LMFDB - Embedded newform 3969.2.a.m.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:	$3$
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			$-1.69963$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.69963 q^{2} +0.888736 q^{4} +3.58836 q^{5} -1.88874 q^{8} +O(q^{10})$	$q+1.69963 q^{2} +0.888736 q^{4} +3.58836 q^{5} -1.88874 q^{8} +6.09888 q^{10} -2.81089 q^{11} -1.00000 q^{13} -4.98762 q^{16} +4.11126 q^{17} +0.888736 q^{19} +3.18911 q^{20} -4.77747 q^{22} +5.87636 q^{23} +7.87636 q^{25} -1.69963 q^{26} -1.69963 q^{29} +6.98762 q^{31} -4.69963 q^{32} +6.98762 q^{34} +4.76509 q^{37} +1.51052 q^{38} -6.77747 q^{40} +5.41164 q^{41} +5.21015 q^{43} -2.49814 q^{44} +9.98762 q^{46} +2.66621 q^{47} +13.3869 q^{50} -0.888736 q^{52} -0.123644 q^{53} -10.0865 q^{55} -2.88874 q^{58} +8.87636 q^{59} -3.87636 q^{61} +11.8764 q^{62} +1.98762 q^{64} -3.58836 q^{65} +12.3090 q^{67} +3.65383 q^{68} -2.87636 q^{71} +10.6414 q^{73} +8.09888 q^{74} +0.789851 q^{76} -7.08650 q^{79} -17.8974 q^{80} +9.19777 q^{82} +4.11126 q^{83} +14.7527 q^{85} +8.85532 q^{86} +5.30903 q^{88} -9.60940 q^{89} +5.22253 q^{92} +4.53156 q^{94} +3.18911 q^{95} -7.32141 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.69963	1.20182	0.600909	−	0.799317i	$-0.294805\pi$
$2$	1.69963	1.20182	0.600909	+	0.799317i	$0.294805\pi$
$3$	0	0
$3$	0	0
$4$	0.888736	0.444368
$5$	3.58836	1.60477	0.802383	−	0.596810i	$-0.203565\pi$
$5$	3.58836	1.60477	0.802383	+	0.596810i	$0.203565\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.88874	−0.667769
$9$	0	0
$10$	6.09888	1.92864
$11$	−2.81089	−0.847516	−0.423758	−	0.905775i	$-0.639289\pi$
$11$	−2.81089	−0.847516	−0.423758	+	0.905775i	$0.639289\pi$
$12$	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
$16$	−4.98762	−1.24691
$17$	4.11126	0.997128	0.498564	−	0.866853i	$-0.333861\pi$
$17$	4.11126	0.997128	0.498564	+	0.866853i	$0.333861\pi$
$18$	0	0
$19$	0.888736	0.203890	0.101945	−	0.994790i	$-0.467493\pi$
$19$	0.888736	0.203890	0.101945	+	0.994790i	$0.467493\pi$
$20$	3.18911	0.713106
$21$	0	0
$22$	−4.77747	−1.01856
$23$	5.87636	1.22530	0.612652	−	0.790352i	$-0.290103\pi$
$23$	5.87636	1.22530	0.612652	+	0.790352i	$0.290103\pi$
$24$	0	0
$25$	7.87636	1.57527
$26$	−1.69963	−0.333325
$27$	0	0
$28$	0	0
$29$	−1.69963	−0.315613	−0.157807	−	0.987470i	$-0.550442\pi$
$29$	−1.69963	−0.315613	−0.157807	+	0.987470i	$0.550442\pi$
$30$	0	0
$31$	6.98762	1.25501	0.627507	−	0.778611i	$-0.284075\pi$
$31$	6.98762	1.25501	0.627507	+	0.778611i	$0.284075\pi$
$32$	−4.69963	−0.830785
$33$	0	0
$34$	6.98762	1.19837
$35$	0	0
$36$	0	0
$37$	4.76509	0.783376	0.391688	−	0.920098i	$-0.371891\pi$
$37$	4.76509	0.783376	0.391688	+	0.920098i	$0.371891\pi$
$38$	1.51052	0.245039
$39$	0	0
$40$	−6.77747	−1.07161
$41$	5.41164	0.845156	0.422578	−	0.906327i	$-0.361125\pi$
$41$	5.41164	0.845156	0.422578	+	0.906327i	$0.361125\pi$
$42$	0	0
$43$	5.21015	0.794540	0.397270	−	0.917702i	$-0.369958\pi$
$43$	5.21015	0.794540	0.397270	+	0.917702i	$0.369958\pi$
$44$	−2.49814	−0.376609
$45$	0	0
$46$	9.98762	1.47259
$47$	2.66621	0.388906	0.194453	−	0.980912i	$-0.437707\pi$
$47$	2.66621	0.388906	0.194453	+	0.980912i	$0.437707\pi$
$48$	0	0
$49$	0	0
$50$	13.3869	1.89319
$51$	0	0
$52$	−0.888736	−0.123245
$53$	−0.123644	−0.0169838	−0.00849190	−	0.999964i	$-0.502703\pi$
$53$	−0.123644	−0.0169838	−0.00849190	+	0.999964i	$0.502703\pi$
$54$	0	0
$55$	−10.0865	−1.36006
$56$	0	0
$57$	0	0
$58$	−2.88874	−0.379310
$59$	8.87636	1.15560	0.577802	−	0.816177i	$-0.303911\pi$
$59$	8.87636	1.15560	0.577802	+	0.816177i	$0.303911\pi$
$60$	0	0
$61$	−3.87636	−0.496317	−0.248158	−	0.968720i	$-0.579825\pi$
$61$	−3.87636	−0.496317	−0.248158	+	0.968720i	$0.579825\pi$
$62$	11.8764	1.50830
$63$	0	0
$64$	1.98762	0.248453
$65$	−3.58836	−0.445082
$66$	0	0
$67$	12.3090	1.50379	0.751894	−	0.659284i	$-0.229141\pi$
$67$	12.3090	1.50379	0.751894	+	0.659284i	$0.229141\pi$
$68$	3.65383	0.443092
$69$	0	0
$70$	0	0
$71$	−2.87636	−0.341361	−0.170680	−	0.985326i	$-0.554597\pi$
$71$	−2.87636	−0.341361	−0.170680	+	0.985326i	$0.554597\pi$
$72$	0	0
$73$	10.6414	1.24549	0.622744	−	0.782426i	$-0.286018\pi$
$73$	10.6414	1.24549	0.622744	+	0.782426i	$0.286018\pi$
$74$	8.09888	0.941476
$75$	0	0
$76$	0.789851	0.0906022
$77$	0	0
$78$	0	0
$79$	−7.08650	−0.797294	−0.398647	−	0.917104i	$-0.630520\pi$
$79$	−7.08650	−0.797294	−0.398647	+	0.917104i	$0.630520\pi$
$80$	−17.8974	−2.00099
$81$	0	0
$82$	9.19777	1.01572
$83$	4.11126	0.451270	0.225635	−	0.974212i	$-0.427554\pi$
$83$	4.11126	0.451270	0.225635	+	0.974212i	$0.427554\pi$
$84$	0	0
$85$	14.7527	1.60016
$86$	8.85532	0.954893
$87$	0	0
$88$	5.30903	0.565945
$89$	−9.60940	−1.01859	−0.509297	−	0.860591i	$-0.670095\pi$
$89$	−9.60940	−1.01859	−0.509297	+	0.860591i	$0.670095\pi$
$90$	0	0
$91$	0	0
$92$	5.22253	0.544486
$93$	0	0
$94$	4.53156	0.467395
$95$	3.18911	0.327196
$96$	0	0
$97$	−7.32141	−0.743377	−0.371688	−	0.928358i	$-0.621221\pi$
$97$	−7.32141	−0.743377	−0.371688	+	0.928358i	$0.621221\pi$
$98$	0	0
$99$	0	0
$100$	7.00000	0.700000
$101$	−3.46472	−0.344753	−0.172376	−	0.985031i	$-0.555144\pi$
$101$	−3.46472	−0.344753	−0.172376	+	0.985031i	$0.555144\pi$
$102$	0	0
$103$	15.8764	1.56434	0.782172	−	0.623063i	$-0.214112\pi$
$103$	15.8764	1.56434	0.782172	+	0.623063i	$0.214112\pi$
$104$	1.88874	0.185206
$105$	0	0
$106$	−0.210149	−0.0204114
$107$	−5.35346	−0.517538	−0.258769	−	0.965939i	$-0.583317\pi$
$107$	−5.35346	−0.517538	−0.258769	+	0.965939i	$0.583317\pi$
$108$	0	0
$109$	−18.8640	−1.80684	−0.903421	−	0.428755i	$-0.858952\pi$
$109$	−18.8640	−1.80684	−0.903421	+	0.428755i	$0.858952\pi$
$110$	−17.1433	−1.63455
$111$	0	0
$112$	0	0
$113$	−18.5512	−1.74515	−0.872576	−	0.488478i	$-0.837552\pi$
$113$	−18.5512	−1.74515	−0.872576	+	0.488478i	$0.837552\pi$
$114$	0	0
$115$	21.0865	1.96633
$116$	−1.51052	−0.140248
$117$	0	0
$118$	15.0865	1.38883
$119$	0	0
$120$	0	0
$121$	−3.09888	−0.281717
$122$	−6.58836	−0.596482
$123$	0	0
$124$	6.21015	0.557688
$125$	10.3214	0.923175
$126$	0	0
$127$	9.98762	0.886258	0.443129	−	0.896458i	$-0.353868\pi$
$127$	9.98762	0.886258	0.443129	+	0.896458i	$0.353868\pi$
$128$	12.7775	1.12938
$129$	0	0
$130$	−6.09888	−0.534908
$131$	−16.0531	−1.40256	−0.701282	−	0.712884i	$-0.747389\pi$
$131$	−16.0531	−1.40256	−0.701282	+	0.712884i	$0.747389\pi$
$132$	0	0
$133$	0	0
$134$	20.9208	1.80728
$135$	0	0
$136$	−7.76509	−0.665851
$137$	−12.9876	−1.10961	−0.554804	−	0.831981i	$-0.687207\pi$
$137$	−12.9876	−1.10961	−0.554804	+	0.831981i	$0.687207\pi$
$138$	0	0
$139$	−1.11126	−0.0942562	−0.0471281	−	0.998889i	$-0.515007\pi$
$139$	−1.11126	−0.0942562	−0.0471281	+	0.998889i	$0.515007\pi$
$140$	0	0
$141$	0	0
$142$	−4.88874	−0.410254
$143$	2.81089	0.235059
$144$	0	0
$145$	−6.09888	−0.506485
$146$	18.0865	1.49685
$147$	0	0
$148$	4.23491	0.348107
$149$	8.43268	0.690832	0.345416	−	0.938450i	$-0.387738\pi$
$149$	8.43268	0.690832	0.345416	+	0.938450i	$0.387738\pi$
$150$	0	0
$151$	−14.8516	−1.20861	−0.604303	−	0.796755i	$-0.706548\pi$
$151$	−14.8516	−1.20861	−0.604303	+	0.796755i	$0.706548\pi$
$152$	−1.67859	−0.136151
$153$	0	0
$154$	0	0
$155$	25.0741	2.01400
$156$	0	0
$157$	−2.88874	−0.230546	−0.115273	−	0.993334i	$-0.536774\pi$
$157$	−2.88874	−0.230546	−0.115273	+	0.993334i	$0.536774\pi$
$158$	−12.0444	−0.958203
$159$	0	0
$160$	−16.8640	−1.33321
$161$	0	0
$162$	0	0
$163$	−10.3090	−0.807466	−0.403733	−	0.914877i	$-0.632287\pi$
$163$	−10.3090	−0.807466	−0.403733	+	0.914877i	$0.632287\pi$
$164$	4.80951	0.375560
$165$	0	0
$166$	6.98762	0.542345
$167$	−12.1520	−0.940348	−0.470174	−	0.882574i	$-0.655809\pi$
$167$	−12.1520	−0.940348	−0.470174	+	0.882574i	$0.655809\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	25.0741	1.92310
$171$	0	0
$172$	4.63045	0.353068
$173$	6.60940	0.502504	0.251252	−	0.967922i	$-0.419158\pi$
$173$	6.60940	0.502504	0.251252	+	0.967922i	$0.419158\pi$
$174$	0	0
$175$	0	0
$176$	14.0197	1.05677
$177$	0	0
$178$	−16.3324	−1.22417
$179$	−3.84294	−0.287234	−0.143617	−	0.989633i	$-0.545873\pi$
$179$	−3.84294	−0.287234	−0.143617	+	0.989633i	$0.545873\pi$
$180$	0	0
$181$	−18.5426	−1.37826	−0.689129	−	0.724639i	$-0.742007\pi$
$181$	−18.5426	−1.37826	−0.689129	+	0.724639i	$0.742007\pi$
$182$	0	0
$183$	0	0
$184$	−11.0989	−0.818221
$185$	17.0989	1.25713
$186$	0	0
$187$	−11.5563	−0.845082
$188$	2.36955	0.172818
$189$	0	0
$190$	5.42030	0.393230
$191$	4.63416	0.335316	0.167658	−	0.985845i	$-0.446380\pi$
$191$	4.63416	0.335316	0.167658	+	0.985845i	$0.446380\pi$
$192$	0	0
$193$	−25.2967	−1.82089	−0.910446	−	0.413627i	$-0.864262\pi$
$193$	−25.2967	−1.82089	−0.910446	+	0.413627i	$0.864262\pi$
$194$	−12.4437	−0.893404
$195$	0	0
$196$	0	0
$197$	10.7207	0.763816	0.381908	−	0.924200i	$-0.375267\pi$
$197$	10.7207	0.763816	0.381908	+	0.924200i	$0.375267\pi$
$198$	0	0
$199$	8.76647	0.621439	0.310719	−	0.950502i	$-0.399430\pi$
$199$	8.76647	0.621439	0.310719	+	0.950502i	$0.399430\pi$
$200$	−14.8764	−1.05192
$201$	0	0
$202$	−5.88874	−0.414330
$203$	0	0
$204$	0	0
$205$	19.4189	1.35628
$206$	26.9839	1.88006
$207$	0	0
$208$	4.98762	0.345829
$209$	−2.49814	−0.172800
$210$	0	0
$211$	10.5302	0.724928	0.362464	−	0.931998i	$-0.381936\pi$
$211$	10.5302	0.724928	0.362464	+	0.931998i	$0.381936\pi$
$212$	−0.109887	−0.00754705
$213$	0	0
$214$	−9.09888	−0.621987
$215$	18.6959	1.27505
$216$	0	0
$217$	0	0
$218$	−32.0617	−2.17150
$219$	0	0
$220$	−8.96424	−0.604369
$221$	−4.11126	−0.276554
$222$	0	0
$223$	5.66758	0.379530	0.189765	−	0.981830i	$-0.439227\pi$
$223$	5.66758	0.379530	0.189765	+	0.981830i	$0.439227\pi$
$224$	0	0
$225$	0	0
$226$	−31.5302	−2.09736
$227$	11.0989	0.736659	0.368329	−	0.929695i	$-0.379930\pi$
$227$	11.0989	0.736659	0.368329	+	0.929695i	$0.379930\pi$
$228$	0	0
$229$	−19.6428	−1.29803	−0.649017	−	0.760774i	$-0.724820\pi$
$229$	−19.6428	−1.29803	−0.649017	+	0.760774i	$0.724820\pi$
$230$	35.8392	2.36317
$231$	0	0
$232$	3.21015	0.210757
$233$	8.96286	0.587177	0.293588	−	0.955932i	$-0.405151\pi$
$233$	8.96286	0.587177	0.293588	+	0.955932i	$0.405151\pi$
$234$	0	0
$235$	9.56732	0.624103
$236$	7.88874	0.513513
$237$	0	0
$238$	0	0
$239$	11.2225	0.725925	0.362963	−	0.931804i	$-0.381765\pi$
$239$	11.2225	0.725925	0.362963	+	0.931804i	$0.381765\pi$
$240$	0	0
$241$	6.98624	0.450023	0.225012	−	0.974356i	$-0.427758\pi$
$241$	6.98624	0.450023	0.225012	+	0.974356i	$0.427758\pi$
$242$	−5.26695	−0.338572
$243$	0	0
$244$	−3.44506	−0.220547
$245$	0	0
$246$	0	0
$247$	−0.888736	−0.0565489
$248$	−13.1978	−0.838059
$249$	0	0
$250$	17.5426	1.10949
$251$	−4.62041	−0.291638	−0.145819	−	0.989311i	$-0.546582\pi$
$251$	−4.62041	−0.291638	−0.145819	+	0.989311i	$0.546582\pi$
$252$	0	0
$253$	−16.5178	−1.03847
$254$	16.9752	1.06512
$255$	0	0
$256$	17.7417	1.10886
$257$	1.42402	0.0888277	0.0444138	−	0.999013i	$-0.485858\pi$
$257$	1.42402	0.0888277	0.0444138	+	0.999013i	$0.485858\pi$
$258$	0	0
$259$	0	0
$260$	−3.18911	−0.197780
$261$	0	0
$262$	−27.2843	−1.68563
$263$	16.2632	1.00283	0.501417	−	0.865206i	$-0.332812\pi$
$263$	16.2632	1.00283	0.501417	+	0.865206i	$0.332812\pi$
$264$	0	0
$265$	−0.443679	−0.0272550
$266$	0	0
$267$	0	0
$268$	10.9395	0.668235
$269$	18.6538	1.13734	0.568672	−	0.822564i	$-0.307457\pi$
$269$	18.6538	1.13734	0.568672	+	0.822564i	$0.307457\pi$
$270$	0	0
$271$	−3.96286	−0.240727	−0.120363	−	0.992730i	$-0.538406\pi$
$271$	−3.96286	−0.240727	−0.120363	+	0.992730i	$0.538406\pi$
$272$	−20.5054	−1.24332
$273$	0	0
$274$	−22.0741	−1.33355
$275$	−22.1396	−1.33507
$276$	0	0
$277$	−2.33379	−0.140224	−0.0701120	−	0.997539i	$-0.522336\pi$
$277$	−2.33379	−0.140224	−0.0701120	+	0.997539i	$0.522336\pi$
$278$	−1.88874	−0.113279
$279$	0	0
$280$	0	0
$281$	27.9949	1.67004	0.835018	−	0.550223i	$-0.185457\pi$
$281$	27.9949	1.67004	0.835018	+	0.550223i	$0.185457\pi$
$282$	0	0
$283$	−10.3200	−0.613462	−0.306731	−	0.951796i	$-0.599235\pi$
$283$	−10.3200	−0.613462	−0.306731	+	0.951796i	$0.599235\pi$
$284$	−2.55632	−0.151690
$285$	0	0
$286$	4.77747	0.282498
$287$	0	0
$288$	0	0
$289$	−0.0975070	−0.00573571
$290$	−10.3658	−0.608703
$291$	0	0
$292$	9.45744	0.553455
$293$	30.6959	1.79327	0.896637	−	0.442766i	$-0.146003\pi$
$293$	30.6959	1.79327	0.896637	+	0.442766i	$0.146003\pi$
$294$	0	0
$295$	31.8516	1.85447
$296$	−9.00000	−0.523114
$297$	0	0
$298$	14.3324	0.830255
$299$	−5.87636	−0.339838
$300$	0	0
$301$	0	0
$302$	−25.2422	−1.45252
$303$	0	0
$304$	−4.43268	−0.254231
$305$	−13.9098	−0.796471
$306$	0	0
$307$	11.4437	0.653125	0.326563	−	0.945176i	$-0.394110\pi$
$307$	11.4437	0.653125	0.326563	+	0.945176i	$0.394110\pi$
$308$	0	0
$309$	0	0
$310$	42.6167	2.42047
$311$	−11.9629	−0.678352	−0.339176	−	0.940723i	$-0.610148\pi$
$311$	−11.9629	−0.678352	−0.339176	+	0.940723i	$0.610148\pi$
$312$	0	0
$313$	13.5439	0.765549	0.382774	−	0.923842i	$-0.374969\pi$
$313$	13.5439	0.765549	0.382774	+	0.923842i	$0.374969\pi$
$314$	−4.90978	−0.277075
$315$	0	0
$316$	−6.29803	−0.354292
$317$	−29.9629	−1.68288	−0.841441	−	0.540349i	$-0.818292\pi$
$317$	−29.9629	−1.68288	−0.841441	+	0.540349i	$0.818292\pi$
$318$	0	0
$319$	4.77747	0.267487
$320$	7.13231	0.398708
$321$	0	0
$322$	0	0
$323$	3.65383	0.203304
$324$	0	0
$325$	−7.87636	−0.436902
$326$	−17.5215	−0.970427
$327$	0	0
$328$	−10.2212	−0.564369
$329$	0	0
$330$	0	0
$331$	2.08650	0.114685	0.0573423	−	0.998355i	$-0.481737\pi$
$331$	2.08650	0.114685	0.0573423	+	0.998355i	$0.481737\pi$
$332$	3.65383	0.200530
$333$	0	0
$334$	−20.6538	−1.13013
$335$	44.1693	2.41323
$336$	0	0
$337$	−16.2088	−0.882948	−0.441474	−	0.897274i	$-0.645544\pi$
$337$	−16.2088	−0.882948	−0.441474	+	0.897274i	$0.645544\pi$
$338$	−20.3955	−1.10937
$339$	0	0
$340$	13.1113	0.711058
$341$	−19.6414	−1.06364
$342$	0	0
$343$	0	0
$344$	−9.84059	−0.530569
$345$	0	0
$346$	11.2335	0.603918
$347$	11.2670	0.604842	0.302421	−	0.953175i	$-0.402205\pi$
$347$	11.2670	0.604842	0.302421	+	0.953175i	$0.402205\pi$
$348$	0	0
$349$	0.197769	0.0105863	0.00529316	−	0.999986i	$-0.498315\pi$
$349$	0.197769	0.0105863	0.00529316	+	0.999986i	$0.498315\pi$
$350$	0	0
$351$	0	0
$352$	13.2101	0.704103
$353$	12.5019	0.665407	0.332703	−	0.943032i	$-0.392039\pi$
$353$	12.5019	0.665407	0.332703	+	0.943032i	$0.392039\pi$
$354$	0	0
$355$	−10.3214	−0.547804
$356$	−8.54022	−0.452631
$357$	0	0
$358$	−6.53156	−0.345204
$359$	20.0197	1.05660	0.528299	−	0.849059i	$-0.322830\pi$
$359$	20.0197	1.05660	0.528299	+	0.849059i	$0.322830\pi$
$360$	0	0
$361$	−18.2101	−0.958429
$362$	−31.5155	−1.65642
$363$	0	0
$364$	0	0
$365$	38.1854	1.99871
$366$	0	0
$367$	−30.0727	−1.56978	−0.784892	−	0.619632i	$-0.787282\pi$
$367$	−30.0727	−1.56978	−0.784892	+	0.619632i	$0.787282\pi$
$368$	−29.3090	−1.52784
$369$	0	0
$370$	29.0617	1.51085
$371$	0	0
$372$	0	0
$373$	7.01238	0.363087	0.181544	−	0.983383i	$-0.441891\pi$
$373$	7.01238	0.363087	0.181544	+	0.983383i	$0.441891\pi$
$374$	−19.6414	−1.01564
$375$	0	0
$376$	−5.03576	−0.259700
$377$	1.69963	0.0875353
$378$	0	0
$379$	−19.0741	−0.979772	−0.489886	−	0.871787i	$-0.662962\pi$
$379$	−19.0741	−0.979772	−0.489886	+	0.871787i	$0.662962\pi$
$380$	2.83427	0.145395
$381$	0	0
$382$	7.87636	0.402989
$383$	−3.21015	−0.164031	−0.0820155	−	0.996631i	$-0.526136\pi$
$383$	−3.21015	−0.164031	−0.0820155	+	0.996631i	$0.526136\pi$
$384$	0	0
$385$	0	0
$386$	−42.9949	−2.18838
$387$	0	0
$388$	−6.50680	−0.330333
$389$	5.13602	0.260407	0.130203	−	0.991487i	$-0.458437\pi$
$389$	5.13602	0.260407	0.130203	+	0.991487i	$0.458437\pi$
$390$	0	0
$391$	24.1593	1.22179
$392$	0	0
$393$	0	0
$394$	18.2212	0.917968
$395$	−25.4290	−1.27947
$396$	0	0
$397$	−22.9381	−1.15123	−0.575615	−	0.817721i	$-0.695237\pi$
$397$	−22.9381	−1.15123	−0.575615	+	0.817721i	$0.695237\pi$
$398$	14.8997	0.746856
$399$	0	0
$400$	−39.2843	−1.96421
$401$	−18.2101	−0.909371	−0.454686	−	0.890652i	$-0.650248\pi$
$401$	−18.2101	−0.909371	−0.454686	+	0.890652i	$0.650248\pi$
$402$	0	0
$403$	−6.98762	−0.348078
$404$	−3.07922	−0.153197
$405$	0	0
$406$	0	0
$407$	−13.3942	−0.663924
$408$	0	0
$409$	15.3324	0.758139	0.379070	−	0.925368i	$-0.376244\pi$
$409$	15.3324	0.758139	0.379070	+	0.925368i	$0.376244\pi$
$410$	33.0049	1.63000
$411$	0	0
$412$	14.1099	0.695144
$413$	0	0
$414$	0	0
$415$	14.7527	0.724182
$416$	4.69963	0.230418
$417$	0	0
$418$	−4.24591	−0.207674
$419$	−10.5687	−0.516315	−0.258157	−	0.966103i	$-0.583115\pi$
$419$	−10.5687	−0.516315	−0.258157	+	0.966103i	$0.583115\pi$
$420$	0	0
$421$	−36.1716	−1.76290	−0.881449	−	0.472280i	$-0.843431\pi$
$421$	−36.1716	−1.76290	−0.881449	+	0.472280i	$0.843431\pi$
$422$	17.8974	0.871232
$423$	0	0
$424$	0.233531	0.0113412
$425$	32.3818	1.57075
$426$	0	0
$427$	0	0
$428$	−4.75781	−0.229977
$429$	0	0
$430$	31.7761	1.53238
$431$	35.0989	1.69065	0.845327	−	0.534249i	$-0.179406\pi$
$431$	35.0989	1.69065	0.845327	+	0.534249i	$0.179406\pi$
$432$	0	0
$433$	41.1730	1.97865	0.989324	−	0.145731i	$-0.0465533\pi$
$433$	41.1730	1.97865	0.989324	+	0.145731i	$0.0465533\pi$
$434$	0	0
$435$	0	0
$436$	−16.7651	−0.802902
$437$	5.22253	0.249827
$438$	0	0
$439$	4.67859	0.223297	0.111648	−	0.993748i	$-0.464387\pi$
$439$	4.67859	0.223297	0.111648	+	0.993748i	$0.464387\pi$
$440$	19.0507	0.908209
$441$	0	0
$442$	−6.98762	−0.332367
$443$	30.1730	1.43356	0.716781	−	0.697298i	$-0.245615\pi$
$443$	30.1730	1.43356	0.716781	+	0.697298i	$0.245615\pi$
$444$	0	0
$445$	−34.4820	−1.63461
$446$	9.63279	0.456126
$447$	0	0
$448$	0	0
$449$	0.333792	0.0157526	0.00787632	−	0.999969i	$-0.497493\pi$
$449$	0.333792	0.0157526	0.00787632	+	0.999969i	$0.497493\pi$
$450$	0	0
$451$	−15.2115	−0.716283
$452$	−16.4871	−0.775490
$453$	0	0
$454$	18.8640	0.885330
$455$	0	0
$456$	0	0
$457$	−19.3090	−0.903238	−0.451619	−	0.892211i	$-0.649153\pi$
$457$	−19.3090	−0.903238	−0.451619	+	0.892211i	$0.649153\pi$
$458$	−33.3855	−1.56000
$459$	0	0
$460$	18.7403	0.873773
$461$	−39.1075	−1.82142	−0.910710	−	0.413046i	$-0.864465\pi$
$461$	−39.1075	−1.82142	−0.910710	+	0.413046i	$0.864465\pi$
$462$	0	0
$463$	21.8764	1.01668	0.508340	−	0.861156i	$-0.330259\pi$
$463$	21.8764	1.01668	0.508340	+	0.861156i	$0.330259\pi$
$464$	8.47710	0.393539
$465$	0	0
$466$	15.2335	0.705680
$467$	12.3200	0.570103	0.285052	−	0.958512i	$-0.407989\pi$
$467$	12.3200	0.570103	0.285052	+	0.958512i	$0.407989\pi$
$468$	0	0
$469$	0	0
$470$	16.2609	0.750059
$471$	0	0
$472$	−16.7651	−0.771676
$473$	−14.6452	−0.673385
$474$	0	0
$475$	7.00000	0.321182
$476$	0	0
$477$	0	0
$478$	19.0741	0.872430
$479$	−13.4895	−0.616350	−0.308175	−	0.951330i	$-0.599718\pi$
$479$	−13.4895	−0.616350	−0.308175	+	0.951330i	$0.599718\pi$
$480$	0	0
$481$	−4.76509	−0.217269
$482$	11.8740	0.540847
$483$	0	0
$484$	−2.75409	−0.125186
$485$	−26.2719	−1.19295
$486$	0	0
$487$	7.54394	0.341849	0.170924	−	0.985284i	$-0.445325\pi$
$487$	7.54394	0.341849	0.170924	+	0.985284i	$0.445325\pi$
$488$	7.32141	0.331425
$489$	0	0
$490$	0	0
$491$	−16.1396	−0.728369	−0.364185	−	0.931327i	$-0.618652\pi$
$491$	−16.1396	−0.728369	−0.364185	+	0.931327i	$0.618652\pi$
$492$	0	0
$493$	−6.98762	−0.314707
$494$	−1.51052	−0.0679615
$495$	0	0
$496$	−34.8516	−1.56488
$497$	0	0
$498$	0	0
$499$	−30.8654	−1.38172	−0.690862	−	0.722987i	$-0.742769\pi$
$499$	−30.8654	−1.38172	−0.690862	+	0.722987i	$0.742769\pi$
$500$	9.17301	0.410229
$501$	0	0
$502$	−7.85297	−0.350495
$503$	−24.6304	−1.09822	−0.549109	−	0.835751i	$-0.685033\pi$
$503$	−24.6304	−1.09822	−0.549109	+	0.835751i	$0.685033\pi$
$504$	0	0
$505$	−12.4327	−0.553247
$506$	−28.0741	−1.24805
$507$	0	0
$508$	8.87636	0.393825
$509$	−13.5897	−0.602355	−0.301177	−	0.953568i	$-0.597380\pi$
$509$	−13.5897	−0.602355	−0.301177	+	0.953568i	$0.597380\pi$
$510$	0	0
$511$	0	0
$512$	4.59937	0.203265
$513$	0	0
$514$	2.42030	0.106755
$515$	56.9701	2.51040
$516$	0	0
$517$	−7.49442	−0.329604
$518$	0	0
$519$	0	0
$520$	6.77747	0.297212
$521$	39.1730	1.71620	0.858100	−	0.513482i	$-0.171645\pi$
$521$	39.1730	1.71620	0.858100	+	0.513482i	$0.171645\pi$
$522$	0	0
$523$	−19.1236	−0.836219	−0.418109	−	0.908397i	$-0.637307\pi$
$523$	−19.1236	−0.836219	−0.418109	+	0.908397i	$0.637307\pi$
$524$	−14.2670	−0.623255
$525$	0	0
$526$	27.6414	1.20522
$527$	28.7280	1.25141
$528$	0	0
$529$	11.5316	0.501372
$530$	−0.754090	−0.0327556
$531$	0	0
$532$	0	0
$533$	−5.41164	−0.234404
$534$	0	0
$535$	−19.2101	−0.830527
$536$	−23.2485	−1.00418
$537$	0	0
$538$	31.7046	1.36688
$539$	0	0
$540$	0	0
$541$	2.53018	0.108781	0.0543906	−	0.998520i	$-0.482678\pi$
$541$	2.53018	0.108781	0.0543906	+	0.998520i	$0.482678\pi$
$542$	−6.73539	−0.289310
$543$	0	0
$544$	−19.3214	−0.828399
$545$	−67.6908	−2.89956
$546$	0	0
$547$	17.8516	0.763279	0.381640	−	0.924311i	$-0.375360\pi$
$547$	17.8516	0.763279	0.381640	+	0.924311i	$0.375360\pi$
$548$	−11.5426	−0.493074
$549$	0	0
$550$	−37.6291	−1.60451
$551$	−1.51052	−0.0643503
$552$	0	0
$553$	0	0
$554$	−3.96658	−0.168524
$555$	0	0
$556$	−0.987620	−0.0418844
$557$	41.3607	1.75251	0.876255	−	0.481847i	$-0.160034\pi$
$557$	41.3607	1.75251	0.876255	+	0.481847i	$0.160034\pi$
$558$	0	0
$559$	−5.21015	−0.220366
$560$	0	0
$561$	0	0
$562$	47.5809	2.00708
$563$	20.7366	0.873944	0.436972	−	0.899475i	$-0.356051\pi$
$563$	20.7366	0.873944	0.436972	+	0.899475i	$0.356051\pi$
$564$	0	0
$565$	−66.5685	−2.80056
$566$	−17.5402	−0.737271
$567$	0	0
$568$	5.43268	0.227950
$569$	−0.268329	−0.0112489	−0.00562446	−	0.999984i	$-0.501790\pi$
$569$	−0.268329	−0.0112489	−0.00562446	+	0.999984i	$0.501790\pi$
$570$	0	0
$571$	35.9367	1.50391	0.751953	−	0.659217i	$-0.229112\pi$
$571$	35.9367	1.50391	0.751953	+	0.659217i	$0.229112\pi$
$572$	2.49814	0.104453
$573$	0	0
$574$	0	0
$575$	46.2843	1.93019
$576$	0	0
$577$	−5.43130	−0.226108	−0.113054	−	0.993589i	$-0.536063\pi$
$577$	−5.43130	−0.226108	−0.113054	+	0.993589i	$0.536063\pi$
$578$	−0.165726	−0.00689328
$579$	0	0
$580$	−5.42030	−0.225066
$581$	0	0
$582$	0	0
$583$	0.347550	0.0143940
$584$	−20.0989	−0.831698
$585$	0	0
$586$	52.1716	2.15519
$587$	−35.1643	−1.45139	−0.725694	−	0.688018i	$-0.758481\pi$
$587$	−35.1643	−1.45139	−0.725694	+	0.688018i	$0.758481\pi$
$588$	0	0
$589$	6.21015	0.255885
$590$	54.1359	2.22874
$591$	0	0
$592$	−23.7665	−0.976796
$593$	33.5068	1.37596	0.687980	−	0.725730i	$-0.258498\pi$
$593$	33.5068	1.37596	0.687980	+	0.725730i	$0.258498\pi$
$594$	0	0
$595$	0	0
$596$	7.49442	0.306983
$597$	0	0
$598$	−9.98762	−0.408424
$599$	6.24729	0.255257	0.127629	−	0.991822i	$-0.459263\pi$
$599$	6.24729	0.255257	0.127629	+	0.991822i	$0.459263\pi$
$600$	0	0
$601$	−22.4079	−0.914038	−0.457019	−	0.889457i	$-0.651083\pi$
$601$	−22.4079	−0.914038	−0.457019	+	0.889457i	$0.651083\pi$
$602$	0	0
$603$	0	0
$604$	−13.1991	−0.537066
$605$	−11.1199	−0.452089
$606$	0	0
$607$	−14.9505	−0.606821	−0.303411	−	0.952860i	$-0.598125\pi$
$607$	−14.9505	−0.606821	−0.303411	+	0.952860i	$0.598125\pi$
$608$	−4.17673	−0.169389
$609$	0	0
$610$	−23.6414	−0.957214
$611$	−2.66621	−0.107863
$612$	0	0
$613$	35.1978	1.42162	0.710812	−	0.703382i	$-0.248328\pi$
$613$	35.1978	1.42162	0.710812	+	0.703382i	$0.248328\pi$
$614$	19.4500	0.784938
$615$	0	0
$616$	0	0
$617$	−2.01238	−0.0810154	−0.0405077	−	0.999179i	$-0.512898\pi$
$617$	−2.01238	−0.0810154	−0.0405077	+	0.999179i	$0.512898\pi$
$618$	0	0
$619$	−39.3818	−1.58289	−0.791444	−	0.611242i	$-0.790670\pi$
$619$	−39.3818	−1.58289	−0.791444	+	0.611242i	$0.790670\pi$
$620$	22.2843	0.894958
$621$	0	0
$622$	−20.3324	−0.815256
$623$	0	0
$624$	0	0
$625$	−2.34479	−0.0937918
$626$	23.0197	0.920051
$627$	0	0
$628$	−2.56732	−0.102447
$629$	19.5906	0.781126
$630$	0	0
$631$	44.3832	1.76687	0.883433	−	0.468558i	$-0.155226\pi$
$631$	44.3832	1.76687	0.883433	+	0.468558i	$0.155226\pi$
$632$	13.3845	0.532408
$633$	0	0
$634$	−50.9257	−2.02252
$635$	35.8392	1.42224
$636$	0	0
$637$	0	0
$638$	8.11993	0.321471
$639$	0	0
$640$	45.8502	1.81239
$641$	−14.9862	−0.591921	−0.295961	−	0.955200i	$-0.595640\pi$
$641$	−14.9862	−0.591921	−0.295961	+	0.955200i	$0.595640\pi$
$642$	0	0
$643$	10.6538	0.420146	0.210073	−	0.977686i	$-0.432630\pi$
$643$	10.6538	0.420146	0.210073	+	0.977686i	$0.432630\pi$
$644$	0	0
$645$	0	0
$646$	6.21015	0.244335
$647$	−2.12955	−0.0837213	−0.0418606	−	0.999123i	$-0.513329\pi$
$647$	−2.12955	−0.0837213	−0.0418606	+	0.999123i	$0.513329\pi$
$648$	0	0
$649$	−24.9505	−0.979392
$650$	−13.3869	−0.525076
$651$	0	0
$652$	−9.16201	−0.358812
$653$	11.1716	0.437180	0.218590	−	0.975817i	$-0.429854\pi$
$653$	11.1716	0.437180	0.218590	+	0.975817i	$0.429854\pi$
$654$	0	0
$655$	−57.6043	−2.25079
$656$	−26.9912	−1.05383
$657$	0	0
$658$	0	0
$659$	−11.3090	−0.440537	−0.220269	−	0.975439i	$-0.570693\pi$
$659$	−11.3090	−0.440537	−0.220269	+	0.975439i	$0.570693\pi$
$660$	0	0
$661$	−32.3570	−1.25854	−0.629271	−	0.777186i	$-0.716646\pi$
$661$	−32.3570	−1.25854	−0.629271	+	0.777186i	$0.716646\pi$
$662$	3.54628	0.137830
$663$	0	0
$664$	−7.76509	−0.301344
$665$	0	0
$666$	0	0
$667$	−9.98762	−0.386722
$668$	−10.7999	−0.417860
$669$	0	0
$670$	75.0714	2.90026
$671$	10.8960	0.420636
$672$	0	0
$673$	−24.1606	−0.931324	−0.465662	−	0.884963i	$-0.654184\pi$
$673$	−24.1606	−0.931324	−0.465662	+	0.884963i	$0.654184\pi$
$674$	−27.5489	−1.06114
$675$	0	0
$676$	−10.6648	−0.410186
$677$	−25.0741	−0.963677	−0.481838	−	0.876260i	$-0.660031\pi$
$677$	−25.0741	−0.963677	−0.481838	+	0.876260i	$0.660031\pi$
$678$	0	0
$679$	0	0
$680$	−27.8640	−1.06853
$681$	0	0
$682$	−33.3832	−1.27831
$683$	−47.6784	−1.82436	−0.912182	−	0.409785i	$-0.865604\pi$
$683$	−47.6784	−1.82436	−0.912182	+	0.409785i	$0.865604\pi$
$684$	0	0
$685$	−46.6043	−1.78066
$686$	0	0
$687$	0	0
$688$	−25.9862	−0.990716
$689$	0.123644	0.00471046
$690$	0	0
$691$	24.6800	0.938870	0.469435	−	0.882967i	$-0.344458\pi$
$691$	24.6800	0.938870	0.469435	+	0.882967i	$0.344458\pi$
$692$	5.87402	0.223297
$693$	0	0
$694$	19.1496	0.726910
$695$	−3.98762	−0.151259
$696$	0	0
$697$	22.2487	0.842728
$698$	0.336134	0.0127228
$699$	0	0
$700$	0	0
$701$	−29.6784	−1.12094	−0.560469	−	0.828175i	$-0.689379\pi$
$701$	−29.6784	−1.12094	−0.560469	+	0.828175i	$0.689379\pi$
$702$	0	0
$703$	4.23491	0.159723
$704$	−5.58699	−0.210567
$705$	0	0
$706$	21.2485	0.799698
$707$	0	0
$708$	0	0
$709$	−29.2581	−1.09881	−0.549406	−	0.835555i	$-0.685146\pi$
$709$	−29.2581	−1.09881	−0.549406	+	0.835555i	$0.685146\pi$
$710$	−17.5426	−0.658361
$711$	0	0
$712$	18.1496	0.680186
$713$	41.0617	1.53777
$714$	0	0
$715$	10.0865	0.377214
$716$	−3.41535	−0.127638
$717$	0	0
$718$	34.0260	1.26984
$719$	−1.07413	−0.0400581	−0.0200291	−	0.999799i	$-0.506376\pi$
$719$	−1.07413	−0.0400581	−0.0200291	+	0.999799i	$0.506376\pi$
$720$	0	0
$721$	0	0
$722$	−30.9505	−1.15186
$723$	0	0
$724$	−16.4794	−0.612454
$725$	−13.3869	−0.497176
$726$	0	0
$727$	−25.4327	−0.943246	−0.471623	−	0.881800i	$-0.656332\pi$
$727$	−25.4327	−0.943246	−0.471623	+	0.881800i	$0.656332\pi$
$728$	0	0
$729$	0	0
$730$	64.9010	2.40209
$731$	21.4203	0.792258
$732$	0	0
$733$	11.3955	0.420904	0.210452	−	0.977604i	$-0.432506\pi$
$733$	11.3955	0.420904	0.210452	+	0.977604i	$0.432506\pi$
$734$	−51.1125	−1.88660
$735$	0	0
$736$	−27.6167	−1.01796
$737$	−34.5994	−1.27448
$738$	0	0
$739$	−29.9395	−1.10134	−0.550671	−	0.834723i	$-0.685628\pi$
$739$	−29.9395	−1.10134	−0.550671	+	0.834723i	$0.685628\pi$
$740$	15.1964	0.558630
$741$	0	0
$742$	0	0
$743$	−19.0014	−0.697093	−0.348546	−	0.937292i	$-0.613325\pi$
$743$	−19.0014	−0.697093	−0.348546	+	0.937292i	$0.613325\pi$
$744$	0	0
$745$	30.2595	1.10862
$746$	11.9184	0.436365
$747$	0	0
$748$	−10.2705	−0.375527
$749$	0	0
$750$	0	0
$751$	0.0261368	0.000953747 0	0.000476873	−	1.00000i	$-0.499848\pi$
$751$	0.0261368	0.000953747 0	0.000476873		1.00000i	$0.499848\pi$
$752$	−13.2980	−0.484929
$753$	0	0
$754$	2.88874	0.105202
$755$	−53.2929	−1.93953
$756$	0	0
$757$	−13.6910	−0.497607	−0.248803	−	0.968554i	$-0.580037\pi$
$757$	−13.6910	−0.497607	−0.248803	+	0.968554i	$0.580037\pi$
$758$	−32.4189	−1.17751
$759$	0	0
$760$	−6.02338	−0.218491
$761$	14.6428	0.530802	0.265401	−	0.964138i	$-0.414496\pi$
$761$	14.6428	0.530802	0.265401	+	0.964138i	$0.414496\pi$
$762$	0	0
$763$	0	0
$764$	4.11855	0.149004
$765$	0	0
$766$	−5.45606	−0.197135
$767$	−8.87636	−0.320507
$768$	0	0
$769$	−49.1345	−1.77184	−0.885918	−	0.463843i	$-0.846470\pi$
$769$	−49.1345	−1.77184	−0.885918	+	0.463843i	$0.846470\pi$
$770$	0	0
$771$	0	0
$772$	−22.4820	−0.809146
$773$	−12.4413	−0.447484	−0.223742	−	0.974648i	$-0.571827\pi$
$773$	−12.4413	−0.447484	−0.223742	+	0.974648i	$0.571827\pi$
$774$	0	0
$775$	55.0370	1.97699
$776$	13.8282	0.496404
$777$	0	0
$778$	8.72933	0.312962
$779$	4.80951	0.172319
$780$	0	0
$781$	8.08513	0.289309
$782$	41.0617	1.46837
$783$	0	0
$784$	0	0
$785$	−10.3658	−0.369973
$786$	0	0
$787$	32.9133	1.17323	0.586617	−	0.809865i	$-0.300459\pi$
$787$	32.9133	1.17323	0.586617	+	0.809865i	$0.300459\pi$
$788$	9.52784	0.339415
$789$	0	0
$790$	−43.2198	−1.53769
$791$	0	0
$792$	0	0
$793$	3.87636	0.137653
$794$	−38.9862	−1.38357
$795$	0	0
$796$	7.79108	0.276147
$797$	−26.3979	−0.935061	−0.467530	−	0.883977i	$-0.654856\pi$
$797$	−26.3979	−0.935061	−0.467530	+	0.883977i	$0.654856\pi$
$798$	0	0
$799$	10.9615	0.387789
$800$	−37.0159	−1.30871
$801$	0	0
$802$	−30.9505	−1.09290
$803$	−29.9120	−1.05557
$804$	0	0
$805$	0	0
$806$	−11.8764	−0.418327
$807$	0	0
$808$	6.54394	0.230215
$809$	35.5919	1.25135	0.625673	−	0.780086i	$-0.284825\pi$
$809$	35.5919	1.25135	0.625673	+	0.780086i	$0.284825\pi$
$810$	0	0
$811$	37.8268	1.32828	0.664140	−	0.747608i	$-0.268798\pi$
$811$	37.8268	1.32828	0.664140	+	0.747608i	$0.268798\pi$
$812$	0	0
$813$	0	0
$814$	−22.7651	−0.797916
$815$	−36.9926	−1.29579
$816$	0	0
$817$	4.63045	0.161999
$818$	26.0594	0.911146
$819$	0	0
$820$	17.2583	0.602686
$821$	−18.3128	−0.639119	−0.319560	−	0.947566i	$-0.603535\pi$
$821$	−18.3128	−0.639119	−0.319560	+	0.947566i	$0.603535\pi$
$822$	0	0
$823$	−36.0000	−1.25488	−0.627441	−	0.778664i	$-0.715897\pi$
$823$	−36.0000	−1.25488	−0.627441	+	0.778664i	$0.715897\pi$
$824$	−29.9862	−1.04462
$825$	0	0
$826$	0	0
$827$	−28.2115	−0.981011	−0.490505	−	0.871438i	$-0.663188\pi$
$827$	−28.2115	−0.981011	−0.490505	+	0.871438i	$0.663188\pi$
$828$	0	0
$829$	11.2843	0.391919	0.195960	−	0.980612i	$-0.437218\pi$
$829$	11.2843	0.391919	0.195960	+	0.980612i	$0.437218\pi$
$830$	25.0741	0.870336
$831$	0	0
$832$	−1.98762	−0.0689083
$833$	0	0
$834$	0	0
$835$	−43.6057	−1.50904
$836$	−2.22019	−0.0767868
$837$	0	0
$838$	−17.9629	−0.620517
$839$	2.04305	0.0705338	0.0352669	−	0.999378i	$-0.488772\pi$
$839$	2.04305	0.0705338	0.0352669	+	0.999378i	$0.488772\pi$
$840$	0	0
$841$	−26.1113	−0.900388
$842$	−61.4783	−2.11868
$843$	0	0
$844$	9.35855	0.322135
$845$	−43.0604	−1.48132
$846$	0	0
$847$	0	0
$848$	0.616689	0.0211772
$849$	0	0
$850$	55.0370	1.88775
$851$	28.0014	0.959875
$852$	0	0
$853$	48.5919	1.66376	0.831878	−	0.554959i	$-0.187266\pi$
$853$	48.5919	1.66376	0.831878	+	0.554959i	$0.187266\pi$
$854$	0	0
$855$	0	0
$856$	10.1113	0.345596
$857$	44.8974	1.53367	0.766833	−	0.641847i	$-0.221831\pi$
$857$	44.8974	1.53367	0.766833	+	0.641847i	$0.221831\pi$
$858$	0	0
$859$	−29.8131	−1.01721	−0.508605	−	0.861000i	$-0.669838\pi$
$859$	−29.8131	−1.01721	−0.508605	+	0.861000i	$0.669838\pi$
$860$	16.6157	0.566592
$861$	0	0
$862$	59.6551	2.03186
$863$	−42.2595	−1.43853	−0.719265	−	0.694736i	$-0.755521\pi$
$863$	−42.2595	−1.43853	−0.719265	+	0.694736i	$0.755521\pi$
$864$	0	0
$865$	23.7170	0.806401
$866$	69.9788	2.37798
$867$	0	0
$868$	0	0
$869$	19.9194	0.675719
$870$	0	0
$871$	−12.3090	−0.417076
$872$	35.6291	1.20655
$873$	0	0
$874$	8.87636	0.300247
$875$	0	0
$876$	0	0
$877$	−30.5316	−1.03098	−0.515489	−	0.856896i	$-0.672390\pi$
$877$	−30.5316	−1.03098	−0.515489	+	0.856896i	$0.672390\pi$
$878$	7.95186	0.268362
$879$	0	0
$880$	50.3077	1.69587
$881$	13.4079	0.451724	0.225862	−	0.974159i	$-0.427480\pi$
$881$	13.4079	0.451724	0.225862	+	0.974159i	$0.427480\pi$
$882$	0	0
$883$	−14.1250	−0.475345	−0.237672	−	0.971345i	$-0.576384\pi$
$883$	−14.1250	−0.475345	−0.237672	+	0.971345i	$0.576384\pi$
$884$	−3.65383	−0.122892
$885$	0	0
$886$	51.2829	1.72288
$887$	39.9432	1.34116	0.670581	−	0.741837i	$-0.266045\pi$
$887$	39.9432	1.34116	0.670581	+	0.741837i	$0.266045\pi$
$888$	0	0
$889$	0	0
$890$	−58.6067	−1.96450
$891$	0	0
$892$	5.03699	0.168651
$893$	2.36955	0.0792941
$894$	0	0
$895$	−13.7899	−0.460944
$896$	0	0
$897$	0	0
$898$	0.567323	0.0189318
$899$	−11.8764	−0.396099
$900$	0	0
$901$	−0.508333	−0.0169350
$902$	−25.8539	−0.860842
$903$	0	0
$904$	35.0384	1.16536
$905$	−66.5375	−2.21178
$906$	0	0
$907$	41.4203	1.37534	0.687669	−	0.726024i	$-0.258634\pi$
$907$	41.4203	1.37534	0.687669	+	0.726024i	$0.258634\pi$
$908$	9.86398	0.327348
$909$	0	0
$910$	0	0
$911$	1.78847	0.0592548	0.0296274	−	0.999561i	$-0.490568\pi$
$911$	1.78847	0.0592548	0.0296274	+	0.999561i	$0.490568\pi$
$912$	0	0
$913$	−11.5563	−0.382458
$914$	−32.8182	−1.08553
$915$	0	0
$916$	−17.4573	−0.576805
$917$	0	0
$918$	0	0
$919$	−57.4683	−1.89570	−0.947852	−	0.318711i	$-0.896750\pi$
$919$	−57.4683	−1.89570	−0.947852	+	0.318711i	$0.896750\pi$
$920$	−39.8268	−1.31305
$921$	0	0
$922$	−66.4683	−2.18902
$923$	2.87636	0.0946764
$924$	0	0
$925$	37.5316	1.23403
$926$	37.1817	1.22187
$927$	0	0
$928$	7.98762	0.262206
$929$	−34.7352	−1.13963	−0.569813	−	0.821774i	$-0.692984\pi$
$929$	−34.7352	−1.13963	−0.569813	+	0.821774i	$0.692984\pi$
$930$	0	0
$931$	0	0
$932$	7.96562	0.260922
$933$	0	0
$934$	20.9395	0.685161
$935$	−41.4683	−1.35616
$936$	0	0
$937$	−11.6662	−0.381118	−0.190559	−	0.981676i	$-0.561030\pi$
$937$	−11.6662	−0.381118	−0.190559	+	0.981676i	$0.561030\pi$
$938$	0	0
$939$	0	0
$940$	8.50282	0.277332
$941$	50.3374	1.64095	0.820475	−	0.571682i	$-0.193709\pi$
$941$	50.3374	1.64095	0.820475	+	0.571682i	$0.193709\pi$
$942$	0	0
$943$	31.8007	1.03557
$944$	−44.2719	−1.44093
$945$	0	0
$946$	−24.8913	−0.809287
$947$	−32.3883	−1.05248	−0.526238	−	0.850337i	$-0.676398\pi$
$947$	−32.3883	−1.05248	−0.526238	+	0.850337i	$0.676398\pi$
$948$	0	0
$949$	−10.6414	−0.345436
$950$	11.8974	0.386003
$951$	0	0
$952$	0	0
$953$	−12.5367	−0.406102	−0.203051	−	0.979168i	$-0.565086\pi$
$953$	−12.5367	−0.406102	−0.203051	+	0.979168i	$0.565086\pi$
$954$	0	0
$955$	16.6291	0.538104
$956$	9.97386	0.322578
$957$	0	0
$958$	−22.9271	−0.740741
$959$	0	0
$960$	0	0
$961$	17.8268	0.575059
$962$	−8.09888	−0.261119
$963$	0	0
$964$	6.20892	0.199976
$965$	−90.7736	−2.92211
$966$	0	0
$967$	−57.9875	−1.86475	−0.932376	−	0.361491i	$-0.882268\pi$
$967$	−57.9875	−1.86475	−0.932376	+	0.361491i	$0.882268\pi$
$968$	5.85297	0.188122
$969$	0	0
$970$	−44.6525	−1.43370
$971$	−28.0370	−0.899750	−0.449875	−	0.893092i	$-0.648531\pi$
$971$	−28.0370	−0.899750	−0.449875	+	0.893092i	$0.648531\pi$
$972$	0	0
$973$	0	0
$974$	12.8219	0.410840
$975$	0	0
$976$	19.3338	0.618860
$977$	9.84059	0.314829	0.157414	−	0.987533i	$-0.449684\pi$
$977$	9.84059	0.314829	0.157414	+	0.987533i	$0.449684\pi$
$978$	0	0
$979$	27.0110	0.863275
$980$	0	0
$981$	0	0
$982$	−27.4313	−0.875368
$983$	48.6894	1.55295	0.776476	−	0.630147i	$-0.217005\pi$
$983$	48.6894	1.55295	0.776476	+	0.630147i	$0.217005\pi$
$984$	0	0
$985$	38.4697	1.22575
$986$	−11.8764	−0.378220
$987$	0	0
$988$	−0.789851	−0.0251285
$989$	30.6167	0.973554
$990$	0	0
$991$	2.86398	0.0909772	0.0454886	−	0.998965i	$-0.485516\pi$
$991$	2.86398	0.0909772	0.0454886	+	0.998965i	$0.485516\pi$
$992$	−32.8392	−1.04265
$993$	0	0
$994$	0	0
$995$	31.4573	0.997263
$996$	0	0
$997$	50.8406	1.61014	0.805069	−	0.593181i	$-0.202128\pi$
$997$	50.8406	1.61014	0.805069	+	0.593181i	$0.202128\pi$
$998$	−52.4596	−1.66058
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.m.1.3		3
3.2	odd	2		3969.2.a.p.1.1		3
7.6	odd	2		567.2.a.d.1.3		3
9.2	odd	6		1323.2.f.c.442.3		6
9.4	even	3		441.2.f.d.295.1		6
9.5	odd	6		1323.2.f.c.883.3		6
9.7	even	3		441.2.f.d.148.1		6
21.20	even	2		567.2.a.g.1.1		3
28.27	even	2		9072.2.a.bq.1.1		3
63.2	odd	6		1323.2.g.b.361.3		6
63.4	even	3		441.2.g.d.79.1		6
63.5	even	6		1323.2.h.d.802.1		6
63.11	odd	6		1323.2.h.e.226.1		6
63.13	odd	6		63.2.f.b.43.1	yes	6
63.16	even	3		441.2.g.d.67.1		6
63.20	even	6		189.2.f.a.64.3		6
63.23	odd	6		1323.2.h.e.802.1		6
63.25	even	3		441.2.h.b.373.3		6
63.31	odd	6		441.2.g.e.79.1		6
63.32	odd	6		1323.2.g.b.667.3		6
63.34	odd	6		63.2.f.b.22.1	✓	6
63.38	even	6		1323.2.h.d.226.1		6
63.40	odd	6		441.2.h.c.214.3		6
63.41	even	6		189.2.f.a.127.3		6
63.47	even	6		1323.2.g.c.361.3		6
63.52	odd	6		441.2.h.c.373.3		6
63.58	even	3		441.2.h.b.214.3		6
63.59	even	6		1323.2.g.c.667.3		6
63.61	odd	6		441.2.g.e.67.1		6
84.83	odd	2		9072.2.a.cd.1.3		3
252.83	odd	6		3024.2.r.g.1009.1		6
252.139	even	6		1008.2.r.k.673.1		6
252.167	odd	6		3024.2.r.g.2017.1		6
252.223	even	6		1008.2.r.k.337.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.f.b.22.1	✓	6	63.34	odd	6
63.2.f.b.43.1	yes	6	63.13	odd	6
189.2.f.a.64.3		6	63.20	even	6
189.2.f.a.127.3		6	63.41	even	6
441.2.f.d.148.1		6	9.7	even	3
441.2.f.d.295.1		6	9.4	even	3
441.2.g.d.67.1		6	63.16	even	3
441.2.g.d.79.1		6	63.4	even	3
441.2.g.e.67.1		6	63.61	odd	6
441.2.g.e.79.1		6	63.31	odd	6
441.2.h.b.214.3		6	63.58	even	3
441.2.h.b.373.3		6	63.25	even	3
441.2.h.c.214.3		6	63.40	odd	6
441.2.h.c.373.3		6	63.52	odd	6
567.2.a.d.1.3		3	7.6	odd	2
567.2.a.g.1.1		3	21.20	even	2
1008.2.r.k.337.1		6	252.223	even	6
1008.2.r.k.673.1		6	252.139	even	6
1323.2.f.c.442.3		6	9.2	odd	6
1323.2.f.c.883.3		6	9.5	odd	6
1323.2.g.b.361.3		6	63.2	odd	6
1323.2.g.b.667.3		6	63.32	odd	6
1323.2.g.c.361.3		6	63.47	even	6
1323.2.g.c.667.3		6	63.59	even	6
1323.2.h.d.226.1		6	63.38	even	6
1323.2.h.d.802.1		6	63.5	even	6
1323.2.h.e.226.1		6	63.11	odd	6
1323.2.h.e.802.1		6	63.23	odd	6
3024.2.r.g.1009.1		6	252.83	odd	6
3024.2.r.g.2017.1		6	252.167	odd	6
3969.2.a.m.1.3		3	1.1	even	1	trivial
3969.2.a.p.1.1		3	3.2	odd	2
9072.2.a.bq.1.1		3	28.27	even	2
9072.2.a.cd.1.3		3	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.69963 q^{2} +0.888736 q^{4} +3.58836 q^{5} -1.88874 q^{8} +O(q^{10})\)	\(q+1.69963 q^{2} +0.888736 q^{4} +3.58836 q^{5} -1.88874 q^{8} +6.09888 q^{10} -2.81089 q^{11} -1.00000 q^{13} -4.98762 q^{16} +4.11126 q^{17} +0.888736 q^{19} +3.18911 q^{20} -4.77747 q^{22} +5.87636 q^{23} +7.87636 q^{25} -1.69963 q^{26} -1.69963 q^{29} +6.98762 q^{31} -4.69963 q^{32} +6.98762 q^{34} +4.76509 q^{37} +1.51052 q^{38} -6.77747 q^{40} +5.41164 q^{41} +5.21015 q^{43} -2.49814 q^{44} +9.98762 q^{46} +2.66621 q^{47} +13.3869 q^{50} -0.888736 q^{52} -0.123644 q^{53} -10.0865 q^{55} -2.88874 q^{58} +8.87636 q^{59} -3.87636 q^{61} +11.8764 q^{62} +1.98762 q^{64} -3.58836 q^{65} +12.3090 q^{67} +3.65383 q^{68} -2.87636 q^{71} +10.6414 q^{73} +8.09888 q^{74} +0.789851 q^{76} -7.08650 q^{79} -17.8974 q^{80} +9.19777 q^{82} +4.11126 q^{83} +14.7527 q^{85} +8.85532 q^{86} +5.30903 q^{88} -9.60940 q^{89} +5.22253 q^{92} +4.53156 q^{94} +3.18911 q^{95} -7.32141 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{4} + 5 q^{5} - 6 q^{8}+O(q^{10}) \) 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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