LMFDB - Embedded newform 6004.2.a.g.1.6

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			1.6
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)

$f(q)$	$=$	$q-2.00223 q^{3} -2.46425 q^{5} +4.06602 q^{7} +1.00893 q^{9} +O(q^{10})$	$q-2.00223 q^{3} -2.46425 q^{5} +4.06602 q^{7} +1.00893 q^{9} -3.25056 q^{11} +4.74938 q^{13} +4.93400 q^{15} -4.01211 q^{17} +1.00000 q^{19} -8.14112 q^{21} -2.85930 q^{23} +1.07253 q^{25} +3.98658 q^{27} -3.94076 q^{29} +2.73532 q^{31} +6.50837 q^{33} -10.0197 q^{35} +2.47454 q^{37} -9.50936 q^{39} +7.55017 q^{41} +2.10768 q^{43} -2.48626 q^{45} -12.7138 q^{47} +9.53253 q^{49} +8.03317 q^{51} +13.6739 q^{53} +8.01019 q^{55} -2.00223 q^{57} +6.69652 q^{59} -14.4098 q^{61} +4.10234 q^{63} -11.7037 q^{65} +2.77431 q^{67} +5.72498 q^{69} -11.2617 q^{71} -9.87501 q^{73} -2.14745 q^{75} -13.2168 q^{77} -1.00000 q^{79} -11.0089 q^{81} -13.5101 q^{83} +9.88684 q^{85} +7.89032 q^{87} -1.17737 q^{89} +19.3111 q^{91} -5.47675 q^{93} -2.46425 q^{95} +7.33863e-5 q^{97} -3.27959 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.00223	−1.15599	−0.577994	−	0.816041i	$-0.696164\pi$
$3$	−2.00223	−1.15599	−0.577994	+	0.816041i	$0.696164\pi$
$4$	0	0
$5$	−2.46425	−1.10205	−0.551023	−	0.834490i	$-0.685762\pi$
$5$	−2.46425	−1.10205	−0.551023	+	0.834490i	$0.685762\pi$
$6$	0	0
$7$	4.06602	1.53681	0.768406	−	0.639963i	$-0.221050\pi$
$7$	4.06602	1.53681	0.768406	+	0.639963i	$0.221050\pi$
$8$	0	0
$9$	1.00893	0.336310
$10$	0	0
$11$	−3.25056	−0.980080	−0.490040	−	0.871700i	$-0.663018\pi$
$11$	−3.25056	−0.980080	−0.490040	+	0.871700i	$0.663018\pi$
$12$	0	0
$13$	4.74938	1.31724	0.658620	−	0.752475i	$-0.271140\pi$
$13$	4.74938	1.31724	0.658620	+	0.752475i	$0.271140\pi$
$14$	0	0
$15$	4.93400	1.27395
$16$	0	0
$17$	−4.01211	−0.973079	−0.486540	−	0.873659i	$-0.661741\pi$
$17$	−4.01211	−0.973079	−0.486540	+	0.873659i	$0.661741\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−8.14112	−1.77654
$22$	0	0
$23$	−2.85930	−0.596205	−0.298102	−	0.954534i	$-0.596354\pi$
$23$	−2.85930	−0.596205	−0.298102	+	0.954534i	$0.596354\pi$
$24$	0	0
$25$	1.07253	0.214506
$26$	0	0
$27$	3.98658	0.767218
$28$	0	0
$29$	−3.94076	−0.731782	−0.365891	−	0.930658i	$-0.619236\pi$
$29$	−3.94076	−0.731782	−0.365891	+	0.930658i	$0.619236\pi$
$30$	0	0
$31$	2.73532	0.491279	0.245639	−	0.969361i	$-0.421002\pi$
$31$	2.73532	0.491279	0.245639	+	0.969361i	$0.421002\pi$
$32$	0	0
$33$	6.50837	1.13296
$34$	0	0
$35$	−10.0197	−1.69364
$36$	0	0
$37$	2.47454	0.406812	0.203406	−	0.979094i	$-0.434799\pi$
$37$	2.47454	0.406812	0.203406	+	0.979094i	$0.434799\pi$
$38$	0	0
$39$	−9.50936	−1.52272
$40$	0	0
$41$	7.55017	1.17914	0.589569	−	0.807718i	$-0.299298\pi$
$41$	7.55017	1.17914	0.589569	+	0.807718i	$0.299298\pi$
$42$	0	0
$43$	2.10768	0.321418	0.160709	−	0.987002i	$-0.448622\pi$
$43$	2.10768	0.321418	0.160709	+	0.987002i	$0.448622\pi$
$44$	0	0
$45$	−2.48626	−0.370630
$46$	0	0
$47$	−12.7138	−1.85450	−0.927252	−	0.374437i	$-0.877836\pi$
$47$	−12.7138	−1.85450	−0.927252	+	0.374437i	$0.877836\pi$
$48$	0	0
$49$	9.53253	1.36179
$50$	0	0
$51$	8.03317	1.12487
$52$	0	0
$53$	13.6739	1.87825	0.939125	−	0.343576i	$-0.111638\pi$
$53$	13.6739	1.87825	0.939125	+	0.343576i	$0.111638\pi$
$54$	0	0
$55$	8.01019	1.08009
$56$	0	0
$57$	−2.00223	−0.265202
$58$	0	0
$59$	6.69652	0.871812	0.435906	−	0.899992i	$-0.356428\pi$
$59$	6.69652	0.871812	0.435906	+	0.899992i	$0.356428\pi$
$60$	0	0
$61$	−14.4098	−1.84498	−0.922492	−	0.386016i	$-0.873851\pi$
$61$	−14.4098	−1.84498	−0.922492	+	0.386016i	$0.873851\pi$
$62$	0	0
$63$	4.10234	0.516846
$64$	0	0
$65$	−11.7037	−1.45166
$66$	0	0
$67$	2.77431	0.338936	0.169468	−	0.985536i	$-0.445795\pi$
$67$	2.77431	0.338936	0.169468	+	0.985536i	$0.445795\pi$
$68$	0	0
$69$	5.72498	0.689206
$70$	0	0
$71$	−11.2617	−1.33651	−0.668257	−	0.743931i	$-0.732959\pi$
$71$	−11.2617	−1.33651	−0.668257	+	0.743931i	$0.732959\pi$
$72$	0	0
$73$	−9.87501	−1.15578	−0.577891	−	0.816114i	$-0.696124\pi$
$73$	−9.87501	−1.15578	−0.577891	+	0.816114i	$0.696124\pi$
$74$	0	0
$75$	−2.14745	−0.247967
$76$	0	0
$77$	−13.2168	−1.50620
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−11.0089	−1.22321
$82$	0	0
$83$	−13.5101	−1.48293	−0.741463	−	0.670994i	$-0.765868\pi$
$83$	−13.5101	−1.48293	−0.741463	+	0.670994i	$0.765868\pi$
$84$	0	0
$85$	9.88684	1.07238
$86$	0	0
$87$	7.89032	0.845932
$88$	0	0
$89$	−1.17737	−0.124801	−0.0624007	−	0.998051i	$-0.519876\pi$
$89$	−1.17737	−0.124801	−0.0624007	+	0.998051i	$0.519876\pi$
$90$	0	0
$91$	19.3111	2.02435
$92$	0	0
$93$	−5.47675	−0.567913
$94$	0	0
$95$	−2.46425	−0.252827
$96$	0	0
$97$	7.33863e−5 0	7.45125e−6 0	3.72562e−6	−	1.00000i	$-0.499999\pi$
$97$	7.33863e−5 0	7.45125e−6 0	3.72562e−6		1.00000i	$0.499999\pi$
$98$	0	0
$99$	−3.27959	−0.329611
$100$	0	0
$101$	17.9605	1.78714	0.893568	−	0.448928i	$-0.148194\pi$
$101$	17.9605	1.78714	0.893568	+	0.448928i	$0.148194\pi$
$102$	0	0
$103$	9.84993	0.970542	0.485271	−	0.874364i	$-0.338721\pi$
$103$	9.84993	0.970542	0.485271	+	0.874364i	$0.338721\pi$
$104$	0	0
$105$	20.0618	1.95783
$106$	0	0
$107$	14.9143	1.44182	0.720908	−	0.693030i	$-0.243725\pi$
$107$	14.9143	1.44182	0.720908	+	0.693030i	$0.243725\pi$
$108$	0	0
$109$	3.56298	0.341272	0.170636	−	0.985334i	$-0.445418\pi$
$109$	3.56298	0.341272	0.170636	+	0.985334i	$0.445418\pi$
$110$	0	0
$111$	−4.95461	−0.470271
$112$	0	0
$113$	11.6812	1.09888	0.549438	−	0.835534i	$-0.314842\pi$
$113$	11.6812	1.09888	0.549438	+	0.835534i	$0.314842\pi$
$114$	0	0
$115$	7.04603	0.657045
$116$	0	0
$117$	4.79180	0.443002
$118$	0	0
$119$	−16.3133	−1.49544
$120$	0	0
$121$	−0.433870	−0.0394427
$122$	0	0
$123$	−15.1172	−1.36307
$124$	0	0
$125$	9.67827	0.865651
$126$	0	0
$127$	15.1253	1.34216	0.671079	−	0.741386i	$-0.265831\pi$
$127$	15.1253	1.34216	0.671079	+	0.741386i	$0.265831\pi$
$128$	0	0
$129$	−4.22007	−0.371556
$130$	0	0
$131$	16.2713	1.42163	0.710816	−	0.703378i	$-0.248326\pi$
$131$	16.2713	1.42163	0.710816	+	0.703378i	$0.248326\pi$
$132$	0	0
$133$	4.06602	0.352569
$134$	0	0
$135$	−9.82393	−0.845510
$136$	0	0
$137$	−5.54441	−0.473691	−0.236846	−	0.971547i	$-0.576114\pi$
$137$	−5.54441	−0.473691	−0.236846	+	0.971547i	$0.576114\pi$
$138$	0	0
$139$	12.8476	1.08972	0.544858	−	0.838528i	$-0.316583\pi$
$139$	12.8476	1.08972	0.544858	+	0.838528i	$0.316583\pi$
$140$	0	0
$141$	25.4561	2.14379
$142$	0	0
$143$	−15.4381	−1.29100
$144$	0	0
$145$	9.71103	0.806457
$146$	0	0
$147$	−19.0863	−1.57421
$148$	0	0
$149$	0.322118	0.0263889	0.0131945	−	0.999913i	$-0.495800\pi$
$149$	0.322118	0.0263889	0.0131945	+	0.999913i	$0.495800\pi$
$150$	0	0
$151$	14.6173	1.18954	0.594770	−	0.803896i	$-0.297243\pi$
$151$	14.6173	1.18954	0.594770	+	0.803896i	$0.297243\pi$
$152$	0	0
$153$	−4.04794	−0.327257
$154$	0	0
$155$	−6.74052	−0.541412
$156$	0	0
$157$	−8.46486	−0.675569	−0.337785	−	0.941223i	$-0.609678\pi$
$157$	−8.46486	−0.675569	−0.337785	+	0.941223i	$0.609678\pi$
$158$	0	0
$159$	−27.3782	−2.17124
$160$	0	0
$161$	−11.6260	−0.916255
$162$	0	0
$163$	−14.4811	−1.13424	−0.567122	−	0.823634i	$-0.691943\pi$
$163$	−14.4811	−1.13424	−0.567122	+	0.823634i	$0.691943\pi$
$164$	0	0
$165$	−16.0383	−1.24858
$166$	0	0
$167$	−17.2913	−1.33804	−0.669022	−	0.743242i	$-0.733287\pi$
$167$	−17.2913	−1.33804	−0.669022	+	0.743242i	$0.733287\pi$
$168$	0	0
$169$	9.55660	0.735123
$170$	0	0
$171$	1.00893	0.0771549
$172$	0	0
$173$	−9.49577	−0.721950	−0.360975	−	0.932575i	$-0.617556\pi$
$173$	−9.49577	−0.721950	−0.360975	+	0.932575i	$0.617556\pi$
$174$	0	0
$175$	4.36093	0.329655
$176$	0	0
$177$	−13.4080	−1.00781
$178$	0	0
$179$	−18.1435	−1.35611	−0.678054	−	0.735012i	$-0.737176\pi$
$179$	−18.1435	−1.35611	−0.678054	+	0.735012i	$0.737176\pi$
$180$	0	0
$181$	12.0309	0.894247	0.447123	−	0.894472i	$-0.352448\pi$
$181$	12.0309	0.894247	0.447123	+	0.894472i	$0.352448\pi$
$182$	0	0
$183$	28.8517	2.13278
$184$	0	0
$185$	−6.09790	−0.448326
$186$	0	0
$187$	13.0416	0.953696
$188$	0	0
$189$	16.2095	1.17907
$190$	0	0
$191$	−11.5961	−0.839062	−0.419531	−	0.907741i	$-0.637805\pi$
$191$	−11.5961	−0.839062	−0.419531	+	0.907741i	$0.637805\pi$
$192$	0	0
$193$	−20.3110	−1.46201	−0.731007	−	0.682370i	$-0.760950\pi$
$193$	−20.3110	−1.46201	−0.731007	+	0.682370i	$0.760950\pi$
$194$	0	0
$195$	23.4334	1.67810
$196$	0	0
$197$	13.4712	0.959781	0.479890	−	0.877329i	$-0.340676\pi$
$197$	13.4712	0.959781	0.479890	+	0.877329i	$0.340676\pi$
$198$	0	0
$199$	−17.9671	−1.27366	−0.636829	−	0.771005i	$-0.719754\pi$
$199$	−17.9671	−1.27366	−0.636829	+	0.771005i	$0.719754\pi$
$200$	0	0
$201$	−5.55482	−0.391807
$202$	0	0
$203$	−16.0232	−1.12461
$204$	0	0
$205$	−18.6055	−1.29946
$206$	0	0
$207$	−2.88484	−0.200510
$208$	0	0
$209$	−3.25056	−0.224846
$210$	0	0
$211$	−0.281244	−0.0193616	−0.00968081	−	0.999953i	$-0.503082\pi$
$211$	−0.281244	−0.0193616	−0.00968081	+	0.999953i	$0.503082\pi$
$212$	0	0
$213$	22.5485	1.54500
$214$	0	0
$215$	−5.19386	−0.354218
$216$	0	0
$217$	11.1219	0.755003
$218$	0	0
$219$	19.7721	1.33607
$220$	0	0
$221$	−19.0550	−1.28178
$222$	0	0
$223$	−26.6025	−1.78143	−0.890716	−	0.454560i	$-0.849797\pi$
$223$	−26.6025	−1.78143	−0.890716	+	0.454560i	$0.849797\pi$
$224$	0	0
$225$	1.08211	0.0721406
$226$	0	0
$227$	−23.1329	−1.53539	−0.767693	−	0.640818i	$-0.778595\pi$
$227$	−23.1329	−1.53539	−0.767693	+	0.640818i	$0.778595\pi$
$228$	0	0
$229$	−1.77613	−0.117370	−0.0586851	−	0.998277i	$-0.518691\pi$
$229$	−1.77613	−0.117370	−0.0586851	+	0.998277i	$0.518691\pi$
$230$	0	0
$231$	26.4632	1.74115
$232$	0	0
$233$	−19.3010	−1.26445	−0.632226	−	0.774784i	$-0.717859\pi$
$233$	−19.3010	−1.26445	−0.632226	+	0.774784i	$0.717859\pi$
$234$	0	0
$235$	31.3301	2.04375
$236$	0	0
$237$	2.00223	0.130059
$238$	0	0
$239$	−2.93699	−0.189978	−0.0949892	−	0.995478i	$-0.530282\pi$
$239$	−2.93699	−0.189978	−0.0949892	+	0.995478i	$0.530282\pi$
$240$	0	0
$241$	−10.6645	−0.686959	−0.343480	−	0.939160i	$-0.611606\pi$
$241$	−10.6645	−0.686959	−0.343480	+	0.939160i	$0.611606\pi$
$242$	0	0
$243$	10.0825	0.646795
$244$	0	0
$245$	−23.4905	−1.50076
$246$	0	0
$247$	4.74938	0.302196
$248$	0	0
$249$	27.0504	1.71425
$250$	0	0
$251$	23.0155	1.45272	0.726362	−	0.687313i	$-0.241210\pi$
$251$	23.0155	1.45272	0.726362	+	0.687313i	$0.241210\pi$
$252$	0	0
$253$	9.29432	0.584329
$254$	0	0
$255$	−19.7957	−1.23966
$256$	0	0
$257$	−18.3997	−1.14774	−0.573870	−	0.818946i	$-0.694559\pi$
$257$	−18.3997	−1.14774	−0.573870	+	0.818946i	$0.694559\pi$
$258$	0	0
$259$	10.0615	0.625194
$260$	0	0
$261$	−3.97596	−0.246106
$262$	0	0
$263$	29.1100	1.79500	0.897500	−	0.441015i	$-0.145381\pi$
$263$	29.1100	1.79500	0.897500	+	0.441015i	$0.145381\pi$
$264$	0	0
$265$	−33.6958	−2.06992
$266$	0	0
$267$	2.35738	0.144269
$268$	0	0
$269$	−21.6268	−1.31861	−0.659306	−	0.751875i	$-0.729150\pi$
$269$	−21.6268	−1.31861	−0.659306	+	0.751875i	$0.729150\pi$
$270$	0	0
$271$	−15.0903	−0.916672	−0.458336	−	0.888779i	$-0.651554\pi$
$271$	−15.0903	−0.916672	−0.458336	+	0.888779i	$0.651554\pi$
$272$	0	0
$273$	−38.6652	−2.34013
$274$	0	0
$275$	−3.48632	−0.210233
$276$	0	0
$277$	−6.41202	−0.385261	−0.192631	−	0.981271i	$-0.561702\pi$
$277$	−6.41202	−0.385261	−0.192631	+	0.981271i	$0.561702\pi$
$278$	0	0
$279$	2.75975	0.165222
$280$	0	0
$281$	−13.4042	−0.799626	−0.399813	−	0.916597i	$-0.630925\pi$
$281$	−13.4042	−0.799626	−0.399813	+	0.916597i	$0.630925\pi$
$282$	0	0
$283$	−0.998663	−0.0593644	−0.0296822	−	0.999559i	$-0.509450\pi$
$283$	−0.998663	−0.0593644	−0.0296822	+	0.999559i	$0.509450\pi$
$284$	0	0
$285$	4.93400	0.292265
$286$	0	0
$287$	30.6992	1.81211
$288$	0	0
$289$	−0.902990	−0.0531171
$290$	0	0
$291$	−0.000146936 0	−8.61356e−6 0
$292$	0	0
$293$	−23.3856	−1.36620	−0.683102	−	0.730323i	$-0.739369\pi$
$293$	−23.3856	−1.36620	−0.683102	+	0.730323i	$0.739369\pi$
$294$	0	0
$295$	−16.5019	−0.960778
$296$	0	0
$297$	−12.9586	−0.751935
$298$	0	0
$299$	−13.5799	−0.785345
$300$	0	0
$301$	8.56988	0.493960
$302$	0	0
$303$	−35.9611	−2.06591
$304$	0	0
$305$	35.5093	2.03326
$306$	0	0
$307$	−14.9284	−0.852006	−0.426003	−	0.904722i	$-0.640079\pi$
$307$	−14.9284	−0.852006	−0.426003	+	0.904722i	$0.640079\pi$
$308$	0	0
$309$	−19.7218	−1.12194
$310$	0	0
$311$	−9.23525	−0.523683	−0.261841	−	0.965111i	$-0.584330\pi$
$311$	−9.23525	−0.523683	−0.261841	+	0.965111i	$0.584330\pi$
$312$	0	0
$313$	−8.52864	−0.482067	−0.241034	−	0.970517i	$-0.577486\pi$
$313$	−8.52864	−0.482067	−0.241034	+	0.970517i	$0.577486\pi$
$314$	0	0
$315$	−10.1092	−0.569588
$316$	0	0
$317$	−7.92737	−0.445245	−0.222623	−	0.974905i	$-0.571462\pi$
$317$	−7.92737	−0.445245	−0.222623	+	0.974905i	$0.571462\pi$
$318$	0	0
$319$	12.8097	0.717205
$320$	0	0
$321$	−29.8618	−1.66672
$322$	0	0
$323$	−4.01211	−0.223240
$324$	0	0
$325$	5.09385	0.282556
$326$	0	0
$327$	−7.13392	−0.394507
$328$	0	0
$329$	−51.6948	−2.85002
$330$	0	0
$331$	−0.769875	−0.0423162	−0.0211581	−	0.999776i	$-0.506735\pi$
$331$	−0.769875	−0.0423162	−0.0211581	+	0.999776i	$0.506735\pi$
$332$	0	0
$333$	2.49664	0.136815
$334$	0	0
$335$	−6.83660	−0.373524
$336$	0	0
$337$	−32.4512	−1.76773	−0.883865	−	0.467743i	$-0.845067\pi$
$337$	−32.4512	−1.76773	−0.883865	+	0.467743i	$0.845067\pi$
$338$	0	0
$339$	−23.3885	−1.27029
$340$	0	0
$341$	−8.89133	−0.481493
$342$	0	0
$343$	10.2973	0.556004
$344$	0	0
$345$	−14.1078	−0.759537
$346$	0	0
$347$	−24.4051	−1.31014	−0.655068	−	0.755570i	$-0.727360\pi$
$347$	−24.4051	−1.31014	−0.655068	+	0.755570i	$0.727360\pi$
$348$	0	0
$349$	−27.5557	−1.47502	−0.737512	−	0.675334i	$-0.763999\pi$
$349$	−27.5557	−1.47502	−0.737512	+	0.675334i	$0.763999\pi$
$350$	0	0
$351$	18.9338	1.01061
$352$	0	0
$353$	−22.7223	−1.20938	−0.604692	−	0.796459i	$-0.706704\pi$
$353$	−22.7223	−1.20938	−0.604692	+	0.796459i	$0.706704\pi$
$354$	0	0
$355$	27.7516	1.47290
$356$	0	0
$357$	32.6630	1.72871
$358$	0	0
$359$	9.45014	0.498759	0.249380	−	0.968406i	$-0.419773\pi$
$359$	9.45014	0.498759	0.249380	+	0.968406i	$0.419773\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.868707	0.0455953
$364$	0	0
$365$	24.3345	1.27373
$366$	0	0
$367$	37.2296	1.94337	0.971686	−	0.236278i	$-0.0759275\pi$
$367$	37.2296	1.94337	0.971686	+	0.236278i	$0.0759275\pi$
$368$	0	0
$369$	7.61760	0.396557
$370$	0	0
$371$	55.5982	2.88652
$372$	0	0
$373$	−12.0405	−0.623433	−0.311716	−	0.950175i	$-0.600904\pi$
$373$	−12.0405	−0.623433	−0.311716	+	0.950175i	$0.600904\pi$
$374$	0	0
$375$	−19.3781	−1.00068
$376$	0	0
$377$	−18.7162	−0.963933
$378$	0	0
$379$	−10.2690	−0.527482	−0.263741	−	0.964593i	$-0.584957\pi$
$379$	−10.2690	−0.527482	−0.263741	+	0.964593i	$0.584957\pi$
$380$	0	0
$381$	−30.2844	−1.55152
$382$	0	0
$383$	−5.19047	−0.265221	−0.132610	−	0.991168i	$-0.542336\pi$
$383$	−5.19047	−0.265221	−0.132610	+	0.991168i	$0.542336\pi$
$384$	0	0
$385$	32.5696	1.65990
$386$	0	0
$387$	2.12651	0.108096
$388$	0	0
$389$	23.4606	1.18950	0.594751	−	0.803910i	$-0.297251\pi$
$389$	23.4606	1.18950	0.594751	+	0.803910i	$0.297251\pi$
$390$	0	0
$391$	11.4718	0.580155
$392$	0	0
$393$	−32.5789	−1.64339
$394$	0	0
$395$	2.46425	0.123990
$396$	0	0
$397$	13.4816	0.676622	0.338311	−	0.941034i	$-0.390144\pi$
$397$	13.4816	0.676622	0.338311	+	0.941034i	$0.390144\pi$
$398$	0	0
$399$	−8.14112	−0.407566
$400$	0	0
$401$	−35.1207	−1.75384	−0.876922	−	0.480634i	$-0.840407\pi$
$401$	−35.1207	−1.75384	−0.876922	+	0.480634i	$0.840407\pi$
$402$	0	0
$403$	12.9911	0.647132
$404$	0	0
$405$	27.1286	1.34803
$406$	0	0
$407$	−8.04365	−0.398709
$408$	0	0
$409$	27.5142	1.36049	0.680244	−	0.732986i	$-0.261874\pi$
$409$	27.5142	1.36049	0.680244	+	0.732986i	$0.261874\pi$
$410$	0	0
$411$	11.1012	0.547582
$412$	0	0
$413$	27.2282	1.33981
$414$	0	0
$415$	33.2923	1.63425
$416$	0	0
$417$	−25.7238	−1.25970
$418$	0	0
$419$	32.8683	1.60572	0.802862	−	0.596166i	$-0.203310\pi$
$419$	32.8683	1.60572	0.802862	+	0.596166i	$0.203310\pi$
$420$	0	0
$421$	8.07512	0.393557	0.196779	−	0.980448i	$-0.436952\pi$
$421$	8.07512	0.393557	0.196779	+	0.980448i	$0.436952\pi$
$422$	0	0
$423$	−12.8274	−0.623689
$424$	0	0
$425$	−4.30311	−0.208731
$426$	0	0
$427$	−58.5905	−2.83539
$428$	0	0
$429$	30.9107	1.49238
$430$	0	0
$431$	9.24355	0.445246	0.222623	−	0.974905i	$-0.428538\pi$
$431$	9.24355	0.445246	0.222623	+	0.974905i	$0.428538\pi$
$432$	0	0
$433$	2.55366	0.122721	0.0613606	−	0.998116i	$-0.480456\pi$
$433$	2.55366	0.122721	0.0613606	+	0.998116i	$0.480456\pi$
$434$	0	0
$435$	−19.4437	−0.932256
$436$	0	0
$437$	−2.85930	−0.136779
$438$	0	0
$439$	−13.6258	−0.650325	−0.325162	−	0.945658i	$-0.605419\pi$
$439$	−13.6258	−0.650325	−0.325162	+	0.945658i	$0.605419\pi$
$440$	0	0
$441$	9.61767	0.457984
$442$	0	0
$443$	−34.2056	−1.62516	−0.812578	−	0.582852i	$-0.801937\pi$
$443$	−34.2056	−1.62516	−0.812578	+	0.582852i	$0.801937\pi$
$444$	0	0
$445$	2.90135	0.137537
$446$	0	0
$447$	−0.644955	−0.0305053
$448$	0	0
$449$	−7.38903	−0.348710	−0.174355	−	0.984683i	$-0.555784\pi$
$449$	−7.38903	−0.348710	−0.174355	+	0.984683i	$0.555784\pi$
$450$	0	0
$451$	−24.5423	−1.15565
$452$	0	0
$453$	−29.2673	−1.37510
$454$	0	0
$455$	−47.5873	−2.23093
$456$	0	0
$457$	9.45614	0.442340	0.221170	−	0.975235i	$-0.429013\pi$
$457$	9.45614	0.442340	0.221170	+	0.975235i	$0.429013\pi$
$458$	0	0
$459$	−15.9946	−0.746564
$460$	0	0
$461$	−41.3261	−1.92475	−0.962374	−	0.271728i	$-0.912405\pi$
$461$	−41.3261	−1.92475	−0.962374	+	0.271728i	$0.912405\pi$
$462$	0	0
$463$	−8.85669	−0.411605	−0.205803	−	0.978594i	$-0.565981\pi$
$463$	−8.85669	−0.411605	−0.205803	+	0.978594i	$0.565981\pi$
$464$	0	0
$465$	13.4961	0.625866
$466$	0	0
$467$	27.5092	1.27297	0.636487	−	0.771287i	$-0.280387\pi$
$467$	27.5092	1.27297	0.636487	+	0.771287i	$0.280387\pi$
$468$	0	0
$469$	11.2804	0.520881
$470$	0	0
$471$	16.9486	0.780950
$472$	0	0
$473$	−6.85114	−0.315016
$474$	0	0
$475$	1.07253	0.0492111
$476$	0	0
$477$	13.7960	0.631675
$478$	0	0
$479$	−27.6175	−1.26188	−0.630938	−	0.775834i	$-0.717330\pi$
$479$	−27.6175	−1.26188	−0.630938	+	0.775834i	$0.717330\pi$
$480$	0	0
$481$	11.7525	0.535870
$482$	0	0
$483$	23.2779	1.05918
$484$	0	0
$485$	−0.000180842 0	−8.21162e−6 0
$486$	0	0
$487$	39.0331	1.76876	0.884380	−	0.466767i	$-0.154582\pi$
$487$	39.0331	1.76876	0.884380	+	0.466767i	$0.154582\pi$
$488$	0	0
$489$	28.9944	1.31117
$490$	0	0
$491$	7.74681	0.349609	0.174804	−	0.984603i	$-0.444071\pi$
$491$	7.74681	0.349609	0.174804	+	0.984603i	$0.444071\pi$
$492$	0	0
$493$	15.8108	0.712081
$494$	0	0
$495$	8.08173	0.363247
$496$	0	0
$497$	−45.7902	−2.05397
$498$	0	0
$499$	−10.3219	−0.462073	−0.231037	−	0.972945i	$-0.574212\pi$
$499$	−10.3219	−0.462073	−0.231037	+	0.972945i	$0.574212\pi$
$500$	0	0
$501$	34.6213	1.54676
$502$	0	0
$503$	36.9413	1.64713	0.823567	−	0.567220i	$-0.191981\pi$
$503$	36.9413	1.64713	0.823567	+	0.567220i	$0.191981\pi$
$504$	0	0
$505$	−44.2592	−1.96951
$506$	0	0
$507$	−19.1345	−0.849794
$508$	0	0
$509$	32.8827	1.45750	0.728751	−	0.684779i	$-0.240101\pi$
$509$	32.8827	1.45750	0.728751	+	0.684779i	$0.240101\pi$
$510$	0	0
$511$	−40.1520	−1.77622
$512$	0	0
$513$	3.98658	0.176012
$514$	0	0
$515$	−24.2727	−1.06958
$516$	0	0
$517$	41.3271	1.81756
$518$	0	0
$519$	19.0127	0.834567
$520$	0	0
$521$	22.6280	0.991353	0.495676	−	0.868507i	$-0.334920\pi$
$521$	22.6280	0.991353	0.495676	+	0.868507i	$0.334920\pi$
$522$	0	0
$523$	19.7042	0.861606	0.430803	−	0.902446i	$-0.358230\pi$
$523$	19.7042	0.861606	0.430803	+	0.902446i	$0.358230\pi$
$524$	0	0
$525$	−8.73159	−0.381078
$526$	0	0
$527$	−10.9744	−0.478053
$528$	0	0
$529$	−14.8244	−0.644540
$530$	0	0
$531$	6.75633	0.293200
$532$	0	0
$533$	35.8586	1.55321
$534$	0	0
$535$	−36.7525	−1.58895
$536$	0	0
$537$	36.3275	1.56765
$538$	0	0
$539$	−30.9861	−1.33466
$540$	0	0
$541$	−43.5457	−1.87217	−0.936087	−	0.351768i	$-0.885581\pi$
$541$	−43.5457	−1.87217	−0.936087	+	0.351768i	$0.885581\pi$
$542$	0	0
$543$	−24.0886	−1.03374
$544$	0	0
$545$	−8.78009	−0.376098
$546$	0	0
$547$	−40.2915	−1.72274	−0.861370	−	0.507978i	$-0.830393\pi$
$547$	−40.2915	−1.72274	−0.861370	+	0.507978i	$0.830393\pi$
$548$	0	0
$549$	−14.5385	−0.620487
$550$	0	0
$551$	−3.94076	−0.167882
$552$	0	0
$553$	−4.06602	−0.172905
$554$	0	0
$555$	12.2094	0.518260
$556$	0	0
$557$	−22.4154	−0.949773	−0.474887	−	0.880047i	$-0.657511\pi$
$557$	−22.4154	−0.949773	−0.474887	+	0.880047i	$0.657511\pi$
$558$	0	0
$559$	10.0102	0.423385
$560$	0	0
$561$	−26.1123	−1.10246
$562$	0	0
$563$	30.7726	1.29691	0.648456	−	0.761252i	$-0.275415\pi$
$563$	30.7726	1.29691	0.648456	+	0.761252i	$0.275415\pi$
$564$	0	0
$565$	−28.7855	−1.21101
$566$	0	0
$567$	−44.7622	−1.87984
$568$	0	0
$569$	−21.1833	−0.888052	−0.444026	−	0.896014i	$-0.646450\pi$
$569$	−21.1833	−0.888052	−0.444026	+	0.896014i	$0.646450\pi$
$570$	0	0
$571$	−29.3932	−1.23007	−0.615034	−	0.788500i	$-0.710858\pi$
$571$	−29.3932	−1.23007	−0.615034	+	0.788500i	$0.710858\pi$
$572$	0	0
$573$	23.2180	0.969946
$574$	0	0
$575$	−3.06668	−0.127890
$576$	0	0
$577$	5.88503	0.244997	0.122499	−	0.992469i	$-0.460909\pi$
$577$	5.88503	0.244997	0.122499	+	0.992469i	$0.460909\pi$
$578$	0	0
$579$	40.6672	1.69007
$580$	0	0
$581$	−54.9324	−2.27898
$582$	0	0
$583$	−44.4477	−1.84084
$584$	0	0
$585$	−11.8082	−0.488208
$586$	0	0
$587$	17.8053	0.734903	0.367452	−	0.930043i	$-0.380230\pi$
$587$	17.8053	0.734903	0.367452	+	0.930043i	$0.380230\pi$
$588$	0	0
$589$	2.73532	0.112707
$590$	0	0
$591$	−26.9724	−1.10950
$592$	0	0
$593$	−8.60392	−0.353321	−0.176660	−	0.984272i	$-0.556529\pi$
$593$	−8.60392	−0.353321	−0.176660	+	0.984272i	$0.556529\pi$
$594$	0	0
$595$	40.2001	1.64804
$596$	0	0
$597$	35.9744	1.47233
$598$	0	0
$599$	33.4644	1.36732	0.683660	−	0.729801i	$-0.260387\pi$
$599$	33.4644	1.36732	0.683660	+	0.729801i	$0.260387\pi$
$600$	0	0
$601$	−10.2715	−0.418983	−0.209491	−	0.977810i	$-0.567181\pi$
$601$	−10.2715	−0.418983	−0.209491	+	0.977810i	$0.567181\pi$
$602$	0	0
$603$	2.79909	0.113988
$604$	0	0
$605$	1.06916	0.0434677
$606$	0	0
$607$	30.0027	1.21777	0.608886	−	0.793258i	$-0.291617\pi$
$607$	30.0027	1.21777	0.608886	+	0.793258i	$0.291617\pi$
$608$	0	0
$609$	32.0822	1.30004
$610$	0	0
$611$	−60.3829	−2.44283
$612$	0	0
$613$	−34.5067	−1.39371	−0.696856	−	0.717211i	$-0.745419\pi$
$613$	−34.5067	−1.39371	−0.696856	+	0.717211i	$0.745419\pi$
$614$	0	0
$615$	37.2525	1.50217
$616$	0	0
$617$	−45.6129	−1.83631	−0.918154	−	0.396224i	$-0.870320\pi$
$617$	−45.6129	−1.83631	−0.918154	+	0.396224i	$0.870320\pi$
$618$	0	0
$619$	11.1456	0.447980	0.223990	−	0.974591i	$-0.428092\pi$
$619$	11.1456	0.447980	0.223990	+	0.974591i	$0.428092\pi$
$620$	0	0
$621$	−11.3988	−0.457419
$622$	0	0
$623$	−4.78723	−0.191796
$624$	0	0
$625$	−29.2123	−1.16849
$626$	0	0
$627$	6.50837	0.259919
$628$	0	0
$629$	−9.92814	−0.395861
$630$	0	0
$631$	49.4569	1.96885	0.984423	−	0.175815i	$-0.0562561\pi$
$631$	49.4569	1.96885	0.984423	+	0.175815i	$0.0562561\pi$
$632$	0	0
$633$	0.563115	0.0223818
$634$	0	0
$635$	−37.2726	−1.47912
$636$	0	0
$637$	45.2736	1.79381
$638$	0	0
$639$	−11.3622	−0.449484
$640$	0	0
$641$	26.8188	1.05928	0.529639	−	0.848223i	$-0.322327\pi$
$641$	26.8188	1.05928	0.529639	+	0.848223i	$0.322327\pi$
$642$	0	0
$643$	−24.3253	−0.959298	−0.479649	−	0.877461i	$-0.659236\pi$
$643$	−24.3253	−0.959298	−0.479649	+	0.877461i	$0.659236\pi$
$644$	0	0
$645$	10.3993	0.409472
$646$	0	0
$647$	−22.2968	−0.876580	−0.438290	−	0.898834i	$-0.644416\pi$
$647$	−22.2968	−0.876580	−0.438290	+	0.898834i	$0.644416\pi$
$648$	0	0
$649$	−21.7674	−0.854446
$650$	0	0
$651$	−22.2686	−0.872775
$652$	0	0
$653$	33.5260	1.31197	0.655987	−	0.754772i	$-0.272252\pi$
$653$	33.5260	1.31197	0.655987	+	0.754772i	$0.272252\pi$
$654$	0	0
$655$	−40.0966	−1.56670
$656$	0	0
$657$	−9.96320	−0.388702
$658$	0	0
$659$	−31.8937	−1.24240	−0.621202	−	0.783651i	$-0.713355\pi$
$659$	−31.8937	−1.24240	−0.621202	+	0.783651i	$0.713355\pi$
$660$	0	0
$661$	8.28363	0.322196	0.161098	−	0.986938i	$-0.448497\pi$
$661$	8.28363	0.322196	0.161098	+	0.986938i	$0.448497\pi$
$662$	0	0
$663$	38.1526	1.48172
$664$	0	0
$665$	−10.0197	−0.388547
$666$	0	0
$667$	11.2678	0.436292
$668$	0	0
$669$	53.2643	2.05932
$670$	0	0
$671$	46.8398	1.80823
$672$	0	0
$673$	40.2386	1.55108	0.775541	−	0.631297i	$-0.217477\pi$
$673$	40.2386	1.55108	0.775541	+	0.631297i	$0.217477\pi$
$674$	0	0
$675$	4.27573	0.164573
$676$	0	0
$677$	−9.54961	−0.367021	−0.183511	−	0.983018i	$-0.558746\pi$
$677$	−9.54961	−0.367021	−0.183511	+	0.983018i	$0.558746\pi$
$678$	0	0
$679$	0.000298390 0	1.14512e−5 0
$680$	0	0
$681$	46.3175	1.77489
$682$	0	0
$683$	13.3845	0.512143	0.256072	−	0.966658i	$-0.417572\pi$
$683$	13.3845	0.512143	0.256072	+	0.966658i	$0.417572\pi$
$684$	0	0
$685$	13.6628	0.522030
$686$	0	0
$687$	3.55623	0.135679
$688$	0	0
$689$	64.9424	2.47411
$690$	0	0
$691$	−42.0952	−1.60138	−0.800689	−	0.599080i	$-0.795533\pi$
$691$	−42.0952	−1.60138	−0.800689	+	0.599080i	$0.795533\pi$
$692$	0	0
$693$	−13.3349	−0.506550
$694$	0	0
$695$	−31.6596	−1.20092
$696$	0	0
$697$	−30.2921	−1.14739
$698$	0	0
$699$	38.6451	1.46169
$700$	0	0
$701$	−16.2520	−0.613831	−0.306916	−	0.951737i	$-0.599297\pi$
$701$	−16.2520	−0.613831	−0.306916	+	0.951737i	$0.599297\pi$
$702$	0	0
$703$	2.47454	0.0933292
$704$	0	0
$705$	−62.7301	−2.36255
$706$	0	0
$707$	73.0278	2.74649
$708$	0	0
$709$	12.5642	0.471859	0.235929	−	0.971770i	$-0.424187\pi$
$709$	12.5642	0.471859	0.235929	+	0.971770i	$0.424187\pi$
$710$	0	0
$711$	−1.00893	−0.0378379
$712$	0	0
$713$	−7.82111	−0.292903
$714$	0	0
$715$	38.0434	1.42274
$716$	0	0
$717$	5.88054	0.219613
$718$	0	0
$719$	27.9035	1.04063	0.520313	−	0.853976i	$-0.325815\pi$
$719$	27.9035	1.04063	0.520313	+	0.853976i	$0.325815\pi$
$720$	0	0
$721$	40.0500	1.49154
$722$	0	0
$723$	21.3527	0.794117
$724$	0	0
$725$	−4.22659	−0.156972
$726$	0	0
$727$	17.8993	0.663850	0.331925	−	0.943306i	$-0.392302\pi$
$727$	17.8993	0.663850	0.331925	+	0.943306i	$0.392302\pi$
$728$	0	0
$729$	12.8390	0.475518
$730$	0	0
$731$	−8.45625	−0.312766
$732$	0	0
$733$	−30.6676	−1.13273	−0.566367	−	0.824153i	$-0.691651\pi$
$733$	−30.6676	−1.13273	−0.566367	+	0.824153i	$0.691651\pi$
$734$	0	0
$735$	47.0335	1.73486
$736$	0	0
$737$	−9.01807	−0.332185
$738$	0	0
$739$	−18.9825	−0.698281	−0.349141	−	0.937070i	$-0.613526\pi$
$739$	−18.9825	−0.698281	−0.349141	+	0.937070i	$0.613526\pi$
$740$	0	0
$741$	−9.50936	−0.349335
$742$	0	0
$743$	−2.74872	−0.100841	−0.0504203	−	0.998728i	$-0.516056\pi$
$743$	−2.74872	−0.100841	−0.0504203	+	0.998728i	$0.516056\pi$
$744$	0	0
$745$	−0.793780	−0.0290818
$746$	0	0
$747$	−13.6308	−0.498724
$748$	0	0
$749$	60.6417	2.21580
$750$	0	0
$751$	−37.3254	−1.36202	−0.681011	−	0.732273i	$-0.738459\pi$
$751$	−37.3254	−1.36202	−0.681011	+	0.732273i	$0.738459\pi$
$752$	0	0
$753$	−46.0823	−1.67933
$754$	0	0
$755$	−36.0207	−1.31093
$756$	0	0
$757$	−12.0263	−0.437105	−0.218553	−	0.975825i	$-0.570133\pi$
$757$	−12.0263	−0.437105	−0.218553	+	0.975825i	$0.570133\pi$
$758$	0	0
$759$	−18.6094	−0.675478
$760$	0	0
$761$	−34.3634	−1.24567	−0.622835	−	0.782353i	$-0.714019\pi$
$761$	−34.3634	−1.24567	−0.622835	+	0.782353i	$0.714019\pi$
$762$	0	0
$763$	14.4872	0.524471
$764$	0	0
$765$	9.97514	0.360652
$766$	0	0
$767$	31.8043	1.14839
$768$	0	0
$769$	28.2503	1.01873	0.509367	−	0.860550i	$-0.329880\pi$
$769$	28.2503	1.01873	0.509367	+	0.860550i	$0.329880\pi$
$770$	0	0
$771$	36.8404	1.32677
$772$	0	0
$773$	−35.8882	−1.29081	−0.645404	−	0.763841i	$-0.723311\pi$
$773$	−35.8882	−1.29081	−0.645404	+	0.763841i	$0.723311\pi$
$774$	0	0
$775$	2.93372	0.105382
$776$	0	0
$777$	−20.1455	−0.722718
$778$	0	0
$779$	7.55017	0.270513
$780$	0	0
$781$	36.6067	1.30989
$782$	0	0
$783$	−15.7102	−0.561436
$784$	0	0
$785$	20.8595	0.744508
$786$	0	0
$787$	−19.3714	−0.690516	−0.345258	−	0.938508i	$-0.612209\pi$
$787$	−19.3714	−0.690516	−0.345258	+	0.938508i	$0.612209\pi$
$788$	0	0
$789$	−58.2850	−2.07500
$790$	0	0
$791$	47.4961	1.68877
$792$	0	0
$793$	−68.4375	−2.43029
$794$	0	0
$795$	67.4669	2.39280
$796$	0	0
$797$	48.8991	1.73209	0.866047	−	0.499963i	$-0.166653\pi$
$797$	48.8991	1.73209	0.866047	+	0.499963i	$0.166653\pi$
$798$	0	0
$799$	51.0093	1.80458
$800$	0	0
$801$	−1.18789	−0.0419720
$802$	0	0
$803$	32.0993	1.13276
$804$	0	0
$805$	28.6493	1.00976
$806$	0	0
$807$	43.3019	1.52430
$808$	0	0
$809$	−27.5666	−0.969192	−0.484596	−	0.874738i	$-0.661033\pi$
$809$	−27.5666	−0.969192	−0.484596	+	0.874738i	$0.661033\pi$
$810$	0	0
$811$	24.5523	0.862148	0.431074	−	0.902317i	$-0.358135\pi$
$811$	24.5523	0.862148	0.431074	+	0.902317i	$0.358135\pi$
$812$	0	0
$813$	30.2143	1.05966
$814$	0	0
$815$	35.6850	1.24999
$816$	0	0
$817$	2.10768	0.0737384
$818$	0	0
$819$	19.4836	0.680810
$820$	0	0
$821$	−18.8845	−0.659075	−0.329537	−	0.944143i	$-0.606893\pi$
$821$	−18.8845	−0.659075	−0.329537	+	0.944143i	$0.606893\pi$
$822$	0	0
$823$	37.3110	1.30058	0.650291	−	0.759685i	$-0.274647\pi$
$823$	37.3110	1.30058	0.650291	+	0.759685i	$0.274647\pi$
$824$	0	0
$825$	6.98042	0.243027
$826$	0	0
$827$	35.4380	1.23230	0.616150	−	0.787629i	$-0.288691\pi$
$827$	35.4380	1.23230	0.616150	+	0.787629i	$0.288691\pi$
$828$	0	0
$829$	21.6258	0.751096	0.375548	−	0.926803i	$-0.377454\pi$
$829$	21.6258	0.751096	0.375548	+	0.926803i	$0.377454\pi$
$830$	0	0
$831$	12.8384	0.445358
$832$	0	0
$833$	−38.2456	−1.32513
$834$	0	0
$835$	42.6102	1.47459
$836$	0	0
$837$	10.9046	0.376918
$838$	0	0
$839$	6.88368	0.237651	0.118825	−	0.992915i	$-0.462087\pi$
$839$	6.88368	0.237651	0.118825	+	0.992915i	$0.462087\pi$
$840$	0	0
$841$	−13.4704	−0.464496
$842$	0	0
$843$	26.8383	0.924359
$844$	0	0
$845$	−23.5499	−0.810140
$846$	0	0
$847$	−1.76412	−0.0606160
$848$	0	0
$849$	1.99955	0.0686245
$850$	0	0
$851$	−7.07546	−0.242544
$852$	0	0
$853$	−20.5202	−0.702597	−0.351298	−	0.936264i	$-0.614260\pi$
$853$	−20.5202	−0.702597	−0.351298	+	0.936264i	$0.614260\pi$
$854$	0	0
$855$	−2.48626	−0.0850283
$856$	0	0
$857$	−48.6493	−1.66183	−0.830914	−	0.556401i	$-0.812182\pi$
$857$	−48.6493	−1.66183	−0.830914	+	0.556401i	$0.812182\pi$
$858$	0	0
$859$	−36.1468	−1.23331	−0.616656	−	0.787232i	$-0.711513\pi$
$859$	−36.1468	−1.23331	−0.616656	+	0.787232i	$0.711513\pi$
$860$	0	0
$861$	−61.4668	−2.09478
$862$	0	0
$863$	−34.4735	−1.17349	−0.586746	−	0.809771i	$-0.699591\pi$
$863$	−34.4735	−1.17349	−0.586746	+	0.809771i	$0.699591\pi$
$864$	0	0
$865$	23.4000	0.795623
$866$	0	0
$867$	1.80800	0.0614027
$868$	0	0
$869$	3.25056	0.110268
$870$	0	0
$871$	13.1763	0.446461
$872$	0	0
$873$	7.40417e−5 0	2.50593e−6 0
$874$	0	0
$875$	39.3521	1.33034
$876$	0	0
$877$	−2.96081	−0.0999794	−0.0499897	−	0.998750i	$-0.515919\pi$
$877$	−2.96081	−0.0999794	−0.0499897	+	0.998750i	$0.515919\pi$
$878$	0	0
$879$	46.8235	1.57932
$880$	0	0
$881$	32.6950	1.10152	0.550761	−	0.834663i	$-0.314338\pi$
$881$	32.6950	1.10152	0.550761	+	0.834663i	$0.314338\pi$
$882$	0	0
$883$	35.3221	1.18868	0.594342	−	0.804212i	$-0.297412\pi$
$883$	35.3221	1.18868	0.594342	+	0.804212i	$0.297412\pi$
$884$	0	0
$885$	33.0406	1.11065
$886$	0	0
$887$	18.4807	0.620520	0.310260	−	0.950652i	$-0.399584\pi$
$887$	18.4807	0.620520	0.310260	+	0.950652i	$0.399584\pi$
$888$	0	0
$889$	61.5000	2.06264
$890$	0	0
$891$	35.7849	1.19884
$892$	0	0
$893$	−12.7138	−0.425453
$894$	0	0
$895$	44.7101	1.49449
$896$	0	0
$897$	27.1901	0.907851
$898$	0	0
$899$	−10.7793	−0.359509
$900$	0	0
$901$	−54.8610	−1.82769
$902$	0	0
$903$	−17.1589	−0.571012
$904$	0	0
$905$	−29.6471	−0.985502
$906$	0	0
$907$	40.0208	1.32887	0.664435	−	0.747346i	$-0.268672\pi$
$907$	40.0208	1.32887	0.664435	+	0.747346i	$0.268672\pi$
$908$	0	0
$909$	18.1209	0.601033
$910$	0	0
$911$	7.01189	0.232314	0.116157	−	0.993231i	$-0.462942\pi$
$911$	7.01189	0.232314	0.116157	+	0.993231i	$0.462942\pi$
$912$	0	0
$913$	43.9154	1.45339
$914$	0	0
$915$	−71.0979	−2.35042
$916$	0	0
$917$	66.1595	2.18478
$918$	0	0
$919$	−18.3016	−0.603715	−0.301858	−	0.953353i	$-0.597607\pi$
$919$	−18.3016	−0.603715	−0.301858	+	0.953353i	$0.597607\pi$
$920$	0	0
$921$	29.8900	0.984910
$922$	0	0
$923$	−53.4859	−1.76051
$924$	0	0
$925$	2.65402	0.0872637
$926$	0	0
$927$	9.93790	0.326403
$928$	0	0
$929$	2.70469	0.0887379	0.0443689	−	0.999015i	$-0.485872\pi$
$929$	2.70469	0.0887379	0.0443689	+	0.999015i	$0.485872\pi$
$930$	0	0
$931$	9.53253	0.312416
$932$	0	0
$933$	18.4911	0.605371
$934$	0	0
$935$	−32.1377	−1.05102
$936$	0	0
$937$	22.1781	0.724528	0.362264	−	0.932075i	$-0.382004\pi$
$937$	22.1781	0.724528	0.362264	+	0.932075i	$0.382004\pi$
$938$	0	0
$939$	17.0763	0.557265
$940$	0	0
$941$	10.3706	0.338071	0.169036	−	0.985610i	$-0.445935\pi$
$941$	10.3706	0.338071	0.169036	+	0.985610i	$0.445935\pi$
$942$	0	0
$943$	−21.5882	−0.703008
$944$	0	0
$945$	−39.9443	−1.29939
$946$	0	0
$947$	−36.9437	−1.20051	−0.600255	−	0.799809i	$-0.704934\pi$
$947$	−36.9437	−1.20051	−0.600255	+	0.799809i	$0.704934\pi$
$948$	0	0
$949$	−46.9001	−1.52244
$950$	0	0
$951$	15.8724	0.514699
$952$	0	0
$953$	−10.5283	−0.341046	−0.170523	−	0.985354i	$-0.554546\pi$
$953$	−10.5283	−0.341046	−0.170523	+	0.985354i	$0.554546\pi$
$954$	0	0
$955$	28.5756	0.924685
$956$	0	0
$957$	−25.6480	−0.829081
$958$	0	0
$959$	−22.5437	−0.727974
$960$	0	0
$961$	−23.5180	−0.758645
$962$	0	0
$963$	15.0475	0.484898
$964$	0	0
$965$	50.0513	1.61121
$966$	0	0
$967$	34.7791	1.11842	0.559210	−	0.829026i	$-0.311105\pi$
$967$	34.7791	1.11842	0.559210	+	0.829026i	$0.311105\pi$
$968$	0	0
$969$	8.03317	0.258063
$970$	0	0
$971$	41.6332	1.33607	0.668036	−	0.744129i	$-0.267135\pi$
$971$	41.6332	1.33607	0.668036	+	0.744129i	$0.267135\pi$
$972$	0	0
$973$	52.2385	1.67469
$974$	0	0
$975$	−10.1991	−0.326632
$976$	0	0
$977$	−14.7360	−0.471448	−0.235724	−	0.971820i	$-0.575746\pi$
$977$	−14.7360	−0.471448	−0.235724	+	0.971820i	$0.575746\pi$
$978$	0	0
$979$	3.82713	0.122315
$980$	0	0
$981$	3.59481	0.114773
$982$	0	0
$983$	28.7435	0.916775	0.458388	−	0.888752i	$-0.348427\pi$
$983$	28.7435	0.916775	0.458388	+	0.888752i	$0.348427\pi$
$984$	0	0
$985$	−33.1963	−1.05772
$986$	0	0
$987$	103.505	3.29460
$988$	0	0
$989$	−6.02649	−0.191631
$990$	0	0
$991$	−53.3287	−1.69404	−0.847022	−	0.531558i	$-0.821607\pi$
$991$	−53.3287	−1.69404	−0.847022	+	0.531558i	$0.821607\pi$
$992$	0	0
$993$	1.54147	0.0489170
$994$	0	0
$995$	44.2756	1.40363
$996$	0	0
$997$	−49.0112	−1.55220	−0.776101	−	0.630609i	$-0.782805\pi$
$997$	−49.0112	−1.55220	−0.776101	+	0.630609i	$0.782805\pi$
$998$	0	0
$999$	9.86497	0.312114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.6	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-2.00223 q^{3} -2.46425 q^{5} +4.06602 q^{7} +1.00893 q^{9} +O(q^{10})\)	\(q-2.00223 q^{3} -2.46425 q^{5} +4.06602 q^{7} +1.00893 q^{9} -3.25056 q^{11} +4.74938 q^{13} +4.93400 q^{15} -4.01211 q^{17} +1.00000 q^{19} -8.14112 q^{21} -2.85930 q^{23} +1.07253 q^{25} +3.98658 q^{27} -3.94076 q^{29} +2.73532 q^{31} +6.50837 q^{33} -10.0197 q^{35} +2.47454 q^{37} -9.50936 q^{39} +7.55017 q^{41} +2.10768 q^{43} -2.48626 q^{45} -12.7138 q^{47} +9.53253 q^{49} +8.03317 q^{51} +13.6739 q^{53} +8.01019 q^{55} -2.00223 q^{57} +6.69652 q^{59} -14.4098 q^{61} +4.10234 q^{63} -11.7037 q^{65} +2.77431 q^{67} +5.72498 q^{69} -11.2617 q^{71} -9.87501 q^{73} -2.14745 q^{75} -13.2168 q^{77} -1.00000 q^{79} -11.0089 q^{81} -13.5101 q^{83} +9.88684 q^{85} +7.89032 q^{87} -1.17737 q^{89} +19.3111 q^{91} -5.47675 q^{93} -2.46425 q^{95} +7.33863e-5 q^{97} -3.27959 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 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

Properties

Related objects

Downloads

Learn more