	0
$981$	0	0
$982$	−2.96225	−0.0945292
$983$	20.3401	0.648748	0.324374	−	0.945929i	$-0.394846\pi$
$983$	20.3401	0.648748	0.324374	+	0.945929i	$0.394846\pi$
$984$	0	0
$985$	−59.8757	−1.90780
$986$	−0.869363	−0.0276861
$987$	0	0
$988$	51.2028	1.62898
$989$	8.50808	0.270541
$990$	0	0
$991$	−12.9655	−0.411863	−0.205932	−	0.978566i	$-0.566022\pi$
$991$	−12.9655	−0.411863	−0.205932	+	0.978566i	$0.566022\pi$
$992$	−18.0447	−0.572919
$993$	0	0
$994$	0	0
$995$	17.4854	0.554324
$996$	0	0
$997$	−49.4816	−1.56710	−0.783548	−	0.621331i	$-0.786592\pi$
$997$	−49.4816	−1.56710	−0.783548	+	0.621331i	$0.786592\pi$
$998$	2.44436	0.0773749
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.be.1.3		6
3.2	odd	2		3969.2.a.bd.1.4		6
7.6	odd	2	inner	3969.2.a.be.1.4		6
9.2	odd	6		1323.2.f.g.442.3		12
9.4	even	3		441.2.f.g.295.3	yes	12
9.5	odd	6		1323.2.f.g.883.3		12
9.7	even	3		441.2.f.g.148.3	✓	12
21.20	even	2		3969.2.a.bd.1.3		6
63.2	odd	6		1323.2.g.g.361.4		12
63.4	even	3		441.2.g.g.79.3		12
63.5	even	6		1323.2.h.g.802.4		12
63.11	odd	6		1323.2.h.g.226.3		12
63.13	odd	6		441.2.f.g.295.4	yes	12
63.16	even	3		441.2.g.g.67.3		12
63.20	even	6		1323.2.f.g.442.4		12
63.23	odd	6		1323.2.h.g.802.3		12
63.25	even	3		441.2.h.g.373.4		12
63.31	odd	6		441.2.g.g.79.4		12
63.32	odd	6		1323.2.g.g.667.4		12
63.34	odd	6		441.2.f.g.148.4	yes	12
63.38	even	6		1323.2.h.g.226.4		12
63.40	odd	6		441.2.h.g.214.3		12
63.41	even	6		1323.2.f.g.883.4		12
63.47	even	6		1323.2.g.g.361.3		12
63.52	odd	6		441.2.h.g.373.3		12
63.58	even	3		441.2.h.g.214.4		12
63.59	even	6		1323.2.g.g.667.3		12
63.61	odd	6		441.2.g.g.67.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.f.g.148.3	✓	12	9.7	even	3
441.2.f.g.148.4	yes	12	63.34	odd	6
441.2.f.g.295.3	yes	12	9.4	even	3
441.2.f.g.295.4	yes	12	63.13	odd	6
441.2.g.g.67.3		12	63.16	even	3
441.2.g.g.67.4		12	63.61	odd	6
441.2.g.g.79.3		12	63.4	even	3
441.2.g.g.79.4		12	63.31	odd	6
441.2.h.g.214.3		12	63.40	odd	6
441.2.h.g.214.4		12	63.58	even	3
441.2.h.g.373.3		12	63.52	odd	6
441.2.h.g.373.4		12	63.25	even	3
1323.2.f.g.442.3		12	9.2	odd	6
1323.2.f.g.442.4		12	63.20	even	6
1323.2.f.g.883.3		12	9.5	odd	6
1323.2.f.g.883.4		12	63.41	even	6
1323.2.g.g.361.3		12	63.47	even	6
1323.2.g.g.361.4		12	63.2	odd	6
1323.2.g.g.667.3		12	63.59	even	6
1323.2.g.g.667.4		12	63.32	odd	6
1323.2.h.g.226.3		12	63.11	odd	6
1323.2.h.g.226.4		12	63.38	even	6
1323.2.h.g.802.3		12	63.23	odd	6
1323.2.h.g.802.4		12	63.5	even	6
3969.2.a.bd.1.3		6	21.20	even	2
3969.2.a.bd.1.4		6	3.2	odd	2
3969.2.a.be.1.3		6	1.1	even	1	trivial
3969.2.a.be.1.4		6	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+0.239123 q^{2} -1.94282 q^{4} -2.59179 q^{5} -0.942820 q^{8} +O(q^{10})\)	\(q+0.239123 q^{2} -1.94282 q^{4} -2.59179 q^{5} -0.942820 q^{8} -0.619757 q^{10} +4.18194 q^{11} +3.68310 q^{13} +3.66019 q^{16} +1.71107 q^{17} -7.15561 q^{19} +5.03538 q^{20} +1.00000 q^{22} -5.12476 q^{23} +1.71737 q^{25} +0.880716 q^{26} -2.12476 q^{29} -6.53585 q^{31} +2.76088 q^{32} +0.409157 q^{34} +1.66019 q^{37} -1.71107 q^{38} +2.44359 q^{40} -10.2190 q^{41} -1.66019 q^{43} -8.12476 q^{44} -1.22545 q^{46} +9.33824 q^{47} +0.410663 q^{50} -7.15561 q^{52} +10.6465 q^{53} -10.8387 q^{55} -0.508080 q^{58} +6.06429 q^{59} -7.98597 q^{61} -1.56287 q^{62} -6.66019 q^{64} -9.54583 q^{65} +8.26320 q^{67} -3.32431 q^{68} +6.23912 q^{71} +7.15561 q^{73} +0.396990 q^{74} +13.9021 q^{76} -9.82846 q^{79} -9.48644 q^{80} -2.44359 q^{82} +6.89465 q^{83} -4.43474 q^{85} -0.396990 q^{86} -3.94282 q^{88} -5.03538 q^{89} +9.95649 q^{92} +2.23299 q^{94} +18.5458 q^{95} +3.06335 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 6 q^{4} + 12 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 6 * q^4 + 12 * q^8	\( 6 q + 2 q^{2} + 6 q^{4} + 12 q^{8} + 8 q^{11} + 6 q^{16} + 6 q^{22} + 4 q^{23} + 12 q^{25} + 22 q^{29} + 16 q^{32} - 6 q^{37} + 6 q^{43} - 14 q^{44} + 12 q^{46} + 56 q^{50} + 28 q^{53} + 18 q^{58} - 24 q^{64} - 6 q^{65} + 38 q^{71} + 36 q^{74} - 6 q^{79} - 30 q^{85} - 36 q^{86} - 6 q^{88} + 62 q^{92} + 60 q^{95}+O(q^{100}) \) 6 * q + 2 * q^2 + 6 * q^4 + 12 * q^8 + 8 * q^11 + 6 * q^16 + 6 * q^22 + 4 * q^23 + 12 * q^25 + 22 * q^29 + 16 * q^32 - 6 * q^37 + 6 * q^43 - 14 * q^44 + 12 * q^46 + 56 * q^50 + 28 * q^53 + 18 * q^58 - 24 * q^64 - 6 * q^65 + 38 * q^71 + 36 * q^74 - 6 * q^79 - 30 * q^85 - 36 * q^86 - 6 * q^88 + 62 * q^92 + 60 * q^95

Introduction

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