LMFDB - Embedded newform 1638.2.a.x.1.1

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:	$0$
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			1.1
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	$-0.677980\pi$
$5$	−2.37228	−1.06092	−0.530458	+	0.847711i	$0.677980\pi$
$6$	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	$-0.383583\pi$
$11$	2.37228	0.715270	0.357635	+	0.933862i	$0.383583\pi$
$12$	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	$-0.322110\pi$
$17$	4.37228	1.06043	0.530217	+	0.847862i	$0.322110\pi$
$18$	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	$-0.376538\pi$
$23$	3.62772	0.756432	0.378216	+	0.925717i	$0.376538\pi$
$24$	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	$-0.298476\pi$
$29$	6.37228	1.18330	0.591651	+	0.806194i	$0.298476\pi$
$30$	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	$-0.260747\pi$
$41$	8.74456	1.36567	0.682836	+	0.730572i	$0.260747\pi$
$42$	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	$-0.470814\pi$
$47$	1.25544	0.183124	0.0915622	+	0.995799i	$0.470814\pi$
$48$	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	$-0.294937\pi$
$53$	8.74456	1.20116	0.600579	+	0.799565i	$0.294937\pi$
$54$	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	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	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	$-0.590845\pi$
$71$	−4.74456	−0.563076	−0.281538	+	0.959550i	$0.590845\pi$
$72$	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	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	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	$-0.555197\pi$
$89$	−3.25544	−0.345076	−0.172538	+	0.985003i	$0.555197\pi$
$90$	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	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	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	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	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	$-0.110150\pi$
$113$	20.0000	1.88144	0.940721	+	0.339182i	$0.110150\pi$
$114$	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	$-0.661414\pi$
$131$	−11.1168	−0.971283	−0.485642	+	0.874158i	$0.661414\pi$
$132$	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	$-0.484808\pi$
$137$	1.11684	0.0954184	0.0477092	+	0.998861i	$0.484808\pi$
$138$	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	$-0.810978\pi$
$149$	−20.2337	−1.65761	−0.828804	+	0.559539i	$0.810978\pi$
$150$	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	$-0.330568\pi$
$167$	13.1168	1.01501	0.507506	+	0.861648i	$0.330568\pi$
$168$	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	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	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	$-0.532706\pi$
$179$	−2.74456	−0.205138	−0.102569	+	0.994726i	$0.532706\pi$
$180$	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	$-0.440732\pi$
$191$	5.11684	0.370242	0.185121	+	0.982716i	$0.440732\pi$
$192$	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	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	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	$-0.375483\pi$
$227$	11.4891	0.762560	0.381280	+	0.924460i	$0.375483\pi$
$228$	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	$-0.515533\pi$
$233$	−1.48913	−0.0975558	−0.0487779	+	0.998810i	$0.515533\pi$
$234$	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	$-0.556811\pi$
$239$	−5.48913	−0.355062	−0.177531	+	0.984115i	$0.556811\pi$
$240$	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	$-0.641517\pi$
$251$	−13.6277	−0.860174	−0.430087	+	0.902787i	$0.641517\pi$
$252$	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	$-0.383344\pi$
$257$	11.4891	0.716672	0.358336	+	0.933593i	$0.383344\pi$
$258$	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	$-0.407888\pi$
$263$	9.25544	0.570715	0.285357	+	0.958421i	$0.407888\pi$
$264$	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	$-0.495042\pi$
$269$	0.510875	0.0311486	0.0155743	+	0.999879i	$0.495042\pi$
$270$	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	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	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	$-0.817233\pi$
$293$	−28.7446	−1.67928	−0.839638	+	0.543147i	$0.817233\pi$
$294$	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	$-0.334853\pi$
$311$	17.4891	0.991717	0.495859	+	0.868403i	$0.334853\pi$
$312$	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	$-0.802600\pi$
$317$	−28.9783	−1.62758	−0.813790	+	0.581159i	$0.802600\pi$
$318$	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	$-0.257418\pi$
$347$	25.7228	1.38087	0.690436	+	0.723393i	$0.257418\pi$
$348$	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	$-0.540299\pi$
$353$	−4.74456	−0.252528	−0.126264	+	0.991997i	$0.540299\pi$
$354$	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	$-0.527379\pi$
$359$	−3.25544	−0.171815	−0.0859077	+	0.996303i	$0.527379\pi$
$360$	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	$-0.446019\pi$
$383$	6.60597	0.337549	0.168775	+	0.985655i	$0.446019\pi$
$384$	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	$-0.473700\pi$
$389$	3.25544	0.165057	0.0825286	+	0.996589i	$0.473700\pi$
$390$	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	$-0.846610\pi$
$401$	−35.4891	−1.77224	−0.886121	+	0.463454i	$0.846610\pi$
$402$	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	$-0.530068\pi$
$419$	−3.86141	−0.188642	−0.0943210	+	0.995542i	$0.530068\pi$
$420$	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	$-0.789207\pi$
$431$	−32.7446	−1.57725	−0.788625	+	0.614874i	$0.789207\pi$
$432$	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	$-0.367852\pi$
$443$	16.9783	0.806661	0.403331	+	0.915054i	$0.367852\pi$
$444$	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	$-0.502796\pi$
$449$	−0.372281	−0.0175690	−0.00878452	+	0.999961i	$0.502796\pi$
$450$	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	$-0.824887\pi$
$461$	−36.6060	−1.70491	−0.852455	+	0.522801i	$0.824887\pi$
$462$	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	$-0.789332\pi$
$467$	−34.0951	−1.57773	−0.788866	+	0.614565i	$0.789332\pi$
$468$	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	$-0.144305\pi$
$479$	39.3505	1.79797	0.898986	+	0.437978i	$0.144305\pi$
$480$	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	$-0.306989\pi$
$491$	25.2554	1.13976	0.569881	+	0.821727i	$0.306989\pi$
$492$	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	$-0.340972\pi$
$503$	21.4891	0.958153	0.479076	+	0.877773i	$0.340972\pi$
$504$	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	$-0.696292\pi$
$509$	−26.0951	−1.15664	−0.578322	+	0.815808i	$0.696292\pi$
$510$	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	$-0.713479\pi$
$521$	−28.3723	−1.24301	−0.621506	+	0.783409i	$0.713479\pi$
$522$	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	$-0.421736\pi$
$557$	11.4891	0.486810	0.243405	+	0.969925i	$0.421736\pi$
$558$	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	$-0.575267\pi$
$563$	−11.1168	−0.468519	−0.234260	+	0.972174i	$0.575267\pi$
$564$	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	$-0.413921\pi$
$569$	12.7446	0.534280	0.267140	+	0.963658i	$0.413921\pi$
$570$	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	$-0.692012\pi$
$587$	−27.4891	−1.13460	−0.567299	+	0.823512i	$0.692012\pi$
$588$	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	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	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	$-0.641982\pi$
$599$	−21.1168	−0.862811	−0.431405	+	0.902158i	$0.641982\pi$
$600$	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	$-0.439445\pi$
$617$	9.39403	0.378189	0.189095	+	0.981959i	$0.439445\pi$
$618$	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	$-0.382745\pi$
$641$	18.2337	0.720187	0.360094	+	0.932916i	$0.382745\pi$
$642$	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	$-0.361183\pi$
$647$	21.4891	0.844825	0.422412	+	0.906404i	$0.361183\pi$
$648$	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	$-0.449094\pi$
$653$	8.13859	0.318488	0.159244	+	0.987239i	$0.449094\pi$
$654$	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	$-0.442303\pi$
$659$	9.25544	0.360541	0.180270	+	0.983617i	$0.442303\pi$
$660$	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	$-0.261120\pi$
$677$	35.4891	1.36396	0.681979	+	0.731372i	$0.261120\pi$
$678$	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	$-0.431778\pi$
$683$	11.1168	0.425374	0.212687	+	0.977120i	$0.431778\pi$
$684$	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	$-0.295534\pi$
$701$	31.7228	1.19815	0.599077	+	0.800691i	$0.295534\pi$
$702$	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	$-0.822331\pi$
$719$	−45.4891	−1.69646	−0.848229	+	0.529630i	$0.822331\pi$
$720$	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	$-0.196366\pi$
$743$	44.4674	1.63135	0.815675	+	0.578511i	$0.196366\pi$
$744$	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	$-0.824295\pi$
$761$	−46.9783	−1.70296	−0.851480	+	0.524387i	$0.824295\pi$
$762$	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	$-0.710101\pi$
$773$	−34.0951	−1.22632	−0.613158	+	0.789961i	$0.710101\pi$
$774$	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	$-0.157116\pi$
$797$	49.7228	1.76127	0.880636	+	0.473793i	$0.157116\pi$
$798$	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	$-0.304761\pi$
$809$	32.7446	1.15124	0.575619	+	0.817718i	$0.304761\pi$
$810$	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	$-0.886838\pi$
$821$	−53.7228	−1.87494	−0.937470	+	0.348067i	$0.886838\pi$
$822$	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	$-0.296113\pi$
$827$	34.3723	1.19524	0.597621	+	0.801779i	$0.296113\pi$
$828$	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	$-0.506899\pi$
$839$	−1.25544	−0.0433425	−0.0216713	+	0.999765i	$0.506899\pi$
$840$	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	$-0.505319\pi$
$857$	−0.978251	−0.0334164	−0.0167082	+	0.999860i	$0.505319\pi$
$858$	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	$-0.604696\pi$
$863$	−18.9783	−0.646027	−0.323014	+	0.946394i	$0.604696\pi$
$864$	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	$-0.576548\pi$
$881$	−14.1386	−0.476341	−0.238171	+	0.971223i	$0.576548\pi$
$882$	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	$-0.148355\pi$
$887$	53.2119	1.78668	0.893341	+	0.449379i	$0.148355\pi$
$888$	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	$-0.376242\pi$
$911$	22.8832	0.758153	0.379076	+	0.925365i	$0.376242\pi$
$912$	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	$-0.419485\pi$
$929$	15.2554	0.500515	0.250257	+	0.968179i	$0.419485\pi$
$930$	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	$-0.367854\pi$
$941$	24.7446	0.806650	0.403325	+	0.915057i	$0.367854\pi$
$942$	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	$-0.542215\pi$
$947$	−8.13859	−0.264469	−0.132234	+	0.991218i	$0.542215\pi$
$948$	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	$-0.763649\pi$
$953$	−45.4891	−1.47354	−0.736769	+	0.676145i	$0.763649\pi$
$954$	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	$-0.631608\pi$
$977$	−25.1168	−0.803559	−0.401780	+	0.915736i	$0.631608\pi$
$978$	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	$-0.309357\pi$
$983$	35.3505	1.12751	0.563753	+	0.825943i	$0.309357\pi$
$984$	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.x.1.1	yes	2
3.2	odd	2		1638.2.a.v.1.2	✓	2

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

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

Fields

Representations

Groups

Database

Properties

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