LMFDB - Embedded newform 3969.2.a.p.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:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.46050$ 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)

$f(q)$	$=$	$q+2.46050 q^{2} +4.05408 q^{4} -2.59358 q^{5} +5.05408 q^{8} +O(q^{10})$	$q+2.46050 q^{2} +4.05408 q^{4} -2.59358 q^{5} +5.05408 q^{8} -6.38151 q^{10} -4.51459 q^{11} -1.00000 q^{13} +4.32743 q^{16} -0.945916 q^{17} +4.05408 q^{19} -10.5146 q^{20} -11.1082 q^{22} +0.273346 q^{23} +1.72665 q^{25} -2.46050 q^{26} -2.46050 q^{29} -2.32743 q^{31} +0.539495 q^{32} -2.32743 q^{34} +1.78074 q^{37} +9.97509 q^{38} -13.1082 q^{40} -6.40642 q^{41} -10.4356 q^{43} -18.3025 q^{44} +0.672570 q^{46} -12.1623 q^{47} +4.24844 q^{50} -4.05408 q^{52} +6.27335 q^{53} +11.7089 q^{55} -6.05408 q^{58} -2.72665 q^{59} +2.27335 q^{61} -5.72665 q^{62} -7.32743 q^{64} +2.59358 q^{65} -15.8171 q^{67} -3.83482 q^{68} -3.27335 q^{71} +1.50739 q^{73} +4.38151 q^{74} +16.4356 q^{76} +14.7089 q^{79} -11.2235 q^{80} -15.7630 q^{82} -0.945916 q^{83} +2.45331 q^{85} -25.6768 q^{86} -22.8171 q^{88} -14.3566 q^{89} +1.10817 q^{92} -29.9253 q^{94} -10.5146 q^{95} +11.4897 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 6 q^{8}+O(q^{10}) $ 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 6 * q^8	$ 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 6 q^{8} + 2 q^{11} - 3 q^{13} + 3 q^{16} - 12 q^{17} + 3 q^{19} - 16 q^{20} - 15 q^{22} + 6 q^{25} - q^{26} - q^{29} + 3 q^{31} + 8 q^{32} + 3 q^{34} - 3 q^{37} + 8 q^{38} - 21 q^{40} - 22 q^{41} - 3 q^{43} - 23 q^{44} + 12 q^{46} - 9 q^{47} - 10 q^{50} - 3 q^{52} + 18 q^{53} + 6 q^{55} - 9 q^{58} - 9 q^{59} + 6 q^{61} - 18 q^{62} - 12 q^{64} + 5 q^{65} + 6 q^{68} - 9 q^{71} - 3 q^{73} - 6 q^{74} + 21 q^{76} + 15 q^{79} + 11 q^{80} - 9 q^{82} - 12 q^{83} + 9 q^{85} - 34 q^{86} - 21 q^{88} - 2 q^{89} - 15 q^{92} - 24 q^{94} - 16 q^{95} - 3 q^{97}+O(q^{100}) $ 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 6 * q^8 + 2 * q^11 - 3 * q^13 + 3 * q^16 - 12 * q^17 + 3 * q^19 - 16 * q^20 - 15 * q^22 + 6 * q^25 - q^26 - q^29 + 3 * q^31 + 8 * q^32 + 3 * q^34 - 3 * q^37 + 8 * q^38 - 21 * q^40 - 22 * q^41 - 3 * q^43 - 23 * q^44 + 12 * q^46 - 9 * q^47 - 10 * q^50 - 3 * q^52 + 18 * q^53 + 6 * q^55 - 9 * q^58 - 9 * q^59 + 6 * q^61 - 18 * q^62 - 12 * q^64 + 5 * q^65 + 6 * q^68 - 9 * q^71 - 3 * q^73 - 6 * q^74 + 21 * q^76 + 15 * q^79 + 11 * q^80 - 9 * q^82 - 12 * q^83 + 9 * q^85 - 34 * q^86 - 21 * q^88 - 2 * q^89 - 15 * q^92 - 24 * q^94 - 16 * q^95 - 3 * 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 $

$2$	2.46050	1.73984	0.869920	−	0.493193i	$-0.164170\pi$
$2$	2.46050	1.73984	0.869920	+	0.493193i	$0.164170\pi$
$3$	0	0
$3$	0	0
$4$	4.05408	2.02704
$5$	−2.59358	−1.15988	−0.579942	−	0.814658i	$-0.696925\pi$
$5$	−2.59358	−1.15988	−0.579942	+	0.814658i	$0.696925\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	5.05408	1.78689
$9$	0	0
$10$	−6.38151	−2.01801
$11$	−4.51459	−1.36120	−0.680600	−	0.732655i	$-0.738281\pi$
$11$	−4.51459	−1.36120	−0.680600	+	0.732655i	$0.738281\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	4.32743	1.08186
$17$	−0.945916	−0.229418	−0.114709	−	0.993399i	$-0.536594\pi$
$17$	−0.945916	−0.229418	−0.114709	+	0.993399i	$0.536594\pi$
$18$	0	0
$19$	4.05408	0.930071	0.465035	−	0.885292i	$-0.346042\pi$
$19$	4.05408	0.930071	0.465035	+	0.885292i	$0.346042\pi$
$20$	−10.5146	−2.35113
$21$	0	0
$22$	−11.1082	−2.36827
$23$	0.273346	0.0569966	0.0284983	−	0.999594i	$-0.490927\pi$
$23$	0.273346	0.0569966	0.0284983	+	0.999594i	$0.490927\pi$
$24$	0	0
$25$	1.72665	0.345331
$26$	−2.46050	−0.482545
$27$	0	0
$28$	0	0
$29$	−2.46050	−0.456904	−0.228452	−	0.973555i	$-0.573366\pi$
$29$	−2.46050	−0.456904	−0.228452	+	0.973555i	$0.573366\pi$
$30$	0	0
$31$	−2.32743	−0.418019	−0.209009	−	0.977914i	$-0.567024\pi$
$31$	−2.32743	−0.418019	−0.209009	+	0.977914i	$0.567024\pi$
$32$	0.539495	0.0953702
$33$	0	0
$34$	−2.32743	−0.399151
$35$	0	0
$36$	0	0
$37$	1.78074	0.292752	0.146376	−	0.989229i	$-0.453239\pi$
$37$	1.78074	0.292752	0.146376	+	0.989229i	$0.453239\pi$
$38$	9.97509	1.61817
$39$	0	0
$40$	−13.1082	−2.07258
$41$	−6.40642	−1.00051	−0.500257	−	0.865877i	$-0.666761\pi$
$41$	−6.40642	−1.00051	−0.500257	+	0.865877i	$0.666761\pi$
$42$	0	0
$43$	−10.4356	−1.59141	−0.795707	−	0.605682i	$-0.792900\pi$
$43$	−10.4356	−1.59141	−0.795707	+	0.605682i	$0.792900\pi$
$44$	−18.3025	−2.75921
$45$	0	0
$46$	0.672570	0.0991650
$47$	−12.1623	−1.77405	−0.887023	−	0.461724i	$-0.847231\pi$
$47$	−12.1623	−1.77405	−0.887023	+	0.461724i	$0.847231\pi$
$48$	0	0
$49$	0	0
$50$	4.24844	0.600820
$51$	0	0
$52$	−4.05408	−0.562200
$53$	6.27335	0.861710	0.430855	−	0.902421i	$-0.358212\pi$
$53$	6.27335	0.861710	0.430855	+	0.902421i	$0.358212\pi$
$54$	0	0
$55$	11.7089	1.57883
$56$	0	0
$57$	0	0
$58$	−6.05408	−0.794940
$59$	−2.72665	−0.354980	−0.177490	−	0.984123i	$-0.556798\pi$
$59$	−2.72665	−0.354980	−0.177490	+	0.984123i	$0.556798\pi$
$60$	0	0
$61$	2.27335	0.291072	0.145536	−	0.989353i	$-0.453509\pi$
$61$	2.27335	0.291072	0.145536	+	0.989353i	$0.453509\pi$
$62$	−5.72665	−0.727286
$63$	0	0
$64$	−7.32743	−0.915929
$65$	2.59358	0.321694
$66$	0	0
$67$	−15.8171	−1.93237	−0.966184	−	0.257854i	$-0.916985\pi$
$67$	−15.8171	−1.93237	−0.966184	+	0.257854i	$0.916985\pi$
$68$	−3.83482	−0.465041
$69$	0	0
$70$	0	0
$71$	−3.27335	−0.388475	−0.194237	−	0.980955i	$-0.562223\pi$
$71$	−3.27335	−0.388475	−0.194237	+	0.980955i	$0.562223\pi$
$72$	0	0
$73$	1.50739	0.176427	0.0882134	−	0.996102i	$-0.471884\pi$
$73$	1.50739	0.176427	0.0882134	+	0.996102i	$0.471884\pi$
$74$	4.38151	0.509341
$75$	0	0
$76$	16.4356	1.88529
$77$	0	0
$78$	0	0
$79$	14.7089	1.65489	0.827443	−	0.561550i	$-0.189795\pi$
$79$	14.7089	1.65489	0.827443	+	0.561550i	$0.189795\pi$
$80$	−11.2235	−1.25483
$81$	0	0
$82$	−15.7630	−1.74074
$83$	−0.945916	−0.103828	−0.0519139	−	0.998652i	$-0.516532\pi$
$83$	−0.945916	−0.103828	−0.0519139	+	0.998652i	$0.516532\pi$
$84$	0	0
$85$	2.45331	0.266099
$86$	−25.6768	−2.76881
$87$	0	0
$88$	−22.8171	−2.43231
$89$	−14.3566	−1.52180	−0.760899	−	0.648871i	$-0.775242\pi$
$89$	−14.3566	−1.52180	−0.760899	+	0.648871i	$0.775242\pi$
$90$	0	0
$91$	0	0
$92$	1.10817	0.115535
$93$	0	0
$94$	−29.9253	−3.08656
$95$	−10.5146	−1.07877
$96$	0	0
$97$	11.4897	1.16660	0.583300	−	0.812257i	$-0.301761\pi$
$97$	11.4897	1.16660	0.583300	+	0.812257i	$0.301761\pi$
$98$	0	0
$99$	0	0
$100$	7.00000	0.700000
$101$	−3.67977	−0.366150	−0.183075	−	0.983099i	$-0.558605\pi$
$101$	−3.67977	−0.366150	−0.183075	+	0.983099i	$0.558605\pi$
$102$	0	0
$103$	9.72665	0.958396	0.479198	−	0.877707i	$-0.340928\pi$
$103$	9.72665	0.958396	0.479198	+	0.877707i	$0.340928\pi$
$104$	−5.05408	−0.495594
$105$	0	0
$106$	15.4356	1.49924
$107$	1.37432	0.132860	0.0664301	−	0.997791i	$-0.478839\pi$
$107$	1.37432	0.132860	0.0664301	+	0.997791i	$0.478839\pi$
$108$	0	0
$109$	−3.39922	−0.325587	−0.162793	−	0.986660i	$-0.552050\pi$
$109$	−3.39922	−0.325587	−0.162793	+	0.986660i	$0.552050\pi$
$110$	28.8099	2.74692
$111$	0	0
$112$	0	0
$113$	−10.3887	−0.977288	−0.488644	−	0.872483i	$-0.662508\pi$
$113$	−10.3887	−0.977288	−0.488644	+	0.872483i	$0.662508\pi$
$114$	0	0
$115$	−0.708945	−0.0661095
$116$	−9.97509	−0.926164
$117$	0	0
$118$	−6.70895	−0.617608
$119$	0	0
$120$	0	0
$121$	9.38151	0.852865
$122$	5.59358	0.506419
$123$	0	0
$124$	−9.43560	−0.847342
$125$	8.48968	0.759340
$126$	0	0
$127$	0.672570	0.0596809	0.0298405	−	0.999555i	$-0.490500\pi$
$127$	0.672570	0.0596809	0.0298405	+	0.999555i	$0.490500\pi$
$128$	−19.1082	−1.68894
$129$	0	0
$130$	6.38151	0.559696
$131$	7.91381	0.691433	0.345717	−	0.938339i	$-0.387636\pi$
$131$	7.91381	0.691433	0.345717	+	0.938339i	$0.387636\pi$
$132$	0	0
$133$	0	0
$134$	−38.9181	−3.36201
$135$	0	0
$136$	−4.78074	−0.409945
$137$	3.67257	0.313769	0.156884	−	0.987617i	$-0.449855\pi$
$137$	3.67257	0.313769	0.156884	+	0.987617i	$0.449855\pi$
$138$	0	0
$139$	2.05408	0.174225	0.0871126	−	0.996198i	$-0.472236\pi$
$139$	2.05408	0.174225	0.0871126	+	0.996198i	$0.472236\pi$
$140$	0	0
$141$	0	0
$142$	−8.05408	−0.675884
$143$	4.51459	0.377529
$144$	0	0
$145$	6.38151	0.529956
$146$	3.70895	0.306954
$147$	0	0
$148$	7.21926	0.593420
$149$	13.5438	1.10955	0.554774	−	0.832001i	$-0.312805\pi$
$149$	13.5438	1.10955	0.554774	+	0.832001i	$0.312805\pi$
$150$	0	0
$151$	9.92821	0.807946	0.403973	−	0.914771i	$-0.367629\pi$
$151$	9.92821	0.807946	0.403973	+	0.914771i	$0.367629\pi$
$152$	20.4897	1.66193
$153$	0	0
$154$	0	0
$155$	6.03638	0.484853
$156$	0	0
$157$	−6.05408	−0.483169	−0.241584	−	0.970380i	$-0.577667\pi$
$157$	−6.05408	−0.483169	−0.241584	+	0.970380i	$0.577667\pi$
$158$	36.1914	2.87924
$159$	0	0
$160$	−1.39922	−0.110618
$161$	0	0
$162$	0	0
$163$	17.8171	1.39554	0.697772	−	0.716320i	$-0.254175\pi$
$163$	17.8171	1.39554	0.697772	+	0.716320i	$0.254175\pi$
$164$	−25.9722	−2.02809
$165$	0	0
$166$	−2.32743	−0.180644
$167$	−8.46770	−0.655250	−0.327625	−	0.944808i	$-0.606248\pi$
$167$	−8.46770	−0.655250	−0.327625	+	0.944808i	$0.606248\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	6.03638	0.462969
$171$	0	0
$172$	−42.3068	−3.22586
$173$	17.3566	1.31960	0.659799	−	0.751442i	$-0.270641\pi$
$173$	17.3566	1.31960	0.659799	+	0.751442i	$0.270641\pi$
$174$	0	0
$175$	0	0
$176$	−19.5366	−1.47262
$177$	0	0
$178$	−35.3245	−2.64768
$179$	11.3494	0.848295	0.424147	−	0.905593i	$-0.360574\pi$
$179$	11.3494	0.848295	0.424147	+	0.905593i	$0.360574\pi$
$180$	0	0
$181$	−21.8889	−1.62699	−0.813495	−	0.581572i	$-0.802438\pi$
$181$	−21.8889	−1.62699	−0.813495	+	0.581572i	$0.802438\pi$
$182$	0	0
$183$	0	0
$184$	1.38151	0.101847
$185$	−4.61849	−0.339558
$186$	0	0
$187$	4.27042	0.312284
$188$	−49.3068	−3.59607
$189$	0	0
$190$	−25.8712	−1.87689
$191$	0.701748	0.0507767	0.0253883	−	0.999678i	$-0.491918\pi$
$191$	0.701748	0.0507767	0.0253883	+	0.999678i	$0.491918\pi$
$192$	0	0
$193$	12.1445	0.874183	0.437092	−	0.899417i	$-0.356009\pi$
$193$	12.1445	0.874183	0.437092	+	0.899417i	$0.356009\pi$
$194$	28.2704	2.02970
$195$	0	0
$196$	0	0
$197$	16.4107	1.16921	0.584607	−	0.811317i	$-0.301249\pi$
$197$	16.4107	1.16921	0.584607	+	0.811317i	$0.301249\pi$
$198$	0	0
$199$	−22.7060	−1.60959	−0.804794	−	0.593555i	$-0.797724\pi$
$199$	−22.7060	−1.60959	−0.804794	+	0.593555i	$0.797724\pi$
$200$	8.72665	0.617068
$201$	0	0
$202$	−9.05408	−0.637043
$203$	0	0
$204$	0	0
$205$	16.6156	1.16048
$206$	23.9325	1.66745
$207$	0	0
$208$	−4.32743	−0.300053
$209$	−18.3025	−1.26601
$210$	0	0
$211$	4.56148	0.314025	0.157012	−	0.987597i	$-0.449814\pi$
$211$	4.56148	0.314025	0.157012	+	0.987597i	$0.449814\pi$
$212$	25.4327	1.74672
$213$	0	0
$214$	3.38151	0.231156
$215$	27.0656	1.84586
$216$	0	0
$217$	0	0
$218$	−8.36381	−0.566468
$219$	0	0
$220$	47.4690	3.20036
$221$	0.945916	0.0636292
$222$	0	0
$223$	−13.3245	−0.892275	−0.446137	−	0.894964i	$-0.647201\pi$
$223$	−13.3245	−0.892275	−0.446137	+	0.894964i	$0.647201\pi$
$224$	0	0
$225$	0	0
$226$	−25.5615	−1.70032
$227$	1.38151	0.0916943	0.0458472	−	0.998948i	$-0.485401\pi$
$227$	1.38151	0.0916943	0.0458472	+	0.998948i	$0.485401\pi$
$228$	0	0
$229$	17.9794	1.18811	0.594055	−	0.804424i	$-0.297526\pi$
$229$	17.9794	1.18811	0.594055	+	0.804424i	$0.297526\pi$
$230$	−1.74436	−0.115020
$231$	0	0
$232$	−12.4356	−0.816437
$233$	18.9823	1.24357	0.621786	−	0.783187i	$-0.286408\pi$
$233$	18.9823	1.24357	0.621786	+	0.783187i	$0.286408\pi$
$234$	0	0
$235$	31.5438	2.05769
$236$	−11.0541	−0.719560
$237$	0	0
$238$	0	0
$239$	−4.89183	−0.316426	−0.158213	−	0.987405i	$-0.550573\pi$
$239$	−4.89183	−0.316426	−0.158213	+	0.987405i	$0.550573\pi$
$240$	0	0
$241$	26.1593	1.68507	0.842535	−	0.538641i	$-0.181062\pi$
$241$	26.1593	1.68507	0.842535	+	0.538641i	$0.181062\pi$
$242$	23.0833	1.48385
$243$	0	0
$244$	9.21634	0.590016
$245$	0	0
$246$	0	0
$247$	−4.05408	−0.257955
$248$	−11.7630	−0.746953
$249$	0	0
$250$	20.8889	1.32113
$251$	18.4576	1.16503	0.582516	−	0.812819i	$-0.302068\pi$
$251$	18.4576	1.16503	0.582516	+	0.812819i	$0.302068\pi$
$252$	0	0
$253$	−1.23405	−0.0775838
$254$	1.65486	0.103835
$255$	0	0
$256$	−32.3609	−2.02256
$257$	−11.7339	−0.731938	−0.365969	−	0.930627i	$-0.619262\pi$
$257$	−11.7339	−0.731938	−0.365969	+	0.930627i	$0.619262\pi$
$258$	0	0
$259$	0	0
$260$	10.5146	0.652087
$261$	0	0
$262$	19.4720	1.20298
$263$	7.52179	0.463813	0.231907	−	0.972738i	$-0.425504\pi$
$263$	7.52179	0.463813	0.231907	+	0.972738i	$0.425504\pi$
$264$	0	0
$265$	−16.2704	−0.999484
$266$	0	0
$267$	0	0
$268$	−64.1239	−3.91699
$269$	−18.8348	−1.14838	−0.574190	−	0.818722i	$-0.694683\pi$
$269$	−18.8348	−1.14838	−0.574190	+	0.818722i	$0.694683\pi$
$270$	0	0
$271$	23.9823	1.45682	0.728410	−	0.685141i	$-0.240260\pi$
$271$	23.9823	1.45682	0.728410	+	0.685141i	$0.240260\pi$
$272$	−4.09338	−0.248198
$273$	0	0
$274$	9.03638	0.545907
$275$	−7.79513	−0.470064
$276$	0	0
$277$	7.16225	0.430338	0.215169	−	0.976577i	$-0.430970\pi$
$277$	7.16225	0.430338	0.215169	+	0.976577i	$0.430970\pi$
$278$	5.05408	0.303124
$279$	0	0
$280$	0	0
$281$	−14.8817	−0.887768	−0.443884	−	0.896084i	$-0.646400\pi$
$281$	−14.8817	−0.887768	−0.443884	+	0.896084i	$0.646400\pi$
$282$	0	0
$283$	−19.9971	−1.18870	−0.594351	−	0.804205i	$-0.702591\pi$
$283$	−19.9971	−1.18870	−0.594351	+	0.804205i	$0.702591\pi$
$284$	−13.2704	−0.787455
$285$	0	0
$286$	11.1082	0.656840
$287$	0	0
$288$	0	0
$289$	−16.1052	−0.947367
$290$	15.7017	0.922038
$291$	0	0
$292$	6.11109	0.357625
$293$	15.0656	0.880139	0.440070	−	0.897964i	$-0.354954\pi$
$293$	15.0656	0.880139	0.440070	+	0.897964i	$0.354954\pi$
$294$	0	0
$295$	7.07179	0.411736
$296$	9.00000	0.523114
$297$	0	0
$298$	33.3245	1.93044
$299$	−0.273346	−0.0158080
$300$	0	0
$301$	0	0
$302$	24.4284	1.40570
$303$	0	0
$304$	17.5438	1.00620
$305$	−5.89610	−0.337610
$306$	0	0
$307$	27.2704	1.55641	0.778203	−	0.628013i	$-0.216132\pi$
$307$	27.2704	1.55641	0.778203	+	0.628013i	$0.216132\pi$
$308$	0	0
$309$	0	0
$310$	14.8525	0.843567
$311$	−15.9823	−0.906273	−0.453136	−	0.891441i	$-0.649695\pi$
$311$	−15.9823	−0.906273	−0.453136	+	0.891441i	$0.649695\pi$
$312$	0	0
$313$	−11.5979	−0.655549	−0.327775	−	0.944756i	$-0.606299\pi$
$313$	−11.5979	−0.655549	−0.327775	+	0.944756i	$0.606299\pi$
$314$	−14.8961	−0.840636
$315$	0	0
$316$	59.6313	3.35452
$317$	2.01771	0.113326	0.0566629	−	0.998393i	$-0.481954\pi$
$317$	2.01771	0.113326	0.0566629	+	0.998393i	$0.481954\pi$
$318$	0	0
$319$	11.1082	0.621938
$320$	19.0043	1.06237
$321$	0	0
$322$	0	0
$323$	−3.83482	−0.213375
$324$	0	0
$325$	−1.72665	−0.0957775
$326$	43.8391	2.42802
$327$	0	0
$328$	−32.3786	−1.78781
$329$	0	0
$330$	0	0
$331$	−19.7089	−1.08330	−0.541651	−	0.840604i	$-0.682200\pi$
$331$	−19.7089	−1.08330	−0.541651	+	0.840604i	$0.682200\pi$
$332$	−3.83482	−0.210463
$333$	0	0
$334$	−20.8348	−1.14003
$335$	41.0229	2.24132
$336$	0	0
$337$	−29.0512	−1.58252	−0.791259	−	0.611481i	$-0.790574\pi$
$337$	−29.0512	−1.58252	−0.791259	+	0.611481i	$0.790574\pi$
$338$	−29.5261	−1.60601
$339$	0	0
$340$	9.94592	0.539393
$341$	10.5074	0.569007
$342$	0	0
$343$	0	0
$344$	−52.7424	−2.84368
$345$	0	0
$346$	42.7060	2.29589
$347$	−29.0833	−1.56127	−0.780636	−	0.624986i	$-0.785105\pi$
$347$	−29.0833	−1.56127	−0.780636	+	0.624986i	$0.785105\pi$
$348$	0	0
$349$	−24.7630	−1.32553	−0.662767	−	0.748825i	$-0.730618\pi$
$349$	−24.7630	−1.32553	−0.662767	+	0.748825i	$0.730618\pi$
$350$	0	0
$351$	0	0
$352$	−2.43560	−0.129818
$353$	−33.3025	−1.77251	−0.886257	−	0.463193i	$-0.846704\pi$
$353$	−33.3025	−1.77251	−0.886257	+	0.463193i	$0.846704\pi$
$354$	0	0
$355$	8.48968	0.450586
$356$	−58.2029	−3.08475
$357$	0	0
$358$	27.9253	1.47590
$359$	−25.5366	−1.34777	−0.673884	−	0.738837i	$-0.735375\pi$
$359$	−25.5366	−1.34777	−0.673884	+	0.738837i	$0.735375\pi$
$360$	0	0
$361$	−2.56440	−0.134968
$362$	−53.8578	−2.83070
$363$	0	0
$364$	0	0
$365$	−3.90954	−0.204635
$366$	0	0
$367$	−27.4504	−1.43290	−0.716449	−	0.697639i	$-0.754234\pi$
$367$	−27.4504	−1.43290	−0.716449	+	0.697639i	$0.754234\pi$
$368$	1.18289	0.0616622
$369$	0	0
$370$	−11.3638	−0.590776
$371$	0	0
$372$	0	0
$373$	16.3274	0.845402	0.422701	−	0.906269i	$-0.361082\pi$
$373$	16.3274	0.845402	0.422701	+	0.906269i	$0.361082\pi$
$374$	10.5074	0.543324
$375$	0	0
$376$	−61.4690	−3.17002
$377$	2.46050	0.126722
$378$	0	0
$379$	12.0364	0.618267	0.309134	−	0.951019i	$-0.399961\pi$
$379$	12.0364	0.618267	0.309134	+	0.951019i	$0.399961\pi$
$380$	−42.6270	−2.18672
$381$	0	0
$382$	1.72665	0.0883433
$383$	−12.4356	−0.635429	−0.317715	−	0.948186i	$-0.602915\pi$
$383$	−12.4356	−0.635429	−0.317715	+	0.948186i	$0.602915\pi$
$384$	0	0
$385$	0	0
$386$	29.8817	1.52094
$387$	0	0
$388$	46.5801	2.36475
$389$	−20.6008	−1.04450	−0.522250	−	0.852792i	$-0.674907\pi$
$389$	−20.6008	−1.04450	−0.522250	+	0.852792i	$0.674907\pi$
$390$	0	0
$391$	−0.258562	−0.0130761
$392$	0	0
$393$	0	0
$394$	40.3786	2.03424
$395$	−38.1488	−1.91948
$396$	0	0
$397$	23.6372	1.18631	0.593157	−	0.805087i	$-0.297881\pi$
$397$	23.6372	1.18631	0.593157	+	0.805087i	$0.297881\pi$
$398$	−55.8683	−2.80042
$399$	0	0
$400$	7.47197	0.373599
$401$	2.56440	0.128060	0.0640300	−	0.997948i	$-0.479605\pi$
$401$	2.56440	0.128060	0.0640300	+	0.997948i	$0.479605\pi$
$402$	0	0
$403$	2.32743	0.115938
$404$	−14.9181	−0.742202
$405$	0	0
$406$	0	0
$407$	−8.03930	−0.398493
$408$	0	0
$409$	34.3245	1.69724	0.848619	−	0.529005i	$-0.177435\pi$
$409$	34.3245	1.69724	0.848619	+	0.529005i	$0.177435\pi$
$410$	40.8827	2.01905
$411$	0	0
$412$	39.4327	1.94271
$413$	0	0
$414$	0	0
$415$	2.45331	0.120428
$416$	−0.539495	−0.0264509
$417$	0	0
$418$	−45.0335	−2.20266
$419$	4.05701	0.198198	0.0990989	−	0.995078i	$-0.468404\pi$
$419$	4.05701	0.198198	0.0990989	+	0.995078i	$0.468404\pi$
$420$	0	0
$421$	−21.0689	−1.02683	−0.513417	−	0.858139i	$-0.671621\pi$
$421$	−21.0689	−1.02683	−0.513417	+	0.858139i	$0.671621\pi$
$422$	11.2235	0.546353
$423$	0	0
$424$	31.7060	1.53978
$425$	−1.63327	−0.0792252
$426$	0	0
$427$	0	0
$428$	5.57160	0.269313
$429$	0	0
$430$	66.5949	3.21149
$431$	−22.6185	−1.08949	−0.544747	−	0.838600i	$-0.683374\pi$
$431$	−22.6185	−1.08949	−0.544747	+	0.838600i	$0.683374\pi$
$432$	0	0
$433$	−2.41789	−0.116196	−0.0580982	−	0.998311i	$-0.518504\pi$
$433$	−2.41789	−0.116196	−0.0580982	+	0.998311i	$0.518504\pi$
$434$	0	0
$435$	0	0
$436$	−13.7807	−0.659978
$437$	1.10817	0.0530109
$438$	0	0
$439$	23.4897	1.12110	0.560551	−	0.828120i	$-0.310589\pi$
$439$	23.4897	1.12110	0.560551	+	0.828120i	$0.310589\pi$
$440$	59.1780	2.82120
$441$	0	0
$442$	2.32743	0.110705
$443$	13.4179	0.637503	0.318752	−	0.947838i	$-0.396736\pi$
$443$	13.4179	0.637503	0.318752	+	0.947838i	$0.396736\pi$
$444$	0	0
$445$	37.2350	1.76511
$446$	−32.7850	−1.55242
$447$	0	0
$448$	0	0
$449$	9.16225	0.432393	0.216197	−	0.976350i	$-0.430635\pi$
$449$	9.16225	0.432393	0.216197	+	0.976350i	$0.430635\pi$
$450$	0	0
$451$	28.9224	1.36190
$452$	−42.1167	−1.98100
$453$	0	0
$454$	3.39922	0.159533
$455$	0	0
$456$	0	0
$457$	8.81711	0.412447	0.206224	−	0.978505i	$-0.433883\pi$
$457$	8.81711	0.412447	0.206224	+	0.978505i	$0.433883\pi$
$458$	44.2383	2.06712
$459$	0	0
$460$	−2.87412	−0.134007
$461$	−5.65913	−0.263572	−0.131786	−	0.991278i	$-0.542071\pi$
$461$	−5.65913	−0.263572	−0.131786	+	0.991278i	$0.542071\pi$
$462$	0	0
$463$	15.7267	0.730880	0.365440	−	0.930835i	$-0.380919\pi$
$463$	15.7267	0.730880	0.365440	+	0.930835i	$0.380919\pi$
$464$	−10.6477	−0.494305
$465$	0	0
$466$	46.7060	2.16361
$467$	−21.9971	−1.01790	−0.508952	−	0.860795i	$-0.669967\pi$
$467$	−21.9971	−1.01790	−0.508952	+	0.860795i	$0.669967\pi$
$468$	0	0
$469$	0	0
$470$	77.6136	3.58005
$471$	0	0
$472$	−13.7807	−0.634310
$473$	47.1124	2.16623
$474$	0	0
$475$	7.00000	0.321182
$476$	0	0
$477$	0	0
$478$	−12.0364	−0.550531
$479$	24.9751	1.14114	0.570571	−	0.821249i	$-0.306722\pi$
$479$	24.9751	1.14114	0.570571	+	0.821249i	$0.306722\pi$
$480$	0	0
$481$	−1.78074	−0.0811947
$482$	64.3652	2.93175
$483$	0	0
$484$	38.0335	1.72879
$485$	−29.7994	−1.35312
$486$	0	0
$487$	−17.5979	−0.797435	−0.398717	−	0.917074i	$-0.630545\pi$
$487$	−17.5979	−0.797435	−0.398717	+	0.917074i	$0.630545\pi$
$488$	11.4897	0.520114
$489$	0	0
$490$	0	0
$491$	−13.7951	−0.622566	−0.311283	−	0.950317i	$-0.600759\pi$
$491$	−13.7951	−0.622566	−0.311283	+	0.950317i	$0.600759\pi$
$492$	0	0
$493$	2.32743	0.104822
$494$	−9.97509	−0.448801
$495$	0	0
$496$	−10.0718	−0.452237
$497$	0	0
$498$	0	0
$499$	13.0875	0.585879	0.292939	−	0.956131i	$-0.405367\pi$
$499$	13.0875	0.585879	0.292939	+	0.956131i	$0.405367\pi$
$500$	34.4179	1.53921
$501$	0	0
$502$	45.4150	2.02697
$503$	−22.3068	−0.994611	−0.497305	−	0.867576i	$-0.665677\pi$
$503$	−22.3068	−0.994611	−0.497305	+	0.867576i	$0.665677\pi$
$504$	0	0
$505$	9.54377	0.424692
$506$	−3.03638	−0.134983
$507$	0	0
$508$	2.72665	0.120976
$509$	−15.8932	−0.704453	−0.352226	−	0.935915i	$-0.614575\pi$
$509$	−15.8932	−0.704453	−0.352226	+	0.935915i	$0.614575\pi$
$510$	0	0
$511$	0	0
$512$	−41.4078	−1.82998
$513$	0	0
$514$	−28.8712	−1.27345
$515$	−25.2268	−1.11163
$516$	0	0
$517$	54.9076	2.41483
$518$	0	0
$519$	0	0
$520$	13.1082	0.574831
$521$	4.41789	0.193551	0.0967756	−	0.995306i	$-0.469147\pi$
$521$	4.41789	0.193551	0.0967756	+	0.995306i	$0.469147\pi$
$522$	0	0
$523$	−25.2733	−1.10513	−0.552563	−	0.833471i	$-0.686350\pi$
$523$	−25.2733	−1.10513	−0.552563	+	0.833471i	$0.686350\pi$
$524$	32.0833	1.40156
$525$	0	0
$526$	18.5074	0.806961
$527$	2.20155	0.0959012
$528$	0	0
$529$	−22.9253	−0.996751
$530$	−40.0335	−1.73894
$531$	0	0
$532$	0	0
$533$	6.40642	0.277493
$534$	0	0
$535$	−3.56440	−0.154103
$536$	−79.9410	−3.45293
$537$	0	0
$538$	−46.3432	−1.99800
$539$	0	0
$540$	0	0
$541$	−3.43852	−0.147834	−0.0739168	−	0.997264i	$-0.523550\pi$
$541$	−3.43852	−0.147834	−0.0739168	+	0.997264i	$0.523550\pi$
$542$	59.0085	2.53463
$543$	0	0
$544$	−0.510317	−0.0218797
$545$	8.81616	0.377643
$546$	0	0
$547$	−6.92821	−0.296229	−0.148114	−	0.988970i	$-0.547320\pi$
$547$	−6.92821	−0.296229	−0.148114	+	0.988970i	$0.547320\pi$
$548$	14.8889	0.636023
$549$	0	0
$550$	−19.1800	−0.817836
$551$	−9.97509	−0.424953
$552$	0	0
$553$	0	0
$554$	17.6228	0.748719
$555$	0	0
$556$	8.32743	0.353162
$557$	−33.5835	−1.42298	−0.711488	−	0.702698i	$-0.751979\pi$
$557$	−33.5835	−1.42298	−0.711488	+	0.702698i	$0.751979\pi$
$558$	0	0
$559$	10.4356	0.441379
$560$	0	0
$561$	0	0
$562$	−36.6165	−1.54457
$563$	42.4792	1.79028	0.895142	−	0.445781i	$-0.147074\pi$
$563$	42.4792	1.79028	0.895142	+	0.445781i	$0.147074\pi$
$564$	0	0
$565$	26.9439	1.13354
$566$	−49.2029	−2.06815
$567$	0	0
$568$	−16.5438	−0.694161
$569$	−10.4035	−0.436137	−0.218069	−	0.975933i	$-0.569976\pi$
$569$	−10.4035	−0.436137	−0.218069	+	0.975933i	$0.569976\pi$
$570$	0	0
$571$	17.8496	0.746983	0.373491	−	0.927634i	$-0.378161\pi$
$571$	17.8496	0.746983	0.373491	+	0.927634i	$0.378161\pi$
$572$	18.3025	0.765267
$573$	0	0
$574$	0	0
$575$	0.471974	0.0196827
$576$	0	0
$577$	−11.9430	−0.497193	−0.248597	−	0.968607i	$-0.579969\pi$
$577$	−11.9430	−0.497193	−0.248597	+	0.968607i	$0.579969\pi$
$578$	−39.6270	−1.64827
$579$	0	0
$580$	25.8712	1.07424
$581$	0	0
$582$	0	0
$583$	−28.3216	−1.17296
$584$	7.61849	0.315255
$585$	0	0
$586$	37.0689	1.53130
$587$	23.8597	0.984796	0.492398	−	0.870370i	$-0.336120\pi$
$587$	23.8597	0.984796	0.492398	+	0.870370i	$0.336120\pi$
$588$	0	0
$589$	−9.43560	−0.388787
$590$	17.4002	0.716354
$591$	0	0
$592$	7.70602	0.316715
$593$	19.5801	0.804060	0.402030	−	0.915626i	$-0.368305\pi$
$593$	19.5801	0.804060	0.402030	+	0.915626i	$0.368305\pi$
$594$	0	0
$595$	0	0
$596$	54.9076	2.24910
$597$	0	0
$598$	−0.672570	−0.0275034
$599$	−18.5467	−0.757797	−0.378899	−	0.925438i	$-0.623697\pi$
$599$	−18.5467	−0.757797	−0.378899	+	0.925438i	$0.623697\pi$
$600$	0	0
$601$	18.1986	0.742338	0.371169	−	0.928565i	$-0.378957\pi$
$601$	18.1986	0.742338	0.371169	+	0.928565i	$0.378957\pi$
$602$	0	0
$603$	0	0
$604$	40.2498	1.63774
$605$	−24.3317	−0.989224
$606$	0	0
$607$	22.3097	0.905524	0.452762	−	0.891631i	$-0.350439\pi$
$607$	22.3097	0.905524	0.452762	+	0.891631i	$0.350439\pi$
$608$	2.18716	0.0887010
$609$	0	0
$610$	−14.5074	−0.587387
$611$	12.1623	0.492032
$612$	0	0
$613$	10.2370	0.413467	0.206734	−	0.978397i	$-0.433717\pi$
$613$	10.2370	0.413467	0.206734	+	0.978397i	$0.433717\pi$
$614$	67.0990	2.70790
$615$	0	0
$616$	0	0
$617$	11.3274	0.456025	0.228013	−	0.973658i	$-0.426777\pi$
$617$	11.3274	0.456025	0.228013	+	0.973658i	$0.426777\pi$
$618$	0	0
$619$	−8.63327	−0.347000	−0.173500	−	0.984834i	$-0.555508\pi$
$619$	−8.63327	−0.347000	−0.173500	+	0.984834i	$0.555508\pi$
$620$	24.4720	0.982818
$621$	0	0
$622$	−39.3245	−1.57677
$623$	0	0
$624$	0	0
$625$	−30.6519	−1.22608
$626$	−28.5366	−1.14055
$627$	0	0
$628$	−24.5438	−0.979403
$629$	−1.68443	−0.0671626
$630$	0	0
$631$	−14.8535	−0.591308	−0.295654	−	0.955295i	$-0.595538\pi$
$631$	−14.8535	−0.591308	−0.295654	+	0.955295i	$0.595538\pi$
$632$	74.3402	2.95710
$633$	0	0
$634$	4.96458	0.197169
$635$	−1.74436	−0.0692229
$636$	0	0
$637$	0	0
$638$	27.3317	1.08207
$639$	0	0
$640$	49.5586	1.95897
$641$	34.1593	1.34921	0.674606	−	0.738178i	$-0.264313\pi$
$641$	34.1593	1.34921	0.674606	+	0.738178i	$0.264313\pi$
$642$	0	0
$643$	10.8348	0.427284	0.213642	−	0.976912i	$-0.431467\pi$
$643$	10.8348	0.427284	0.213642	+	0.976912i	$0.431467\pi$
$644$	0	0
$645$	0	0
$646$	−9.43560	−0.371239
$647$	32.9692	1.29615	0.648077	−	0.761575i	$-0.275573\pi$
$647$	32.9692	1.29615	0.648077	+	0.761575i	$0.275573\pi$
$648$	0	0
$649$	12.3097	0.483199
$650$	−4.24844	−0.166638
$651$	0	0
$652$	72.2321	2.82883
$653$	3.93113	0.153837	0.0769185	−	0.997037i	$-0.475492\pi$
$653$	3.93113	0.153837	0.0769185	+	0.997037i	$0.475492\pi$
$654$	0	0
$655$	−20.5251	−0.801982
$656$	−27.7233	−1.08241
$657$	0	0
$658$	0	0
$659$	−16.8171	−0.655102	−0.327551	−	0.944834i	$-0.606223\pi$
$659$	−16.8171	−0.655102	−0.327551	+	0.944834i	$0.606223\pi$
$660$	0	0
$661$	17.0216	0.662063	0.331032	−	0.943620i	$-0.392603\pi$
$661$	17.0216	0.662063	0.331032	+	0.943620i	$0.392603\pi$
$662$	−48.4940	−1.88477
$663$	0	0
$664$	−4.78074	−0.185529
$665$	0	0
$666$	0	0
$667$	−0.672570	−0.0260420
$668$	−34.3288	−1.32822
$669$	0	0
$670$	100.937	3.89954
$671$	−10.2632	−0.396207
$672$	0	0
$673$	28.7453	1.10805	0.554025	−	0.832500i	$-0.313091\pi$
$673$	28.7453	1.10805	0.554025	+	0.832500i	$0.313091\pi$
$674$	−71.4805	−2.75333
$675$	0	0
$676$	−48.6490	−1.87112
$677$	−6.03638	−0.231997	−0.115998	−	0.993249i	$-0.537007\pi$
$677$	−6.03638	−0.231997	−0.115998	+	0.993249i	$0.537007\pi$
$678$	0	0
$679$	0	0
$680$	12.3992	0.475489
$681$	0	0
$682$	25.8535	0.989981
$683$	−20.5113	−0.784842	−0.392421	−	0.919786i	$-0.628362\pi$
$683$	−20.5113	−0.784842	−0.392421	+	0.919786i	$0.628362\pi$
$684$	0	0
$685$	−9.52510	−0.363935
$686$	0	0
$687$	0	0
$688$	−45.1593	−1.72168
$689$	−6.27335	−0.238995
$690$	0	0
$691$	15.0029	0.570738	0.285369	−	0.958418i	$-0.407884\pi$
$691$	15.0029	0.570738	0.285369	+	0.958418i	$0.407884\pi$
$692$	70.3652	2.67488
$693$	0	0
$694$	−71.5595	−2.71636
$695$	−5.32743	−0.202081
$696$	0	0
$697$	6.05993	0.229536
$698$	−60.9296	−2.30622
$699$	0	0
$700$	0	0
$701$	−38.5113	−1.45455	−0.727275	−	0.686346i	$-0.759214\pi$
$701$	−38.5113	−1.45455	−0.727275	+	0.686346i	$0.759214\pi$
$702$	0	0
$703$	7.21926	0.272280
$704$	33.0803	1.24676
$705$	0	0
$706$	−81.9410	−3.08389
$707$	0	0
$708$	0	0
$709$	7.64008	0.286929	0.143465	−	0.989655i	$-0.454176\pi$
$709$	7.64008	0.286929	0.143465	+	0.989655i	$0.454176\pi$
$710$	20.8889	0.783947
$711$	0	0
$712$	−72.5595	−2.71928
$713$	−0.636194	−0.0238257
$714$	0	0
$715$	−11.7089	−0.437890
$716$	46.0115	1.71953
$717$	0	0
$718$	−62.8329	−2.34490
$719$	−30.0364	−1.12017	−0.560084	−	0.828436i	$-0.689231\pi$
$719$	−30.0364	−1.12017	−0.560084	+	0.828436i	$0.689231\pi$
$720$	0	0
$721$	0	0
$722$	−6.30972	−0.234824
$723$	0	0
$724$	−88.7395	−3.29798
$725$	−4.24844	−0.157783
$726$	0	0
$727$	−3.45623	−0.128185	−0.0640923	−	0.997944i	$-0.520415\pi$
$727$	−3.45623	−0.128185	−0.0640923	+	0.997944i	$0.520415\pi$
$728$	0	0
$729$	0	0
$730$	−9.61944	−0.356032
$731$	9.87120	0.365099
$732$	0	0
$733$	−38.5261	−1.42299	−0.711496	−	0.702690i	$-0.751982\pi$
$733$	−38.5261	−1.42299	−0.711496	+	0.702690i	$0.751982\pi$
$734$	−67.5418	−2.49301
$735$	0	0
$736$	0.147469	0.00543578
$737$	71.4078	2.63034
$738$	0	0
$739$	45.1239	1.65991	0.829955	−	0.557830i	$-0.188366\pi$
$739$	45.1239	1.65991	0.829955	+	0.557830i	$0.188366\pi$
$740$	−18.7237	−0.688298
$741$	0	0
$742$	0	0
$743$	−9.48676	−0.348035	−0.174018	−	0.984743i	$-0.555675\pi$
$743$	−9.48676	−0.348035	−0.174018	+	0.984743i	$0.555675\pi$
$744$	0	0
$745$	−35.1268	−1.28695
$746$	40.1737	1.47086
$747$	0	0
$748$	17.3126	0.633013
$749$	0	0
$750$	0	0
$751$	−9.83190	−0.358771	−0.179386	−	0.983779i	$-0.557411\pi$
$751$	−9.83190	−0.358771	−0.179386	+	0.983779i	$0.557411\pi$
$752$	−52.6313	−1.91927
$753$	0	0
$754$	6.05408	0.220477
$755$	−25.7496	−0.937124
$756$	0	0
$757$	−41.8171	−1.51987	−0.759934	−	0.650000i	$-0.774769\pi$
$757$	−41.8171	−1.51987	−0.759934	+	0.650000i	$0.774769\pi$
$758$	29.6156	1.07569
$759$	0	0
$760$	−53.1416	−1.92765
$761$	22.9794	0.833001	0.416501	−	0.909135i	$-0.363256\pi$
$761$	22.9794	0.833001	0.416501	+	0.909135i	$0.363256\pi$
$762$	0	0
$763$	0	0
$764$	2.84494	0.102926
$765$	0	0
$766$	−30.5979	−1.10555
$767$	2.72665	0.0984538
$768$	0	0
$769$	−6.08658	−0.219488	−0.109744	−	0.993960i	$-0.535003\pi$
$769$	−6.08658	−0.219488	−0.109744	+	0.993960i	$0.535003\pi$
$770$	0	0
$771$	0	0
$772$	49.2350	1.77201
$773$	−41.8214	−1.50421	−0.752105	−	0.659043i	$-0.770962\pi$
$773$	−41.8214	−1.50421	−0.752105	+	0.659043i	$0.770962\pi$
$774$	0	0
$775$	−4.01867	−0.144355
$776$	58.0698	2.08459
$777$	0	0
$778$	−50.6883	−1.81726
$779$	−25.9722	−0.930550
$780$	0	0
$781$	14.7778	0.528792
$782$	−0.636194	−0.0227503
$783$	0	0
$784$	0	0
$785$	15.7017	0.560419
$786$	0	0
$787$	−32.2920	−1.15109	−0.575543	−	0.817772i	$-0.695209\pi$
$787$	−32.2920	−1.15109	−0.575543	+	0.817772i	$0.695209\pi$
$788$	66.5303	2.37004
$789$	0	0
$790$	−93.8653	−3.33958
$791$	0	0
$792$	0	0
$793$	−2.27335	−0.0807289
$794$	58.1593	2.06400
$795$	0	0
$796$	−92.0521	−3.26270
$797$	46.5657	1.64944	0.824722	−	0.565539i	$-0.191332\pi$
$797$	46.5657	1.64944	0.824722	+	0.565539i	$0.191332\pi$
$798$	0	0
$799$	11.5045	0.406999
$800$	0.931521	0.0329343
$801$	0	0
$802$	6.30972	0.222804
$803$	−6.80525	−0.240152
$804$	0	0
$805$	0	0
$806$	5.72665	0.201713
$807$	0	0
$808$	−18.5979	−0.654270
$809$	10.8023	0.379790	0.189895	−	0.981804i	$-0.439185\pi$
$809$	10.8023	0.379790	0.189895	+	0.981804i	$0.439185\pi$
$810$	0	0
$811$	−5.58307	−0.196048	−0.0980240	−	0.995184i	$-0.531252\pi$
$811$	−5.58307	−0.196048	−0.0980240	+	0.995184i	$0.531252\pi$
$812$	0	0
$813$	0	0
$814$	−19.7807	−0.693315
$815$	−46.2101	−1.61867
$816$	0	0
$817$	−42.3068	−1.48013
$818$	84.4556	2.95292
$819$	0	0
$820$	67.3609	2.35234
$821$	31.7879	1.10941	0.554703	−	0.832048i	$-0.312832\pi$
$821$	31.7879	1.10941	0.554703	+	0.832048i	$0.312832\pi$
$822$	0	0
$823$	−36.0000	−1.25488	−0.627441	−	0.778664i	$-0.715897\pi$
$823$	−36.0000	−1.25488	−0.627441	+	0.778664i	$0.715897\pi$
$824$	49.1593	1.71255
$825$	0	0
$826$	0	0
$827$	−15.9224	−0.553675	−0.276837	−	0.960917i	$-0.589286\pi$
$827$	−15.9224	−0.553675	−0.276837	+	0.960917i	$0.589286\pi$
$828$	0	0
$829$	−35.4720	−1.23199	−0.615996	−	0.787749i	$-0.711246\pi$
$829$	−35.4720	−1.23199	−0.615996	+	0.787749i	$0.711246\pi$
$830$	6.03638	0.209526
$831$	0	0
$832$	7.32743	0.254033
$833$	0	0
$834$	0	0
$835$	21.9617	0.760014
$836$	−74.2000	−2.56626
$837$	0	0
$838$	9.98229	0.344833
$839$	−54.6782	−1.88770	−0.943850	−	0.330373i	$-0.892825\pi$
$839$	−54.6782	−1.88770	−0.943850	+	0.330373i	$0.892825\pi$
$840$	0	0
$841$	−22.9459	−0.791238
$842$	−51.8401	−1.78653
$843$	0	0
$844$	18.4926	0.636542
$845$	31.1230	1.07066
$846$	0	0
$847$	0	0
$848$	27.1475	0.932248
$849$	0	0
$850$	−4.01867	−0.137839
$851$	0.486758	0.0166858
$852$	0	0
$853$	2.19767	0.0752468	0.0376234	−	0.999292i	$-0.488021\pi$
$853$	2.19767	0.0752468	0.0376234	+	0.999292i	$0.488021\pi$
$854$	0	0
$855$	0	0
$856$	6.94592	0.237407
$857$	−15.7765	−0.538914	−0.269457	−	0.963012i	$-0.586844\pi$
$857$	−15.7765	−0.538914	−0.269457	+	0.963012i	$0.586844\pi$
$858$	0	0
$859$	−5.57626	−0.190260	−0.0951298	−	0.995465i	$-0.530327\pi$
$859$	−5.57626	−0.190260	−0.0951298	+	0.995465i	$0.530327\pi$
$860$	109.726	3.74163
$861$	0	0
$862$	−55.6529	−1.89555
$863$	−23.1268	−0.787247	−0.393623	−	0.919272i	$-0.628779\pi$
$863$	−23.1268	−0.787247	−0.393623	+	0.919272i	$0.628779\pi$
$864$	0	0
$865$	−45.0157	−1.53058
$866$	−5.94923	−0.202163
$867$	0	0
$868$	0	0
$869$	−66.4048	−2.25263
$870$	0	0
$871$	15.8171	0.535942
$872$	−17.1800	−0.581787
$873$	0	0
$874$	2.72665	0.0922304
$875$	0	0
$876$	0	0
$877$	3.92528	0.132547	0.0662737	−	0.997801i	$-0.478889\pi$
$877$	3.92528	0.132547	0.0662737	+	0.997801i	$0.478889\pi$
$878$	57.7965	1.95054
$879$	0	0
$880$	50.6696	1.70807
$881$	27.1986	0.916345	0.458173	−	0.888863i	$-0.348504\pi$
$881$	27.1986	0.916345	0.458173	+	0.888863i	$0.348504\pi$
$882$	0	0
$883$	8.21341	0.276403	0.138202	−	0.990404i	$-0.455868\pi$
$883$	8.21341	0.276403	0.138202	+	0.990404i	$0.455868\pi$
$884$	3.83482	0.128979
$885$	0	0
$886$	33.0148	1.10915
$887$	−6.48114	−0.217615	−0.108808	−	0.994063i	$-0.534703\pi$
$887$	−6.48114	−0.217615	−0.108808	+	0.994063i	$0.534703\pi$
$888$	0	0
$889$	0	0
$890$	91.6169	3.07101
$891$	0	0
$892$	−54.0187	−1.80868
$893$	−49.3068	−1.64999
$894$	0	0
$895$	−29.4356	−0.983924
$896$	0	0
$897$	0	0
$898$	22.5438	0.752295
$899$	5.72665	0.190995
$900$	0	0
$901$	−5.93406	−0.197692
$902$	71.1636	2.36949
$903$	0	0
$904$	−52.5054	−1.74630
$905$	56.7706	1.88712
$906$	0	0
$907$	10.1288	0.336321	0.168161	−	0.985760i	$-0.446217\pi$
$907$	10.1288	0.336321	0.168161	+	0.985760i	$0.446217\pi$
$908$	5.60078	0.185868
$909$	0	0
$910$	0	0
$911$	−45.9224	−1.52148	−0.760738	−	0.649059i	$-0.775163\pi$
$911$	−45.9224	−1.52148	−0.760738	+	0.649059i	$0.775163\pi$
$912$	0	0
$913$	4.27042	0.141330
$914$	21.6946	0.717592
$915$	0	0
$916$	72.8899	2.40835
$917$	0	0
$918$	0	0
$919$	−4.92432	−0.162438	−0.0812192	−	0.996696i	$-0.525881\pi$
$919$	−4.92432	−0.162438	−0.0812192	+	0.996696i	$0.525881\pi$
$920$	−3.58307	−0.118130
$921$	0	0
$922$	−13.9243	−0.458573
$923$	3.27335	0.107744
$924$	0	0
$925$	3.07472	0.101096
$926$	38.6955	1.27161
$927$	0	0
$928$	−1.32743	−0.0435750
$929$	0.00758649	0.000248905 0	0.000124452	−	1.00000i	$-0.499960\pi$
$929$	0.00758649	0.000248905 0	0.000124452		1.00000i	$0.499960\pi$
$930$	0	0
$931$	0	0
$932$	76.9558	2.52077
$933$	0	0
$934$	−54.1239	−1.77099
$935$	−11.0757	−0.362213
$936$	0	0
$937$	−21.1623	−0.691341	−0.345670	−	0.938356i	$-0.612348\pi$
$937$	−21.1623	−0.691341	−0.345670	+	0.938356i	$0.612348\pi$
$938$	0	0
$939$	0	0
$940$	127.881	4.17102
$941$	4.55816	0.148592	0.0742959	−	0.997236i	$-0.476329\pi$
$941$	4.55816	0.148592	0.0742959	+	0.997236i	$0.476329\pi$
$942$	0	0
$943$	−1.75117	−0.0570260
$944$	−11.7994	−0.384038
$945$	0	0
$946$	115.920	3.76890
$947$	−13.7352	−0.446334	−0.223167	−	0.974780i	$-0.571640\pi$
$947$	−13.7352	−0.446334	−0.223167	+	0.974780i	$0.571640\pi$
$948$	0	0
$949$	−1.50739	−0.0489320
$950$	17.2235	0.558805
$951$	0	0
$952$	0	0
$953$	−8.80699	−0.285286	−0.142643	−	0.989774i	$-0.545560\pi$
$953$	−8.80699	−0.285286	−0.142643	+	0.989774i	$0.545560\pi$
$954$	0	0
$955$	−1.82004	−0.0588951
$956$	−19.8319	−0.641409
$957$	0	0
$958$	61.4513	1.98540
$959$	0	0
$960$	0	0
$961$	−25.5831	−0.825260
$962$	−4.38151	−0.141266
$963$	0	0
$964$	106.052	3.41571
$965$	−31.4978	−1.01395
$966$	0	0
$967$	38.3284	1.23256	0.616279	−	0.787528i	$-0.288639\pi$
$967$	38.3284	1.23256	0.616279	+	0.787528i	$0.288639\pi$
$968$	47.4150	1.52397
$969$	0	0
$970$	−73.3216	−2.35421
$971$	−31.0187	−0.995436	−0.497718	−	0.867339i	$-0.665829\pi$
$971$	−31.0187	−0.995436	−0.497718	+	0.867339i	$0.665829\pi$
$972$	0	0
$973$	0	0
$974$	−43.2996	−1.38741
$975$	0	0
$976$	9.83775	0.314899
$977$	52.7424	1.68738	0.843689	−	0.536832i	$-0.180379\pi$
$977$	52.7424	1.68738	0.843689	+	0.536832i	$0.180379\pi$
$978$	0	0
$979$	64.8142	2.07147
$980$	0	0
$981$	0	0
$982$	−33.9430	−1.08316
$983$	−18.3029	−0.583772	−0.291886	−	0.956453i	$-0.594283\pi$
$983$	−18.3029	−0.583772	−0.291886	+	0.956453i	$0.594283\pi$
$984$	0	0
$985$	−42.5624	−1.35615
$986$	5.72665	0.182374
$987$	0	0
$988$	−16.4356	−0.522886
$989$	−2.85253	−0.0907052
$990$	0	0
$991$	−12.6008	−0.400277	−0.200138	−	0.979768i	$-0.564139\pi$
$991$	−12.6008	−0.400277	−0.200138	+	0.979768i	$0.564139\pi$
$992$	−1.25564	−0.0398665
$993$	0	0
$994$	0	0
$995$	58.8899	1.86693
$996$	0	0
$997$	−11.7424	−0.371885	−0.185943	−	0.982561i	$-0.559534\pi$
$997$	−11.7424	−0.371885	−0.185943	+	0.982561i	$0.559534\pi$
$998$	32.2019	1.01933
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.p.1.3		3
3.2	odd	2		3969.2.a.m.1.1		3
7.6	odd	2		567.2.a.g.1.3		3
9.2	odd	6		441.2.f.d.148.3		6
9.4	even	3		1323.2.f.c.883.1		6
9.5	odd	6		441.2.f.d.295.3		6
9.7	even	3		1323.2.f.c.442.1		6
21.20	even	2		567.2.a.d.1.1		3
28.27	even	2		9072.2.a.cd.1.2		3
63.2	odd	6		441.2.g.d.67.3		6
63.4	even	3		1323.2.g.b.667.1		6
63.5	even	6		441.2.h.c.214.1		6
63.11	odd	6		441.2.h.b.373.1		6
63.13	odd	6		189.2.f.a.127.1		6
63.16	even	3		1323.2.g.b.361.1		6
63.20	even	6		63.2.f.b.22.3	✓	6
63.23	odd	6		441.2.h.b.214.1		6
63.25	even	3		1323.2.h.e.226.3		6
63.31	odd	6		1323.2.g.c.667.1		6
63.32	odd	6		441.2.g.d.79.3		6
63.34	odd	6		189.2.f.a.64.1		6
63.38	even	6		441.2.h.c.373.1		6
63.40	odd	6		1323.2.h.d.802.3		6
63.41	even	6		63.2.f.b.43.3	yes	6
63.47	even	6		441.2.g.e.67.3		6
63.52	odd	6		1323.2.h.d.226.3		6
63.58	even	3		1323.2.h.e.802.3		6
63.59	even	6		441.2.g.e.79.3		6
63.61	odd	6		1323.2.g.c.361.1		6
84.83	odd	2		9072.2.a.bq.1.2		3
252.83	odd	6		1008.2.r.k.337.3		6
252.139	even	6		3024.2.r.g.2017.2		6
252.167	odd	6		1008.2.r.k.673.3		6
252.223	even	6		3024.2.r.g.1009.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.f.b.22.3	✓	6	63.20	even	6
63.2.f.b.43.3	yes	6	63.41	even	6
189.2.f.a.64.1		6	63.34	odd	6
189.2.f.a.127.1		6	63.13	odd	6
441.2.f.d.148.3		6	9.2	odd	6
441.2.f.d.295.3		6	9.5	odd	6
441.2.g.d.67.3		6	63.2	odd	6
441.2.g.d.79.3		6	63.32	odd	6
441.2.g.e.67.3		6	63.47	even	6
441.2.g.e.79.3		6	63.59	even	6
441.2.h.b.214.1		6	63.23	odd	6
441.2.h.b.373.1		6	63.11	odd	6
441.2.h.c.214.1		6	63.5	even	6
441.2.h.c.373.1		6	63.38	even	6
567.2.a.d.1.1		3	21.20	even	2
567.2.a.g.1.3		3	7.6	odd	2
1008.2.r.k.337.3		6	252.83	odd	6
1008.2.r.k.673.3		6	252.167	odd	6
1323.2.f.c.442.1		6	9.7	even	3
1323.2.f.c.883.1		6	9.4	even	3
1323.2.g.b.361.1		6	63.16	even	3
1323.2.g.b.667.1		6	63.4	even	3
1323.2.g.c.361.1		6	63.61	odd	6
1323.2.g.c.667.1		6	63.31	odd	6
1323.2.h.d.226.3		6	63.52	odd	6
1323.2.h.d.802.3		6	63.40	odd	6
1323.2.h.e.226.3		6	63.25	even	3
1323.2.h.e.802.3		6	63.58	even	3
3024.2.r.g.1009.2		6	252.223	even	6
3024.2.r.g.2017.2		6	252.139	even	6
3969.2.a.m.1.1		3	3.2	odd	2
3969.2.a.p.1.3		3	1.1	even	1	trivial
9072.2.a.bq.1.2		3	84.83	odd	2
9072.2.a.cd.1.2		3	28.27	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)

\(f(q)\)	\(=\)	\(q+2.46050 q^{2} +4.05408 q^{4} -2.59358 q^{5} +5.05408 q^{8} +O(q^{10})\)	\(q+2.46050 q^{2} +4.05408 q^{4} -2.59358 q^{5} +5.05408 q^{8} -6.38151 q^{10} -4.51459 q^{11} -1.00000 q^{13} +4.32743 q^{16} -0.945916 q^{17} +4.05408 q^{19} -10.5146 q^{20} -11.1082 q^{22} +0.273346 q^{23} +1.72665 q^{25} -2.46050 q^{26} -2.46050 q^{29} -2.32743 q^{31} +0.539495 q^{32} -2.32743 q^{34} +1.78074 q^{37} +9.97509 q^{38} -13.1082 q^{40} -6.40642 q^{41} -10.4356 q^{43} -18.3025 q^{44} +0.672570 q^{46} -12.1623 q^{47} +4.24844 q^{50} -4.05408 q^{52} +6.27335 q^{53} +11.7089 q^{55} -6.05408 q^{58} -2.72665 q^{59} +2.27335 q^{61} -5.72665 q^{62} -7.32743 q^{64} +2.59358 q^{65} -15.8171 q^{67} -3.83482 q^{68} -3.27335 q^{71} +1.50739 q^{73} +4.38151 q^{74} +16.4356 q^{76} +14.7089 q^{79} -11.2235 q^{80} -15.7630 q^{82} -0.945916 q^{83} +2.45331 q^{85} -25.6768 q^{86} -22.8171 q^{88} -14.3566 q^{89} +1.10817 q^{92} -29.9253 q^{94} -10.5146 q^{95} +11.4897 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 6 q^{8}+O(q^{10}) \) 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 6 * q^8	\( 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 6 q^{8} + 2 q^{11} - 3 q^{13} + 3 q^{16} - 12 q^{17} + 3 q^{19} - 16 q^{20} - 15 q^{22} + 6 q^{25} - q^{26} - q^{29} + 3 q^{31} + 8 q^{32} + 3 q^{34} - 3 q^{37} + 8 q^{38} - 21 q^{40} - 22 q^{41} - 3 q^{43} - 23 q^{44} + 12 q^{46} - 9 q^{47} - 10 q^{50} - 3 q^{52} + 18 q^{53} + 6 q^{55} - 9 q^{58} - 9 q^{59} + 6 q^{61} - 18 q^{62} - 12 q^{64} + 5 q^{65} + 6 q^{68} - 9 q^{71} - 3 q^{73} - 6 q^{74} + 21 q^{76} + 15 q^{79} + 11 q^{80} - 9 q^{82} - 12 q^{83} + 9 q^{85} - 34 q^{86} - 21 q^{88} - 2 q^{89} - 15 q^{92} - 24 q^{94} - 16 q^{95} - 3 q^{97}+O(q^{100}) \) 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 6 * q^8 + 2 * q^11 - 3 * q^13 + 3 * q^16 - 12 * q^17 + 3 * q^19 - 16 * q^20 - 15 * q^22 + 6 * q^25 - q^26 - q^29 + 3 * q^31 + 8 * q^32 + 3 * q^34 - 3 * q^37 + 8 * q^38 - 21 * q^40 - 22 * q^41 - 3 * q^43 - 23 * q^44 + 12 * q^46 - 9 * q^47 - 10 * q^50 - 3 * q^52 + 18 * q^53 + 6 * q^55 - 9 * q^58 - 9 * q^59 + 6 * q^61 - 18 * q^62 - 12 * q^64 + 5 * q^65 + 6 * q^68 - 9 * q^71 - 3 * q^73 - 6 * q^74 + 21 * q^76 + 15 * q^79 + 11 * q^80 - 9 * q^82 - 12 * q^83 + 9 * q^85 - 34 * q^86 - 21 * q^88 - 2 * q^89 - 15 * q^92 - 24 * q^94 - 16 * q^95 - 3 * q^97

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