LMFDB - Embedded newform 668.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [668,2,Mod(1,668)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(668, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("668.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 668 = 2^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	668.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:	$5.33400685502$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.826865.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 6x - 1 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 6x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.75474$ of defining polynomial
Character	$\chi$	$=$	668.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83385 q^{3} +2.00000 q^{5} +1.53681 q^{7} +0.363011 q^{9} +O(q^{10})$	$q-1.83385 q^{3} +2.00000 q^{5} +1.53681 q^{7} +0.363011 q^{9} +6.05178 q^{11} -5.50948 q^{13} -3.66770 q^{15} +1.11576 q^{17} -2.87249 q^{19} -2.81828 q^{21} +6.59409 q^{23} -1.00000 q^{25} +4.83585 q^{27} +3.97267 q^{29} -1.23351 q^{31} -11.0981 q^{33} +3.07362 q^{35} +11.1772 q^{37} +10.1036 q^{39} +2.55195 q^{41} +11.0190 q^{43} +0.726021 q^{45} +8.14647 q^{47} -4.63822 q^{49} -2.04613 q^{51} -6.62523 q^{53} +12.1036 q^{55} +5.26772 q^{57} -12.3391 q^{59} -0.918459 q^{61} +0.557878 q^{63} -11.0190 q^{65} +12.9768 q^{67} -12.0926 q^{69} -11.3775 q^{71} +5.47833 q^{73} +1.83385 q^{75} +9.30043 q^{77} +10.1457 q^{79} -9.95726 q^{81} +14.5284 q^{83} +2.23151 q^{85} -7.28529 q^{87} +4.26971 q^{89} -8.46701 q^{91} +2.26207 q^{93} -5.74498 q^{95} -16.8392 q^{97} +2.19686 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} + 10 q^{5} + 9 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 + 10 * q^5 + 9 * q^7 + 6 * q^9	$ 5 q + 3 q^{3} + 10 q^{5} + 9 q^{7} + 6 q^{9} + 5 q^{11} - 4 q^{13} + 6 q^{15} - 2 q^{17} + 5 q^{19} - 4 q^{21} + 6 q^{23} - 5 q^{25} + 12 q^{27} - 5 q^{29} + 9 q^{31} - 8 q^{33} + 18 q^{35} + 8 q^{37} - 4 q^{41} + 8 q^{43} + 12 q^{45} + 13 q^{47} + 14 q^{49} - 2 q^{53} + 10 q^{55} - 12 q^{57} + 4 q^{59} + 11 q^{61} - q^{63} - 8 q^{65} + 28 q^{67} - 16 q^{69} + 2 q^{71} + 8 q^{73} - 3 q^{75} - 12 q^{77} - 10 q^{79} - 15 q^{81} + 2 q^{83} - 4 q^{85} + 4 q^{87} - 17 q^{89} - 12 q^{91} - 12 q^{93} + 10 q^{95} - 27 q^{97} + 3 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 + 10 * q^5 + 9 * q^7 + 6 * q^9 + 5 * q^11 - 4 * q^13 + 6 * q^15 - 2 * q^17 + 5 * q^19 - 4 * q^21 + 6 * q^23 - 5 * q^25 + 12 * q^27 - 5 * q^29 + 9 * q^31 - 8 * q^33 + 18 * q^35 + 8 * q^37 - 4 * q^41 + 8 * q^43 + 12 * q^45 + 13 * q^47 + 14 * q^49 - 2 * q^53 + 10 * q^55 - 12 * q^57 + 4 * q^59 + 11 * q^61 - q^63 - 8 * q^65 + 28 * q^67 - 16 * q^69 + 2 * q^71 + 8 * q^73 - 3 * q^75 - 12 * q^77 - 10 * q^79 - 15 * q^81 + 2 * q^83 - 4 * q^85 + 4 * q^87 - 17 * q^89 - 12 * q^91 - 12 * q^93 + 10 * q^95 - 27 * q^97 + 3 * 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
$3$	−1.83385	−1.05877	−0.529387	−	0.848380i	$-0.677578\pi$
$3$	−1.83385	−1.05877	−0.529387	+	0.848380i	$0.677578\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	1.53681	0.580859	0.290429	−	0.956896i	$-0.406202\pi$
$7$	1.53681	0.580859	0.290429	+	0.956896i	$0.406202\pi$
$8$	0	0
$9$	0.363011	0.121004
$10$	0	0
$11$	6.05178	1.82468	0.912341	−	0.409432i	$-0.134273\pi$
$11$	6.05178	1.82468	0.912341	+	0.409432i	$0.134273\pi$
$12$	0	0
$13$	−5.50948	−1.52805	−0.764027	−	0.645184i	$-0.776781\pi$
$13$	−5.50948	−1.52805	−0.764027	+	0.645184i	$0.776781\pi$
$14$	0	0
$15$	−3.66770	−0.946997
$16$	0	0
$17$	1.11576	0.270610	0.135305	−	0.990804i	$-0.456799\pi$
$17$	1.11576	0.270610	0.135305	+	0.990804i	$0.456799\pi$
$18$	0	0
$19$	−2.87249	−0.658994	−0.329497	−	0.944157i	$-0.606879\pi$
$19$	−2.87249	−0.658994	−0.329497	+	0.944157i	$0.606879\pi$
$20$	0	0
$21$	−2.81828	−0.614999
$22$	0	0
$23$	6.59409	1.37496	0.687481	−	0.726202i	$-0.258716\pi$
$23$	6.59409	1.37496	0.687481	+	0.726202i	$0.258716\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	4.83585	0.930659
$28$	0	0
$29$	3.97267	0.737707	0.368853	−	0.929488i	$-0.379750\pi$
$29$	3.97267	0.737707	0.368853	+	0.929488i	$0.379750\pi$
$30$	0	0
$31$	−1.23351	−0.221544	−0.110772	−	0.993846i	$-0.535332\pi$
$31$	−1.23351	−0.221544	−0.110772	+	0.993846i	$0.535332\pi$
$32$	0	0
$33$	−11.0981	−1.93193
$34$	0	0
$35$	3.07362	0.519536
$36$	0	0
$37$	11.1772	1.83752	0.918759	−	0.394819i	$-0.129193\pi$
$37$	11.1772	1.83752	0.918759	+	0.394819i	$0.129193\pi$
$38$	0	0
$39$	10.1036	1.61787
$40$	0	0
$41$	2.55195	0.398547	0.199274	−	0.979944i	$-0.436142\pi$
$41$	2.55195	0.398547	0.199274	+	0.979944i	$0.436142\pi$
$42$	0	0
$43$	11.0190	1.68038	0.840188	−	0.542296i	$-0.182445\pi$
$43$	11.0190	1.68038	0.840188	+	0.542296i	$0.182445\pi$
$44$	0	0
$45$	0.726021	0.108229
$46$	0	0
$47$	8.14647	1.18828	0.594142	−	0.804360i	$-0.297492\pi$
$47$	8.14647	1.18828	0.594142	+	0.804360i	$0.297492\pi$
$48$	0	0
$49$	−4.63822	−0.662603
$50$	0	0
$51$	−2.04613	−0.286515
$52$	0	0
$53$	−6.62523	−0.910046	−0.455023	−	0.890480i	$-0.650369\pi$
$53$	−6.62523	−0.910046	−0.455023	+	0.890480i	$0.650369\pi$
$54$	0	0
$55$	12.1036	1.63204
$56$	0	0
$57$	5.26772	0.697727
$58$	0	0
$59$	−12.3391	−1.60641	−0.803205	−	0.595703i	$-0.796874\pi$
$59$	−12.3391	−1.60641	−0.803205	+	0.595703i	$0.796874\pi$
$60$	0	0
$61$	−0.918459	−0.117597	−0.0587983	−	0.998270i	$-0.518727\pi$
$61$	−0.918459	−0.117597	−0.0587983	+	0.998270i	$0.518727\pi$
$62$	0	0
$63$	0.557878	0.0702860
$64$	0	0
$65$	−11.0190	−1.36673
$66$	0	0
$67$	12.9768	1.58537	0.792685	−	0.609631i	$-0.208682\pi$
$67$	12.9768	1.58537	0.792685	+	0.609631i	$0.208682\pi$
$68$	0	0
$69$	−12.0926	−1.45577
$70$	0	0
$71$	−11.3775	−1.35027	−0.675133	−	0.737696i	$-0.735914\pi$
$71$	−11.3775	−1.35027	−0.675133	+	0.737696i	$0.735914\pi$
$72$	0	0
$73$	5.47833	0.641190	0.320595	−	0.947216i	$-0.396117\pi$
$73$	5.47833	0.641190	0.320595	+	0.947216i	$0.396117\pi$
$74$	0	0
$75$	1.83385	0.211755
$76$	0	0
$77$	9.30043	1.05988
$78$	0	0
$79$	10.1457	1.14148	0.570741	−	0.821130i	$-0.306656\pi$
$79$	10.1457	1.14148	0.570741	+	0.821130i	$0.306656\pi$
$80$	0	0
$81$	−9.95726	−1.10636
$82$	0	0
$83$	14.5284	1.59470	0.797352	−	0.603515i	$-0.206234\pi$
$83$	14.5284	1.59470	0.797352	+	0.603515i	$0.206234\pi$
$84$	0	0
$85$	2.23151	0.242041
$86$	0	0
$87$	−7.28529	−0.781065
$88$	0	0
$89$	4.26971	0.452589	0.226294	−	0.974059i	$-0.427339\pi$
$89$	4.26971	0.452589	0.226294	+	0.974059i	$0.427339\pi$
$90$	0	0
$91$	−8.46701	−0.887584
$92$	0	0
$93$	2.26207	0.234565
$94$	0	0
$95$	−5.74498	−0.589423
$96$	0	0
$97$	−16.8392	−1.70976	−0.854882	−	0.518822i	$-0.826371\pi$
$97$	−16.8392	−1.70976	−0.854882	+	0.518822i	$0.826371\pi$
$98$	0	0
$99$	2.19686	0.220793
$100$	0	0
$101$	−1.81063	−0.180164	−0.0900821	−	0.995934i	$-0.528713\pi$
$101$	−1.81063	−0.180164	−0.0900821	+	0.995934i	$0.528713\pi$
$102$	0	0
$103$	−13.0031	−1.28123	−0.640617	−	0.767860i	$-0.721321\pi$
$103$	−13.0031	−1.28123	−0.640617	+	0.767860i	$0.721321\pi$
$104$	0	0
$105$	−5.63655	−0.550071
$106$	0	0
$107$	−4.69470	−0.453854	−0.226927	−	0.973912i	$-0.572868\pi$
$107$	−4.69470	−0.453854	−0.226927	+	0.973912i	$0.572868\pi$
$108$	0	0
$109$	4.03514	0.386496	0.193248	−	0.981150i	$-0.438098\pi$
$109$	4.03514	0.386496	0.193248	+	0.981150i	$0.438098\pi$
$110$	0	0
$111$	−20.4973	−1.94552
$112$	0	0
$113$	−10.0153	−0.942160	−0.471080	−	0.882091i	$-0.656136\pi$
$113$	−10.0153	−0.942160	−0.471080	+	0.882091i	$0.656136\pi$
$114$	0	0
$115$	13.1882	1.22980
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	1.71470	0.157186
$120$	0	0
$121$	25.6241	2.32946
$122$	0	0
$123$	−4.67989	−0.421972
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−4.19901	−0.372602	−0.186301	−	0.982493i	$-0.559650\pi$
$127$	−4.19901	−0.372602	−0.186301	+	0.982493i	$0.559650\pi$
$128$	0	0
$129$	−20.2071	−1.77914
$130$	0	0
$131$	−10.7874	−0.942504	−0.471252	−	0.881999i	$-0.656198\pi$
$131$	−10.7874	−0.942504	−0.471252	+	0.881999i	$0.656198\pi$
$132$	0	0
$133$	−4.41447	−0.382783
$134$	0	0
$135$	9.67169	0.832407
$136$	0	0
$137$	12.0711	1.03130	0.515651	−	0.856799i	$-0.327550\pi$
$137$	12.0711	1.03130	0.515651	+	0.856799i	$0.327550\pi$
$138$	0	0
$139$	−13.4237	−1.13858	−0.569291	−	0.822136i	$-0.692782\pi$
$139$	−13.4237	−1.13858	−0.569291	+	0.822136i	$0.692782\pi$
$140$	0	0
$141$	−14.9394	−1.25813
$142$	0	0
$143$	−33.3422	−2.78821
$144$	0	0
$145$	7.94534	0.659825
$146$	0	0
$147$	8.50581	0.701547
$148$	0	0
$149$	−12.7874	−1.04759	−0.523794	−	0.851845i	$-0.675484\pi$
$149$	−12.7874	−1.04759	−0.523794	+	0.851845i	$0.675484\pi$
$150$	0	0
$151$	−21.2035	−1.72551	−0.862757	−	0.505619i	$-0.831264\pi$
$151$	−21.2035	−1.72551	−0.862757	+	0.505619i	$0.831264\pi$
$152$	0	0
$153$	0.405031	0.0327448
$154$	0	0
$155$	−2.46701	−0.198155
$156$	0	0
$157$	7.71917	0.616057	0.308028	−	0.951377i	$-0.400331\pi$
$157$	7.71917	0.616057	0.308028	+	0.951377i	$0.400331\pi$
$158$	0	0
$159$	12.1497	0.963534
$160$	0	0
$161$	10.1338	0.798659
$162$	0	0
$163$	2.25901	0.176939	0.0884696	−	0.996079i	$-0.471802\pi$
$163$	2.25901	0.176939	0.0884696	+	0.996079i	$0.471802\pi$
$164$	0	0
$165$	−22.1961	−1.72797
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	17.3544	1.33495
$170$	0	0
$171$	−1.04274	−0.0797407
$172$	0	0
$173$	−5.80099	−0.441041	−0.220520	−	0.975382i	$-0.570776\pi$
$173$	−5.80099	−0.441041	−0.220520	+	0.975382i	$0.570776\pi$
$174$	0	0
$175$	−1.53681	−0.116172
$176$	0	0
$177$	22.6280	1.70083
$178$	0	0
$179$	2.31568	0.173082	0.0865412	−	0.996248i	$-0.472419\pi$
$179$	2.31568	0.173082	0.0865412	+	0.996248i	$0.472419\pi$
$180$	0	0
$181$	12.3820	0.920345	0.460172	−	0.887830i	$-0.347788\pi$
$181$	12.3820	0.920345	0.460172	+	0.887830i	$0.347788\pi$
$182$	0	0
$183$	1.68432	0.124508
$184$	0	0
$185$	22.3544	1.64353
$186$	0	0
$187$	6.75231	0.493778
$188$	0	0
$189$	7.43177	0.540582
$190$	0	0
$191$	16.1682	1.16989	0.584944	−	0.811073i	$-0.301116\pi$
$191$	16.1682	1.16989	0.584944	+	0.811073i	$0.301116\pi$
$192$	0	0
$193$	−20.4552	−1.47239	−0.736197	−	0.676767i	$-0.763380\pi$
$193$	−20.4552	−1.47239	−0.736197	+	0.676767i	$0.763380\pi$
$194$	0	0
$195$	20.2071	1.44706
$196$	0	0
$197$	1.98415	0.141365	0.0706824	−	0.997499i	$-0.477482\pi$
$197$	1.98415	0.141365	0.0706824	+	0.997499i	$0.477482\pi$
$198$	0	0
$199$	−2.07107	−0.146814	−0.0734071	−	0.997302i	$-0.523387\pi$
$199$	−2.07107	−0.146814	−0.0734071	+	0.997302i	$0.523387\pi$
$200$	0	0
$201$	−23.7976	−1.67855
$202$	0	0
$203$	6.10523	0.428503
$204$	0	0
$205$	5.10389	0.356471
$206$	0	0
$207$	2.39372	0.166375
$208$	0	0
$209$	−17.3837	−1.20245
$210$	0	0
$211$	−10.3374	−0.711656	−0.355828	−	0.934551i	$-0.615801\pi$
$211$	−10.3374	−0.711656	−0.355828	+	0.934551i	$0.615801\pi$
$212$	0	0
$213$	20.8647	1.42963
$214$	0	0
$215$	22.0379	1.50297
$216$	0	0
$217$	−1.89566	−0.128686
$218$	0	0
$219$	−10.0464	−0.678876
$220$	0	0
$221$	−6.14723	−0.413508
$222$	0	0
$223$	2.25043	0.150700	0.0753499	−	0.997157i	$-0.475993\pi$
$223$	2.25043	0.150700	0.0753499	+	0.997157i	$0.475993\pi$
$224$	0	0
$225$	−0.363011	−0.0242007
$226$	0	0
$227$	20.4704	1.35867	0.679336	−	0.733828i	$-0.262268\pi$
$227$	20.4704	1.35867	0.679336	+	0.733828i	$0.262268\pi$
$228$	0	0
$229$	−11.3081	−0.747259	−0.373629	−	0.927578i	$-0.621887\pi$
$229$	−11.3081	−0.747259	−0.373629	+	0.927578i	$0.621887\pi$
$230$	0	0
$231$	−17.0556	−1.12218
$232$	0	0
$233$	−13.1712	−0.862875	−0.431437	−	0.902143i	$-0.641993\pi$
$233$	−13.1712	−0.862875	−0.431437	+	0.902143i	$0.641993\pi$
$234$	0	0
$235$	16.2929	1.06283
$236$	0	0
$237$	−18.6057	−1.20857
$238$	0	0
$239$	−20.9212	−1.35328	−0.676641	−	0.736313i	$-0.736565\pi$
$239$	−20.9212	−1.35328	−0.676641	+	0.736313i	$0.736565\pi$
$240$	0	0
$241$	15.1265	0.974385	0.487192	−	0.873295i	$-0.338021\pi$
$241$	15.1265	0.974385	0.487192	+	0.873295i	$0.338021\pi$
$242$	0	0
$243$	3.75259	0.240729
$244$	0	0
$245$	−9.27644	−0.592650
$246$	0	0
$247$	15.8259	1.00698
$248$	0	0
$249$	−26.6430	−1.68843
$250$	0	0
$251$	13.4275	0.847536	0.423768	−	0.905771i	$-0.360707\pi$
$251$	13.4275	0.847536	0.423768	+	0.905771i	$0.360707\pi$
$252$	0	0
$253$	39.9060	2.50887
$254$	0	0
$255$	−4.09226	−0.256267
$256$	0	0
$257$	−18.8519	−1.17595	−0.587974	−	0.808880i	$-0.700074\pi$
$257$	−18.8519	−1.17595	−0.587974	+	0.808880i	$0.700074\pi$
$258$	0	0
$259$	17.1772	1.06734
$260$	0	0
$261$	1.44212	0.0892651
$262$	0	0
$263$	−13.9726	−0.861585	−0.430792	−	0.902451i	$-0.641766\pi$
$263$	−13.9726	−0.861585	−0.430792	+	0.902451i	$0.641766\pi$
$264$	0	0
$265$	−13.2505	−0.813970
$266$	0	0
$267$	−7.83002	−0.479190
$268$	0	0
$269$	10.4209	0.635372	0.317686	−	0.948196i	$-0.397094\pi$
$269$	10.4209	0.635372	0.317686	+	0.948196i	$0.397094\pi$
$270$	0	0
$271$	−5.01896	−0.304880	−0.152440	−	0.988313i	$-0.548713\pi$
$271$	−5.01896	−0.304880	−0.152440	+	0.988313i	$0.548713\pi$
$272$	0	0
$273$	15.5272	0.939751
$274$	0	0
$275$	−6.05178	−0.364936
$276$	0	0
$277$	−7.14603	−0.429364	−0.214682	−	0.976684i	$-0.568871\pi$
$277$	−7.14603	−0.429364	−0.214682	+	0.976684i	$0.568871\pi$
$278$	0	0
$279$	−0.447776	−0.0268076
$280$	0	0
$281$	−14.1005	−0.841165	−0.420583	−	0.907254i	$-0.638174\pi$
$281$	−14.1005	−0.841165	−0.420583	+	0.907254i	$0.638174\pi$
$282$	0	0
$283$	21.7546	1.29318	0.646589	−	0.762838i	$-0.276195\pi$
$283$	21.7546	1.29318	0.646589	+	0.762838i	$0.276195\pi$
$284$	0	0
$285$	10.5354	0.624066
$286$	0	0
$287$	3.92185	0.231500
$288$	0	0
$289$	−15.7551	−0.926770
$290$	0	0
$291$	30.8806	1.81026
$292$	0	0
$293$	25.7571	1.50474	0.752372	−	0.658738i	$-0.228909\pi$
$293$	25.7571	1.50474	0.752372	+	0.658738i	$0.228909\pi$
$294$	0	0
$295$	−24.6781	−1.43682
$296$	0	0
$297$	29.2655	1.69816
$298$	0	0
$299$	−36.3300	−2.10102
$300$	0	0
$301$	16.9340	0.976061
$302$	0	0
$303$	3.32042	0.190753
$304$	0	0
$305$	−1.83692	−0.105182
$306$	0	0
$307$	−15.5551	−0.887774	−0.443887	−	0.896083i	$-0.646401\pi$
$307$	−15.5551	−0.887774	−0.443887	+	0.896083i	$0.646401\pi$
$308$	0	0
$309$	23.8458	1.35654
$310$	0	0
$311$	9.27245	0.525793	0.262896	−	0.964824i	$-0.415322\pi$
$311$	9.27245	0.525793	0.262896	+	0.964824i	$0.415322\pi$
$312$	0	0
$313$	−27.3312	−1.54485	−0.772425	−	0.635106i	$-0.780956\pi$
$313$	−27.3312	−1.54485	−0.772425	+	0.635106i	$0.780956\pi$
$314$	0	0
$315$	1.11576	0.0628657
$316$	0	0
$317$	21.1414	1.18742	0.593711	−	0.804678i	$-0.297662\pi$
$317$	21.1414	1.18742	0.593711	+	0.804678i	$0.297662\pi$
$318$	0	0
$319$	24.0417	1.34608
$320$	0	0
$321$	8.60939	0.480529
$322$	0	0
$323$	−3.20500	−0.178331
$324$	0	0
$325$	5.50948	0.305611
$326$	0	0
$327$	−7.39984	−0.409212
$328$	0	0
$329$	12.5196	0.690226
$330$	0	0
$331$	29.1730	1.60349	0.801745	−	0.597666i	$-0.203905\pi$
$331$	29.1730	1.60349	0.801745	+	0.597666i	$0.203905\pi$
$332$	0	0
$333$	4.05744	0.222346
$334$	0	0
$335$	25.9536	1.41800
$336$	0	0
$337$	−22.1811	−1.20828	−0.604139	−	0.796879i	$-0.706483\pi$
$337$	−22.1811	−1.20828	−0.604139	+	0.796879i	$0.706483\pi$
$338$	0	0
$339$	18.3666	0.997535
$340$	0	0
$341$	−7.46491	−0.404248
$342$	0	0
$343$	−17.8857	−0.965738
$344$	0	0
$345$	−24.1851	−1.30208
$346$	0	0
$347$	−26.7862	−1.43796	−0.718980	−	0.695030i	$-0.755391\pi$
$347$	−26.7862	−1.43796	−0.718980	+	0.695030i	$0.755391\pi$
$348$	0	0
$349$	6.53266	0.349685	0.174843	−	0.984596i	$-0.444058\pi$
$349$	6.53266	0.349685	0.174843	+	0.984596i	$0.444058\pi$
$350$	0	0
$351$	−26.6430	−1.42210
$352$	0	0
$353$	−0.267135	−0.0142182	−0.00710908	−	0.999975i	$-0.502263\pi$
$353$	−0.267135	−0.0142182	−0.00710908	+	0.999975i	$0.502263\pi$
$354$	0	0
$355$	−22.7551	−1.20771
$356$	0	0
$357$	−3.14451	−0.166425
$358$	0	0
$359$	−18.1019	−0.955382	−0.477691	−	0.878528i	$-0.658526\pi$
$359$	−18.1019	−0.955382	−0.477691	+	0.878528i	$0.658526\pi$
$360$	0	0
$361$	−10.7488	−0.565726
$362$	0	0
$363$	−46.9908	−2.46637
$364$	0	0
$365$	10.9567	0.573498
$366$	0	0
$367$	−4.55517	−0.237778	−0.118889	−	0.992908i	$-0.537933\pi$
$367$	−4.55517	−0.237778	−0.118889	+	0.992908i	$0.537933\pi$
$368$	0	0
$369$	0.926384	0.0482256
$370$	0	0
$371$	−10.1817	−0.528608
$372$	0	0
$373$	9.89643	0.512418	0.256209	−	0.966621i	$-0.417526\pi$
$373$	9.89643	0.512418	0.256209	+	0.966621i	$0.417526\pi$
$374$	0	0
$375$	22.0062	1.13640
$376$	0	0
$377$	−21.8874	−1.12726
$378$	0	0
$379$	25.1072	1.28967	0.644836	−	0.764321i	$-0.276926\pi$
$379$	25.1072	1.28967	0.644836	+	0.764321i	$0.276926\pi$
$380$	0	0
$381$	7.70037	0.394502
$382$	0	0
$383$	−20.1530	−1.02977	−0.514884	−	0.857260i	$-0.672165\pi$
$383$	−20.1530	−1.02977	−0.514884	+	0.857260i	$0.672165\pi$
$384$	0	0
$385$	18.6009	0.947987
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	11.2040	0.568067	0.284033	−	0.958814i	$-0.408327\pi$
$389$	11.2040	0.568067	0.284033	+	0.958814i	$0.408327\pi$
$390$	0	0
$391$	7.35739	0.372079
$392$	0	0
$393$	19.7826	0.997899
$394$	0	0
$395$	20.2914	1.02097
$396$	0	0
$397$	12.5098	0.627847	0.313923	−	0.949448i	$-0.398357\pi$
$397$	12.5098	0.627847	0.313923	+	0.949448i	$0.398357\pi$
$398$	0	0
$399$	8.09547	0.405281
$400$	0	0
$401$	5.26912	0.263127	0.131564	−	0.991308i	$-0.458000\pi$
$401$	5.26912	0.263127	0.131564	+	0.991308i	$0.458000\pi$
$402$	0	0
$403$	6.79598	0.338532
$404$	0	0
$405$	−19.9145	−0.989560
$406$	0	0
$407$	67.6419	3.35288
$408$	0	0
$409$	8.34625	0.412695	0.206348	−	0.978479i	$-0.433842\pi$
$409$	8.34625	0.412695	0.206348	+	0.978479i	$0.433842\pi$
$410$	0	0
$411$	−22.1365	−1.09192
$412$	0	0
$413$	−18.9628	−0.933097
$414$	0	0
$415$	29.0569	1.42635
$416$	0	0
$417$	24.6170	1.20550
$418$	0	0
$419$	−12.8854	−0.629491	−0.314745	−	0.949176i	$-0.601919\pi$
$419$	−12.8854	−0.629491	−0.314745	+	0.949176i	$0.601919\pi$
$420$	0	0
$421$	3.27748	0.159735	0.0798674	−	0.996805i	$-0.474550\pi$
$421$	3.27748	0.159735	0.0798674	+	0.996805i	$0.474550\pi$
$422$	0	0
$423$	2.95726	0.143787
$424$	0	0
$425$	−1.11576	−0.0541221
$426$	0	0
$427$	−1.41149	−0.0683070
$428$	0	0
$429$	61.1446	2.95209
$430$	0	0
$431$	−6.89917	−0.332321	−0.166161	−	0.986099i	$-0.553137\pi$
$431$	−6.89917	−0.332321	−0.166161	+	0.986099i	$0.553137\pi$
$432$	0	0
$433$	−9.36148	−0.449884	−0.224942	−	0.974372i	$-0.572219\pi$
$433$	−9.36148	−0.449884	−0.224942	+	0.974372i	$0.572219\pi$
$434$	0	0
$435$	−14.5706	−0.698606
$436$	0	0
$437$	−18.9414	−0.906092
$438$	0	0
$439$	20.0639	0.957597	0.478799	−	0.877925i	$-0.341072\pi$
$439$	20.0639	0.957597	0.478799	+	0.877925i	$0.341072\pi$
$440$	0	0
$441$	−1.68372	−0.0801773
$442$	0	0
$443$	−39.1949	−1.86221	−0.931104	−	0.364754i	$-0.881153\pi$
$443$	−39.1949	−1.86221	−0.931104	+	0.364754i	$0.881153\pi$
$444$	0	0
$445$	8.53943	0.404808
$446$	0	0
$447$	23.4503	1.10916
$448$	0	0
$449$	−30.4199	−1.43560	−0.717802	−	0.696248i	$-0.754852\pi$
$449$	−30.4199	−1.43560	−0.717802	+	0.696248i	$0.754852\pi$
$450$	0	0
$451$	15.4438	0.727222
$452$	0	0
$453$	38.8840	1.82693
$454$	0	0
$455$	−16.9340	−0.793879
$456$	0	0
$457$	−36.0801	−1.68775	−0.843877	−	0.536537i	$-0.819732\pi$
$457$	−36.0801	−1.68775	−0.843877	+	0.536537i	$0.819732\pi$
$458$	0	0
$459$	5.39562	0.251846
$460$	0	0
$461$	−17.5701	−0.818323	−0.409162	−	0.912462i	$-0.634179\pi$
$461$	−17.5701	−0.818323	−0.409162	+	0.912462i	$0.634179\pi$
$462$	0	0
$463$	40.3895	1.87706	0.938530	−	0.345199i	$-0.112188\pi$
$463$	40.3895	1.87706	0.938530	+	0.345199i	$0.112188\pi$
$464$	0	0
$465$	4.52413	0.209802
$466$	0	0
$467$	18.3549	0.849361	0.424681	−	0.905343i	$-0.360386\pi$
$467$	18.3549	0.849361	0.424681	+	0.905343i	$0.360386\pi$
$468$	0	0
$469$	19.9429	0.920877
$470$	0	0
$471$	−14.1558	−0.652265
$472$	0	0
$473$	66.6844	3.06615
$474$	0	0
$475$	2.87249	0.131799
$476$	0	0
$477$	−2.40503	−0.110119
$478$	0	0
$479$	9.30912	0.425344	0.212672	−	0.977124i	$-0.431783\pi$
$479$	9.30912	0.425344	0.212672	+	0.977124i	$0.431783\pi$
$480$	0	0
$481$	−61.5805	−2.80783
$482$	0	0
$483$	−18.5840	−0.845600
$484$	0	0
$485$	−33.6785	−1.52926
$486$	0	0
$487$	30.5893	1.38613	0.693067	−	0.720873i	$-0.256259\pi$
$487$	30.5893	1.38613	0.693067	+	0.720873i	$0.256259\pi$
$488$	0	0
$489$	−4.14269	−0.187339
$490$	0	0
$491$	5.65120	0.255035	0.127517	−	0.991836i	$-0.459299\pi$
$491$	5.65120	0.255035	0.127517	+	0.991836i	$0.459299\pi$
$492$	0	0
$493$	4.43253	0.199631
$494$	0	0
$495$	4.39372	0.197483
$496$	0	0
$497$	−17.4851	−0.784314
$498$	0	0
$499$	−12.8497	−0.575234	−0.287617	−	0.957746i	$-0.592863\pi$
$499$	−12.8497	−0.575234	−0.287617	+	0.957746i	$0.592863\pi$
$500$	0	0
$501$	1.83385	0.0819304
$502$	0	0
$503$	22.0032	0.981072	0.490536	−	0.871421i	$-0.336801\pi$
$503$	22.0032	0.981072	0.490536	+	0.871421i	$0.336801\pi$
$504$	0	0
$505$	−3.62126	−0.161144
$506$	0	0
$507$	−31.8253	−1.41341
$508$	0	0
$509$	−42.0580	−1.86419	−0.932094	−	0.362216i	$-0.882020\pi$
$509$	−42.0580	−1.86419	−0.932094	+	0.362216i	$0.882020\pi$
$510$	0	0
$511$	8.41914	0.372441
$512$	0	0
$513$	−13.8909	−0.613299
$514$	0	0
$515$	−26.0062	−1.14597
$516$	0	0
$517$	49.3007	2.16824
$518$	0	0
$519$	10.6381	0.466963
$520$	0	0
$521$	−2.84758	−0.124755	−0.0623774	−	0.998053i	$-0.519868\pi$
$521$	−2.84758	−0.124755	−0.0623774	+	0.998053i	$0.519868\pi$
$522$	0	0
$523$	28.3973	1.24173	0.620864	−	0.783919i	$-0.286782\pi$
$523$	28.3973	1.24173	0.620864	+	0.783919i	$0.286782\pi$
$524$	0	0
$525$	2.81828	0.123000
$526$	0	0
$527$	−1.37629	−0.0599522
$528$	0	0
$529$	20.4820	0.890521
$530$	0	0
$531$	−4.47921	−0.194381
$532$	0	0
$533$	−14.0599	−0.609002
$534$	0	0
$535$	−9.38941	−0.405939
$536$	0	0
$537$	−4.24662	−0.183255
$538$	0	0
$539$	−28.0695	−1.20904
$540$	0	0
$541$	−26.9524	−1.15878	−0.579388	−	0.815052i	$-0.696708\pi$
$541$	−26.9524	−1.15878	−0.579388	+	0.815052i	$0.696708\pi$
$542$	0	0
$543$	−22.7067	−0.974437
$544$	0	0
$545$	8.07028	0.345693
$546$	0	0
$547$	−9.65949	−0.413010	−0.206505	−	0.978446i	$-0.566209\pi$
$547$	−9.65949	−0.413010	−0.206505	+	0.978446i	$0.566209\pi$
$548$	0	0
$549$	−0.333410	−0.0142296
$550$	0	0
$551$	−11.4115	−0.486145
$552$	0	0
$553$	15.5920	0.663039
$554$	0	0
$555$	−40.9946	−1.74012
$556$	0	0
$557$	26.3948	1.11839	0.559193	−	0.829038i	$-0.311111\pi$
$557$	26.3948	1.11839	0.559193	+	0.829038i	$0.311111\pi$
$558$	0	0
$559$	−60.7087	−2.56771
$560$	0	0
$561$	−12.3827	−0.522799
$562$	0	0
$563$	3.99989	0.168575	0.0842877	−	0.996441i	$-0.473139\pi$
$563$	3.99989	0.168575	0.0842877	+	0.996441i	$0.473139\pi$
$564$	0	0
$565$	−20.0306	−0.842693
$566$	0	0
$567$	−15.3024	−0.642640
$568$	0	0
$569$	33.9714	1.42416	0.712078	−	0.702101i	$-0.247754\pi$
$569$	33.9714	1.42416	0.712078	+	0.702101i	$0.247754\pi$
$570$	0	0
$571$	−4.18108	−0.174973	−0.0874863	−	0.996166i	$-0.527883\pi$
$571$	−4.18108	−0.174973	−0.0874863	+	0.996166i	$0.527883\pi$
$572$	0	0
$573$	−29.6500	−1.23865
$574$	0	0
$575$	−6.59409	−0.274992
$576$	0	0
$577$	4.85476	0.202106	0.101053	−	0.994881i	$-0.467779\pi$
$577$	4.85476	0.202106	0.101053	+	0.994881i	$0.467779\pi$
$578$	0	0
$579$	37.5117	1.55893
$580$	0	0
$581$	22.3274	0.926297
$582$	0	0
$583$	−40.0945	−1.66054
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	−41.1977	−1.70041	−0.850206	−	0.526450i	$-0.823523\pi$
$587$	−41.1977	−1.70041	−0.850206	+	0.526450i	$0.823523\pi$
$588$	0	0
$589$	3.54323	0.145996
$590$	0	0
$591$	−3.63863	−0.149673
$592$	0	0
$593$	−37.4546	−1.53808	−0.769038	−	0.639203i	$-0.779264\pi$
$593$	−37.4546	−1.53808	−0.769038	+	0.639203i	$0.779264\pi$
$594$	0	0
$595$	3.42940	0.140592
$596$	0	0
$597$	3.79803	0.155443
$598$	0	0
$599$	24.8799	1.01656	0.508282	−	0.861191i	$-0.330281\pi$
$599$	24.8799	1.01656	0.508282	+	0.861191i	$0.330281\pi$
$600$	0	0
$601$	−12.4360	−0.507276	−0.253638	−	0.967299i	$-0.581627\pi$
$601$	−12.4360	−0.507276	−0.253638	+	0.967299i	$0.581627\pi$
$602$	0	0
$603$	4.71072	0.191836
$604$	0	0
$605$	51.2482	2.08353
$606$	0	0
$607$	−30.0532	−1.21982	−0.609911	−	0.792470i	$-0.708795\pi$
$607$	−30.0532	−1.21982	−0.609911	+	0.792470i	$0.708795\pi$
$608$	0	0
$609$	−11.1961	−0.453688
$610$	0	0
$611$	−44.8828	−1.81576
$612$	0	0
$613$	29.2271	1.18047	0.590236	−	0.807231i	$-0.299035\pi$
$613$	29.2271	1.18047	0.590236	+	0.807231i	$0.299035\pi$
$614$	0	0
$615$	−9.35978	−0.377423
$616$	0	0
$617$	17.1003	0.688432	0.344216	−	0.938891i	$-0.388145\pi$
$617$	17.1003	0.688432	0.344216	+	0.938891i	$0.388145\pi$
$618$	0	0
$619$	1.58589	0.0637422	0.0318711	−	0.999492i	$-0.489853\pi$
$619$	1.58589	0.0637422	0.0318711	+	0.999492i	$0.489853\pi$
$620$	0	0
$621$	31.8880	1.27962
$622$	0	0
$623$	6.56173	0.262890
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	31.8791	1.27313
$628$	0	0
$629$	12.4710	0.497251
$630$	0	0
$631$	−18.7899	−0.748012	−0.374006	−	0.927426i	$-0.622016\pi$
$631$	−18.7899	−0.748012	−0.374006	+	0.927426i	$0.622016\pi$
$632$	0	0
$633$	18.9573	0.753483
$634$	0	0
$635$	−8.39803	−0.333266
$636$	0	0
$637$	25.5542	1.01249
$638$	0	0
$639$	−4.13017	−0.163387
$640$	0	0
$641$	12.9249	0.510501	0.255251	−	0.966875i	$-0.417842\pi$
$641$	12.9249	0.510501	0.255251	+	0.966875i	$0.417842\pi$
$642$	0	0
$643$	1.09799	0.0433007	0.0216503	−	0.999766i	$-0.493108\pi$
$643$	1.09799	0.0433007	0.0216503	+	0.999766i	$0.493108\pi$
$644$	0	0
$645$	−40.4143	−1.59131
$646$	0	0
$647$	22.9563	0.902506	0.451253	−	0.892396i	$-0.350977\pi$
$647$	22.9563	0.902506	0.451253	+	0.892396i	$0.350977\pi$
$648$	0	0
$649$	−74.6734	−2.93119
$650$	0	0
$651$	3.47636	0.136249
$652$	0	0
$653$	15.4333	0.603952	0.301976	−	0.953315i	$-0.402354\pi$
$653$	15.4333	0.603952	0.301976	+	0.953315i	$0.402354\pi$
$654$	0	0
$655$	−21.5749	−0.843001
$656$	0	0
$657$	1.98869	0.0775863
$658$	0	0
$659$	−8.26179	−0.321834	−0.160917	−	0.986968i	$-0.551445\pi$
$659$	−8.26179	−0.321834	−0.160917	+	0.986968i	$0.551445\pi$
$660$	0	0
$661$	−28.7197	−1.11707	−0.558534	−	0.829482i	$-0.688636\pi$
$661$	−28.7197	−1.11707	−0.558534	+	0.829482i	$0.688636\pi$
$662$	0	0
$663$	11.2731	0.437811
$664$	0	0
$665$	−8.82893	−0.342371
$666$	0	0
$667$	26.1961	1.01432
$668$	0	0
$669$	−4.12695	−0.159557
$670$	0	0
$671$	−5.55831	−0.214576
$672$	0	0
$673$	35.9750	1.38674	0.693368	−	0.720584i	$-0.256126\pi$
$673$	35.9750	1.38674	0.693368	+	0.720584i	$0.256126\pi$
$674$	0	0
$675$	−4.83585	−0.186132
$676$	0	0
$677$	20.6402	0.793268	0.396634	−	0.917977i	$-0.370178\pi$
$677$	20.6402	0.793268	0.396634	+	0.917977i	$0.370178\pi$
$678$	0	0
$679$	−25.8787	−0.993132
$680$	0	0
$681$	−37.5398	−1.43853
$682$	0	0
$683$	−13.5046	−0.516740	−0.258370	−	0.966046i	$-0.583185\pi$
$683$	−13.5046	−0.516740	−0.258370	+	0.966046i	$0.583185\pi$
$684$	0	0
$685$	24.1421	0.922424
$686$	0	0
$687$	20.7373	0.791179
$688$	0	0
$689$	36.5016	1.39060
$690$	0	0
$691$	−5.99213	−0.227951	−0.113976	−	0.993484i	$-0.536359\pi$
$691$	−5.99213	−0.227951	−0.113976	+	0.993484i	$0.536359\pi$
$692$	0	0
$693$	3.37615	0.128250
$694$	0	0
$695$	−26.8473	−1.01838
$696$	0	0
$697$	2.84735	0.107851
$698$	0	0
$699$	24.1540	0.913590
$700$	0	0
$701$	−37.3266	−1.40981	−0.704903	−	0.709304i	$-0.749009\pi$
$701$	−37.3266	−1.40981	−0.704903	+	0.709304i	$0.749009\pi$
$702$	0	0
$703$	−32.1063	−1.21091
$704$	0	0
$705$	−29.8788	−1.12530
$706$	0	0
$707$	−2.78259	−0.104650
$708$	0	0
$709$	−0.250714	−0.00941576	−0.00470788	−	0.999989i	$-0.501499\pi$
$709$	−0.250714	−0.00941576	−0.00470788	+	0.999989i	$0.501499\pi$
$710$	0	0
$711$	3.68300	0.138123
$712$	0	0
$713$	−8.13384	−0.304615
$714$	0	0
$715$	−66.6844	−2.49385
$716$	0	0
$717$	38.3664	1.43282
$718$	0	0
$719$	−16.7688	−0.625371	−0.312685	−	0.949857i	$-0.601229\pi$
$719$	−16.7688	−0.625371	−0.312685	+	0.949857i	$0.601229\pi$
$720$	0	0
$721$	−19.9833	−0.744216
$722$	0	0
$723$	−27.7398	−1.03165
$724$	0	0
$725$	−3.97267	−0.147541
$726$	0	0
$727$	27.8678	1.03356	0.516780	−	0.856118i	$-0.327130\pi$
$727$	27.8678	1.03356	0.516780	+	0.856118i	$0.327130\pi$
$728$	0	0
$729$	22.9901	0.851484
$730$	0	0
$731$	12.2945	0.454727
$732$	0	0
$733$	5.46236	0.201757	0.100878	−	0.994899i	$-0.467835\pi$
$733$	5.46236	0.201757	0.100878	+	0.994899i	$0.467835\pi$
$734$	0	0
$735$	17.0116	0.627483
$736$	0	0
$737$	78.5329	2.89280
$738$	0	0
$739$	−36.4399	−1.34046	−0.670232	−	0.742151i	$-0.733806\pi$
$739$	−36.4399	−1.34046	−0.670232	+	0.742151i	$0.733806\pi$
$740$	0	0
$741$	−29.0224	−1.06616
$742$	0	0
$743$	−13.8614	−0.508525	−0.254262	−	0.967135i	$-0.581833\pi$
$743$	−13.8614	−0.508525	−0.254262	+	0.967135i	$0.581833\pi$
$744$	0	0
$745$	−25.5749	−0.936992
$746$	0	0
$747$	5.27398	0.192965
$748$	0	0
$749$	−7.21486	−0.263625
$750$	0	0
$751$	32.9296	1.20162	0.600809	−	0.799393i	$-0.294845\pi$
$751$	32.9296	1.20162	0.600809	+	0.799393i	$0.294845\pi$
$752$	0	0
$753$	−24.6240	−0.897349
$754$	0	0
$755$	−42.4069	−1.54335
$756$	0	0
$757$	−14.9978	−0.545103	−0.272552	−	0.962141i	$-0.587868\pi$
$757$	−14.9978	−0.545103	−0.272552	+	0.962141i	$0.587868\pi$
$758$	0	0
$759$	−73.1816	−2.65633
$760$	0	0
$761$	−13.4658	−0.488133	−0.244067	−	0.969758i	$-0.578482\pi$
$761$	−13.4658	−0.488133	−0.244067	+	0.969758i	$0.578482\pi$
$762$	0	0
$763$	6.20123	0.224500
$764$	0	0
$765$	0.810062	0.0292879
$766$	0	0
$767$	67.9818	2.45468
$768$	0	0
$769$	26.9048	0.970211	0.485106	−	0.874456i	$-0.338781\pi$
$769$	26.9048	0.970211	0.485106	+	0.874456i	$0.338781\pi$
$770$	0	0
$771$	34.5716	1.24506
$772$	0	0
$773$	−18.2551	−0.656590	−0.328295	−	0.944575i	$-0.606474\pi$
$773$	−18.2551	−0.656590	−0.328295	+	0.944575i	$0.606474\pi$
$774$	0	0
$775$	1.23351	0.0443088
$776$	0	0
$777$	−31.5004	−1.13007
$778$	0	0
$779$	−7.33044	−0.262640
$780$	0	0
$781$	−68.8544	−2.46381
$782$	0	0
$783$	19.2112	0.686553
$784$	0	0
$785$	15.4383	0.551018
$786$	0	0
$787$	16.4967	0.588042	0.294021	−	0.955799i	$-0.405006\pi$
$787$	16.4967	0.588042	0.294021	+	0.955799i	$0.405006\pi$
$788$	0	0
$789$	25.6236	0.912224
$790$	0	0
$791$	−15.3916	−0.547262
$792$	0	0
$793$	5.06023	0.179694
$794$	0	0
$795$	24.2994	0.861811
$796$	0	0
$797$	−11.1958	−0.396576	−0.198288	−	0.980144i	$-0.563538\pi$
$797$	−11.1958	−0.396576	−0.198288	+	0.980144i	$0.563538\pi$
$798$	0	0
$799$	9.08947	0.321562
$800$	0	0
$801$	1.54995	0.0547649
$802$	0	0
$803$	33.1537	1.16997
$804$	0	0
$805$	20.2677	0.714342
$806$	0	0
$807$	−19.1103	−0.672716
$808$	0	0
$809$	49.7828	1.75027	0.875135	−	0.483878i	$-0.160772\pi$
$809$	49.7828	1.75027	0.875135	+	0.483878i	$0.160772\pi$
$810$	0	0
$811$	7.70131	0.270430	0.135215	−	0.990816i	$-0.456828\pi$
$811$	7.70131	0.270430	0.135215	+	0.990816i	$0.456828\pi$
$812$	0	0
$813$	9.20402	0.322799
$814$	0	0
$815$	4.51802	0.158259
$816$	0	0
$817$	−31.6519	−1.10736
$818$	0	0
$819$	−3.07362	−0.107401
$820$	0	0
$821$	13.9531	0.486966	0.243483	−	0.969905i	$-0.421710\pi$
$821$	13.9531	0.486966	0.243483	+	0.969905i	$0.421710\pi$
$822$	0	0
$823$	25.0452	0.873022	0.436511	−	0.899699i	$-0.356214\pi$
$823$	25.0452	0.873022	0.436511	+	0.899699i	$0.356214\pi$
$824$	0	0
$825$	11.0981	0.386385
$826$	0	0
$827$	12.6564	0.440106	0.220053	−	0.975488i	$-0.429377\pi$
$827$	12.6564	0.440106	0.220053	+	0.975488i	$0.429377\pi$
$828$	0	0
$829$	34.6964	1.20506	0.602529	−	0.798097i	$-0.294160\pi$
$829$	34.6964	1.20506	0.602529	+	0.798097i	$0.294160\pi$
$830$	0	0
$831$	13.1048	0.454599
$832$	0	0
$833$	−5.17512	−0.179307
$834$	0	0
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	−5.96504	−0.206182
$838$	0	0
$839$	−23.1380	−0.798813	−0.399406	−	0.916774i	$-0.630784\pi$
$839$	−23.1380	−0.798813	−0.399406	+	0.916774i	$0.630784\pi$
$840$	0	0
$841$	−13.2179	−0.455789
$842$	0	0
$843$	25.8582	0.890604
$844$	0	0
$845$	34.7087	1.19402
$846$	0	0
$847$	39.3793	1.35309
$848$	0	0
$849$	−39.8947	−1.36918
$850$	0	0
$851$	73.7033	2.52652
$852$	0	0
$853$	−29.6187	−1.01412	−0.507062	−	0.861909i	$-0.669269\pi$
$853$	−29.6187	−1.01412	−0.507062	+	0.861909i	$0.669269\pi$
$854$	0	0
$855$	−2.08549	−0.0713222
$856$	0	0
$857$	21.7803	0.744002	0.372001	−	0.928232i	$-0.378672\pi$
$857$	21.7803	0.744002	0.372001	+	0.928232i	$0.378672\pi$
$858$	0	0
$859$	14.4256	0.492195	0.246097	−	0.969245i	$-0.420852\pi$
$859$	14.4256	0.492195	0.246097	+	0.969245i	$0.420852\pi$
$860$	0	0
$861$	−7.19209	−0.245106
$862$	0	0
$863$	0.0664423	0.00226172	0.00113086	−	0.999999i	$-0.499640\pi$
$863$	0.0664423	0.00226172	0.00113086	+	0.999999i	$0.499640\pi$
$864$	0	0
$865$	−11.6020	−0.394479
$866$	0	0
$867$	28.8925	0.981240
$868$	0	0
$869$	61.3996	2.08284
$870$	0	0
$871$	−71.4955	−2.42253
$872$	0	0
$873$	−6.11282	−0.206888
$874$	0	0
$875$	−18.4417	−0.623443
$876$	0	0
$877$	2.91486	0.0984279	0.0492139	−	0.998788i	$-0.484328\pi$
$877$	2.91486	0.0984279	0.0492139	+	0.998788i	$0.484328\pi$
$878$	0	0
$879$	−47.2347	−1.59319
$880$	0	0
$881$	−9.91063	−0.333898	−0.166949	−	0.985966i	$-0.553391\pi$
$881$	−9.91063	−0.333898	−0.166949	+	0.985966i	$0.553391\pi$
$882$	0	0
$883$	−12.3842	−0.416760	−0.208380	−	0.978048i	$-0.566819\pi$
$883$	−12.3842	−0.416760	−0.208380	+	0.978048i	$0.566819\pi$
$884$	0	0
$885$	45.2560	1.52126
$886$	0	0
$887$	−30.6773	−1.03004	−0.515021	−	0.857178i	$-0.672216\pi$
$887$	−30.6773	−1.03004	−0.515021	+	0.857178i	$0.672216\pi$
$888$	0	0
$889$	−6.45308	−0.216429
$890$	0	0
$891$	−60.2592	−2.01876
$892$	0	0
$893$	−23.4007	−0.783073
$894$	0	0
$895$	4.63137	0.154810
$896$	0	0
$897$	66.6238	2.22450
$898$	0	0
$899$	−4.90031	−0.163435
$900$	0	0
$901$	−7.39214	−0.246268
$902$	0	0
$903$	−31.0545	−1.03343
$904$	0	0
$905$	24.7639	0.823181
$906$	0	0
$907$	15.7014	0.521355	0.260678	−	0.965426i	$-0.416054\pi$
$907$	15.7014	0.521355	0.260678	+	0.965426i	$0.416054\pi$
$908$	0	0
$909$	−0.657278	−0.0218005
$910$	0	0
$911$	10.6183	0.351801	0.175901	−	0.984408i	$-0.443716\pi$
$911$	10.6183	0.351801	0.175901	+	0.984408i	$0.443716\pi$
$912$	0	0
$913$	87.9230	2.90983
$914$	0	0
$915$	3.36863	0.111364
$916$	0	0
$917$	−16.5782	−0.547462
$918$	0	0
$919$	32.4157	1.06929	0.534647	−	0.845075i	$-0.320444\pi$
$919$	32.4157	1.06929	0.534647	+	0.845075i	$0.320444\pi$
$920$	0	0
$921$	28.5257	0.939952
$922$	0	0
$923$	62.6844	2.06328
$924$	0	0
$925$	−11.1772	−0.367503
$926$	0	0
$927$	−4.72027	−0.155034
$928$	0	0
$929$	0.0892981	0.00292977	0.00146489	−	0.999999i	$-0.499534\pi$
$929$	0.0892981	0.00292977	0.00146489	+	0.999999i	$0.499534\pi$
$930$	0	0
$931$	13.3232	0.436652
$932$	0	0
$933$	−17.0043	−0.556696
$934$	0	0
$935$	13.5046	0.441648
$936$	0	0
$937$	−6.80760	−0.222395	−0.111197	−	0.993798i	$-0.535469\pi$
$937$	−6.80760	−0.222395	−0.111197	+	0.993798i	$0.535469\pi$
$938$	0	0
$939$	50.1213	1.63565
$940$	0	0
$941$	−19.2599	−0.627856	−0.313928	−	0.949447i	$-0.601645\pi$
$941$	−19.2599	−0.627856	−0.313928	+	0.949447i	$0.601645\pi$
$942$	0	0
$943$	16.8278	0.547987
$944$	0	0
$945$	14.8635	0.483511
$946$	0	0
$947$	−37.2451	−1.21030	−0.605151	−	0.796110i	$-0.706887\pi$
$947$	−37.2451	−1.21030	−0.605151	+	0.796110i	$0.706887\pi$
$948$	0	0
$949$	−30.1828	−0.979774
$950$	0	0
$951$	−38.7703	−1.25721
$952$	0	0
$953$	0.648190	0.0209969	0.0104985	−	0.999945i	$-0.496658\pi$
$953$	0.648190	0.0209969	0.0104985	+	0.999945i	$0.496658\pi$
$954$	0	0
$955$	32.3364	1.04638
$956$	0	0
$957$	−44.0890	−1.42519
$958$	0	0
$959$	18.5509	0.599040
$960$	0	0
$961$	−29.4785	−0.950918
$962$	0	0
$963$	−1.70423	−0.0549180
$964$	0	0
$965$	−40.9103	−1.31695
$966$	0	0
$967$	35.5793	1.14415	0.572076	−	0.820201i	$-0.306138\pi$
$967$	35.5793	1.14415	0.572076	+	0.820201i	$0.306138\pi$
$968$	0	0
$969$	5.87749	0.188812
$970$	0	0
$971$	16.3660	0.525210	0.262605	−	0.964903i	$-0.415418\pi$
$971$	16.3660	0.525210	0.262605	+	0.964903i	$0.415418\pi$
$972$	0	0
$973$	−20.6296	−0.661355
$974$	0	0
$975$	−10.1036	−0.323573
$976$	0	0
$977$	21.8152	0.697929	0.348964	−	0.937136i	$-0.386533\pi$
$977$	21.8152	0.697929	0.348964	+	0.937136i	$0.386533\pi$
$978$	0	0
$979$	25.8394	0.825830
$980$	0	0
$981$	1.46480	0.0467674
$982$	0	0
$983$	47.7594	1.52329	0.761644	−	0.647996i	$-0.224393\pi$
$983$	47.7594	1.52329	0.761644	+	0.647996i	$0.224393\pi$
$984$	0	0
$985$	3.96830	0.126440
$986$	0	0
$987$	−22.9590	−0.730793
$988$	0	0
$989$	72.6600	2.31045
$990$	0	0
$991$	19.3121	0.613470	0.306735	−	0.951795i	$-0.400763\pi$
$991$	19.3121	0.613470	0.306735	+	0.951795i	$0.400763\pi$
$992$	0	0
$993$	−53.4989	−1.69774
$994$	0	0
$995$	−4.14214	−0.131315
$996$	0	0
$997$	8.21884	0.260293	0.130147	−	0.991495i	$-0.458455\pi$
$997$	8.21884	0.260293	0.130147	+	0.991495i	$0.458455\pi$
$998$	0	0
$999$	54.0511	1.71010

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.a.b.1.1	✓	5
3.2	odd	2		6012.2.a.d.1.3		5
4.3	odd	2		2672.2.a.j.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.b.1.1	✓	5	1.1	even	1	trivial
2672.2.a.j.1.5		5	4.3	odd	2
6012.2.a.d.1.3		5	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:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.a (trivial)

\(f(q)\)	\(=\)	\(q-1.83385 q^{3} +2.00000 q^{5} +1.53681 q^{7} +0.363011 q^{9} +O(q^{10})\)	\(q-1.83385 q^{3} +2.00000 q^{5} +1.53681 q^{7} +0.363011 q^{9} +6.05178 q^{11} -5.50948 q^{13} -3.66770 q^{15} +1.11576 q^{17} -2.87249 q^{19} -2.81828 q^{21} +6.59409 q^{23} -1.00000 q^{25} +4.83585 q^{27} +3.97267 q^{29} -1.23351 q^{31} -11.0981 q^{33} +3.07362 q^{35} +11.1772 q^{37} +10.1036 q^{39} +2.55195 q^{41} +11.0190 q^{43} +0.726021 q^{45} +8.14647 q^{47} -4.63822 q^{49} -2.04613 q^{51} -6.62523 q^{53} +12.1036 q^{55} +5.26772 q^{57} -12.3391 q^{59} -0.918459 q^{61} +0.557878 q^{63} -11.0190 q^{65} +12.9768 q^{67} -12.0926 q^{69} -11.3775 q^{71} +5.47833 q^{73} +1.83385 q^{75} +9.30043 q^{77} +10.1457 q^{79} -9.95726 q^{81} +14.5284 q^{83} +2.23151 q^{85} -7.28529 q^{87} +4.26971 q^{89} -8.46701 q^{91} +2.26207 q^{93} -5.74498 q^{95} -16.8392 q^{97} +2.19686 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} + 10 q^{5} + 9 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 + 10 * q^5 + 9 * q^7 + 6 * q^9	\( 5 q + 3 q^{3} + 10 q^{5} + 9 q^{7} + 6 q^{9} + 5 q^{11} - 4 q^{13} + 6 q^{15} - 2 q^{17} + 5 q^{19} - 4 q^{21} + 6 q^{23} - 5 q^{25} + 12 q^{27} - 5 q^{29} + 9 q^{31} - 8 q^{33} + 18 q^{35} + 8 q^{37} - 4 q^{41} + 8 q^{43} + 12 q^{45} + 13 q^{47} + 14 q^{49} - 2 q^{53} + 10 q^{55} - 12 q^{57} + 4 q^{59} + 11 q^{61} - q^{63} - 8 q^{65} + 28 q^{67} - 16 q^{69} + 2 q^{71} + 8 q^{73} - 3 q^{75} - 12 q^{77} - 10 q^{79} - 15 q^{81} + 2 q^{83} - 4 q^{85} + 4 q^{87} - 17 q^{89} - 12 q^{91} - 12 q^{93} + 10 q^{95} - 27 q^{97} + 3 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 + 10 * q^5 + 9 * q^7 + 6 * q^9 + 5 * q^11 - 4 * q^13 + 6 * q^15 - 2 * q^17 + 5 * q^19 - 4 * q^21 + 6 * q^23 - 5 * q^25 + 12 * q^27 - 5 * q^29 + 9 * q^31 - 8 * q^33 + 18 * q^35 + 8 * q^37 - 4 * q^41 + 8 * q^43 + 12 * q^45 + 13 * q^47 + 14 * q^49 - 2 * q^53 + 10 * q^55 - 12 * q^57 + 4 * q^59 + 11 * q^61 - q^63 - 8 * q^65 + 28 * q^67 - 16 * q^69 + 2 * q^71 + 8 * q^73 - 3 * q^75 - 12 * q^77 - 10 * q^79 - 15 * q^81 + 2 * q^83 - 4 * q^85 + 4 * q^87 - 17 * q^89 - 12 * q^91 - 12 * q^93 + 10 * q^95 - 27 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more