LMFDB - Embedded newform 1568.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(1,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, 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("1568.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.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:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1568.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23607 q^{3} +1.23607 q^{5} +7.47214 q^{9} +O(q^{10})$	$q+3.23607 q^{3} +1.23607 q^{5} +7.47214 q^{9} +2.47214 q^{11} -5.23607 q^{13} +4.00000 q^{15} +4.47214 q^{17} -3.23607 q^{19} +4.00000 q^{23} -3.47214 q^{25} +14.4721 q^{27} +4.47214 q^{29} -6.47214 q^{31} +8.00000 q^{33} +4.47214 q^{37} -16.9443 q^{39} -0.472136 q^{41} -2.47214 q^{43} +9.23607 q^{45} -1.52786 q^{47} +14.4721 q^{51} -10.0000 q^{53} +3.05573 q^{55} -10.4721 q^{57} +4.76393 q^{59} -6.76393 q^{61} -6.47214 q^{65} +4.00000 q^{67} +12.9443 q^{69} +12.9443 q^{71} -14.9443 q^{73} -11.2361 q^{75} -4.94427 q^{79} +24.4164 q^{81} +4.76393 q^{83} +5.52786 q^{85} +14.4721 q^{87} +6.00000 q^{89} -20.9443 q^{93} -4.00000 q^{95} -3.52786 q^{97} +18.4721 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 6 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 6 q^{9} - 4 q^{11} - 6 q^{13} + 8 q^{15} - 2 q^{19} + 8 q^{23} + 2 q^{25} + 20 q^{27} - 4 q^{31} + 16 q^{33} - 16 q^{39} + 8 q^{41} + 4 q^{43} + 14 q^{45} - 12 q^{47} + 20 q^{51} - 20 q^{53} + 24 q^{55} - 12 q^{57} + 14 q^{59} - 18 q^{61} - 4 q^{65} + 8 q^{67} + 8 q^{69} + 8 q^{71} - 12 q^{73} - 18 q^{75} + 8 q^{79} + 22 q^{81} + 14 q^{83} + 20 q^{85} + 20 q^{87} + 12 q^{89} - 24 q^{93} - 8 q^{95} - 16 q^{97} + 28 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 6 * q^9 - 4 * q^11 - 6 * q^13 + 8 * q^15 - 2 * q^19 + 8 * q^23 + 2 * q^25 + 20 * q^27 - 4 * q^31 + 16 * q^33 - 16 * q^39 + 8 * q^41 + 4 * q^43 + 14 * q^45 - 12 * q^47 + 20 * q^51 - 20 * q^53 + 24 * q^55 - 12 * q^57 + 14 * q^59 - 18 * q^61 - 4 * q^65 + 8 * q^67 + 8 * q^69 + 8 * q^71 - 12 * q^73 - 18 * q^75 + 8 * q^79 + 22 * q^81 + 14 * q^83 + 20 * q^85 + 20 * q^87 + 12 * q^89 - 24 * q^93 - 8 * q^95 - 16 * q^97 + 28 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	3.23607	1.86834	0.934172	−	0.356822i	$-0.116140\pi$
$3$	3.23607	1.86834	0.934172	+	0.356822i	$0.116140\pi$
$4$	0	0
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.47214	2.49071
$10$	0	0
$11$	2.47214	0.745377	0.372689	−	0.927957i	$-0.378436\pi$
$11$	2.47214	0.745377	0.372689	+	0.927957i	$0.378436\pi$
$12$	0	0
$13$	−5.23607	−1.45222	−0.726112	−	0.687576i	$-0.758675\pi$
$13$	−5.23607	−1.45222	−0.726112	+	0.687576i	$0.758675\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	−3.23607	−0.742405	−0.371202	−	0.928552i	$-0.621054\pi$
$19$	−3.23607	−0.742405	−0.371202	+	0.928552i	$0.621054\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	14.4721	2.78516
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	0	0
$33$	8.00000	1.39262
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	0	0
$39$	−16.9443	−2.71325
$40$	0	0
$41$	−0.472136	−0.0737352	−0.0368676	−	0.999320i	$-0.511738\pi$
$41$	−0.472136	−0.0737352	−0.0368676	+	0.999320i	$0.511738\pi$
$42$	0	0
$43$	−2.47214	−0.376997	−0.188499	−	0.982073i	$-0.560362\pi$
$43$	−2.47214	−0.376997	−0.188499	+	0.982073i	$0.560362\pi$
$44$	0	0
$45$	9.23607	1.37683
$46$	0	0
$47$	−1.52786	−0.222862	−0.111431	−	0.993772i	$-0.535543\pi$
$47$	−1.52786	−0.222862	−0.111431	+	0.993772i	$0.535543\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	14.4721	2.02650
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	3.05573	0.412034
$56$	0	0
$57$	−10.4721	−1.38707
$58$	0	0
$59$	4.76393	0.620211	0.310106	−	0.950702i	$-0.399636\pi$
$59$	4.76393	0.620211	0.310106	+	0.950702i	$0.399636\pi$
$60$	0	0
$61$	−6.76393	−0.866033	−0.433016	−	0.901386i	$-0.642551\pi$
$61$	−6.76393	−0.866033	−0.433016	+	0.901386i	$0.642551\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.47214	−0.802770
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	12.9443	1.55831
$70$	0	0
$71$	12.9443	1.53620	0.768101	−	0.640328i	$-0.221202\pi$
$71$	12.9443	1.53620	0.768101	+	0.640328i	$0.221202\pi$
$72$	0	0
$73$	−14.9443	−1.74909	−0.874547	−	0.484940i	$-0.838841\pi$
$73$	−14.9443	−1.74909	−0.874547	+	0.484940i	$0.838841\pi$
$74$	0	0
$75$	−11.2361	−1.29743
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.94427	−0.556274	−0.278137	−	0.960541i	$-0.589717\pi$
$79$	−4.94427	−0.556274	−0.278137	+	0.960541i	$0.589717\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	4.76393	0.522909	0.261455	−	0.965216i	$-0.415798\pi$
$83$	4.76393	0.522909	0.261455	+	0.965216i	$0.415798\pi$
$84$	0	0
$85$	5.52786	0.599581
$86$	0	0
$87$	14.4721	1.55158
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−20.9443	−2.17182
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−3.52786	−0.358200	−0.179100	−	0.983831i	$-0.557319\pi$
$97$	−3.52786	−0.358200	−0.179100	+	0.983831i	$0.557319\pi$
$98$	0	0
$99$	18.4721	1.85652
$100$	0	0
$101$	−11.7082	−1.16501	−0.582505	−	0.812827i	$-0.697927\pi$
$101$	−11.7082	−1.16501	−0.582505	+	0.812827i	$0.697927\pi$
$102$	0	0
$103$	14.4721	1.42598	0.712991	−	0.701173i	$-0.247340\pi$
$103$	14.4721	1.42598	0.712991	+	0.701173i	$0.247340\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.94427	−0.864675	−0.432338	−	0.901712i	$-0.642311\pi$
$107$	−8.94427	−0.864675	−0.432338	+	0.901712i	$0.642311\pi$
$108$	0	0
$109$	−0.472136	−0.0452224	−0.0226112	−	0.999744i	$-0.507198\pi$
$109$	−0.472136	−0.0452224	−0.0226112	+	0.999744i	$0.507198\pi$
$110$	0	0
$111$	14.4721	1.37363
$112$	0	0
$113$	−3.52786	−0.331874	−0.165937	−	0.986136i	$-0.553065\pi$
$113$	−3.52786	−0.331874	−0.165937	+	0.986136i	$0.553065\pi$
$114$	0	0
$115$	4.94427	0.461056
$116$	0	0
$117$	−39.1246	−3.61707
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.88854	−0.444413
$122$	0	0
$123$	−1.52786	−0.137763
$124$	0	0
$125$	−10.4721	−0.936656
$126$	0	0
$127$	−8.94427	−0.793676	−0.396838	−	0.917889i	$-0.629892\pi$
$127$	−8.94427	−0.793676	−0.396838	+	0.917889i	$0.629892\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	1.70820	0.149246	0.0746232	−	0.997212i	$-0.476225\pi$
$131$	1.70820	0.149246	0.0746232	+	0.997212i	$0.476225\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	17.8885	1.53960
$136$	0	0
$137$	−2.94427	−0.251546	−0.125773	−	0.992059i	$-0.540141\pi$
$137$	−2.94427	−0.251546	−0.125773	+	0.992059i	$0.540141\pi$
$138$	0	0
$139$	3.23607	0.274480	0.137240	−	0.990538i	$-0.456177\pi$
$139$	3.23607	0.274480	0.137240	+	0.990538i	$0.456177\pi$
$140$	0	0
$141$	−4.94427	−0.416383
$142$	0	0
$143$	−12.9443	−1.08245
$144$	0	0
$145$	5.52786	0.459064
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.9443	−1.22428	−0.612141	−	0.790748i	$-0.709692\pi$
$149$	−14.9443	−1.22428	−0.612141	+	0.790748i	$0.709692\pi$
$150$	0	0
$151$	8.94427	0.727875	0.363937	−	0.931423i	$-0.381432\pi$
$151$	8.94427	0.727875	0.363937	+	0.931423i	$0.381432\pi$
$152$	0	0
$153$	33.4164	2.70156
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	−5.23607	−0.417884	−0.208942	−	0.977928i	$-0.567002\pi$
$157$	−5.23607	−0.417884	−0.208942	+	0.977928i	$0.567002\pi$
$158$	0	0
$159$	−32.3607	−2.56637
$160$	0	0
$161$	0	0
$162$	0	0
$163$	23.4164	1.83411	0.917057	−	0.398755i	$-0.130558\pi$
$163$	23.4164	1.83411	0.917057	+	0.398755i	$0.130558\pi$
$164$	0	0
$165$	9.88854	0.769822
$166$	0	0
$167$	3.41641	0.264370	0.132185	−	0.991225i	$-0.457801\pi$
$167$	3.41641	0.264370	0.132185	+	0.991225i	$0.457801\pi$
$168$	0	0
$169$	14.4164	1.10895
$170$	0	0
$171$	−24.1803	−1.84912
$172$	0	0
$173$	7.70820	0.586044	0.293022	−	0.956106i	$-0.405339\pi$
$173$	7.70820	0.586044	0.293022	+	0.956106i	$0.405339\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	15.4164	1.15877
$178$	0	0
$179$	−24.9443	−1.86442	−0.932211	−	0.361915i	$-0.882123\pi$
$179$	−24.9443	−1.86442	−0.932211	+	0.361915i	$0.882123\pi$
$180$	0	0
$181$	−10.1803	−0.756699	−0.378349	−	0.925663i	$-0.623508\pi$
$181$	−10.1803	−0.756699	−0.378349	+	0.925663i	$0.623508\pi$
$182$	0	0
$183$	−21.8885	−1.61805
$184$	0	0
$185$	5.52786	0.406417
$186$	0	0
$187$	11.0557	0.808475
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.9443	−0.936615	−0.468307	−	0.883566i	$-0.655136\pi$
$191$	−12.9443	−0.936615	−0.468307	+	0.883566i	$0.655136\pi$
$192$	0	0
$193$	−8.47214	−0.609838	−0.304919	−	0.952378i	$-0.598629\pi$
$193$	−8.47214	−0.609838	−0.304919	+	0.952378i	$0.598629\pi$
$194$	0	0
$195$	−20.9443	−1.49985
$196$	0	0
$197$	−6.94427	−0.494759	−0.247379	−	0.968919i	$-0.579569\pi$
$197$	−6.94427	−0.494759	−0.247379	+	0.968919i	$0.579569\pi$
$198$	0	0
$199$	11.4164	0.809288	0.404644	−	0.914474i	$-0.367395\pi$
$199$	11.4164	0.809288	0.404644	+	0.914474i	$0.367395\pi$
$200$	0	0
$201$	12.9443	0.913019
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.583592	−0.0407598
$206$	0	0
$207$	29.8885	2.07740
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	41.8885	2.87016
$214$	0	0
$215$	−3.05573	−0.208399
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−48.3607	−3.26791
$220$	0	0
$221$	−23.4164	−1.57516
$222$	0	0
$223$	4.94427	0.331093	0.165546	−	0.986202i	$-0.447061\pi$
$223$	4.94427	0.331093	0.165546	+	0.986202i	$0.447061\pi$
$224$	0	0
$225$	−25.9443	−1.72962
$226$	0	0
$227$	12.7639	0.847172	0.423586	−	0.905856i	$-0.360771\pi$
$227$	12.7639	0.847172	0.423586	+	0.905856i	$0.360771\pi$
$228$	0	0
$229$	25.5967	1.69148	0.845740	−	0.533595i	$-0.179159\pi$
$229$	25.5967	1.69148	0.845740	+	0.533595i	$0.179159\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.8885	1.30294	0.651471	−	0.758674i	$-0.274152\pi$
$233$	19.8885	1.30294	0.651471	+	0.758674i	$0.274152\pi$
$234$	0	0
$235$	−1.88854	−0.123195
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	21.8885	1.41585	0.707926	−	0.706287i	$-0.249631\pi$
$239$	21.8885	1.41585	0.707926	+	0.706287i	$0.249631\pi$
$240$	0	0
$241$	−3.52786	−0.227250	−0.113625	−	0.993524i	$-0.536246\pi$
$241$	−3.52786	−0.227250	−0.113625	+	0.993524i	$0.536246\pi$
$242$	0	0
$243$	35.5967	2.28353
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.9443	1.07814
$248$	0	0
$249$	15.4164	0.976975
$250$	0	0
$251$	−17.7082	−1.11773	−0.558866	−	0.829258i	$-0.688763\pi$
$251$	−17.7082	−1.11773	−0.558866	+	0.829258i	$0.688763\pi$
$252$	0	0
$253$	9.88854	0.621687
$254$	0	0
$255$	17.8885	1.12022
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	33.4164	2.06842
$262$	0	0
$263$	28.9443	1.78478	0.892390	−	0.451265i	$-0.149027\pi$
$263$	28.9443	1.78478	0.892390	+	0.451265i	$0.149027\pi$
$264$	0	0
$265$	−12.3607	−0.759311
$266$	0	0
$267$	19.4164	1.18826
$268$	0	0
$269$	−18.1803	−1.10847	−0.554237	−	0.832359i	$-0.686990\pi$
$269$	−18.1803	−1.10847	−0.554237	+	0.832359i	$0.686990\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.58359	−0.517610
$276$	0	0
$277$	−27.8885	−1.67566	−0.837830	−	0.545931i	$-0.816176\pi$
$277$	−27.8885	−1.67566	−0.837830	+	0.545931i	$0.816176\pi$
$278$	0	0
$279$	−48.3607	−2.89528
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	−16.1803	−0.961821	−0.480911	−	0.876770i	$-0.659694\pi$
$283$	−16.1803	−0.961821	−0.480911	+	0.876770i	$0.659694\pi$
$284$	0	0
$285$	−12.9443	−0.766752
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	−11.4164	−0.669242
$292$	0	0
$293$	17.2361	1.00694	0.503471	−	0.864012i	$-0.332056\pi$
$293$	17.2361	1.00694	0.503471	+	0.864012i	$0.332056\pi$
$294$	0	0
$295$	5.88854	0.342844
$296$	0	0
$297$	35.7771	2.07600
$298$	0	0
$299$	−20.9443	−1.21124
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−37.8885	−2.17664
$304$	0	0
$305$	−8.36068	−0.478731
$306$	0	0
$307$	−24.1803	−1.38004	−0.690022	−	0.723788i	$-0.742399\pi$
$307$	−24.1803	−1.38004	−0.690022	+	0.723788i	$0.742399\pi$
$308$	0	0
$309$	46.8328	2.66423
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−0.472136	−0.0266867	−0.0133434	−	0.999911i	$-0.504247\pi$
$313$	−0.472136	−0.0266867	−0.0133434	+	0.999911i	$0.504247\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.9443	1.51334	0.756671	−	0.653796i	$-0.226825\pi$
$317$	26.9443	1.51334	0.756671	+	0.653796i	$0.226825\pi$
$318$	0	0
$319$	11.0557	0.619002
$320$	0	0
$321$	−28.9443	−1.61551
$322$	0	0
$323$	−14.4721	−0.805251
$324$	0	0
$325$	18.1803	1.00846
$326$	0	0
$327$	−1.52786	−0.0844911
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.5279	0.743559	0.371779	−	0.928321i	$-0.378748\pi$
$331$	13.5279	0.743559	0.371779	+	0.928321i	$0.378748\pi$
$332$	0	0
$333$	33.4164	1.83121
$334$	0	0
$335$	4.94427	0.270134
$336$	0	0
$337$	−34.3607	−1.87175	−0.935873	−	0.352338i	$-0.885387\pi$
$337$	−34.3607	−1.87175	−0.935873	+	0.352338i	$0.885387\pi$
$338$	0	0
$339$	−11.4164	−0.620054
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	0	0
$344$	0	0
$345$	16.0000	0.861411
$346$	0	0
$347$	−2.47214	−0.132711	−0.0663556	−	0.997796i	$-0.521137\pi$
$347$	−2.47214	−0.132711	−0.0663556	+	0.997796i	$0.521137\pi$
$348$	0	0
$349$	4.65248	0.249041	0.124521	−	0.992217i	$-0.460261\pi$
$349$	4.65248	0.249041	0.124521	+	0.992217i	$0.460261\pi$
$350$	0	0
$351$	−75.7771	−4.04468
$352$	0	0
$353$	−19.8885	−1.05856	−0.529280	−	0.848447i	$-0.677538\pi$
$353$	−19.8885	−1.05856	−0.529280	+	0.848447i	$0.677538\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.944272	0.0498368	0.0249184	−	0.999689i	$-0.492067\pi$
$359$	0.944272	0.0498368	0.0249184	+	0.999689i	$0.492067\pi$
$360$	0	0
$361$	−8.52786	−0.448835
$362$	0	0
$363$	−15.8197	−0.830317
$364$	0	0
$365$	−18.4721	−0.966876
$366$	0	0
$367$	−30.8328	−1.60946	−0.804730	−	0.593641i	$-0.797690\pi$
$367$	−30.8328	−1.60946	−0.804730	+	0.593641i	$0.797690\pi$
$368$	0	0
$369$	−3.52786	−0.183653
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−14.9443	−0.773785	−0.386893	−	0.922125i	$-0.626452\pi$
$373$	−14.9443	−0.773785	−0.386893	+	0.922125i	$0.626452\pi$
$374$	0	0
$375$	−33.8885	−1.75000
$376$	0	0
$377$	−23.4164	−1.20601
$378$	0	0
$379$	−31.4164	−1.61375	−0.806876	−	0.590721i	$-0.798844\pi$
$379$	−31.4164	−1.61375	−0.806876	+	0.590721i	$0.798844\pi$
$380$	0	0
$381$	−28.9443	−1.48286
$382$	0	0
$383$	−11.4164	−0.583351	−0.291676	−	0.956517i	$-0.594213\pi$
$383$	−11.4164	−0.583351	−0.291676	+	0.956517i	$0.594213\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−18.4721	−0.938991
$388$	0	0
$389$	4.47214	0.226746	0.113373	−	0.993552i	$-0.463834\pi$
$389$	4.47214	0.226746	0.113373	+	0.993552i	$0.463834\pi$
$390$	0	0
$391$	17.8885	0.904663
$392$	0	0
$393$	5.52786	0.278844
$394$	0	0
$395$	−6.11146	−0.307501
$396$	0	0
$397$	10.7639	0.540226	0.270113	−	0.962829i	$-0.412939\pi$
$397$	10.7639	0.540226	0.270113	+	0.962829i	$0.412939\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−32.4721	−1.62158	−0.810791	−	0.585336i	$-0.800962\pi$
$401$	−32.4721	−1.62158	−0.810791	+	0.585336i	$0.800962\pi$
$402$	0	0
$403$	33.8885	1.68811
$404$	0	0
$405$	30.1803	1.49967
$406$	0	0
$407$	11.0557	0.548012
$408$	0	0
$409$	−5.41641	−0.267824	−0.133912	−	0.990993i	$-0.542754\pi$
$409$	−5.41641	−0.267824	−0.133912	+	0.990993i	$0.542754\pi$
$410$	0	0
$411$	−9.52786	−0.469975
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5.88854	0.289057
$416$	0	0
$417$	10.4721	0.512823
$418$	0	0
$419$	0.180340	0.00881018	0.00440509	−	0.999990i	$-0.498598\pi$
$419$	0.180340	0.00881018	0.00440509	+	0.999990i	$0.498598\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	−11.4164	−0.555085
$424$	0	0
$425$	−15.5279	−0.753212
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−41.8885	−2.02240
$430$	0	0
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	0	0
$433$	17.4164	0.836979	0.418490	−	0.908222i	$-0.362560\pi$
$433$	17.4164	0.836979	0.418490	+	0.908222i	$0.362560\pi$
$434$	0	0
$435$	17.8885	0.857690
$436$	0	0
$437$	−12.9443	−0.619208
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.8885	1.03996	0.519978	−	0.854180i	$-0.325940\pi$
$443$	21.8885	1.03996	0.519978	+	0.854180i	$0.325940\pi$
$444$	0	0
$445$	7.41641	0.351571
$446$	0	0
$447$	−48.3607	−2.28738
$448$	0	0
$449$	27.8885	1.31614	0.658071	−	0.752956i	$-0.271373\pi$
$449$	27.8885	1.31614	0.658071	+	0.752956i	$0.271373\pi$
$450$	0	0
$451$	−1.16718	−0.0549606
$452$	0	0
$453$	28.9443	1.35992
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.4721	−0.770534	−0.385267	−	0.922805i	$-0.625891\pi$
$457$	−16.4721	−0.770534	−0.385267	+	0.922805i	$0.625891\pi$
$458$	0	0
$459$	64.7214	3.02093
$460$	0	0
$461$	−8.29180	−0.386187	−0.193094	−	0.981180i	$-0.561852\pi$
$461$	−8.29180	−0.386187	−0.193094	+	0.981180i	$0.561852\pi$
$462$	0	0
$463$	35.7771	1.66270	0.831351	−	0.555748i	$-0.187568\pi$
$463$	35.7771	1.66270	0.831351	+	0.555748i	$0.187568\pi$
$464$	0	0
$465$	−25.8885	−1.20055
$466$	0	0
$467$	−26.0689	−1.20632	−0.603162	−	0.797619i	$-0.706093\pi$
$467$	−26.0689	−1.20632	−0.603162	+	0.797619i	$0.706093\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−16.9443	−0.780751
$472$	0	0
$473$	−6.11146	−0.281005
$474$	0	0
$475$	11.2361	0.515546
$476$	0	0
$477$	−74.7214	−3.42126
$478$	0	0
$479$	35.4164	1.61822	0.809108	−	0.587659i	$-0.199950\pi$
$479$	35.4164	1.61822	0.809108	+	0.587659i	$0.199950\pi$
$480$	0	0
$481$	−23.4164	−1.06770
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.36068	−0.198008
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	0	0
$489$	75.7771	3.42676
$490$	0	0
$491$	2.11146	0.0952887	0.0476443	−	0.998864i	$-0.484829\pi$
$491$	2.11146	0.0952887	0.0476443	+	0.998864i	$0.484829\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	22.8328	1.02626
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13.8885	−0.621737	−0.310868	−	0.950453i	$-0.600620\pi$
$499$	−13.8885	−0.621737	−0.310868	+	0.950453i	$0.600620\pi$
$500$	0	0
$501$	11.0557	0.493934
$502$	0	0
$503$	−12.9443	−0.577157	−0.288578	−	0.957456i	$-0.593183\pi$
$503$	−12.9443	−0.577157	−0.288578	+	0.957456i	$0.593183\pi$
$504$	0	0
$505$	−14.4721	−0.644002
$506$	0	0
$507$	46.6525	2.07191
$508$	0	0
$509$	0.875388	0.0388009	0.0194004	−	0.999812i	$-0.493824\pi$
$509$	0.875388	0.0388009	0.0194004	+	0.999812i	$0.493824\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−46.8328	−2.06772
$514$	0	0
$515$	17.8885	0.788263
$516$	0	0
$517$	−3.77709	−0.166116
$518$	0	0
$519$	24.9443	1.09493
$520$	0	0
$521$	33.4164	1.46400	0.732000	−	0.681305i	$-0.238587\pi$
$521$	33.4164	1.46400	0.732000	+	0.681305i	$0.238587\pi$
$522$	0	0
$523$	17.7082	0.774326	0.387163	−	0.922011i	$-0.373455\pi$
$523$	17.7082	0.774326	0.387163	+	0.922011i	$0.373455\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28.9443	−1.26083
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	35.5967	1.54477
$532$	0	0
$533$	2.47214	0.107080
$534$	0	0
$535$	−11.0557	−0.477981
$536$	0	0
$537$	−80.7214	−3.48338
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22.9443	−0.986451	−0.493226	−	0.869901i	$-0.664182\pi$
$541$	−22.9443	−0.986451	−0.493226	+	0.869901i	$0.664182\pi$
$542$	0	0
$543$	−32.9443	−1.41377
$544$	0	0
$545$	−0.583592	−0.0249983
$546$	0	0
$547$	−31.4164	−1.34327	−0.671634	−	0.740883i	$-0.734407\pi$
$547$	−31.4164	−1.34327	−0.671634	+	0.740883i	$0.734407\pi$
$548$	0	0
$549$	−50.5410	−2.15704
$550$	0	0
$551$	−14.4721	−0.616534
$552$	0	0
$553$	0	0
$554$	0	0
$555$	17.8885	0.759326
$556$	0	0
$557$	26.9443	1.14167	0.570833	−	0.821066i	$-0.306620\pi$
$557$	26.9443	1.14167	0.570833	+	0.821066i	$0.306620\pi$
$558$	0	0
$559$	12.9443	0.547484
$560$	0	0
$561$	35.7771	1.51051
$562$	0	0
$563$	40.1803	1.69340	0.846700	−	0.532071i	$-0.178586\pi$
$563$	40.1803	1.69340	0.846700	+	0.532071i	$0.178586\pi$
$564$	0	0
$565$	−4.36068	−0.183455
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.3607	−0.769720	−0.384860	−	0.922975i	$-0.625750\pi$
$569$	−18.3607	−0.769720	−0.384860	+	0.922975i	$0.625750\pi$
$570$	0	0
$571$	−5.52786	−0.231334	−0.115667	−	0.993288i	$-0.536901\pi$
$571$	−5.52786	−0.231334	−0.115667	+	0.993288i	$0.536901\pi$
$572$	0	0
$573$	−41.8885	−1.74992
$574$	0	0
$575$	−13.8885	−0.579192
$576$	0	0
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	0	0
$579$	−27.4164	−1.13939
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−24.7214	−1.02385
$584$	0	0
$585$	−48.3607	−1.99947
$586$	0	0
$587$	−9.70820	−0.400700	−0.200350	−	0.979724i	$-0.564208\pi$
$587$	−9.70820	−0.400700	−0.200350	+	0.979724i	$0.564208\pi$
$588$	0	0
$589$	20.9443	0.862994
$590$	0	0
$591$	−22.4721	−0.924380
$592$	0	0
$593$	20.8328	0.855501	0.427751	−	0.903897i	$-0.359306\pi$
$593$	20.8328	0.855501	0.427751	+	0.903897i	$0.359306\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	36.9443	1.51203
$598$	0	0
$599$	−17.8885	−0.730906	−0.365453	−	0.930830i	$-0.619086\pi$
$599$	−17.8885	−0.730906	−0.365453	+	0.930830i	$0.619086\pi$
$600$	0	0
$601$	41.7771	1.70412	0.852061	−	0.523442i	$-0.175352\pi$
$601$	41.7771	1.70412	0.852061	+	0.523442i	$0.175352\pi$
$602$	0	0
$603$	29.8885	1.21716
$604$	0	0
$605$	−6.04257	−0.245666
$606$	0	0
$607$	25.8885	1.05078	0.525392	−	0.850860i	$-0.323919\pi$
$607$	25.8885	1.05078	0.525392	+	0.850860i	$0.323919\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	2.58359	0.104350	0.0521752	−	0.998638i	$-0.483385\pi$
$613$	2.58359	0.104350	0.0521752	+	0.998638i	$0.483385\pi$
$614$	0	0
$615$	−1.88854	−0.0761534
$616$	0	0
$617$	−10.3607	−0.417105	−0.208553	−	0.978011i	$-0.566875\pi$
$617$	−10.3607	−0.417105	−0.208553	+	0.978011i	$0.566875\pi$
$618$	0	0
$619$	−10.0689	−0.404703	−0.202351	−	0.979313i	$-0.564858\pi$
$619$	−10.0689	−0.404703	−0.202351	+	0.979313i	$0.564858\pi$
$620$	0	0
$621$	57.8885	2.32299
$622$	0	0
$623$	0	0
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	−25.8885	−1.03389
$628$	0	0
$629$	20.0000	0.797452
$630$	0	0
$631$	−27.0557	−1.07707	−0.538536	−	0.842603i	$-0.681022\pi$
$631$	−27.0557	−1.07707	−0.538536	+	0.842603i	$0.681022\pi$
$632$	0	0
$633$	−38.8328	−1.54347
$634$	0	0
$635$	−11.0557	−0.438733
$636$	0	0
$637$	0	0
$638$	0	0
$639$	96.7214	3.82624
$640$	0	0
$641$	41.4164	1.63585	0.817925	−	0.575325i	$-0.195124\pi$
$641$	41.4164	1.63585	0.817925	+	0.575325i	$0.195124\pi$
$642$	0	0
$643$	30.2918	1.19459	0.597296	−	0.802021i	$-0.296242\pi$
$643$	30.2918	1.19459	0.597296	+	0.802021i	$0.296242\pi$
$644$	0	0
$645$	−9.88854	−0.389361
$646$	0	0
$647$	−24.3607	−0.957717	−0.478859	−	0.877892i	$-0.658949\pi$
$647$	−24.3607	−0.957717	−0.478859	+	0.877892i	$0.658949\pi$
$648$	0	0
$649$	11.7771	0.462291
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.4164	0.681557	0.340778	−	0.940144i	$-0.389309\pi$
$653$	17.4164	0.681557	0.340778	+	0.940144i	$0.389309\pi$
$654$	0	0
$655$	2.11146	0.0825014
$656$	0	0
$657$	−111.666	−4.35649
$658$	0	0
$659$	−25.3050	−0.985741	−0.492870	−	0.870103i	$-0.664052\pi$
$659$	−25.3050	−0.985741	−0.492870	+	0.870103i	$0.664052\pi$
$660$	0	0
$661$	−8.65248	−0.336542	−0.168271	−	0.985741i	$-0.553818\pi$
$661$	−8.65248	−0.336542	−0.168271	+	0.985741i	$0.553818\pi$
$662$	0	0
$663$	−75.7771	−2.94294
$664$	0	0
$665$	0	0
$666$	0	0
$667$	17.8885	0.692647
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−16.7214	−0.645521
$672$	0	0
$673$	14.9443	0.576059	0.288030	−	0.957621i	$-0.407000\pi$
$673$	14.9443	0.576059	0.288030	+	0.957621i	$0.407000\pi$
$674$	0	0
$675$	−50.2492	−1.93409
$676$	0	0
$677$	−42.1803	−1.62112	−0.810561	−	0.585654i	$-0.800838\pi$
$677$	−42.1803	−1.62112	−0.810561	+	0.585654i	$0.800838\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	41.3050	1.58281
$682$	0	0
$683$	−47.7771	−1.82814	−0.914070	−	0.405557i	$-0.867078\pi$
$683$	−47.7771	−1.82814	−0.914070	+	0.405557i	$0.867078\pi$
$684$	0	0
$685$	−3.63932	−0.139051
$686$	0	0
$687$	82.8328	3.16027
$688$	0	0
$689$	52.3607	1.99478
$690$	0	0
$691$	−8.18034	−0.311195	−0.155597	−	0.987821i	$-0.549730\pi$
$691$	−8.18034	−0.311195	−0.155597	+	0.987821i	$0.549730\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	−2.11146	−0.0799771
$698$	0	0
$699$	64.3607	2.43434
$700$	0	0
$701$	−14.5836	−0.550815	−0.275407	−	0.961328i	$-0.588813\pi$
$701$	−14.5836	−0.550815	−0.275407	+	0.961328i	$0.588813\pi$
$702$	0	0
$703$	−14.4721	−0.545827
$704$	0	0
$705$	−6.11146	−0.230171
$706$	0	0
$707$	0	0
$708$	0	0
$709$	46.3607	1.74111	0.870556	−	0.492069i	$-0.163759\pi$
$709$	46.3607	1.74111	0.870556	+	0.492069i	$0.163759\pi$
$710$	0	0
$711$	−36.9443	−1.38552
$712$	0	0
$713$	−25.8885	−0.969534
$714$	0	0
$715$	−16.0000	−0.598366
$716$	0	0
$717$	70.8328	2.64530
$718$	0	0
$719$	47.1935	1.76002	0.880010	−	0.474955i	$-0.157536\pi$
$719$	47.1935	1.76002	0.880010	+	0.474955i	$0.157536\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−11.4164	−0.424581
$724$	0	0
$725$	−15.5279	−0.576690
$726$	0	0
$727$	32.3607	1.20019	0.600096	−	0.799928i	$-0.295129\pi$
$727$	32.3607	1.20019	0.600096	+	0.799928i	$0.295129\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	−11.0557	−0.408911
$732$	0	0
$733$	9.23607	0.341142	0.170571	−	0.985345i	$-0.445439\pi$
$733$	9.23607	0.341142	0.170571	+	0.985345i	$0.445439\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.88854	0.364249
$738$	0	0
$739$	23.4164	0.861386	0.430693	−	0.902498i	$-0.358269\pi$
$739$	23.4164	0.861386	0.430693	+	0.902498i	$0.358269\pi$
$740$	0	0
$741$	54.8328	2.01433
$742$	0	0
$743$	7.05573	0.258850	0.129425	−	0.991589i	$-0.458687\pi$
$743$	7.05573	0.258850	0.129425	+	0.991589i	$0.458687\pi$
$744$	0	0
$745$	−18.4721	−0.676767
$746$	0	0
$747$	35.5967	1.30242
$748$	0	0
$749$	0	0
$750$	0	0
$751$	36.0000	1.31366	0.656829	−	0.754039i	$-0.271897\pi$
$751$	36.0000	1.31366	0.656829	+	0.754039i	$0.271897\pi$
$752$	0	0
$753$	−57.3050	−2.08831
$754$	0	0
$755$	11.0557	0.402359
$756$	0	0
$757$	−23.3050	−0.847033	−0.423516	−	0.905888i	$-0.639204\pi$
$757$	−23.3050	−0.847033	−0.423516	+	0.905888i	$0.639204\pi$
$758$	0	0
$759$	32.0000	1.16153
$760$	0	0
$761$	12.4721	0.452115	0.226057	−	0.974114i	$-0.427416\pi$
$761$	12.4721	0.452115	0.226057	+	0.974114i	$0.427416\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	41.3050	1.49338
$766$	0	0
$767$	−24.9443	−0.900685
$768$	0	0
$769$	−26.3607	−0.950590	−0.475295	−	0.879826i	$-0.657659\pi$
$769$	−26.3607	−0.950590	−0.475295	+	0.879826i	$0.657659\pi$
$770$	0	0
$771$	45.3050	1.63162
$772$	0	0
$773$	−17.8197	−0.640929	−0.320464	−	0.947261i	$-0.603839\pi$
$773$	−17.8197	−0.640929	−0.320464	+	0.947261i	$0.603839\pi$
$774$	0	0
$775$	22.4721	0.807223
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.52786	0.0547414
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	64.7214	2.31295
$784$	0	0
$785$	−6.47214	−0.231000
$786$	0	0
$787$	25.7082	0.916399	0.458199	−	0.888850i	$-0.348495\pi$
$787$	25.7082	0.916399	0.458199	+	0.888850i	$0.348495\pi$
$788$	0	0
$789$	93.6656	3.33458
$790$	0	0
$791$	0	0
$792$	0	0
$793$	35.4164	1.25767
$794$	0	0
$795$	−40.0000	−1.41865
$796$	0	0
$797$	29.8197	1.05627	0.528133	−	0.849161i	$-0.322892\pi$
$797$	29.8197	1.05627	0.528133	+	0.849161i	$0.322892\pi$
$798$	0	0
$799$	−6.83282	−0.241728
$800$	0	0
$801$	44.8328	1.58409
$802$	0	0
$803$	−36.9443	−1.30374
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−58.8328	−2.07101
$808$	0	0
$809$	9.41641	0.331063	0.165532	−	0.986204i	$-0.447066\pi$
$809$	9.41641	0.331063	0.165532	+	0.986204i	$0.447066\pi$
$810$	0	0
$811$	−23.0132	−0.808101	−0.404051	−	0.914737i	$-0.632398\pi$
$811$	−23.0132	−0.808101	−0.404051	+	0.914737i	$0.632398\pi$
$812$	0	0
$813$	77.6656	2.72385
$814$	0	0
$815$	28.9443	1.01387
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.8885	1.39212	0.696060	−	0.717984i	$-0.254935\pi$
$821$	39.8885	1.39212	0.696060	+	0.717984i	$0.254935\pi$
$822$	0	0
$823$	51.7771	1.80484	0.902418	−	0.430862i	$-0.141790\pi$
$823$	51.7771	1.80484	0.902418	+	0.430862i	$0.141790\pi$
$824$	0	0
$825$	−27.7771	−0.967074
$826$	0	0
$827$	32.9443	1.14558	0.572792	−	0.819701i	$-0.305860\pi$
$827$	32.9443	1.14558	0.572792	+	0.819701i	$0.305860\pi$
$828$	0	0
$829$	−6.76393	−0.234921	−0.117461	−	0.993078i	$-0.537475\pi$
$829$	−6.76393	−0.234921	−0.117461	+	0.993078i	$0.537475\pi$
$830$	0	0
$831$	−90.2492	−3.13071
$832$	0	0
$833$	0	0
$834$	0	0
$835$	4.22291	0.146140
$836$	0	0
$837$	−93.6656	−3.23756
$838$	0	0
$839$	30.4721	1.05201	0.526007	−	0.850480i	$-0.323688\pi$
$839$	30.4721	1.05201	0.526007	+	0.850480i	$0.323688\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	84.1378	2.89786
$844$	0	0
$845$	17.8197	0.613015
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−52.3607	−1.79701
$850$	0	0
$851$	17.8885	0.613211
$852$	0	0
$853$	−0.291796	−0.00999091	−0.00499545	−	0.999988i	$-0.501590\pi$
$853$	−0.291796	−0.00999091	−0.00499545	+	0.999988i	$0.501590\pi$
$854$	0	0
$855$	−29.8885	−1.02217
$856$	0	0
$857$	36.4721	1.24586	0.622932	−	0.782276i	$-0.285941\pi$
$857$	36.4721	1.24586	0.622932	+	0.782276i	$0.285941\pi$
$858$	0	0
$859$	56.1803	1.91685	0.958424	−	0.285347i	$-0.0921089\pi$
$859$	56.1803	1.91685	0.958424	+	0.285347i	$0.0921089\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.05573	−0.104018	−0.0520091	−	0.998647i	$-0.516562\pi$
$863$	−3.05573	−0.104018	−0.0520091	+	0.998647i	$0.516562\pi$
$864$	0	0
$865$	9.52786	0.323957
$866$	0	0
$867$	9.70820	0.329708
$868$	0	0
$869$	−12.2229	−0.414634
$870$	0	0
$871$	−20.9443	−0.709670
$872$	0	0
$873$	−26.3607	−0.892174
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.4164	−0.453040	−0.226520	−	0.974007i	$-0.572735\pi$
$877$	−13.4164	−0.453040	−0.226520	+	0.974007i	$0.572735\pi$
$878$	0	0
$879$	55.7771	1.88131
$880$	0	0
$881$	28.8328	0.971402	0.485701	−	0.874125i	$-0.338564\pi$
$881$	28.8328	0.971402	0.485701	+	0.874125i	$0.338564\pi$
$882$	0	0
$883$	40.9443	1.37788	0.688942	−	0.724816i	$-0.258075\pi$
$883$	40.9443	1.37788	0.688942	+	0.724816i	$0.258075\pi$
$884$	0	0
$885$	19.0557	0.640551
$886$	0	0
$887$	−29.3050	−0.983964	−0.491982	−	0.870605i	$-0.663727\pi$
$887$	−29.3050	−0.983964	−0.491982	+	0.870605i	$0.663727\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	60.3607	2.02216
$892$	0	0
$893$	4.94427	0.165454
$894$	0	0
$895$	−30.8328	−1.03063
$896$	0	0
$897$	−67.7771	−2.26301
$898$	0	0
$899$	−28.9443	−0.965346
$900$	0	0
$901$	−44.7214	−1.48988
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.5836	−0.418293
$906$	0	0
$907$	−16.9443	−0.562625	−0.281313	−	0.959616i	$-0.590770\pi$
$907$	−16.9443	−0.562625	−0.281313	+	0.959616i	$0.590770\pi$
$908$	0	0
$909$	−87.4853	−2.90170
$910$	0	0
$911$	18.8328	0.623959	0.311980	−	0.950089i	$-0.399008\pi$
$911$	18.8328	0.623959	0.311980	+	0.950089i	$0.399008\pi$
$912$	0	0
$913$	11.7771	0.389765
$914$	0	0
$915$	−27.0557	−0.894435
$916$	0	0
$917$	0	0
$918$	0	0
$919$	35.7771	1.18018	0.590089	−	0.807338i	$-0.299093\pi$
$919$	35.7771	1.18018	0.590089	+	0.807338i	$0.299093\pi$
$920$	0	0
$921$	−78.2492	−2.57840
$922$	0	0
$923$	−67.7771	−2.23091
$924$	0	0
$925$	−15.5279	−0.510553
$926$	0	0
$927$	108.138	3.55171
$928$	0	0
$929$	−15.3050	−0.502139	−0.251070	−	0.967969i	$-0.580782\pi$
$929$	−15.3050	−0.502139	−0.251070	+	0.967969i	$0.580782\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−25.8885	−0.847553
$934$	0	0
$935$	13.6656	0.446914
$936$	0	0
$937$	26.9443	0.880231	0.440115	−	0.897941i	$-0.354937\pi$
$937$	26.9443	0.880231	0.440115	+	0.897941i	$0.354937\pi$
$938$	0	0
$939$	−1.52786	−0.0498600
$940$	0	0
$941$	−13.5967	−0.443241	−0.221621	−	0.975133i	$-0.571135\pi$
$941$	−13.5967	−0.443241	−0.221621	+	0.975133i	$0.571135\pi$
$942$	0	0
$943$	−1.88854	−0.0614994
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31.4164	−1.02090	−0.510448	−	0.859909i	$-0.670520\pi$
$947$	−31.4164	−1.02090	−0.510448	+	0.859909i	$0.670520\pi$
$948$	0	0
$949$	78.2492	2.54008
$950$	0	0
$951$	87.1935	2.82744
$952$	0	0
$953$	16.1115	0.521901	0.260951	−	0.965352i	$-0.415964\pi$
$953$	16.1115	0.521901	0.260951	+	0.965352i	$0.415964\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	0	0
$957$	35.7771	1.15651
$958$	0	0
$959$	0	0
$960$	0	0
$961$	10.8885	0.351243
$962$	0	0
$963$	−66.8328	−2.15366
$964$	0	0
$965$	−10.4721	−0.337110
$966$	0	0
$967$	−5.88854	−0.189363	−0.0946814	−	0.995508i	$-0.530183\pi$
$967$	−5.88854	−0.189363	−0.0946814	+	0.995508i	$0.530183\pi$
$968$	0	0
$969$	−46.8328	−1.50449
$970$	0	0
$971$	−27.2361	−0.874047	−0.437024	−	0.899450i	$-0.643967\pi$
$971$	−27.2361	−0.874047	−0.437024	+	0.899450i	$0.643967\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	58.8328	1.88416
$976$	0	0
$977$	40.8328	1.30636	0.653179	−	0.757204i	$-0.273435\pi$
$977$	40.8328	1.30636	0.653179	+	0.757204i	$0.273435\pi$
$978$	0	0
$979$	14.8328	0.474059
$980$	0	0
$981$	−3.52786	−0.112636
$982$	0	0
$983$	14.4721	0.461589	0.230795	−	0.973002i	$-0.425867\pi$
$983$	14.4721	0.461589	0.230795	+	0.973002i	$0.425867\pi$
$984$	0	0
$985$	−8.58359	−0.273496
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.88854	−0.314437
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	0	0
$993$	43.7771	1.38922
$994$	0	0
$995$	14.1115	0.447363
$996$	0	0
$997$	40.0689	1.26899	0.634497	−	0.772925i	$-0.281207\pi$
$997$	40.0689	1.26899	0.634497	+	0.772925i	$0.281207\pi$
$998$	0	0
$999$	64.7214	2.04769

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.a.v.1.2		2
4.3	odd	2		1568.2.a.k.1.1		2
7.2	even	3		1568.2.i.n.1537.1		4
7.3	odd	6		1568.2.i.v.961.2		4
7.4	even	3		1568.2.i.n.961.1		4
7.5	odd	6		1568.2.i.v.1537.2		4
7.6	odd	2		224.2.a.c.1.1	✓	2
8.3	odd	2		3136.2.a.by.1.2		2
8.5	even	2		3136.2.a.bf.1.1		2
21.20	even	2		2016.2.a.r.1.2		2
28.3	even	6		1568.2.i.m.961.1		4
28.11	odd	6		1568.2.i.w.961.2		4
28.19	even	6		1568.2.i.m.1537.1		4
28.23	odd	6		1568.2.i.w.1537.2		4
28.27	even	2		224.2.a.d.1.2	yes	2
35.34	odd	2		5600.2.a.bk.1.2		2
56.13	odd	2		448.2.a.j.1.2		2
56.27	even	2		448.2.a.i.1.1		2
84.83	odd	2		2016.2.a.o.1.2		2
112.13	odd	4		1792.2.b.k.897.1		4
112.27	even	4		1792.2.b.m.897.1		4
112.69	odd	4		1792.2.b.k.897.4		4
112.83	even	4		1792.2.b.m.897.4		4
140.139	even	2		5600.2.a.z.1.1		2
168.83	odd	2		4032.2.a.bv.1.1		2
168.125	even	2		4032.2.a.bw.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.a.c.1.1	✓	2	7.6	odd	2
224.2.a.d.1.2	yes	2	28.27	even	2
448.2.a.i.1.1		2	56.27	even	2
448.2.a.j.1.2		2	56.13	odd	2
1568.2.a.k.1.1		2	4.3	odd	2
1568.2.a.v.1.2		2	1.1	even	1	trivial
1568.2.i.m.961.1		4	28.3	even	6
1568.2.i.m.1537.1		4	28.19	even	6
1568.2.i.n.961.1		4	7.4	even	3
1568.2.i.n.1537.1		4	7.2	even	3
1568.2.i.v.961.2		4	7.3	odd	6
1568.2.i.v.1537.2		4	7.5	odd	6
1568.2.i.w.961.2		4	28.11	odd	6
1568.2.i.w.1537.2		4	28.23	odd	6
1792.2.b.k.897.1		4	112.13	odd	4
1792.2.b.k.897.4		4	112.69	odd	4
1792.2.b.m.897.1		4	112.27	even	4
1792.2.b.m.897.4		4	112.83	even	4
2016.2.a.o.1.2		2	84.83	odd	2
2016.2.a.r.1.2		2	21.20	even	2
3136.2.a.bf.1.1		2	8.5	even	2
3136.2.a.by.1.2		2	8.3	odd	2
4032.2.a.bv.1.1		2	168.83	odd	2
4032.2.a.bw.1.1		2	168.125	even	2
5600.2.a.z.1.1		2	140.139	even	2
5600.2.a.bk.1.2		2	35.34	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.a (trivial)

\(f(q)\)	\(=\)	\(q+3.23607 q^{3} +1.23607 q^{5} +7.47214 q^{9} +O(q^{10})\)	\(q+3.23607 q^{3} +1.23607 q^{5} +7.47214 q^{9} +2.47214 q^{11} -5.23607 q^{13} +4.00000 q^{15} +4.47214 q^{17} -3.23607 q^{19} +4.00000 q^{23} -3.47214 q^{25} +14.4721 q^{27} +4.47214 q^{29} -6.47214 q^{31} +8.00000 q^{33} +4.47214 q^{37} -16.9443 q^{39} -0.472136 q^{41} -2.47214 q^{43} +9.23607 q^{45} -1.52786 q^{47} +14.4721 q^{51} -10.0000 q^{53} +3.05573 q^{55} -10.4721 q^{57} +4.76393 q^{59} -6.76393 q^{61} -6.47214 q^{65} +4.00000 q^{67} +12.9443 q^{69} +12.9443 q^{71} -14.9443 q^{73} -11.2361 q^{75} -4.94427 q^{79} +24.4164 q^{81} +4.76393 q^{83} +5.52786 q^{85} +14.4721 q^{87} +6.00000 q^{89} -20.9443 q^{93} -4.00000 q^{95} -3.52786 q^{97} +18.4721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 6 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 6 q^{9} - 4 q^{11} - 6 q^{13} + 8 q^{15} - 2 q^{19} + 8 q^{23} + 2 q^{25} + 20 q^{27} - 4 q^{31} + 16 q^{33} - 16 q^{39} + 8 q^{41} + 4 q^{43} + 14 q^{45} - 12 q^{47} + 20 q^{51} - 20 q^{53} + 24 q^{55} - 12 q^{57} + 14 q^{59} - 18 q^{61} - 4 q^{65} + 8 q^{67} + 8 q^{69} + 8 q^{71} - 12 q^{73} - 18 q^{75} + 8 q^{79} + 22 q^{81} + 14 q^{83} + 20 q^{85} + 20 q^{87} + 12 q^{89} - 24 q^{93} - 8 q^{95} - 16 q^{97} + 28 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 6 * q^9 - 4 * q^11 - 6 * q^13 + 8 * q^15 - 2 * q^19 + 8 * q^23 + 2 * q^25 + 20 * q^27 - 4 * q^31 + 16 * q^33 - 16 * q^39 + 8 * q^41 + 4 * q^43 + 14 * q^45 - 12 * q^47 + 20 * q^51 - 20 * q^53 + 24 * q^55 - 12 * q^57 + 14 * q^59 - 18 * q^61 - 4 * q^65 + 8 * q^67 + 8 * q^69 + 8 * q^71 - 12 * q^73 - 18 * q^75 + 8 * q^79 + 22 * q^81 + 14 * q^83 + 20 * q^85 + 20 * q^87 + 12 * q^89 - 24 * q^93 - 8 * q^95 - 16 * q^97 + 28 * q^99

Introduction

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