LMFDB - Embedded newform 3969.2.a.t.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:	$4$
Coefficient field:	4.4.14013.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 6x + 3 $ x^4 - x^3 - 6x^2 + 6x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 567)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.372845$ 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)

$f(q)$	$=$	$q+0.372845 q^{2} -1.86099 q^{4} +1.42143 q^{5} -1.43955 q^{8} +O(q^{10})$	$q+0.372845 q^{2} -1.86099 q^{4} +1.42143 q^{5} -1.43955 q^{8} +0.529976 q^{10} -3.76381 q^{11} -1.86099 q^{13} +3.18524 q^{16} -7.53437 q^{17} +1.67574 q^{19} -2.64527 q^{20} -1.40332 q^{22} +0.511859 q^{23} -2.97952 q^{25} -0.693860 q^{26} +8.72197 q^{29} +2.93329 q^{31} +4.06671 q^{32} -2.80916 q^{34} +4.33101 q^{37} +0.624794 q^{38} -2.04623 q^{40} +6.84051 q^{41} -4.53673 q^{43} +7.00439 q^{44} +0.190844 q^{46} -7.43955 q^{47} -1.11090 q^{50} +3.46327 q^{52} +0.832874 q^{53} -5.35001 q^{55} +3.25195 q^{58} +13.0225 q^{59} +10.3038 q^{61} +1.09367 q^{62} -4.85423 q^{64} -2.64527 q^{65} +6.51625 q^{67} +14.0214 q^{68} +4.11854 q^{71} -5.69474 q^{73} +1.61480 q^{74} -3.11854 q^{76} +0.264305 q^{79} +4.52761 q^{80} +2.55045 q^{82} +7.90046 q^{83} -10.7096 q^{85} -1.69150 q^{86} +5.41819 q^{88} -0.796641 q^{89} -0.952563 q^{92} -2.77380 q^{94} +2.38196 q^{95} +12.8800 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8}+O(q^{10}) $ 4 * q - q^2 + 5 * q^4 + 2 * q^5 + 3 * q^8	$ 4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8} + 7 q^{10} - 5 q^{11} + 5 q^{13} - q^{16} + 6 q^{17} + 8 q^{19} - 8 q^{20} - 7 q^{22} + 12 q^{23} + 8 q^{25} + q^{26} + 10 q^{29} + 18 q^{31} + 10 q^{32} - 20 q^{38} + 18 q^{40} - 5 q^{41} - 7 q^{43} - 13 q^{44} + 12 q^{46} - 21 q^{47} - 38 q^{50} + 25 q^{52} + 12 q^{53} + 26 q^{55} - 7 q^{58} + 6 q^{59} + 20 q^{61} + 18 q^{62} - 23 q^{64} - 8 q^{65} - 5 q^{67} + 51 q^{68} + 9 q^{71} + 6 q^{73} - 5 q^{76} - 10 q^{79} - 2 q^{80} + 35 q^{82} + 9 q^{83} - 9 q^{85} + 22 q^{86} + 18 q^{88} - 22 q^{89} + 36 q^{92} + 15 q^{94} + 16 q^{95} + 9 q^{97}+O(q^{100}) $ 4 * q - q^2 + 5 * q^4 + 2 * q^5 + 3 * q^8 + 7 * q^10 - 5 * q^11 + 5 * q^13 - q^16 + 6 * q^17 + 8 * q^19 - 8 * q^20 - 7 * q^22 + 12 * q^23 + 8 * q^25 + q^26 + 10 * q^29 + 18 * q^31 + 10 * q^32 - 20 * q^38 + 18 * q^40 - 5 * q^41 - 7 * q^43 - 13 * q^44 + 12 * q^46 - 21 * q^47 - 38 * q^50 + 25 * q^52 + 12 * q^53 + 26 * q^55 - 7 * q^58 + 6 * q^59 + 20 * q^61 + 18 * q^62 - 23 * q^64 - 8 * q^65 - 5 * q^67 + 51 * q^68 + 9 * q^71 + 6 * q^73 - 5 * q^76 - 10 * q^79 - 2 * q^80 + 35 * q^82 + 9 * q^83 - 9 * q^85 + 22 * q^86 + 18 * q^88 - 22 * q^89 + 36 * q^92 + 15 * q^94 + 16 * q^95 + 9 * 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 $

$2$	0.372845	0.263642	0.131821	−	0.991274i	$-0.457918\pi$
$2$	0.372845	0.263642	0.131821	+	0.991274i	$0.457918\pi$
$3$	0	0
$3$	0	0
$4$	−1.86099	−0.930493
$5$	1.42143	0.635685	0.317843	−	0.948144i	$-0.397042\pi$
$5$	1.42143	0.635685	0.317843	+	0.948144i	$0.397042\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.43955	−0.508958
$9$	0	0
$10$	0.529976	0.167593
$11$	−3.76381	−1.13483	−0.567415	−	0.823432i	$-0.692057\pi$
$11$	−3.76381	−1.13483	−0.567415	+	0.823432i	$0.692057\pi$
$12$	0	0
$13$	−1.86099	−0.516145	−0.258072	−	0.966126i	$-0.583087\pi$
$13$	−1.86099	−0.516145	−0.258072	+	0.966126i	$0.583087\pi$
$14$	0	0
$15$	0	0
$16$	3.18524	0.796311
$17$	−7.53437	−1.82735	−0.913676	−	0.406442i	$-0.866769\pi$
$17$	−7.53437	−1.82735	−0.913676	+	0.406442i	$0.866769\pi$
$18$	0	0
$19$	1.67574	0.384442	0.192221	−	0.981352i	$-0.438431\pi$
$19$	1.67574	0.384442	0.192221	+	0.981352i	$0.438431\pi$
$20$	−2.64527	−0.591501
$21$	0	0
$22$	−1.40332	−0.299189
$23$	0.511859	0.106730	0.0533650	−	0.998575i	$-0.483005\pi$
$23$	0.511859	0.106730	0.0533650	+	0.998575i	$0.483005\pi$
$24$	0	0
$25$	−2.97952	−0.595905
$26$	−0.693860	−0.136077
$27$	0	0
$28$	0	0
$29$	8.72197	1.61963	0.809815	−	0.586686i	$-0.199568\pi$
$29$	8.72197	1.61963	0.809815	+	0.586686i	$0.199568\pi$
$30$	0	0
$31$	2.93329	0.526835	0.263418	−	0.964682i	$-0.415150\pi$
$31$	2.93329	0.526835	0.263418	+	0.964682i	$0.415150\pi$
$32$	4.06671	0.718899
$33$	0	0
$34$	−2.80916	−0.481766
$35$	0	0
$36$	0	0
$37$	4.33101	0.712014	0.356007	−	0.934483i	$-0.384138\pi$
$37$	4.33101	0.712014	0.356007	+	0.934483i	$0.384138\pi$
$38$	0.624794	0.101355
$39$	0	0
$40$	−2.04623	−0.323537
$41$	6.84051	1.06831	0.534154	−	0.845387i	$-0.320630\pi$
$41$	6.84051	1.06831	0.534154	+	0.845387i	$0.320630\pi$
$42$	0	0
$43$	−4.53673	−0.691845	−0.345922	−	0.938263i	$-0.612434\pi$
$43$	−4.53673	−0.691845	−0.345922	+	0.938263i	$0.612434\pi$
$44$	7.00439	1.05595
$45$	0	0
$46$	0.190844	0.0281385
$47$	−7.43955	−1.08517	−0.542585	−	0.840001i	$-0.682554\pi$
$47$	−7.43955	−1.08517	−0.542585	+	0.840001i	$0.682554\pi$
$48$	0	0
$49$	0	0
$50$	−1.11090	−0.157105
$51$	0	0
$52$	3.46327	0.480269
$53$	0.832874	0.114404	0.0572020	−	0.998363i	$-0.481782\pi$
$53$	0.832874	0.114404	0.0572020	+	0.998363i	$0.481782\pi$
$54$	0	0
$55$	−5.35001	−0.721395
$56$	0	0
$57$	0	0
$58$	3.25195	0.427002
$59$	13.0225	1.69539	0.847693	−	0.530487i	$-0.177991\pi$
$59$	13.0225	1.69539	0.847693	+	0.530487i	$0.177991\pi$
$60$	0	0
$61$	10.3038	1.31926	0.659632	−	0.751589i	$-0.270712\pi$
$61$	10.3038	1.31926	0.659632	+	0.751589i	$0.270712\pi$
$62$	1.09367	0.138896
$63$	0	0
$64$	−4.85423	−0.606779
$65$	−2.64527	−0.328105
$66$	0	0
$67$	6.51625	0.796087	0.398043	−	0.917367i	$-0.369689\pi$
$67$	6.51625	0.796087	0.398043	+	0.917367i	$0.369689\pi$
$68$	14.0214	1.70034
$69$	0	0
$70$	0	0
$71$	4.11854	0.488780	0.244390	−	0.969677i	$-0.421412\pi$
$71$	4.11854	0.488780	0.244390	+	0.969677i	$0.421412\pi$
$72$	0	0
$73$	−5.69474	−0.666519	−0.333259	−	0.942835i	$-0.608149\pi$
$73$	−5.69474	−0.666519	−0.333259	+	0.942835i	$0.608149\pi$
$74$	1.61480	0.187716
$75$	0	0
$76$	−3.11854	−0.357721
$77$	0	0
$78$	0	0
$79$	0.264305	0.0297366	0.0148683	−	0.999889i	$-0.495267\pi$
$79$	0.264305	0.0297366	0.0148683	+	0.999889i	$0.495267\pi$
$80$	4.52761	0.506203
$81$	0	0
$82$	2.55045	0.281650
$83$	7.90046	0.867188	0.433594	−	0.901108i	$-0.357245\pi$
$83$	7.90046	0.867188	0.433594	+	0.901108i	$0.357245\pi$
$84$	0	0
$85$	−10.7096	−1.16162
$86$	−1.69150	−0.182399
$87$	0	0
$88$	5.41819	0.577581
$89$	−0.796641	−0.0844438	−0.0422219	−	0.999108i	$-0.513444\pi$
$89$	−0.796641	−0.0844438	−0.0422219	+	0.999108i	$0.513444\pi$
$90$	0	0
$91$	0	0
$92$	−0.952563	−0.0993116
$93$	0	0
$94$	−2.77380	−0.286096
$95$	2.38196	0.244384
$96$	0	0
$97$	12.8800	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$97$	12.8800	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$98$	0	0
$99$	0	0
$100$	5.54485	0.554485
$101$	−12.8586	−1.27948	−0.639741	−	0.768591i	$-0.720958\pi$
$101$	−12.8586	−1.27948	−0.639741	+	0.768591i	$0.720958\pi$
$102$	0	0
$103$	19.7558	1.94660	0.973301	−	0.229534i	$-0.0737203\pi$
$103$	19.7558	1.94660	0.973301	+	0.229534i	$0.0737203\pi$
$104$	2.67899	0.262696
$105$	0	0
$106$	0.310533	0.0301617
$107$	−7.23059	−0.699008	−0.349504	−	0.936935i	$-0.613650\pi$
$107$	−7.23059	−0.699008	−0.349504	+	0.936935i	$0.613650\pi$
$108$	0	0
$109$	9.76820	0.935624	0.467812	−	0.883828i	$-0.345042\pi$
$109$	9.76820	0.935624	0.467812	+	0.883828i	$0.345042\pi$
$110$	−1.99473	−0.190190
$111$	0	0
$112$	0	0
$113$	−16.9072	−1.59050	−0.795248	−	0.606284i	$-0.792660\pi$
$113$	−16.9072	−1.59050	−0.795248	+	0.606284i	$0.792660\pi$
$114$	0	0
$115$	0.727575	0.0678467
$116$	−16.2315	−1.50705
$117$	0	0
$118$	4.85538	0.446974
$119$	0	0
$120$	0	0
$121$	3.16625	0.287841
$122$	3.84172	0.347813
$123$	0	0
$124$	−5.45882	−0.490217
$125$	−11.3424	−1.01449
$126$	0	0
$127$	−16.1715	−1.43499	−0.717495	−	0.696563i	$-0.754711\pi$
$127$	−16.1715	−1.43499	−0.717495	+	0.696563i	$0.754711\pi$
$128$	−9.94329	−0.878871
$129$	0	0
$130$	−0.986277	−0.0865022
$131$	13.6216	1.19012	0.595060	−	0.803681i	$-0.297128\pi$
$131$	13.6216	1.19012	0.595060	+	0.803681i	$0.297128\pi$
$132$	0	0
$133$	0	0
$134$	2.42956	0.209882
$135$	0	0
$136$	10.8461	0.930046
$137$	−6.51186	−0.556346	−0.278173	−	0.960531i	$-0.589729\pi$
$137$	−6.51186	−0.556346	−0.278173	+	0.960531i	$0.589729\pi$
$138$	0	0
$139$	17.9072	1.51887	0.759435	−	0.650583i	$-0.225475\pi$
$139$	17.9072	1.51887	0.759435	+	0.650583i	$0.225475\pi$
$140$	0	0
$141$	0	0
$142$	1.53558	0.128863
$143$	7.00439	0.585737
$144$	0	0
$145$	12.3977	1.02957
$146$	−2.12326	−0.175722
$147$	0	0
$148$	−8.05995	−0.662524
$149$	19.9692	1.63594	0.817970	−	0.575261i	$-0.195099\pi$
$149$	19.9692	1.63594	0.817970	+	0.575261i	$0.195099\pi$
$150$	0	0
$151$	6.31865	0.514205	0.257102	−	0.966384i	$-0.417232\pi$
$151$	6.31865	0.514205	0.257102	+	0.966384i	$0.417232\pi$
$152$	−2.41232	−0.195665
$153$	0	0
$154$	0	0
$155$	4.16949	0.334901
$156$	0	0
$157$	6.49578	0.518419	0.259210	−	0.965821i	$-0.416538\pi$
$157$	6.49578	0.518419	0.259210	+	0.965821i	$0.416538\pi$
$158$	0.0985449	0.00783981
$159$	0	0
$160$	5.78056	0.456993
$161$	0	0
$162$	0	0
$163$	−19.2326	−1.50642	−0.753208	−	0.657783i	$-0.771494\pi$
$163$	−19.2326	−1.50642	−0.753208	+	0.657783i	$0.771494\pi$
$164$	−12.7301	−0.994053
$165$	0	0
$166$	2.94565	0.228627
$167$	−10.3772	−0.803015	−0.401507	−	0.915856i	$-0.631514\pi$
$167$	−10.3772	−0.803015	−0.401507	+	0.915856i	$0.631514\pi$
$168$	0	0
$169$	−9.53673	−0.733595
$170$	−3.99303	−0.306252
$171$	0	0
$172$	8.44279	0.643757
$173$	−13.2201	−1.00511	−0.502553	−	0.864546i	$-0.667606\pi$
$173$	−13.2201	−1.00511	−0.502553	+	0.864546i	$0.667606\pi$
$174$	0	0
$175$	0	0
$176$	−11.9886	−0.903678
$177$	0	0
$178$	−0.297024	−0.0222629
$179$	11.4949	0.859169	0.429584	−	0.903027i	$-0.358660\pi$
$179$	11.4949	0.859169	0.429584	+	0.903027i	$0.358660\pi$
$180$	0	0
$181$	13.7040	1.01861	0.509306	−	0.860586i	$-0.329902\pi$
$181$	13.7040	1.01861	0.509306	+	0.860586i	$0.329902\pi$
$182$	0	0
$183$	0	0
$184$	−0.736848	−0.0543211
$185$	6.15625	0.452616
$186$	0	0
$187$	28.3579	2.07374
$188$	13.8449	1.00974
$189$	0	0
$190$	0.888103	0.0644298
$191$	7.74333	0.560288	0.280144	−	0.959958i	$-0.409618\pi$
$191$	7.74333	0.560288	0.280144	+	0.959958i	$0.409618\pi$
$192$	0	0
$193$	15.8200	1.13875	0.569375	−	0.822078i	$-0.307185\pi$
$193$	15.8200	1.13875	0.569375	+	0.822078i	$0.307185\pi$
$194$	4.80224	0.344781
$195$	0	0
$196$	0	0
$197$	3.64966	0.260028	0.130014	−	0.991512i	$-0.458498\pi$
$197$	3.64966	0.260028	0.130014	+	0.991512i	$0.458498\pi$
$198$	0	0
$199$	20.7164	1.46854	0.734272	−	0.678855i	$-0.237524\pi$
$199$	20.7164	1.46854	0.734272	+	0.678855i	$0.237524\pi$
$200$	4.28918	0.303291
$201$	0	0
$202$	−4.79428	−0.337324
$203$	0	0
$204$	0	0
$205$	9.72334	0.679107
$206$	7.36588	0.513205
$207$	0	0
$208$	−5.92769	−0.411011
$209$	−6.30718	−0.436277
$210$	0	0
$211$	19.8539	1.36680	0.683400	−	0.730045i	$-0.260501\pi$
$211$	19.8539	1.36680	0.683400	+	0.730045i	$0.260501\pi$
$212$	−1.54997	−0.106452
$213$	0	0
$214$	−2.69589	−0.184287
$215$	−6.44867	−0.439795
$216$	0	0
$217$	0	0
$218$	3.64203	0.246669
$219$	0	0
$220$	9.95629	0.671253
$221$	14.0214	0.943179
$222$	0	0
$223$	−16.0720	−1.07626	−0.538130	−	0.842862i	$-0.680869\pi$
$223$	−16.0720	−1.07626	−0.538130	+	0.842862i	$0.680869\pi$
$224$	0	0
$225$	0	0
$226$	−6.30378	−0.419321
$227$	0.857745	0.0569305	0.0284653	−	0.999595i	$-0.490938\pi$
$227$	0.857745	0.0569305	0.0284653	+	0.999595i	$0.490938\pi$
$228$	0	0
$229$	−5.75584	−0.380357	−0.190178	−	0.981750i	$-0.560907\pi$
$229$	−5.75584	−0.380357	−0.190178	+	0.981750i	$0.560907\pi$
$230$	0.271273	0.0178872
$231$	0	0
$232$	−12.5557	−0.824324
$233$	−7.38860	−0.484043	−0.242022	−	0.970271i	$-0.577811\pi$
$233$	−7.38860	−0.484043	−0.242022	+	0.970271i	$0.577811\pi$
$234$	0	0
$235$	−10.5748	−0.689826
$236$	−24.2347	−1.57755
$237$	0	0
$238$	0	0
$239$	−13.8010	−0.892715	−0.446357	−	0.894855i	$-0.647279\pi$
$239$	−13.8010	−0.892715	−0.446357	+	0.894855i	$0.647279\pi$
$240$	0	0
$241$	19.7558	1.27259	0.636293	−	0.771448i	$-0.280467\pi$
$241$	19.7558	1.27259	0.636293	+	0.771448i	$0.280467\pi$
$242$	1.18052	0.0758867
$243$	0	0
$244$	−19.1752	−1.22757
$245$	0	0
$246$	0	0
$247$	−3.11854	−0.198428
$248$	−4.22263	−0.268137
$249$	0	0
$250$	−4.22895	−0.267462
$251$	9.45207	0.596609	0.298305	−	0.954471i	$-0.403579\pi$
$251$	9.45207	0.596609	0.298305	+	0.954471i	$0.403579\pi$
$252$	0	0
$253$	−1.92654	−0.121121
$254$	−6.02948	−0.378323
$255$	0	0
$256$	6.00115	0.375072
$257$	7.47327	0.466169	0.233085	−	0.972456i	$-0.425118\pi$
$257$	7.47327	0.466169	0.233085	+	0.972456i	$0.425118\pi$
$258$	0	0
$259$	0	0
$260$	4.92281	0.305300
$261$	0	0
$262$	5.07873	0.313765
$263$	−25.8112	−1.59159	−0.795793	−	0.605569i	$-0.792946\pi$
$263$	−25.8112	−1.59159	−0.795793	+	0.605569i	$0.792946\pi$
$264$	0	0
$265$	1.18388	0.0727249
$266$	0	0
$267$	0	0
$268$	−12.1267	−0.740753
$269$	−6.48699	−0.395519	−0.197759	−	0.980251i	$-0.563366\pi$
$269$	−6.48699	−0.395519	−0.197759	+	0.980251i	$0.563366\pi$
$270$	0	0
$271$	18.2396	1.10798	0.553988	−	0.832525i	$-0.313105\pi$
$271$	18.2396	1.10798	0.553988	+	0.832525i	$0.313105\pi$
$272$	−23.9988	−1.45514
$273$	0	0
$274$	−2.42792	−0.146676
$275$	11.2143	0.676251
$276$	0	0
$277$	−3.04656	−0.183050	−0.0915249	−	0.995803i	$-0.529174\pi$
$277$	−3.04656	−0.183050	−0.0915249	+	0.995803i	$0.529174\pi$
$278$	6.67662	0.400437
$279$	0	0
$280$	0	0
$281$	28.5445	1.70282	0.851412	−	0.524498i	$-0.175747\pi$
$281$	28.5445	1.70282	0.851412	+	0.524498i	$0.175747\pi$
$282$	0	0
$283$	9.54085	0.567145	0.283572	−	0.958951i	$-0.408480\pi$
$283$	9.54085	0.567145	0.283572	+	0.958951i	$0.408480\pi$
$284$	−7.66454	−0.454807
$285$	0	0
$286$	2.61156	0.154425
$287$	0	0
$288$	0	0
$289$	39.7667	2.33922
$290$	4.62243	0.271439
$291$	0	0
$292$	10.5978	0.620191
$293$	12.6307	0.737891	0.368946	−	0.929451i	$-0.379719\pi$
$293$	12.6307	0.737891	0.368946	+	0.929451i	$0.379719\pi$
$294$	0	0
$295$	18.5107	1.07773
$296$	−6.23471	−0.362385
$297$	0	0
$298$	7.44543	0.431302
$299$	−0.952563	−0.0550881
$300$	0	0
$301$	0	0
$302$	2.35588	0.135566
$303$	0	0
$304$	5.33765	0.306135
$305$	14.6462	0.838636
$306$	0	0
$307$	10.9319	0.623918	0.311959	−	0.950096i	$-0.399015\pi$
$307$	10.9319	0.623918	0.311959	+	0.950096i	$0.399015\pi$
$308$	0	0
$309$	0	0
$310$	1.55457	0.0882939
$311$	−4.76057	−0.269947	−0.134973	−	0.990849i	$-0.543095\pi$
$311$	−4.76057	−0.269947	−0.134973	+	0.990849i	$0.543095\pi$
$312$	0	0
$313$	−7.21099	−0.407589	−0.203795	−	0.979014i	$-0.565328\pi$
$313$	−7.21099	−0.407589	−0.203795	+	0.979014i	$0.565328\pi$
$314$	2.42192	0.136677
$315$	0	0
$316$	−0.491868	−0.0276697
$317$	−7.28918	−0.409401	−0.204700	−	0.978825i	$-0.565622\pi$
$317$	−7.28918	−0.409401	−0.204700	+	0.978825i	$0.565622\pi$
$318$	0	0
$319$	−32.8278	−1.83801
$320$	−6.89997	−0.385720
$321$	0	0
$322$	0	0
$323$	−12.6257	−0.702511
$324$	0	0
$325$	5.54485	0.307573
$326$	−7.17080	−0.397154
$327$	0	0
$328$	−9.84726	−0.543724
$329$	0	0
$330$	0	0
$331$	−1.21911	−0.0670086	−0.0335043	−	0.999439i	$-0.510667\pi$
$331$	−1.21911	−0.0670086	−0.0335043	+	0.999439i	$0.510667\pi$
$332$	−14.7026	−0.806913
$333$	0	0
$334$	−3.86911	−0.211708
$335$	9.26243	0.506061
$336$	0	0
$337$	−14.9140	−0.812416	−0.406208	−	0.913781i	$-0.633149\pi$
$337$	−14.9140	−0.812416	−0.406208	+	0.913781i	$0.633149\pi$
$338$	−3.55573	−0.193406
$339$	0	0
$340$	19.9304	1.08088
$341$	−11.0404	−0.597869
$342$	0	0
$343$	0	0
$344$	6.53086	0.352120
$345$	0	0
$346$	−4.92906	−0.264988
$347$	5.15828	0.276911	0.138456	−	0.990369i	$-0.455786\pi$
$347$	5.15828	0.276911	0.138456	+	0.990369i	$0.455786\pi$
$348$	0	0
$349$	24.4507	1.30882	0.654408	−	0.756142i	$-0.272918\pi$
$349$	24.4507	1.30882	0.654408	+	0.756142i	$0.272918\pi$
$350$	0	0
$351$	0	0
$352$	−15.3063	−0.815828
$353$	14.9095	0.793551	0.396775	−	0.917916i	$-0.370129\pi$
$353$	14.9095	0.793551	0.396775	+	0.917916i	$0.370129\pi$
$354$	0	0
$355$	5.85423	0.310710
$356$	1.48254	0.0785744
$357$	0	0
$358$	4.28582	0.226513
$359$	2.14780	0.113357	0.0566783	−	0.998392i	$-0.481949\pi$
$359$	2.14780	0.113357	0.0566783	+	0.998392i	$0.481949\pi$
$360$	0	0
$361$	−16.1919	−0.852204
$362$	5.10948	0.268548
$363$	0	0
$364$	0	0
$365$	−8.09470	−0.423696
$366$	0	0
$367$	15.8610	0.827937	0.413968	−	0.910291i	$-0.364142\pi$
$367$	15.8610	0.827937	0.413968	+	0.910291i	$0.364142\pi$
$368$	1.63040	0.0849903
$369$	0	0
$370$	2.29533	0.119329
$371$	0	0
$372$	0	0
$373$	11.7368	0.607711	0.303855	−	0.952718i	$-0.401726\pi$
$373$	11.7368	0.607711	0.303855	+	0.952718i	$0.401726\pi$
$374$	10.5731	0.546723
$375$	0	0
$376$	10.7096	0.552306
$377$	−16.2315	−0.835963
$378$	0	0
$379$	18.9400	0.972885	0.486442	−	0.873713i	$-0.338294\pi$
$379$	18.9400	0.972885	0.486442	+	0.873713i	$0.338294\pi$
$380$	−4.43280	−0.227398
$381$	0	0
$382$	2.88707	0.147715
$383$	29.6812	1.51664	0.758319	−	0.651884i	$-0.226021\pi$
$383$	29.6812	1.51664	0.758319	+	0.651884i	$0.226021\pi$
$384$	0	0
$385$	0	0
$386$	5.89843	0.300222
$387$	0	0
$388$	−23.9695	−1.21687
$389$	1.27819	0.0648066	0.0324033	−	0.999475i	$-0.489684\pi$
$389$	1.27819	0.0648066	0.0324033	+	0.999475i	$0.489684\pi$
$390$	0	0
$391$	−3.85654	−0.195033
$392$	0	0
$393$	0	0
$394$	1.36076	0.0685541
$395$	0.375692	0.0189031
$396$	0	0
$397$	6.43467	0.322947	0.161473	−	0.986877i	$-0.448375\pi$
$397$	6.43467	0.322947	0.161473	+	0.986877i	$0.448375\pi$
$398$	7.72401	0.387169
$399$	0	0
$400$	−9.49050	−0.474525
$401$	−27.6910	−1.38282	−0.691412	−	0.722461i	$-0.743011\pi$
$401$	−27.6910	−1.38282	−0.691412	+	0.722461i	$0.743011\pi$
$402$	0	0
$403$	−5.45882	−0.271923
$404$	23.9297	1.19055
$405$	0	0
$406$	0	0
$407$	−16.3011	−0.808015
$408$	0	0
$409$	12.0272	0.594708	0.297354	−	0.954767i	$-0.403896\pi$
$409$	12.0272	0.594708	0.297354	+	0.954767i	$0.403896\pi$
$410$	3.62530	0.179041
$411$	0	0
$412$	−36.7654	−1.81130
$413$	0	0
$414$	0	0
$415$	11.2300	0.551259
$416$	−7.56808	−0.371056
$417$	0	0
$418$	−2.35160	−0.115021
$419$	−18.0190	−0.880286	−0.440143	−	0.897928i	$-0.645072\pi$
$419$	−18.0190	−0.880286	−0.440143	+	0.897928i	$0.645072\pi$
$420$	0	0
$421$	−17.2572	−0.841066	−0.420533	−	0.907277i	$-0.638157\pi$
$421$	−17.2572	−0.841066	−0.420533	+	0.907277i	$0.638157\pi$
$422$	7.40244	0.360345
$423$	0	0
$424$	−1.19896	−0.0582269
$425$	22.4488	1.08893
$426$	0	0
$427$	0	0
$428$	13.4560	0.650422
$429$	0	0
$430$	−2.40436	−0.115948
$431$	−8.56736	−0.412675	−0.206338	−	0.978481i	$-0.566155\pi$
$431$	−8.56736	−0.412675	−0.206338	+	0.978481i	$0.566155\pi$
$432$	0	0
$433$	−1.58971	−0.0763967	−0.0381984	−	0.999270i	$-0.512162\pi$
$433$	−1.58971	−0.0763967	−0.0381984	+	0.999270i	$0.512162\pi$
$434$	0	0
$435$	0	0
$436$	−18.1785	−0.870592
$437$	0.857745	0.0410315
$438$	0	0
$439$	−22.4436	−1.07118	−0.535588	−	0.844480i	$-0.679910\pi$
$439$	−22.4436	−1.07118	−0.535588	+	0.844480i	$0.679910\pi$
$440$	7.70161	0.367160
$441$	0	0
$442$	5.22780	0.248661
$443$	24.3210	1.15553	0.577763	−	0.816204i	$-0.303926\pi$
$443$	24.3210	1.15553	0.577763	+	0.816204i	$0.303926\pi$
$444$	0	0
$445$	−1.13237	−0.0536797
$446$	−5.99237	−0.283747
$447$	0	0
$448$	0	0
$449$	−12.9016	−0.608865	−0.304432	−	0.952534i	$-0.598467\pi$
$449$	−12.9016	−0.608865	−0.304432	+	0.952534i	$0.598467\pi$
$450$	0	0
$451$	−25.7464	−1.21235
$452$	31.4641	1.47995
$453$	0	0
$454$	0.319806	0.0150093
$455$	0	0
$456$	0	0
$457$	1.18776	0.0555611	0.0277806	−	0.999614i	$-0.491156\pi$
$457$	1.18776	0.0555611	0.0277806	+	0.999614i	$0.491156\pi$
$458$	−2.14604	−0.100278
$459$	0	0
$460$	−1.35401	−0.0631309
$461$	16.1674	0.752991	0.376495	−	0.926418i	$-0.377129\pi$
$461$	16.1674	0.752991	0.376495	+	0.926418i	$0.377129\pi$
$462$	0	0
$463$	−0.531128	−0.0246836	−0.0123418	−	0.999924i	$-0.503929\pi$
$463$	−0.531128	−0.0246836	−0.0123418	+	0.999924i	$0.503929\pi$
$464$	27.7816	1.28973
$465$	0	0
$466$	−2.75481	−0.127614
$467$	−21.4787	−0.993916	−0.496958	−	0.867775i	$-0.665550\pi$
$467$	−21.4787	−0.993916	−0.496958	+	0.867775i	$0.665550\pi$
$468$	0	0
$469$	0	0
$470$	−3.94278	−0.181867
$471$	0	0
$472$	−18.7466	−0.862881
$473$	17.0754	0.785127
$474$	0	0
$475$	−4.99292	−0.229091
$476$	0	0
$477$	0	0
$478$	−5.14565	−0.235357
$479$	−28.3529	−1.29548	−0.647739	−	0.761862i	$-0.724285\pi$
$479$	−28.3529	−1.29548	−0.647739	+	0.761862i	$0.724285\pi$
$480$	0	0
$481$	−8.05995	−0.367502
$482$	7.36588	0.335507
$483$	0	0
$484$	−5.89234	−0.267834
$485$	18.3081	0.831326
$486$	0	0
$487$	7.87026	0.356635	0.178318	−	0.983973i	$-0.442935\pi$
$487$	7.87026	0.356635	0.178318	+	0.983973i	$0.442935\pi$
$488$	−14.8328	−0.671450
$489$	0	0
$490$	0	0
$491$	28.3183	1.27799	0.638994	−	0.769212i	$-0.279351\pi$
$491$	28.3183	1.27799	0.638994	+	0.769212i	$0.279351\pi$
$492$	0	0
$493$	−65.7146	−2.95963
$494$	−1.16273	−0.0523138
$495$	0	0
$496$	9.34325	0.419524
$497$	0	0
$498$	0	0
$499$	6.84187	0.306284	0.153142	−	0.988204i	$-0.451061\pi$
$499$	6.84187	0.306284	0.153142	+	0.988204i	$0.451061\pi$
$500$	21.1080	0.943978
$501$	0	0
$502$	3.52416	0.157291
$503$	−30.5760	−1.36332	−0.681658	−	0.731671i	$-0.738741\pi$
$503$	−30.5760	−1.36332	−0.681658	+	0.731671i	$0.738741\pi$
$504$	0	0
$505$	−18.2777	−0.813347
$506$	−0.718302	−0.0319324
$507$	0	0
$508$	30.0950	1.33525
$509$	−34.1123	−1.51200	−0.756001	−	0.654570i	$-0.772850\pi$
$509$	−34.1123	−1.51200	−0.756001	+	0.654570i	$0.772850\pi$
$510$	0	0
$511$	0	0
$512$	22.1241	0.977756
$513$	0	0
$514$	2.78637	0.122902
$515$	28.0816	1.23743
$516$	0	0
$517$	28.0010	1.23148
$518$	0	0
$519$	0	0
$520$	3.80800	0.166992
$521$	−14.7443	−0.645961	−0.322980	−	0.946406i	$-0.604685\pi$
$521$	−14.7443	−0.645961	−0.322980	+	0.946406i	$0.604685\pi$
$522$	0	0
$523$	−4.07906	−0.178365	−0.0891825	−	0.996015i	$-0.528425\pi$
$523$	−4.07906	−0.178365	−0.0891825	+	0.996015i	$0.528425\pi$
$524$	−25.3495	−1.10740
$525$	0	0
$526$	−9.62359	−0.419608
$527$	−22.1005	−0.962714
$528$	0	0
$529$	−22.7380	−0.988609
$530$	0.441403	0.0191733
$531$	0	0
$532$	0	0
$533$	−12.7301	−0.551402
$534$	0	0
$535$	−10.2778	−0.444349
$536$	−9.38048	−0.405175
$537$	0	0
$538$	−2.41864	−0.104275
$539$	0	0
$540$	0	0
$541$	3.80104	0.163419	0.0817096	−	0.996656i	$-0.473962\pi$
$541$	3.80104	0.163419	0.0817096	+	0.996656i	$0.473962\pi$
$542$	6.80055	0.292109
$543$	0	0
$544$	−30.6401	−1.31368
$545$	13.8849	0.594762
$546$	0	0
$547$	−10.4778	−0.447999	−0.224000	−	0.974589i	$-0.571911\pi$
$547$	−10.4778	−0.447999	−0.224000	+	0.974589i	$0.571911\pi$
$548$	12.1185	0.517676
$549$	0	0
$550$	4.18122	0.178288
$551$	14.6158	0.622654
$552$	0	0
$553$	0	0
$554$	−1.13589	−0.0482596
$555$	0	0
$556$	−33.3251	−1.41330
$557$	−34.9949	−1.48278	−0.741392	−	0.671072i	$-0.765834\pi$
$557$	−34.9949	−1.48278	−0.741392	+	0.671072i	$0.765834\pi$
$558$	0	0
$559$	8.44279	0.357092
$560$	0	0
$561$	0	0
$562$	10.6427	0.448935
$563$	−29.0643	−1.22492	−0.612458	−	0.790503i	$-0.709819\pi$
$563$	−29.0643	−1.22492	−0.612458	+	0.790503i	$0.709819\pi$
$564$	0	0
$565$	−24.0325	−1.01106
$566$	3.55726	0.149523
$567$	0	0
$568$	−5.92884	−0.248769
$569$	−22.0327	−0.923657	−0.461829	−	0.886969i	$-0.652806\pi$
$569$	−22.0327	−0.923657	−0.461829	+	0.886969i	$0.652806\pi$
$570$	0	0
$571$	2.02827	0.0848804	0.0424402	−	0.999099i	$-0.486487\pi$
$571$	2.02827	0.0848804	0.0424402	+	0.999099i	$0.486487\pi$
$572$	−13.0351	−0.545024
$573$	0	0
$574$	0	0
$575$	−1.52510	−0.0636009
$576$	0	0
$577$	4.94120	0.205705	0.102852	−	0.994697i	$-0.467203\pi$
$577$	4.94120	0.205705	0.102852	+	0.994697i	$0.467203\pi$
$578$	14.8268	0.616715
$579$	0	0
$580$	−23.0720	−0.958012
$581$	0	0
$582$	0	0
$583$	−3.13478	−0.129829
$584$	8.19787	0.339230
$585$	0	0
$586$	4.70929	0.194539
$587$	32.4983	1.34135	0.670674	−	0.741752i	$-0.266005\pi$
$587$	32.4983	1.34135	0.670674	+	0.741752i	$0.266005\pi$
$588$	0	0
$589$	4.91545	0.202538
$590$	6.90161	0.284135
$591$	0	0
$592$	13.7953	0.566984
$593$	−15.5472	−0.638447	−0.319224	−	0.947679i	$-0.603422\pi$
$593$	−15.5472	−0.638447	−0.319224	+	0.947679i	$0.603422\pi$
$594$	0	0
$595$	0	0
$596$	−37.1624	−1.52223
$597$	0	0
$598$	−0.355159	−0.0145235
$599$	−23.9883	−0.980136	−0.490068	−	0.871684i	$-0.663028\pi$
$599$	−23.9883	−0.980136	−0.490068	+	0.871684i	$0.663028\pi$
$600$	0	0
$601$	0.887066	0.0361842	0.0180921	−	0.999836i	$-0.494241\pi$
$601$	0.887066	0.0361842	0.0180921	+	0.999836i	$0.494241\pi$
$602$	0	0
$603$	0	0
$604$	−11.7589	−0.478464
$605$	4.50061	0.182976
$606$	0	0
$607$	11.7206	0.475725	0.237862	−	0.971299i	$-0.423553\pi$
$607$	11.7206	0.475725	0.237862	+	0.971299i	$0.423553\pi$
$608$	6.81476	0.276375
$609$	0	0
$610$	5.46075	0.221099
$611$	13.8449	0.560105
$612$	0	0
$613$	−9.67690	−0.390846	−0.195423	−	0.980719i	$-0.562608\pi$
$613$	−9.67690	−0.390846	−0.195423	+	0.980719i	$0.562608\pi$
$614$	4.07592	0.164491
$615$	0	0
$616$	0	0
$617$	9.29027	0.374012	0.187006	−	0.982359i	$-0.440122\pi$
$617$	9.29027	0.374012	0.187006	+	0.982359i	$0.440122\pi$
$618$	0	0
$619$	30.2696	1.21664	0.608319	−	0.793693i	$-0.291844\pi$
$619$	30.2696	1.21664	0.608319	+	0.793693i	$0.291844\pi$
$620$	−7.75936	−0.311623
$621$	0	0
$622$	−1.77496	−0.0711692
$623$	0	0
$624$	0	0
$625$	−1.22483	−0.0489932
$626$	−2.68859	−0.107458
$627$	0	0
$628$	−12.0885	−0.482386
$629$	−32.6314	−1.30110
$630$	0	0
$631$	30.5921	1.21785	0.608926	−	0.793227i	$-0.291600\pi$
$631$	30.5921	1.21785	0.608926	+	0.793227i	$0.291600\pi$
$632$	−0.380480	−0.0151347
$633$	0	0
$634$	−2.71774	−0.107935
$635$	−22.9868	−0.912202
$636$	0	0
$637$	0	0
$638$	−12.2397	−0.484575
$639$	0	0
$640$	−14.1337	−0.558685
$641$	−32.8876	−1.29898	−0.649491	−	0.760369i	$-0.725018\pi$
$641$	−32.8876	−1.29898	−0.649491	+	0.760369i	$0.725018\pi$
$642$	0	0
$643$	−5.37197	−0.211850	−0.105925	−	0.994374i	$-0.533780\pi$
$643$	−5.37197	−0.211850	−0.105925	+	0.994374i	$0.533780\pi$
$644$	0	0
$645$	0	0
$646$	−4.70743	−0.185211
$647$	13.8608	0.544925	0.272463	−	0.962166i	$-0.412162\pi$
$647$	13.8608	0.544925	0.272463	+	0.962166i	$0.412162\pi$
$648$	0	0
$649$	−49.0142	−1.92398
$650$	2.06737	0.0810890
$651$	0	0
$652$	35.7916	1.40171
$653$	−33.2740	−1.30211	−0.651056	−	0.759030i	$-0.725674\pi$
$653$	−33.2740	−1.30211	−0.651056	+	0.759030i	$0.725674\pi$
$654$	0	0
$655$	19.3622	0.756542
$656$	21.7887	0.850705
$657$	0	0
$658$	0	0
$659$	−12.7890	−0.498189	−0.249094	−	0.968479i	$-0.580133\pi$
$659$	−12.7890	−0.498189	−0.249094	+	0.968479i	$0.580133\pi$
$660$	0	0
$661$	−41.1004	−1.59862	−0.799309	−	0.600920i	$-0.794801\pi$
$661$	−41.1004	−1.59862	−0.799309	+	0.600920i	$0.794801\pi$
$662$	−0.454541	−0.0176662
$663$	0	0
$664$	−11.3731	−0.441363
$665$	0	0
$666$	0	0
$667$	4.46442	0.172863
$668$	19.3119	0.747200
$669$	0	0
$670$	3.45346	0.133419
$671$	−38.7814	−1.49714
$672$	0	0
$673$	28.6891	1.10589	0.552943	−	0.833219i	$-0.313505\pi$
$673$	28.6891	1.10589	0.552943	+	0.833219i	$0.313505\pi$
$674$	−5.56061	−0.214187
$675$	0	0
$676$	17.7477	0.682605
$677$	49.5122	1.90291	0.951454	−	0.307790i	$-0.0995896\pi$
$677$	49.5122	1.90291	0.951454	+	0.307790i	$0.0995896\pi$
$678$	0	0
$679$	0	0
$680$	15.4170	0.591217
$681$	0	0
$682$	−4.11635	−0.157623
$683$	32.5071	1.24385	0.621926	−	0.783076i	$-0.286351\pi$
$683$	32.5071	1.24385	0.621926	+	0.783076i	$0.286351\pi$
$684$	0	0
$685$	−9.25618	−0.353661
$686$	0	0
$687$	0	0
$688$	−14.4506	−0.550923
$689$	−1.54997	−0.0590490
$690$	0	0
$691$	14.1729	0.539162	0.269581	−	0.962978i	$-0.413115\pi$
$691$	14.1729	0.539162	0.269581	+	0.962978i	$0.413115\pi$
$692$	24.6024	0.935244
$693$	0	0
$694$	1.92324	0.0730053
$695$	25.4539	0.965523
$696$	0	0
$697$	−51.5389	−1.95218
$698$	9.11633	0.345058
$699$	0	0
$700$	0	0
$701$	−28.1485	−1.06316	−0.531578	−	0.847010i	$-0.678401\pi$
$701$	−28.1485	−1.06316	−0.531578	+	0.847010i	$0.678401\pi$
$702$	0	0
$703$	7.25766	0.273728
$704$	18.2704	0.688591
$705$	0	0
$706$	5.55893	0.209213
$707$	0	0
$708$	0	0
$709$	24.4669	0.918875	0.459438	−	0.888210i	$-0.348051\pi$
$709$	24.4669	0.918875	0.459438	+	0.888210i	$0.348051\pi$
$710$	2.18272	0.0819162
$711$	0	0
$712$	1.14681	0.0429784
$713$	1.50143	0.0562291
$714$	0	0
$715$	9.95629	0.372344
$716$	−21.3918	−0.799451
$717$	0	0
$718$	0.800798	0.0298855
$719$	16.5546	0.617384	0.308692	−	0.951162i	$-0.400109\pi$
$719$	16.5546	0.617384	0.308692	+	0.951162i	$0.400109\pi$
$720$	0	0
$721$	0	0
$722$	−6.03707	−0.224676
$723$	0	0
$724$	−25.5030	−0.947811
$725$	−25.9873	−0.965145
$726$	0	0
$727$	34.2381	1.26982	0.634911	−	0.772586i	$-0.281037\pi$
$727$	34.2381	1.26982	0.634911	+	0.772586i	$0.281037\pi$
$728$	0	0
$729$	0	0
$730$	−3.01807	−0.111704
$731$	34.1814	1.26424
$732$	0	0
$733$	12.2889	0.453901	0.226951	−	0.973906i	$-0.427124\pi$
$733$	12.2889	0.453901	0.226951	+	0.973906i	$0.427124\pi$
$734$	5.91370	0.218279
$735$	0	0
$736$	2.08158	0.0767281
$737$	−24.5259	−0.903424
$738$	0	0
$739$	−43.2992	−1.59279	−0.796394	−	0.604779i	$-0.793262\pi$
$739$	−43.2992	−1.59279	−0.796394	+	0.604779i	$0.793262\pi$
$740$	−11.4567	−0.421157
$741$	0	0
$742$	0	0
$743$	23.9478	0.878561	0.439281	−	0.898350i	$-0.355233\pi$
$743$	23.9478	0.878561	0.439281	+	0.898350i	$0.355233\pi$
$744$	0	0
$745$	28.3849	1.03994
$746$	4.37603	0.160218
$747$	0	0
$748$	−52.7737	−1.92960
$749$	0	0
$750$	0	0
$751$	−2.90298	−0.105931	−0.0529656	−	0.998596i	$-0.516867\pi$
$751$	−2.90298	−0.105931	−0.0529656	+	0.998596i	$0.516867\pi$
$752$	−23.6968	−0.864132
$753$	0	0
$754$	−6.05183	−0.220395
$755$	8.98156	0.326872
$756$	0	0
$757$	−8.99407	−0.326895	−0.163448	−	0.986552i	$-0.552261\pi$
$757$	−8.99407	−0.326895	−0.163448	+	0.986552i	$0.552261\pi$
$758$	7.06171	0.256493
$759$	0	0
$760$	−3.42896	−0.124381
$761$	−38.0509	−1.37934	−0.689672	−	0.724122i	$-0.742245\pi$
$761$	−38.0509	−1.37934	−0.689672	+	0.724122i	$0.742245\pi$
$762$	0	0
$763$	0	0
$764$	−14.4102	−0.521344
$765$	0	0
$766$	11.0665	0.399849
$767$	−24.2347	−0.875065
$768$	0	0
$769$	−22.8131	−0.822660	−0.411330	−	0.911486i	$-0.634936\pi$
$769$	−22.8131	−0.822660	−0.411330	+	0.911486i	$0.634936\pi$
$770$	0	0
$771$	0	0
$772$	−29.4409	−1.05960
$773$	−6.12886	−0.220440	−0.110220	−	0.993907i	$-0.535156\pi$
$773$	−6.12886	−0.220440	−0.110220	+	0.993907i	$0.535156\pi$
$774$	0	0
$775$	−8.73982	−0.313943
$776$	−18.5414	−0.665597
$777$	0	0
$778$	0.476566	0.0170857
$779$	11.4629	0.410703
$780$	0	0
$781$	−15.5014	−0.554683
$782$	−1.43789	−0.0514189
$783$	0	0
$784$	0	0
$785$	9.23332	0.329551
$786$	0	0
$787$	46.4627	1.65622	0.828109	−	0.560568i	$-0.189417\pi$
$787$	46.4627	1.65622	0.828109	+	0.560568i	$0.189417\pi$
$788$	−6.79198	−0.241954
$789$	0	0
$790$	0.140075	0.00498365
$791$	0	0
$792$	0	0
$793$	−19.1752	−0.680931
$794$	2.39914	0.0851422
$795$	0	0
$796$	−38.5529	−1.36647
$797$	−7.63879	−0.270580	−0.135290	−	0.990806i	$-0.543197\pi$
$797$	−7.63879	−0.270580	−0.135290	+	0.990806i	$0.543197\pi$
$798$	0	0
$799$	56.0523	1.98299
$800$	−12.1168	−0.428395
$801$	0	0
$802$	−10.3245	−0.364570
$803$	21.4339	0.756386
$804$	0	0
$805$	0	0
$806$	−2.03530	−0.0716903
$807$	0	0
$808$	18.5107	0.651202
$809$	27.6632	0.972587	0.486293	−	0.873796i	$-0.338349\pi$
$809$	27.6632	0.972587	0.486293	+	0.873796i	$0.338349\pi$
$810$	0	0
$811$	34.0746	1.19652	0.598261	−	0.801301i	$-0.295859\pi$
$811$	34.0746	1.19652	0.598261	+	0.801301i	$0.295859\pi$
$812$	0	0
$813$	0	0
$814$	−6.07779	−0.213026
$815$	−27.3379	−0.957606
$816$	0	0
$817$	−7.60240	−0.265974
$818$	4.48430	0.156790
$819$	0	0
$820$	−18.0950	−0.631905
$821$	7.27446	0.253880	0.126940	−	0.991910i	$-0.459484\pi$
$821$	7.27446	0.253880	0.126940	+	0.991910i	$0.459484\pi$
$822$	0	0
$823$	−5.66202	−0.197366	−0.0986828	−	0.995119i	$-0.531463\pi$
$823$	−5.66202	−0.197366	−0.0986828	+	0.995119i	$0.531463\pi$
$824$	−28.4396	−0.990739
$825$	0	0
$826$	0	0
$827$	35.3143	1.22800	0.614000	−	0.789306i	$-0.289560\pi$
$827$	35.3143	1.22800	0.614000	+	0.789306i	$0.289560\pi$
$828$	0	0
$829$	−29.7368	−1.03280	−0.516402	−	0.856346i	$-0.672729\pi$
$829$	−29.7368	−1.03280	−0.516402	+	0.856346i	$0.672729\pi$
$830$	4.18705	0.145335
$831$	0	0
$832$	9.03366	0.313186
$833$	0	0
$834$	0	0
$835$	−14.7506	−0.510465
$836$	11.7376	0.405952
$837$	0	0
$838$	−6.71830	−0.232080
$839$	−5.25058	−0.181270	−0.0906351	−	0.995884i	$-0.528890\pi$
$839$	−5.25058	−0.181270	−0.0906351	+	0.995884i	$0.528890\pi$
$840$	0	0
$841$	47.0728	1.62320
$842$	−6.43428	−0.221740
$843$	0	0
$844$	−36.9478	−1.27180
$845$	−13.5558	−0.466335
$846$	0	0
$847$	0	0
$848$	2.65291	0.0911012
$849$	0	0
$850$	8.36994	0.287087
$851$	2.21687	0.0759933
$852$	0	0
$853$	−22.4915	−0.770096	−0.385048	−	0.922897i	$-0.625815\pi$
$853$	−22.4915	−0.770096	−0.385048	+	0.922897i	$0.625815\pi$
$854$	0	0
$855$	0	0
$856$	10.4088	0.355766
$857$	55.2956	1.88886	0.944432	−	0.328708i	$-0.106613\pi$
$857$	55.2956	1.88886	0.944432	+	0.328708i	$0.106613\pi$
$858$	0	0
$859$	−51.2704	−1.74932	−0.874662	−	0.484734i	$-0.838917\pi$
$859$	−51.2704	−1.74932	−0.874662	+	0.484734i	$0.838917\pi$
$860$	12.0009	0.409227
$861$	0	0
$862$	−3.19430	−0.108798
$863$	−29.1815	−0.993350	−0.496675	−	0.867937i	$-0.665446\pi$
$863$	−29.1815	−0.993350	−0.496675	+	0.867937i	$0.665446\pi$
$864$	0	0
$865$	−18.7915	−0.638931
$866$	−0.592717	−0.0201414
$867$	0	0
$868$	0	0
$869$	−0.994792	−0.0337460
$870$	0	0
$871$	−12.1267	−0.410896
$872$	−14.0618	−0.476194
$873$	0	0
$874$	0.319806	0.0108176
$875$	0	0
$876$	0	0
$877$	17.9605	0.606484	0.303242	−	0.952914i	$-0.401931\pi$
$877$	17.9605	0.606484	0.303242	+	0.952914i	$0.401931\pi$
$878$	−8.36800	−0.282406
$879$	0	0
$880$	−17.0411	−0.574454
$881$	45.7619	1.54176	0.770880	−	0.636981i	$-0.219817\pi$
$881$	45.7619	1.54176	0.770880	+	0.636981i	$0.219817\pi$
$882$	0	0
$883$	40.8060	1.37323	0.686615	−	0.727021i	$-0.259096\pi$
$883$	40.8060	1.37323	0.686615	+	0.727021i	$0.259096\pi$
$884$	−26.0936	−0.877621
$885$	0	0
$886$	9.06798	0.304645
$887$	−38.8683	−1.30507	−0.652535	−	0.757759i	$-0.726294\pi$
$887$	−38.8683	−1.30507	−0.652535	+	0.757759i	$0.726294\pi$
$888$	0	0
$889$	0	0
$890$	−0.422200	−0.0141522
$891$	0	0
$892$	29.9097	1.00145
$893$	−12.4668	−0.417185
$894$	0	0
$895$	16.3392	0.546161
$896$	0	0
$897$	0	0
$898$	−4.81031	−0.160522
$899$	25.5841	0.853278
$900$	0	0
$901$	−6.27518	−0.209057
$902$	−9.59941	−0.319626
$903$	0	0
$904$	24.3388	0.809497
$905$	19.4794	0.647516
$906$	0	0
$907$	−28.0570	−0.931617	−0.465809	−	0.884886i	$-0.654236\pi$
$907$	−28.0570	−0.931617	−0.465809	+	0.884886i	$0.654236\pi$
$908$	−1.59625	−0.0529735
$909$	0	0
$910$	0	0
$911$	12.3839	0.410296	0.205148	−	0.978731i	$-0.434232\pi$
$911$	12.3839	0.410296	0.205148	+	0.978731i	$0.434232\pi$
$912$	0	0
$913$	−29.7358	−0.984112
$914$	0.442851	0.0146482
$915$	0	0
$916$	10.7115	0.353919
$917$	0	0
$918$	0	0
$919$	−54.6059	−1.80128	−0.900641	−	0.434565i	$-0.856902\pi$
$919$	−54.6059	−1.80128	−0.900641	+	0.434565i	$0.856902\pi$
$920$	−1.04738	−0.0345311
$921$	0	0
$922$	6.02794	0.198520
$923$	−7.66454	−0.252281
$924$	0	0
$925$	−12.9043	−0.424292
$926$	−0.198029	−0.00650763
$927$	0	0
$928$	35.4697	1.16435
$929$	12.8071	0.420188	0.210094	−	0.977681i	$-0.432623\pi$
$929$	12.8071	0.420188	0.210094	+	0.977681i	$0.432623\pi$
$930$	0	0
$931$	0	0
$932$	13.7501	0.450399
$933$	0	0
$934$	−8.00824	−0.262037
$935$	40.3089	1.31824
$936$	0	0
$937$	−37.9601	−1.24010	−0.620051	−	0.784562i	$-0.712888\pi$
$937$	−37.9601	−1.24010	−0.620051	+	0.784562i	$0.712888\pi$
$938$	0	0
$939$	0	0
$940$	19.6796	0.641879
$941$	−18.2697	−0.595576	−0.297788	−	0.954632i	$-0.596249\pi$
$941$	−18.2697	−0.595576	−0.297788	+	0.954632i	$0.596249\pi$
$942$	0	0
$943$	3.50138	0.114021
$944$	41.4798	1.35005
$945$	0	0
$946$	6.36648	0.206992
$947$	41.5519	1.35026	0.675128	−	0.737700i	$-0.264088\pi$
$947$	41.5519	1.35026	0.675128	+	0.737700i	$0.264088\pi$
$948$	0	0
$949$	10.5978	0.344020
$950$	−1.86159	−0.0603978
$951$	0	0
$952$	0	0
$953$	−26.0298	−0.843186	−0.421593	−	0.906785i	$-0.638529\pi$
$953$	−26.0298	−0.843186	−0.421593	+	0.906785i	$0.638529\pi$
$954$	0	0
$955$	11.0066	0.356167
$956$	25.6835	0.830665
$957$	0	0
$958$	−10.5713	−0.341542
$959$	0	0
$960$	0	0
$961$	−22.3958	−0.722445
$962$	−3.00512	−0.0968888
$963$	0	0
$964$	−36.7654	−1.18413
$965$	22.4871	0.723887
$966$	0	0
$967$	−14.6507	−0.471135	−0.235567	−	0.971858i	$-0.575695\pi$
$967$	−14.6507	−0.471135	−0.235567	+	0.971858i	$0.575695\pi$
$968$	−4.55797	−0.146499
$969$	0	0
$970$	6.82608	0.219172
$971$	39.0014	1.25162	0.625808	−	0.779977i	$-0.284769\pi$
$971$	39.0014	1.25162	0.625808	+	0.779977i	$0.284769\pi$
$972$	0	0
$973$	0	0
$974$	2.93439	0.0940239
$975$	0	0
$976$	32.8200	1.05054
$977$	−19.1068	−0.611282	−0.305641	−	0.952147i	$-0.598871\pi$
$977$	−19.1068	−0.611282	−0.305641	+	0.952147i	$0.598871\pi$
$978$	0	0
$979$	2.99840	0.0958294
$980$	0	0
$981$	0	0
$982$	10.5584	0.336931
$983$	12.8877	0.411055	0.205527	−	0.978651i	$-0.434109\pi$
$983$	12.8877	0.411055	0.205527	+	0.978651i	$0.434109\pi$
$984$	0	0
$985$	5.18776	0.165296
$986$	−24.5014	−0.780283
$987$	0	0
$988$	5.80355	0.184636
$989$	−2.32217	−0.0738406
$990$	0	0
$991$	−2.97288	−0.0944367	−0.0472184	−	0.998885i	$-0.515036\pi$
$991$	−2.97288	−0.0944367	−0.0472184	+	0.998885i	$0.515036\pi$
$992$	11.9288	0.378741
$993$	0	0
$994$	0	0
$995$	29.4470	0.933532
$996$	0	0
$997$	−13.6689	−0.432898	−0.216449	−	0.976294i	$-0.569447\pi$
$997$	−13.6689	−0.432898	−0.216449	+	0.976294i	$0.569447\pi$
$998$	2.55096	0.0807493
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.t.1.3		4
3.2	odd	2		3969.2.a.w.1.2		4
7.3	odd	6		567.2.e.d.163.2	yes	8
7.5	odd	6		567.2.e.d.487.2	yes	8
7.6	odd	2		3969.2.a.s.1.3		4
21.5	even	6		567.2.e.c.487.3	yes	8
21.17	even	6		567.2.e.c.163.3	✓	8
21.20	even	2		3969.2.a.x.1.2		4
63.5	even	6		567.2.h.k.298.2		8
63.31	odd	6		567.2.g.k.541.2		8
63.38	even	6		567.2.h.k.352.2		8
63.40	odd	6		567.2.h.j.298.3		8
63.47	even	6		567.2.g.j.109.3		8
63.52	odd	6		567.2.h.j.352.3		8
63.59	even	6		567.2.g.j.541.3		8
63.61	odd	6		567.2.g.k.109.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.e.c.163.3	✓	8	21.17	even	6
567.2.e.c.487.3	yes	8	21.5	even	6
567.2.e.d.163.2	yes	8	7.3	odd	6
567.2.e.d.487.2	yes	8	7.5	odd	6
567.2.g.j.109.3		8	63.47	even	6
567.2.g.j.541.3		8	63.59	even	6
567.2.g.k.109.2		8	63.61	odd	6
567.2.g.k.541.2		8	63.31	odd	6
567.2.h.j.298.3		8	63.40	odd	6
567.2.h.j.352.3		8	63.52	odd	6
567.2.h.k.298.2		8	63.5	even	6
567.2.h.k.352.2		8	63.38	even	6
3969.2.a.s.1.3		4	7.6	odd	2
3969.2.a.t.1.3		4	1.1	even	1	trivial
3969.2.a.w.1.2		4	3.2	odd	2
3969.2.a.x.1.2		4	21.20	even	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)

\(f(q)\)	\(=\)	\(q+0.372845 q^{2} -1.86099 q^{4} +1.42143 q^{5} -1.43955 q^{8} +O(q^{10})\)	\(q+0.372845 q^{2} -1.86099 q^{4} +1.42143 q^{5} -1.43955 q^{8} +0.529976 q^{10} -3.76381 q^{11} -1.86099 q^{13} +3.18524 q^{16} -7.53437 q^{17} +1.67574 q^{19} -2.64527 q^{20} -1.40332 q^{22} +0.511859 q^{23} -2.97952 q^{25} -0.693860 q^{26} +8.72197 q^{29} +2.93329 q^{31} +4.06671 q^{32} -2.80916 q^{34} +4.33101 q^{37} +0.624794 q^{38} -2.04623 q^{40} +6.84051 q^{41} -4.53673 q^{43} +7.00439 q^{44} +0.190844 q^{46} -7.43955 q^{47} -1.11090 q^{50} +3.46327 q^{52} +0.832874 q^{53} -5.35001 q^{55} +3.25195 q^{58} +13.0225 q^{59} +10.3038 q^{61} +1.09367 q^{62} -4.85423 q^{64} -2.64527 q^{65} +6.51625 q^{67} +14.0214 q^{68} +4.11854 q^{71} -5.69474 q^{73} +1.61480 q^{74} -3.11854 q^{76} +0.264305 q^{79} +4.52761 q^{80} +2.55045 q^{82} +7.90046 q^{83} -10.7096 q^{85} -1.69150 q^{86} +5.41819 q^{88} -0.796641 q^{89} -0.952563 q^{92} -2.77380 q^{94} +2.38196 q^{95} +12.8800 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8}+O(q^{10}) \) 4 * q - q^2 + 5 * q^4 + 2 * q^5 + 3 * q^8	\( 4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8} + 7 q^{10} - 5 q^{11} + 5 q^{13} - q^{16} + 6 q^{17} + 8 q^{19} - 8 q^{20} - 7 q^{22} + 12 q^{23} + 8 q^{25} + q^{26} + 10 q^{29} + 18 q^{31} + 10 q^{32} - 20 q^{38} + 18 q^{40} - 5 q^{41} - 7 q^{43} - 13 q^{44} + 12 q^{46} - 21 q^{47} - 38 q^{50} + 25 q^{52} + 12 q^{53} + 26 q^{55} - 7 q^{58} + 6 q^{59} + 20 q^{61} + 18 q^{62} - 23 q^{64} - 8 q^{65} - 5 q^{67} + 51 q^{68} + 9 q^{71} + 6 q^{73} - 5 q^{76} - 10 q^{79} - 2 q^{80} + 35 q^{82} + 9 q^{83} - 9 q^{85} + 22 q^{86} + 18 q^{88} - 22 q^{89} + 36 q^{92} + 15 q^{94} + 16 q^{95} + 9 q^{97}+O(q^{100}) \) 4 * q - q^2 + 5 * q^4 + 2 * q^5 + 3 * q^8 + 7 * q^10 - 5 * q^11 + 5 * q^13 - q^16 + 6 * q^17 + 8 * q^19 - 8 * q^20 - 7 * q^22 + 12 * q^23 + 8 * q^25 + q^26 + 10 * q^29 + 18 * q^31 + 10 * q^32 - 20 * q^38 + 18 * q^40 - 5 * q^41 - 7 * q^43 - 13 * q^44 + 12 * q^46 - 21 * q^47 - 38 * q^50 + 25 * q^52 + 12 * q^53 + 26 * q^55 - 7 * q^58 + 6 * q^59 + 20 * q^61 + 18 * q^62 - 23 * q^64 - 8 * q^65 - 5 * q^67 + 51 * q^68 + 9 * q^71 + 6 * q^73 - 5 * q^76 - 10 * q^79 - 2 * q^80 + 35 * q^82 + 9 * q^83 - 9 * q^85 + 22 * q^86 + 18 * q^88 - 22 * q^89 + 36 * q^92 + 15 * q^94 + 16 * q^95 + 9 * q^97

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