LMFDB - Embedded newform 1386.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(1,1386)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1386.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 2 $ x^3 - 11*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.22168$ of defining polynomial
Character	$\chi$	$=$	1386.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.80044 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.80044 q^{5} +1.00000 q^{7} -1.00000 q^{8} +3.80044 q^{10} -1.00000 q^{11} -6.44335 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.80044 q^{17} +6.64291 q^{19} -3.80044 q^{20} +1.00000 q^{22} -0.842472 q^{23} +9.44335 q^{25} +6.44335 q^{26} +1.00000 q^{28} +8.44335 q^{29} -9.40132 q^{31} -1.00000 q^{32} +3.80044 q^{34} -3.80044 q^{35} +9.60088 q^{37} -6.64291 q^{38} +3.80044 q^{40} +3.80044 q^{41} -8.04424 q^{43} -1.00000 q^{44} +0.842472 q^{46} -6.24380 q^{47} +1.00000 q^{49} -9.44335 q^{50} -6.44335 q^{52} +1.60088 q^{53} +3.80044 q^{55} -1.00000 q^{56} -8.44335 q^{58} +9.28583 q^{59} +8.75841 q^{61} +9.40132 q^{62} +1.00000 q^{64} +24.4876 q^{65} +1.15753 q^{67} -3.80044 q^{68} +3.80044 q^{70} +6.75841 q^{71} -7.80044 q^{73} -9.60088 q^{74} +6.64291 q^{76} -1.00000 q^{77} +2.84247 q^{79} -3.80044 q^{80} -3.80044 q^{82} +9.80044 q^{83} +14.4434 q^{85} +8.04424 q^{86} +1.00000 q^{88} +5.15753 q^{89} -6.44335 q^{91} -0.842472 q^{92} +6.24380 q^{94} -25.2460 q^{95} +18.8867 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} - 3 q^{8} + 2 q^{10} - 3 q^{11} - 3 q^{14} + 3 q^{16} - 2 q^{17} + 10 q^{19} - 2 q^{20} + 3 q^{22} - 2 q^{23} + 9 q^{25} + 3 q^{28} + 6 q^{29} - 3 q^{32} + 2 q^{34} - 2 q^{35} + 10 q^{37} - 10 q^{38} + 2 q^{40} + 2 q^{41} + 14 q^{43} - 3 q^{44} + 2 q^{46} + 10 q^{47} + 3 q^{49} - 9 q^{50} - 14 q^{53} + 2 q^{55} - 3 q^{56} - 6 q^{58} + 8 q^{59} + 8 q^{61} + 3 q^{64} + 16 q^{65} + 4 q^{67} - 2 q^{68} + 2 q^{70} + 2 q^{71} - 14 q^{73} - 10 q^{74} + 10 q^{76} - 3 q^{77} + 8 q^{79} - 2 q^{80} - 2 q^{82} + 20 q^{83} + 24 q^{85} - 14 q^{86} + 3 q^{88} + 16 q^{89} - 2 q^{92} - 10 q^{94} + 18 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 - 3 * q^8 + 2 * q^10 - 3 * q^11 - 3 * q^14 + 3 * q^16 - 2 * q^17 + 10 * q^19 - 2 * q^20 + 3 * q^22 - 2 * q^23 + 9 * q^25 + 3 * q^28 + 6 * q^29 - 3 * q^32 + 2 * q^34 - 2 * q^35 + 10 * q^37 - 10 * q^38 + 2 * q^40 + 2 * q^41 + 14 * q^43 - 3 * q^44 + 2 * q^46 + 10 * q^47 + 3 * q^49 - 9 * q^50 - 14 * q^53 + 2 * q^55 - 3 * q^56 - 6 * q^58 + 8 * q^59 + 8 * q^61 + 3 * q^64 + 16 * q^65 + 4 * q^67 - 2 * q^68 + 2 * q^70 + 2 * q^71 - 14 * q^73 - 10 * q^74 + 10 * q^76 - 3 * q^77 + 8 * q^79 - 2 * q^80 - 2 * q^82 + 20 * q^83 + 24 * q^85 - 14 * q^86 + 3 * q^88 + 16 * q^89 - 2 * q^92 - 10 * q^94 + 18 * q^97 - 3 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.80044	−1.69961	−0.849805	−	0.527098i	$-0.823280\pi$
$5$	−3.80044	−1.69961	−0.849805	+	0.527098i	$0.823280\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.80044	1.20181
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.44335	−1.78707	−0.893533	−	0.448998i	$-0.851781\pi$
$13$	−6.44335	−1.78707	−0.893533	+	0.448998i	$0.851781\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.80044	−0.921742	−0.460871	−	0.887467i	$-0.652463\pi$
$17$	−3.80044	−0.921742	−0.460871	+	0.887467i	$0.652463\pi$
$18$	0	0
$19$	6.64291	1.52399	0.761994	−	0.647584i	$-0.224220\pi$
$19$	6.64291	1.52399	0.761994	+	0.647584i	$0.224220\pi$
$20$	−3.80044	−0.849805
$21$	0	0
$22$	1.00000	0.213201
$23$	−0.842472	−0.175668	−0.0878338	−	0.996135i	$-0.527994\pi$
$23$	−0.842472	−0.175668	−0.0878338	+	0.996135i	$0.527994\pi$
$24$	0	0
$25$	9.44335	1.88867
$26$	6.44335	1.26365
$27$	0	0
$28$	1.00000	0.188982
$29$	8.44335	1.56789	0.783946	−	0.620829i	$-0.213204\pi$
$29$	8.44335	1.56789	0.783946	+	0.620829i	$0.213204\pi$
$30$	0	0
$31$	−9.40132	−1.68853	−0.844264	−	0.535928i	$-0.819962\pi$
$31$	−9.40132	−1.68853	−0.844264	+	0.535928i	$0.819962\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.80044	0.651770
$35$	−3.80044	−0.642392
$36$	0	0
$37$	9.60088	1.57838	0.789188	−	0.614152i	$-0.210502\pi$
$37$	9.60088	1.57838	0.789188	+	0.614152i	$0.210502\pi$
$38$	−6.64291	−1.07762
$39$	0	0
$40$	3.80044	0.600903
$41$	3.80044	0.593529	0.296765	−	0.954951i	$-0.404092\pi$
$41$	3.80044	0.593529	0.296765	+	0.954951i	$0.404092\pi$
$42$	0	0
$43$	−8.04424	−1.22673	−0.613367	−	0.789798i	$-0.710185\pi$
$43$	−8.04424	−1.22673	−0.613367	+	0.789798i	$0.710185\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	0.842472	0.124216
$47$	−6.24380	−0.910751	−0.455376	−	0.890299i	$-0.650495\pi$
$47$	−6.24380	−0.910751	−0.455376	+	0.890299i	$0.650495\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−9.44335	−1.33549
$51$	0	0
$52$	−6.44335	−0.893533
$53$	1.60088	0.219898	0.109949	−	0.993937i	$-0.464931\pi$
$53$	1.60088	0.219898	0.109949	+	0.993937i	$0.464931\pi$
$54$	0	0
$55$	3.80044	0.512451
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−8.44335	−1.10867
$59$	9.28583	1.20891	0.604456	−	0.796639i	$-0.293391\pi$
$59$	9.28583	1.20891	0.604456	+	0.796639i	$0.293391\pi$
$60$	0	0
$61$	8.75841	1.12140	0.560700	−	0.828019i	$-0.310532\pi$
$61$	8.75841	1.12140	0.560700	+	0.828019i	$0.310532\pi$
$62$	9.40132	1.19397
$63$	0	0
$64$	1.00000	0.125000
$65$	24.4876	3.03731
$66$	0	0
$67$	1.15753	0.141415	0.0707073	−	0.997497i	$-0.477474\pi$
$67$	1.15753	0.141415	0.0707073	+	0.997497i	$0.477474\pi$
$68$	−3.80044	−0.460871
$69$	0	0
$70$	3.80044	0.454240
$71$	6.75841	0.802076	0.401038	−	0.916061i	$-0.368650\pi$
$71$	6.75841	0.802076	0.401038	+	0.916061i	$0.368650\pi$
$72$	0	0
$73$	−7.80044	−0.912973	−0.456486	−	0.889730i	$-0.650892\pi$
$73$	−7.80044	−0.912973	−0.456486	+	0.889730i	$0.650892\pi$
$74$	−9.60088	−1.11608
$75$	0	0
$76$	6.64291	0.761994
$77$	−1.00000	−0.113961
$78$	0	0
$79$	2.84247	0.319803	0.159902	−	0.987133i	$-0.448882\pi$
$79$	2.84247	0.319803	0.159902	+	0.987133i	$0.448882\pi$
$80$	−3.80044	−0.424902
$81$	0	0
$82$	−3.80044	−0.419689
$83$	9.80044	1.07574	0.537869	−	0.843028i	$-0.319229\pi$
$83$	9.80044	1.07574	0.537869	+	0.843028i	$0.319229\pi$
$84$	0	0
$85$	14.4434	1.56660
$86$	8.04424	0.867432
$87$	0	0
$88$	1.00000	0.106600
$89$	5.15753	0.546697	0.273348	−	0.961915i	$-0.411869\pi$
$89$	5.15753	0.546697	0.273348	+	0.961915i	$0.411869\pi$
$90$	0	0
$91$	−6.44335	−0.675447
$92$	−0.842472	−0.0878338
$93$	0	0
$94$	6.24380	0.643998
$95$	−25.2460	−2.59019
$96$	0	0
$97$	18.8867	1.91765	0.958827	−	0.283989i	$-0.0916581\pi$
$97$	18.8867	1.91765	0.958827	+	0.283989i	$0.0916581\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	9.44335	0.944335
$101$	−10.8867	−1.08327	−0.541634	−	0.840614i	$-0.682194\pi$
$101$	−10.8867	−1.08327	−0.541634	+	0.840614i	$0.682194\pi$
$102$	0	0
$103$	5.80044	0.571534	0.285767	−	0.958299i	$-0.407752\pi$
$103$	5.80044	0.571534	0.285767	+	0.958299i	$0.407752\pi$
$104$	6.44335	0.631823
$105$	0	0
$106$	−1.60088	−0.155491
$107$	7.60088	0.734805	0.367403	−	0.930062i	$-0.380247\pi$
$107$	7.60088	0.734805	0.367403	+	0.930062i	$0.380247\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	−3.80044	−0.362358
$111$	0	0
$112$	1.00000	0.0944911
$113$	−0.758411	−0.0713453	−0.0356726	−	0.999364i	$-0.511357\pi$
$113$	−0.758411	−0.0713453	−0.0356726	+	0.999364i	$0.511357\pi$
$114$	0	0
$115$	3.20177	0.298566
$116$	8.44335	0.783946
$117$	0	0
$118$	−9.28583	−0.854830
$119$	−3.80044	−0.348386
$120$	0	0
$121$	1.00000	0.0909091
$122$	−8.75841	−0.792949
$123$	0	0
$124$	−9.40132	−0.844264
$125$	−16.8867	−1.51039
$126$	0	0
$127$	2.84247	0.252229	0.126114	−	0.992016i	$-0.459749\pi$
$127$	2.84247	0.252229	0.126114	+	0.992016i	$0.459749\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−24.4876	−2.14770
$131$	14.1996	1.24062	0.620311	−	0.784356i	$-0.287007\pi$
$131$	14.1996	1.24062	0.620311	+	0.784356i	$0.287007\pi$
$132$	0	0
$133$	6.64291	0.576014
$134$	−1.15753	−0.0999952
$135$	0	0
$136$	3.80044	0.325885
$137$	−12.7584	−1.09002	−0.545012	−	0.838428i	$-0.683475\pi$
$137$	−12.7584	−1.09002	−0.545012	+	0.838428i	$0.683475\pi$
$138$	0	0
$139$	−8.55885	−0.725952	−0.362976	−	0.931798i	$-0.618239\pi$
$139$	−8.55885	−0.725952	−0.362976	+	0.931798i	$0.618239\pi$
$140$	−3.80044	−0.321196
$141$	0	0
$142$	−6.75841	−0.567153
$143$	6.44335	0.538820
$144$	0	0
$145$	−32.0885	−2.66480
$146$	7.80044	0.645569
$147$	0	0
$148$	9.60088	0.789188
$149$	−10.8867	−0.891874	−0.445937	−	0.895064i	$-0.647129\pi$
$149$	−10.8867	−0.891874	−0.445937	+	0.895064i	$0.647129\pi$
$150$	0	0
$151$	6.31506	0.513912	0.256956	−	0.966423i	$-0.417280\pi$
$151$	6.31506	0.513912	0.256956	+	0.966423i	$0.417280\pi$
$152$	−6.64291	−0.538811
$153$	0	0
$154$	1.00000	0.0805823
$155$	35.7292	2.86984
$156$	0	0
$157$	−12.1996	−0.973631	−0.486815	−	0.873505i	$-0.661842\pi$
$157$	−12.1996	−0.973631	−0.486815	+	0.873505i	$0.661842\pi$
$158$	−2.84247	−0.226135
$159$	0	0
$160$	3.80044	0.300451
$161$	−0.842472	−0.0663961
$162$	0	0
$163$	−5.68494	−0.445279	−0.222640	−	0.974901i	$-0.571467\pi$
$163$	−5.68494	−0.445279	−0.222640	+	0.974901i	$0.571467\pi$
$164$	3.80044	0.296765
$165$	0	0
$166$	−9.80044	−0.760662
$167$	24.4876	1.89491	0.947453	−	0.319894i	$-0.103647\pi$
$167$	24.4876	1.89491	0.947453	+	0.319894i	$0.103647\pi$
$168$	0	0
$169$	28.5168	2.19360
$170$	−14.4434	−1.10775
$171$	0	0
$172$	−8.04424	−0.613367
$173$	15.2858	1.16216	0.581080	−	0.813846i	$-0.302630\pi$
$173$	15.2858	1.16216	0.581080	+	0.813846i	$0.302630\pi$
$174$	0	0
$175$	9.44335	0.713851
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−5.15753	−0.386573
$179$	25.2460	1.88697	0.943487	−	0.331408i	$-0.107524\pi$
$179$	25.2460	1.88697	0.943487	+	0.331408i	$0.107524\pi$
$180$	0	0
$181$	−7.80044	−0.579802	−0.289901	−	0.957057i	$-0.593622\pi$
$181$	−7.80044	−0.579802	−0.289901	+	0.957057i	$0.593622\pi$
$182$	6.44335	0.477613
$183$	0	0
$184$	0.842472	0.0621079
$185$	−36.4876	−2.68262
$186$	0	0
$187$	3.80044	0.277916
$188$	−6.24380	−0.455376
$189$	0	0
$190$	25.2460	1.83154
$191$	−20.9309	−1.51451	−0.757255	−	0.653119i	$-0.773460\pi$
$191$	−20.9309	−1.51451	−0.757255	+	0.653119i	$0.773460\pi$
$192$	0	0
$193$	5.20177	0.374431	0.187216	−	0.982319i	$-0.440054\pi$
$193$	5.20177	0.374431	0.187216	+	0.982319i	$0.440054\pi$
$194$	−18.8867	−1.35599
$195$	0	0
$196$	1.00000	0.0714286
$197$	−13.3301	−0.949728	−0.474864	−	0.880059i	$-0.657503\pi$
$197$	−13.3301	−0.949728	−0.474864	+	0.880059i	$0.657503\pi$
$198$	0	0
$199$	−13.8004	−0.978287	−0.489144	−	0.872203i	$-0.662691\pi$
$199$	−13.8004	−0.978287	−0.489144	+	0.872203i	$0.662691\pi$
$200$	−9.44335	−0.667746
$201$	0	0
$202$	10.8867	0.765986
$203$	8.44335	0.592607
$204$	0	0
$205$	−14.4434	−1.00877
$206$	−5.80044	−0.404136
$207$	0	0
$208$	−6.44335	−0.446766
$209$	−6.64291	−0.459500
$210$	0	0
$211$	13.7292	0.945156	0.472578	−	0.881289i	$-0.343324\pi$
$211$	13.7292	0.945156	0.472578	+	0.881289i	$0.343324\pi$
$212$	1.60088	0.109949
$213$	0	0
$214$	−7.60088	−0.519586
$215$	30.5717	2.08497
$216$	0	0
$217$	−9.40132	−0.638203
$218$	−8.00000	−0.541828
$219$	0	0
$220$	3.80044	0.256226
$221$	24.4876	1.64721
$222$	0	0
$223$	−24.6031	−1.64754	−0.823772	−	0.566921i	$-0.808135\pi$
$223$	−24.6031	−1.64754	−0.823772	+	0.566921i	$0.808135\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	0.758411	0.0504487
$227$	25.0022	1.65945	0.829727	−	0.558169i	$-0.188496\pi$
$227$	25.0022	1.65945	0.829727	+	0.558169i	$0.188496\pi$
$228$	0	0
$229$	16.6872	1.10272	0.551359	−	0.834268i	$-0.314109\pi$
$229$	16.6872	1.10272	0.551359	+	0.834268i	$0.314109\pi$
$230$	−3.20177	−0.211118
$231$	0	0
$232$	−8.44335	−0.554333
$233$	27.7734	1.81950	0.909749	−	0.415160i	$-0.136274\pi$
$233$	27.7734	1.81950	0.909749	+	0.415160i	$0.136274\pi$
$234$	0	0
$235$	23.7292	1.54792
$236$	9.28583	0.604456
$237$	0	0
$238$	3.80044	0.246346
$239$	−1.68494	−0.108990	−0.0544950	−	0.998514i	$-0.517355\pi$
$239$	−1.68494	−0.108990	−0.0544950	+	0.998514i	$0.517355\pi$
$240$	0	0
$241$	−15.4013	−0.992087	−0.496043	−	0.868298i	$-0.665214\pi$
$241$	−15.4013	−0.992087	−0.496043	+	0.868298i	$0.665214\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	8.75841	0.560700
$245$	−3.80044	−0.242801
$246$	0	0
$247$	−42.8026	−2.72347
$248$	9.40132	0.596985
$249$	0	0
$250$	16.8867	1.06801
$251$	−17.9159	−1.13084	−0.565422	−	0.824802i	$-0.691287\pi$
$251$	−17.9159	−1.13084	−0.565422	+	0.824802i	$0.691287\pi$
$252$	0	0
$253$	0.842472	0.0529658
$254$	−2.84247	−0.178353
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.0442	0.626542	0.313271	−	0.949664i	$-0.398575\pi$
$257$	10.0442	0.626542	0.313271	+	0.949664i	$0.398575\pi$
$258$	0	0
$259$	9.60088	0.596570
$260$	24.4876	1.51866
$261$	0	0
$262$	−14.1996	−0.877252
$263$	−11.7292	−0.723252	−0.361626	−	0.932323i	$-0.617778\pi$
$263$	−11.7292	−0.723252	−0.361626	+	0.932323i	$0.617778\pi$
$264$	0	0
$265$	−6.08406	−0.373741
$266$	−6.64291	−0.407303
$267$	0	0
$268$	1.15753	0.0707073
$269$	−20.1996	−1.23159	−0.615794	−	0.787907i	$-0.711165\pi$
$269$	−20.1996	−1.23159	−0.615794	+	0.787907i	$0.711165\pi$
$270$	0	0
$271$	−8.39912	−0.510210	−0.255105	−	0.966913i	$-0.582110\pi$
$271$	−8.39912	−0.510210	−0.255105	+	0.966913i	$0.582110\pi$
$272$	−3.80044	−0.230436
$273$	0	0
$274$	12.7584	0.770764
$275$	−9.44335	−0.569456
$276$	0	0
$277$	−8.88671	−0.533951	−0.266975	−	0.963703i	$-0.586024\pi$
$277$	−8.88671	−0.533951	−0.266975	+	0.963703i	$0.586024\pi$
$278$	8.55885	0.513326
$279$	0	0
$280$	3.80044	0.227120
$281$	4.31506	0.257415	0.128707	−	0.991683i	$-0.458917\pi$
$281$	4.31506	0.257415	0.128707	+	0.991683i	$0.458917\pi$
$282$	0	0
$283$	−2.15532	−0.128121	−0.0640603	−	0.997946i	$-0.520405\pi$
$283$	−2.15532	−0.128121	−0.0640603	+	0.997946i	$0.520405\pi$
$284$	6.75841	0.401038
$285$	0	0
$286$	−6.44335	−0.381004
$287$	3.80044	0.224333
$288$	0	0
$289$	−2.55665	−0.150391
$290$	32.0885	1.88430
$291$	0	0
$292$	−7.80044	−0.456486
$293$	1.60088	0.0935246	0.0467623	−	0.998906i	$-0.485110\pi$
$293$	1.60088	0.0935246	0.0467623	+	0.998906i	$0.485110\pi$
$294$	0	0
$295$	−35.2902	−2.05468
$296$	−9.60088	−0.558040
$297$	0	0
$298$	10.8867	0.630650
$299$	5.42835	0.313929
$300$	0	0
$301$	−8.04424	−0.463662
$302$	−6.31506	−0.363391
$303$	0	0
$304$	6.64291	0.380997
$305$	−33.2858	−1.90594
$306$	0	0
$307$	−7.52962	−0.429738	−0.214869	−	0.976643i	$-0.568933\pi$
$307$	−7.52962	−0.429738	−0.214869	+	0.976643i	$0.568933\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−35.7292	−2.02928
$311$	−6.24380	−0.354053	−0.177027	−	0.984206i	$-0.556648\pi$
$311$	−6.24380	−0.354053	−0.177027	+	0.984206i	$0.556648\pi$
$312$	0	0
$313$	−0.714173	−0.0403675	−0.0201837	−	0.999796i	$-0.506425\pi$
$313$	−0.714173	−0.0403675	−0.0201837	+	0.999796i	$0.506425\pi$
$314$	12.1996	0.688461
$315$	0	0
$316$	2.84247	0.159902
$317$	−8.71417	−0.489437	−0.244718	−	0.969594i	$-0.578696\pi$
$317$	−8.71417	−0.489437	−0.244718	+	0.969594i	$0.578696\pi$
$318$	0	0
$319$	−8.44335	−0.472737
$320$	−3.80044	−0.212451
$321$	0	0
$322$	0.842472	0.0469491
$323$	−25.2460	−1.40473
$324$	0	0
$325$	−60.8469	−3.37518
$326$	5.68494	0.314860
$327$	0	0
$328$	−3.80044	−0.209844
$329$	−6.24380	−0.344232
$330$	0	0
$331$	6.04424	0.332221	0.166111	−	0.986107i	$-0.446879\pi$
$331$	6.04424	0.332221	0.166111	+	0.986107i	$0.446879\pi$
$332$	9.80044	0.537869
$333$	0	0
$334$	−24.4876	−1.33990
$335$	−4.39912	−0.240349
$336$	0	0
$337$	−34.4876	−1.87866	−0.939329	−	0.343016i	$-0.888551\pi$
$337$	−34.4876	−1.87866	−0.939329	+	0.343016i	$0.888551\pi$
$338$	−28.5168	−1.55111
$339$	0	0
$340$	14.4434	0.783301
$341$	9.40132	0.509110
$342$	0	0
$343$	1.00000	0.0539949
$344$	8.04424	0.433716
$345$	0	0
$346$	−15.2858	−0.821771
$347$	7.60088	0.408037	0.204018	−	0.978967i	$-0.434600\pi$
$347$	7.60088	0.408037	0.204018	+	0.978967i	$0.434600\pi$
$348$	0	0
$349$	14.8425	0.794499	0.397250	−	0.917711i	$-0.369965\pi$
$349$	14.8425	0.794499	0.397250	+	0.917711i	$0.369965\pi$
$350$	−9.44335	−0.504769
$351$	0	0
$352$	1.00000	0.0533002
$353$	10.0442	0.534601	0.267300	−	0.963613i	$-0.413868\pi$
$353$	10.0442	0.534601	0.267300	+	0.963613i	$0.413868\pi$
$354$	0	0
$355$	−25.6849	−1.36322
$356$	5.15753	0.273348
$357$	0	0
$358$	−25.2460	−1.33429
$359$	−3.47258	−0.183276	−0.0916380	−	0.995792i	$-0.529210\pi$
$359$	−3.47258	−0.183276	−0.0916380	+	0.995792i	$0.529210\pi$
$360$	0	0
$361$	25.1283	1.32254
$362$	7.80044	0.409982
$363$	0	0
$364$	−6.44335	−0.337724
$365$	29.6451	1.55170
$366$	0	0
$367$	1.40132	0.0731485	0.0365743	−	0.999331i	$-0.488355\pi$
$367$	1.40132	0.0731485	0.0365743	+	0.999331i	$0.488355\pi$
$368$	−0.842472	−0.0439169
$369$	0	0
$370$	36.4876	1.89690
$371$	1.60088	0.0831137
$372$	0	0
$373$	13.6451	0.706518	0.353259	−	0.935526i	$-0.385074\pi$
$373$	13.6451	0.706518	0.353259	+	0.935526i	$0.385074\pi$
$374$	−3.80044	−0.196516
$375$	0	0
$376$	6.24380	0.321999
$377$	−54.4035	−2.80192
$378$	0	0
$379$	9.51682	0.488846	0.244423	−	0.969669i	$-0.421401\pi$
$379$	9.51682	0.488846	0.244423	+	0.969669i	$0.421401\pi$
$380$	−25.2460	−1.29509
$381$	0	0
$382$	20.9309	1.07092
$383$	−15.0420	−0.768612	−0.384306	−	0.923206i	$-0.625559\pi$
$383$	−15.0420	−0.768612	−0.384306	+	0.923206i	$0.625559\pi$
$384$	0	0
$385$	3.80044	0.193688
$386$	−5.20177	−0.264763
$387$	0	0
$388$	18.8867	0.958827
$389$	−2.63011	−0.133352	−0.0666760	−	0.997775i	$-0.521239\pi$
$389$	−2.63011	−0.133352	−0.0666760	+	0.997775i	$0.521239\pi$
$390$	0	0
$391$	3.20177	0.161920
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	13.3301	0.671559
$395$	−10.8026	−0.543540
$396$	0	0
$397$	−2.91373	−0.146236	−0.0731180	−	0.997323i	$-0.523295\pi$
$397$	−2.91373	−0.146236	−0.0731180	+	0.997323i	$0.523295\pi$
$398$	13.8004	0.691754
$399$	0	0
$400$	9.44335	0.472168
$401$	−12.7584	−0.637125	−0.318562	−	0.947902i	$-0.603200\pi$
$401$	−12.7584	−0.637125	−0.318562	+	0.947902i	$0.603200\pi$
$402$	0	0
$403$	60.5761	3.01751
$404$	−10.8867	−0.541634
$405$	0	0
$406$	−8.44335	−0.419037
$407$	−9.60088	−0.475898
$408$	0	0
$409$	−4.59868	−0.227390	−0.113695	−	0.993516i	$-0.536269\pi$
$409$	−4.59868	−0.227390	−0.113695	+	0.993516i	$0.536269\pi$
$410$	14.4434	0.713306
$411$	0	0
$412$	5.80044	0.285767
$413$	9.28583	0.456926
$414$	0	0
$415$	−37.2460	−1.82833
$416$	6.44335	0.315911
$417$	0	0
$418$	6.64291	0.324916
$419$	−21.7734	−1.06370	−0.531851	−	0.846838i	$-0.678503\pi$
$419$	−21.7734	−1.06370	−0.531851	+	0.846838i	$0.678503\pi$
$420$	0	0
$421$	29.2018	1.42321	0.711603	−	0.702581i	$-0.247969\pi$
$421$	29.2018	1.42321	0.711603	+	0.702581i	$0.247969\pi$
$422$	−13.7292	−0.668326
$423$	0	0
$424$	−1.60088	−0.0777457
$425$	−35.8889	−1.74087
$426$	0	0
$427$	8.75841	0.423849
$428$	7.60088	0.367403
$429$	0	0
$430$	−30.5717	−1.47430
$431$	−22.0442	−1.06183	−0.530917	−	0.847424i	$-0.678152\pi$
$431$	−22.0442	−1.06183	−0.530917	+	0.847424i	$0.678152\pi$
$432$	0	0
$433$	−16.4035	−0.788303	−0.394152	−	0.919045i	$-0.628962\pi$
$433$	−16.4035	−0.788303	−0.394152	+	0.919045i	$0.628962\pi$
$434$	9.40132	0.451278
$435$	0	0
$436$	8.00000	0.383131
$437$	−5.59647	−0.267715
$438$	0	0
$439$	17.2858	0.825008	0.412504	−	0.910956i	$-0.364654\pi$
$439$	17.2858	0.825008	0.412504	+	0.910956i	$0.364654\pi$
$440$	−3.80044	−0.181179
$441$	0	0
$442$	−24.4876	−1.16476
$443$	1.95576	0.0929211	0.0464605	−	0.998920i	$-0.485206\pi$
$443$	1.95576	0.0929211	0.0464605	+	0.998920i	$0.485206\pi$
$444$	0	0
$445$	−19.6009	−0.929171
$446$	24.6031	1.16499
$447$	0	0
$448$	1.00000	0.0472456
$449$	−24.7584	−1.16842	−0.584211	−	0.811602i	$-0.698596\pi$
$449$	−24.7584	−1.16842	−0.584211	+	0.811602i	$0.698596\pi$
$450$	0	0
$451$	−3.80044	−0.178956
$452$	−0.758411	−0.0356726
$453$	0	0
$454$	−25.0022	−1.17341
$455$	24.4876	1.14800
$456$	0	0
$457$	22.0885	1.03326	0.516628	−	0.856210i	$-0.327187\pi$
$457$	22.0885	1.03326	0.516628	+	0.856210i	$0.327187\pi$
$458$	−16.6872	−0.779739
$459$	0	0
$460$	3.20177	0.149283
$461$	39.7734	1.85243	0.926216	−	0.376992i	$-0.123042\pi$
$461$	39.7734	1.85243	0.926216	+	0.376992i	$0.123042\pi$
$462$	0	0
$463$	−7.20177	−0.334694	−0.167347	−	0.985898i	$-0.553520\pi$
$463$	−7.20177	−0.334694	−0.167347	+	0.985898i	$0.553520\pi$
$464$	8.44335	0.391973
$465$	0	0
$466$	−27.7734	−1.28658
$467$	−1.68494	−0.0779699	−0.0389850	−	0.999240i	$-0.512412\pi$
$467$	−1.68494	−0.0779699	−0.0389850	+	0.999240i	$0.512412\pi$
$468$	0	0
$469$	1.15753	0.0534497
$470$	−23.7292	−1.09455
$471$	0	0
$472$	−9.28583	−0.427415
$473$	8.04424	0.369874
$474$	0	0
$475$	62.7314	2.87831
$476$	−3.80044	−0.174193
$477$	0	0
$478$	1.68494	0.0770675
$479$	27.2018	1.24288	0.621440	−	0.783462i	$-0.286548\pi$
$479$	27.2018	1.24288	0.621440	+	0.783462i	$0.286548\pi$
$480$	0	0
$481$	−61.8619	−2.82066
$482$	15.4013	0.701511
$483$	0	0
$484$	1.00000	0.0454545
$485$	−71.7778	−3.25926
$486$	0	0
$487$	13.9159	0.630591	0.315296	−	0.948993i	$-0.397896\pi$
$487$	13.9159	0.630591	0.315296	+	0.948993i	$0.397896\pi$
$488$	−8.75841	−0.396475
$489$	0	0
$490$	3.80044	0.171686
$491$	−3.20177	−0.144494	−0.0722468	−	0.997387i	$-0.523017\pi$
$491$	−3.20177	−0.144494	−0.0722468	+	0.997387i	$0.523017\pi$
$492$	0	0
$493$	−32.0885	−1.44519
$494$	42.8026	1.92578
$495$	0	0
$496$	−9.40132	−0.422132
$497$	6.75841	0.303156
$498$	0	0
$499$	−17.2460	−0.772037	−0.386019	−	0.922491i	$-0.626150\pi$
$499$	−17.2460	−0.772037	−0.386019	+	0.922491i	$0.626150\pi$
$500$	−16.8867	−0.755197
$501$	0	0
$502$	17.9159	0.799627
$503$	0.487592	0.0217407	0.0108703	−	0.999941i	$-0.496540\pi$
$503$	0.487592	0.0217407	0.0108703	+	0.999941i	$0.496540\pi$
$504$	0	0
$505$	41.3743	1.84113
$506$	−0.842472	−0.0374525
$507$	0	0
$508$	2.84247	0.126114
$509$	28.7756	1.27546	0.637729	−	0.770261i	$-0.279874\pi$
$509$	28.7756	1.27546	0.637729	+	0.770261i	$0.279874\pi$
$510$	0	0
$511$	−7.80044	−0.345071
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−10.0442	−0.443032
$515$	−22.0442	−0.971385
$516$	0	0
$517$	6.24380	0.274602
$518$	−9.60088	−0.421839
$519$	0	0
$520$	−24.4876	−1.07385
$521$	25.2460	1.10605	0.553024	−	0.833165i	$-0.313474\pi$
$521$	25.2460	1.10605	0.553024	+	0.833165i	$0.313474\pi$
$522$	0	0
$523$	37.0464	1.61993	0.809964	−	0.586480i	$-0.199487\pi$
$523$	37.0464	1.61993	0.809964	+	0.586480i	$0.199487\pi$
$524$	14.1996	0.620311
$525$	0	0
$526$	11.7292	0.511417
$527$	35.7292	1.55639
$528$	0	0
$529$	−22.2902	−0.969141
$530$	6.08406	0.264275
$531$	0	0
$532$	6.64291	0.288007
$533$	−24.4876	−1.06068
$534$	0	0
$535$	−28.8867	−1.24888
$536$	−1.15753	−0.0499976
$537$	0	0
$538$	20.1996	0.870865
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	30.5318	1.31267	0.656333	−	0.754471i	$-0.272107\pi$
$541$	30.5318	1.31267	0.656333	+	0.754471i	$0.272107\pi$
$542$	8.39912	0.360773
$543$	0	0
$544$	3.80044	0.162943
$545$	−30.4035	−1.30234
$546$	0	0
$547$	45.1619	1.93099	0.965493	−	0.260430i	$-0.0838645\pi$
$547$	45.1619	1.93099	0.965493	+	0.260430i	$0.0838645\pi$
$548$	−12.7584	−0.545012
$549$	0	0
$550$	9.44335	0.402666
$551$	56.0885	2.38945
$552$	0	0
$553$	2.84247	0.120874
$554$	8.88671	0.377560
$555$	0	0
$556$	−8.55885	−0.362976
$557$	9.96018	0.422026	0.211013	−	0.977483i	$-0.432324\pi$
$557$	9.96018	0.422026	0.211013	+	0.977483i	$0.432324\pi$
$558$	0	0
$559$	51.8319	2.19225
$560$	−3.80044	−0.160598
$561$	0	0
$562$	−4.31506	−0.182020
$563$	33.8004	1.42452	0.712259	−	0.701916i	$-0.247672\pi$
$563$	33.8004	1.42452	0.712259	+	0.701916i	$0.247672\pi$
$564$	0	0
$565$	2.88230	0.121259
$566$	2.15532	0.0905949
$567$	0	0
$568$	−6.75841	−0.283577
$569$	−31.6849	−1.32830	−0.664151	−	0.747598i	$-0.731207\pi$
$569$	−31.6849	−1.32830	−0.664151	+	0.747598i	$0.731207\pi$
$570$	0	0
$571$	−0.443355	−0.0185538	−0.00927691	−	0.999957i	$-0.502953\pi$
$571$	−0.443355	−0.0185538	−0.00927691	+	0.999957i	$0.502953\pi$
$572$	6.44335	0.269410
$573$	0	0
$574$	−3.80044	−0.158627
$575$	−7.95576	−0.331778
$576$	0	0
$577$	−0.714173	−0.0297314	−0.0148657	−	0.999889i	$-0.504732\pi$
$577$	−0.714173	−0.0297314	−0.0148657	+	0.999889i	$0.504732\pi$
$578$	2.55665	0.106342
$579$	0	0
$580$	−32.0885	−1.33240
$581$	9.80044	0.406591
$582$	0	0
$583$	−1.60088	−0.0663018
$584$	7.80044	0.322785
$585$	0	0
$586$	−1.60088	−0.0661319
$587$	−4.39912	−0.181571	−0.0907855	−	0.995870i	$-0.528938\pi$
$587$	−4.39912	−0.181571	−0.0907855	+	0.995870i	$0.528938\pi$
$588$	0	0
$589$	−62.4522	−2.57330
$590$	35.2902	1.45288
$591$	0	0
$592$	9.60088	0.394594
$593$	34.2040	1.40459	0.702294	−	0.711887i	$-0.252159\pi$
$593$	34.2040	1.40459	0.702294	+	0.711887i	$0.252159\pi$
$594$	0	0
$595$	14.4434	0.592120
$596$	−10.8867	−0.445937
$597$	0	0
$598$	−5.42835	−0.221982
$599$	−5.24159	−0.214166	−0.107083	−	0.994250i	$-0.534151\pi$
$599$	−5.24159	−0.214166	−0.107083	+	0.994250i	$0.534151\pi$
$600$	0	0
$601$	16.0314	0.653936	0.326968	−	0.945035i	$-0.393973\pi$
$601$	16.0314	0.653936	0.326968	+	0.945035i	$0.393973\pi$
$602$	8.04424	0.327859
$603$	0	0
$604$	6.31506	0.256956
$605$	−3.80044	−0.154510
$606$	0	0
$607$	31.2902	1.27003	0.635016	−	0.772499i	$-0.280994\pi$
$607$	31.2902	1.27003	0.635016	+	0.772499i	$0.280994\pi$
$608$	−6.64291	−0.269406
$609$	0	0
$610$	33.2858	1.34770
$611$	40.2310	1.62757
$612$	0	0
$613$	15.8717	0.641052	0.320526	−	0.947240i	$-0.396140\pi$
$613$	15.8717	0.641052	0.320526	+	0.947240i	$0.396140\pi$
$614$	7.52962	0.303871
$615$	0	0
$616$	1.00000	0.0402911
$617$	22.3151	0.898370	0.449185	−	0.893439i	$-0.351714\pi$
$617$	22.3151	0.898370	0.449185	+	0.893439i	$0.351714\pi$
$618$	0	0
$619$	−5.68494	−0.228497	−0.114249	−	0.993452i	$-0.536446\pi$
$619$	−5.68494	−0.228497	−0.114249	+	0.993452i	$0.536446\pi$
$620$	35.7292	1.43492
$621$	0	0
$622$	6.24380	0.250353
$623$	5.15753	0.206632
$624$	0	0
$625$	16.9602	0.678407
$626$	0.714173	0.0285441
$627$	0	0
$628$	−12.1996	−0.486815
$629$	−36.4876	−1.45486
$630$	0	0
$631$	46.1725	1.83810	0.919050	−	0.394141i	$-0.128958\pi$
$631$	46.1725	1.83810	0.919050	+	0.394141i	$0.128958\pi$
$632$	−2.84247	−0.113067
$633$	0	0
$634$	8.71417	0.346084
$635$	−10.8026	−0.428690
$636$	0	0
$637$	−6.44335	−0.255295
$638$	8.44335	0.334276
$639$	0	0
$640$	3.80044	0.150226
$641$	−35.4584	−1.40052	−0.700261	−	0.713887i	$-0.746933\pi$
$641$	−35.4584	−1.40052	−0.700261	+	0.713887i	$0.746933\pi$
$642$	0	0
$643$	−6.71417	−0.264781	−0.132391	−	0.991198i	$-0.542265\pi$
$643$	−6.71417	−0.264781	−0.132391	+	0.991198i	$0.542265\pi$
$644$	−0.842472	−0.0331980
$645$	0	0
$646$	25.2460	0.993291
$647$	−30.2438	−1.18901	−0.594503	−	0.804093i	$-0.702651\pi$
$647$	−30.2438	−1.18901	−0.594503	+	0.804093i	$0.702651\pi$
$648$	0	0
$649$	−9.28583	−0.364501
$650$	60.8469	2.38661
$651$	0	0
$652$	−5.68494	−0.222640
$653$	−15.7734	−0.617262	−0.308631	−	0.951182i	$-0.599871\pi$
$653$	−15.7734	−0.617262	−0.308631	+	0.951182i	$0.599871\pi$
$654$	0	0
$655$	−53.9646	−2.10857
$656$	3.80044	0.148382
$657$	0	0
$658$	6.24380	0.243409
$659$	39.8575	1.55263	0.776314	−	0.630347i	$-0.217087\pi$
$659$	39.8575	1.55263	0.776314	+	0.630347i	$0.217087\pi$
$660$	0	0
$661$	35.2588	1.37141	0.685704	−	0.727880i	$-0.259494\pi$
$661$	35.2588	1.37141	0.685704	+	0.727880i	$0.259494\pi$
$662$	−6.04424	−0.234916
$663$	0	0
$664$	−9.80044	−0.380331
$665$	−25.2460	−0.978998
$666$	0	0
$667$	−7.11329	−0.275428
$668$	24.4876	0.947453
$669$	0	0
$670$	4.39912	0.169953
$671$	−8.75841	−0.338115
$672$	0	0
$673$	19.9159	0.767703	0.383852	−	0.923395i	$-0.374597\pi$
$673$	19.9159	0.767703	0.383852	+	0.923395i	$0.374597\pi$
$674$	34.4876	1.32841
$675$	0	0
$676$	28.5168	1.09680
$677$	−45.6894	−1.75598	−0.877992	−	0.478675i	$-0.841117\pi$
$677$	−45.6894	−1.75598	−0.877992	+	0.478675i	$0.841117\pi$
$678$	0	0
$679$	18.8867	0.724805
$680$	−14.4434	−0.553877
$681$	0	0
$682$	−9.40132	−0.359995
$683$	20.3593	0.779027	0.389513	−	0.921021i	$-0.372643\pi$
$683$	20.3593	0.779027	0.389513	+	0.921021i	$0.372643\pi$
$684$	0	0
$685$	48.4876	1.85262
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−8.04424	−0.306684
$689$	−10.3151	−0.392972
$690$	0	0
$691$	24.2310	0.921790	0.460895	−	0.887455i	$-0.347528\pi$
$691$	24.2310	0.921790	0.460895	+	0.887455i	$0.347528\pi$
$692$	15.2858	0.581080
$693$	0	0
$694$	−7.60088	−0.288526
$695$	32.5274	1.23384
$696$	0	0
$697$	−14.4434	−0.547081
$698$	−14.8425	−0.561796
$699$	0	0
$700$	9.44335	0.356925
$701$	14.2566	0.538464	0.269232	−	0.963075i	$-0.413230\pi$
$701$	14.2566	0.538464	0.269232	+	0.963075i	$0.413230\pi$
$702$	0	0
$703$	63.7778	2.40543
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−10.0442	−0.378020
$707$	−10.8867	−0.409437
$708$	0	0
$709$	−10.6557	−0.400184	−0.200092	−	0.979777i	$-0.564124\pi$
$709$	−10.6557	−0.400184	−0.200092	+	0.979777i	$0.564124\pi$
$710$	25.6849	0.963939
$711$	0	0
$712$	−5.15753	−0.193287
$713$	7.92035	0.296620
$714$	0	0
$715$	−24.4876	−0.915784
$716$	25.2460	0.943487
$717$	0	0
$718$	3.47258	0.129596
$719$	−28.5589	−1.06507	−0.532533	−	0.846409i	$-0.678760\pi$
$719$	−28.5589	−1.06507	−0.532533	+	0.846409i	$0.678760\pi$
$720$	0	0
$721$	5.80044	0.216020
$722$	−25.1283	−0.935178
$723$	0	0
$724$	−7.80044	−0.289901
$725$	79.7336	2.96123
$726$	0	0
$727$	−6.03144	−0.223694	−0.111847	−	0.993725i	$-0.535677\pi$
$727$	−6.03144	−0.223694	−0.111847	+	0.993725i	$0.535677\pi$
$728$	6.44335	0.238807
$729$	0	0
$730$	−29.6451	−1.09722
$731$	30.5717	1.13073
$732$	0	0
$733$	3.33006	0.122999	0.0614994	−	0.998107i	$-0.480412\pi$
$733$	3.33006	0.122999	0.0614994	+	0.998107i	$0.480412\pi$
$734$	−1.40132	−0.0517238
$735$	0	0
$736$	0.842472	0.0310539
$737$	−1.15753	−0.0426381
$738$	0	0
$739$	6.66994	0.245358	0.122679	−	0.992446i	$-0.460852\pi$
$739$	6.66994	0.245358	0.122679	+	0.992446i	$0.460852\pi$
$740$	−36.4876	−1.34131
$741$	0	0
$742$	−1.60088	−0.0587703
$743$	−15.2018	−0.557699	−0.278849	−	0.960335i	$-0.589953\pi$
$743$	−15.2018	−0.557699	−0.278849	+	0.960335i	$0.589953\pi$
$744$	0	0
$745$	41.3743	1.51584
$746$	−13.6451	−0.499583
$747$	0	0
$748$	3.80044	0.138958
$749$	7.60088	0.277730
$750$	0	0
$751$	−8.23099	−0.300353	−0.150177	−	0.988659i	$-0.547984\pi$
$751$	−8.23099	−0.300353	−0.150177	+	0.988659i	$0.547984\pi$
$752$	−6.24380	−0.227688
$753$	0	0
$754$	54.4035	1.98126
$755$	−24.0000	−0.873449
$756$	0	0
$757$	−49.6894	−1.80599	−0.902995	−	0.429651i	$-0.858637\pi$
$757$	−49.6894	−1.80599	−0.902995	+	0.429651i	$0.858637\pi$
$758$	−9.51682	−0.345667
$759$	0	0
$760$	25.2460	0.915769
$761$	0.111083	0.00402677	0.00201338	−	0.999998i	$-0.499359\pi$
$761$	0.111083	0.00402677	0.00201338	+	0.999998i	$0.499359\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	−20.9309	−0.757255
$765$	0	0
$766$	15.0420	0.543491
$767$	−59.8319	−2.16040
$768$	0	0
$769$	45.5739	1.64344	0.821718	−	0.569895i	$-0.193016\pi$
$769$	45.5739	1.64344	0.821718	+	0.569895i	$0.193016\pi$
$770$	−3.80044	−0.136958
$771$	0	0
$772$	5.20177	0.187216
$773$	−22.2040	−0.798621	−0.399311	−	0.916816i	$-0.630751\pi$
$773$	−22.2040	−0.798621	−0.399311	+	0.916816i	$0.630751\pi$
$774$	0	0
$775$	−88.7800	−3.18907
$776$	−18.8867	−0.677993
$777$	0	0
$778$	2.63011	0.0942941
$779$	25.2460	0.904532
$780$	0	0
$781$	−6.75841	−0.241835
$782$	−3.20177	−0.114495
$783$	0	0
$784$	1.00000	0.0357143
$785$	46.3637	1.65479
$786$	0	0
$787$	−51.1305	−1.82261	−0.911303	−	0.411737i	$-0.864922\pi$
$787$	−51.1305	−1.82261	−0.911303	+	0.411737i	$0.864922\pi$
$788$	−13.3301	−0.474864
$789$	0	0
$790$	10.8026	0.384341
$791$	−0.758411	−0.0269660
$792$	0	0
$793$	−56.4335	−2.00401
$794$	2.91373	0.103404
$795$	0	0
$796$	−13.8004	−0.489144
$797$	−32.6872	−1.15784	−0.578919	−	0.815385i	$-0.696525\pi$
$797$	−32.6872	−1.15784	−0.578919	+	0.815385i	$0.696525\pi$
$798$	0	0
$799$	23.7292	0.839478
$800$	−9.44335	−0.333873
$801$	0	0
$802$	12.7584	0.450515
$803$	7.80044	0.275272
$804$	0	0
$805$	3.20177	0.112847
$806$	−60.5761	−2.13370
$807$	0	0
$808$	10.8867	0.382993
$809$	0.0840613	0.00295544	0.00147772	−	0.999999i	$-0.499530\pi$
$809$	0.0840613	0.00295544	0.00147772	+	0.999999i	$0.499530\pi$
$810$	0	0
$811$	11.6977	0.410763	0.205382	−	0.978682i	$-0.434156\pi$
$811$	11.6977	0.410763	0.205382	+	0.978682i	$0.434156\pi$
$812$	8.44335	0.296304
$813$	0	0
$814$	9.60088	0.336511
$815$	21.6053	0.756801
$816$	0	0
$817$	−53.4372	−1.86953
$818$	4.59868	0.160789
$819$	0	0
$820$	−14.4434	−0.504384
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	46.1725	1.60947	0.804737	−	0.593632i	$-0.202306\pi$
$823$	46.1725	1.60947	0.804737	+	0.593632i	$0.202306\pi$
$824$	−5.80044	−0.202068
$825$	0	0
$826$	−9.28583	−0.323095
$827$	27.2018	0.945898	0.472949	−	0.881090i	$-0.343189\pi$
$827$	27.2018	0.945898	0.472949	+	0.881090i	$0.343189\pi$
$828$	0	0
$829$	−19.2588	−0.668886	−0.334443	−	0.942416i	$-0.608548\pi$
$829$	−19.2588	−0.668886	−0.334443	+	0.942416i	$0.608548\pi$
$830$	37.2460	1.29283
$831$	0	0
$832$	−6.44335	−0.223383
$833$	−3.80044	−0.131677
$834$	0	0
$835$	−93.0637	−3.22060
$836$	−6.64291	−0.229750
$837$	0	0
$838$	21.7734	0.752150
$839$	−28.5589	−0.985961	−0.492981	−	0.870040i	$-0.664093\pi$
$839$	−28.5589	−0.985961	−0.492981	+	0.870040i	$0.664093\pi$
$840$	0	0
$841$	42.2902	1.45828
$842$	−29.2018	−1.00636
$843$	0	0
$844$	13.7292	0.472578
$845$	−108.377	−3.72827
$846$	0	0
$847$	1.00000	0.0343604
$848$	1.60088	0.0549745
$849$	0	0
$850$	35.8889	1.23098
$851$	−8.08848	−0.277269
$852$	0	0
$853$	10.2752	0.351817	0.175909	−	0.984406i	$-0.443714\pi$
$853$	10.2752	0.351817	0.175909	+	0.984406i	$0.443714\pi$
$854$	−8.75841	−0.299707
$855$	0	0
$856$	−7.60088	−0.259793
$857$	−57.1747	−1.95305	−0.976526	−	0.215399i	$-0.930895\pi$
$857$	−57.1747	−1.95305	−0.976526	+	0.215399i	$0.930895\pi$
$858$	0	0
$859$	5.82746	0.198830	0.0994152	−	0.995046i	$-0.468303\pi$
$859$	5.82746	0.198830	0.0994152	+	0.995046i	$0.468303\pi$
$860$	30.5717	1.04248
$861$	0	0
$862$	22.0442	0.750830
$863$	46.4478	1.58110	0.790550	−	0.612397i	$-0.209795\pi$
$863$	46.4478	1.58110	0.790550	+	0.612397i	$0.209795\pi$
$864$	0	0
$865$	−58.0929	−1.97522
$866$	16.4035	0.557415
$867$	0	0
$868$	−9.40132	−0.319102
$869$	−2.84247	−0.0964243
$870$	0	0
$871$	−7.45836	−0.252717
$872$	−8.00000	−0.270914
$873$	0	0
$874$	5.59647	0.189303
$875$	−16.8867	−0.570875
$876$	0	0
$877$	11.2018	0.378257	0.189128	−	0.981952i	$-0.439434\pi$
$877$	11.2018	0.378257	0.189128	+	0.981952i	$0.439434\pi$
$878$	−17.2858	−0.583368
$879$	0	0
$880$	3.80044	0.128113
$881$	34.7000	1.16907	0.584536	−	0.811368i	$-0.301277\pi$
$881$	34.7000	1.16907	0.584536	+	0.811368i	$0.301277\pi$
$882$	0	0
$883$	39.9204	1.34343	0.671713	−	0.740811i	$-0.265559\pi$
$883$	39.9204	1.34343	0.671713	+	0.740811i	$0.265559\pi$
$884$	24.4876	0.823607
$885$	0	0
$886$	−1.95576	−0.0657051
$887$	8.08848	0.271584	0.135792	−	0.990737i	$-0.456642\pi$
$887$	8.08848	0.271584	0.135792	+	0.990737i	$0.456642\pi$
$888$	0	0
$889$	2.84247	0.0953335
$890$	19.6009	0.657023
$891$	0	0
$892$	−24.6031	−0.823772
$893$	−41.4770	−1.38797
$894$	0	0
$895$	−95.9460	−3.20712
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	24.7584	0.826199
$899$	−79.3787	−2.64743
$900$	0	0
$901$	−6.08406	−0.202689
$902$	3.80044	0.126541
$903$	0	0
$904$	0.758411	0.0252244
$905$	29.6451	0.985437
$906$	0	0
$907$	−5.95576	−0.197758	−0.0988789	−	0.995099i	$-0.531526\pi$
$907$	−5.95576	−0.197758	−0.0988789	+	0.995099i	$0.531526\pi$
$908$	25.0022	0.829727
$909$	0	0
$910$	−24.4876	−0.811756
$911$	22.4478	0.743728	0.371864	−	0.928287i	$-0.378719\pi$
$911$	22.4478	0.743728	0.371864	+	0.928287i	$0.378719\pi$
$912$	0	0
$913$	−9.80044	−0.324347
$914$	−22.0885	−0.730622
$915$	0	0
$916$	16.6872	0.551359
$917$	14.1996	0.468911
$918$	0	0
$919$	24.8867	0.820937	0.410468	−	0.911875i	$-0.365365\pi$
$919$	24.8867	0.820937	0.410468	+	0.911875i	$0.365365\pi$
$920$	−3.20177	−0.105559
$921$	0	0
$922$	−39.7734	−1.30987
$923$	−43.5468	−1.43336
$924$	0	0
$925$	90.6645	2.98103
$926$	7.20177	0.236665
$927$	0	0
$928$	−8.44335	−0.277167
$929$	−15.4726	−0.507639	−0.253820	−	0.967252i	$-0.581687\pi$
$929$	−15.4726	−0.507639	−0.253820	+	0.967252i	$0.581687\pi$
$930$	0	0
$931$	6.64291	0.217713
$932$	27.7734	0.909749
$933$	0	0
$934$	1.68494	0.0551331
$935$	−14.4434	−0.472348
$936$	0	0
$937$	17.8845	0.584261	0.292131	−	0.956378i	$-0.405636\pi$
$937$	17.8845	0.584261	0.292131	+	0.956378i	$0.405636\pi$
$938$	−1.15753	−0.0377946
$939$	0	0
$940$	23.7292	0.773961
$941$	6.65572	0.216970	0.108485	−	0.994098i	$-0.465400\pi$
$941$	6.65572	0.216970	0.108485	+	0.994098i	$0.465400\pi$
$942$	0	0
$943$	−3.20177	−0.104264
$944$	9.28583	0.302228
$945$	0	0
$946$	−8.04424	−0.261541
$947$	32.2566	1.04820	0.524099	−	0.851657i	$-0.324402\pi$
$947$	32.2566	1.04820	0.524099	+	0.851657i	$0.324402\pi$
$948$	0	0
$949$	50.2610	1.63154
$950$	−62.7314	−2.03528
$951$	0	0
$952$	3.80044	0.123173
$953$	−37.6009	−1.21801	−0.609006	−	0.793166i	$-0.708431\pi$
$953$	−37.6009	−1.21801	−0.609006	+	0.793166i	$0.708431\pi$
$954$	0	0
$955$	79.5468	2.57408
$956$	−1.68494	−0.0544950
$957$	0	0
$958$	−27.2018	−0.878849
$959$	−12.7584	−0.411991
$960$	0	0
$961$	57.3849	1.85113
$962$	61.8619	1.99451
$963$	0	0
$964$	−15.4013	−0.496043
$965$	−19.7690	−0.636387
$966$	0	0
$967$	−2.48318	−0.0798536	−0.0399268	−	0.999203i	$-0.512712\pi$
$967$	−2.48318	−0.0798536	−0.0399268	+	0.999203i	$0.512712\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	71.7778	2.30465
$971$	−13.6849	−0.439171	−0.219585	−	0.975593i	$-0.570470\pi$
$971$	−13.6849	−0.439171	−0.219585	+	0.975593i	$0.570470\pi$
$972$	0	0
$973$	−8.55885	−0.274384
$974$	−13.9159	−0.445895
$975$	0	0
$976$	8.75841	0.280350
$977$	13.6849	0.437820	0.218910	−	0.975745i	$-0.429750\pi$
$977$	13.6849	0.437820	0.218910	+	0.975745i	$0.429750\pi$
$978$	0	0
$979$	−5.15753	−0.164835
$980$	−3.80044	−0.121401
$981$	0	0
$982$	3.20177	0.102172
$983$	−31.2190	−0.995731	−0.497865	−	0.867254i	$-0.665883\pi$
$983$	−31.2190	−0.995731	−0.497865	+	0.867254i	$0.665883\pi$
$984$	0	0
$985$	50.6601	1.61417
$986$	32.0885	1.02191
$987$	0	0
$988$	−42.8026	−1.36173
$989$	6.77705	0.215498
$990$	0	0
$991$	9.51682	0.302312	0.151156	−	0.988510i	$-0.451700\pi$
$991$	9.51682	0.302312	0.151156	+	0.988510i	$0.451700\pi$
$992$	9.40132	0.298492
$993$	0	0
$994$	−6.75841	−0.214364
$995$	52.4478	1.66271
$996$	0	0
$997$	−17.9558	−0.568665	−0.284332	−	0.958726i	$-0.591772\pi$
$997$	−17.9558	−0.568665	−0.284332	+	0.958726i	$0.591772\pi$
$998$	17.2460	0.545913
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.a.q.1.1	✓	3
3.2	odd	2		1386.2.a.r.1.3	yes	3
7.6	odd	2		9702.2.a.dx.1.3		3
21.20	even	2		9702.2.a.dy.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.a.q.1.1	✓	3	1.1	even	1	trivial
1386.2.a.r.1.3	yes	3	3.2	odd	2
9702.2.a.dx.1.3		3	7.6	odd	2
9702.2.a.dy.1.1		3	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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.80044 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.80044 q^{5} +1.00000 q^{7} -1.00000 q^{8} +3.80044 q^{10} -1.00000 q^{11} -6.44335 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.80044 q^{17} +6.64291 q^{19} -3.80044 q^{20} +1.00000 q^{22} -0.842472 q^{23} +9.44335 q^{25} +6.44335 q^{26} +1.00000 q^{28} +8.44335 q^{29} -9.40132 q^{31} -1.00000 q^{32} +3.80044 q^{34} -3.80044 q^{35} +9.60088 q^{37} -6.64291 q^{38} +3.80044 q^{40} +3.80044 q^{41} -8.04424 q^{43} -1.00000 q^{44} +0.842472 q^{46} -6.24380 q^{47} +1.00000 q^{49} -9.44335 q^{50} -6.44335 q^{52} +1.60088 q^{53} +3.80044 q^{55} -1.00000 q^{56} -8.44335 q^{58} +9.28583 q^{59} +8.75841 q^{61} +9.40132 q^{62} +1.00000 q^{64} +24.4876 q^{65} +1.15753 q^{67} -3.80044 q^{68} +3.80044 q^{70} +6.75841 q^{71} -7.80044 q^{73} -9.60088 q^{74} +6.64291 q^{76} -1.00000 q^{77} +2.84247 q^{79} -3.80044 q^{80} -3.80044 q^{82} +9.80044 q^{83} +14.4434 q^{85} +8.04424 q^{86} +1.00000 q^{88} +5.15753 q^{89} -6.44335 q^{91} -0.842472 q^{92} +6.24380 q^{94} -25.2460 q^{95} +18.8867 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} - 3 q^{8} + 2 q^{10} - 3 q^{11} - 3 q^{14} + 3 q^{16} - 2 q^{17} + 10 q^{19} - 2 q^{20} + 3 q^{22} - 2 q^{23} + 9 q^{25} + 3 q^{28} + 6 q^{29} - 3 q^{32} + 2 q^{34} - 2 q^{35} + 10 q^{37} - 10 q^{38} + 2 q^{40} + 2 q^{41} + 14 q^{43} - 3 q^{44} + 2 q^{46} + 10 q^{47} + 3 q^{49} - 9 q^{50} - 14 q^{53} + 2 q^{55} - 3 q^{56} - 6 q^{58} + 8 q^{59} + 8 q^{61} + 3 q^{64} + 16 q^{65} + 4 q^{67} - 2 q^{68} + 2 q^{70} + 2 q^{71} - 14 q^{73} - 10 q^{74} + 10 q^{76} - 3 q^{77} + 8 q^{79} - 2 q^{80} - 2 q^{82} + 20 q^{83} + 24 q^{85} - 14 q^{86} + 3 q^{88} + 16 q^{89} - 2 q^{92} - 10 q^{94} + 18 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 - 3 * q^8 + 2 * q^10 - 3 * q^11 - 3 * q^14 + 3 * q^16 - 2 * q^17 + 10 * q^19 - 2 * q^20 + 3 * q^22 - 2 * q^23 + 9 * q^25 + 3 * q^28 + 6 * q^29 - 3 * q^32 + 2 * q^34 - 2 * q^35 + 10 * q^37 - 10 * q^38 + 2 * q^40 + 2 * q^41 + 14 * q^43 - 3 * q^44 + 2 * q^46 + 10 * q^47 + 3 * q^49 - 9 * q^50 - 14 * q^53 + 2 * q^55 - 3 * q^56 - 6 * q^58 + 8 * q^59 + 8 * q^61 + 3 * q^64 + 16 * q^65 + 4 * q^67 - 2 * q^68 + 2 * q^70 + 2 * q^71 - 14 * q^73 - 10 * q^74 + 10 * q^76 - 3 * q^77 + 8 * q^79 - 2 * q^80 - 2 * q^82 + 20 * q^83 + 24 * q^85 - 14 * q^86 + 3 * q^88 + 16 * q^89 - 2 * q^92 - 10 * q^94 + 18 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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