LMFDB - Embedded newform 1040.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(1,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.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:	$8.30444181021$
Analytic rank:	$1$
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 520)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607 q^{3} -1.00000 q^{5} -1.47214 q^{9} +O(q^{10})$	$q+1.23607 q^{3} -1.00000 q^{5} -1.47214 q^{9} -5.23607 q^{11} -1.00000 q^{13} -1.23607 q^{15} +2.00000 q^{17} -5.23607 q^{19} -1.23607 q^{23} +1.00000 q^{25} -5.52786 q^{27} -0.472136 q^{29} -7.70820 q^{31} -6.47214 q^{33} +0.472136 q^{37} -1.23607 q^{39} +6.94427 q^{41} -1.23607 q^{43} +1.47214 q^{45} +4.94427 q^{47} -7.00000 q^{49} +2.47214 q^{51} +6.94427 q^{53} +5.23607 q^{55} -6.47214 q^{57} -7.70820 q^{59} +4.47214 q^{61} +1.00000 q^{65} -1.52786 q^{67} -1.52786 q^{69} +5.23607 q^{71} -16.4721 q^{73} +1.23607 q^{75} +2.47214 q^{79} -2.41641 q^{81} -4.00000 q^{83} -2.00000 q^{85} -0.583592 q^{87} +10.0000 q^{89} -9.52786 q^{93} +5.23607 q^{95} -10.0000 q^{97} +7.70820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{11} - 2 q^{13} + 2 q^{15} + 4 q^{17} - 6 q^{19} + 2 q^{23} + 2 q^{25} - 20 q^{27} + 8 q^{29} - 2 q^{31} - 4 q^{33} - 8 q^{37} + 2 q^{39} - 4 q^{41} + 2 q^{43} - 6 q^{45} - 8 q^{47} - 14 q^{49} - 4 q^{51} - 4 q^{53} + 6 q^{55} - 4 q^{57} - 2 q^{59} + 2 q^{65} - 12 q^{67} - 12 q^{69} + 6 q^{71} - 24 q^{73} - 2 q^{75} - 4 q^{79} + 22 q^{81} - 8 q^{83} - 4 q^{85} - 28 q^{87} + 20 q^{89} - 28 q^{93} + 6 q^{95} - 20 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 - 6 * q^11 - 2 * q^13 + 2 * q^15 + 4 * q^17 - 6 * q^19 + 2 * q^23 + 2 * q^25 - 20 * q^27 + 8 * q^29 - 2 * q^31 - 4 * q^33 - 8 * q^37 + 2 * q^39 - 4 * q^41 + 2 * q^43 - 6 * q^45 - 8 * q^47 - 14 * q^49 - 4 * q^51 - 4 * q^53 + 6 * q^55 - 4 * q^57 - 2 * q^59 + 2 * q^65 - 12 * q^67 - 12 * q^69 + 6 * q^71 - 24 * q^73 - 2 * q^75 - 4 * q^79 + 22 * q^81 - 8 * q^83 - 4 * q^85 - 28 * q^87 + 20 * q^89 - 28 * q^93 + 6 * q^95 - 20 * q^97 + 2 * 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
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	−5.23607	−1.57873	−0.789367	−	0.613922i	$-0.789591\pi$
$11$	−5.23607	−1.57873	−0.789367	+	0.613922i	$0.789591\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.23607	−0.319151
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−5.23607	−1.20124	−0.600618	−	0.799536i	$-0.705079\pi$
$19$	−5.23607	−1.20124	−0.600618	+	0.799536i	$0.705079\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.23607	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.52786	−1.06384
$28$	0	0
$29$	−0.472136	−0.0876734	−0.0438367	−	0.999039i	$-0.513958\pi$
$29$	−0.472136	−0.0876734	−0.0438367	+	0.999039i	$0.513958\pi$
$30$	0	0
$31$	−7.70820	−1.38443	−0.692217	−	0.721689i	$-0.743366\pi$
$31$	−7.70820	−1.38443	−0.692217	+	0.721689i	$0.743366\pi$
$32$	0	0
$33$	−6.47214	−1.12665
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.472136	0.0776187	0.0388093	−	0.999247i	$-0.487644\pi$
$37$	0.472136	0.0776187	0.0388093	+	0.999247i	$0.487644\pi$
$38$	0	0
$39$	−1.23607	−0.197929
$40$	0	0
$41$	6.94427	1.08451	0.542257	−	0.840213i	$-0.317570\pi$
$41$	6.94427	1.08451	0.542257	+	0.840213i	$0.317570\pi$
$42$	0	0
$43$	−1.23607	−0.188499	−0.0942493	−	0.995549i	$-0.530045\pi$
$43$	−1.23607	−0.188499	−0.0942493	+	0.995549i	$0.530045\pi$
$44$	0	0
$45$	1.47214	0.219453
$46$	0	0
$47$	4.94427	0.721196	0.360598	−	0.932721i	$-0.382573\pi$
$47$	4.94427	0.721196	0.360598	+	0.932721i	$0.382573\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	2.47214	0.346168
$52$	0	0
$53$	6.94427	0.953869	0.476935	−	0.878939i	$-0.341748\pi$
$53$	6.94427	0.953869	0.476935	+	0.878939i	$0.341748\pi$
$54$	0	0
$55$	5.23607	0.706031
$56$	0	0
$57$	−6.47214	−0.857255
$58$	0	0
$59$	−7.70820	−1.00352	−0.501761	−	0.865006i	$-0.667314\pi$
$59$	−7.70820	−1.00352	−0.501761	+	0.865006i	$0.667314\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−1.52786	−0.186658	−0.0933292	−	0.995635i	$-0.529751\pi$
$67$	−1.52786	−0.186658	−0.0933292	+	0.995635i	$0.529751\pi$
$68$	0	0
$69$	−1.52786	−0.183933
$70$	0	0
$71$	5.23607	0.621407	0.310703	−	0.950507i	$-0.399435\pi$
$71$	5.23607	0.621407	0.310703	+	0.950507i	$0.399435\pi$
$72$	0	0
$73$	−16.4721	−1.92792	−0.963959	−	0.266051i	$-0.914281\pi$
$73$	−16.4721	−1.92792	−0.963959	+	0.266051i	$0.914281\pi$
$74$	0	0
$75$	1.23607	0.142729
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.47214	0.278137	0.139069	−	0.990283i	$-0.455589\pi$
$79$	2.47214	0.278137	0.139069	+	0.990283i	$0.455589\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−0.583592	−0.0625676
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.52786	−0.987993
$94$	0	0
$95$	5.23607	0.537209
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	7.70820	0.774704
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	19.7082	1.94191	0.970954	−	0.239268i	$-0.0769075\pi$
$103$	19.7082	1.94191	0.970954	+	0.239268i	$0.0769075\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.18034	−0.597476	−0.298738	−	0.954335i	$-0.596566\pi$
$107$	−6.18034	−0.597476	−0.298738	+	0.954335i	$0.596566\pi$
$108$	0	0
$109$	−18.9443	−1.81453	−0.907266	−	0.420557i	$-0.861835\pi$
$109$	−18.9443	−1.81453	−0.907266	+	0.420557i	$0.861835\pi$
$110$	0	0
$111$	0.583592	0.0553921
$112$	0	0
$113$	−15.8885	−1.49467	−0.747334	−	0.664448i	$-0.768667\pi$
$113$	−15.8885	−1.49467	−0.747334	+	0.664448i	$0.768667\pi$
$114$	0	0
$115$	1.23607	0.115264
$116$	0	0
$117$	1.47214	0.136099
$118$	0	0
$119$	0	0
$120$	0	0
$121$	16.4164	1.49240
$122$	0	0
$123$	8.58359	0.773956
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.70820	−0.329050	−0.164525	−	0.986373i	$-0.552609\pi$
$127$	−3.70820	−0.329050	−0.164525	+	0.986373i	$0.552609\pi$
$128$	0	0
$129$	−1.52786	−0.134521
$130$	0	0
$131$	0.944272	0.0825014	0.0412507	−	0.999149i	$-0.486866\pi$
$131$	0.944272	0.0825014	0.0412507	+	0.999149i	$0.486866\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.52786	0.475763
$136$	0	0
$137$	15.8885	1.35745	0.678725	−	0.734393i	$-0.262533\pi$
$137$	15.8885	1.35745	0.678725	+	0.734393i	$0.262533\pi$
$138$	0	0
$139$	−22.4721	−1.90606	−0.953031	−	0.302873i	$-0.902054\pi$
$139$	−22.4721	−1.90606	−0.953031	+	0.302873i	$0.902054\pi$
$140$	0	0
$141$	6.11146	0.514677
$142$	0	0
$143$	5.23607	0.437862
$144$	0	0
$145$	0.472136	0.0392088
$146$	0	0
$147$	−8.65248	−0.713644
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	20.6525	1.68067	0.840337	−	0.542064i	$-0.182357\pi$
$151$	20.6525	1.68067	0.840337	+	0.542064i	$0.182357\pi$
$152$	0	0
$153$	−2.94427	−0.238030
$154$	0	0
$155$	7.70820	0.619138
$156$	0	0
$157$	6.94427	0.554213	0.277107	−	0.960839i	$-0.410624\pi$
$157$	6.94427	0.554213	0.277107	+	0.960839i	$0.410624\pi$
$158$	0	0
$159$	8.58359	0.680723
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1.52786	0.119672	0.0598358	−	0.998208i	$-0.480942\pi$
$163$	1.52786	0.119672	0.0598358	+	0.998208i	$0.480942\pi$
$164$	0	0
$165$	6.47214	0.503855
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.70820	0.589461
$172$	0	0
$173$	6.94427	0.527963	0.263982	−	0.964528i	$-0.414964\pi$
$173$	6.94427	0.527963	0.263982	+	0.964528i	$0.414964\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.52786	−0.716158
$178$	0	0
$179$	0.944272	0.0705782	0.0352891	−	0.999377i	$-0.488765\pi$
$179$	0.944272	0.0705782	0.0352891	+	0.999377i	$0.488765\pi$
$180$	0	0
$181$	−8.47214	−0.629729	−0.314864	−	0.949137i	$-0.601959\pi$
$181$	−8.47214	−0.629729	−0.314864	+	0.949137i	$0.601959\pi$
$182$	0	0
$183$	5.52786	0.408631
$184$	0	0
$185$	−0.472136	−0.0347121
$186$	0	0
$187$	−10.4721	−0.765798
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.8885	−1.29437	−0.647185	−	0.762333i	$-0.724054\pi$
$191$	−17.8885	−1.29437	−0.647185	+	0.762333i	$0.724054\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	1.23607	0.0885167
$196$	0	0
$197$	−15.8885	−1.13201	−0.566006	−	0.824401i	$-0.691512\pi$
$197$	−15.8885	−1.13201	−0.566006	+	0.824401i	$0.691512\pi$
$198$	0	0
$199$	−3.05573	−0.216615	−0.108307	−	0.994117i	$-0.534543\pi$
$199$	−3.05573	−0.216615	−0.108307	+	0.994117i	$0.534543\pi$
$200$	0	0
$201$	−1.88854	−0.133208
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.94427	−0.485009
$206$	0	0
$207$	1.81966	0.126475
$208$	0	0
$209$	27.4164	1.89643
$210$	0	0
$211$	8.94427	0.615749	0.307875	−	0.951427i	$-0.400382\pi$
$211$	8.94427	0.615749	0.307875	+	0.951427i	$0.400382\pi$
$212$	0	0
$213$	6.47214	0.443463
$214$	0	0
$215$	1.23607	0.0842991
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−20.3607	−1.37585
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	−25.8885	−1.73363	−0.866813	−	0.498634i	$-0.833835\pi$
$223$	−25.8885	−1.73363	−0.866813	+	0.498634i	$0.833835\pi$
$224$	0	0
$225$	−1.47214	−0.0981424
$226$	0	0
$227$	−9.52786	−0.632387	−0.316193	−	0.948695i	$-0.602405\pi$
$227$	−9.52786	−0.632387	−0.316193	+	0.948695i	$0.602405\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.94427	0.454934	0.227467	−	0.973786i	$-0.426956\pi$
$233$	6.94427	0.454934	0.227467	+	0.973786i	$0.426956\pi$
$234$	0	0
$235$	−4.94427	−0.322529
$236$	0	0
$237$	3.05573	0.198491
$238$	0	0
$239$	28.6525	1.85337	0.926687	−	0.375833i	$-0.122643\pi$
$239$	28.6525	1.85337	0.926687	+	0.375833i	$0.122643\pi$
$240$	0	0
$241$	19.8885	1.28113	0.640567	−	0.767902i	$-0.278699\pi$
$241$	19.8885	1.28113	0.640567	+	0.767902i	$0.278699\pi$
$242$	0	0
$243$	13.5967	0.872232
$244$	0	0
$245$	7.00000	0.447214
$246$	0	0
$247$	5.23607	0.333163
$248$	0	0
$249$	−4.94427	−0.313331
$250$	0	0
$251$	6.47214	0.408518	0.204259	−	0.978917i	$-0.434522\pi$
$251$	6.47214	0.408518	0.204259	+	0.978917i	$0.434522\pi$
$252$	0	0
$253$	6.47214	0.406900
$254$	0	0
$255$	−2.47214	−0.154811
$256$	0	0
$257$	22.9443	1.43122	0.715612	−	0.698498i	$-0.246148\pi$
$257$	22.9443	1.43122	0.715612	+	0.698498i	$0.246148\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0.695048	0.0430224
$262$	0	0
$263$	−11.7082	−0.721959	−0.360979	−	0.932574i	$-0.617558\pi$
$263$	−11.7082	−0.721959	−0.360979	+	0.932574i	$0.617558\pi$
$264$	0	0
$265$	−6.94427	−0.426583
$266$	0	0
$267$	12.3607	0.756461
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	12.6525	0.768583	0.384292	−	0.923212i	$-0.374446\pi$
$271$	12.6525	0.768583	0.384292	+	0.923212i	$0.374446\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.23607	−0.315747
$276$	0	0
$277$	−10.9443	−0.657578	−0.328789	−	0.944403i	$-0.606640\pi$
$277$	−10.9443	−0.657578	−0.328789	+	0.944403i	$0.606640\pi$
$278$	0	0
$279$	11.3475	0.679359
$280$	0	0
$281$	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
$283$	0.652476	0.0387857	0.0193928	−	0.999812i	$-0.493827\pi$
$283$	0.652476	0.0387857	0.0193928	+	0.999812i	$0.493827\pi$
$284$	0	0
$285$	6.47214	0.383376
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−12.3607	−0.724596
$292$	0	0
$293$	−22.3607	−1.30632	−0.653162	−	0.757218i	$-0.726558\pi$
$293$	−22.3607	−1.30632	−0.653162	+	0.757218i	$0.726558\pi$
$294$	0	0
$295$	7.70820	0.448789
$296$	0	0
$297$	28.9443	1.67952
$298$	0	0
$299$	1.23607	0.0714837
$300$	0	0
$301$	0	0
$302$	0	0
$303$	17.3050	0.994143
$304$	0	0
$305$	−4.47214	−0.256074
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	24.3607	1.38583
$310$	0	0
$311$	15.4164	0.874184	0.437092	−	0.899417i	$-0.356008\pi$
$311$	15.4164	0.874184	0.437092	+	0.899417i	$0.356008\pi$
$312$	0	0
$313$	32.8328	1.85582	0.927910	−	0.372804i	$-0.121604\pi$
$313$	32.8328	1.85582	0.927910	+	0.372804i	$0.121604\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.3607	−0.806576	−0.403288	−	0.915073i	$-0.632133\pi$
$317$	−14.3607	−0.806576	−0.403288	+	0.915073i	$0.632133\pi$
$318$	0	0
$319$	2.47214	0.138413
$320$	0	0
$321$	−7.63932	−0.426385
$322$	0	0
$323$	−10.4721	−0.582685
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−23.4164	−1.29493
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5.23607	0.287800	0.143900	−	0.989592i	$-0.454036\pi$
$331$	5.23607	0.287800	0.143900	+	0.989592i	$0.454036\pi$
$332$	0	0
$333$	−0.695048	−0.0380884
$334$	0	0
$335$	1.52786	0.0834761
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	0	0
$339$	−19.6393	−1.06666
$340$	0	0
$341$	40.3607	2.18565
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.52786	0.0822574
$346$	0	0
$347$	−25.2361	−1.35474	−0.677372	−	0.735641i	$-0.736881\pi$
$347$	−25.2361	−1.35474	−0.677372	+	0.735641i	$0.736881\pi$
$348$	0	0
$349$	−2.94427	−0.157603	−0.0788016	−	0.996890i	$-0.525109\pi$
$349$	−2.94427	−0.157603	−0.0788016	+	0.996890i	$0.525109\pi$
$350$	0	0
$351$	5.52786	0.295056
$352$	0	0
$353$	−3.52786	−0.187769	−0.0938846	−	0.995583i	$-0.529928\pi$
$353$	−3.52786	−0.187769	−0.0938846	+	0.995583i	$0.529928\pi$
$354$	0	0
$355$	−5.23607	−0.277902
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.76393	−0.145875	−0.0729374	−	0.997337i	$-0.523237\pi$
$359$	−2.76393	−0.145875	−0.0729374	+	0.997337i	$0.523237\pi$
$360$	0	0
$361$	8.41641	0.442969
$362$	0	0
$363$	20.2918	1.06504
$364$	0	0
$365$	16.4721	0.862191
$366$	0	0
$367$	1.81966	0.0949855	0.0474927	−	0.998872i	$-0.484877\pi$
$367$	1.81966	0.0949855	0.0474927	+	0.998872i	$0.484877\pi$
$368$	0	0
$369$	−10.2229	−0.532184
$370$	0	0
$371$	0	0
$372$	0	0
$373$	14.9443	0.773785	0.386893	−	0.922125i	$-0.373548\pi$
$373$	14.9443	0.773785	0.386893	+	0.922125i	$0.373548\pi$
$374$	0	0
$375$	−1.23607	−0.0638303
$376$	0	0
$377$	0.472136	0.0243162
$378$	0	0
$379$	−21.2361	−1.09082	−0.545412	−	0.838168i	$-0.683627\pi$
$379$	−21.2361	−1.09082	−0.545412	+	0.838168i	$0.683627\pi$
$380$	0	0
$381$	−4.58359	−0.234825
$382$	0	0
$383$	−17.8885	−0.914062	−0.457031	−	0.889451i	$-0.651087\pi$
$383$	−17.8885	−0.914062	−0.457031	+	0.889451i	$0.651087\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.81966	0.0924985
$388$	0	0
$389$	−3.88854	−0.197157	−0.0985785	−	0.995129i	$-0.531430\pi$
$389$	−3.88854	−0.197157	−0.0985785	+	0.995129i	$0.531430\pi$
$390$	0	0
$391$	−2.47214	−0.125021
$392$	0	0
$393$	1.16718	0.0588767
$394$	0	0
$395$	−2.47214	−0.124387
$396$	0	0
$397$	−33.4164	−1.67712	−0.838561	−	0.544808i	$-0.816602\pi$
$397$	−33.4164	−1.67712	−0.838561	+	0.544808i	$0.816602\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.8885	1.39269	0.696344	−	0.717708i	$-0.254809\pi$
$401$	27.8885	1.39269	0.696344	+	0.717708i	$0.254809\pi$
$402$	0	0
$403$	7.70820	0.383973
$404$	0	0
$405$	2.41641	0.120072
$406$	0	0
$407$	−2.47214	−0.122539
$408$	0	0
$409$	−20.8328	−1.03012	−0.515058	−	0.857155i	$-0.672230\pi$
$409$	−20.8328	−1.03012	−0.515058	+	0.857155i	$0.672230\pi$
$410$	0	0
$411$	19.6393	0.968736
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	−27.7771	−1.36025
$418$	0	0
$419$	27.4164	1.33938	0.669690	−	0.742641i	$-0.266427\pi$
$419$	27.4164	1.33938	0.669690	+	0.742641i	$0.266427\pi$
$420$	0	0
$421$	−31.8885	−1.55415	−0.777076	−	0.629406i	$-0.783298\pi$
$421$	−31.8885	−1.55415	−0.777076	+	0.629406i	$0.783298\pi$
$422$	0	0
$423$	−7.27864	−0.353900
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6.47214	0.312478
$430$	0	0
$431$	20.0689	0.966684	0.483342	−	0.875432i	$-0.339423\pi$
$431$	20.0689	0.966684	0.483342	+	0.875432i	$0.339423\pi$
$432$	0	0
$433$	−28.8328	−1.38562	−0.692808	−	0.721122i	$-0.743627\pi$
$433$	−28.8328	−1.38562	−0.692808	+	0.721122i	$0.743627\pi$
$434$	0	0
$435$	0.583592	0.0279811
$436$	0	0
$437$	6.47214	0.309604
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	10.3050	0.490712
$442$	0	0
$443$	−8.65248	−0.411092	−0.205546	−	0.978648i	$-0.565897\pi$
$443$	−8.65248	−0.411092	−0.205546	+	0.978648i	$0.565897\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	−7.41641	−0.350784
$448$	0	0
$449$	−26.9443	−1.27158	−0.635789	−	0.771863i	$-0.719325\pi$
$449$	−26.9443	−1.27158	−0.635789	+	0.771863i	$0.719325\pi$
$450$	0	0
$451$	−36.3607	−1.71216
$452$	0	0
$453$	25.5279	1.19940
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.8885	−1.30457	−0.652286	−	0.757973i	$-0.726190\pi$
$457$	−27.8885	−1.30457	−0.652286	+	0.757973i	$0.726190\pi$
$458$	0	0
$459$	−11.0557	−0.516037
$460$	0	0
$461$	−39.8885	−1.85779	−0.928897	−	0.370337i	$-0.879242\pi$
$461$	−39.8885	−1.85779	−0.928897	+	0.370337i	$0.879242\pi$
$462$	0	0
$463$	−7.41641	−0.344670	−0.172335	−	0.985038i	$-0.555131\pi$
$463$	−7.41641	−0.344670	−0.172335	+	0.985038i	$0.555131\pi$
$464$	0	0
$465$	9.52786	0.441844
$466$	0	0
$467$	−11.7082	−0.541791	−0.270896	−	0.962609i	$-0.587320\pi$
$467$	−11.7082	−0.541791	−0.270896	+	0.962609i	$0.587320\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	8.58359	0.395511
$472$	0	0
$473$	6.47214	0.297589
$474$	0	0
$475$	−5.23607	−0.240247
$476$	0	0
$477$	−10.2229	−0.468075
$478$	0	0
$479$	31.1246	1.42212	0.711060	−	0.703131i	$-0.248215\pi$
$479$	31.1246	1.42212	0.711060	+	0.703131i	$0.248215\pi$
$480$	0	0
$481$	−0.472136	−0.0215275
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.0000	0.454077
$486$	0	0
$487$	33.3050	1.50919	0.754596	−	0.656190i	$-0.227833\pi$
$487$	33.3050	1.50919	0.754596	+	0.656190i	$0.227833\pi$
$488$	0	0
$489$	1.88854	0.0854029
$490$	0	0
$491$	25.5279	1.15206	0.576028	−	0.817430i	$-0.304602\pi$
$491$	25.5279	1.15206	0.576028	+	0.817430i	$0.304602\pi$
$492$	0	0
$493$	−0.944272	−0.0425279
$494$	0	0
$495$	−7.70820	−0.346458
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−24.2918	−1.08745	−0.543725	−	0.839263i	$-0.682987\pi$
$499$	−24.2918	−1.08745	−0.543725	+	0.839263i	$0.682987\pi$
$500$	0	0
$501$	−9.88854	−0.441788
$502$	0	0
$503$	−27.7082	−1.23545	−0.617724	−	0.786395i	$-0.711945\pi$
$503$	−27.7082	−1.23545	−0.617724	+	0.786395i	$0.711945\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	1.23607	0.0548957
$508$	0	0
$509$	−26.9443	−1.19428	−0.597142	−	0.802136i	$-0.703697\pi$
$509$	−26.9443	−1.19428	−0.597142	+	0.802136i	$0.703697\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	28.9443	1.27792
$514$	0	0
$515$	−19.7082	−0.868447
$516$	0	0
$517$	−25.8885	−1.13858
$518$	0	0
$519$	8.58359	0.376778
$520$	0	0
$521$	−33.4164	−1.46400	−0.732000	−	0.681305i	$-0.761413\pi$
$521$	−33.4164	−1.46400	−0.732000	+	0.681305i	$0.761413\pi$
$522$	0	0
$523$	−29.5967	−1.29418	−0.647088	−	0.762416i	$-0.724013\pi$
$523$	−29.5967	−1.29418	−0.647088	+	0.762416i	$0.724013\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.4164	−0.671549
$528$	0	0
$529$	−21.4721	−0.933571
$530$	0	0
$531$	11.3475	0.492441
$532$	0	0
$533$	−6.94427	−0.300790
$534$	0	0
$535$	6.18034	0.267199
$536$	0	0
$537$	1.16718	0.0503677
$538$	0	0
$539$	36.6525	1.57873
$540$	0	0
$541$	−34.9443	−1.50237	−0.751186	−	0.660091i	$-0.770518\pi$
$541$	−34.9443	−1.50237	−0.751186	+	0.660091i	$0.770518\pi$
$542$	0	0
$543$	−10.4721	−0.449402
$544$	0	0
$545$	18.9443	0.811483
$546$	0	0
$547$	13.5967	0.581355	0.290677	−	0.956821i	$-0.406119\pi$
$547$	13.5967	0.581355	0.290677	+	0.956821i	$0.406119\pi$
$548$	0	0
$549$	−6.58359	−0.280981
$550$	0	0
$551$	2.47214	0.105317
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.583592	−0.0247721
$556$	0	0
$557$	−1.41641	−0.0600151	−0.0300076	−	0.999550i	$-0.509553\pi$
$557$	−1.41641	−0.0600151	−0.0300076	+	0.999550i	$0.509553\pi$
$558$	0	0
$559$	1.23607	0.0522801
$560$	0	0
$561$	−12.9443	−0.546508
$562$	0	0
$563$	27.7082	1.16776	0.583881	−	0.811839i	$-0.301534\pi$
$563$	27.7082	1.16776	0.583881	+	0.811839i	$0.301534\pi$
$564$	0	0
$565$	15.8885	0.668436
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.5836	0.611376	0.305688	−	0.952132i	$-0.401114\pi$
$569$	14.5836	0.611376	0.305688	+	0.952132i	$0.401114\pi$
$570$	0	0
$571$	22.4721	0.940430	0.470215	−	0.882552i	$-0.344176\pi$
$571$	22.4721	0.940430	0.470215	+	0.882552i	$0.344176\pi$
$572$	0	0
$573$	−22.1115	−0.923719
$574$	0	0
$575$	−1.23607	−0.0515476
$576$	0	0
$577$	−14.5836	−0.607123	−0.303561	−	0.952812i	$-0.598176\pi$
$577$	−14.5836	−0.607123	−0.303561	+	0.952812i	$0.598176\pi$
$578$	0	0
$579$	7.41641	0.308215
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−36.3607	−1.50591
$584$	0	0
$585$	−1.47214	−0.0608653
$586$	0	0
$587$	22.4721	0.927524	0.463762	−	0.885960i	$-0.346499\pi$
$587$	22.4721	0.927524	0.463762	+	0.885960i	$0.346499\pi$
$588$	0	0
$589$	40.3607	1.66303
$590$	0	0
$591$	−19.6393	−0.807854
$592$	0	0
$593$	−35.8885	−1.47377	−0.736883	−	0.676020i	$-0.763703\pi$
$593$	−35.8885	−1.47377	−0.736883	+	0.676020i	$0.763703\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.77709	−0.154586
$598$	0	0
$599$	−2.47214	−0.101009	−0.0505044	−	0.998724i	$-0.516083\pi$
$599$	−2.47214	−0.101009	−0.0505044	+	0.998724i	$0.516083\pi$
$600$	0	0
$601$	19.8885	0.811271	0.405635	−	0.914035i	$-0.367050\pi$
$601$	19.8885	0.811271	0.405635	+	0.914035i	$0.367050\pi$
$602$	0	0
$603$	2.24922	0.0915955
$604$	0	0
$605$	−16.4164	−0.667422
$606$	0	0
$607$	−24.0689	−0.976926	−0.488463	−	0.872585i	$-0.662442\pi$
$607$	−24.0689	−0.976926	−0.488463	+	0.872585i	$0.662442\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.94427	−0.200024
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	0	0
$615$	−8.58359	−0.346124
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−28.6525	−1.15164	−0.575820	−	0.817576i	$-0.695317\pi$
$619$	−28.6525	−1.15164	−0.575820	+	0.817576i	$0.695317\pi$
$620$	0	0
$621$	6.83282	0.274191
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	33.8885	1.35338
$628$	0	0
$629$	0.944272	0.0376506
$630$	0	0
$631$	−33.0132	−1.31423	−0.657116	−	0.753789i	$-0.728224\pi$
$631$	−33.0132	−1.31423	−0.657116	+	0.753789i	$0.728224\pi$
$632$	0	0
$633$	11.0557	0.439426
$634$	0	0
$635$	3.70820	0.147156
$636$	0	0
$637$	7.00000	0.277350
$638$	0	0
$639$	−7.70820	−0.304932
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	0	0
$643$	−32.9443	−1.29920	−0.649598	−	0.760278i	$-0.725063\pi$
$643$	−32.9443	−1.29920	−0.649598	+	0.760278i	$0.725063\pi$
$644$	0	0
$645$	1.52786	0.0601596
$646$	0	0
$647$	−14.7639	−0.580430	−0.290215	−	0.956961i	$-0.593727\pi$
$647$	−14.7639	−0.580430	−0.290215	+	0.956961i	$0.593727\pi$
$648$	0	0
$649$	40.3607	1.58430
$650$	0	0
$651$	0	0
$652$	0	0
$653$	40.8328	1.59791	0.798956	−	0.601390i	$-0.205386\pi$
$653$	40.8328	1.59791	0.798956	+	0.601390i	$0.205386\pi$
$654$	0	0
$655$	−0.944272	−0.0368958
$656$	0	0
$657$	24.2492	0.946052
$658$	0	0
$659$	−29.3050	−1.14156	−0.570779	−	0.821103i	$-0.693359\pi$
$659$	−29.3050	−1.14156	−0.570779	+	0.821103i	$0.693359\pi$
$660$	0	0
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	0	0
$663$	−2.47214	−0.0960098
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.583592	0.0225968
$668$	0	0
$669$	−32.0000	−1.23719
$670$	0	0
$671$	−23.4164	−0.903980
$672$	0	0
$673$	8.11146	0.312674	0.156337	−	0.987704i	$-0.450031\pi$
$673$	8.11146	0.312674	0.156337	+	0.987704i	$0.450031\pi$
$674$	0	0
$675$	−5.52786	−0.212768
$676$	0	0
$677$	40.8328	1.56933	0.784666	−	0.619918i	$-0.212834\pi$
$677$	40.8328	1.56933	0.784666	+	0.619918i	$0.212834\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−11.7771	−0.451299
$682$	0	0
$683$	−0.360680	−0.0138010	−0.00690051	−	0.999976i	$-0.502197\pi$
$683$	−0.360680	−0.0138010	−0.00690051	+	0.999976i	$0.502197\pi$
$684$	0	0
$685$	−15.8885	−0.607070
$686$	0	0
$687$	12.3607	0.471589
$688$	0	0
$689$	−6.94427	−0.264556
$690$	0	0
$691$	33.0132	1.25588	0.627940	−	0.778262i	$-0.283898\pi$
$691$	33.0132	1.25588	0.627940	+	0.778262i	$0.283898\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22.4721	0.852417
$696$	0	0
$697$	13.8885	0.526066
$698$	0	0
$699$	8.58359	0.324661
$700$	0	0
$701$	−16.1115	−0.608521	−0.304261	−	0.952589i	$-0.598409\pi$
$701$	−16.1115	−0.608521	−0.304261	+	0.952589i	$0.598409\pi$
$702$	0	0
$703$	−2.47214	−0.0932384
$704$	0	0
$705$	−6.11146	−0.230171
$706$	0	0
$707$	0	0
$708$	0	0
$709$	19.8885	0.746930	0.373465	−	0.927644i	$-0.378170\pi$
$709$	19.8885	0.746930	0.373465	+	0.927644i	$0.378170\pi$
$710$	0	0
$711$	−3.63932	−0.136485
$712$	0	0
$713$	9.52786	0.356821
$714$	0	0
$715$	−5.23607	−0.195818
$716$	0	0
$717$	35.4164	1.32265
$718$	0	0
$719$	−6.83282	−0.254821	−0.127411	−	0.991850i	$-0.540667\pi$
$719$	−6.83282	−0.254821	−0.127411	+	0.991850i	$0.540667\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	24.5836	0.914274
$724$	0	0
$725$	−0.472136	−0.0175347
$726$	0	0
$727$	12.2918	0.455877	0.227939	−	0.973675i	$-0.426801\pi$
$727$	12.2918	0.455877	0.227939	+	0.973675i	$0.426801\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	−2.47214	−0.0914353
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	0	0
$735$	8.65248	0.319151
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	41.0132	1.50869	0.754347	−	0.656476i	$-0.227954\pi$
$739$	41.0132	1.50869	0.754347	+	0.656476i	$0.227954\pi$
$740$	0	0
$741$	6.47214	0.237760
$742$	0	0
$743$	−41.8885	−1.53674	−0.768371	−	0.640005i	$-0.778932\pi$
$743$	−41.8885	−1.53674	−0.768371	+	0.640005i	$0.778932\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	5.88854	0.215451
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−37.5279	−1.36941	−0.684706	−	0.728820i	$-0.740069\pi$
$751$	−37.5279	−1.36941	−0.684706	+	0.728820i	$0.740069\pi$
$752$	0	0
$753$	8.00000	0.291536
$754$	0	0
$755$	−20.6525	−0.751621
$756$	0	0
$757$	24.8328	0.902564	0.451282	−	0.892381i	$-0.350967\pi$
$757$	24.8328	0.902564	0.451282	+	0.892381i	$0.350967\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.94427	0.106450
$766$	0	0
$767$	7.70820	0.278327
$768$	0	0
$769$	−2.94427	−0.106173	−0.0530866	−	0.998590i	$-0.516906\pi$
$769$	−2.94427	−0.106173	−0.0530866	+	0.998590i	$0.516906\pi$
$770$	0	0
$771$	28.3607	1.02138
$772$	0	0
$773$	52.2492	1.87927	0.939637	−	0.342173i	$-0.111163\pi$
$773$	52.2492	1.87927	0.939637	+	0.342173i	$0.111163\pi$
$774$	0	0
$775$	−7.70820	−0.276887
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−36.3607	−1.30276
$780$	0	0
$781$	−27.4164	−0.981036
$782$	0	0
$783$	2.60990	0.0932703
$784$	0	0
$785$	−6.94427	−0.247852
$786$	0	0
$787$	−13.8885	−0.495073	−0.247537	−	0.968879i	$-0.579621\pi$
$787$	−13.8885	−0.495073	−0.247537	+	0.968879i	$0.579621\pi$
$788$	0	0
$789$	−14.4721	−0.515222
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.47214	−0.158810
$794$	0	0
$795$	−8.58359	−0.304429
$796$	0	0
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	9.88854	0.349832
$800$	0	0
$801$	−14.7214	−0.520154
$802$	0	0
$803$	86.2492	3.04367
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.41641	0.261070
$808$	0	0
$809$	−4.47214	−0.157232	−0.0786160	−	0.996905i	$-0.525050\pi$
$809$	−4.47214	−0.157232	−0.0786160	+	0.996905i	$0.525050\pi$
$810$	0	0
$811$	−34.7639	−1.22073	−0.610363	−	0.792122i	$-0.708977\pi$
$811$	−34.7639	−1.22073	−0.610363	+	0.792122i	$0.708977\pi$
$812$	0	0
$813$	15.6393	0.548495
$814$	0	0
$815$	−1.52786	−0.0535187
$816$	0	0
$817$	6.47214	0.226431
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.9443	0.800761	0.400380	−	0.916349i	$-0.368878\pi$
$821$	22.9443	0.800761	0.400380	+	0.916349i	$0.368878\pi$
$822$	0	0
$823$	−19.1246	−0.666642	−0.333321	−	0.942813i	$-0.608169\pi$
$823$	−19.1246	−0.666642	−0.333321	+	0.942813i	$0.608169\pi$
$824$	0	0
$825$	−6.47214	−0.225331
$826$	0	0
$827$	26.8328	0.933068	0.466534	−	0.884503i	$-0.345502\pi$
$827$	26.8328	0.933068	0.466534	+	0.884503i	$0.345502\pi$
$828$	0	0
$829$	10.5836	0.367583	0.183792	−	0.982965i	$-0.441163\pi$
$829$	10.5836	0.367583	0.183792	+	0.982965i	$0.441163\pi$
$830$	0	0
$831$	−13.5279	−0.469276
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	42.6099	1.47281
$838$	0	0
$839$	19.3475	0.667951	0.333975	−	0.942582i	$-0.391610\pi$
$839$	19.3475	0.667951	0.333975	+	0.942582i	$0.391610\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	12.3607	0.425724
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0.806504	0.0276792
$850$	0	0
$851$	−0.583592	−0.0200053
$852$	0	0
$853$	3.52786	0.120792	0.0603959	−	0.998175i	$-0.480764\pi$
$853$	3.52786	0.120792	0.0603959	+	0.998175i	$0.480764\pi$
$854$	0	0
$855$	−7.70820	−0.263615
$856$	0	0
$857$	38.9443	1.33031	0.665155	−	0.746705i	$-0.268365\pi$
$857$	38.9443	1.33031	0.665155	+	0.746705i	$0.268365\pi$
$858$	0	0
$859$	−56.3607	−1.92300	−0.961501	−	0.274802i	$-0.911388\pi$
$859$	−56.3607	−1.92300	−0.961501	+	0.274802i	$0.911388\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.9443	−1.25760	−0.628799	−	0.777568i	$-0.716453\pi$
$863$	−36.9443	−1.25760	−0.628799	+	0.777568i	$0.716453\pi$
$864$	0	0
$865$	−6.94427	−0.236112
$866$	0	0
$867$	−16.0689	−0.545728
$868$	0	0
$869$	−12.9443	−0.439104
$870$	0	0
$871$	1.52786	0.0517697
$872$	0	0
$873$	14.7214	0.498243
$874$	0	0
$875$	0	0
$876$	0	0
$877$	53.7771	1.81592	0.907962	−	0.419053i	$-0.137638\pi$
$877$	53.7771	1.81592	0.907962	+	0.419053i	$0.137638\pi$
$878$	0	0
$879$	−27.6393	−0.932251
$880$	0	0
$881$	−54.3607	−1.83146	−0.915729	−	0.401797i	$-0.868386\pi$
$881$	−54.3607	−1.83146	−0.915729	+	0.401797i	$0.868386\pi$
$882$	0	0
$883$	27.7082	0.932455	0.466228	−	0.884665i	$-0.345613\pi$
$883$	27.7082	0.932455	0.466228	+	0.884665i	$0.345613\pi$
$884$	0	0
$885$	9.52786	0.320276
$886$	0	0
$887$	−9.23607	−0.310117	−0.155058	−	0.987905i	$-0.549557\pi$
$887$	−9.23607	−0.310117	−0.155058	+	0.987905i	$0.549557\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	12.6525	0.423874
$892$	0	0
$893$	−25.8885	−0.866327
$894$	0	0
$895$	−0.944272	−0.0315635
$896$	0	0
$897$	1.52786	0.0510139
$898$	0	0
$899$	3.63932	0.121378
$900$	0	0
$901$	13.8885	0.462694
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.47214	0.281623
$906$	0	0
$907$	28.2918	0.939414	0.469707	−	0.882822i	$-0.344360\pi$
$907$	28.2918	0.939414	0.469707	+	0.882822i	$0.344360\pi$
$908$	0	0
$909$	−20.6099	−0.683587
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	20.9443	0.693154
$914$	0	0
$915$	−5.52786	−0.182746
$916$	0	0
$917$	0	0
$918$	0	0
$919$	29.5279	0.974034	0.487017	−	0.873392i	$-0.338085\pi$
$919$	29.5279	0.974034	0.487017	+	0.873392i	$0.338085\pi$
$920$	0	0
$921$	14.8328	0.488758
$922$	0	0
$923$	−5.23607	−0.172347
$924$	0	0
$925$	0.472136	0.0155237
$926$	0	0
$927$	−29.0132	−0.952917
$928$	0	0
$929$	−1.05573	−0.0346373	−0.0173187	−	0.999850i	$-0.505513\pi$
$929$	−1.05573	−0.0346373	−0.0173187	+	0.999850i	$0.505513\pi$
$930$	0	0
$931$	36.6525	1.20124
$932$	0	0
$933$	19.0557	0.623857
$934$	0	0
$935$	10.4721	0.342475
$936$	0	0
$937$	8.83282	0.288556	0.144278	−	0.989537i	$-0.453914\pi$
$937$	8.83282	0.288556	0.144278	+	0.989537i	$0.453914\pi$
$938$	0	0
$939$	40.5836	1.32440
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	−8.58359	−0.279520
$944$	0	0
$945$	0	0
$946$	0	0
$947$	56.9443	1.85044	0.925220	−	0.379431i	$-0.123880\pi$
$947$	56.9443	1.85044	0.925220	+	0.379431i	$0.123880\pi$
$948$	0	0
$949$	16.4721	0.534708
$950$	0	0
$951$	−17.7508	−0.575608
$952$	0	0
$953$	−60.8328	−1.97057	−0.985284	−	0.170925i	$-0.945325\pi$
$953$	−60.8328	−1.97057	−0.985284	+	0.170925i	$0.945325\pi$
$954$	0	0
$955$	17.8885	0.578860
$956$	0	0
$957$	3.05573	0.0987777
$958$	0	0
$959$	0	0
$960$	0	0
$961$	28.4164	0.916658
$962$	0	0
$963$	9.09830	0.293189
$964$	0	0
$965$	−6.00000	−0.193147
$966$	0	0
$967$	56.1378	1.80527	0.902634	−	0.430408i	$-0.141630\pi$
$967$	56.1378	1.80527	0.902634	+	0.430408i	$0.141630\pi$
$968$	0	0
$969$	−12.9443	−0.415830
$970$	0	0
$971$	26.8328	0.861106	0.430553	−	0.902565i	$-0.358319\pi$
$971$	26.8328	0.861106	0.430553	+	0.902565i	$0.358319\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.23607	−0.0395859
$976$	0	0
$977$	−24.4721	−0.782933	−0.391466	−	0.920192i	$-0.628032\pi$
$977$	−24.4721	−0.782933	−0.391466	+	0.920192i	$0.628032\pi$
$978$	0	0
$979$	−52.3607	−1.67345
$980$	0	0
$981$	27.8885	0.890413
$982$	0	0
$983$	−52.3607	−1.67005	−0.835023	−	0.550215i	$-0.814546\pi$
$983$	−52.3607	−1.67005	−0.835023	+	0.550215i	$0.814546\pi$
$984$	0	0
$985$	15.8885	0.506251
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.52786	0.0485833
$990$	0	0
$991$	20.9443	0.665317	0.332658	−	0.943047i	$-0.392054\pi$
$991$	20.9443	0.665317	0.332658	+	0.943047i	$0.392054\pi$
$992$	0	0
$993$	6.47214	0.205387
$994$	0	0
$995$	3.05573	0.0968731
$996$	0	0
$997$	−52.8328	−1.67323	−0.836616	−	0.547790i	$-0.815469\pi$
$997$	−52.8328	−1.67323	−0.836616	+	0.547790i	$0.815469\pi$
$998$	0	0
$999$	−2.60990	−0.0825737

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.a.i.1.2		2
3.2	odd	2		9360.2.a.cs.1.2		2
4.3	odd	2		520.2.a.g.1.1	✓	2
5.4	even	2		5200.2.a.bz.1.1		2
8.3	odd	2		4160.2.a.x.1.2		2
8.5	even	2		4160.2.a.bm.1.1		2
12.11	even	2		4680.2.a.bc.1.1		2
20.3	even	4		2600.2.d.j.1249.2		4
20.7	even	4		2600.2.d.j.1249.3		4
20.19	odd	2		2600.2.a.o.1.2		2
52.51	odd	2		6760.2.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.g.1.1	✓	2	4.3	odd	2
1040.2.a.i.1.2		2	1.1	even	1	trivial
2600.2.a.o.1.2		2	20.19	odd	2
2600.2.d.j.1249.2		4	20.3	even	4
2600.2.d.j.1249.3		4	20.7	even	4
4160.2.a.x.1.2		2	8.3	odd	2
4160.2.a.bm.1.1		2	8.5	even	2
4680.2.a.bc.1.1		2	12.11	even	2
5200.2.a.bz.1.1		2	5.4	even	2
6760.2.a.t.1.1		2	52.51	odd	2
9360.2.a.cs.1.2		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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{3} -1.00000 q^{5} -1.47214 q^{9} +O(q^{10})\)	\(q+1.23607 q^{3} -1.00000 q^{5} -1.47214 q^{9} -5.23607 q^{11} -1.00000 q^{13} -1.23607 q^{15} +2.00000 q^{17} -5.23607 q^{19} -1.23607 q^{23} +1.00000 q^{25} -5.52786 q^{27} -0.472136 q^{29} -7.70820 q^{31} -6.47214 q^{33} +0.472136 q^{37} -1.23607 q^{39} +6.94427 q^{41} -1.23607 q^{43} +1.47214 q^{45} +4.94427 q^{47} -7.00000 q^{49} +2.47214 q^{51} +6.94427 q^{53} +5.23607 q^{55} -6.47214 q^{57} -7.70820 q^{59} +4.47214 q^{61} +1.00000 q^{65} -1.52786 q^{67} -1.52786 q^{69} +5.23607 q^{71} -16.4721 q^{73} +1.23607 q^{75} +2.47214 q^{79} -2.41641 q^{81} -4.00000 q^{83} -2.00000 q^{85} -0.583592 q^{87} +10.0000 q^{89} -9.52786 q^{93} +5.23607 q^{95} -10.0000 q^{97} +7.70820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9} - 6 q^{11} - 2 q^{13} + 2 q^{15} + 4 q^{17} - 6 q^{19} + 2 q^{23} + 2 q^{25} - 20 q^{27} + 8 q^{29} - 2 q^{31} - 4 q^{33} - 8 q^{37} + 2 q^{39} - 4 q^{41} + 2 q^{43} - 6 q^{45} - 8 q^{47} - 14 q^{49} - 4 q^{51} - 4 q^{53} + 6 q^{55} - 4 q^{57} - 2 q^{59} + 2 q^{65} - 12 q^{67} - 12 q^{69} + 6 q^{71} - 24 q^{73} - 2 q^{75} - 4 q^{79} + 22 q^{81} - 8 q^{83} - 4 q^{85} - 28 q^{87} + 20 q^{89} - 28 q^{93} + 6 q^{95} - 20 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 - 6 * q^11 - 2 * q^13 + 2 * q^15 + 4 * q^17 - 6 * q^19 + 2 * q^23 + 2 * q^25 - 20 * q^27 + 8 * q^29 - 2 * q^31 - 4 * q^33 - 8 * q^37 + 2 * q^39 - 4 * q^41 + 2 * q^43 - 6 * q^45 - 8 * q^47 - 14 * q^49 - 4 * q^51 - 4 * q^53 + 6 * q^55 - 4 * q^57 - 2 * q^59 + 2 * q^65 - 12 * q^67 - 12 * q^69 + 6 * q^71 - 24 * q^73 - 2 * q^75 - 4 * q^79 + 22 * q^81 - 8 * q^83 - 4 * q^85 - 28 * q^87 + 20 * q^89 - 28 * q^93 + 6 * q^95 - 20 * q^97 + 2 * q^99

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