LMFDB - Embedded newform 6028.2.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82460 q^{3} +0.0657942 q^{5} -4.70856 q^{7} +4.97835 q^{9} +O(q^{10})$	$q-2.82460 q^{3} +0.0657942 q^{5} -4.70856 q^{7} +4.97835 q^{9} -1.00000 q^{11} +3.06256 q^{13} -0.185842 q^{15} +2.19697 q^{17} +0.519839 q^{19} +13.2998 q^{21} -3.30178 q^{23} -4.99567 q^{25} -5.58803 q^{27} +2.06524 q^{29} -8.48545 q^{31} +2.82460 q^{33} -0.309796 q^{35} -5.97305 q^{37} -8.65049 q^{39} +9.46109 q^{41} -11.2345 q^{43} +0.327546 q^{45} +10.1460 q^{47} +15.1705 q^{49} -6.20555 q^{51} +5.27740 q^{53} -0.0657942 q^{55} -1.46833 q^{57} +10.2593 q^{59} +6.96458 q^{61} -23.4408 q^{63} +0.201498 q^{65} +2.27842 q^{67} +9.32618 q^{69} +4.83869 q^{71} +1.71205 q^{73} +14.1108 q^{75} +4.70856 q^{77} +6.01610 q^{79} +0.848892 q^{81} +5.10189 q^{83} +0.144548 q^{85} -5.83346 q^{87} +14.5717 q^{89} -14.4202 q^{91} +23.9680 q^{93} +0.0342024 q^{95} -3.05645 q^{97} -4.97835 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) $ 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9	$ 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) $ 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9 - 27 * q^11 - 6 * q^15 - 21 * q^17 - 3 * q^19 - 4 * q^21 - 44 * q^23 + 38 * q^25 - 18 * q^27 + q^29 - 8 * q^31 + 6 * q^33 - 33 * q^35 + 11 * q^37 - 13 * q^39 - 19 * q^41 - 11 * q^43 + 17 * q^45 - 37 * q^47 + 41 * q^49 - 49 * q^51 - 12 * q^53 - q^55 - 50 * q^57 - 14 * q^59 + 12 * q^61 - 53 * q^63 - 55 * q^65 - 5 * q^67 + 14 * q^69 - 67 * q^71 - 27 * q^73 - 70 * q^75 + 14 * q^77 - 31 * q^79 - 5 * q^81 - 55 * q^83 - 3 * q^85 - 31 * q^87 + 11 * q^89 - 11 * q^91 - 24 * q^93 - 47 * q^95 - q^97 - 33 * 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
$3$	−2.82460	−1.63078	−0.815391	−	0.578911i	$-0.803478\pi$
$3$	−2.82460	−1.63078	−0.815391	+	0.578911i	$0.803478\pi$
$4$	0	0
$5$	0.0657942	0.0294240	0.0147120	−	0.999892i	$-0.495317\pi$
$5$	0.0657942	0.0294240	0.0147120	+	0.999892i	$0.495317\pi$
$6$	0	0
$7$	−4.70856	−1.77967	−0.889834	−	0.456285i	$-0.849180\pi$
$7$	−4.70856	−1.77967	−0.889834	+	0.456285i	$0.849180\pi$
$8$	0	0
$9$	4.97835	1.65945
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.06256	0.849401	0.424700	−	0.905334i	$-0.360379\pi$
$13$	3.06256	0.849401	0.424700	+	0.905334i	$0.360379\pi$
$14$	0	0
$15$	−0.185842	−0.0479842
$16$	0	0
$17$	2.19697	0.532843	0.266422	−	0.963857i	$-0.414159\pi$
$17$	2.19697	0.532843	0.266422	+	0.963857i	$0.414159\pi$
$18$	0	0
$19$	0.519839	0.119259	0.0596296	−	0.998221i	$-0.481008\pi$
$19$	0.519839	0.119259	0.0596296	+	0.998221i	$0.481008\pi$
$20$	0	0
$21$	13.2998	2.90225
$22$	0	0
$23$	−3.30178	−0.688468	−0.344234	−	0.938884i	$-0.611861\pi$
$23$	−3.30178	−0.688468	−0.344234	+	0.938884i	$0.611861\pi$
$24$	0	0
$25$	−4.99567	−0.999134
$26$	0	0
$27$	−5.58803	−1.07542
$28$	0	0
$29$	2.06524	0.383505	0.191752	−	0.981443i	$-0.438583\pi$
$29$	2.06524	0.383505	0.191752	+	0.981443i	$0.438583\pi$
$30$	0	0
$31$	−8.48545	−1.52403	−0.762016	−	0.647558i	$-0.775790\pi$
$31$	−8.48545	−1.52403	−0.762016	+	0.647558i	$0.775790\pi$
$32$	0	0
$33$	2.82460	0.491699
$34$	0	0
$35$	−0.309796	−0.0523650
$36$	0	0
$37$	−5.97305	−0.981963	−0.490982	−	0.871170i	$-0.663362\pi$
$37$	−5.97305	−0.981963	−0.490982	+	0.871170i	$0.663362\pi$
$38$	0	0
$39$	−8.65049	−1.38519
$40$	0	0
$41$	9.46109	1.47757	0.738787	−	0.673939i	$-0.235399\pi$
$41$	9.46109	1.47757	0.738787	+	0.673939i	$0.235399\pi$
$42$	0	0
$43$	−11.2345	−1.71324	−0.856622	−	0.515945i	$-0.827441\pi$
$43$	−11.2345	−1.71324	−0.856622	+	0.515945i	$0.827441\pi$
$44$	0	0
$45$	0.327546	0.0488277
$46$	0	0
$47$	10.1460	1.47995	0.739975	−	0.672635i	$-0.234838\pi$
$47$	10.1460	1.47995	0.739975	+	0.672635i	$0.234838\pi$
$48$	0	0
$49$	15.1705	2.16722
$50$	0	0
$51$	−6.20555	−0.868951
$52$	0	0
$53$	5.27740	0.724906	0.362453	−	0.932002i	$-0.381939\pi$
$53$	5.27740	0.724906	0.362453	+	0.932002i	$0.381939\pi$
$54$	0	0
$55$	−0.0657942	−0.00887168
$56$	0	0
$57$	−1.46833	−0.194486
$58$	0	0
$59$	10.2593	1.33565	0.667823	−	0.744320i	$-0.267226\pi$
$59$	10.2593	1.33565	0.667823	+	0.744320i	$0.267226\pi$
$60$	0	0
$61$	6.96458	0.891723	0.445862	−	0.895102i	$-0.352897\pi$
$61$	6.96458	0.891723	0.445862	+	0.895102i	$0.352897\pi$
$62$	0	0
$63$	−23.4408	−2.95327
$64$	0	0
$65$	0.201498	0.0249928
$66$	0	0
$67$	2.27842	0.278354	0.139177	−	0.990268i	$-0.455554\pi$
$67$	2.27842	0.278354	0.139177	+	0.990268i	$0.455554\pi$
$68$	0	0
$69$	9.32618	1.12274
$70$	0	0
$71$	4.83869	0.574247	0.287123	−	0.957894i	$-0.407301\pi$
$71$	4.83869	0.574247	0.287123	+	0.957894i	$0.407301\pi$
$72$	0	0
$73$	1.71205	0.200380	0.100190	−	0.994968i	$-0.468055\pi$
$73$	1.71205	0.200380	0.100190	+	0.994968i	$0.468055\pi$
$74$	0	0
$75$	14.1108	1.62937
$76$	0	0
$77$	4.70856	0.536590
$78$	0	0
$79$	6.01610	0.676865	0.338432	−	0.940991i	$-0.390103\pi$
$79$	6.01610	0.676865	0.338432	+	0.940991i	$0.390103\pi$
$80$	0	0
$81$	0.848892	0.0943213
$82$	0	0
$83$	5.10189	0.560005	0.280003	−	0.959999i	$-0.409665\pi$
$83$	5.10189	0.560005	0.280003	+	0.959999i	$0.409665\pi$
$84$	0	0
$85$	0.144548	0.0156784
$86$	0	0
$87$	−5.83346	−0.625412
$88$	0	0
$89$	14.5717	1.54460	0.772298	−	0.635260i	$-0.219107\pi$
$89$	14.5717	1.54460	0.772298	+	0.635260i	$0.219107\pi$
$90$	0	0
$91$	−14.4202	−1.51165
$92$	0	0
$93$	23.9680	2.48536
$94$	0	0
$95$	0.0342024	0.00350909
$96$	0	0
$97$	−3.05645	−0.310336	−0.155168	−	0.987888i	$-0.549592\pi$
$97$	−3.05645	−0.310336	−0.155168	+	0.987888i	$0.549592\pi$
$98$	0	0
$99$	−4.97835	−0.500343
$100$	0	0
$101$	−12.0503	−1.19905	−0.599524	−	0.800357i	$-0.704643\pi$
$101$	−12.0503	−1.19905	−0.599524	+	0.800357i	$0.704643\pi$
$102$	0	0
$103$	16.2717	1.60330	0.801649	−	0.597795i	$-0.203956\pi$
$103$	16.2717	1.60330	0.801649	+	0.597795i	$0.203956\pi$
$104$	0	0
$105$	0.875048	0.0853959
$106$	0	0
$107$	4.55865	0.440702	0.220351	−	0.975421i	$-0.429280\pi$
$107$	4.55865	0.440702	0.220351	+	0.975421i	$0.429280\pi$
$108$	0	0
$109$	−4.71392	−0.451512	−0.225756	−	0.974184i	$-0.572485\pi$
$109$	−4.71392	−0.451512	−0.225756	+	0.974184i	$0.572485\pi$
$110$	0	0
$111$	16.8715	1.60137
$112$	0	0
$113$	−18.9141	−1.77929	−0.889646	−	0.456651i	$-0.849049\pi$
$113$	−18.9141	−1.77929	−0.889646	+	0.456651i	$0.849049\pi$
$114$	0	0
$115$	−0.217238	−0.0202575
$116$	0	0
$117$	15.2465	1.40954
$118$	0	0
$119$	−10.3446	−0.948284
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−26.7238	−2.40960
$124$	0	0
$125$	−0.657657	−0.0588226
$126$	0	0
$127$	10.6791	0.947621	0.473810	−	0.880627i	$-0.342878\pi$
$127$	10.6791	0.947621	0.473810	+	0.880627i	$0.342878\pi$
$128$	0	0
$129$	31.7329	2.79393
$130$	0	0
$131$	11.3300	0.989907	0.494953	−	0.868920i	$-0.335185\pi$
$131$	11.3300	0.989907	0.494953	+	0.868920i	$0.335185\pi$
$132$	0	0
$133$	−2.44769	−0.212242
$134$	0	0
$135$	−0.367660	−0.0316431
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	−6.00909	−0.509685	−0.254842	−	0.966983i	$-0.582024\pi$
$139$	−6.00909	−0.509685	−0.254842	+	0.966983i	$0.582024\pi$
$140$	0	0
$141$	−28.6584	−2.41347
$142$	0	0
$143$	−3.06256	−0.256104
$144$	0	0
$145$	0.135880	0.0112843
$146$	0	0
$147$	−42.8506	−3.53426
$148$	0	0
$149$	3.60510	0.295342	0.147671	−	0.989037i	$-0.452822\pi$
$149$	3.60510	0.295342	0.147671	+	0.989037i	$0.452822\pi$
$150$	0	0
$151$	15.2216	1.23872	0.619359	−	0.785108i	$-0.287393\pi$
$151$	15.2216	1.23872	0.619359	+	0.785108i	$0.287393\pi$
$152$	0	0
$153$	10.9373	0.884226
$154$	0	0
$155$	−0.558293	−0.0448432
$156$	0	0
$157$	−22.3402	−1.78294	−0.891472	−	0.453076i	$-0.850327\pi$
$157$	−22.3402	−1.78294	−0.891472	+	0.453076i	$0.850327\pi$
$158$	0	0
$159$	−14.9065	−1.18216
$160$	0	0
$161$	15.5466	1.22524
$162$	0	0
$163$	−15.5291	−1.21633	−0.608165	−	0.793810i	$-0.708094\pi$
$163$	−15.5291	−1.21633	−0.608165	+	0.793810i	$0.708094\pi$
$164$	0	0
$165$	0.185842	0.0144678
$166$	0	0
$167$	−13.6732	−1.05806	−0.529032	−	0.848602i	$-0.677445\pi$
$167$	−13.6732	−1.05806	−0.529032	+	0.848602i	$0.677445\pi$
$168$	0	0
$169$	−3.62074	−0.278518
$170$	0	0
$171$	2.58794	0.197905
$172$	0	0
$173$	9.77782	0.743394	0.371697	−	0.928354i	$-0.378776\pi$
$173$	9.77782	0.743394	0.371697	+	0.928354i	$0.378776\pi$
$174$	0	0
$175$	23.5224	1.77813
$176$	0	0
$177$	−28.9784	−2.17815
$178$	0	0
$179$	−5.01666	−0.374963	−0.187481	−	0.982268i	$-0.560032\pi$
$179$	−5.01666	−0.374963	−0.187481	+	0.982268i	$0.560032\pi$
$180$	0	0
$181$	−25.9423	−1.92828	−0.964139	−	0.265399i	$-0.914496\pi$
$181$	−25.9423	−1.92828	−0.964139	+	0.265399i	$0.914496\pi$
$182$	0	0
$183$	−19.6721	−1.45421
$184$	0	0
$185$	−0.392992	−0.0288933
$186$	0	0
$187$	−2.19697	−0.160658
$188$	0	0
$189$	26.3116	1.91388
$190$	0	0
$191$	−5.05233	−0.365574	−0.182787	−	0.983153i	$-0.558512\pi$
$191$	−5.05233	−0.365574	−0.182787	+	0.983153i	$0.558512\pi$
$192$	0	0
$193$	−26.2481	−1.88938	−0.944690	−	0.327965i	$-0.893637\pi$
$193$	−26.2481	−1.88938	−0.944690	+	0.327965i	$0.893637\pi$
$194$	0	0
$195$	−0.569152	−0.0407578
$196$	0	0
$197$	15.5098	1.10502	0.552512	−	0.833505i	$-0.313669\pi$
$197$	15.5098	1.10502	0.552512	+	0.833505i	$0.313669\pi$
$198$	0	0
$199$	13.5410	0.959897	0.479949	−	0.877297i	$-0.340655\pi$
$199$	13.5410	0.959897	0.479949	+	0.877297i	$0.340655\pi$
$200$	0	0
$201$	−6.43563	−0.453934
$202$	0	0
$203$	−9.72428	−0.682511
$204$	0	0
$205$	0.622485	0.0434762
$206$	0	0
$207$	−16.4374	−1.14248
$208$	0	0
$209$	−0.519839	−0.0359580
$210$	0	0
$211$	−13.7578	−0.947126	−0.473563	−	0.880760i	$-0.657032\pi$
$211$	−13.7578	−0.947126	−0.473563	+	0.880760i	$0.657032\pi$
$212$	0	0
$213$	−13.6673	−0.936471
$214$	0	0
$215$	−0.739164	−0.0504106
$216$	0	0
$217$	39.9542	2.71227
$218$	0	0
$219$	−4.83584	−0.326776
$220$	0	0
$221$	6.72834	0.452597
$222$	0	0
$223$	12.8127	0.858003	0.429002	−	0.903304i	$-0.358865\pi$
$223$	12.8127	0.858003	0.429002	+	0.903304i	$0.358865\pi$
$224$	0	0
$225$	−24.8702	−1.65801
$226$	0	0
$227$	−27.5863	−1.83097	−0.915485	−	0.402352i	$-0.868193\pi$
$227$	−27.5863	−1.83097	−0.915485	+	0.402352i	$0.868193\pi$
$228$	0	0
$229$	−20.5867	−1.36041	−0.680205	−	0.733022i	$-0.738109\pi$
$229$	−20.5867	−1.36041	−0.680205	+	0.733022i	$0.738109\pi$
$230$	0	0
$231$	−13.2998	−0.875061
$232$	0	0
$233$	−24.1718	−1.58355	−0.791775	−	0.610813i	$-0.790843\pi$
$233$	−24.1718	−1.58355	−0.791775	+	0.610813i	$0.790843\pi$
$234$	0	0
$235$	0.667549	0.0435461
$236$	0	0
$237$	−16.9931	−1.10382
$238$	0	0
$239$	−29.5737	−1.91296	−0.956482	−	0.291791i	$-0.905749\pi$
$239$	−29.5737	−1.91296	−0.956482	+	0.291791i	$0.905749\pi$
$240$	0	0
$241$	22.0844	1.42258	0.711292	−	0.702897i	$-0.248110\pi$
$241$	22.0844	1.42258	0.711292	+	0.702897i	$0.248110\pi$
$242$	0	0
$243$	14.3663	0.921599
$244$	0	0
$245$	0.998131	0.0637683
$246$	0	0
$247$	1.59204	0.101299
$248$	0	0
$249$	−14.4108	−0.913246
$250$	0	0
$251$	6.18726	0.390537	0.195268	−	0.980750i	$-0.437442\pi$
$251$	6.18726	0.390537	0.195268	+	0.980750i	$0.437442\pi$
$252$	0	0
$253$	3.30178	0.207581
$254$	0	0
$255$	−0.408289	−0.0255681
$256$	0	0
$257$	2.72164	0.169771	0.0848855	−	0.996391i	$-0.472948\pi$
$257$	2.72164	0.169771	0.0848855	+	0.996391i	$0.472948\pi$
$258$	0	0
$259$	28.1244	1.74757
$260$	0	0
$261$	10.2815	0.636406
$262$	0	0
$263$	22.1124	1.36351	0.681756	−	0.731580i	$-0.261217\pi$
$263$	22.1124	1.36351	0.681756	+	0.731580i	$0.261217\pi$
$264$	0	0
$265$	0.347222	0.0213297
$266$	0	0
$267$	−41.1591	−2.51890
$268$	0	0
$269$	25.9959	1.58500	0.792499	−	0.609873i	$-0.208780\pi$
$269$	25.9959	1.58500	0.792499	+	0.609873i	$0.208780\pi$
$270$	0	0
$271$	−12.9571	−0.787087	−0.393544	−	0.919306i	$-0.628751\pi$
$271$	−12.9571	−0.787087	−0.393544	+	0.919306i	$0.628751\pi$
$272$	0	0
$273$	40.7313	2.46517
$274$	0	0
$275$	4.99567	0.301250
$276$	0	0
$277$	25.0377	1.50437	0.752184	−	0.658953i	$-0.229000\pi$
$277$	25.0377	1.50437	0.752184	+	0.658953i	$0.229000\pi$
$278$	0	0
$279$	−42.2435	−2.52905
$280$	0	0
$281$	28.6973	1.71194	0.855970	−	0.517026i	$-0.172961\pi$
$281$	28.6973	1.71194	0.855970	+	0.517026i	$0.172961\pi$
$282$	0	0
$283$	−2.84262	−0.168976	−0.0844881	−	0.996424i	$-0.526926\pi$
$283$	−2.84262	−0.168976	−0.0844881	+	0.996424i	$0.526926\pi$
$284$	0	0
$285$	−0.0966079	−0.00572256
$286$	0	0
$287$	−44.5481	−2.62959
$288$	0	0
$289$	−12.1733	−0.716078
$290$	0	0
$291$	8.63324	0.506090
$292$	0	0
$293$	−10.9960	−0.642391	−0.321196	−	0.947013i	$-0.604085\pi$
$293$	−10.9960	−0.642391	−0.321196	+	0.947013i	$0.604085\pi$
$294$	0	0
$295$	0.675002	0.0393001
$296$	0	0
$297$	5.58803	0.324250
$298$	0	0
$299$	−10.1119	−0.584785
$300$	0	0
$301$	52.8982	3.04900
$302$	0	0
$303$	34.0372	1.95539
$304$	0	0
$305$	0.458229	0.0262381
$306$	0	0
$307$	6.87415	0.392328	0.196164	−	0.980571i	$-0.437151\pi$
$307$	6.87415	0.392328	0.196164	+	0.980571i	$0.437151\pi$
$308$	0	0
$309$	−45.9610	−2.61463
$310$	0	0
$311$	−23.3431	−1.32367	−0.661833	−	0.749652i	$-0.730221\pi$
$311$	−23.3431	−1.32367	−0.661833	+	0.749652i	$0.730221\pi$
$312$	0	0
$313$	−10.5592	−0.596842	−0.298421	−	0.954434i	$-0.596460\pi$
$313$	−10.5592	−0.596842	−0.298421	+	0.954434i	$0.596460\pi$
$314$	0	0
$315$	−1.54227	−0.0868971
$316$	0	0
$317$	10.8117	0.607244	0.303622	−	0.952793i	$-0.401804\pi$
$317$	10.8117	0.607244	0.303622	+	0.952793i	$0.401804\pi$
$318$	0	0
$319$	−2.06524	−0.115631
$320$	0	0
$321$	−12.8764	−0.718688
$322$	0	0
$323$	1.14207	0.0635465
$324$	0	0
$325$	−15.2995	−0.848665
$326$	0	0
$327$	13.3149	0.736318
$328$	0	0
$329$	−47.7731	−2.63382
$330$	0	0
$331$	22.5792	1.24106	0.620532	−	0.784181i	$-0.286917\pi$
$331$	22.5792	1.24106	0.620532	+	0.784181i	$0.286917\pi$
$332$	0	0
$333$	−29.7359	−1.62952
$334$	0	0
$335$	0.149907	0.00819029
$336$	0	0
$337$	−12.4164	−0.676366	−0.338183	−	0.941080i	$-0.609812\pi$
$337$	−12.4164	−0.676366	−0.338183	+	0.941080i	$0.609812\pi$
$338$	0	0
$339$	53.4248	2.90164
$340$	0	0
$341$	8.48545	0.459513
$342$	0	0
$343$	−38.4713	−2.07726
$344$	0	0
$345$	0.613609	0.0330356
$346$	0	0
$347$	34.3777	1.84549	0.922746	−	0.385408i	$-0.125939\pi$
$347$	34.3777	1.84549	0.922746	+	0.385408i	$0.125939\pi$
$348$	0	0
$349$	−20.0540	−1.07347	−0.536733	−	0.843752i	$-0.680342\pi$
$349$	−20.0540	−1.07347	−0.536733	+	0.843752i	$0.680342\pi$
$350$	0	0
$351$	−17.1137	−0.913460
$352$	0	0
$353$	−27.8416	−1.48186	−0.740931	−	0.671582i	$-0.765615\pi$
$353$	−27.8416	−1.48186	−0.740931	+	0.671582i	$0.765615\pi$
$354$	0	0
$355$	0.318358	0.0168967
$356$	0	0
$357$	29.2192	1.54644
$358$	0	0
$359$	−22.8848	−1.20781	−0.603907	−	0.797055i	$-0.706390\pi$
$359$	−22.8848	−1.20781	−0.603907	+	0.797055i	$0.706390\pi$
$360$	0	0
$361$	−18.7298	−0.985777
$362$	0	0
$363$	−2.82460	−0.148253
$364$	0	0
$365$	0.112643	0.00589598
$366$	0	0
$367$	−25.8849	−1.35118	−0.675589	−	0.737278i	$-0.736111\pi$
$367$	−25.8849	−1.35118	−0.675589	+	0.737278i	$0.736111\pi$
$368$	0	0
$369$	47.1006	2.45196
$370$	0	0
$371$	−24.8489	−1.29009
$372$	0	0
$373$	−8.92304	−0.462017	−0.231009	−	0.972952i	$-0.574203\pi$
$373$	−8.92304	−0.462017	−0.231009	+	0.972952i	$0.574203\pi$
$374$	0	0
$375$	1.85762	0.0959269
$376$	0	0
$377$	6.32490	0.325749
$378$	0	0
$379$	32.2869	1.65847	0.829234	−	0.558901i	$-0.188777\pi$
$379$	32.2869	1.65847	0.829234	+	0.558901i	$0.188777\pi$
$380$	0	0
$381$	−30.1643	−1.54536
$382$	0	0
$383$	−31.9415	−1.63213	−0.816067	−	0.577957i	$-0.803850\pi$
$383$	−31.9415	−1.63213	−0.816067	+	0.577957i	$0.803850\pi$
$384$	0	0
$385$	0.309796	0.0157886
$386$	0	0
$387$	−55.9292	−2.84304
$388$	0	0
$389$	22.1158	1.12131	0.560657	−	0.828048i	$-0.310549\pi$
$389$	22.1158	1.12131	0.560657	+	0.828048i	$0.310549\pi$
$390$	0	0
$391$	−7.25390	−0.366845
$392$	0	0
$393$	−32.0027	−1.61432
$394$	0	0
$395$	0.395825	0.0199161
$396$	0	0
$397$	27.7310	1.39178	0.695889	−	0.718149i	$-0.255011\pi$
$397$	27.7310	1.39178	0.695889	+	0.718149i	$0.255011\pi$
$398$	0	0
$399$	6.91374	0.346120
$400$	0	0
$401$	3.76274	0.187902	0.0939511	−	0.995577i	$-0.470050\pi$
$401$	3.76274	0.187902	0.0939511	+	0.995577i	$0.470050\pi$
$402$	0	0
$403$	−25.9872	−1.29451
$404$	0	0
$405$	0.0558521	0.00277531
$406$	0	0
$407$	5.97305	0.296073
$408$	0	0
$409$	−28.6533	−1.41681	−0.708406	−	0.705805i	$-0.750586\pi$
$409$	−28.6533	−1.41681	−0.708406	+	0.705805i	$0.750586\pi$
$410$	0	0
$411$	−2.82460	−0.139327
$412$	0	0
$413$	−48.3065	−2.37701
$414$	0	0
$415$	0.335675	0.0164776
$416$	0	0
$417$	16.9733	0.831184
$418$	0	0
$419$	−13.7177	−0.670155	−0.335077	−	0.942191i	$-0.608762\pi$
$419$	−13.7177	−0.670155	−0.335077	+	0.942191i	$0.608762\pi$
$420$	0	0
$421$	30.4567	1.48437	0.742184	−	0.670196i	$-0.233790\pi$
$421$	30.4567	1.48437	0.742184	+	0.670196i	$0.233790\pi$
$422$	0	0
$423$	50.5104	2.45590
$424$	0	0
$425$	−10.9753	−0.532382
$426$	0	0
$427$	−32.7931	−1.58697
$428$	0	0
$429$	8.65049	0.417650
$430$	0	0
$431$	−11.3431	−0.546377	−0.273189	−	0.961960i	$-0.588078\pi$
$431$	−11.3431	−0.546377	−0.273189	+	0.961960i	$0.588078\pi$
$432$	0	0
$433$	−23.6497	−1.13653	−0.568265	−	0.822845i	$-0.692385\pi$
$433$	−23.6497	−1.13653	−0.568265	+	0.822845i	$0.692385\pi$
$434$	0	0
$435$	−0.383807	−0.0184022
$436$	0	0
$437$	−1.71639	−0.0821061
$438$	0	0
$439$	6.40327	0.305611	0.152806	−	0.988256i	$-0.451169\pi$
$439$	6.40327	0.305611	0.152806	+	0.988256i	$0.451169\pi$
$440$	0	0
$441$	75.5240	3.59638
$442$	0	0
$443$	17.4386	0.828533	0.414267	−	0.910156i	$-0.364038\pi$
$443$	17.4386	0.828533	0.414267	+	0.910156i	$0.364038\pi$
$444$	0	0
$445$	0.958732	0.0454483
$446$	0	0
$447$	−10.1830	−0.481638
$448$	0	0
$449$	12.0074	0.566666	0.283333	−	0.959022i	$-0.408560\pi$
$449$	12.0074	0.566666	0.283333	+	0.959022i	$0.408560\pi$
$450$	0	0
$451$	−9.46109	−0.445505
$452$	0	0
$453$	−42.9949	−2.02008
$454$	0	0
$455$	−0.948767	−0.0444789
$456$	0	0
$457$	−4.55883	−0.213253	−0.106626	−	0.994299i	$-0.534005\pi$
$457$	−4.55883	−0.213253	−0.106626	+	0.994299i	$0.534005\pi$
$458$	0	0
$459$	−12.2767	−0.573029
$460$	0	0
$461$	4.76564	0.221958	0.110979	−	0.993823i	$-0.464601\pi$
$461$	4.76564	0.221958	0.110979	+	0.993823i	$0.464601\pi$
$462$	0	0
$463$	−0.290028	−0.0134788	−0.00673938	−	0.999977i	$-0.502145\pi$
$463$	−0.290028	−0.0134788	−0.00673938	+	0.999977i	$0.502145\pi$
$464$	0	0
$465$	1.57695	0.0731295
$466$	0	0
$467$	−20.3172	−0.940168	−0.470084	−	0.882622i	$-0.655776\pi$
$467$	−20.3172	−0.940168	−0.470084	+	0.882622i	$0.655776\pi$
$468$	0	0
$469$	−10.7281	−0.495377
$470$	0	0
$471$	63.1021	2.90759
$472$	0	0
$473$	11.2345	0.516562
$474$	0	0
$475$	−2.59694	−0.119156
$476$	0	0
$477$	26.2727	1.20294
$478$	0	0
$479$	1.09196	0.0498930	0.0249465	−	0.999689i	$-0.492058\pi$
$479$	1.09196	0.0498930	0.0249465	+	0.999689i	$0.492058\pi$
$480$	0	0
$481$	−18.2928	−0.834081
$482$	0	0
$483$	−43.9129	−1.99810
$484$	0	0
$485$	−0.201097	−0.00913133
$486$	0	0
$487$	−22.8228	−1.03420	−0.517100	−	0.855925i	$-0.672988\pi$
$487$	−22.8228	−1.03420	−0.517100	+	0.855925i	$0.672988\pi$
$488$	0	0
$489$	43.8633	1.98357
$490$	0	0
$491$	16.4596	0.742810	0.371405	−	0.928471i	$-0.378876\pi$
$491$	16.4596	0.742810	0.371405	+	0.928471i	$0.378876\pi$
$492$	0	0
$493$	4.53726	0.204348
$494$	0	0
$495$	−0.327546	−0.0147221
$496$	0	0
$497$	−22.7832	−1.02197
$498$	0	0
$499$	−20.8291	−0.932437	−0.466219	−	0.884670i	$-0.654384\pi$
$499$	−20.8291	−0.932437	−0.466219	+	0.884670i	$0.654384\pi$
$500$	0	0
$501$	38.6213	1.72547
$502$	0	0
$503$	25.1443	1.12113	0.560564	−	0.828111i	$-0.310584\pi$
$503$	25.1443	1.12113	0.560564	+	0.828111i	$0.310584\pi$
$504$	0	0
$505$	−0.792839	−0.0352809
$506$	0	0
$507$	10.2271	0.454203
$508$	0	0
$509$	−7.15511	−0.317145	−0.158572	−	0.987347i	$-0.550689\pi$
$509$	−7.15511	−0.317145	−0.158572	+	0.987347i	$0.550689\pi$
$510$	0	0
$511$	−8.06126	−0.356609
$512$	0	0
$513$	−2.90487	−0.128253
$514$	0	0
$515$	1.07058	0.0471755
$516$	0	0
$517$	−10.1460	−0.446221
$518$	0	0
$519$	−27.6184	−1.21231
$520$	0	0
$521$	−35.3671	−1.54946	−0.774730	−	0.632292i	$-0.782114\pi$
$521$	−35.3671	−1.54946	−0.774730	+	0.632292i	$0.782114\pi$
$522$	0	0
$523$	−30.9797	−1.35465	−0.677324	−	0.735685i	$-0.736860\pi$
$523$	−30.9797	−1.35465	−0.677324	+	0.735685i	$0.736860\pi$
$524$	0	0
$525$	−66.4413	−2.89974
$526$	0	0
$527$	−18.6423	−0.812070
$528$	0	0
$529$	−12.0983	−0.526012
$530$	0	0
$531$	51.0743	2.21644
$532$	0	0
$533$	28.9751	1.25505
$534$	0	0
$535$	0.299933	0.0129672
$536$	0	0
$537$	14.1700	0.611482
$538$	0	0
$539$	−15.1705	−0.653440
$540$	0	0
$541$	−2.62972	−0.113061	−0.0565303	−	0.998401i	$-0.518004\pi$
$541$	−2.62972	−0.113061	−0.0565303	+	0.998401i	$0.518004\pi$
$542$	0	0
$543$	73.2766	3.14460
$544$	0	0
$545$	−0.310149	−0.0132853
$546$	0	0
$547$	−1.83917	−0.0786371	−0.0393186	−	0.999227i	$-0.512519\pi$
$547$	−1.83917	−0.0786371	−0.0393186	+	0.999227i	$0.512519\pi$
$548$	0	0
$549$	34.6721	1.47977
$550$	0	0
$551$	1.07359	0.0457364
$552$	0	0
$553$	−28.3272	−1.20459
$554$	0	0
$555$	1.11004	0.0471187
$556$	0	0
$557$	−5.69739	−0.241406	−0.120703	−	0.992689i	$-0.538515\pi$
$557$	−5.69739	−0.241406	−0.120703	+	0.992689i	$0.538515\pi$
$558$	0	0
$559$	−34.4063	−1.45523
$560$	0	0
$561$	6.20555	0.261999
$562$	0	0
$563$	27.5025	1.15909	0.579546	−	0.814939i	$-0.303230\pi$
$563$	27.5025	1.15909	0.579546	+	0.814939i	$0.303230\pi$
$564$	0	0
$565$	−1.24444	−0.0523540
$566$	0	0
$567$	−3.99706	−0.167861
$568$	0	0
$569$	11.3055	0.473953	0.236977	−	0.971515i	$-0.423844\pi$
$569$	11.3055	0.473953	0.236977	+	0.971515i	$0.423844\pi$
$570$	0	0
$571$	−18.8199	−0.787588	−0.393794	−	0.919199i	$-0.628838\pi$
$571$	−18.8199	−0.787588	−0.393794	+	0.919199i	$0.628838\pi$
$572$	0	0
$573$	14.2708	0.596171
$574$	0	0
$575$	16.4946	0.687872
$576$	0	0
$577$	−1.45416	−0.0605374	−0.0302687	−	0.999542i	$-0.509636\pi$
$577$	−1.45416	−0.0605374	−0.0302687	+	0.999542i	$0.509636\pi$
$578$	0	0
$579$	74.1403	3.08117
$580$	0	0
$581$	−24.0225	−0.996623
$582$	0	0
$583$	−5.27740	−0.218567
$584$	0	0
$585$	1.00313	0.0414743
$586$	0	0
$587$	−13.4922	−0.556882	−0.278441	−	0.960453i	$-0.589818\pi$
$587$	−13.4922	−0.556882	−0.278441	+	0.960453i	$0.589818\pi$
$588$	0	0
$589$	−4.41107	−0.181755
$590$	0	0
$591$	−43.8088	−1.80205
$592$	0	0
$593$	−34.8203	−1.42990	−0.714949	−	0.699177i	$-0.753550\pi$
$593$	−34.8203	−1.42990	−0.714949	+	0.699177i	$0.753550\pi$
$594$	0	0
$595$	−0.680611	−0.0279023
$596$	0	0
$597$	−38.2479	−1.56538
$598$	0	0
$599$	22.6562	0.925707	0.462853	−	0.886435i	$-0.346826\pi$
$599$	22.6562	0.925707	0.462853	+	0.886435i	$0.346826\pi$
$600$	0	0
$601$	25.2913	1.03166	0.515828	−	0.856692i	$-0.327485\pi$
$601$	25.2913	1.03166	0.515828	+	0.856692i	$0.327485\pi$
$602$	0	0
$603$	11.3428	0.461914
$604$	0	0
$605$	0.0657942	0.00267491
$606$	0	0
$607$	0.462889	0.0187881	0.00939404	−	0.999956i	$-0.497010\pi$
$607$	0.462889	0.0187881	0.00939404	+	0.999956i	$0.497010\pi$
$608$	0	0
$609$	27.4672	1.11303
$610$	0	0
$611$	31.0728	1.25707
$612$	0	0
$613$	4.35775	0.176008	0.0880039	−	0.996120i	$-0.471951\pi$
$613$	4.35775	0.176008	0.0880039	+	0.996120i	$0.471951\pi$
$614$	0	0
$615$	−1.75827	−0.0709002
$616$	0	0
$617$	−41.0931	−1.65434	−0.827172	−	0.561949i	$-0.810052\pi$
$617$	−41.0931	−1.65434	−0.827172	+	0.561949i	$0.810052\pi$
$618$	0	0
$619$	23.5459	0.946388	0.473194	−	0.880958i	$-0.343101\pi$
$619$	23.5459	0.946388	0.473194	+	0.880958i	$0.343101\pi$
$620$	0	0
$621$	18.4504	0.740390
$622$	0	0
$623$	−68.6116	−2.74887
$624$	0	0
$625$	24.9351	0.997403
$626$	0	0
$627$	1.46833	0.0586397
$628$	0	0
$629$	−13.1226	−0.523233
$630$	0	0
$631$	11.1293	0.443052	0.221526	−	0.975155i	$-0.428896\pi$
$631$	11.1293	0.443052	0.221526	+	0.975155i	$0.428896\pi$
$632$	0	0
$633$	38.8603	1.54456
$634$	0	0
$635$	0.702625	0.0278828
$636$	0	0
$637$	46.4606	1.84083
$638$	0	0
$639$	24.0887	0.952933
$640$	0	0
$641$	−30.8352	−1.21792	−0.608958	−	0.793203i	$-0.708412\pi$
$641$	−30.8352	−1.21792	−0.608958	+	0.793203i	$0.708412\pi$
$642$	0	0
$643$	−37.0864	−1.46254	−0.731272	−	0.682086i	$-0.761073\pi$
$643$	−37.0864	−1.46254	−0.731272	+	0.682086i	$0.761073\pi$
$644$	0	0
$645$	2.08784	0.0822086
$646$	0	0
$647$	−29.3418	−1.15355	−0.576773	−	0.816904i	$-0.695688\pi$
$647$	−29.3418	−1.15355	−0.576773	+	0.816904i	$0.695688\pi$
$648$	0	0
$649$	−10.2593	−0.402713
$650$	0	0
$651$	−112.855	−4.42312
$652$	0	0
$653$	28.5869	1.11869	0.559347	−	0.828934i	$-0.311052\pi$
$653$	28.5869	1.11869	0.559347	+	0.828934i	$0.311052\pi$
$654$	0	0
$655$	0.745448	0.0291271
$656$	0	0
$657$	8.52316	0.332520
$658$	0	0
$659$	−25.8490	−1.00693	−0.503466	−	0.864015i	$-0.667942\pi$
$659$	−25.8490	−1.00693	−0.503466	+	0.864015i	$0.667942\pi$
$660$	0	0
$661$	−30.5787	−1.18937	−0.594687	−	0.803957i	$-0.702724\pi$
$661$	−30.5787	−1.18937	−0.594687	+	0.803957i	$0.702724\pi$
$662$	0	0
$663$	−19.0049	−0.738088
$664$	0	0
$665$	−0.161044	−0.00624501
$666$	0	0
$667$	−6.81894	−0.264031
$668$	0	0
$669$	−36.1908	−1.39922
$670$	0	0
$671$	−6.96458	−0.268865
$672$	0	0
$673$	23.9784	0.924299	0.462150	−	0.886802i	$-0.347078\pi$
$673$	23.9784	0.924299	0.462150	+	0.886802i	$0.347078\pi$
$674$	0	0
$675$	27.9160	1.07449
$676$	0	0
$677$	15.7425	0.605033	0.302516	−	0.953144i	$-0.402173\pi$
$677$	15.7425	0.605033	0.302516	+	0.953144i	$0.402173\pi$
$678$	0	0
$679$	14.3915	0.552294
$680$	0	0
$681$	77.9203	2.98591
$682$	0	0
$683$	−5.56860	−0.213077	−0.106538	−	0.994309i	$-0.533977\pi$
$683$	−5.56860	−0.213077	−0.106538	+	0.994309i	$0.533977\pi$
$684$	0	0
$685$	0.0657942	0.00251387
$686$	0	0
$687$	58.1492	2.21853
$688$	0	0
$689$	16.1623	0.615736
$690$	0	0
$691$	48.0754	1.82888	0.914438	−	0.404727i	$-0.132633\pi$
$691$	48.0754	1.82888	0.914438	+	0.404727i	$0.132633\pi$
$692$	0	0
$693$	23.4408	0.890443
$694$	0	0
$695$	−0.395363	−0.0149970
$696$	0	0
$697$	20.7857	0.787315
$698$	0	0
$699$	68.2757	2.58242
$700$	0	0
$701$	9.94729	0.375704	0.187852	−	0.982197i	$-0.439847\pi$
$701$	9.94729	0.375704	0.187852	+	0.982197i	$0.439847\pi$
$702$	0	0
$703$	−3.10502	−0.117108
$704$	0	0
$705$	−1.88556	−0.0710142
$706$	0	0
$707$	56.7395	2.13391
$708$	0	0
$709$	−26.0001	−0.976454	−0.488227	−	0.872717i	$-0.662356\pi$
$709$	−26.0001	−0.976454	−0.488227	+	0.872717i	$0.662356\pi$
$710$	0	0
$711$	29.9502	1.12322
$712$	0	0
$713$	28.0171	1.04925
$714$	0	0
$715$	−0.201498	−0.00753562
$716$	0	0
$717$	83.5338	3.11963
$718$	0	0
$719$	15.3946	0.574121	0.287060	−	0.957912i	$-0.407322\pi$
$719$	15.3946	0.574121	0.287060	+	0.957912i	$0.407322\pi$
$720$	0	0
$721$	−76.6162	−2.85334
$722$	0	0
$723$	−62.3796	−2.31992
$724$	0	0
$725$	−10.3172	−0.383173
$726$	0	0
$727$	4.37514	0.162265	0.0811324	−	0.996703i	$-0.474146\pi$
$727$	4.37514	0.162265	0.0811324	+	0.996703i	$0.474146\pi$
$728$	0	0
$729$	−43.1257	−1.59725
$730$	0	0
$731$	−24.6818	−0.912890
$732$	0	0
$733$	−16.0434	−0.592576	−0.296288	−	0.955099i	$-0.595749\pi$
$733$	−16.0434	−0.592576	−0.296288	+	0.955099i	$0.595749\pi$
$734$	0	0
$735$	−2.81932	−0.103992
$736$	0	0
$737$	−2.27842	−0.0839268
$738$	0	0
$739$	49.9248	1.83651	0.918257	−	0.395986i	$-0.129597\pi$
$739$	49.9248	1.83651	0.918257	+	0.395986i	$0.129597\pi$
$740$	0	0
$741$	−4.49686	−0.165196
$742$	0	0
$743$	21.1073	0.774353	0.387176	−	0.922006i	$-0.373450\pi$
$743$	21.1073	0.774353	0.387176	+	0.922006i	$0.373450\pi$
$744$	0	0
$745$	0.237195	0.00869015
$746$	0	0
$747$	25.3990	0.929300
$748$	0	0
$749$	−21.4647	−0.784302
$750$	0	0
$751$	−12.6344	−0.461037	−0.230518	−	0.973068i	$-0.574042\pi$
$751$	−12.6344	−0.461037	−0.230518	+	0.973068i	$0.574042\pi$
$752$	0	0
$753$	−17.4765	−0.636880
$754$	0	0
$755$	1.00149	0.0364481
$756$	0	0
$757$	36.8702	1.34007	0.670034	−	0.742330i	$-0.266279\pi$
$757$	36.8702	1.34007	0.670034	+	0.742330i	$0.266279\pi$
$758$	0	0
$759$	−9.32618	−0.338519
$760$	0	0
$761$	10.3812	0.376317	0.188159	−	0.982139i	$-0.439748\pi$
$761$	10.3812	0.376317	0.188159	+	0.982139i	$0.439748\pi$
$762$	0	0
$763$	22.1958	0.803541
$764$	0	0
$765$	0.719609	0.0260175
$766$	0	0
$767$	31.4197	1.13450
$768$	0	0
$769$	−7.43628	−0.268159	−0.134079	−	0.990971i	$-0.542808\pi$
$769$	−7.43628	−0.268159	−0.134079	+	0.990971i	$0.542808\pi$
$770$	0	0
$771$	−7.68752	−0.276859
$772$	0	0
$773$	35.5758	1.27957	0.639787	−	0.768552i	$-0.279023\pi$
$773$	35.5758	1.27957	0.639787	+	0.768552i	$0.279023\pi$
$774$	0	0
$775$	42.3905	1.52271
$776$	0	0
$777$	−79.4402	−2.84990
$778$	0	0
$779$	4.91824	0.176214
$780$	0	0
$781$	−4.83869	−0.173142
$782$	0	0
$783$	−11.5406	−0.412427
$784$	0	0
$785$	−1.46986	−0.0524614
$786$	0	0
$787$	−53.1674	−1.89521	−0.947606	−	0.319440i	$-0.896505\pi$
$787$	−53.1674	−1.89521	−0.947606	+	0.319440i	$0.896505\pi$
$788$	0	0
$789$	−62.4587	−2.22359
$790$	0	0
$791$	89.0583	3.16655
$792$	0	0
$793$	21.3294	0.757430
$794$	0	0
$795$	−0.980762	−0.0347840
$796$	0	0
$797$	−14.7086	−0.521005	−0.260503	−	0.965473i	$-0.583888\pi$
$797$	−14.7086	−0.521005	−0.260503	+	0.965473i	$0.583888\pi$
$798$	0	0
$799$	22.2905	0.788581
$800$	0	0
$801$	72.5429	2.56318
$802$	0	0
$803$	−1.71205	−0.0604168
$804$	0	0
$805$	1.02288	0.0360516
$806$	0	0
$807$	−73.4279	−2.58479
$808$	0	0
$809$	−52.9934	−1.86315	−0.931575	−	0.363549i	$-0.881565\pi$
$809$	−52.9934	−1.86315	−0.931575	+	0.363549i	$0.881565\pi$
$810$	0	0
$811$	−31.9911	−1.12336	−0.561679	−	0.827355i	$-0.689844\pi$
$811$	−31.9911	−1.12336	−0.561679	+	0.827355i	$0.689844\pi$
$812$	0	0
$813$	36.5986	1.28357
$814$	0	0
$815$	−1.02172	−0.0357894
$816$	0	0
$817$	−5.84012	−0.204320
$818$	0	0
$819$	−71.7889	−2.50851
$820$	0	0
$821$	−20.0170	−0.698598	−0.349299	−	0.937011i	$-0.613580\pi$
$821$	−20.0170	−0.698598	−0.349299	+	0.937011i	$0.613580\pi$
$822$	0	0
$823$	−5.16222	−0.179944	−0.0899719	−	0.995944i	$-0.528678\pi$
$823$	−5.16222	−0.179944	−0.0899719	+	0.995944i	$0.528678\pi$
$824$	0	0
$825$	−14.1108	−0.491273
$826$	0	0
$827$	−53.9489	−1.87599	−0.937994	−	0.346652i	$-0.887319\pi$
$827$	−53.9489	−1.87599	−0.937994	+	0.346652i	$0.887319\pi$
$828$	0	0
$829$	38.2100	1.32709	0.663544	−	0.748137i	$-0.269052\pi$
$829$	38.2100	1.32709	0.663544	+	0.748137i	$0.269052\pi$
$830$	0	0
$831$	−70.7214	−2.45330
$832$	0	0
$833$	33.3291	1.15479
$834$	0	0
$835$	−0.899617	−0.0311325
$836$	0	0
$837$	47.4170	1.63897
$838$	0	0
$839$	−10.6826	−0.368804	−0.184402	−	0.982851i	$-0.559035\pi$
$839$	−10.6826	−0.368804	−0.184402	+	0.982851i	$0.559035\pi$
$840$	0	0
$841$	−24.7348	−0.852924
$842$	0	0
$843$	−81.0584	−2.79180
$844$	0	0
$845$	−0.238223	−0.00819514
$846$	0	0
$847$	−4.70856	−0.161788
$848$	0	0
$849$	8.02926	0.275563
$850$	0	0
$851$	19.7217	0.676050
$852$	0	0
$853$	−41.8024	−1.43129	−0.715643	−	0.698466i	$-0.753866\pi$
$853$	−41.8024	−1.43129	−0.715643	+	0.698466i	$0.753866\pi$
$854$	0	0
$855$	0.170271	0.00582315
$856$	0	0
$857$	22.3220	0.762504	0.381252	−	0.924471i	$-0.375493\pi$
$857$	22.3220	0.762504	0.381252	+	0.924471i	$0.375493\pi$
$858$	0	0
$859$	−28.3039	−0.965718	−0.482859	−	0.875698i	$-0.660402\pi$
$859$	−28.3039	−0.965718	−0.482859	+	0.875698i	$0.660402\pi$
$860$	0	0
$861$	125.830	4.28829
$862$	0	0
$863$	29.3935	1.00057	0.500284	−	0.865861i	$-0.333229\pi$
$863$	29.3935	1.00057	0.500284	+	0.865861i	$0.333229\pi$
$864$	0	0
$865$	0.643324	0.0218737
$866$	0	0
$867$	34.3847	1.16777
$868$	0	0
$869$	−6.01610	−0.204082
$870$	0	0
$871$	6.97780	0.236434
$872$	0	0
$873$	−15.2161	−0.514986
$874$	0	0
$875$	3.09662	0.104685
$876$	0	0
$877$	−7.70909	−0.260317	−0.130159	−	0.991493i	$-0.541549\pi$
$877$	−7.70909	−0.260317	−0.130159	+	0.991493i	$0.541549\pi$
$878$	0	0
$879$	31.0592	1.04760
$880$	0	0
$881$	−22.2493	−0.749599	−0.374799	−	0.927106i	$-0.622288\pi$
$881$	−22.2493	−0.749599	−0.374799	+	0.927106i	$0.622288\pi$
$882$	0	0
$883$	−13.1243	−0.441669	−0.220835	−	0.975311i	$-0.570878\pi$
$883$	−13.1243	−0.441669	−0.220835	+	0.975311i	$0.570878\pi$
$884$	0	0
$885$	−1.90661	−0.0640899
$886$	0	0
$887$	−3.66595	−0.123091	−0.0615454	−	0.998104i	$-0.519603\pi$
$887$	−3.66595	−0.123091	−0.0615454	+	0.998104i	$0.519603\pi$
$888$	0	0
$889$	−50.2833	−1.68645
$890$	0	0
$891$	−0.848892	−0.0284389
$892$	0	0
$893$	5.27430	0.176498
$894$	0	0
$895$	−0.330067	−0.0110329
$896$	0	0
$897$	28.5620	0.953657
$898$	0	0
$899$	−17.5245	−0.584473
$900$	0	0
$901$	11.5943	0.386261
$902$	0	0
$903$	−149.416	−4.97226
$904$	0	0
$905$	−1.70685	−0.0567377
$906$	0	0
$907$	33.1212	1.09977	0.549886	−	0.835240i	$-0.314671\pi$
$907$	33.1212	1.09977	0.549886	+	0.835240i	$0.314671\pi$
$908$	0	0
$909$	−59.9905	−1.98976
$910$	0	0
$911$	25.8451	0.856287	0.428144	−	0.903711i	$-0.359168\pi$
$911$	25.8451	0.856287	0.428144	+	0.903711i	$0.359168\pi$
$912$	0	0
$913$	−5.10189	−0.168848
$914$	0	0
$915$	−1.29431	−0.0427886
$916$	0	0
$917$	−53.3480	−1.76170
$918$	0	0
$919$	−13.6774	−0.451176	−0.225588	−	0.974223i	$-0.572430\pi$
$919$	−13.6774	−0.451176	−0.225588	+	0.974223i	$0.572430\pi$
$920$	0	0
$921$	−19.4167	−0.639802
$922$	0	0
$923$	14.8188	0.487766
$924$	0	0
$925$	29.8394	0.981113
$926$	0	0
$927$	81.0062	2.66059
$928$	0	0
$929$	3.35441	0.110055	0.0550274	−	0.998485i	$-0.482475\pi$
$929$	3.35441	0.110055	0.0550274	+	0.998485i	$0.482475\pi$
$930$	0	0
$931$	7.88622	0.258460
$932$	0	0
$933$	65.9348	2.15861
$934$	0	0
$935$	−0.144548	−0.00472722
$936$	0	0
$937$	10.1300	0.330934	0.165467	−	0.986215i	$-0.447087\pi$
$937$	10.1300	0.330934	0.165467	+	0.986215i	$0.447087\pi$
$938$	0	0
$939$	29.8255	0.973319
$940$	0	0
$941$	3.13453	0.102183	0.0510914	−	0.998694i	$-0.483730\pi$
$941$	3.13453	0.102183	0.0510914	+	0.998694i	$0.483730\pi$
$942$	0	0
$943$	−31.2384	−1.01726
$944$	0	0
$945$	1.73115	0.0563142
$946$	0	0
$947$	−49.2381	−1.60002	−0.800012	−	0.599984i	$-0.795174\pi$
$947$	−49.2381	−1.60002	−0.800012	+	0.599984i	$0.795174\pi$
$948$	0	0
$949$	5.24324	0.170203
$950$	0	0
$951$	−30.5386	−0.990283
$952$	0	0
$953$	37.3472	1.20979	0.604897	−	0.796304i	$-0.293214\pi$
$953$	37.3472	1.20979	0.604897	+	0.796304i	$0.293214\pi$
$954$	0	0
$955$	−0.332414	−0.0107567
$956$	0	0
$957$	5.83346	0.188569
$958$	0	0
$959$	−4.70856	−0.152047
$960$	0	0
$961$	41.0029	1.32268
$962$	0	0
$963$	22.6945	0.731322
$964$	0	0
$965$	−1.72697	−0.0555932
$966$	0	0
$967$	−31.1265	−1.00096	−0.500480	−	0.865748i	$-0.666843\pi$
$967$	−31.1265	−1.00096	−0.500480	+	0.865748i	$0.666843\pi$
$968$	0	0
$969$	−3.22589	−0.103630
$970$	0	0
$971$	18.8222	0.604032	0.302016	−	0.953303i	$-0.402340\pi$
$971$	18.8222	0.604032	0.302016	+	0.953303i	$0.402340\pi$
$972$	0	0
$973$	28.2942	0.907069
$974$	0	0
$975$	43.2150	1.38399
$976$	0	0
$977$	−8.43921	−0.269994	−0.134997	−	0.990846i	$-0.543103\pi$
$977$	−8.43921	−0.269994	−0.134997	+	0.990846i	$0.543103\pi$
$978$	0	0
$979$	−14.5717	−0.465713
$980$	0	0
$981$	−23.4675	−0.749261
$982$	0	0
$983$	36.9340	1.17801	0.589006	−	0.808129i	$-0.299519\pi$
$983$	36.9340	1.17801	0.589006	+	0.808129i	$0.299519\pi$
$984$	0	0
$985$	1.02045	0.0325143
$986$	0	0
$987$	134.940	4.29518
$988$	0	0
$989$	37.0938	1.17951
$990$	0	0
$991$	−35.8238	−1.13798	−0.568991	−	0.822344i	$-0.692666\pi$
$991$	−35.8238	−1.13798	−0.568991	+	0.822344i	$0.692666\pi$
$992$	0	0
$993$	−63.7771	−2.02390
$994$	0	0
$995$	0.890920	0.0282441
$996$	0	0
$997$	−12.8591	−0.407253	−0.203627	−	0.979049i	$-0.565273\pi$
$997$	−12.8591	−0.407253	−0.203627	+	0.979049i	$0.565273\pi$
$998$	0	0
$999$	33.3776	1.05602

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.d.1.5	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.d.1.5	✓	27	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82460 q^{3} +0.0657942 q^{5} -4.70856 q^{7} +4.97835 q^{9} +O(q^{10})\)	\(q-2.82460 q^{3} +0.0657942 q^{5} -4.70856 q^{7} +4.97835 q^{9} -1.00000 q^{11} +3.06256 q^{13} -0.185842 q^{15} +2.19697 q^{17} +0.519839 q^{19} +13.2998 q^{21} -3.30178 q^{23} -4.99567 q^{25} -5.58803 q^{27} +2.06524 q^{29} -8.48545 q^{31} +2.82460 q^{33} -0.309796 q^{35} -5.97305 q^{37} -8.65049 q^{39} +9.46109 q^{41} -11.2345 q^{43} +0.327546 q^{45} +10.1460 q^{47} +15.1705 q^{49} -6.20555 q^{51} +5.27740 q^{53} -0.0657942 q^{55} -1.46833 q^{57} +10.2593 q^{59} +6.96458 q^{61} -23.4408 q^{63} +0.201498 q^{65} +2.27842 q^{67} +9.32618 q^{69} +4.83869 q^{71} +1.71205 q^{73} +14.1108 q^{75} +4.70856 q^{77} +6.01610 q^{79} +0.848892 q^{81} +5.10189 q^{83} +0.144548 q^{85} -5.83346 q^{87} +14.5717 q^{89} -14.4202 q^{91} +23.9680 q^{93} +0.0342024 q^{95} -3.05645 q^{97} -4.97835 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) \) 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) \) 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9 - 27 * q^11 - 6 * q^15 - 21 * q^17 - 3 * q^19 - 4 * q^21 - 44 * q^23 + 38 * q^25 - 18 * q^27 + q^29 - 8 * q^31 + 6 * q^33 - 33 * q^35 + 11 * q^37 - 13 * q^39 - 19 * q^41 - 11 * q^43 + 17 * q^45 - 37 * q^47 + 41 * q^49 - 49 * q^51 - 12 * q^53 - q^55 - 50 * q^57 - 14 * q^59 + 12 * q^61 - 53 * q^63 - 55 * q^65 - 5 * q^67 + 14 * q^69 - 67 * q^71 - 27 * q^73 - 70 * q^75 + 14 * q^77 - 31 * q^79 - 5 * q^81 - 55 * q^83 - 3 * q^85 - 31 * q^87 + 11 * q^89 - 11 * q^91 - 24 * q^93 - 47 * q^95 - q^97 - 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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