LMFDB - Embedded newform 6004.2.a.g.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.21945 q^{3} +1.36200 q^{5} -2.61900 q^{7} -1.51294 q^{9} +O(q^{10})$	$q+1.21945 q^{3} +1.36200 q^{5} -2.61900 q^{7} -1.51294 q^{9} -3.78429 q^{11} +1.85777 q^{13} +1.66089 q^{15} +2.70157 q^{17} +1.00000 q^{19} -3.19374 q^{21} +7.26110 q^{23} -3.14496 q^{25} -5.50331 q^{27} +1.91435 q^{29} +5.73815 q^{31} -4.61475 q^{33} -3.56707 q^{35} -4.70501 q^{37} +2.26546 q^{39} -10.8631 q^{41} -12.3830 q^{43} -2.06062 q^{45} +8.24286 q^{47} -0.140862 q^{49} +3.29443 q^{51} +8.29716 q^{53} -5.15419 q^{55} +1.21945 q^{57} -8.48032 q^{59} -0.427688 q^{61} +3.96238 q^{63} +2.53028 q^{65} -7.29027 q^{67} +8.85456 q^{69} -0.159689 q^{71} +2.28441 q^{73} -3.83512 q^{75} +9.91103 q^{77} -1.00000 q^{79} -2.17220 q^{81} +12.6225 q^{83} +3.67954 q^{85} +2.33446 q^{87} -12.5491 q^{89} -4.86548 q^{91} +6.99740 q^{93} +1.36200 q^{95} -14.6462 q^{97} +5.72539 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * 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
$3$	1.21945	0.704051	0.352025	−	0.935991i	$-0.385493\pi$
$3$	1.21945	0.704051	0.352025	+	0.935991i	$0.385493\pi$
$4$	0	0
$5$	1.36200	0.609105	0.304552	−	0.952496i	$-0.401493\pi$
$5$	1.36200	0.609105	0.304552	+	0.952496i	$0.401493\pi$
$6$	0	0
$7$	−2.61900	−0.989887	−0.494944	−	0.868925i	$-0.664811\pi$
$7$	−2.61900	−0.989887	−0.494944	+	0.868925i	$0.664811\pi$
$8$	0	0
$9$	−1.51294	−0.504313
$10$	0	0
$11$	−3.78429	−1.14100	−0.570502	−	0.821296i	$-0.693252\pi$
$11$	−3.78429	−1.14100	−0.570502	+	0.821296i	$0.693252\pi$
$12$	0	0
$13$	1.85777	0.515252	0.257626	−	0.966245i	$-0.417060\pi$
$13$	1.85777	0.515252	0.257626	+	0.966245i	$0.417060\pi$
$14$	0	0
$15$	1.66089	0.428840
$16$	0	0
$17$	2.70157	0.655227	0.327613	−	0.944812i	$-0.393756\pi$
$17$	2.70157	0.655227	0.327613	+	0.944812i	$0.393756\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.19374	−0.696931
$22$	0	0
$23$	7.26110	1.51404	0.757022	−	0.653389i	$-0.226653\pi$
$23$	7.26110	1.51404	0.757022	+	0.653389i	$0.226653\pi$
$24$	0	0
$25$	−3.14496	−0.628992
$26$	0	0
$27$	−5.50331	−1.05911
$28$	0	0
$29$	1.91435	0.355486	0.177743	−	0.984077i	$-0.443120\pi$
$29$	1.91435	0.355486	0.177743	+	0.984077i	$0.443120\pi$
$30$	0	0
$31$	5.73815	1.03060	0.515301	−	0.857009i	$-0.327680\pi$
$31$	5.73815	1.03060	0.515301	+	0.857009i	$0.327680\pi$
$32$	0	0
$33$	−4.61475	−0.803325
$34$	0	0
$35$	−3.56707	−0.602945
$36$	0	0
$37$	−4.70501	−0.773500	−0.386750	−	0.922185i	$-0.626402\pi$
$37$	−4.70501	−0.773500	−0.386750	+	0.922185i	$0.626402\pi$
$38$	0	0
$39$	2.26546	0.362763
$40$	0	0
$41$	−10.8631	−1.69653	−0.848267	−	0.529569i	$-0.822354\pi$
$41$	−10.8631	−1.69653	−0.848267	+	0.529569i	$0.822354\pi$
$42$	0	0
$43$	−12.3830	−1.88839	−0.944195	−	0.329386i	$-0.893158\pi$
$43$	−12.3830	−1.88839	−0.944195	+	0.329386i	$0.893158\pi$
$44$	0	0
$45$	−2.06062	−0.307179
$46$	0	0
$47$	8.24286	1.20234	0.601172	−	0.799119i	$-0.294701\pi$
$47$	8.24286	1.20234	0.601172	+	0.799119i	$0.294701\pi$
$48$	0	0
$49$	−0.140862	−0.0201231
$50$	0	0
$51$	3.29443	0.461313
$52$	0	0
$53$	8.29716	1.13970	0.569851	−	0.821748i	$-0.307001\pi$
$53$	8.29716	1.13970	0.569851	+	0.821748i	$0.307001\pi$
$54$	0	0
$55$	−5.15419	−0.694991
$56$	0	0
$57$	1.21945	0.161520
$58$	0	0
$59$	−8.48032	−1.10404	−0.552022	−	0.833829i	$-0.686144\pi$
$59$	−8.48032	−1.10404	−0.552022	+	0.833829i	$0.686144\pi$
$60$	0	0
$61$	−0.427688	−0.0547599	−0.0273799	−	0.999625i	$-0.508716\pi$
$61$	−0.427688	−0.0547599	−0.0273799	+	0.999625i	$0.508716\pi$
$62$	0	0
$63$	3.96238	0.499213
$64$	0	0
$65$	2.53028	0.313842
$66$	0	0
$67$	−7.29027	−0.890649	−0.445324	−	0.895369i	$-0.646912\pi$
$67$	−7.29027	−0.890649	−0.445324	+	0.895369i	$0.646912\pi$
$68$	0	0
$69$	8.85456	1.06596
$70$	0	0
$71$	−0.159689	−0.0189516	−0.00947580	−	0.999955i	$-0.503016\pi$
$71$	−0.159689	−0.0189516	−0.00947580	+	0.999955i	$0.503016\pi$
$72$	0	0
$73$	2.28441	0.267370	0.133685	−	0.991024i	$-0.457319\pi$
$73$	2.28441	0.267370	0.133685	+	0.991024i	$0.457319\pi$
$74$	0	0
$75$	−3.83512	−0.442842
$76$	0	0
$77$	9.91103	1.12947
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−2.17220	−0.241356
$82$	0	0
$83$	12.6225	1.38550	0.692750	−	0.721178i	$-0.256399\pi$
$83$	12.6225	1.38550	0.692750	+	0.721178i	$0.256399\pi$
$84$	0	0
$85$	3.67954	0.399102
$86$	0	0
$87$	2.33446	0.250280
$88$	0	0
$89$	−12.5491	−1.33021	−0.665103	−	0.746751i	$-0.731613\pi$
$89$	−12.5491	−1.33021	−0.665103	+	0.746751i	$0.731613\pi$
$90$	0	0
$91$	−4.86548	−0.510041
$92$	0	0
$93$	6.99740	0.725596
$94$	0	0
$95$	1.36200	0.139738
$96$	0	0
$97$	−14.6462	−1.48709	−0.743547	−	0.668684i	$-0.766858\pi$
$97$	−14.6462	−1.48709	−0.743547	+	0.668684i	$0.766858\pi$
$98$	0	0
$99$	5.72539	0.575424
$100$	0	0
$101$	5.65996	0.563187	0.281593	−	0.959534i	$-0.409137\pi$
$101$	5.65996	0.563187	0.281593	+	0.959534i	$0.409137\pi$
$102$	0	0
$103$	−13.5375	−1.33389	−0.666946	−	0.745106i	$-0.732399\pi$
$103$	−13.5375	−1.33389	−0.666946	+	0.745106i	$0.732399\pi$
$104$	0	0
$105$	−4.34987	−0.424504
$106$	0	0
$107$	−8.87228	−0.857715	−0.428858	−	0.903372i	$-0.641084\pi$
$107$	−8.87228	−0.857715	−0.428858	+	0.903372i	$0.641084\pi$
$108$	0	0
$109$	−9.11056	−0.872634	−0.436317	−	0.899793i	$-0.643717\pi$
$109$	−9.11056	−0.872634	−0.436317	+	0.899793i	$0.643717\pi$
$110$	0	0
$111$	−5.73754	−0.544583
$112$	0	0
$113$	−19.9146	−1.87341	−0.936703	−	0.350126i	$-0.886139\pi$
$113$	−19.9146	−1.87341	−0.936703	+	0.350126i	$0.886139\pi$
$114$	0	0
$115$	9.88961	0.922211
$116$	0	0
$117$	−2.81069	−0.259848
$118$	0	0
$119$	−7.07540	−0.648601
$120$	0	0
$121$	3.32082	0.301892
$122$	0	0
$123$	−13.2470	−1.19445
$124$	0	0
$125$	−11.0934	−0.992226
$126$	0	0
$127$	8.55669	0.759283	0.379642	−	0.925134i	$-0.376047\pi$
$127$	8.55669	0.759283	0.379642	+	0.925134i	$0.376047\pi$
$128$	0	0
$129$	−15.1005	−1.32952
$130$	0	0
$131$	16.2350	1.41846	0.709228	−	0.704980i	$-0.249044\pi$
$131$	16.2350	1.41846	0.709228	+	0.704980i	$0.249044\pi$
$132$	0	0
$133$	−2.61900	−0.227096
$134$	0	0
$135$	−7.49550	−0.645110
$136$	0	0
$137$	−11.1743	−0.954686	−0.477343	−	0.878717i	$-0.658400\pi$
$137$	−11.1743	−0.954686	−0.477343	+	0.878717i	$0.658400\pi$
$138$	0	0
$139$	−13.9349	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$139$	−13.9349	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$140$	0	0
$141$	10.0518	0.846511
$142$	0	0
$143$	−7.03032	−0.587905
$144$	0	0
$145$	2.60734	0.216528
$146$	0	0
$147$	−0.171774	−0.0141677
$148$	0	0
$149$	−21.8523	−1.79021	−0.895107	−	0.445852i	$-0.852901\pi$
$149$	−21.8523	−1.79021	−0.895107	+	0.445852i	$0.852901\pi$
$150$	0	0
$151$	5.52135	0.449321	0.224661	−	0.974437i	$-0.427873\pi$
$151$	5.52135	0.449321	0.224661	+	0.974437i	$0.427873\pi$
$152$	0	0
$153$	−4.08731	−0.330439
$154$	0	0
$155$	7.81536	0.627745
$156$	0	0
$157$	−13.8482	−1.10521	−0.552604	−	0.833444i	$-0.686366\pi$
$157$	−13.8482	−1.10521	−0.552604	+	0.833444i	$0.686366\pi$
$158$	0	0
$159$	10.1180	0.802408
$160$	0	0
$161$	−19.0168	−1.49873
$162$	0	0
$163$	5.21181	0.408220	0.204110	−	0.978948i	$-0.434570\pi$
$163$	5.21181	0.408220	0.204110	+	0.978948i	$0.434570\pi$
$164$	0	0
$165$	−6.28529	−0.489309
$166$	0	0
$167$	−4.54694	−0.351853	−0.175926	−	0.984403i	$-0.556292\pi$
$167$	−4.54694	−0.351853	−0.175926	+	0.984403i	$0.556292\pi$
$168$	0	0
$169$	−9.54870	−0.734516
$170$	0	0
$171$	−1.51294	−0.115697
$172$	0	0
$173$	−17.0432	−1.29577	−0.647885	−	0.761738i	$-0.724346\pi$
$173$	−17.0432	−1.29577	−0.647885	+	0.761738i	$0.724346\pi$
$174$	0	0
$175$	8.23663	0.622631
$176$	0	0
$177$	−10.3413	−0.777303
$178$	0	0
$179$	11.7846	0.880821	0.440410	−	0.897797i	$-0.354833\pi$
$179$	11.7846	0.880821	0.440410	+	0.897797i	$0.354833\pi$
$180$	0	0
$181$	24.5013	1.82117	0.910583	−	0.413325i	$-0.135633\pi$
$181$	24.5013	1.82117	0.910583	+	0.413325i	$0.135633\pi$
$182$	0	0
$183$	−0.521545	−0.0385537
$184$	0	0
$185$	−6.40823	−0.471142
$186$	0	0
$187$	−10.2235	−0.747617
$188$	0	0
$189$	14.4131	1.04840
$190$	0	0
$191$	−3.46352	−0.250612	−0.125306	−	0.992118i	$-0.539991\pi$
$191$	−3.46352	−0.250612	−0.125306	+	0.992118i	$0.539991\pi$
$192$	0	0
$193$	−13.3788	−0.963029	−0.481515	−	0.876438i	$-0.659913\pi$
$193$	−13.3788	−0.963029	−0.481515	+	0.876438i	$0.659913\pi$
$194$	0	0
$195$	3.08555	0.220961
$196$	0	0
$197$	7.98328	0.568785	0.284393	−	0.958708i	$-0.408208\pi$
$197$	7.98328	0.568785	0.284393	+	0.958708i	$0.408208\pi$
$198$	0	0
$199$	1.35809	0.0962722	0.0481361	−	0.998841i	$-0.484672\pi$
$199$	1.35809	0.0962722	0.0481361	+	0.998841i	$0.484672\pi$
$200$	0	0
$201$	−8.89013	−0.627062
$202$	0	0
$203$	−5.01367	−0.351891
$204$	0	0
$205$	−14.7956	−1.03337
$206$	0	0
$207$	−10.9856	−0.763552
$208$	0	0
$209$	−3.78429	−0.261764
$210$	0	0
$211$	5.30977	0.365540	0.182770	−	0.983156i	$-0.441494\pi$
$211$	5.30977	0.365540	0.182770	+	0.983156i	$0.441494\pi$
$212$	0	0
$213$	−0.194733	−0.0133429
$214$	0	0
$215$	−16.8656	−1.15023
$216$	0	0
$217$	−15.0282	−1.02018
$218$	0	0
$219$	2.78573	0.188242
$220$	0	0
$221$	5.01889	0.337607
$222$	0	0
$223$	7.59874	0.508849	0.254425	−	0.967093i	$-0.418114\pi$
$223$	7.59874	0.508849	0.254425	+	0.967093i	$0.418114\pi$
$224$	0	0
$225$	4.75813	0.317209
$226$	0	0
$227$	13.9932	0.928764	0.464382	−	0.885635i	$-0.346277\pi$
$227$	13.9932	0.928764	0.464382	+	0.885635i	$0.346277\pi$
$228$	0	0
$229$	4.43286	0.292932	0.146466	−	0.989216i	$-0.453210\pi$
$229$	4.43286	0.292932	0.146466	+	0.989216i	$0.453210\pi$
$230$	0	0
$231$	12.0860	0.795201
$232$	0	0
$233$	−14.6083	−0.957018	−0.478509	−	0.878083i	$-0.658823\pi$
$233$	−14.6083	−0.957018	−0.478509	+	0.878083i	$0.658823\pi$
$234$	0	0
$235$	11.2268	0.732354
$236$	0	0
$237$	−1.21945	−0.0792119
$238$	0	0
$239$	−9.23501	−0.597363	−0.298682	−	0.954353i	$-0.596547\pi$
$239$	−9.23501	−0.597363	−0.298682	+	0.954353i	$0.596547\pi$
$240$	0	0
$241$	12.7048	0.818391	0.409196	−	0.912447i	$-0.365809\pi$
$241$	12.7048	0.818391	0.409196	+	0.912447i	$0.365809\pi$
$242$	0	0
$243$	13.8610	0.889186
$244$	0	0
$245$	−0.191854	−0.0122571
$246$	0	0
$247$	1.85777	0.118207
$248$	0	0
$249$	15.3925	0.975461
$250$	0	0
$251$	−18.8283	−1.18843	−0.594217	−	0.804305i	$-0.702538\pi$
$251$	−18.8283	−1.18843	−0.594217	+	0.804305i	$0.702538\pi$
$252$	0	0
$253$	−27.4781	−1.72753
$254$	0	0
$255$	4.48701	0.280988
$256$	0	0
$257$	11.4559	0.714601	0.357300	−	0.933989i	$-0.383697\pi$
$257$	11.4559	0.714601	0.357300	+	0.933989i	$0.383697\pi$
$258$	0	0
$259$	12.3224	0.765677
$260$	0	0
$261$	−2.89629	−0.179276
$262$	0	0
$263$	−14.5254	−0.895675	−0.447837	−	0.894115i	$-0.647806\pi$
$263$	−14.5254	−0.895675	−0.447837	+	0.894115i	$0.647806\pi$
$264$	0	0
$265$	11.3007	0.694198
$266$	0	0
$267$	−15.3031	−0.936533
$268$	0	0
$269$	11.6789	0.712076	0.356038	−	0.934472i	$-0.384127\pi$
$269$	11.6789	0.712076	0.356038	+	0.934472i	$0.384127\pi$
$270$	0	0
$271$	12.1117	0.735734	0.367867	−	0.929878i	$-0.380088\pi$
$271$	12.1117	0.735734	0.367867	+	0.929878i	$0.380088\pi$
$272$	0	0
$273$	−5.93322	−0.359095
$274$	0	0
$275$	11.9014	0.717683
$276$	0	0
$277$	−5.98590	−0.359658	−0.179829	−	0.983698i	$-0.557554\pi$
$277$	−5.98590	−0.359658	−0.179829	+	0.983698i	$0.557554\pi$
$278$	0	0
$279$	−8.68147	−0.519746
$280$	0	0
$281$	26.0410	1.55348	0.776739	−	0.629822i	$-0.216872\pi$
$281$	26.0410	1.55348	0.776739	+	0.629822i	$0.216872\pi$
$282$	0	0
$283$	−3.60003	−0.213999	−0.107000	−	0.994259i	$-0.534124\pi$
$283$	−3.60003	−0.213999	−0.107000	+	0.994259i	$0.534124\pi$
$284$	0	0
$285$	1.66089	0.0983827
$286$	0	0
$287$	28.4504	1.67938
$288$	0	0
$289$	−9.70152	−0.570678
$290$	0	0
$291$	−17.8603	−1.04699
$292$	0	0
$293$	−17.8937	−1.04536	−0.522680	−	0.852529i	$-0.675068\pi$
$293$	−17.8937	−1.04536	−0.522680	+	0.852529i	$0.675068\pi$
$294$	0	0
$295$	−11.5502	−0.672478
$296$	0	0
$297$	20.8261	1.20845
$298$	0	0
$299$	13.4894	0.780114
$300$	0	0
$301$	32.4310	1.86929
$302$	0	0
$303$	6.90204	0.396512
$304$	0	0
$305$	−0.582511	−0.0333545
$306$	0	0
$307$	−6.87629	−0.392451	−0.196225	−	0.980559i	$-0.562868\pi$
$307$	−6.87629	−0.392451	−0.196225	+	0.980559i	$0.562868\pi$
$308$	0	0
$309$	−16.5083	−0.939127
$310$	0	0
$311$	6.53075	0.370325	0.185162	−	0.982708i	$-0.440719\pi$
$311$	6.53075	0.370325	0.185162	+	0.982708i	$0.440719\pi$
$312$	0	0
$313$	14.5647	0.823248	0.411624	−	0.911354i	$-0.364962\pi$
$313$	14.5647	0.823248	0.411624	+	0.911354i	$0.364962\pi$
$314$	0	0
$315$	5.39676	0.304073
$316$	0	0
$317$	29.5927	1.66209	0.831046	−	0.556203i	$-0.187742\pi$
$317$	29.5927	1.66209	0.831046	+	0.556203i	$0.187742\pi$
$318$	0	0
$319$	−7.24444	−0.405611
$320$	0	0
$321$	−10.8193	−0.603875
$322$	0	0
$323$	2.70157	0.150319
$324$	0	0
$325$	−5.84260	−0.324089
$326$	0	0
$327$	−11.1099	−0.614378
$328$	0	0
$329$	−21.5880	−1.19019
$330$	0	0
$331$	−3.87299	−0.212879	−0.106439	−	0.994319i	$-0.533945\pi$
$331$	−3.87299	−0.212879	−0.106439	+	0.994319i	$0.533945\pi$
$332$	0	0
$333$	7.11840	0.390086
$334$	0	0
$335$	−9.92935	−0.542498
$336$	0	0
$337$	−4.66535	−0.254138	−0.127069	−	0.991894i	$-0.540557\pi$
$337$	−4.66535	−0.254138	−0.127069	+	0.991894i	$0.540557\pi$
$338$	0	0
$339$	−24.2849	−1.31897
$340$	0	0
$341$	−21.7148	−1.17592
$342$	0	0
$343$	18.7019	1.00981
$344$	0	0
$345$	12.0599	0.649283
$346$	0	0
$347$	23.6391	1.26901	0.634507	−	0.772917i	$-0.281203\pi$
$347$	23.6391	1.26901	0.634507	+	0.772917i	$0.281203\pi$
$348$	0	0
$349$	−33.9618	−1.81794	−0.908968	−	0.416866i	$-0.863129\pi$
$349$	−33.9618	−1.81794	−0.908968	+	0.416866i	$0.863129\pi$
$350$	0	0
$351$	−10.2239	−0.545709
$352$	0	0
$353$	13.2591	0.705709	0.352854	−	0.935678i	$-0.385211\pi$
$353$	13.2591	0.705709	0.352854	+	0.935678i	$0.385211\pi$
$354$	0	0
$355$	−0.217496	−0.0115435
$356$	0	0
$357$	−8.62810	−0.456648
$358$	0	0
$359$	−0.700947	−0.0369946	−0.0184973	−	0.999829i	$-0.505888\pi$
$359$	−0.700947	−0.0369946	−0.0184973	+	0.999829i	$0.505888\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	4.04957	0.212547
$364$	0	0
$365$	3.11137	0.162856
$366$	0	0
$367$	−3.59649	−0.187735	−0.0938675	−	0.995585i	$-0.529923\pi$
$367$	−3.59649	−0.187735	−0.0938675	+	0.995585i	$0.529923\pi$
$368$	0	0
$369$	16.4352	0.855584
$370$	0	0
$371$	−21.7302	−1.12818
$372$	0	0
$373$	7.36913	0.381559	0.190780	−	0.981633i	$-0.438898\pi$
$373$	7.36913	0.381559	0.190780	+	0.981633i	$0.438898\pi$
$374$	0	0
$375$	−13.5279	−0.698577
$376$	0	0
$377$	3.55641	0.183165
$378$	0	0
$379$	−23.6465	−1.21464	−0.607319	−	0.794458i	$-0.707755\pi$
$379$	−23.6465	−1.21464	−0.607319	+	0.794458i	$0.707755\pi$
$380$	0	0
$381$	10.4345	0.534574
$382$	0	0
$383$	−15.7699	−0.805804	−0.402902	−	0.915243i	$-0.631998\pi$
$383$	−15.7699	−0.805804	−0.402902	+	0.915243i	$0.631998\pi$
$384$	0	0
$385$	13.4988	0.687963
$386$	0	0
$387$	18.7347	0.952340
$388$	0	0
$389$	11.6307	0.589699	0.294849	−	0.955544i	$-0.404731\pi$
$389$	11.6307	0.589699	0.294849	+	0.955544i	$0.404731\pi$
$390$	0	0
$391$	19.6164	0.992043
$392$	0	0
$393$	19.7977	0.998664
$394$	0	0
$395$	−1.36200	−0.0685296
$396$	0	0
$397$	27.2316	1.36672	0.683358	−	0.730084i	$-0.260519\pi$
$397$	27.2316	1.36672	0.683358	+	0.730084i	$0.260519\pi$
$398$	0	0
$399$	−3.19374	−0.159887
$400$	0	0
$401$	18.2105	0.909391	0.454695	−	0.890647i	$-0.349748\pi$
$401$	18.2105	0.909391	0.454695	+	0.890647i	$0.349748\pi$
$402$	0	0
$403$	10.6601	0.531020
$404$	0	0
$405$	−2.95854	−0.147011
$406$	0	0
$407$	17.8051	0.882567
$408$	0	0
$409$	8.91520	0.440828	0.220414	−	0.975406i	$-0.429259\pi$
$409$	8.91520	0.440828	0.220414	+	0.975406i	$0.429259\pi$
$410$	0	0
$411$	−13.6265	−0.672147
$412$	0	0
$413$	22.2099	1.09288
$414$	0	0
$415$	17.1918	0.843914
$416$	0	0
$417$	−16.9930	−0.832150
$418$	0	0
$419$	−3.03648	−0.148342	−0.0741710	−	0.997246i	$-0.523631\pi$
$419$	−3.03648	−0.148342	−0.0741710	+	0.997246i	$0.523631\pi$
$420$	0	0
$421$	−16.7166	−0.814717	−0.407359	−	0.913268i	$-0.633550\pi$
$421$	−16.7166	−0.814717	−0.407359	+	0.913268i	$0.633550\pi$
$422$	0	0
$423$	−12.4709	−0.606358
$424$	0	0
$425$	−8.49632	−0.412132
$426$	0	0
$427$	1.12011	0.0542061
$428$	0	0
$429$	−8.57313	−0.413915
$430$	0	0
$431$	6.04079	0.290974	0.145487	−	0.989360i	$-0.453525\pi$
$431$	6.04079	0.290974	0.145487	+	0.989360i	$0.453525\pi$
$432$	0	0
$433$	−7.42407	−0.356778	−0.178389	−	0.983960i	$-0.557089\pi$
$433$	−7.42407	−0.356778	−0.178389	+	0.983960i	$0.557089\pi$
$434$	0	0
$435$	3.17953	0.152447
$436$	0	0
$437$	7.26110	0.347346
$438$	0	0
$439$	20.3642	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$439$	20.3642	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$440$	0	0
$441$	0.213115	0.0101483
$442$	0	0
$443$	−7.73863	−0.367673	−0.183837	−	0.982957i	$-0.558852\pi$
$443$	−7.73863	−0.367673	−0.183837	+	0.982957i	$0.558852\pi$
$444$	0	0
$445$	−17.0919	−0.810235
$446$	0	0
$447$	−26.6479	−1.26040
$448$	0	0
$449$	34.4893	1.62765	0.813826	−	0.581109i	$-0.197381\pi$
$449$	34.4893	1.62765	0.813826	+	0.581109i	$0.197381\pi$
$450$	0	0
$451$	41.1091	1.93575
$452$	0	0
$453$	6.73302	0.316345
$454$	0	0
$455$	−6.62678	−0.310668
$456$	0	0
$457$	−35.5876	−1.66472	−0.832360	−	0.554235i	$-0.813011\pi$
$457$	−35.5876	−1.66472	−0.832360	+	0.554235i	$0.813011\pi$
$458$	0	0
$459$	−14.8676	−0.693959
$460$	0	0
$461$	−8.84719	−0.412055	−0.206027	−	0.978546i	$-0.566054\pi$
$461$	−8.84719	−0.412055	−0.206027	+	0.978546i	$0.566054\pi$
$462$	0	0
$463$	11.8800	0.552112	0.276056	−	0.961142i	$-0.410972\pi$
$463$	11.8800	0.552112	0.276056	+	0.961142i	$0.410972\pi$
$464$	0	0
$465$	9.53045	0.441964
$466$	0	0
$467$	20.8715	0.965819	0.482910	−	0.875670i	$-0.339580\pi$
$467$	20.8715	0.965819	0.482910	+	0.875670i	$0.339580\pi$
$468$	0	0
$469$	19.0932	0.881642
$470$	0	0
$471$	−16.8872	−0.778123
$472$	0	0
$473$	46.8608	2.15466
$474$	0	0
$475$	−3.14496	−0.144301
$476$	0	0
$477$	−12.5531	−0.574766
$478$	0	0
$479$	−27.8227	−1.27125	−0.635627	−	0.771996i	$-0.719258\pi$
$479$	−27.8227	−1.27125	−0.635627	+	0.771996i	$0.719258\pi$
$480$	0	0
$481$	−8.74082	−0.398547
$482$	0	0
$483$	−23.1901	−1.05518
$484$	0	0
$485$	−19.9481	−0.905795
$486$	0	0
$487$	14.6047	0.661803	0.330902	−	0.943665i	$-0.392647\pi$
$487$	14.6047	0.661803	0.330902	+	0.943665i	$0.392647\pi$
$488$	0	0
$489$	6.35555	0.287408
$490$	0	0
$491$	34.4556	1.55496	0.777479	−	0.628908i	$-0.216498\pi$
$491$	34.4556	1.55496	0.777479	+	0.628908i	$0.216498\pi$
$492$	0	0
$493$	5.17175	0.232924
$494$	0	0
$495$	7.79798	0.350493
$496$	0	0
$497$	0.418225	0.0187599
$498$	0	0
$499$	−4.14844	−0.185710	−0.0928548	−	0.995680i	$-0.529599\pi$
$499$	−4.14844	−0.185710	−0.0928548	+	0.995680i	$0.529599\pi$
$500$	0	0
$501$	−5.54477	−0.247722
$502$	0	0
$503$	0.439866	0.0196126	0.00980632	−	0.999952i	$-0.496879\pi$
$503$	0.439866	0.0196126	0.00980632	+	0.999952i	$0.496879\pi$
$504$	0	0
$505$	7.70886	0.343040
$506$	0	0
$507$	−11.6442	−0.517136
$508$	0	0
$509$	−31.3792	−1.39086	−0.695429	−	0.718595i	$-0.744786\pi$
$509$	−31.3792	−1.39086	−0.695429	+	0.718595i	$0.744786\pi$
$510$	0	0
$511$	−5.98286	−0.264666
$512$	0	0
$513$	−5.50331	−0.242977
$514$	0	0
$515$	−18.4381	−0.812479
$516$	0	0
$517$	−31.1933	−1.37188
$518$	0	0
$519$	−20.7833	−0.912288
$520$	0	0
$521$	17.3052	0.758153	0.379076	−	0.925365i	$-0.376242\pi$
$521$	17.3052	0.758153	0.379076	+	0.925365i	$0.376242\pi$
$522$	0	0
$523$	4.61804	0.201933	0.100966	−	0.994890i	$-0.467807\pi$
$523$	4.61804	0.201933	0.100966	+	0.994890i	$0.467807\pi$
$524$	0	0
$525$	10.0442	0.438364
$526$	0	0
$527$	15.5020	0.675279
$528$	0	0
$529$	29.7236	1.29233
$530$	0	0
$531$	12.8302	0.556784
$532$	0	0
$533$	−20.1811	−0.874142
$534$	0	0
$535$	−12.0840	−0.522438
$536$	0	0
$537$	14.3707	0.620142
$538$	0	0
$539$	0.533061	0.0229606
$540$	0	0
$541$	18.0702	0.776899	0.388449	−	0.921470i	$-0.373011\pi$
$541$	18.0702	0.776899	0.388449	+	0.921470i	$0.373011\pi$
$542$	0	0
$543$	29.8781	1.28219
$544$	0	0
$545$	−12.4086	−0.531525
$546$	0	0
$547$	−19.3446	−0.827114	−0.413557	−	0.910478i	$-0.635714\pi$
$547$	−19.3446	−0.827114	−0.413557	+	0.910478i	$0.635714\pi$
$548$	0	0
$549$	0.647066	0.0276161
$550$	0	0
$551$	1.91435	0.0815540
$552$	0	0
$553$	2.61900	0.111371
$554$	0	0
$555$	−7.81452	−0.331708
$556$	0	0
$557$	−43.4593	−1.84143	−0.920714	−	0.390237i	$-0.872393\pi$
$557$	−43.4593	−1.84143	−0.920714	+	0.390237i	$0.872393\pi$
$558$	0	0
$559$	−23.0047	−0.972997
$560$	0	0
$561$	−12.4671	−0.526360
$562$	0	0
$563$	−3.97092	−0.167354	−0.0836771	−	0.996493i	$-0.526666\pi$
$563$	−3.97092	−0.167354	−0.0836771	+	0.996493i	$0.526666\pi$
$564$	0	0
$565$	−27.1236	−1.14110
$566$	0	0
$567$	5.68898	0.238915
$568$	0	0
$569$	−17.6084	−0.738184	−0.369092	−	0.929393i	$-0.620331\pi$
$569$	−17.6084	−0.738184	−0.369092	+	0.929393i	$0.620331\pi$
$570$	0	0
$571$	−9.73458	−0.407380	−0.203690	−	0.979035i	$-0.565293\pi$
$571$	−9.73458	−0.407380	−0.203690	+	0.979035i	$0.565293\pi$
$572$	0	0
$573$	−4.22360	−0.176443
$574$	0	0
$575$	−22.8359	−0.952321
$576$	0	0
$577$	−1.86715	−0.0777305	−0.0388653	−	0.999244i	$-0.512374\pi$
$577$	−1.86715	−0.0777305	−0.0388653	+	0.999244i	$0.512374\pi$
$578$	0	0
$579$	−16.3148	−0.678021
$580$	0	0
$581$	−33.0583	−1.37149
$582$	0	0
$583$	−31.3988	−1.30041
$584$	0	0
$585$	−3.82815	−0.158275
$586$	0	0
$587$	9.36420	0.386502	0.193251	−	0.981149i	$-0.438097\pi$
$587$	9.36420	0.386502	0.193251	+	0.981149i	$0.438097\pi$
$588$	0	0
$589$	5.73815	0.236436
$590$	0	0
$591$	9.73522	0.400454
$592$	0	0
$593$	0.510412	0.0209601	0.0104801	−	0.999945i	$-0.496664\pi$
$593$	0.510412	0.0209601	0.0104801	+	0.999945i	$0.496664\pi$
$594$	0	0
$595$	−9.63669	−0.395066
$596$	0	0
$597$	1.65612	0.0677805
$598$	0	0
$599$	2.11801	0.0865396	0.0432698	−	0.999063i	$-0.486222\pi$
$599$	2.11801	0.0865396	0.0432698	+	0.999063i	$0.486222\pi$
$600$	0	0
$601$	1.74605	0.0712228	0.0356114	−	0.999366i	$-0.488662\pi$
$601$	1.74605	0.0712228	0.0356114	+	0.999366i	$0.488662\pi$
$602$	0	0
$603$	11.0297	0.449166
$604$	0	0
$605$	4.52295	0.183884
$606$	0	0
$607$	0.971749	0.0394421	0.0197210	−	0.999806i	$-0.493722\pi$
$607$	0.971749	0.0394421	0.0197210	+	0.999806i	$0.493722\pi$
$608$	0	0
$609$	−6.11393	−0.247749
$610$	0	0
$611$	15.3133	0.619510
$612$	0	0
$613$	−1.53615	−0.0620446	−0.0310223	−	0.999519i	$-0.509876\pi$
$613$	−1.53615	−0.0620446	−0.0310223	+	0.999519i	$0.509876\pi$
$614$	0	0
$615$	−18.0425	−0.727542
$616$	0	0
$617$	−35.3033	−1.42126	−0.710630	−	0.703566i	$-0.751590\pi$
$617$	−35.3033	−1.42126	−0.710630	+	0.703566i	$0.751590\pi$
$618$	0	0
$619$	20.4109	0.820381	0.410191	−	0.912000i	$-0.365462\pi$
$619$	20.4109	0.820381	0.410191	+	0.912000i	$0.365462\pi$
$620$	0	0
$621$	−39.9601	−1.60354
$622$	0	0
$623$	32.8662	1.31675
$624$	0	0
$625$	0.615553	0.0246221
$626$	0	0
$627$	−4.61475	−0.184295
$628$	0	0
$629$	−12.7109	−0.506818
$630$	0	0
$631$	39.1700	1.55933	0.779667	−	0.626194i	$-0.215388\pi$
$631$	39.1700	1.55933	0.779667	+	0.626194i	$0.215388\pi$
$632$	0	0
$633$	6.47501	0.257358
$634$	0	0
$635$	11.6542	0.462483
$636$	0	0
$637$	−0.261688	−0.0103685
$638$	0	0
$639$	0.241600	0.00955754
$640$	0	0
$641$	16.3103	0.644218	0.322109	−	0.946703i	$-0.395608\pi$
$641$	16.3103	0.644218	0.322109	+	0.946703i	$0.395608\pi$
$642$	0	0
$643$	−36.1924	−1.42729	−0.713644	−	0.700508i	$-0.752957\pi$
$643$	−36.1924	−1.42729	−0.713644	+	0.700508i	$0.752957\pi$
$644$	0	0
$645$	−20.5668	−0.809818
$646$	0	0
$647$	−29.4837	−1.15912	−0.579561	−	0.814929i	$-0.696776\pi$
$647$	−29.4837	−1.15912	−0.579561	+	0.814929i	$0.696776\pi$
$648$	0	0
$649$	32.0920	1.25972
$650$	0	0
$651$	−18.3262	−0.718259
$652$	0	0
$653$	34.0928	1.33415	0.667076	−	0.744989i	$-0.267546\pi$
$653$	34.0928	1.33415	0.667076	+	0.744989i	$0.267546\pi$
$654$	0	0
$655$	22.1120	0.863988
$656$	0	0
$657$	−3.45617	−0.134838
$658$	0	0
$659$	0.611377	0.0238159	0.0119079	−	0.999929i	$-0.496209\pi$
$659$	0.611377	0.0238159	0.0119079	+	0.999929i	$0.496209\pi$
$660$	0	0
$661$	4.42674	0.172180	0.0860901	−	0.996287i	$-0.472563\pi$
$661$	4.42674	0.172180	0.0860901	+	0.996287i	$0.472563\pi$
$662$	0	0
$663$	6.12029	0.237692
$664$	0	0
$665$	−3.56707	−0.138325
$666$	0	0
$667$	13.9003	0.538221
$668$	0	0
$669$	9.26630	0.358256
$670$	0	0
$671$	1.61849	0.0624813
$672$	0	0
$673$	−22.0602	−0.850359	−0.425179	−	0.905109i	$-0.639789\pi$
$673$	−22.0602	−0.850359	−0.425179	+	0.905109i	$0.639789\pi$
$674$	0	0
$675$	17.3077	0.666173
$676$	0	0
$677$	−44.4958	−1.71011	−0.855056	−	0.518535i	$-0.826478\pi$
$677$	−44.4958	−1.71011	−0.855056	+	0.518535i	$0.826478\pi$
$678$	0	0
$679$	38.3583	1.47205
$680$	0	0
$681$	17.0641	0.653897
$682$	0	0
$683$	−24.5295	−0.938594	−0.469297	−	0.883040i	$-0.655493\pi$
$683$	−24.5295	−0.938594	−0.469297	+	0.883040i	$0.655493\pi$
$684$	0	0
$685$	−15.2194	−0.581503
$686$	0	0
$687$	5.40566	0.206239
$688$	0	0
$689$	15.4142	0.587233
$690$	0	0
$691$	−5.65506	−0.215129	−0.107564	−	0.994198i	$-0.534305\pi$
$691$	−5.65506	−0.215129	−0.107564	+	0.994198i	$0.534305\pi$
$692$	0	0
$693$	−14.9948	−0.569604
$694$	0	0
$695$	−18.9794	−0.719929
$696$	0	0
$697$	−29.3475	−1.11161
$698$	0	0
$699$	−17.8140	−0.673789
$700$	0	0
$701$	30.8542	1.16535	0.582674	−	0.812706i	$-0.302006\pi$
$701$	30.8542	1.16535	0.582674	+	0.812706i	$0.302006\pi$
$702$	0	0
$703$	−4.70501	−0.177453
$704$	0	0
$705$	13.6905	0.515614
$706$	0	0
$707$	−14.8234	−0.557492
$708$	0	0
$709$	6.09169	0.228778	0.114389	−	0.993436i	$-0.463509\pi$
$709$	6.09169	0.228778	0.114389	+	0.993436i	$0.463509\pi$
$710$	0	0
$711$	1.51294	0.0567396
$712$	0	0
$713$	41.6653	1.56038
$714$	0	0
$715$	−9.57529	−0.358095
$716$	0	0
$717$	−11.2616	−0.420574
$718$	0	0
$719$	14.3430	0.534905	0.267452	−	0.963571i	$-0.413818\pi$
$719$	14.3430	0.534905	0.267452	+	0.963571i	$0.413818\pi$
$720$	0	0
$721$	35.4547	1.32040
$722$	0	0
$723$	15.4929	0.576189
$724$	0	0
$725$	−6.02055	−0.223598
$726$	0	0
$727$	18.7363	0.694891	0.347446	−	0.937700i	$-0.387049\pi$
$727$	18.7363	0.694891	0.347446	+	0.937700i	$0.387049\pi$
$728$	0	0
$729$	23.4195	0.867387
$730$	0	0
$731$	−33.4536	−1.23732
$732$	0	0
$733$	30.1935	1.11522	0.557611	−	0.830102i	$-0.311718\pi$
$733$	30.1935	1.11522	0.557611	+	0.830102i	$0.311718\pi$
$734$	0	0
$735$	−0.233956	−0.00862961
$736$	0	0
$737$	27.5885	1.01623
$738$	0	0
$739$	2.96383	0.109026	0.0545132	−	0.998513i	$-0.482639\pi$
$739$	2.96383	0.109026	0.0545132	+	0.998513i	$0.482639\pi$
$740$	0	0
$741$	2.26546	0.0832236
$742$	0	0
$743$	17.6841	0.648767	0.324383	−	0.945926i	$-0.394843\pi$
$743$	17.6841	0.648767	0.324383	+	0.945926i	$0.394843\pi$
$744$	0	0
$745$	−29.7629	−1.09043
$746$	0	0
$747$	−19.0971	−0.698725
$748$	0	0
$749$	23.2365	0.849042
$750$	0	0
$751$	−8.39421	−0.306309	−0.153155	−	0.988202i	$-0.548943\pi$
$751$	−8.39421	−0.306309	−0.153155	+	0.988202i	$0.548943\pi$
$752$	0	0
$753$	−22.9602	−0.836717
$754$	0	0
$755$	7.52007	0.273684
$756$	0	0
$757$	37.5428	1.36451	0.682257	−	0.731112i	$-0.260998\pi$
$757$	37.5428	1.36451	0.682257	+	0.731112i	$0.260998\pi$
$758$	0	0
$759$	−33.5082	−1.21627
$760$	0	0
$761$	−22.1452	−0.802765	−0.401382	−	0.915911i	$-0.631470\pi$
$761$	−22.1452	−0.802765	−0.401382	+	0.915911i	$0.631470\pi$
$762$	0	0
$763$	23.8605	0.863809
$764$	0	0
$765$	−5.56691	−0.201272
$766$	0	0
$767$	−15.7545	−0.568861
$768$	0	0
$769$	51.0082	1.83940	0.919701	−	0.392619i	$-0.128431\pi$
$769$	51.0082	1.83940	0.919701	+	0.392619i	$0.128431\pi$
$770$	0	0
$771$	13.9699	0.503115
$772$	0	0
$773$	−20.6703	−0.743460	−0.371730	−	0.928341i	$-0.621235\pi$
$773$	−20.6703	−0.743460	−0.371730	+	0.928341i	$0.621235\pi$
$774$	0	0
$775$	−18.0463	−0.648240
$776$	0	0
$777$	15.0266	0.539076
$778$	0	0
$779$	−10.8631	−0.389211
$780$	0	0
$781$	0.604309	0.0216239
$782$	0	0
$783$	−10.5353	−0.376499
$784$	0	0
$785$	−18.8613	−0.673188
$786$	0	0
$787$	−9.92806	−0.353897	−0.176948	−	0.984220i	$-0.556623\pi$
$787$	−9.92806	−0.353897	−0.176948	+	0.984220i	$0.556623\pi$
$788$	0	0
$789$	−17.7130	−0.630600
$790$	0	0
$791$	52.1562	1.85446
$792$	0	0
$793$	−0.794545	−0.0282151
$794$	0	0
$795$	13.7807	0.488750
$796$	0	0
$797$	21.3227	0.755288	0.377644	−	0.925951i	$-0.376734\pi$
$797$	21.3227	0.755288	0.377644	+	0.925951i	$0.376734\pi$
$798$	0	0
$799$	22.2687	0.787809
$800$	0	0
$801$	18.9861	0.670840
$802$	0	0
$803$	−8.64487	−0.305071
$804$	0	0
$805$	−25.9009	−0.912885
$806$	0	0
$807$	14.2419	0.501337
$808$	0	0
$809$	25.8046	0.907242	0.453621	−	0.891195i	$-0.350132\pi$
$809$	25.8046	0.907242	0.453621	+	0.891195i	$0.350132\pi$
$810$	0	0
$811$	3.73107	0.131016	0.0655078	−	0.997852i	$-0.479133\pi$
$811$	3.73107	0.131016	0.0655078	+	0.997852i	$0.479133\pi$
$812$	0	0
$813$	14.7696	0.517994
$814$	0	0
$815$	7.09848	0.248649
$816$	0	0
$817$	−12.3830	−0.433227
$818$	0	0
$819$	7.36118	0.257220
$820$	0	0
$821$	32.6861	1.14075	0.570376	−	0.821384i	$-0.306798\pi$
$821$	32.6861	1.14075	0.570376	+	0.821384i	$0.306798\pi$
$822$	0	0
$823$	7.20042	0.250991	0.125495	−	0.992094i	$-0.459948\pi$
$823$	7.20042	0.250991	0.125495	+	0.992094i	$0.459948\pi$
$824$	0	0
$825$	14.5132	0.505285
$826$	0	0
$827$	12.4961	0.434531	0.217265	−	0.976113i	$-0.430286\pi$
$827$	12.4961	0.434531	0.217265	+	0.976113i	$0.430286\pi$
$828$	0	0
$829$	0.797994	0.0277155	0.0138577	−	0.999904i	$-0.495589\pi$
$829$	0.797994	0.0277155	0.0138577	+	0.999904i	$0.495589\pi$
$830$	0	0
$831$	−7.29952	−0.253217
$832$	0	0
$833$	−0.380548	−0.0131852
$834$	0	0
$835$	−6.19292	−0.214315
$836$	0	0
$837$	−31.5788	−1.09152
$838$	0	0
$839$	−32.3364	−1.11638	−0.558188	−	0.829714i	$-0.688503\pi$
$839$	−32.3364	−1.11638	−0.558188	+	0.829714i	$0.688503\pi$
$840$	0	0
$841$	−25.3353	−0.873630
$842$	0	0
$843$	31.7558	1.09373
$844$	0	0
$845$	−13.0053	−0.447397
$846$	0	0
$847$	−8.69720	−0.298839
$848$	0	0
$849$	−4.39006	−0.150666
$850$	0	0
$851$	−34.1636	−1.17111
$852$	0	0
$853$	−33.9277	−1.16166	−0.580831	−	0.814024i	$-0.697272\pi$
$853$	−33.9277	−1.16166	−0.580831	+	0.814024i	$0.697272\pi$
$854$	0	0
$855$	−2.06062	−0.0704718
$856$	0	0
$857$	−15.4278	−0.527003	−0.263501	−	0.964659i	$-0.584877\pi$
$857$	−15.4278	−0.527003	−0.263501	+	0.964659i	$0.584877\pi$
$858$	0	0
$859$	−47.6354	−1.62530	−0.812649	−	0.582754i	$-0.801975\pi$
$859$	−47.6354	−1.62530	−0.812649	+	0.582754i	$0.801975\pi$
$860$	0	0
$861$	34.6939	1.18237
$862$	0	0
$863$	0.806285	0.0274463	0.0137231	−	0.999906i	$-0.495632\pi$
$863$	0.806285	0.0274463	0.0137231	+	0.999906i	$0.495632\pi$
$864$	0	0
$865$	−23.2128	−0.789259
$866$	0	0
$867$	−11.8305	−0.401786
$868$	0	0
$869$	3.78429	0.128373
$870$	0	0
$871$	−13.5436	−0.458908
$872$	0	0
$873$	22.1588	0.749960
$874$	0	0
$875$	29.0536	0.982192
$876$	0	0
$877$	−51.0019	−1.72221	−0.861106	−	0.508426i	$-0.830228\pi$
$877$	−51.0019	−1.72221	−0.861106	+	0.508426i	$0.830228\pi$
$878$	0	0
$879$	−21.8205	−0.735986
$880$	0	0
$881$	−39.4980	−1.33072	−0.665362	−	0.746521i	$-0.731723\pi$
$881$	−39.4980	−1.33072	−0.665362	+	0.746521i	$0.731723\pi$
$882$	0	0
$883$	33.7284	1.13505	0.567525	−	0.823356i	$-0.307901\pi$
$883$	33.7284	1.13505	0.567525	+	0.823356i	$0.307901\pi$
$884$	0	0
$885$	−14.0849	−0.473459
$886$	0	0
$887$	0.829776	0.0278611	0.0139306	−	0.999903i	$-0.495566\pi$
$887$	0.829776	0.0278611	0.0139306	+	0.999903i	$0.495566\pi$
$888$	0	0
$889$	−22.4099	−0.751605
$890$	0	0
$891$	8.22023	0.275388
$892$	0	0
$893$	8.24286	0.275837
$894$	0	0
$895$	16.0506	0.536512
$896$	0	0
$897$	16.4497	0.549240
$898$	0	0
$899$	10.9848	0.366365
$900$	0	0
$901$	22.4153	0.746763
$902$	0	0
$903$	39.5481	1.31608
$904$	0	0
$905$	33.3707	1.10928
$906$	0	0
$907$	16.7705	0.556856	0.278428	−	0.960457i	$-0.410187\pi$
$907$	16.7705	0.556856	0.278428	+	0.960457i	$0.410187\pi$
$908$	0	0
$909$	−8.56317	−0.284022
$910$	0	0
$911$	−29.3158	−0.971275	−0.485638	−	0.874160i	$-0.661413\pi$
$911$	−29.3158	−0.971275	−0.485638	+	0.874160i	$0.661413\pi$
$912$	0	0
$913$	−47.7671	−1.58086
$914$	0	0
$915$	−0.710344	−0.0234832
$916$	0	0
$917$	−42.5193	−1.40411
$918$	0	0
$919$	36.8420	1.21530	0.607652	−	0.794203i	$-0.292111\pi$
$919$	36.8420	1.21530	0.607652	+	0.794203i	$0.292111\pi$
$920$	0	0
$921$	−8.38530	−0.276305
$922$	0	0
$923$	−0.296665	−0.00976484
$924$	0	0
$925$	14.7971	0.486525
$926$	0	0
$927$	20.4814	0.672699
$928$	0	0
$929$	−53.9870	−1.77126	−0.885628	−	0.464396i	$-0.846271\pi$
$929$	−53.9870	−1.77126	−0.885628	+	0.464396i	$0.846271\pi$
$930$	0	0
$931$	−0.140862	−0.00461656
$932$	0	0
$933$	7.96393	0.260727
$934$	0	0
$935$	−13.9244	−0.455377
$936$	0	0
$937$	12.8031	0.418260	0.209130	−	0.977888i	$-0.432937\pi$
$937$	12.8031	0.418260	0.209130	+	0.977888i	$0.432937\pi$
$938$	0	0
$939$	17.7610	0.579608
$940$	0	0
$941$	16.5443	0.539327	0.269664	−	0.962955i	$-0.413087\pi$
$941$	16.5443	0.539327	0.269664	+	0.962955i	$0.413087\pi$
$942$	0	0
$943$	−78.8782	−2.56863
$944$	0	0
$945$	19.6307	0.638586
$946$	0	0
$947$	51.9125	1.68693	0.843465	−	0.537184i	$-0.180512\pi$
$947$	51.9125	1.68693	0.843465	+	0.537184i	$0.180512\pi$
$948$	0	0
$949$	4.24390	0.137763
$950$	0	0
$951$	36.0869	1.17020
$952$	0	0
$953$	19.3129	0.625605	0.312803	−	0.949818i	$-0.398732\pi$
$953$	19.3129	0.625605	0.312803	+	0.949818i	$0.398732\pi$
$954$	0	0
$955$	−4.71732	−0.152649
$956$	0	0
$957$	−8.83425	−0.285571
$958$	0	0
$959$	29.2655	0.945031
$960$	0	0
$961$	1.92640	0.0621419
$962$	0	0
$963$	13.4232	0.432557
$964$	0	0
$965$	−18.2220	−0.586586
$966$	0	0
$967$	−23.1426	−0.744215	−0.372107	−	0.928190i	$-0.621365\pi$
$967$	−23.1426	−0.744215	−0.372107	+	0.928190i	$0.621365\pi$
$968$	0	0
$969$	3.29443	0.105832
$970$	0	0
$971$	34.8990	1.11996	0.559981	−	0.828505i	$-0.310808\pi$
$971$	34.8990	1.11996	0.559981	+	0.828505i	$0.310808\pi$
$972$	0	0
$973$	36.4956	1.16999
$974$	0	0
$975$	−7.12476	−0.228175
$976$	0	0
$977$	28.4296	0.909545	0.454772	−	0.890608i	$-0.349721\pi$
$977$	28.4296	0.909545	0.454772	+	0.890608i	$0.349721\pi$
$978$	0	0
$979$	47.4896	1.51777
$980$	0	0
$981$	13.7837	0.440080
$982$	0	0
$983$	10.6863	0.340839	0.170419	−	0.985372i	$-0.445488\pi$
$983$	10.6863	0.340839	0.170419	+	0.985372i	$0.445488\pi$
$984$	0	0
$985$	10.8732	0.346450
$986$	0	0
$987$	−26.3255	−0.837951
$988$	0	0
$989$	−89.9143	−2.85911
$990$	0	0
$991$	−5.47077	−0.173785	−0.0868924	−	0.996218i	$-0.527694\pi$
$991$	−5.47077	−0.173785	−0.0868924	+	0.996218i	$0.527694\pi$
$992$	0	0
$993$	−4.72293	−0.149878
$994$	0	0
$995$	1.84971	0.0586398
$996$	0	0
$997$	55.1101	1.74536	0.872678	−	0.488296i	$-0.162381\pi$
$997$	55.1101	1.74536	0.872678	+	0.488296i	$0.162381\pi$
$998$	0	0
$999$	25.8931	0.819223

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.19	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.19	✓	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.21945 q^{3} +1.36200 q^{5} -2.61900 q^{7} -1.51294 q^{9} +O(q^{10})\)	\(q+1.21945 q^{3} +1.36200 q^{5} -2.61900 q^{7} -1.51294 q^{9} -3.78429 q^{11} +1.85777 q^{13} +1.66089 q^{15} +2.70157 q^{17} +1.00000 q^{19} -3.19374 q^{21} +7.26110 q^{23} -3.14496 q^{25} -5.50331 q^{27} +1.91435 q^{29} +5.73815 q^{31} -4.61475 q^{33} -3.56707 q^{35} -4.70501 q^{37} +2.26546 q^{39} -10.8631 q^{41} -12.3830 q^{43} -2.06062 q^{45} +8.24286 q^{47} -0.140862 q^{49} +3.29443 q^{51} +8.29716 q^{53} -5.15419 q^{55} +1.21945 q^{57} -8.48032 q^{59} -0.427688 q^{61} +3.96238 q^{63} +2.53028 q^{65} -7.29027 q^{67} +8.85456 q^{69} -0.159689 q^{71} +2.28441 q^{73} -3.83512 q^{75} +9.91103 q^{77} -1.00000 q^{79} -2.17220 q^{81} +12.6225 q^{83} +3.67954 q^{85} +2.33446 q^{87} -12.5491 q^{89} -4.86548 q^{91} +6.99740 q^{93} +1.36200 q^{95} -14.6462 q^{97} +5.72539 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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