LMFDB - Embedded newform 1040.2.a.i.1.1

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			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-3.23607 q^{3} -1.00000 q^{5} +7.47214 q^{9} +O(q^{10})$	$q-3.23607 q^{3} -1.00000 q^{5} +7.47214 q^{9} -0.763932 q^{11} -1.00000 q^{13} +3.23607 q^{15} +2.00000 q^{17} -0.763932 q^{19} +3.23607 q^{23} +1.00000 q^{25} -14.4721 q^{27} +8.47214 q^{29} +5.70820 q^{31} +2.47214 q^{33} -8.47214 q^{37} +3.23607 q^{39} -10.9443 q^{41} +3.23607 q^{43} -7.47214 q^{45} -12.9443 q^{47} -7.00000 q^{49} -6.47214 q^{51} -10.9443 q^{53} +0.763932 q^{55} +2.47214 q^{57} +5.70820 q^{59} -4.47214 q^{61} +1.00000 q^{65} -10.4721 q^{67} -10.4721 q^{69} +0.763932 q^{71} -7.52786 q^{73} -3.23607 q^{75} -6.47214 q^{79} +24.4164 q^{81} -4.00000 q^{83} -2.00000 q^{85} -27.4164 q^{87} +10.0000 q^{89} -18.4721 q^{93} +0.763932 q^{95} -10.0000 q^{97} -5.70820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.23607	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−3.23607	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	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
$9$	7.47214	2.49071
$10$	0	0
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	3.23607	0.835549
$16$	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
$19$	−0.763932	−0.175258	−0.0876290	−	0.996153i	$-0.527929\pi$
$19$	−0.763932	−0.175258	−0.0876290	+	0.996153i	$0.527929\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.23607	0.674767	0.337383	−	0.941367i	$-0.390458\pi$
$23$	3.23607	0.674767	0.337383	+	0.941367i	$0.390458\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−14.4721	−2.78516
$28$	0	0
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	5.70820	1.02522	0.512612	−	0.858620i	$-0.328678\pi$
$31$	5.70820	1.02522	0.512612	+	0.858620i	$0.328678\pi$
$32$	0	0
$33$	2.47214	0.430344
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.47214	−1.39281	−0.696405	−	0.717649i	$-0.745218\pi$
$37$	−8.47214	−1.39281	−0.696405	+	0.717649i	$0.745218\pi$
$38$	0	0
$39$	3.23607	0.518186
$40$	0	0
$41$	−10.9443	−1.70921	−0.854604	−	0.519280i	$-0.826200\pi$
$41$	−10.9443	−1.70921	−0.854604	+	0.519280i	$0.826200\pi$
$42$	0	0
$43$	3.23607	0.493496	0.246748	−	0.969080i	$-0.420638\pi$
$43$	3.23607	0.493496	0.246748	+	0.969080i	$0.420638\pi$
$44$	0	0
$45$	−7.47214	−1.11388
$46$	0	0
$47$	−12.9443	−1.88812	−0.944058	−	0.329779i	$-0.893026\pi$
$47$	−12.9443	−1.88812	−0.944058	+	0.329779i	$0.893026\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−6.47214	−0.906280
$52$	0	0
$53$	−10.9443	−1.50331	−0.751656	−	0.659556i	$-0.770744\pi$
$53$	−10.9443	−1.50331	−0.751656	+	0.659556i	$0.770744\pi$
$54$	0	0
$55$	0.763932	0.103009
$56$	0	0
$57$	2.47214	0.327442
$58$	0	0
$59$	5.70820	0.743145	0.371572	−	0.928404i	$-0.378819\pi$
$59$	5.70820	0.743145	0.371572	+	0.928404i	$0.378819\pi$
$60$	0	0
$61$	−4.47214	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−10.4721	−1.27938	−0.639688	−	0.768635i	$-0.720936\pi$
$67$	−10.4721	−1.27938	−0.639688	+	0.768635i	$0.720936\pi$
$68$	0	0
$69$	−10.4721	−1.26070
$70$	0	0
$71$	0.763932	0.0906621	0.0453310	−	0.998972i	$-0.485566\pi$
$71$	0.763932	0.0906621	0.0453310	+	0.998972i	$0.485566\pi$
$72$	0	0
$73$	−7.52786	−0.881070	−0.440535	−	0.897735i	$-0.645211\pi$
$73$	−7.52786	−0.881070	−0.440535	+	0.897735i	$0.645211\pi$
$74$	0	0
$75$	−3.23607	−0.373669
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.47214	−0.728172	−0.364086	−	0.931365i	$-0.618619\pi$
$79$	−6.47214	−0.728172	−0.364086	+	0.931365i	$0.618619\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	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
$87$	−27.4164	−2.93935
$88$	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
$93$	−18.4721	−1.91547
$94$	0	0
$95$	0.763932	0.0783778
$96$	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
$99$	−5.70820	−0.573696
$100$	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
$103$	6.29180	0.619949	0.309975	−	0.950745i	$-0.399679\pi$
$103$	6.29180	0.619949	0.309975	+	0.950745i	$0.399679\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.1803	1.56421	0.782106	−	0.623145i	$-0.214145\pi$
$107$	16.1803	1.56421	0.782106	+	0.623145i	$0.214145\pi$
$108$	0	0
$109$	−1.05573	−0.101120	−0.0505602	−	0.998721i	$-0.516101\pi$
$109$	−1.05573	−0.101120	−0.0505602	+	0.998721i	$0.516101\pi$
$110$	0	0
$111$	27.4164	2.60225
$112$	0	0
$113$	19.8885	1.87096	0.935478	−	0.353384i	$-0.114969\pi$
$113$	19.8885	1.87096	0.935478	+	0.353384i	$0.114969\pi$
$114$	0	0
$115$	−3.23607	−0.301765
$116$	0	0
$117$	−7.47214	−0.690799
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.4164	−0.946946
$122$	0	0
$123$	35.4164	3.19339
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.70820	0.861464	0.430732	−	0.902480i	$-0.358255\pi$
$127$	9.70820	0.861464	0.430732	+	0.902480i	$0.358255\pi$
$128$	0	0
$129$	−10.4721	−0.922020
$130$	0	0
$131$	−16.9443	−1.48043	−0.740214	−	0.672371i	$-0.765276\pi$
$131$	−16.9443	−1.48043	−0.740214	+	0.672371i	$0.765276\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	14.4721	1.24556
$136$	0	0
$137$	−19.8885	−1.69919	−0.849596	−	0.527433i	$-0.823154\pi$
$137$	−19.8885	−1.69919	−0.849596	+	0.527433i	$0.823154\pi$
$138$	0	0
$139$	−13.5279	−1.14742	−0.573709	−	0.819059i	$-0.694496\pi$
$139$	−13.5279	−1.14742	−0.573709	+	0.819059i	$0.694496\pi$
$140$	0	0
$141$	41.8885	3.52765
$142$	0	0
$143$	0.763932	0.0638832
$144$	0	0
$145$	−8.47214	−0.703573
$146$	0	0
$147$	22.6525	1.86834
$148$	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
$151$	−10.6525	−0.866886	−0.433443	−	0.901181i	$-0.642701\pi$
$151$	−10.6525	−0.866886	−0.433443	+	0.901181i	$0.642701\pi$
$152$	0	0
$153$	14.9443	1.20817
$154$	0	0
$155$	−5.70820	−0.458494
$156$	0	0
$157$	−10.9443	−0.873448	−0.436724	−	0.899596i	$-0.643861\pi$
$157$	−10.9443	−0.873448	−0.436724	+	0.899596i	$0.643861\pi$
$158$	0	0
$159$	35.4164	2.80870
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.4721	0.820241	0.410120	−	0.912031i	$-0.365487\pi$
$163$	10.4721	0.820241	0.410120	+	0.912031i	$0.365487\pi$
$164$	0	0
$165$	−2.47214	−0.192456
$166$	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
$171$	−5.70820	−0.436517
$172$	0	0
$173$	−10.9443	−0.832078	−0.416039	−	0.909347i	$-0.636582\pi$
$173$	−10.9443	−0.832078	−0.416039	+	0.909347i	$0.636582\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−18.4721	−1.38845
$178$	0	0
$179$	−16.9443	−1.26647	−0.633237	−	0.773958i	$-0.718274\pi$
$179$	−16.9443	−1.26647	−0.633237	+	0.773958i	$0.718274\pi$
$180$	0	0
$181$	0.472136	0.0350936	0.0175468	−	0.999846i	$-0.494414\pi$
$181$	0.472136	0.0350936	0.0175468	+	0.999846i	$0.494414\pi$
$182$	0	0
$183$	14.4721	1.06981
$184$	0	0
$185$	8.47214	0.622884
$186$	0	0
$187$	−1.52786	−0.111728
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.8885	1.29437	0.647185	−	0.762333i	$-0.275946\pi$
$191$	17.8885	1.29437	0.647185	+	0.762333i	$0.275946\pi$
$192$	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
$195$	−3.23607	−0.231740
$196$	0	0
$197$	19.8885	1.41700	0.708500	−	0.705711i	$-0.249372\pi$
$197$	19.8885	1.41700	0.708500	+	0.705711i	$0.249372\pi$
$198$	0	0
$199$	−20.9443	−1.48470	−0.742350	−	0.670012i	$-0.766289\pi$
$199$	−20.9443	−1.48470	−0.742350	+	0.670012i	$0.766289\pi$
$200$	0	0
$201$	33.8885	2.39031
$202$	0	0
$203$	0	0
$204$	0	0
$205$	10.9443	0.764381
$206$	0	0
$207$	24.1803	1.68065
$208$	0	0
$209$	0.583592	0.0403679
$210$	0	0
$211$	−8.94427	−0.615749	−0.307875	−	0.951427i	$-0.599618\pi$
$211$	−8.94427	−0.615749	−0.307875	+	0.951427i	$0.599618\pi$
$212$	0	0
$213$	−2.47214	−0.169388
$214$	0	0
$215$	−3.23607	−0.220698
$216$	0	0
$217$	0	0
$218$	0	0
$219$	24.3607	1.64614
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	9.88854	0.662186	0.331093	−	0.943598i	$-0.392583\pi$
$223$	9.88854	0.662186	0.331093	+	0.943598i	$0.392583\pi$
$224$	0	0
$225$	7.47214	0.498142
$226$	0	0
$227$	−18.4721	−1.22604	−0.613019	−	0.790068i	$-0.710045\pi$
$227$	−18.4721	−1.22604	−0.613019	+	0.790068i	$0.710045\pi$
$228$	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
$233$	−10.9443	−0.716983	−0.358492	−	0.933533i	$-0.616709\pi$
$233$	−10.9443	−0.716983	−0.358492	+	0.933533i	$0.616709\pi$
$234$	0	0
$235$	12.9443	0.844391
$236$	0	0
$237$	20.9443	1.36048
$238$	0	0
$239$	−2.65248	−0.171574	−0.0857872	−	0.996313i	$-0.527341\pi$
$239$	−2.65248	−0.171574	−0.0857872	+	0.996313i	$0.527341\pi$
$240$	0	0
$241$	−15.8885	−1.02347	−0.511736	−	0.859143i	$-0.670997\pi$
$241$	−15.8885	−1.02347	−0.511736	+	0.859143i	$0.670997\pi$
$242$	0	0
$243$	−35.5967	−2.28353
$244$	0	0
$245$	7.00000	0.447214
$246$	0	0
$247$	0.763932	0.0486078
$248$	0	0
$249$	12.9443	0.820310
$250$	0	0
$251$	−2.47214	−0.156040	−0.0780199	−	0.996952i	$-0.524860\pi$
$251$	−2.47214	−0.156040	−0.0780199	+	0.996952i	$0.524860\pi$
$252$	0	0
$253$	−2.47214	−0.155422
$254$	0	0
$255$	6.47214	0.405301
$256$	0	0
$257$	5.05573	0.315368	0.157684	−	0.987490i	$-0.449597\pi$
$257$	5.05573	0.315368	0.157684	+	0.987490i	$0.449597\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	63.3050	3.91848
$262$	0	0
$263$	1.70820	0.105332	0.0526662	−	0.998612i	$-0.483228\pi$
$263$	1.70820	0.105332	0.0526662	+	0.998612i	$0.483228\pi$
$264$	0	0
$265$	10.9443	0.672301
$266$	0	0
$267$	−32.3607	−1.98044
$268$	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
$271$	−18.6525	−1.13306	−0.566529	−	0.824042i	$-0.691714\pi$
$271$	−18.6525	−1.13306	−0.566529	+	0.824042i	$0.691714\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.763932	−0.0460668
$276$	0	0
$277$	6.94427	0.417241	0.208620	−	0.977997i	$-0.433103\pi$
$277$	6.94427	0.417241	0.208620	+	0.977997i	$0.433103\pi$
$278$	0	0
$279$	42.6525	2.55354
$280$	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
$283$	−30.6525	−1.82210	−0.911050	−	0.412295i	$-0.864727\pi$
$283$	−30.6525	−1.82210	−0.911050	+	0.412295i	$0.864727\pi$
$284$	0	0
$285$	−2.47214	−0.146437
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	32.3607	1.89702
$292$	0	0
$293$	22.3607	1.30632	0.653162	−	0.757218i	$-0.273442\pi$
$293$	22.3607	1.30632	0.653162	+	0.757218i	$0.273442\pi$
$294$	0	0
$295$	−5.70820	−0.332344
$296$	0	0
$297$	11.0557	0.641518
$298$	0	0
$299$	−3.23607	−0.187147
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−45.3050	−2.60270
$304$	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
$309$	−20.3607	−1.15828
$310$	0	0
$311$	−11.4164	−0.647365	−0.323683	−	0.946166i	$-0.604921\pi$
$311$	−11.4164	−0.647365	−0.323683	+	0.946166i	$0.604921\pi$
$312$	0	0
$313$	−20.8328	−1.17754	−0.588770	−	0.808300i	$-0.700388\pi$
$313$	−20.8328	−1.17754	−0.588770	+	0.808300i	$0.700388\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.3607	1.70523	0.852613	−	0.522543i	$-0.175017\pi$
$317$	30.3607	1.70523	0.852613	+	0.522543i	$0.175017\pi$
$318$	0	0
$319$	−6.47214	−0.362370
$320$	0	0
$321$	−52.3607	−2.92249
$322$	0	0
$323$	−1.52786	−0.0850126
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	3.41641	0.188928
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0.763932	0.0419895	0.0209948	−	0.999780i	$-0.493317\pi$
$331$	0.763932	0.0419895	0.0209948	+	0.999780i	$0.493317\pi$
$332$	0	0
$333$	−63.3050	−3.46909
$334$	0	0
$335$	10.4721	0.572154
$336$	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
$339$	−64.3607	−3.49559
$340$	0	0
$341$	−4.36068	−0.236144
$342$	0	0
$343$	0	0
$344$	0	0
$345$	10.4721	0.563801
$346$	0	0
$347$	−20.7639	−1.11467	−0.557333	−	0.830289i	$-0.688175\pi$
$347$	−20.7639	−1.11467	−0.557333	+	0.830289i	$0.688175\pi$
$348$	0	0
$349$	14.9443	0.799949	0.399974	−	0.916526i	$-0.369019\pi$
$349$	14.9443	0.799949	0.399974	+	0.916526i	$0.369019\pi$
$350$	0	0
$351$	14.4721	0.772465
$352$	0	0
$353$	−12.4721	−0.663825	−0.331912	−	0.943310i	$-0.607694\pi$
$353$	−12.4721	−0.663825	−0.331912	+	0.943310i	$0.607694\pi$
$354$	0	0
$355$	−0.763932	−0.0405453
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.23607	−0.381905	−0.190953	−	0.981599i	$-0.561158\pi$
$359$	−7.23607	−0.381905	−0.190953	+	0.981599i	$0.561158\pi$
$360$	0	0
$361$	−18.4164	−0.969285
$362$	0	0
$363$	33.7082	1.76922
$364$	0	0
$365$	7.52786	0.394026
$366$	0	0
$367$	24.1803	1.26220	0.631102	−	0.775700i	$-0.282603\pi$
$367$	24.1803	1.26220	0.631102	+	0.775700i	$0.282603\pi$
$368$	0	0
$369$	−81.7771	−4.25715
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.94427	−0.152449	−0.0762243	−	0.997091i	$-0.524287\pi$
$373$	−2.94427	−0.152449	−0.0762243	+	0.997091i	$0.524287\pi$
$374$	0	0
$375$	3.23607	0.167110
$376$	0	0
$377$	−8.47214	−0.436337
$378$	0	0
$379$	−16.7639	−0.861105	−0.430553	−	0.902565i	$-0.641681\pi$
$379$	−16.7639	−0.861105	−0.430553	+	0.902565i	$0.641681\pi$
$380$	0	0
$381$	−31.4164	−1.60951
$382$	0	0
$383$	17.8885	0.914062	0.457031	−	0.889451i	$-0.348913\pi$
$383$	17.8885	0.914062	0.457031	+	0.889451i	$0.348913\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	24.1803	1.22916
$388$	0	0
$389$	31.8885	1.61681	0.808407	−	0.588624i	$-0.200330\pi$
$389$	31.8885	1.61681	0.808407	+	0.588624i	$0.200330\pi$
$390$	0	0
$391$	6.47214	0.327310
$392$	0	0
$393$	54.8328	2.76595
$394$	0	0
$395$	6.47214	0.325649
$396$	0	0
$397$	−6.58359	−0.330421	−0.165211	−	0.986258i	$-0.552830\pi$
$397$	−6.58359	−0.330421	−0.165211	+	0.986258i	$0.552830\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.88854	−0.393935	−0.196968	−	0.980410i	$-0.563109\pi$
$401$	−7.88854	−0.393935	−0.196968	+	0.980410i	$0.563109\pi$
$402$	0	0
$403$	−5.70820	−0.284346
$404$	0	0
$405$	−24.4164	−1.21326
$406$	0	0
$407$	6.47214	0.320812
$408$	0	0
$409$	32.8328	1.62348	0.811739	−	0.584020i	$-0.198521\pi$
$409$	32.8328	1.62348	0.811739	+	0.584020i	$0.198521\pi$
$410$	0	0
$411$	64.3607	3.17468
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	43.7771	2.14377
$418$	0	0
$419$	0.583592	0.0285103	0.0142552	−	0.999898i	$-0.495462\pi$
$419$	0.583592	0.0285103	0.0142552	+	0.999898i	$0.495462\pi$
$420$	0	0
$421$	3.88854	0.189516	0.0947580	−	0.995500i	$-0.469792\pi$
$421$	3.88854	0.189516	0.0947580	+	0.995500i	$0.469792\pi$
$422$	0	0
$423$	−96.7214	−4.70275
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.47214	−0.119356
$430$	0	0
$431$	−38.0689	−1.83371	−0.916857	−	0.399216i	$-0.869282\pi$
$431$	−38.0689	−1.83371	−0.916857	+	0.399216i	$0.869282\pi$
$432$	0	0
$433$	24.8328	1.19339	0.596694	−	0.802469i	$-0.296480\pi$
$433$	24.8328	1.19339	0.596694	+	0.802469i	$0.296480\pi$
$434$	0	0
$435$	27.4164	1.31452
$436$	0	0
$437$	−2.47214	−0.118258
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−52.3050	−2.49071
$442$	0	0
$443$	22.6525	1.07625	0.538126	−	0.842865i	$-0.319133\pi$
$443$	22.6525	1.07625	0.538126	+	0.842865i	$0.319133\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	19.4164	0.918365
$448$	0	0
$449$	−9.05573	−0.427366	−0.213683	−	0.976903i	$-0.568546\pi$
$449$	−9.05573	−0.427366	−0.213683	+	0.976903i	$0.568546\pi$
$450$	0	0
$451$	8.36068	0.393689
$452$	0	0
$453$	34.4721	1.61964
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.88854	0.369011	0.184505	−	0.982832i	$-0.440932\pi$
$457$	7.88854	0.369011	0.184505	+	0.982832i	$0.440932\pi$
$458$	0	0
$459$	−28.9443	−1.35100
$460$	0	0
$461$	−4.11146	−0.191490	−0.0957448	−	0.995406i	$-0.530523\pi$
$461$	−4.11146	−0.191490	−0.0957448	+	0.995406i	$0.530523\pi$
$462$	0	0
$463$	19.4164	0.902357	0.451178	−	0.892434i	$-0.351004\pi$
$463$	19.4164	0.902357	0.451178	+	0.892434i	$0.351004\pi$
$464$	0	0
$465$	18.4721	0.856625
$466$	0	0
$467$	1.70820	0.0790463	0.0395231	−	0.999219i	$-0.487416\pi$
$467$	1.70820	0.0790463	0.0395231	+	0.999219i	$0.487416\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	35.4164	1.63190
$472$	0	0
$473$	−2.47214	−0.113669
$474$	0	0
$475$	−0.763932	−0.0350516
$476$	0	0
$477$	−81.7771	−3.74432
$478$	0	0
$479$	−9.12461	−0.416914	−0.208457	−	0.978032i	$-0.566844\pi$
$479$	−9.12461	−0.416914	−0.208457	+	0.978032i	$0.566844\pi$
$480$	0	0
$481$	8.47214	0.386296
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.0000	0.454077
$486$	0	0
$487$	−29.3050	−1.32793	−0.663967	−	0.747762i	$-0.731128\pi$
$487$	−29.3050	−1.32793	−0.663967	+	0.747762i	$0.731128\pi$
$488$	0	0
$489$	−33.8885	−1.53249
$490$	0	0
$491$	34.4721	1.55571	0.777853	−	0.628446i	$-0.216309\pi$
$491$	34.4721	1.55571	0.777853	+	0.628446i	$0.216309\pi$
$492$	0	0
$493$	16.9443	0.763132
$494$	0	0
$495$	5.70820	0.256565
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−37.7082	−1.68805	−0.844026	−	0.536303i	$-0.819820\pi$
$499$	−37.7082	−1.68805	−0.844026	+	0.536303i	$0.819820\pi$
$500$	0	0
$501$	25.8885	1.15661
$502$	0	0
$503$	−14.2918	−0.637240	−0.318620	−	0.947883i	$-0.603219\pi$
$503$	−14.2918	−0.637240	−0.318620	+	0.947883i	$0.603219\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	−3.23607	−0.143719
$508$	0	0
$509$	−9.05573	−0.401388	−0.200694	−	0.979654i	$-0.564320\pi$
$509$	−9.05573	−0.401388	−0.200694	+	0.979654i	$0.564320\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	11.0557	0.488122
$514$	0	0
$515$	−6.29180	−0.277250
$516$	0	0
$517$	9.88854	0.434898
$518$	0	0
$519$	35.4164	1.55461
$520$	0	0
$521$	−6.58359	−0.288432	−0.144216	−	0.989546i	$-0.546066\pi$
$521$	−6.58359	−0.288432	−0.144216	+	0.989546i	$0.546066\pi$
$522$	0	0
$523$	19.5967	0.856906	0.428453	−	0.903564i	$-0.359059\pi$
$523$	19.5967	0.856906	0.428453	+	0.903564i	$0.359059\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.4164	0.497307
$528$	0	0
$529$	−12.5279	−0.544690
$530$	0	0
$531$	42.6525	1.85096
$532$	0	0
$533$	10.9443	0.474049
$534$	0	0
$535$	−16.1803	−0.699537
$536$	0	0
$537$	54.8328	2.36621
$538$	0	0
$539$	5.34752	0.230334
$540$	0	0
$541$	−17.0557	−0.733283	−0.366642	−	0.930362i	$-0.619492\pi$
$541$	−17.0557	−0.733283	−0.366642	+	0.930362i	$0.619492\pi$
$542$	0	0
$543$	−1.52786	−0.0655669
$544$	0	0
$545$	1.05573	0.0452224
$546$	0	0
$547$	−35.5967	−1.52201	−0.761004	−	0.648748i	$-0.775293\pi$
$547$	−35.5967	−1.52201	−0.761004	+	0.648748i	$0.775293\pi$
$548$	0	0
$549$	−33.4164	−1.42618
$550$	0	0
$551$	−6.47214	−0.275722
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−27.4164	−1.16376
$556$	0	0
$557$	25.4164	1.07693	0.538464	−	0.842649i	$-0.319005\pi$
$557$	25.4164	1.07693	0.538464	+	0.842649i	$0.319005\pi$
$558$	0	0
$559$	−3.23607	−0.136871
$560$	0	0
$561$	4.94427	0.208747
$562$	0	0
$563$	14.2918	0.602327	0.301164	−	0.953572i	$-0.402625\pi$
$563$	14.2918	0.602327	0.301164	+	0.953572i	$0.402625\pi$
$564$	0	0
$565$	−19.8885	−0.836717
$566$	0	0
$567$	0	0
$568$	0	0
$569$	41.4164	1.73627	0.868133	−	0.496332i	$-0.165320\pi$
$569$	41.4164	1.73627	0.868133	+	0.496332i	$0.165320\pi$
$570$	0	0
$571$	13.5279	0.566123	0.283062	−	0.959102i	$-0.408650\pi$
$571$	13.5279	0.566123	0.283062	+	0.959102i	$0.408650\pi$
$572$	0	0
$573$	−57.8885	−2.41833
$574$	0	0
$575$	3.23607	0.134953
$576$	0	0
$577$	−41.4164	−1.72419	−0.862094	−	0.506749i	$-0.830847\pi$
$577$	−41.4164	−1.72419	−0.862094	+	0.506749i	$0.830847\pi$
$578$	0	0
$579$	−19.4164	−0.806918
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8.36068	0.346264
$584$	0	0
$585$	7.47214	0.308935
$586$	0	0
$587$	13.5279	0.558355	0.279177	−	0.960240i	$-0.409938\pi$
$587$	13.5279	0.558355	0.279177	+	0.960240i	$0.409938\pi$
$588$	0	0
$589$	−4.36068	−0.179679
$590$	0	0
$591$	−64.3607	−2.64744
$592$	0	0
$593$	−0.111456	−0.00457696	−0.00228848	−	0.999997i	$-0.500728\pi$
$593$	−0.111456	−0.00457696	−0.00228848	+	0.999997i	$0.500728\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	67.7771	2.77393
$598$	0	0
$599$	6.47214	0.264444	0.132222	−	0.991220i	$-0.457789\pi$
$599$	6.47214	0.264444	0.132222	+	0.991220i	$0.457789\pi$
$600$	0	0
$601$	−15.8885	−0.648107	−0.324054	−	0.946039i	$-0.605046\pi$
$601$	−15.8885	−0.648107	−0.324054	+	0.946039i	$0.605046\pi$
$602$	0	0
$603$	−78.2492	−3.18655
$604$	0	0
$605$	10.4164	0.423487
$606$	0	0
$607$	34.0689	1.38281	0.691407	−	0.722466i	$-0.256991\pi$
$607$	34.0689	1.38281	0.691407	+	0.722466i	$0.256991\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.9443	0.523669
$612$	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
$615$	−35.4164	−1.42813
$616$	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
$619$	2.65248	0.106612	0.0533060	−	0.998578i	$-0.483024\pi$
$619$	2.65248	0.106612	0.0533060	+	0.998578i	$0.483024\pi$
$620$	0	0
$621$	−46.8328	−1.87934
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.88854	−0.0754212
$628$	0	0
$629$	−16.9443	−0.675612
$630$	0	0
$631$	43.0132	1.71233	0.856163	−	0.516705i	$-0.172842\pi$
$631$	43.0132	1.71233	0.856163	+	0.516705i	$0.172842\pi$
$632$	0	0
$633$	28.9443	1.15043
$634$	0	0
$635$	−9.70820	−0.385258
$636$	0	0
$637$	7.00000	0.277350
$638$	0	0
$639$	5.70820	0.225813
$640$	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
$643$	−15.0557	−0.593740	−0.296870	−	0.954918i	$-0.595943\pi$
$643$	−15.0557	−0.593740	−0.296870	+	0.954918i	$0.595943\pi$
$644$	0	0
$645$	10.4721	0.412340
$646$	0	0
$647$	−19.2361	−0.756248	−0.378124	−	0.925755i	$-0.623431\pi$
$647$	−19.2361	−0.756248	−0.378124	+	0.925755i	$0.623431\pi$
$648$	0	0
$649$	−4.36068	−0.171172
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.8328	−0.502187	−0.251093	−	0.967963i	$-0.580790\pi$
$653$	−12.8328	−0.502187	−0.251093	+	0.967963i	$0.580790\pi$
$654$	0	0
$655$	16.9443	0.662067
$656$	0	0
$657$	−56.2492	−2.19449
$658$	0	0
$659$	33.3050	1.29738	0.648688	−	0.761054i	$-0.275318\pi$
$659$	33.3050	1.29738	0.648688	+	0.761054i	$0.275318\pi$
$660$	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
$663$	6.47214	0.251357
$664$	0	0
$665$	0	0
$666$	0	0
$667$	27.4164	1.06157
$668$	0	0
$669$	−32.0000	−1.23719
$670$	0	0
$671$	3.41641	0.131889
$672$	0	0
$673$	43.8885	1.69178	0.845890	−	0.533358i	$-0.179070\pi$
$673$	43.8885	1.69178	0.845890	+	0.533358i	$0.179070\pi$
$674$	0	0
$675$	−14.4721	−0.557033
$676$	0	0
$677$	−12.8328	−0.493205	−0.246603	−	0.969117i	$-0.579314\pi$
$677$	−12.8328	−0.493205	−0.246603	+	0.969117i	$0.579314\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	59.7771	2.29066
$682$	0	0
$683$	44.3607	1.69741	0.848707	−	0.528863i	$-0.177382\pi$
$683$	44.3607	1.69741	0.848707	+	0.528863i	$0.177382\pi$
$684$	0	0
$685$	19.8885	0.759902
$686$	0	0
$687$	−32.3607	−1.23464
$688$	0	0
$689$	10.9443	0.416944
$690$	0	0
$691$	−43.0132	−1.63630	−0.818149	−	0.575007i	$-0.804999\pi$
$691$	−43.0132	−1.63630	−0.818149	+	0.575007i	$0.804999\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.5279	0.513141
$696$	0	0
$697$	−21.8885	−0.829088
$698$	0	0
$699$	35.4164	1.33957
$700$	0	0
$701$	−51.8885	−1.95980	−0.979902	−	0.199481i	$-0.936074\pi$
$701$	−51.8885	−1.95980	−0.979902	+	0.199481i	$0.936074\pi$
$702$	0	0
$703$	6.47214	0.244101
$704$	0	0
$705$	−41.8885	−1.57761
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−15.8885	−0.596707	−0.298353	−	0.954455i	$-0.596437\pi$
$709$	−15.8885	−0.596707	−0.298353	+	0.954455i	$0.596437\pi$
$710$	0	0
$711$	−48.3607	−1.81367
$712$	0	0
$713$	18.4721	0.691787
$714$	0	0
$715$	−0.763932	−0.0285694
$716$	0	0
$717$	8.58359	0.320560
$718$	0	0
$719$	46.8328	1.74657	0.873285	−	0.487210i	$-0.161985\pi$
$719$	46.8328	1.74657	0.873285	+	0.487210i	$0.161985\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	51.4164	1.91220
$724$	0	0
$725$	8.47214	0.314647
$726$	0	0
$727$	25.7082	0.953465	0.476732	−	0.879049i	$-0.341821\pi$
$727$	25.7082	0.953465	0.476732	+	0.879049i	$0.341821\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	6.47214	0.239381
$732$	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
$735$	−22.6525	−0.835549
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	−35.0132	−1.28798	−0.643990	−	0.765034i	$-0.722722\pi$
$739$	−35.0132	−1.28798	−0.643990	+	0.765034i	$0.722722\pi$
$740$	0	0
$741$	−2.47214	−0.0908162
$742$	0	0
$743$	−6.11146	−0.224208	−0.112104	−	0.993697i	$-0.535759\pi$
$743$	−6.11146	−0.224208	−0.112104	+	0.993697i	$0.535759\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	−29.8885	−1.09356
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−46.4721	−1.69579	−0.847896	−	0.530162i	$-0.822131\pi$
$751$	−46.4721	−1.69579	−0.847896	+	0.530162i	$0.822131\pi$
$752$	0	0
$753$	8.00000	0.291536
$754$	0	0
$755$	10.6525	0.387683
$756$	0	0
$757$	−28.8328	−1.04795	−0.523973	−	0.851735i	$-0.675551\pi$
$757$	−28.8328	−1.04795	−0.523973	+	0.851735i	$0.675551\pi$
$758$	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
$765$	−14.9443	−0.540311
$766$	0	0
$767$	−5.70820	−0.206111
$768$	0	0
$769$	14.9443	0.538904	0.269452	−	0.963014i	$-0.413157\pi$
$769$	14.9443	0.538904	0.269452	+	0.963014i	$0.413157\pi$
$770$	0	0
$771$	−16.3607	−0.589215
$772$	0	0
$773$	−28.2492	−1.01605	−0.508027	−	0.861341i	$-0.669625\pi$
$773$	−28.2492	−1.01605	−0.508027	+	0.861341i	$0.669625\pi$
$774$	0	0
$775$	5.70820	0.205045
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.36068	0.299552
$780$	0	0
$781$	−0.583592	−0.0208826
$782$	0	0
$783$	−122.610	−4.38172
$784$	0	0
$785$	10.9443	0.390618
$786$	0	0
$787$	21.8885	0.780242	0.390121	−	0.920764i	$-0.372433\pi$
$787$	21.8885	0.780242	0.390121	+	0.920764i	$0.372433\pi$
$788$	0	0
$789$	−5.52786	−0.196797
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.47214	0.158810
$794$	0	0
$795$	−35.4164	−1.25609
$796$	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
$799$	−25.8885	−0.915871
$800$	0	0
$801$	74.7214	2.64015
$802$	0	0
$803$	5.75078	0.202940
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19.4164	−0.683490
$808$	0	0
$809$	4.47214	0.157232	0.0786160	−	0.996905i	$-0.474950\pi$
$809$	4.47214	0.157232	0.0786160	+	0.996905i	$0.474950\pi$
$810$	0	0
$811$	−39.2361	−1.37776	−0.688882	−	0.724873i	$-0.741898\pi$
$811$	−39.2361	−1.37776	−0.688882	+	0.724873i	$0.741898\pi$
$812$	0	0
$813$	60.3607	2.11694
$814$	0	0
$815$	−10.4721	−0.366823
$816$	0	0
$817$	−2.47214	−0.0864891
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.05573	0.176446	0.0882231	−	0.996101i	$-0.471881\pi$
$821$	5.05573	0.176446	0.0882231	+	0.996101i	$0.471881\pi$
$822$	0	0
$823$	21.1246	0.736358	0.368179	−	0.929755i	$-0.379981\pi$
$823$	21.1246	0.736358	0.368179	+	0.929755i	$0.379981\pi$
$824$	0	0
$825$	2.47214	0.0860687
$826$	0	0
$827$	−26.8328	−0.933068	−0.466534	−	0.884503i	$-0.654498\pi$
$827$	−26.8328	−0.933068	−0.466534	+	0.884503i	$0.654498\pi$
$828$	0	0
$829$	37.4164	1.29953	0.649763	−	0.760137i	$-0.274868\pi$
$829$	37.4164	1.29953	0.649763	+	0.760137i	$0.274868\pi$
$830$	0	0
$831$	−22.4721	−0.779550
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	−82.6099	−2.85542
$838$	0	0
$839$	50.6525	1.74872	0.874359	−	0.485280i	$-0.161282\pi$
$839$	50.6525	1.74872	0.874359	+	0.485280i	$0.161282\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	−32.3607	−1.11456
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	0	0
$848$	0	0
$849$	99.1935	3.40431
$850$	0	0
$851$	−27.4164	−0.939822
$852$	0	0
$853$	12.4721	0.427038	0.213519	−	0.976939i	$-0.431508\pi$
$853$	12.4721	0.427038	0.213519	+	0.976939i	$0.431508\pi$
$854$	0	0
$855$	5.70820	0.195216
$856$	0	0
$857$	21.0557	0.719250	0.359625	−	0.933097i	$-0.382905\pi$
$857$	21.0557	0.719250	0.359625	+	0.933097i	$0.382905\pi$
$858$	0	0
$859$	−11.6393	−0.397128	−0.198564	−	0.980088i	$-0.563628\pi$
$859$	−11.6393	−0.397128	−0.198564	+	0.980088i	$0.563628\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.0557	−0.648665	−0.324332	−	0.945943i	$-0.605140\pi$
$863$	−19.0557	−0.648665	−0.324332	+	0.945943i	$0.605140\pi$
$864$	0	0
$865$	10.9443	0.372116
$866$	0	0
$867$	42.0689	1.42873
$868$	0	0
$869$	4.94427	0.167723
$870$	0	0
$871$	10.4721	0.354835
$872$	0	0
$873$	−74.7214	−2.52893
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−17.7771	−0.600290	−0.300145	−	0.953894i	$-0.597035\pi$
$877$	−17.7771	−0.600290	−0.300145	+	0.953894i	$0.597035\pi$
$878$	0	0
$879$	−72.3607	−2.44067
$880$	0	0
$881$	−9.63932	−0.324757	−0.162378	−	0.986729i	$-0.551917\pi$
$881$	−9.63932	−0.324757	−0.162378	+	0.986729i	$0.551917\pi$
$882$	0	0
$883$	14.2918	0.480957	0.240479	−	0.970654i	$-0.422696\pi$
$883$	14.2918	0.480957	0.240479	+	0.970654i	$0.422696\pi$
$884$	0	0
$885$	18.4721	0.620934
$886$	0	0
$887$	−4.76393	−0.159957	−0.0799786	−	0.996797i	$-0.525485\pi$
$887$	−4.76393	−0.159957	−0.0799786	+	0.996797i	$0.525485\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−18.6525	−0.624881
$892$	0	0
$893$	9.88854	0.330908
$894$	0	0
$895$	16.9443	0.566385
$896$	0	0
$897$	10.4721	0.349654
$898$	0	0
$899$	48.3607	1.61292
$900$	0	0
$901$	−21.8885	−0.729213
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.472136	−0.0156943
$906$	0	0
$907$	41.7082	1.38490	0.692449	−	0.721467i	$-0.256532\pi$
$907$	41.7082	1.38490	0.692449	+	0.721467i	$0.256532\pi$
$908$	0	0
$909$	104.610	3.46969
$910$	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
$913$	3.05573	0.101130
$914$	0	0
$915$	−14.4721	−0.478434
$916$	0	0
$917$	0	0
$918$	0	0
$919$	38.4721	1.26908	0.634539	−	0.772891i	$-0.281190\pi$
$919$	38.4721	1.26908	0.634539	+	0.772891i	$0.281190\pi$
$920$	0	0
$921$	−38.8328	−1.27958
$922$	0	0
$923$	−0.763932	−0.0251451
$924$	0	0
$925$	−8.47214	−0.278562
$926$	0	0
$927$	47.0132	1.54411
$928$	0	0
$929$	−18.9443	−0.621541	−0.310771	−	0.950485i	$-0.600587\pi$
$929$	−18.9443	−0.621541	−0.310771	+	0.950485i	$0.600587\pi$
$930$	0	0
$931$	5.34752	0.175258
$932$	0	0
$933$	36.9443	1.20950
$934$	0	0
$935$	1.52786	0.0499665
$936$	0	0
$937$	−44.8328	−1.46462	−0.732312	−	0.680969i	$-0.761559\pi$
$937$	−44.8328	−1.46462	−0.732312	+	0.680969i	$0.761559\pi$
$938$	0	0
$939$	67.4164	2.20005
$940$	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
$943$	−35.4164	−1.15332
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.0557	1.26914	0.634570	−	0.772865i	$-0.281177\pi$
$947$	39.0557	1.26914	0.634570	+	0.772865i	$0.281177\pi$
$948$	0	0
$949$	7.52786	0.244365
$950$	0	0
$951$	−98.2492	−3.18595
$952$	0	0
$953$	−7.16718	−0.232168	−0.116084	−	0.993239i	$-0.537034\pi$
$953$	−7.16718	−0.232168	−0.116084	+	0.993239i	$0.537034\pi$
$954$	0	0
$955$	−17.8885	−0.578860
$956$	0	0
$957$	20.9443	0.677032
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.58359	0.0510836
$962$	0	0
$963$	120.902	3.89600
$964$	0	0
$965$	−6.00000	−0.193147
$966$	0	0
$967$	−60.1378	−1.93390	−0.966950	−	0.254966i	$-0.917936\pi$
$967$	−60.1378	−1.93390	−0.966950	+	0.254966i	$0.917936\pi$
$968$	0	0
$969$	4.94427	0.158833
$970$	0	0
$971$	−26.8328	−0.861106	−0.430553	−	0.902565i	$-0.641681\pi$
$971$	−26.8328	−0.861106	−0.430553	+	0.902565i	$0.641681\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	3.23607	0.103637
$976$	0	0
$977$	−15.5279	−0.496780	−0.248390	−	0.968660i	$-0.579902\pi$
$977$	−15.5279	−0.496780	−0.248390	+	0.968660i	$0.579902\pi$
$978$	0	0
$979$	−7.63932	−0.244154
$980$	0	0
$981$	−7.88854	−0.251862
$982$	0	0
$983$	−7.63932	−0.243656	−0.121828	−	0.992551i	$-0.538876\pi$
$983$	−7.63932	−0.243656	−0.121828	+	0.992551i	$0.538876\pi$
$984$	0	0
$985$	−19.8885	−0.633702
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.4721	0.332995
$990$	0	0
$991$	3.05573	0.0970684	0.0485342	−	0.998822i	$-0.484545\pi$
$991$	3.05573	0.0970684	0.0485342	+	0.998822i	$0.484545\pi$
$992$	0	0
$993$	−2.47214	−0.0784509
$994$	0	0
$995$	20.9443	0.663978
$996$	0	0
$997$	0.832816	0.0263755	0.0131878	−	0.999913i	$-0.495802\pi$
$997$	0.832816	0.0263755	0.0131878	+	0.999913i	$0.495802\pi$
$998$	0	0
$999$	122.610	3.87921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.a.i.1.1		2
3.2	odd	2		9360.2.a.cs.1.1		2
4.3	odd	2		520.2.a.g.1.2	✓	2
5.4	even	2		5200.2.a.bz.1.2		2
8.3	odd	2		4160.2.a.x.1.1		2
8.5	even	2		4160.2.a.bm.1.2		2
12.11	even	2		4680.2.a.bc.1.2		2
20.3	even	4		2600.2.d.j.1249.4		4
20.7	even	4		2600.2.d.j.1249.1		4
20.19	odd	2		2600.2.a.o.1.1		2
52.51	odd	2		6760.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.g.1.2	✓	2	4.3	odd	2
1040.2.a.i.1.1		2	1.1	even	1	trivial
2600.2.a.o.1.1		2	20.19	odd	2
2600.2.d.j.1249.1		4	20.7	even	4
2600.2.d.j.1249.4		4	20.3	even	4
4160.2.a.x.1.1		2	8.3	odd	2
4160.2.a.bm.1.2		2	8.5	even	2
4680.2.a.bc.1.2		2	12.11	even	2
5200.2.a.bz.1.2		2	5.4	even	2
6760.2.a.t.1.2		2	52.51	odd	2
9360.2.a.cs.1.1		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)

\(f(q)\)	\(=\)	\(q-3.23607 q^{3} -1.00000 q^{5} +7.47214 q^{9} +O(q^{10})\)	\(q-3.23607 q^{3} -1.00000 q^{5} +7.47214 q^{9} -0.763932 q^{11} -1.00000 q^{13} +3.23607 q^{15} +2.00000 q^{17} -0.763932 q^{19} +3.23607 q^{23} +1.00000 q^{25} -14.4721 q^{27} +8.47214 q^{29} +5.70820 q^{31} +2.47214 q^{33} -8.47214 q^{37} +3.23607 q^{39} -10.9443 q^{41} +3.23607 q^{43} -7.47214 q^{45} -12.9443 q^{47} -7.00000 q^{49} -6.47214 q^{51} -10.9443 q^{53} +0.763932 q^{55} +2.47214 q^{57} +5.70820 q^{59} -4.47214 q^{61} +1.00000 q^{65} -10.4721 q^{67} -10.4721 q^{69} +0.763932 q^{71} -7.52786 q^{73} -3.23607 q^{75} -6.47214 q^{79} +24.4164 q^{81} -4.00000 q^{83} -2.00000 q^{85} -27.4164 q^{87} +10.0000 q^{89} -18.4721 q^{93} +0.763932 q^{95} -10.0000 q^{97} -5.70820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) 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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