LMFDB - Embedded newform 3969.2.a.bd.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:	$1$
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	$-0.526943\pi$
$2$	−0.239123	−0.169086	−0.0845428	+	0.996420i	$0.526943\pi$
$3$	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	$-0.717130\pi$
$11$	−4.18194	−1.26090	−0.630452	+	0.776228i	$0.717130\pi$
$12$	0	0
$13$	−3.68310	−1.02151	−0.510755	−	0.859726i	$-0.670634\pi$
$13$	−3.68310	−1.02151	−0.510755	+	0.859726i	$0.670634\pi$
$14$	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	$-0.193525\pi$
$19$	7.15561	1.64161	0.820805	+	0.571209i	$0.193525\pi$
$20$	5.03538	1.12595
$21$	0	0
$22$	1.00000	0.213201
$23$	5.12476	1.06859	0.534294	−	0.845299i	$-0.320578\pi$
$23$	5.12476	1.06859	0.534294	+	0.845299i	$0.320578\pi$
$24$	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	$-0.436789\pi$
$29$	2.12476	0.394559	0.197279	+	0.980347i	$0.436789\pi$
$30$	0	0
$31$	6.53585	1.17387	0.586937	−	0.809633i	$-0.300334\pi$
$31$	6.53585	1.17387	0.586937	+	0.809633i	$0.300334\pi$
$32$	−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	$-0.761042\pi$
$53$	−10.6465	−1.46241	−0.731206	+	0.682157i	$0.761042\pi$
$54$	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	$-0.329183\pi$
$61$	7.98597	1.02250	0.511249	+	0.859433i	$0.329183\pi$
$62$	−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	$-0.620719\pi$
$71$	−6.23912	−0.740448	−0.370224	+	0.928943i	$0.620719\pi$
$72$	0	0
$73$	−7.15561	−0.837501	−0.418750	−	0.908101i	$-0.637532\pi$
$73$	−7.15561	−0.837501	−0.418750	+	0.908101i	$0.637532\pi$
$74$	−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	$-0.549705\pi$
$97$	−3.06335	−0.311036	−0.155518	+	0.987833i	$0.549705\pi$
$98$	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	$-0.628715\pi$
$103$	−7.98597	−0.786881	−0.393440	+	0.919350i	$0.628715\pi$
$104$	−3.47250	−0.340507
$105$	0	0
$106$	2.54583	0.247273
$107$	−3.95649	−0.382489	−0.191244	−	0.981542i	$-0.561252\pi$
$107$	−3.95649	−0.382489	−0.191244	+	0.981542i	$0.561252\pi$
$108$	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	$-0.605674\pi$
$113$	−6.92915	−0.651839	−0.325920	+	0.945397i	$0.605674\pi$
$114$	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	$-0.842145\pi$
$137$	−20.5893	−1.75907	−0.879533	+	0.475838i	$0.842145\pi$
$138$	0	0
$139$	15.7613	1.33686	0.668429	−	0.743776i	$-0.266967\pi$
$139$	15.7613	1.33686	0.668429	+	0.743776i	$0.266967\pi$
$140$	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	$-0.579950\pi$
$149$	−6.06758	−0.497076	−0.248538	+	0.968622i	$0.579950\pi$
$150$	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	$-0.513073\pi$
$157$	−1.02891	−0.0821163	−0.0410582	+	0.999157i	$0.513073\pi$
$158$	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	$-0.591409\pi$
$179$	−7.57893	−0.566476	−0.283238	+	0.959050i	$0.591409\pi$
$180$	0	0
$181$	−0.409157	−0.0304124	−0.0152062	−	0.999884i	$-0.504840\pi$
$181$	−0.409157	−0.0304124	−0.0152062	+	0.999884i	$0.504840\pi$
$182$	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	$-0.696842\pi$
$191$	−16.0241	−1.15946	−0.579731	+	0.814808i	$0.696842\pi$
$192$	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	$-0.807688\pi$
$197$	−23.1021	−1.64595	−0.822977	+	0.568075i	$0.807688\pi$
$198$	0	0
$199$	6.74645	0.478243	0.239122	−	0.970990i	$-0.423141\pi$
$199$	6.74645	0.478243	0.239122	+	0.970990i	$0.423141\pi$
$200$	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	$-0.451841\pi$
$223$	4.50142	0.301437	0.150719	+	0.988577i	$0.451841\pi$
$224$	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	$-0.381042\pi$
$229$	11.0493	0.730159	0.365080	+	0.930976i	$0.381042\pi$
$230$	3.17611	0.209426
$231$	0	0
$232$	2.00327	0.131521
$233$	8.13844	0.533167	0.266583	−	0.963812i	$-0.414105\pi$
$233$	8.13844	0.533167	0.266583	+	0.963812i	$0.414105\pi$
$234$	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	$-0.739955\pi$
$239$	−21.1625	−1.36889	−0.684445	+	0.729065i	$0.739955\pi$
$240$	0	0
$241$	−13.6915	−0.881945	−0.440972	−	0.897521i	$-0.645366\pi$
$241$	−13.6915	−0.881945	−0.440972	+	0.897521i	$0.645366\pi$
$242$	−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	$-0.570255\pi$
$263$	−7.10069	−0.437847	−0.218924	+	0.975742i	$0.570255\pi$
$264$	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	$-0.626031\pi$
$271$	−12.6980	−0.771348	−0.385674	+	0.922635i	$0.626031\pi$
$272$	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	$-0.549760\pi$
$281$	−5.21969	−0.311381	−0.155690	+	0.987806i	$0.549760\pi$
$282$	0	0
$283$	7.35417	0.437160	0.218580	−	0.975819i	$-0.429858\pi$
$283$	7.35417	0.437160	0.218580	+	0.975819i	$0.429858\pi$
$284$	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	$-0.723955\pi$
$307$	−22.6709	−1.29390	−0.646948	+	0.762534i	$0.723955\pi$
$308$	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	$-0.258791\pi$
$313$	24.3196	1.37462	0.687312	+	0.726362i	$0.258791\pi$
$314$	0.246037	0.0138847
$315$	0	0
$316$	19.0949	1.07417
$317$	−5.13844	−0.288603	−0.144302	−	0.989534i	$-0.546094\pi$
$317$	−5.13844	−0.288603	−0.144302	+	0.989534i	$0.546094\pi$
$318$	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	$-0.761916\pi$
$347$	−27.3114	−1.46615	−0.733075	+	0.680148i	$0.761916\pi$
$348$	0	0
$349$	−22.9169	−1.22672	−0.613358	−	0.789805i	$-0.710182\pi$
$349$	−22.9169	−1.22672	−0.613358	+	0.789805i	$0.710182\pi$
$350$	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	$-0.585883\pi$
$359$	−10.1007	−0.533094	−0.266547	+	0.963822i	$0.585883\pi$
$360$	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	$-0.434952\pi$
$367$	7.77537	0.405871	0.202935	+	0.979192i	$0.434952\pi$
$368$	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	$-0.585332\pi$
$389$	−10.4484	−0.529756	−0.264878	+	0.964282i	$0.585332\pi$
$390$	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	$-0.496732\pi$
$397$	0.409157	0.0205350	0.0102675	+	0.999947i	$0.496732\pi$
$398$	−1.61323	−0.0808641
$399$	0	0
$400$	6.28590	0.314295
$401$	−15.2528	−0.761688	−0.380844	−	0.924639i	$-0.624367\pi$
$401$	−15.2528	−0.761688	−0.380844	+	0.924639i	$0.624367\pi$
$402$	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	$-0.451598\pi$
$409$	6.12670	0.302946	0.151473	+	0.988461i	$0.451598\pi$
$410$	−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	$-0.507711\pi$
$431$	−1.00576	−0.0484456	−0.0242228	+	0.999707i	$0.507711\pi$
$432$	0	0
$433$	−13.1071	−0.629889	−0.314945	−	0.949110i	$-0.601986\pi$
$433$	−13.1071	−0.629889	−0.314945	+	0.949110i	$0.601986\pi$
$434$	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	$-0.646512\pi$
$439$	−18.6141	−0.888402	−0.444201	+	0.895927i	$0.646512\pi$
$440$	10.2190	0.487170
$441$	0	0
$442$	1.50697	0.0716792
$443$	1.11901	0.0531656	0.0265828	−	0.999647i	$-0.491537\pi$
$443$	1.11901	0.0531656	0.0265828	+	0.999647i	$0.491537\pi$
$444$	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	$-0.119209\pi$
$449$	39.4419	1.86138	0.930689	+	0.365813i	$0.119209\pi$
$450$	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	$-0.409821\pi$
$491$	12.3880	0.559061	0.279530	+	0.960137i	$0.409821\pi$
$492$	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	$-0.248161\pi$
$523$	32.5282	1.42236	0.711179	+	0.703011i	$0.248161\pi$
$524$	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	$-0.699357\pi$
$557$	−27.6673	−1.17230	−0.586151	+	0.810202i	$0.699357\pi$
$558$	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	$-0.592907\pi$
$569$	−13.7278	−0.575498	−0.287749	+	0.957706i	$0.592907\pi$
$570$	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	$-0.898827\pi$
$577$	−45.6353	−1.89982	−0.949912	+	0.312518i	$0.898827\pi$
$578$	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	$-0.250345\pi$
$599$	34.5746	1.41268	0.706339	+	0.707874i	$0.250345\pi$
$600$	0	0
$601$	−38.8414	−1.58437	−0.792187	−	0.610279i	$-0.791057\pi$
$601$	−38.8414	−1.58437	−0.792187	+	0.610279i	$0.791057\pi$
$602$	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	$-0.816752\pi$
$607$	−41.3325	−1.67763	−0.838817	+	0.544414i	$0.816752\pi$
$608$	−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	$-0.262661\pi$
$617$	33.7037	1.35686	0.678430	+	0.734665i	$0.262661\pi$
$618$	0	0
$619$	1.43807	0.0578010	0.0289005	−	0.999582i	$-0.490799\pi$
$619$	1.43807	0.0578010	0.0289005	+	0.999582i	$0.490799\pi$
$620$	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	$-0.558407\pi$
$641$	−9.23912	−0.364923	−0.182462	+	0.983213i	$0.558407\pi$
$642$	0	0
$643$	−25.5591	−1.00795	−0.503976	−	0.863718i	$-0.668130\pi$
$643$	−25.5591	−1.00795	−0.503976	+	0.863718i	$0.668130\pi$
$644$	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	$-0.447759\pi$
$653$	8.35021	0.326769	0.163385	+	0.986562i	$0.447759\pi$
$654$	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	$-0.273133\pi$
$659$	33.5724	1.30779	0.653897	+	0.756583i	$0.273133\pi$
$660$	0	0
$661$	−16.9534	−0.659410	−0.329705	−	0.944084i	$-0.606949\pi$
$661$	−16.9534	−0.659410	−0.329705	+	0.944084i	$0.606949\pi$
$662$	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	$-0.287887\pi$
$683$	32.3092	1.23628	0.618138	+	0.786069i	$0.287887\pi$
$684$	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	$-0.685957\pi$
$691$	−28.9962	−1.10307	−0.551533	+	0.834153i	$0.685957\pi$
$692$	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	$-0.333924\pi$
$701$	26.3912	0.996783	0.498392	+	0.866952i	$0.333924\pi$
$702$	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	$-0.289634\pi$
$727$	33.1005	1.22763	0.613814	+	0.789451i	$0.289634\pi$
$728$	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	$-0.807651\pi$
$733$	−44.5589	−1.64582	−0.822911	+	0.568170i	$0.807651\pi$
$734$	−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	$-0.563131\pi$
$743$	−10.7414	−0.394065	−0.197033	+	0.980397i	$0.563131\pi$
$744$	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	$-0.739070\pi$
$769$	−37.8479	−1.36483	−0.682415	+	0.730965i	$0.739070\pi$
$770$	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	$-0.398313\pi$
$787$	17.6206	0.628107	0.314053	+	0.949405i	$0.398313\pi$
$788$	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	$-0.189025\pi$
$809$	47.1469	1.65760	0.828799	+	0.559546i	$0.189025\pi$
$810$	0	0
$811$	21.0577	0.739435	0.369717	−	0.929144i	$-0.379454\pi$
$811$	21.0577	0.739435	0.369717	+	0.929144i	$0.379454\pi$
$812$	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	$-0.562389\pi$
$821$	−11.1604	−0.389499	−0.194750	+	0.980853i	$0.562389\pi$
$822$	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	$-0.597045\pi$
$827$	−17.2646	−0.600348	−0.300174	+	0.953884i	$0.597045\pi$
$828$	0	0
$829$	−48.4526	−1.68283	−0.841415	−	0.540390i	$-0.818277\pi$
$829$	−48.4526	−1.68283	−0.841415	+	0.540390i	$0.818277\pi$
$830$	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	$-0.456279\pi$
$853$	7.99801	0.273847	0.136923	+	0.990582i	$0.456279\pi$
$854$	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	$-0.513273\pi$
$859$	−2.44359	−0.0833742	−0.0416871	+	0.999131i	$0.513273\pi$
$860$	−8.35969	−0.285063
$861$	0	0
$862$	0.240500	0.00819146
$863$	−25.7187	−0.875476	−0.437738	−	0.899103i	$-0.644220\pi$
$863$	−25.7187	−0.875476	−0.437738	+	0.899103i	$0.644220\pi$
$864$	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	$-0.403296\pi$
$911$	18.0586	0.598307	0.299153	+	0.954205i	$0.403296\pi$
$912$	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	$-0.816182\pi$
$937$	−51.2933	−1.67568	−0.837840	+	0.545915i	$0.816182\pi$
$938$	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	$-0.476741\pi$
$947$	4.49330	0.146013	0.0730063	+	0.997331i	$0.476741\pi$
$948$	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	$-0.505910\pi$
$953$	−1.14635	−0.0371340	−0.0185670	+	0.999828i	$0.505910\pi$
$954$	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	$-0.666301\pi$
$977$	−31.1948	−0.998011	−0.499006	+	0.866599i	$0.666301\pi$
$978$	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	$-0.213408\pi$
$997$	49.4816	1.56710	0.783548	+	0.621331i	$0.213408\pi$
$998$	−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.bd.1.3		6
3.2	odd	2		3969.2.a.be.1.4		6
7.6	odd	2	inner	3969.2.a.bd.1.4		6
9.2	odd	6		441.2.f.g.148.4	yes	12
9.4	even	3		1323.2.f.g.883.4		12
9.5	odd	6		441.2.f.g.295.4	yes	12
9.7	even	3		1323.2.f.g.442.4		12
21.20	even	2		3969.2.a.be.1.3		6
63.2	odd	6		441.2.g.g.67.4		12
63.4	even	3		1323.2.g.g.667.3		12
63.5	even	6		441.2.h.g.214.4		12
63.11	odd	6		441.2.h.g.373.3		12
63.13	odd	6		1323.2.f.g.883.3		12
63.16	even	3		1323.2.g.g.361.3		12
63.20	even	6		441.2.f.g.148.3	✓	12
63.23	odd	6		441.2.h.g.214.3		12
63.25	even	3		1323.2.h.g.226.4		12
63.31	odd	6		1323.2.g.g.667.4		12
63.32	odd	6		441.2.g.g.79.4		12
63.34	odd	6		1323.2.f.g.442.3		12
63.38	even	6		441.2.h.g.373.4		12
63.40	odd	6		1323.2.h.g.802.3		12
63.41	even	6		441.2.f.g.295.3	yes	12
63.47	even	6		441.2.g.g.67.3		12
63.52	odd	6		1323.2.h.g.226.3		12
63.58	even	3		1323.2.h.g.802.4		12
63.59	even	6		441.2.g.g.79.3		12
63.61	odd	6		1323.2.g.g.361.4		12

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

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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