LMFDB - Embedded newform 1638.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.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:	$13.0794958511$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} -2.37228 q^{10} -2.37228 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.37228 q^{17} +1.62772 q^{19} +2.37228 q^{20} +2.37228 q^{22} -3.62772 q^{23} +0.627719 q^{25} +1.00000 q^{26} -1.00000 q^{28} -6.37228 q^{29} -4.74456 q^{31} -1.00000 q^{32} +4.37228 q^{34} -2.37228 q^{35} -4.37228 q^{37} -1.62772 q^{38} -2.37228 q^{40} -8.74456 q^{41} +11.1168 q^{43} -2.37228 q^{44} +3.62772 q^{46} -1.25544 q^{47} +1.00000 q^{49} -0.627719 q^{50} -1.00000 q^{52} -8.74456 q^{53} -5.62772 q^{55} +1.00000 q^{56} +6.37228 q^{58} -2.00000 q^{59} +5.11684 q^{61} +4.74456 q^{62} +1.00000 q^{64} -2.37228 q^{65} +9.48913 q^{67} -4.37228 q^{68} +2.37228 q^{70} +4.74456 q^{71} -8.37228 q^{73} +4.37228 q^{74} +1.62772 q^{76} +2.37228 q^{77} +4.74456 q^{79} +2.37228 q^{80} +8.74456 q^{82} +6.00000 q^{83} -10.3723 q^{85} -11.1168 q^{86} +2.37228 q^{88} +3.25544 q^{89} +1.00000 q^{91} -3.62772 q^{92} +1.25544 q^{94} +3.86141 q^{95} +7.48913 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8} + q^{10} + q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} + 9 q^{19} - q^{20} - q^{22} - 13 q^{23} + 7 q^{25} + 2 q^{26} - 2 q^{28} - 7 q^{29} + 2 q^{31} - 2 q^{32} + 3 q^{34} + q^{35} - 3 q^{37} - 9 q^{38} + q^{40} - 6 q^{41} + 5 q^{43} + q^{44} + 13 q^{46} - 14 q^{47} + 2 q^{49} - 7 q^{50} - 2 q^{52} - 6 q^{53} - 17 q^{55} + 2 q^{56} + 7 q^{58} - 4 q^{59} - 7 q^{61} - 2 q^{62} + 2 q^{64} + q^{65} - 4 q^{67} - 3 q^{68} - q^{70} - 2 q^{71} - 11 q^{73} + 3 q^{74} + 9 q^{76} - q^{77} - 2 q^{79} - q^{80} + 6 q^{82} + 12 q^{83} - 15 q^{85} - 5 q^{86} - q^{88} + 18 q^{89} + 2 q^{91} - 13 q^{92} + 14 q^{94} - 21 q^{95} - 8 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8 + q^10 + q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 + 9 * q^19 - q^20 - q^22 - 13 * q^23 + 7 * q^25 + 2 * q^26 - 2 * q^28 - 7 * q^29 + 2 * q^31 - 2 * q^32 + 3 * q^34 + q^35 - 3 * q^37 - 9 * q^38 + q^40 - 6 * q^41 + 5 * q^43 + q^44 + 13 * q^46 - 14 * q^47 + 2 * q^49 - 7 * q^50 - 2 * q^52 - 6 * q^53 - 17 * q^55 + 2 * q^56 + 7 * q^58 - 4 * q^59 - 7 * q^61 - 2 * q^62 + 2 * q^64 + q^65 - 4 * q^67 - 3 * q^68 - q^70 - 2 * q^71 - 11 * q^73 + 3 * q^74 + 9 * q^76 - q^77 - 2 * q^79 - q^80 + 6 * q^82 + 12 * q^83 - 15 * q^85 - 5 * q^86 - q^88 + 18 * q^89 + 2 * q^91 - 13 * q^92 + 14 * q^94 - 21 * q^95 - 8 * q^97 - 2 * 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
$5$	2.37228	1.06092	0.530458	−	0.847711i	$-0.322020\pi$
$5$	2.37228	1.06092	0.530458	+	0.847711i	$0.322020\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.37228	−0.750181
$11$	−2.37228	−0.715270	−0.357635	−	0.933862i	$-0.616417\pi$
$11$	−2.37228	−0.715270	−0.357635	+	0.933862i	$0.616417\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.37228	−1.06043	−0.530217	−	0.847862i	$-0.677890\pi$
$17$	−4.37228	−1.06043	−0.530217	+	0.847862i	$0.677890\pi$
$18$	0	0
$19$	1.62772	0.373424	0.186712	−	0.982415i	$-0.440217\pi$
$19$	1.62772	0.373424	0.186712	+	0.982415i	$0.440217\pi$
$20$	2.37228	0.530458
$21$	0	0
$22$	2.37228	0.505772
$23$	−3.62772	−0.756432	−0.378216	−	0.925717i	$-0.623462\pi$
$23$	−3.62772	−0.756432	−0.378216	+	0.925717i	$0.623462\pi$
$24$	0	0
$25$	0.627719	0.125544
$26$	1.00000	0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−6.37228	−1.18330	−0.591651	−	0.806194i	$-0.701524\pi$
$29$	−6.37228	−1.18330	−0.591651	+	0.806194i	$0.701524\pi$
$30$	0	0
$31$	−4.74456	−0.852149	−0.426074	−	0.904688i	$-0.640104\pi$
$31$	−4.74456	−0.852149	−0.426074	+	0.904688i	$0.640104\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.37228	0.749840
$35$	−2.37228	−0.400989
$36$	0	0
$37$	−4.37228	−0.718799	−0.359399	−	0.933184i	$-0.617018\pi$
$37$	−4.37228	−0.718799	−0.359399	+	0.933184i	$0.617018\pi$
$38$	−1.62772	−0.264051
$39$	0	0
$40$	−2.37228	−0.375091
$41$	−8.74456	−1.36567	−0.682836	−	0.730572i	$-0.739253\pi$
$41$	−8.74456	−1.36567	−0.682836	+	0.730572i	$0.739253\pi$
$42$	0	0
$43$	11.1168	1.69530	0.847651	−	0.530554i	$-0.178016\pi$
$43$	11.1168	1.69530	0.847651	+	0.530554i	$0.178016\pi$
$44$	−2.37228	−0.357635
$45$	0	0
$46$	3.62772	0.534878
$47$	−1.25544	−0.183124	−0.0915622	−	0.995799i	$-0.529186\pi$
$47$	−1.25544	−0.183124	−0.0915622	+	0.995799i	$0.529186\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.627719	−0.0887728
$51$	0	0
$52$	−1.00000	−0.138675
$53$	−8.74456	−1.20116	−0.600579	−	0.799565i	$-0.705063\pi$
$53$	−8.74456	−1.20116	−0.600579	+	0.799565i	$0.705063\pi$
$54$	0	0
$55$	−5.62772	−0.758841
$56$	1.00000	0.133631
$57$	0	0
$58$	6.37228	0.836722
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	5.11684	0.655145	0.327572	−	0.944826i	$-0.393769\pi$
$61$	5.11684	0.655145	0.327572	+	0.944826i	$0.393769\pi$
$62$	4.74456	0.602560
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.37228	−0.294245
$66$	0	0
$67$	9.48913	1.15928	0.579641	−	0.814872i	$-0.303193\pi$
$67$	9.48913	1.15928	0.579641	+	0.814872i	$0.303193\pi$
$68$	−4.37228	−0.530217
$69$	0	0
$70$	2.37228	0.283542
$71$	4.74456	0.563076	0.281538	−	0.959550i	$-0.409155\pi$
$71$	4.74456	0.563076	0.281538	+	0.959550i	$0.409155\pi$
$72$	0	0
$73$	−8.37228	−0.979901	−0.489951	−	0.871750i	$-0.662985\pi$
$73$	−8.37228	−0.979901	−0.489951	+	0.871750i	$0.662985\pi$
$74$	4.37228	0.508267
$75$	0	0
$76$	1.62772	0.186712
$77$	2.37228	0.270347
$78$	0	0
$79$	4.74456	0.533805	0.266903	−	0.963724i	$-0.414000\pi$
$79$	4.74456	0.533805	0.266903	+	0.963724i	$0.414000\pi$
$80$	2.37228	0.265229
$81$	0	0
$82$	8.74456	0.965675
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−10.3723	−1.12503
$86$	−11.1168	−1.19876
$87$	0	0
$88$	2.37228	0.252886
$89$	3.25544	0.345076	0.172538	−	0.985003i	$-0.444803\pi$
$89$	3.25544	0.345076	0.172538	+	0.985003i	$0.444803\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−3.62772	−0.378216
$93$	0	0
$94$	1.25544	0.129488
$95$	3.86141	0.396172
$96$	0	0
$97$	7.48913	0.760405	0.380203	−	0.924903i	$-0.375854\pi$
$97$	7.48913	0.760405	0.380203	+	0.924903i	$0.375854\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	0.627719	0.0627719
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−11.8614	−1.16874	−0.584370	−	0.811488i	$-0.698658\pi$
$103$	−11.8614	−1.16874	−0.584370	+	0.811488i	$0.698658\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	8.74456	0.849347
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	−3.62772	−0.347472	−0.173736	−	0.984792i	$-0.555584\pi$
$109$	−3.62772	−0.347472	−0.173736	+	0.984792i	$0.555584\pi$
$110$	5.62772	0.536582
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−20.0000	−1.88144	−0.940721	−	0.339182i	$-0.889850\pi$
$113$	−20.0000	−1.88144	−0.940721	+	0.339182i	$0.889850\pi$
$114$	0	0
$115$	−8.60597	−0.802511
$116$	−6.37228	−0.591651
$117$	0	0
$118$	2.00000	0.184115
$119$	4.37228	0.400806
$120$	0	0
$121$	−5.37228	−0.488389
$122$	−5.11684	−0.463257
$123$	0	0
$124$	−4.74456	−0.426074
$125$	−10.3723	−0.927725
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.37228	0.208063
$131$	11.1168	0.971283	0.485642	−	0.874158i	$-0.338586\pi$
$131$	11.1168	0.971283	0.485642	+	0.874158i	$0.338586\pi$
$132$	0	0
$133$	−1.62772	−0.141141
$134$	−9.48913	−0.819736
$135$	0	0
$136$	4.37228	0.374920
$137$	−1.11684	−0.0954184	−0.0477092	−	0.998861i	$-0.515192\pi$
$137$	−1.11684	−0.0954184	−0.0477092	+	0.998861i	$0.515192\pi$
$138$	0	0
$139$	0.744563	0.0631530	0.0315765	−	0.999501i	$-0.489947\pi$
$139$	0.744563	0.0631530	0.0315765	+	0.999501i	$0.489947\pi$
$140$	−2.37228	−0.200494
$141$	0	0
$142$	−4.74456	−0.398155
$143$	2.37228	0.198380
$144$	0	0
$145$	−15.1168	−1.25539
$146$	8.37228	0.692895
$147$	0	0
$148$	−4.37228	−0.359399
$149$	20.2337	1.65761	0.828804	−	0.559539i	$-0.189022\pi$
$149$	20.2337	1.65761	0.828804	+	0.559539i	$0.189022\pi$
$150$	0	0
$151$	19.1168	1.55571	0.777853	−	0.628446i	$-0.216309\pi$
$151$	19.1168	1.55571	0.777853	+	0.628446i	$0.216309\pi$
$152$	−1.62772	−0.132025
$153$	0	0
$154$	−2.37228	−0.191164
$155$	−11.2554	−0.904058
$156$	0	0
$157$	−20.3723	−1.62589	−0.812943	−	0.582344i	$-0.802136\pi$
$157$	−20.3723	−1.62589	−0.812943	+	0.582344i	$0.802136\pi$
$158$	−4.74456	−0.377457
$159$	0	0
$160$	−2.37228	−0.187545
$161$	3.62772	0.285904
$162$	0	0
$163$	−18.2337	−1.42817	−0.714086	−	0.700058i	$-0.753158\pi$
$163$	−18.2337	−1.42817	−0.714086	+	0.700058i	$0.753158\pi$
$164$	−8.74456	−0.682836
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−13.1168	−1.01501	−0.507506	−	0.861648i	$-0.669432\pi$
$167$	−13.1168	−1.01501	−0.507506	+	0.861648i	$0.669432\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	10.3723	0.795518
$171$	0	0
$172$	11.1168	0.847651
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	−0.627719	−0.0474511
$176$	−2.37228	−0.178817
$177$	0	0
$178$	−3.25544	−0.244005
$179$	2.74456	0.205138	0.102569	−	0.994726i	$-0.467294\pi$
$179$	2.74456	0.205138	0.102569	+	0.994726i	$0.467294\pi$
$180$	0	0
$181$	−23.4891	−1.74593	−0.872966	−	0.487780i	$-0.837807\pi$
$181$	−23.4891	−1.74593	−0.872966	+	0.487780i	$0.837807\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	3.62772	0.267439
$185$	−10.3723	−0.762585
$186$	0	0
$187$	10.3723	0.758496
$188$	−1.25544	−0.0915622
$189$	0	0
$190$	−3.86141	−0.280136
$191$	−5.11684	−0.370242	−0.185121	−	0.982716i	$-0.559268\pi$
$191$	−5.11684	−0.370242	−0.185121	+	0.982716i	$0.559268\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−7.48913	−0.537688
$195$	0	0
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	11.8614	0.840833	0.420416	−	0.907331i	$-0.361884\pi$
$199$	11.8614	0.840833	0.420416	+	0.907331i	$0.361884\pi$
$200$	−0.627719	−0.0443864
$201$	0	0
$202$	−2.00000	−0.140720
$203$	6.37228	0.447246
$204$	0	0
$205$	−20.7446	−1.44886
$206$	11.8614	0.826423
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−3.86141	−0.267099
$210$	0	0
$211$	−17.6277	−1.21354	−0.606771	−	0.794877i	$-0.707536\pi$
$211$	−17.6277	−1.21354	−0.606771	+	0.794877i	$0.707536\pi$
$212$	−8.74456	−0.600579
$213$	0	0
$214$	2.00000	0.136717
$215$	26.3723	1.79857
$216$	0	0
$217$	4.74456	0.322082
$218$	3.62772	0.245700
$219$	0	0
$220$	−5.62772	−0.379421
$221$	4.37228	0.294111
$222$	0	0
$223$	−17.4891	−1.17116	−0.585579	−	0.810615i	$-0.699133\pi$
$223$	−17.4891	−1.17116	−0.585579	+	0.810615i	$0.699133\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	20.0000	1.33038
$227$	−11.4891	−0.762560	−0.381280	−	0.924460i	$-0.624517\pi$
$227$	−11.4891	−0.762560	−0.381280	+	0.924460i	$0.624517\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	8.60597	0.567461
$231$	0	0
$232$	6.37228	0.418361
$233$	1.48913	0.0975558	0.0487779	−	0.998810i	$-0.484467\pi$
$233$	1.48913	0.0975558	0.0487779	+	0.998810i	$0.484467\pi$
$234$	0	0
$235$	−2.97825	−0.194280
$236$	−2.00000	−0.130189
$237$	0	0
$238$	−4.37228	−0.283413
$239$	5.48913	0.355062	0.177531	−	0.984115i	$-0.443189\pi$
$239$	5.48913	0.355062	0.177531	+	0.984115i	$0.443189\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	5.37228	0.345343
$243$	0	0
$244$	5.11684	0.327572
$245$	2.37228	0.151559
$246$	0	0
$247$	−1.62772	−0.103569
$248$	4.74456	0.301280
$249$	0	0
$250$	10.3723	0.656001
$251$	13.6277	0.860174	0.430087	−	0.902787i	$-0.358483\pi$
$251$	13.6277	0.860174	0.430087	+	0.902787i	$0.358483\pi$
$252$	0	0
$253$	8.60597	0.541053
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.4891	−0.716672	−0.358336	−	0.933593i	$-0.616656\pi$
$257$	−11.4891	−0.716672	−0.358336	+	0.933593i	$0.616656\pi$
$258$	0	0
$259$	4.37228	0.271680
$260$	−2.37228	−0.147123
$261$	0	0
$262$	−11.1168	−0.686801
$263$	−9.25544	−0.570715	−0.285357	−	0.958421i	$-0.592112\pi$
$263$	−9.25544	−0.570715	−0.285357	+	0.958421i	$0.592112\pi$
$264$	0	0
$265$	−20.7446	−1.27433
$266$	1.62772	0.0998018
$267$	0	0
$268$	9.48913	0.579641
$269$	−0.510875	−0.0311486	−0.0155743	−	0.999879i	$-0.504958\pi$
$269$	−0.510875	−0.0311486	−0.0155743	+	0.999879i	$0.504958\pi$
$270$	0	0
$271$	30.2337	1.83657	0.918283	−	0.395925i	$-0.129576\pi$
$271$	30.2337	1.83657	0.918283	+	0.395925i	$0.129576\pi$
$272$	−4.37228	−0.265108
$273$	0	0
$274$	1.11684	0.0674710
$275$	−1.48913	−0.0897976
$276$	0	0
$277$	26.7446	1.60693	0.803463	−	0.595355i	$-0.202989\pi$
$277$	26.7446	1.60693	0.803463	+	0.595355i	$0.202989\pi$
$278$	−0.744563	−0.0446559
$279$	0	0
$280$	2.37228	0.141771
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	14.9783	0.890365	0.445182	−	0.895440i	$-0.353139\pi$
$283$	14.9783	0.890365	0.445182	+	0.895440i	$0.353139\pi$
$284$	4.74456	0.281538
$285$	0	0
$286$	−2.37228	−0.140276
$287$	8.74456	0.516175
$288$	0	0
$289$	2.11684	0.124520
$290$	15.1168	0.887692
$291$	0	0
$292$	−8.37228	−0.489951
$293$	28.7446	1.67928	0.839638	−	0.543147i	$-0.182767\pi$
$293$	28.7446	1.67928	0.839638	+	0.543147i	$0.182767\pi$
$294$	0	0
$295$	−4.74456	−0.276239
$296$	4.37228	0.254134
$297$	0	0
$298$	−20.2337	−1.17211
$299$	3.62772	0.209796
$300$	0	0
$301$	−11.1168	−0.640764
$302$	−19.1168	−1.10005
$303$	0	0
$304$	1.62772	0.0933561
$305$	12.1386	0.695054
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	2.37228	0.135173
$309$	0	0
$310$	11.2554	0.639266
$311$	−17.4891	−0.991717	−0.495859	−	0.868403i	$-0.665147\pi$
$311$	−17.4891	−0.991717	−0.495859	+	0.868403i	$0.665147\pi$
$312$	0	0
$313$	12.9783	0.733574	0.366787	−	0.930305i	$-0.380458\pi$
$313$	12.9783	0.733574	0.366787	+	0.930305i	$0.380458\pi$
$314$	20.3723	1.14967
$315$	0	0
$316$	4.74456	0.266903
$317$	28.9783	1.62758	0.813790	−	0.581159i	$-0.197400\pi$
$317$	28.9783	1.62758	0.813790	+	0.581159i	$0.197400\pi$
$318$	0	0
$319$	15.1168	0.846381
$320$	2.37228	0.132615
$321$	0	0
$322$	−3.62772	−0.202165
$323$	−7.11684	−0.395992
$324$	0	0
$325$	−0.627719	−0.0348196
$326$	18.2337	1.00987
$327$	0	0
$328$	8.74456	0.482838
$329$	1.25544	0.0692145
$330$	0	0
$331$	16.7446	0.920364	0.460182	−	0.887824i	$-0.347784\pi$
$331$	16.7446	0.920364	0.460182	+	0.887824i	$0.347784\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	13.1168	0.717722
$335$	22.5109	1.22990
$336$	0	0
$337$	−1.86141	−0.101397	−0.0506986	−	0.998714i	$-0.516145\pi$
$337$	−1.86141	−0.101397	−0.0506986	+	0.998714i	$0.516145\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	−10.3723	−0.562516
$341$	11.2554	0.609516
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−11.1168	−0.599380
$345$	0	0
$346$	−10.0000	−0.537603
$347$	−25.7228	−1.38087	−0.690436	−	0.723393i	$-0.742582\pi$
$347$	−25.7228	−1.38087	−0.690436	+	0.723393i	$0.742582\pi$
$348$	0	0
$349$	−22.7446	−1.21749	−0.608744	−	0.793367i	$-0.708326\pi$
$349$	−22.7446	−1.21749	−0.608744	+	0.793367i	$0.708326\pi$
$350$	0.627719	0.0335530
$351$	0	0
$352$	2.37228	0.126443
$353$	4.74456	0.252528	0.126264	−	0.991997i	$-0.459701\pi$
$353$	4.74456	0.252528	0.126264	+	0.991997i	$0.459701\pi$
$354$	0	0
$355$	11.2554	0.597377
$356$	3.25544	0.172538
$357$	0	0
$358$	−2.74456	−0.145055
$359$	3.25544	0.171815	0.0859077	−	0.996303i	$-0.472621\pi$
$359$	3.25544	0.171815	0.0859077	+	0.996303i	$0.472621\pi$
$360$	0	0
$361$	−16.3505	−0.860554
$362$	23.4891	1.23456
$363$	0	0
$364$	1.00000	0.0524142
$365$	−19.8614	−1.03959
$366$	0	0
$367$	2.51087	0.131067	0.0655333	−	0.997850i	$-0.479125\pi$
$367$	2.51087	0.131067	0.0655333	+	0.997850i	$0.479125\pi$
$368$	−3.62772	−0.189108
$369$	0	0
$370$	10.3723	0.539229
$371$	8.74456	0.453995
$372$	0	0
$373$	−15.4891	−0.801997	−0.400998	−	0.916079i	$-0.631337\pi$
$373$	−15.4891	−0.801997	−0.400998	+	0.916079i	$0.631337\pi$
$374$	−10.3723	−0.536338
$375$	0	0
$376$	1.25544	0.0647442
$377$	6.37228	0.328189
$378$	0	0
$379$	−28.7446	−1.47651	−0.738255	−	0.674522i	$-0.764350\pi$
$379$	−28.7446	−1.47651	−0.738255	+	0.674522i	$0.764350\pi$
$380$	3.86141	0.198086
$381$	0	0
$382$	5.11684	0.261801
$383$	−6.60597	−0.337549	−0.168775	−	0.985655i	$-0.553981\pi$
$383$	−6.60597	−0.337549	−0.168775	+	0.985655i	$0.553981\pi$
$384$	0	0
$385$	5.62772	0.286815
$386$	−2.00000	−0.101797
$387$	0	0
$388$	7.48913	0.380203
$389$	−3.25544	−0.165057	−0.0825286	−	0.996589i	$-0.526300\pi$
$389$	−3.25544	−0.165057	−0.0825286	+	0.996589i	$0.526300\pi$
$390$	0	0
$391$	15.8614	0.802146
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	6.00000	0.302276
$395$	11.2554	0.566323
$396$	0	0
$397$	5.25544	0.263763	0.131881	−	0.991265i	$-0.457898\pi$
$397$	5.25544	0.263763	0.131881	+	0.991265i	$0.457898\pi$
$398$	−11.8614	−0.594559
$399$	0	0
$400$	0.627719	0.0313859
$401$	35.4891	1.77224	0.886121	−	0.463454i	$-0.153390\pi$
$401$	35.4891	1.77224	0.886121	+	0.463454i	$0.153390\pi$
$402$	0	0
$403$	4.74456	0.236343
$404$	2.00000	0.0995037
$405$	0	0
$406$	−6.37228	−0.316251
$407$	10.3723	0.514135
$408$	0	0
$409$	25.8614	1.27876	0.639382	−	0.768889i	$-0.279190\pi$
$409$	25.8614	1.27876	0.639382	+	0.768889i	$0.279190\pi$
$410$	20.7446	1.02450
$411$	0	0
$412$	−11.8614	−0.584370
$413$	2.00000	0.0984136
$414$	0	0
$415$	14.2337	0.698704
$416$	1.00000	0.0490290
$417$	0	0
$418$	3.86141	0.188868
$419$	3.86141	0.188642	0.0943210	−	0.995542i	$-0.469932\pi$
$419$	3.86141	0.188642	0.0943210	+	0.995542i	$0.469932\pi$
$420$	0	0
$421$	−7.48913	−0.364998	−0.182499	−	0.983206i	$-0.558419\pi$
$421$	−7.48913	−0.364998	−0.182499	+	0.983206i	$0.558419\pi$
$422$	17.6277	0.858104
$423$	0	0
$424$	8.74456	0.424674
$425$	−2.74456	−0.133131
$426$	0	0
$427$	−5.11684	−0.247621
$428$	−2.00000	−0.0966736
$429$	0	0
$430$	−26.3723	−1.27178
$431$	32.7446	1.57725	0.788625	−	0.614874i	$-0.210793\pi$
$431$	32.7446	1.57725	0.788625	+	0.614874i	$0.210793\pi$
$432$	0	0
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	−4.74456	−0.227746
$435$	0	0
$436$	−3.62772	−0.173736
$437$	−5.90491	−0.282470
$438$	0	0
$439$	9.62772	0.459506	0.229753	−	0.973249i	$-0.426208\pi$
$439$	9.62772	0.459506	0.229753	+	0.973249i	$0.426208\pi$
$440$	5.62772	0.268291
$441$	0	0
$442$	−4.37228	−0.207968
$443$	−16.9783	−0.806661	−0.403331	−	0.915054i	$-0.632148\pi$
$443$	−16.9783	−0.806661	−0.403331	+	0.915054i	$0.632148\pi$
$444$	0	0
$445$	7.72281	0.366096
$446$	17.4891	0.828134
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	0.372281	0.0175690	0.00878452	−	0.999961i	$-0.497204\pi$
$449$	0.372281	0.0175690	0.00878452	+	0.999961i	$0.497204\pi$
$450$	0	0
$451$	20.7446	0.976823
$452$	−20.0000	−0.940721
$453$	0	0
$454$	11.4891	0.539211
$455$	2.37228	0.111214
$456$	0	0
$457$	12.2337	0.572268	0.286134	−	0.958190i	$-0.407630\pi$
$457$	12.2337	0.572268	0.286134	+	0.958190i	$0.407630\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	−8.60597	−0.401255
$461$	36.6060	1.70491	0.852455	−	0.522801i	$-0.175113\pi$
$461$	36.6060	1.70491	0.852455	+	0.522801i	$0.175113\pi$
$462$	0	0
$463$	−17.3505	−0.806348	−0.403174	−	0.915123i	$-0.632093\pi$
$463$	−17.3505	−0.806348	−0.403174	+	0.915123i	$0.632093\pi$
$464$	−6.37228	−0.295826
$465$	0	0
$466$	−1.48913	−0.0689824
$467$	34.0951	1.57773	0.788866	−	0.614565i	$-0.210668\pi$
$467$	34.0951	1.57773	0.788866	+	0.614565i	$0.210668\pi$
$468$	0	0
$469$	−9.48913	−0.438167
$470$	2.97825	0.137376
$471$	0	0
$472$	2.00000	0.0920575
$473$	−26.3723	−1.21260
$474$	0	0
$475$	1.02175	0.0468811
$476$	4.37228	0.200403
$477$	0	0
$478$	−5.48913	−0.251067
$479$	−39.3505	−1.79797	−0.898986	−	0.437978i	$-0.855695\pi$
$479$	−39.3505	−1.79797	−0.898986	+	0.437978i	$0.855695\pi$
$480$	0	0
$481$	4.37228	0.199359
$482$	6.00000	0.273293
$483$	0	0
$484$	−5.37228	−0.244195
$485$	17.7663	0.806727
$486$	0	0
$487$	−32.4674	−1.47124	−0.735619	−	0.677396i	$-0.763108\pi$
$487$	−32.4674	−1.47124	−0.735619	+	0.677396i	$0.763108\pi$
$488$	−5.11684	−0.231629
$489$	0	0
$490$	−2.37228	−0.107169
$491$	−25.2554	−1.13976	−0.569881	−	0.821727i	$-0.693011\pi$
$491$	−25.2554	−1.13976	−0.569881	+	0.821727i	$0.693011\pi$
$492$	0	0
$493$	27.8614	1.25481
$494$	1.62772	0.0732345
$495$	0	0
$496$	−4.74456	−0.213037
$497$	−4.74456	−0.212823
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−10.3723	−0.463863
$501$	0	0
$502$	−13.6277	−0.608235
$503$	−21.4891	−0.958153	−0.479076	−	0.877773i	$-0.659028\pi$
$503$	−21.4891	−0.958153	−0.479076	+	0.877773i	$0.659028\pi$
$504$	0	0
$505$	4.74456	0.211130
$506$	−8.60597	−0.382582
$507$	0	0
$508$	−8.00000	−0.354943
$509$	26.0951	1.15664	0.578322	−	0.815808i	$-0.303708\pi$
$509$	26.0951	1.15664	0.578322	+	0.815808i	$0.303708\pi$
$510$	0	0
$511$	8.37228	0.370368
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	11.4891	0.506764
$515$	−28.1386	−1.23993
$516$	0	0
$517$	2.97825	0.130983
$518$	−4.37228	−0.192107
$519$	0	0
$520$	2.37228	0.104031
$521$	28.3723	1.24301	0.621506	−	0.783409i	$-0.286521\pi$
$521$	28.3723	1.24301	0.621506	+	0.783409i	$0.286521\pi$
$522$	0	0
$523$	2.23369	0.0976724	0.0488362	−	0.998807i	$-0.484449\pi$
$523$	2.23369	0.0976724	0.0488362	+	0.998807i	$0.484449\pi$
$524$	11.1168	0.485642
$525$	0	0
$526$	9.25544	0.403556
$527$	20.7446	0.903647
$528$	0	0
$529$	−9.83966	−0.427811
$530$	20.7446	0.901086
$531$	0	0
$532$	−1.62772	−0.0705706
$533$	8.74456	0.378769
$534$	0	0
$535$	−4.74456	−0.205125
$536$	−9.48913	−0.409868
$537$	0	0
$538$	0.510875	0.0220254
$539$	−2.37228	−0.102181
$540$	0	0
$541$	23.3505	1.00392	0.501959	−	0.864891i	$-0.332613\pi$
$541$	23.3505	1.00392	0.501959	+	0.864891i	$0.332613\pi$
$542$	−30.2337	−1.29865
$543$	0	0
$544$	4.37228	0.187460
$545$	−8.60597	−0.368639
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−1.11684	−0.0477092
$549$	0	0
$550$	1.48913	0.0634965
$551$	−10.3723	−0.441874
$552$	0	0
$553$	−4.74456	−0.201759
$554$	−26.7446	−1.13627
$555$	0	0
$556$	0.744563	0.0315765
$557$	−11.4891	−0.486810	−0.243405	−	0.969925i	$-0.578264\pi$
$557$	−11.4891	−0.486810	−0.243405	+	0.969925i	$0.578264\pi$
$558$	0	0
$559$	−11.1168	−0.470192
$560$	−2.37228	−0.100247
$561$	0	0
$562$	10.0000	0.421825
$563$	11.1168	0.468519	0.234260	−	0.972174i	$-0.424733\pi$
$563$	11.1168	0.468519	0.234260	+	0.972174i	$0.424733\pi$
$564$	0	0
$565$	−47.4456	−1.99605
$566$	−14.9783	−0.629583
$567$	0	0
$568$	−4.74456	−0.199077
$569$	−12.7446	−0.534280	−0.267140	−	0.963658i	$-0.586079\pi$
$569$	−12.7446	−0.534280	−0.267140	+	0.963658i	$0.586079\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	2.37228	0.0991901
$573$	0	0
$574$	−8.74456	−0.364991
$575$	−2.27719	−0.0949653
$576$	0	0
$577$	−0.510875	−0.0212680	−0.0106340	−	0.999943i	$-0.503385\pi$
$577$	−0.510875	−0.0212680	−0.0106340	+	0.999943i	$0.503385\pi$
$578$	−2.11684	−0.0880491
$579$	0	0
$580$	−15.1168	−0.627693
$581$	−6.00000	−0.248922
$582$	0	0
$583$	20.7446	0.859152
$584$	8.37228	0.346447
$585$	0	0
$586$	−28.7446	−1.18743
$587$	27.4891	1.13460	0.567299	−	0.823512i	$-0.307988\pi$
$587$	27.4891	1.13460	0.567299	+	0.823512i	$0.307988\pi$
$588$	0	0
$589$	−7.72281	−0.318213
$590$	4.74456	0.195331
$591$	0	0
$592$	−4.37228	−0.179700
$593$	4.00000	0.164260	0.0821302	−	0.996622i	$-0.473828\pi$
$593$	4.00000	0.164260	0.0821302	+	0.996622i	$0.473828\pi$
$594$	0	0
$595$	10.3723	0.425222
$596$	20.2337	0.828804
$597$	0	0
$598$	−3.62772	−0.148348
$599$	21.1168	0.862811	0.431405	−	0.902158i	$-0.358018\pi$
$599$	21.1168	0.862811	0.431405	+	0.902158i	$0.358018\pi$
$600$	0	0
$601$	−42.4674	−1.73228	−0.866140	−	0.499801i	$-0.833406\pi$
$601$	−42.4674	−1.73228	−0.866140	+	0.499801i	$0.833406\pi$
$602$	11.1168	0.453089
$603$	0	0
$604$	19.1168	0.777853
$605$	−12.7446	−0.518140
$606$	0	0
$607$	19.1168	0.775929	0.387964	−	0.921674i	$-0.373178\pi$
$607$	19.1168	0.775929	0.387964	+	0.921674i	$0.373178\pi$
$608$	−1.62772	−0.0660127
$609$	0	0
$610$	−12.1386	−0.491477
$611$	1.25544	0.0507896
$612$	0	0
$613$	−6.13859	−0.247935	−0.123968	−	0.992286i	$-0.539562\pi$
$613$	−6.13859	−0.247935	−0.123968	+	0.992286i	$0.539562\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	−2.37228	−0.0955819
$617$	−9.39403	−0.378189	−0.189095	−	0.981959i	$-0.560555\pi$
$617$	−9.39403	−0.378189	−0.189095	+	0.981959i	$0.560555\pi$
$618$	0	0
$619$	11.1168	0.446824	0.223412	−	0.974724i	$-0.428281\pi$
$619$	11.1168	0.446824	0.223412	+	0.974724i	$0.428281\pi$
$620$	−11.2554	−0.452029
$621$	0	0
$622$	17.4891	0.701250
$623$	−3.25544	−0.130426
$624$	0	0
$625$	−27.7446	−1.10978
$626$	−12.9783	−0.518715
$627$	0	0
$628$	−20.3723	−0.812943
$629$	19.1168	0.762238
$630$	0	0
$631$	16.6060	0.661073	0.330537	−	0.943793i	$-0.392770\pi$
$631$	16.6060	0.661073	0.330537	+	0.943793i	$0.392770\pi$
$632$	−4.74456	−0.188729
$633$	0	0
$634$	−28.9783	−1.15087
$635$	−18.9783	−0.753129
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	−15.1168	−0.598482
$639$	0	0
$640$	−2.37228	−0.0937727
$641$	−18.2337	−0.720187	−0.360094	−	0.932916i	$-0.617255\pi$
$641$	−18.2337	−0.720187	−0.360094	+	0.932916i	$0.617255\pi$
$642$	0	0
$643$	−0.138593	−0.00546559	−0.00273279	−	0.999996i	$-0.500870\pi$
$643$	−0.138593	−0.00546559	−0.00273279	+	0.999996i	$0.500870\pi$
$644$	3.62772	0.142952
$645$	0	0
$646$	7.11684	0.280008
$647$	−21.4891	−0.844825	−0.422412	−	0.906404i	$-0.638817\pi$
$647$	−21.4891	−0.844825	−0.422412	+	0.906404i	$0.638817\pi$
$648$	0	0
$649$	4.74456	0.186240
$650$	0.627719	0.0246212
$651$	0	0
$652$	−18.2337	−0.714086
$653$	−8.13859	−0.318488	−0.159244	−	0.987239i	$-0.550906\pi$
$653$	−8.13859	−0.318488	−0.159244	+	0.987239i	$0.550906\pi$
$654$	0	0
$655$	26.3723	1.03045
$656$	−8.74456	−0.341418
$657$	0	0
$658$	−1.25544	−0.0489420
$659$	−9.25544	−0.360541	−0.180270	−	0.983617i	$-0.557697\pi$
$659$	−9.25544	−0.360541	−0.180270	+	0.983617i	$0.557697\pi$
$660$	0	0
$661$	10.7446	0.417915	0.208958	−	0.977925i	$-0.432993\pi$
$661$	10.7446	0.417915	0.208958	+	0.977925i	$0.432993\pi$
$662$	−16.7446	−0.650796
$663$	0	0
$664$	−6.00000	−0.232845
$665$	−3.86141	−0.149739
$666$	0	0
$667$	23.1168	0.895088
$668$	−13.1168	−0.507506
$669$	0	0
$670$	−22.5109	−0.869671
$671$	−12.1386	−0.468605
$672$	0	0
$673$	16.0951	0.620420	0.310210	−	0.950668i	$-0.399601\pi$
$673$	16.0951	0.620420	0.310210	+	0.950668i	$0.399601\pi$
$674$	1.86141	0.0716987
$675$	0	0
$676$	1.00000	0.0384615
$677$	−35.4891	−1.36396	−0.681979	−	0.731372i	$-0.738880\pi$
$677$	−35.4891	−1.36396	−0.681979	+	0.731372i	$0.738880\pi$
$678$	0	0
$679$	−7.48913	−0.287406
$680$	10.3723	0.397759
$681$	0	0
$682$	−11.2554	−0.430993
$683$	−11.1168	−0.425374	−0.212687	−	0.977120i	$-0.568222\pi$
$683$	−11.1168	−0.425374	−0.212687	+	0.977120i	$0.568222\pi$
$684$	0	0
$685$	−2.64947	−0.101231
$686$	1.00000	0.0381802
$687$	0	0
$688$	11.1168	0.423826
$689$	8.74456	0.333141
$690$	0	0
$691$	45.4891	1.73049	0.865244	−	0.501351i	$-0.167163\pi$
$691$	45.4891	1.73049	0.865244	+	0.501351i	$0.167163\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	25.7228	0.976425
$695$	1.76631	0.0670000
$696$	0	0
$697$	38.2337	1.44820
$698$	22.7446	0.860894
$699$	0	0
$700$	−0.627719	−0.0237255
$701$	−31.7228	−1.19815	−0.599077	−	0.800691i	$-0.704466\pi$
$701$	−31.7228	−1.19815	−0.599077	+	0.800691i	$0.704466\pi$
$702$	0	0
$703$	−7.11684	−0.268417
$704$	−2.37228	−0.0894087
$705$	0	0
$706$	−4.74456	−0.178564
$707$	−2.00000	−0.0752177
$708$	0	0
$709$	−28.5109	−1.07075	−0.535374	−	0.844615i	$-0.679829\pi$
$709$	−28.5109	−1.07075	−0.535374	+	0.844615i	$0.679829\pi$
$710$	−11.2554	−0.422409
$711$	0	0
$712$	−3.25544	−0.122003
$713$	17.2119	0.644592
$714$	0	0
$715$	5.62772	0.210465
$716$	2.74456	0.102569
$717$	0	0
$718$	−3.25544	−0.121492
$719$	45.4891	1.69646	0.848229	−	0.529630i	$-0.177669\pi$
$719$	45.4891	1.69646	0.848229	+	0.529630i	$0.177669\pi$
$720$	0	0
$721$	11.8614	0.441742
$722$	16.3505	0.608504
$723$	0	0
$724$	−23.4891	−0.872966
$725$	−4.00000	−0.148556
$726$	0	0
$727$	46.0951	1.70957	0.854786	−	0.518980i	$-0.173688\pi$
$727$	46.0951	1.70957	0.854786	+	0.518980i	$0.173688\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	19.8614	0.735104
$731$	−48.6060	−1.79776
$732$	0	0
$733$	−30.7446	−1.13558	−0.567788	−	0.823175i	$-0.692201\pi$
$733$	−30.7446	−1.13558	−0.567788	+	0.823175i	$0.692201\pi$
$734$	−2.51087	−0.0926781
$735$	0	0
$736$	3.62772	0.133719
$737$	−22.5109	−0.829199
$738$	0	0
$739$	31.7228	1.16694	0.583471	−	0.812134i	$-0.301694\pi$
$739$	31.7228	1.16694	0.583471	+	0.812134i	$0.301694\pi$
$740$	−10.3723	−0.381293
$741$	0	0
$742$	−8.74456	−0.321023
$743$	−44.4674	−1.63135	−0.815675	−	0.578511i	$-0.803634\pi$
$743$	−44.4674	−1.63135	−0.815675	+	0.578511i	$0.803634\pi$
$744$	0	0
$745$	48.0000	1.75858
$746$	15.4891	0.567097
$747$	0	0
$748$	10.3723	0.379248
$749$	2.00000	0.0730784
$750$	0	0
$751$	−10.9783	−0.400602	−0.200301	−	0.979734i	$-0.564192\pi$
$751$	−10.9783	−0.400602	−0.200301	+	0.979734i	$0.564192\pi$
$752$	−1.25544	−0.0457811
$753$	0	0
$754$	−6.37228	−0.232065
$755$	45.3505	1.65047
$756$	0	0
$757$	−43.2119	−1.57056	−0.785282	−	0.619138i	$-0.787482\pi$
$757$	−43.2119	−1.57056	−0.785282	+	0.619138i	$0.787482\pi$
$758$	28.7446	1.04405
$759$	0	0
$760$	−3.86141	−0.140068
$761$	46.9783	1.70296	0.851480	−	0.524387i	$-0.175705\pi$
$761$	46.9783	1.70296	0.851480	+	0.524387i	$0.175705\pi$
$762$	0	0
$763$	3.62772	0.131332
$764$	−5.11684	−0.185121
$765$	0	0
$766$	6.60597	0.238683
$767$	2.00000	0.0722158
$768$	0	0
$769$	−19.6277	−0.707794	−0.353897	−	0.935284i	$-0.615144\pi$
$769$	−19.6277	−0.707794	−0.353897	+	0.935284i	$0.615144\pi$
$770$	−5.62772	−0.202809
$771$	0	0
$772$	2.00000	0.0719816
$773$	34.0951	1.22632	0.613158	−	0.789961i	$-0.289899\pi$
$773$	34.0951	1.22632	0.613158	+	0.789961i	$0.289899\pi$
$774$	0	0
$775$	−2.97825	−0.106982
$776$	−7.48913	−0.268844
$777$	0	0
$778$	3.25544	0.116713
$779$	−14.2337	−0.509975
$780$	0	0
$781$	−11.2554	−0.402751
$782$	−15.8614	−0.567203
$783$	0	0
$784$	1.00000	0.0357143
$785$	−48.3288	−1.72493
$786$	0	0
$787$	27.1168	0.966611	0.483306	−	0.875452i	$-0.339436\pi$
$787$	27.1168	0.966611	0.483306	+	0.875452i	$0.339436\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−11.2554	−0.400450
$791$	20.0000	0.711118
$792$	0	0
$793$	−5.11684	−0.181704
$794$	−5.25544	−0.186508
$795$	0	0
$796$	11.8614	0.420416
$797$	−49.7228	−1.76127	−0.880636	−	0.473793i	$-0.842884\pi$
$797$	−49.7228	−1.76127	−0.880636	+	0.473793i	$0.842884\pi$
$798$	0	0
$799$	5.48913	0.194191
$800$	−0.627719	−0.0221932
$801$	0	0
$802$	−35.4891	−1.25316
$803$	19.8614	0.700894
$804$	0	0
$805$	8.60597	0.303321
$806$	−4.74456	−0.167120
$807$	0	0
$808$	−2.00000	−0.0703598
$809$	−32.7446	−1.15124	−0.575619	−	0.817718i	$-0.695239\pi$
$809$	−32.7446	−1.15124	−0.575619	+	0.817718i	$0.695239\pi$
$810$	0	0
$811$	3.11684	0.109447	0.0547236	−	0.998502i	$-0.482572\pi$
$811$	3.11684	0.109447	0.0547236	+	0.998502i	$0.482572\pi$
$812$	6.37228	0.223623
$813$	0	0
$814$	−10.3723	−0.363548
$815$	−43.2554	−1.51517
$816$	0	0
$817$	18.0951	0.633067
$818$	−25.8614	−0.904223
$819$	0	0
$820$	−20.7446	−0.724432
$821$	53.7228	1.87494	0.937470	−	0.348067i	$-0.113162\pi$
$821$	53.7228	1.87494	0.937470	+	0.348067i	$0.113162\pi$
$822$	0	0
$823$	−31.7228	−1.10579	−0.552894	−	0.833252i	$-0.686477\pi$
$823$	−31.7228	−1.10579	−0.552894	+	0.833252i	$0.686477\pi$
$824$	11.8614	0.413212
$825$	0	0
$826$	−2.00000	−0.0695889
$827$	−34.3723	−1.19524	−0.597621	−	0.801779i	$-0.703887\pi$
$827$	−34.3723	−1.19524	−0.597621	+	0.801779i	$0.703887\pi$
$828$	0	0
$829$	−24.3723	−0.846484	−0.423242	−	0.906017i	$-0.639108\pi$
$829$	−24.3723	−0.846484	−0.423242	+	0.906017i	$0.639108\pi$
$830$	−14.2337	−0.494059
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	−4.37228	−0.151491
$834$	0	0
$835$	−31.1168	−1.07684
$836$	−3.86141	−0.133550
$837$	0	0
$838$	−3.86141	−0.133390
$839$	1.25544	0.0433425	0.0216713	−	0.999765i	$-0.493101\pi$
$839$	1.25544	0.0433425	0.0216713	+	0.999765i	$0.493101\pi$
$840$	0	0
$841$	11.6060	0.400206
$842$	7.48913	0.258092
$843$	0	0
$844$	−17.6277	−0.606771
$845$	2.37228	0.0816090
$846$	0	0
$847$	5.37228	0.184594
$848$	−8.74456	−0.300290
$849$	0	0
$850$	2.74456	0.0941377
$851$	15.8614	0.543722
$852$	0	0
$853$	−39.2119	−1.34259	−0.671296	−	0.741190i	$-0.734262\pi$
$853$	−39.2119	−1.34259	−0.671296	+	0.741190i	$0.734262\pi$
$854$	5.11684	0.175095
$855$	0	0
$856$	2.00000	0.0683586
$857$	0.978251	0.0334164	0.0167082	−	0.999860i	$-0.494681\pi$
$857$	0.978251	0.0334164	0.0167082	+	0.999860i	$0.494681\pi$
$858$	0	0
$859$	−0.744563	−0.0254041	−0.0127021	−	0.999919i	$-0.504043\pi$
$859$	−0.744563	−0.0254041	−0.0127021	+	0.999919i	$0.504043\pi$
$860$	26.3723	0.899287
$861$	0	0
$862$	−32.7446	−1.11528
$863$	18.9783	0.646027	0.323014	−	0.946394i	$-0.395304\pi$
$863$	18.9783	0.646027	0.323014	+	0.946394i	$0.395304\pi$
$864$	0	0
$865$	23.7228	0.806600
$866$	−10.0000	−0.339814
$867$	0	0
$868$	4.74456	0.161041
$869$	−11.2554	−0.381815
$870$	0	0
$871$	−9.48913	−0.321527
$872$	3.62772	0.122850
$873$	0	0
$874$	5.90491	0.199736
$875$	10.3723	0.350647
$876$	0	0
$877$	−52.9783	−1.78895	−0.894474	−	0.447120i	$-0.852450\pi$
$877$	−52.9783	−1.78895	−0.894474	+	0.447120i	$0.852450\pi$
$878$	−9.62772	−0.324920
$879$	0	0
$880$	−5.62772	−0.189710
$881$	14.1386	0.476341	0.238171	−	0.971223i	$-0.423452\pi$
$881$	14.1386	0.476341	0.238171	+	0.971223i	$0.423452\pi$
$882$	0	0
$883$	−30.3723	−1.02211	−0.511054	−	0.859548i	$-0.670745\pi$
$883$	−30.3723	−1.02211	−0.511054	+	0.859548i	$0.670745\pi$
$884$	4.37228	0.147056
$885$	0	0
$886$	16.9783	0.570395
$887$	−53.2119	−1.78668	−0.893341	−	0.449379i	$-0.851645\pi$
$887$	−53.2119	−1.78668	−0.893341	+	0.449379i	$0.851645\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	−7.72281	−0.258869
$891$	0	0
$892$	−17.4891	−0.585579
$893$	−2.04350	−0.0683831
$894$	0	0
$895$	6.51087	0.217635
$896$	1.00000	0.0334077
$897$	0	0
$898$	−0.372281	−0.0124232
$899$	30.2337	1.00835
$900$	0	0
$901$	38.2337	1.27375
$902$	−20.7446	−0.690718
$903$	0	0
$904$	20.0000	0.665190
$905$	−55.7228	−1.85229
$906$	0	0
$907$	−57.9565	−1.92441	−0.962207	−	0.272319i	$-0.912209\pi$
$907$	−57.9565	−1.92441	−0.962207	+	0.272319i	$0.912209\pi$
$908$	−11.4891	−0.381280
$909$	0	0
$910$	−2.37228	−0.0786404
$911$	−22.8832	−0.758153	−0.379076	−	0.925365i	$-0.623758\pi$
$911$	−22.8832	−0.758153	−0.379076	+	0.925365i	$0.623758\pi$
$912$	0	0
$913$	−14.2337	−0.471066
$914$	−12.2337	−0.404654
$915$	0	0
$916$	6.00000	0.198246
$917$	−11.1168	−0.367111
$918$	0	0
$919$	−49.2119	−1.62335	−0.811676	−	0.584108i	$-0.801444\pi$
$919$	−49.2119	−1.62335	−0.811676	+	0.584108i	$0.801444\pi$
$920$	8.60597	0.283730
$921$	0	0
$922$	−36.6060	−1.20555
$923$	−4.74456	−0.156169
$924$	0	0
$925$	−2.74456	−0.0902407
$926$	17.3505	0.570174
$927$	0	0
$928$	6.37228	0.209180
$929$	−15.2554	−0.500515	−0.250257	−	0.968179i	$-0.580515\pi$
$929$	−15.2554	−0.500515	−0.250257	+	0.968179i	$0.580515\pi$
$930$	0	0
$931$	1.62772	0.0533463
$932$	1.48913	0.0487779
$933$	0	0
$934$	−34.0951	−1.11563
$935$	24.6060	0.804701
$936$	0	0
$937$	24.5109	0.800735	0.400368	−	0.916355i	$-0.368882\pi$
$937$	24.5109	0.800735	0.400368	+	0.916355i	$0.368882\pi$
$938$	9.48913	0.309831
$939$	0	0
$940$	−2.97825	−0.0971398
$941$	−24.7446	−0.806650	−0.403325	−	0.915057i	$-0.632146\pi$
$941$	−24.7446	−0.806650	−0.403325	+	0.915057i	$0.632146\pi$
$942$	0	0
$943$	31.7228	1.03304
$944$	−2.00000	−0.0650945
$945$	0	0
$946$	26.3723	0.857437
$947$	8.13859	0.264469	0.132234	−	0.991218i	$-0.457785\pi$
$947$	8.13859	0.264469	0.132234	+	0.991218i	$0.457785\pi$
$948$	0	0
$949$	8.37228	0.271776
$950$	−1.02175	−0.0331499
$951$	0	0
$952$	−4.37228	−0.141706
$953$	45.4891	1.47354	0.736769	−	0.676145i	$-0.236351\pi$
$953$	45.4891	1.47354	0.736769	+	0.676145i	$0.236351\pi$
$954$	0	0
$955$	−12.1386	−0.392796
$956$	5.48913	0.177531
$957$	0	0
$958$	39.3505	1.27136
$959$	1.11684	0.0360648
$960$	0	0
$961$	−8.48913	−0.273843
$962$	−4.37228	−0.140968
$963$	0	0
$964$	−6.00000	−0.193247
$965$	4.74456	0.152733
$966$	0	0
$967$	24.8832	0.800188	0.400094	−	0.916474i	$-0.368977\pi$
$967$	24.8832	0.800188	0.400094	+	0.916474i	$0.368977\pi$
$968$	5.37228	0.172672
$969$	0	0
$970$	−17.7663	−0.570442
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−0.744563	−0.0238696
$974$	32.4674	1.04032
$975$	0	0
$976$	5.11684	0.163786
$977$	25.1168	0.803559	0.401780	−	0.915736i	$-0.368392\pi$
$977$	25.1168	0.803559	0.401780	+	0.915736i	$0.368392\pi$
$978$	0	0
$979$	−7.72281	−0.246822
$980$	2.37228	0.0757797
$981$	0	0
$982$	25.2554	0.805933
$983$	−35.3505	−1.12751	−0.563753	−	0.825943i	$-0.690643\pi$
$983$	−35.3505	−1.12751	−0.563753	+	0.825943i	$0.690643\pi$
$984$	0	0
$985$	−14.2337	−0.453523
$986$	−27.8614	−0.887288
$987$	0	0
$988$	−1.62772	−0.0517846
$989$	−40.3288	−1.28238
$990$	0	0
$991$	14.2337	0.452148	0.226074	−	0.974110i	$-0.427411\pi$
$991$	14.2337	0.452148	0.226074	+	0.974110i	$0.427411\pi$
$992$	4.74456	0.150640
$993$	0	0
$994$	4.74456	0.150488
$995$	28.1386	0.892053
$996$	0	0
$997$	20.5109	0.649586	0.324793	−	0.945785i	$-0.394705\pi$
$997$	20.5109	0.649586	0.324793	+	0.945785i	$0.394705\pi$
$998$	20.0000	0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.v.1.2	✓	2
3.2	odd	2		1638.2.a.x.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1638.2.a.v.1.2	✓	2	1.1	even	1	trivial
1638.2.a.x.1.1	yes	2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} -2.37228 q^{10} -2.37228 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.37228 q^{17} +1.62772 q^{19} +2.37228 q^{20} +2.37228 q^{22} -3.62772 q^{23} +0.627719 q^{25} +1.00000 q^{26} -1.00000 q^{28} -6.37228 q^{29} -4.74456 q^{31} -1.00000 q^{32} +4.37228 q^{34} -2.37228 q^{35} -4.37228 q^{37} -1.62772 q^{38} -2.37228 q^{40} -8.74456 q^{41} +11.1168 q^{43} -2.37228 q^{44} +3.62772 q^{46} -1.25544 q^{47} +1.00000 q^{49} -0.627719 q^{50} -1.00000 q^{52} -8.74456 q^{53} -5.62772 q^{55} +1.00000 q^{56} +6.37228 q^{58} -2.00000 q^{59} +5.11684 q^{61} +4.74456 q^{62} +1.00000 q^{64} -2.37228 q^{65} +9.48913 q^{67} -4.37228 q^{68} +2.37228 q^{70} +4.74456 q^{71} -8.37228 q^{73} +4.37228 q^{74} +1.62772 q^{76} +2.37228 q^{77} +4.74456 q^{79} +2.37228 q^{80} +8.74456 q^{82} +6.00000 q^{83} -10.3723 q^{85} -11.1168 q^{86} +2.37228 q^{88} +3.25544 q^{89} +1.00000 q^{91} -3.62772 q^{92} +1.25544 q^{94} +3.86141 q^{95} +7.48913 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8} + q^{10} + q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} + 9 q^{19} - q^{20} - q^{22} - 13 q^{23} + 7 q^{25} + 2 q^{26} - 2 q^{28} - 7 q^{29} + 2 q^{31} - 2 q^{32} + 3 q^{34} + q^{35} - 3 q^{37} - 9 q^{38} + q^{40} - 6 q^{41} + 5 q^{43} + q^{44} + 13 q^{46} - 14 q^{47} + 2 q^{49} - 7 q^{50} - 2 q^{52} - 6 q^{53} - 17 q^{55} + 2 q^{56} + 7 q^{58} - 4 q^{59} - 7 q^{61} - 2 q^{62} + 2 q^{64} + q^{65} - 4 q^{67} - 3 q^{68} - q^{70} - 2 q^{71} - 11 q^{73} + 3 q^{74} + 9 q^{76} - q^{77} - 2 q^{79} - q^{80} + 6 q^{82} + 12 q^{83} - 15 q^{85} - 5 q^{86} - q^{88} + 18 q^{89} + 2 q^{91} - 13 q^{92} + 14 q^{94} - 21 q^{95} - 8 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8 + q^10 + q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 + 9 * q^19 - q^20 - q^22 - 13 * q^23 + 7 * q^25 + 2 * q^26 - 2 * q^28 - 7 * q^29 + 2 * q^31 - 2 * q^32 + 3 * q^34 + q^35 - 3 * q^37 - 9 * q^38 + q^40 - 6 * q^41 + 5 * q^43 + q^44 + 13 * q^46 - 14 * q^47 + 2 * q^49 - 7 * q^50 - 2 * q^52 - 6 * q^53 - 17 * q^55 + 2 * q^56 + 7 * q^58 - 4 * q^59 - 7 * q^61 - 2 * q^62 + 2 * q^64 + q^65 - 4 * q^67 - 3 * q^68 - q^70 - 2 * q^71 - 11 * q^73 + 3 * q^74 + 9 * q^76 - q^77 - 2 * q^79 - q^80 + 6 * q^82 + 12 * q^83 - 15 * q^85 - 5 * q^86 - q^88 + 18 * q^89 + 2 * q^91 - 13 * q^92 + 14 * q^94 - 21 * q^95 - 8 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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