LMFDB - Embedded newform 1620.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1620.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.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:	$95.5830942093$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.560145.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 90x + 192 $ x^3 - x^2 - 90*x + 192
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.76746$ of defining polynomial
Character	$\chi$	$=$	1620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} +0.949584 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +0.949584 q^{7} -62.6552 q^{11} +6.94958 q^{13} +103.411 q^{17} +73.8064 q^{19} -86.1591 q^{23} +25.0000 q^{25} +55.5624 q^{29} +199.865 q^{31} -4.74792 q^{35} -18.9070 q^{37} +57.3025 q^{41} +296.032 q^{43} -57.5384 q^{47} -342.098 q^{49} -672.005 q^{53} +313.276 q^{55} +546.644 q^{59} -791.559 q^{61} -34.7479 q^{65} +631.560 q^{67} -102.942 q^{71} -200.652 q^{73} -59.4963 q^{77} -665.931 q^{79} -1347.41 q^{83} -517.056 q^{85} +12.6866 q^{89} +6.59922 q^{91} -369.032 q^{95} -637.379 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} + 15 q^{7}+O(q^{10}) $ 3 * q - 15 * q^5 + 15 * q^7	$ 3 q - 15 q^{5} + 15 q^{7} - 24 q^{11} + 33 q^{13} - 42 q^{17} + 21 q^{19} + 33 q^{23} + 75 q^{25} - 222 q^{29} + 132 q^{31} - 75 q^{35} + 174 q^{37} + 99 q^{41} - 120 q^{43} - 537 q^{47} + 492 q^{49} - 267 q^{53} + 120 q^{55} + 225 q^{59} - 480 q^{61} - 165 q^{65} + 12 q^{67} - 570 q^{71} + 1062 q^{73} - 312 q^{77} - 1026 q^{79} - 702 q^{83} + 210 q^{85} - 1140 q^{89} + 1611 q^{91} - 105 q^{95} - 2556 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 + 15 * q^7 - 24 * q^11 + 33 * q^13 - 42 * q^17 + 21 * q^19 + 33 * q^23 + 75 * q^25 - 222 * q^29 + 132 * q^31 - 75 * q^35 + 174 * q^37 + 99 * q^41 - 120 * q^43 - 537 * q^47 + 492 * q^49 - 267 * q^53 + 120 * q^55 + 225 * q^59 - 480 * q^61 - 165 * q^65 + 12 * q^67 - 570 * q^71 + 1062 * q^73 - 312 * q^77 - 1026 * q^79 - 702 * q^83 + 210 * q^85 - 1140 * q^89 + 1611 * q^91 - 105 * q^95 - 2556 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0.949584	0.0512727	0.0256364	−	0.999671i	$-0.491839\pi$
$7$	0.949584	0.0512727	0.0256364	+	0.999671i	$0.491839\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−62.6552	−1.71739	−0.858693	−	0.512491i	$-0.828723\pi$
$11$	−62.6552	−1.71739	−0.858693	+	0.512491i	$0.828723\pi$
$12$	0	0
$13$	6.94958	0.148267	0.0741334	−	0.997248i	$-0.476381\pi$
$13$	6.94958	0.148267	0.0741334	+	0.997248i	$0.476381\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	103.411	1.47535	0.737673	−	0.675158i	$-0.235925\pi$
$17$	103.411	1.47535	0.737673	+	0.675158i	$0.235925\pi$
$18$	0	0
$19$	73.8064	0.891176	0.445588	−	0.895238i	$-0.352995\pi$
$19$	73.8064	0.891176	0.445588	+	0.895238i	$0.352995\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−86.1591	−0.781105	−0.390552	−	0.920581i	$-0.627716\pi$
$23$	−86.1591	−0.781105	−0.390552	+	0.920581i	$0.627716\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	55.5624	0.355782	0.177891	−	0.984050i	$-0.443073\pi$
$29$	55.5624	0.355782	0.177891	+	0.984050i	$0.443073\pi$
$30$	0	0
$31$	199.865	1.15796	0.578980	−	0.815342i	$-0.303451\pi$
$31$	199.865	1.15796	0.578980	+	0.815342i	$0.303451\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.74792	−0.0229299
$36$	0	0
$37$	−18.9070	−0.0840078	−0.0420039	−	0.999117i	$-0.513374\pi$
$37$	−18.9070	−0.0840078	−0.0420039	+	0.999117i	$0.513374\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	57.3025	0.218272	0.109136	−	0.994027i	$-0.465192\pi$
$41$	57.3025	0.218272	0.109136	+	0.994027i	$0.465192\pi$
$42$	0	0
$43$	296.032	1.04987	0.524935	−	0.851142i	$-0.324090\pi$
$43$	296.032	1.04987	0.524935	+	0.851142i	$0.324090\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−57.5384	−0.178571	−0.0892856	−	0.996006i	$-0.528458\pi$
$47$	−57.5384	−0.178571	−0.0892856	+	0.996006i	$0.528458\pi$
$48$	0	0
$49$	−342.098	−0.997371
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−672.005	−1.74164	−0.870821	−	0.491600i	$-0.836412\pi$
$53$	−672.005	−1.74164	−0.870821	+	0.491600i	$0.836412\pi$
$54$	0	0
$55$	313.276	0.768038
$56$	0	0
$57$	0	0
$58$	0	0
$59$	546.644	1.20622	0.603110	−	0.797658i	$-0.293928\pi$
$59$	546.644	1.20622	0.603110	+	0.797658i	$0.293928\pi$
$60$	0	0
$61$	−791.559	−1.66146	−0.830728	−	0.556679i	$-0.812075\pi$
$61$	−791.559	−1.66146	−0.830728	+	0.556679i	$0.812075\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−34.7479	−0.0663069
$66$	0	0
$67$	631.560	1.15160	0.575801	−	0.817590i	$-0.304690\pi$
$67$	631.560	1.15160	0.575801	+	0.817590i	$0.304690\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−102.942	−0.172069	−0.0860346	−	0.996292i	$-0.527420\pi$
$71$	−102.942	−0.172069	−0.0860346	+	0.996292i	$0.527420\pi$
$72$	0	0
$73$	−200.652	−0.321707	−0.160853	−	0.986978i	$-0.551425\pi$
$73$	−200.652	−0.321707	−0.160853	+	0.986978i	$0.551425\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−59.4963	−0.0880550
$78$	0	0
$79$	−665.931	−0.948394	−0.474197	−	0.880419i	$-0.657262\pi$
$79$	−665.931	−0.948394	−0.474197	+	0.880419i	$0.657262\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1347.41	−1.78189	−0.890946	−	0.454109i	$-0.849958\pi$
$83$	−1347.41	−1.78189	−0.890946	+	0.454109i	$0.849958\pi$
$84$	0	0
$85$	−517.056	−0.659795
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.6866	0.0151099	0.00755496	−	0.999971i	$-0.497595\pi$
$89$	12.6866	0.0151099	0.00755496	+	0.999971i	$0.497595\pi$
$90$	0	0
$91$	6.59922	0.00760204
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−369.032	−0.398546
$96$	0	0
$97$	−637.379	−0.667175	−0.333588	−	0.942719i	$-0.608259\pi$
$97$	−637.379	−0.667175	−0.333588	+	0.942719i	$0.608259\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	736.903	0.725986	0.362993	−	0.931792i	$-0.381755\pi$
$101$	736.903	0.725986	0.362993	+	0.931792i	$0.381755\pi$
$102$	0	0
$103$	−1396.64	−1.33607	−0.668034	−	0.744130i	$-0.732864\pi$
$103$	−1396.64	−1.33607	−0.668034	+	0.744130i	$0.732864\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	181.225	0.163735	0.0818677	−	0.996643i	$-0.473911\pi$
$107$	181.225	0.163735	0.0818677	+	0.996643i	$0.473911\pi$
$108$	0	0
$109$	1495.62	1.31426	0.657132	−	0.753775i	$-0.271769\pi$
$109$	1495.62	1.31426	0.657132	+	0.753775i	$0.271769\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2238.50	−1.86354	−0.931771	−	0.363048i	$-0.881736\pi$
$113$	−2238.50	−1.86354	−0.931771	+	0.363048i	$0.881736\pi$
$114$	0	0
$115$	430.795	0.349321
$116$	0	0
$117$	0	0
$118$	0	0
$119$	98.1976	0.0756450
$120$	0	0
$121$	2594.67	1.94941
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1163.69	0.813075	0.406537	−	0.913634i	$-0.366736\pi$
$127$	1163.69	0.813075	0.406537	+	0.913634i	$0.366736\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	98.9947	0.0660245	0.0330122	−	0.999455i	$-0.489490\pi$
$131$	98.9947	0.0660245	0.0330122	+	0.999455i	$0.489490\pi$
$132$	0	0
$133$	70.0854	0.0456930
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−174.907	−0.109076	−0.0545378	−	0.998512i	$-0.517369\pi$
$137$	−174.907	−0.109076	−0.0545378	+	0.998512i	$0.517369\pi$
$138$	0	0
$139$	981.514	0.598928	0.299464	−	0.954108i	$-0.403192\pi$
$139$	981.514	0.598928	0.299464	+	0.954108i	$0.403192\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−435.427	−0.254631
$144$	0	0
$145$	−277.812	−0.159111
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2306.94	−1.26840	−0.634200	−	0.773169i	$-0.718670\pi$
$149$	−2306.94	−1.26840	−0.634200	+	0.773169i	$0.718670\pi$
$150$	0	0
$151$	−1981.80	−1.06806	−0.534029	−	0.845466i	$-0.679323\pi$
$151$	−1981.80	−1.06806	−0.534029	+	0.845466i	$0.679323\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−999.323	−0.517855
$156$	0	0
$157$	−433.147	−0.220184	−0.110092	−	0.993921i	$-0.535115\pi$
$157$	−433.147	−0.220184	−0.110092	+	0.993921i	$0.535115\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−81.8153	−0.0400494
$162$	0	0
$163$	3447.72	1.65673	0.828363	−	0.560192i	$-0.189273\pi$
$163$	3447.72	1.65673	0.828363	+	0.560192i	$0.189273\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1054.89	−0.488801	−0.244401	−	0.969674i	$-0.578591\pi$
$167$	−1054.89	−0.488801	−0.244401	+	0.969674i	$0.578591\pi$
$168$	0	0
$169$	−2148.70	−0.978017
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−632.144	−0.277809	−0.138905	−	0.990306i	$-0.544358\pi$
$173$	−632.144	−0.277809	−0.138905	+	0.990306i	$0.544358\pi$
$174$	0	0
$175$	23.7396	0.0102545
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1329.33	−0.555078	−0.277539	−	0.960714i	$-0.589519\pi$
$179$	−1329.33	−0.555078	−0.277539	+	0.960714i	$0.589519\pi$
$180$	0	0
$181$	1468.52	0.603062	0.301531	−	0.953456i	$-0.402502\pi$
$181$	1468.52	0.603062	0.301531	+	0.953456i	$0.402502\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	94.5349	0.0375694
$186$	0	0
$187$	−6479.24	−2.53374
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−482.390	−0.182746	−0.0913732	−	0.995817i	$-0.529126\pi$
$191$	−482.390	−0.182746	−0.0913732	+	0.995817i	$0.529126\pi$
$192$	0	0
$193$	−343.353	−0.128057	−0.0640287	−	0.997948i	$-0.520395\pi$
$193$	−343.353	−0.128057	−0.0640287	+	0.997948i	$0.520395\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4507.54	−1.63020	−0.815099	−	0.579321i	$-0.803318\pi$
$197$	−4507.54	−1.63020	−0.815099	+	0.579321i	$0.803318\pi$
$198$	0	0
$199$	−1750.03	−0.623397	−0.311699	−	0.950181i	$-0.600898\pi$
$199$	−1750.03	−0.623397	−0.311699	+	0.950181i	$0.600898\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	52.7612	0.0182419
$204$	0	0
$205$	−286.512	−0.0976141
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4624.35	−1.53049
$210$	0	0
$211$	−5200.05	−1.69662	−0.848308	−	0.529502i	$-0.822379\pi$
$211$	−5200.05	−1.69662	−0.848308	+	0.529502i	$0.822379\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1480.16	−0.469516
$216$	0	0
$217$	189.788	0.0593717
$218$	0	0
$219$	0	0
$220$	0	0
$221$	718.664	0.218745
$222$	0	0
$223$	−2905.47	−0.872489	−0.436244	−	0.899828i	$-0.643692\pi$
$223$	−2905.47	−0.872489	−0.436244	+	0.899828i	$0.643692\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1734.02	−0.507009	−0.253504	−	0.967334i	$-0.581583\pi$
$227$	−1734.02	−0.507009	−0.253504	+	0.967334i	$0.581583\pi$
$228$	0	0
$229$	3472.97	1.00218	0.501092	−	0.865394i	$-0.332932\pi$
$229$	3472.97	1.00218	0.501092	+	0.865394i	$0.332932\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	595.216	0.167356	0.0836779	−	0.996493i	$-0.473333\pi$
$233$	595.216	0.167356	0.0836779	+	0.996493i	$0.473333\pi$
$234$	0	0
$235$	287.692	0.0798595
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2179.16	−0.589784	−0.294892	−	0.955531i	$-0.595284\pi$
$239$	−2179.16	−0.589784	−0.294892	+	0.955531i	$0.595284\pi$
$240$	0	0
$241$	−4362.16	−1.16594	−0.582970	−	0.812494i	$-0.698109\pi$
$241$	−4362.16	−1.16594	−0.582970	+	0.812494i	$0.698109\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1710.49	0.446038
$246$	0	0
$247$	512.924	0.132132
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2738.00	−0.688531	−0.344266	−	0.938872i	$-0.611872\pi$
$251$	−2738.00	−0.688531	−0.344266	+	0.938872i	$0.611872\pi$
$252$	0	0
$253$	5398.31	1.34146
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1995.09	0.484243	0.242122	−	0.970246i	$-0.422157\pi$
$257$	1995.09	0.484243	0.242122	+	0.970246i	$0.422157\pi$
$258$	0	0
$259$	−17.9538	−0.00430731
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2553.56	0.598705	0.299352	−	0.954143i	$-0.403229\pi$
$263$	2553.56	0.598705	0.299352	+	0.954143i	$0.403229\pi$
$264$	0	0
$265$	3360.03	0.778886
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4534.66	−1.02782	−0.513909	−	0.857845i	$-0.671803\pi$
$269$	−4534.66	−1.02782	−0.513909	+	0.857845i	$0.671803\pi$
$270$	0	0
$271$	4390.49	0.984145	0.492073	−	0.870554i	$-0.336239\pi$
$271$	4390.49	0.984145	0.492073	+	0.870554i	$0.336239\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1566.38	−0.343477
$276$	0	0
$277$	6726.79	1.45911	0.729555	−	0.683922i	$-0.239727\pi$
$277$	6726.79	1.45911	0.729555	+	0.683922i	$0.239727\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7238.73	−1.53675	−0.768374	−	0.640001i	$-0.778934\pi$
$281$	−7238.73	−1.53675	−0.768374	+	0.640001i	$0.778934\pi$
$282$	0	0
$283$	7453.35	1.56557	0.782784	−	0.622293i	$-0.213799\pi$
$283$	7453.35	1.56557	0.782784	+	0.622293i	$0.213799\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	54.4135	0.0111914
$288$	0	0
$289$	5780.86	1.17665
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3064.73	−0.611071	−0.305535	−	0.952181i	$-0.598835\pi$
$293$	−3064.73	−0.611071	−0.305535	+	0.952181i	$0.598835\pi$
$294$	0	0
$295$	−2733.22	−0.539438
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−598.770	−0.115812
$300$	0	0
$301$	281.107	0.0538297
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3957.80	0.743026
$306$	0	0
$307$	7427.82	1.38087	0.690437	−	0.723393i	$-0.257418\pi$
$307$	7427.82	1.38087	0.690437	+	0.723393i	$0.257418\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7574.79	−1.38111	−0.690557	−	0.723278i	$-0.742635\pi$
$311$	−7574.79	−1.38111	−0.690557	+	0.723278i	$0.742635\pi$
$312$	0	0
$313$	−8628.60	−1.55820	−0.779101	−	0.626898i	$-0.784324\pi$
$313$	−8628.60	−1.55820	−0.779101	+	0.626898i	$0.784324\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2946.09	−0.521983	−0.260992	−	0.965341i	$-0.584049\pi$
$317$	−2946.09	−0.521983	−0.260992	+	0.965341i	$0.584049\pi$
$318$	0	0
$319$	−3481.27	−0.611015
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7632.40	1.31479
$324$	0	0
$325$	173.740	0.0296534
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−54.6376	−0.00915583
$330$	0	0
$331$	3137.45	0.520996	0.260498	−	0.965474i	$-0.416113\pi$
$331$	3137.45	0.520996	0.260498	+	0.965474i	$0.416113\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3157.80	−0.515012
$336$	0	0
$337$	−3715.74	−0.600621	−0.300311	−	0.953841i	$-0.597090\pi$
$337$	−3715.74	−0.600621	−0.300311	+	0.953841i	$0.597090\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−12522.5	−1.98866
$342$	0	0
$343$	−650.559	−0.102411
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6060.02	−0.937518	−0.468759	−	0.883326i	$-0.655299\pi$
$347$	−6060.02	−0.937518	−0.468759	+	0.883326i	$0.655299\pi$
$348$	0	0
$349$	−5634.65	−0.864229	−0.432115	−	0.901819i	$-0.642232\pi$
$349$	−5634.65	−0.864229	−0.432115	+	0.901819i	$0.642232\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4557.89	−0.687229	−0.343614	−	0.939111i	$-0.611651\pi$
$353$	−4557.89	−0.687229	−0.343614	+	0.939111i	$0.611651\pi$
$354$	0	0
$355$	514.708	0.0769517
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3206.54	0.471406	0.235703	−	0.971825i	$-0.424261\pi$
$359$	3206.54	0.471406	0.235703	+	0.971825i	$0.424261\pi$
$360$	0	0
$361$	−1411.62	−0.205805
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1003.26	0.143872
$366$	0	0
$367$	−8700.21	−1.23746	−0.618730	−	0.785604i	$-0.712352\pi$
$367$	−8700.21	−1.23746	−0.618730	+	0.785604i	$0.712352\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−638.126	−0.0892987
$372$	0	0
$373$	12722.5	1.76607	0.883035	−	0.469308i	$-0.155497\pi$
$373$	12722.5	1.76607	0.883035	+	0.469308i	$0.155497\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	386.135	0.0527506
$378$	0	0
$379$	1339.34	0.181523	0.0907613	−	0.995873i	$-0.471070\pi$
$379$	1339.34	0.181523	0.0907613	+	0.995873i	$0.471070\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4729.50	0.630982	0.315491	−	0.948928i	$-0.397831\pi$
$383$	4729.50	0.630982	0.315491	+	0.948928i	$0.397831\pi$
$384$	0	0
$385$	297.482	0.0393794
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4838.38	0.630631	0.315316	−	0.948987i	$-0.397890\pi$
$389$	4838.38	0.630631	0.315316	+	0.948987i	$0.397890\pi$
$390$	0	0
$391$	−8909.81	−1.15240
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3329.66	0.424135
$396$	0	0
$397$	−7768.23	−0.982056	−0.491028	−	0.871144i	$-0.663379\pi$
$397$	−7768.23	−0.982056	−0.491028	+	0.871144i	$0.663379\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5207.07	0.648450	0.324225	−	0.945980i	$-0.394896\pi$
$401$	5207.07	0.648450	0.324225	+	0.945980i	$0.394896\pi$
$402$	0	0
$403$	1388.98	0.171687
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1184.62	0.144274
$408$	0	0
$409$	−7831.43	−0.946795	−0.473398	−	0.880849i	$-0.656973\pi$
$409$	−7831.43	−0.946795	−0.473398	+	0.880849i	$0.656973\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	519.085	0.0618462
$414$	0	0
$415$	6737.03	0.796886
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11700.2	1.36419	0.682093	−	0.731265i	$-0.261070\pi$
$419$	11700.2	1.36419	0.682093	+	0.731265i	$0.261070\pi$
$420$	0	0
$421$	−3235.99	−0.374614	−0.187307	−	0.982301i	$-0.559976\pi$
$421$	−3235.99	−0.374614	−0.187307	+	0.982301i	$0.559976\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2585.28	0.295069
$426$	0	0
$427$	−751.652	−0.0851874
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7828.48	−0.874907	−0.437453	−	0.899241i	$-0.644119\pi$
$431$	−7828.48	−0.874907	−0.437453	+	0.899241i	$0.644119\pi$
$432$	0	0
$433$	−11016.3	−1.22265	−0.611327	−	0.791378i	$-0.709364\pi$
$433$	−11016.3	−1.22265	−0.611327	+	0.791378i	$0.709364\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6359.09	−0.696102
$438$	0	0
$439$	−648.915	−0.0705491	−0.0352745	−	0.999378i	$-0.511231\pi$
$439$	−648.915	−0.0705491	−0.0352745	+	0.999378i	$0.511231\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−914.423	−0.0980712	−0.0490356	−	0.998797i	$-0.515615\pi$
$443$	−914.423	−0.0980712	−0.0490356	+	0.998797i	$0.515615\pi$
$444$	0	0
$445$	−63.4332	−0.00675736
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9663.37	−1.01569	−0.507843	−	0.861450i	$-0.669557\pi$
$449$	−9663.37	−1.01569	−0.507843	+	0.861450i	$0.669557\pi$
$450$	0	0
$451$	−3590.30	−0.374857
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−32.9961	−0.00339974
$456$	0	0
$457$	8945.84	0.915686	0.457843	−	0.889033i	$-0.348622\pi$
$457$	8945.84	0.915686	0.457843	+	0.889033i	$0.348622\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6388.10	0.645388	0.322694	−	0.946503i	$-0.395412\pi$
$461$	6388.10	0.645388	0.322694	+	0.946503i	$0.395412\pi$
$462$	0	0
$463$	−189.765	−0.0190478	−0.00952389	−	0.999955i	$-0.503032\pi$
$463$	−189.765	−0.0190478	−0.00952389	+	0.999955i	$0.503032\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17087.3	−1.69317	−0.846583	−	0.532257i	$-0.821344\pi$
$467$	−17087.3	−1.69317	−0.846583	+	0.532257i	$0.821344\pi$
$468$	0	0
$469$	599.719	0.0590458
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−18547.9	−1.80303
$474$	0	0
$475$	1845.16	0.178235
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7748.19	0.739089	0.369545	−	0.929213i	$-0.379514\pi$
$479$	7748.19	0.739089	0.369545	+	0.929213i	$0.379514\pi$
$480$	0	0
$481$	−131.396	−0.0124556
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3186.89	0.298370
$486$	0	0
$487$	5330.93	0.496032	0.248016	−	0.968756i	$-0.420221\pi$
$487$	5330.93	0.496032	0.248016	+	0.968756i	$0.420221\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14497.8	1.33254	0.666269	−	0.745712i	$-0.267890\pi$
$491$	14497.8	1.33254	0.666269	+	0.745712i	$0.267890\pi$
$492$	0	0
$493$	5745.77	0.524901
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−97.7516	−0.00882246
$498$	0	0
$499$	−3062.84	−0.274772	−0.137386	−	0.990518i	$-0.543870\pi$
$499$	−3062.84	−0.274772	−0.137386	+	0.990518i	$0.543870\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10683.8	0.947055	0.473527	−	0.880779i	$-0.342980\pi$
$503$	10683.8	0.947055	0.473527	+	0.880779i	$0.342980\pi$
$504$	0	0
$505$	−3684.52	−0.324671
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14374.4	−1.25174	−0.625868	−	0.779929i	$-0.715255\pi$
$509$	−14374.4	−1.25174	−0.625868	+	0.779929i	$0.715255\pi$
$510$	0	0
$511$	−190.536	−0.0164948
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6983.20	0.597508
$516$	0	0
$517$	3605.08	0.306676
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9386.82	0.789336	0.394668	−	0.918824i	$-0.370860\pi$
$521$	9386.82	0.789336	0.394668	+	0.918824i	$0.370860\pi$
$522$	0	0
$523$	15352.7	1.28361	0.641805	−	0.766868i	$-0.278186\pi$
$523$	15352.7	1.28361	0.641805	+	0.766868i	$0.278186\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20668.2	1.70839
$528$	0	0
$529$	−4743.62	−0.389876
$530$	0	0
$531$	0	0
$532$	0	0
$533$	398.229	0.0323625
$534$	0	0
$535$	−906.125	−0.0732247
$536$	0	0
$537$	0	0
$538$	0	0
$539$	21434.2	1.71287
$540$	0	0
$541$	−23730.6	−1.88588	−0.942938	−	0.332968i	$-0.891950\pi$
$541$	−23730.6	−1.88588	−0.942938	+	0.332968i	$0.891950\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7478.12	−0.587757
$546$	0	0
$547$	−13140.6	−1.02715	−0.513577	−	0.858043i	$-0.671680\pi$
$547$	−13140.6	−1.02715	−0.513577	+	0.858043i	$0.671680\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4100.86	0.317064
$552$	0	0
$553$	−632.358	−0.0486268
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19848.3	1.50987	0.754935	−	0.655800i	$-0.227668\pi$
$557$	19848.3	1.50987	0.754935	+	0.655800i	$0.227668\pi$
$558$	0	0
$559$	2057.30	0.155661
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18601.9	−1.39250	−0.696250	−	0.717800i	$-0.745149\pi$
$563$	−18601.9	−1.39250	−0.696250	+	0.717800i	$0.745149\pi$
$564$	0	0
$565$	11192.5	0.833401
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19378.0	−1.42771	−0.713857	−	0.700292i	$-0.753053\pi$
$569$	−19378.0	−1.42771	−0.713857	+	0.700292i	$0.753053\pi$
$570$	0	0
$571$	4682.33	0.343169	0.171584	−	0.985169i	$-0.445111\pi$
$571$	4682.33	0.343169	0.171584	+	0.985169i	$0.445111\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2153.98	−0.156221
$576$	0	0
$577$	−20543.6	−1.48222	−0.741111	−	0.671382i	$-0.765701\pi$
$577$	−20543.6	−1.48222	−0.741111	+	0.671382i	$0.765701\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1279.48	−0.0913625
$582$	0	0
$583$	42104.6	2.99107
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3698.34	0.260046	0.130023	−	0.991511i	$-0.458495\pi$
$587$	3698.34	0.260046	0.130023	+	0.991511i	$0.458495\pi$
$588$	0	0
$589$	14751.3	1.03195
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9685.31	0.670705	0.335353	−	0.942093i	$-0.391145\pi$
$593$	9685.31	0.670705	0.335353	+	0.942093i	$0.391145\pi$
$594$	0	0
$595$	−490.988	−0.0338295
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14009.7	0.955629	0.477814	−	0.878461i	$-0.341429\pi$
$599$	14009.7	0.955629	0.477814	+	0.878461i	$0.341429\pi$
$600$	0	0
$601$	−3243.57	−0.220146	−0.110073	−	0.993923i	$-0.535108\pi$
$601$	−3243.57	−0.220146	−0.110073	+	0.993923i	$0.535108\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−12973.3	−0.871804
$606$	0	0
$607$	−11726.6	−0.784135	−0.392068	−	0.919936i	$-0.628240\pi$
$607$	−11726.6	−0.784135	−0.392068	+	0.919936i	$0.628240\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−399.868	−0.0264762
$612$	0	0
$613$	20343.9	1.34043	0.670215	−	0.742167i	$-0.266202\pi$
$613$	20343.9	1.34043	0.670215	+	0.742167i	$0.266202\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21893.5	1.42852	0.714260	−	0.699880i	$-0.246763\pi$
$617$	21893.5	1.42852	0.714260	+	0.699880i	$0.246763\pi$
$618$	0	0
$619$	29767.0	1.93285	0.966426	−	0.256944i	$-0.0827157\pi$
$619$	29767.0	1.93285	0.966426	+	0.256944i	$0.0827157\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.0470	0.000774726 0
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1955.19	−0.123941
$630$	0	0
$631$	−6594.67	−0.416054	−0.208027	−	0.978123i	$-0.566704\pi$
$631$	−6594.67	−0.416054	−0.208027	+	0.978123i	$0.566704\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5818.43	−0.363618
$636$	0	0
$637$	−2377.44	−0.147877
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16161.4	0.995847	0.497923	−	0.867221i	$-0.334096\pi$
$641$	16161.4	0.995847	0.497923	+	0.867221i	$0.334096\pi$
$642$	0	0
$643$	−15876.2	−0.973710	−0.486855	−	0.873483i	$-0.661856\pi$
$643$	−15876.2	−0.973710	−0.486855	+	0.873483i	$0.661856\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−624.258	−0.0379322	−0.0189661	−	0.999820i	$-0.506037\pi$
$647$	−624.258	−0.0379322	−0.0189661	+	0.999820i	$0.506037\pi$
$648$	0	0
$649$	−34250.1	−2.07155
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10146.4	0.608053	0.304027	−	0.952664i	$-0.401669\pi$
$653$	10146.4	0.608053	0.304027	+	0.952664i	$0.401669\pi$
$654$	0	0
$655$	−494.973	−0.0295270
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24050.3	1.42165	0.710826	−	0.703368i	$-0.248321\pi$
$659$	24050.3	1.42165	0.710826	+	0.703368i	$0.248321\pi$
$660$	0	0
$661$	−986.004	−0.0580198	−0.0290099	−	0.999579i	$-0.509235\pi$
$661$	−986.004	−0.0580198	−0.0290099	+	0.999579i	$0.509235\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−350.427	−0.0204345
$666$	0	0
$667$	−4787.20	−0.277903
$668$	0	0
$669$	0	0
$670$	0	0
$671$	49595.3	2.85336
$672$	0	0
$673$	30055.5	1.72148	0.860739	−	0.509046i	$-0.170002\pi$
$673$	30055.5	1.72148	0.860739	+	0.509046i	$0.170002\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22370.6	1.26997	0.634986	−	0.772523i	$-0.281006\pi$
$677$	22370.6	1.26997	0.634986	+	0.772523i	$0.281006\pi$
$678$	0	0
$679$	−605.245	−0.0342079
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27040.9	1.51492	0.757460	−	0.652881i	$-0.226440\pi$
$683$	27040.9	1.51492	0.757460	+	0.652881i	$0.226440\pi$
$684$	0	0
$685$	874.537	0.0487801
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4670.16	−0.258228
$690$	0	0
$691$	−22601.5	−1.24429	−0.622143	−	0.782904i	$-0.713738\pi$
$691$	−22601.5	−1.24429	−0.622143	+	0.782904i	$0.713738\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4907.57	−0.267849
$696$	0	0
$697$	5925.72	0.322027
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−20238.5	−1.09044	−0.545219	−	0.838294i	$-0.683554\pi$
$701$	−20238.5	−1.09044	−0.545219	+	0.838294i	$0.683554\pi$
$702$	0	0
$703$	−1395.46	−0.0748658
$704$	0	0
$705$	0	0
$706$	0	0
$707$	699.752	0.0372233
$708$	0	0
$709$	7959.77	0.421630	0.210815	−	0.977526i	$-0.432388\pi$
$709$	7959.77	0.421630	0.210815	+	0.977526i	$0.432388\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−17220.1	−0.904488
$714$	0	0
$715$	2177.14	0.113875
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22376.4	−1.16064	−0.580320	−	0.814389i	$-0.697072\pi$
$719$	−22376.4	−1.16064	−0.580320	+	0.814389i	$0.697072\pi$
$720$	0	0
$721$	−1326.23	−0.0685039
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1389.06	0.0711564
$726$	0	0
$727$	13278.2	0.677389	0.338694	−	0.940896i	$-0.390015\pi$
$727$	13278.2	0.677389	0.338694	+	0.940896i	$0.390015\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	30613.0	1.54892
$732$	0	0
$733$	−11726.7	−0.590906	−0.295453	−	0.955357i	$-0.595471\pi$
$733$	−11726.7	−0.590906	−0.295453	+	0.955357i	$0.595471\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−39570.5	−1.97774
$738$	0	0
$739$	−10822.8	−0.538733	−0.269366	−	0.963038i	$-0.586814\pi$
$739$	−10822.8	−0.538733	−0.269366	+	0.963038i	$0.586814\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−15108.6	−0.746003	−0.373002	−	0.927831i	$-0.621671\pi$
$743$	−15108.6	−0.746003	−0.373002	+	0.927831i	$0.621671\pi$
$744$	0	0
$745$	11534.7	0.567246
$746$	0	0
$747$	0	0
$748$	0	0
$749$	172.088	0.00839516
$750$	0	0
$751$	−3929.51	−0.190932	−0.0954659	−	0.995433i	$-0.530434\pi$
$751$	−3929.51	−0.190932	−0.0954659	+	0.995433i	$0.530434\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	9909.01	0.477650
$756$	0	0
$757$	−32082.2	−1.54035	−0.770176	−	0.637831i	$-0.779832\pi$
$757$	−32082.2	−1.54035	−0.770176	+	0.637831i	$0.779832\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29348.1	1.39799	0.698994	−	0.715128i	$-0.253632\pi$
$761$	29348.1	1.39799	0.698994	+	0.715128i	$0.253632\pi$
$762$	0	0
$763$	1420.22	0.0673860
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3798.95	0.178842
$768$	0	0
$769$	15550.8	0.729230	0.364615	−	0.931158i	$-0.381201\pi$
$769$	15550.8	0.729230	0.364615	+	0.931158i	$0.381201\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	29954.5	1.39378	0.696888	−	0.717180i	$-0.254567\pi$
$773$	29954.5	1.39378	0.696888	+	0.717180i	$0.254567\pi$
$774$	0	0
$775$	4996.62	0.231592
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4229.29	0.194519
$780$	0	0
$781$	6449.82	0.295509
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2165.73	0.0984692
$786$	0	0
$787$	−15602.2	−0.706683	−0.353341	−	0.935494i	$-0.614955\pi$
$787$	−15602.2	−0.706683	−0.353341	+	0.935494i	$0.614955\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2125.64	−0.0955488
$792$	0	0
$793$	−5501.01	−0.246339
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30628.9	1.36127	0.680635	−	0.732623i	$-0.261704\pi$
$797$	30628.9	1.36127	0.680635	+	0.732623i	$0.261704\pi$
$798$	0	0
$799$	−5950.12	−0.263454
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12571.9	0.552494
$804$	0	0
$805$	409.076	0.0179106
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19959.7	0.867425	0.433713	−	0.901051i	$-0.357203\pi$
$809$	19959.7	0.867425	0.433713	+	0.901051i	$0.357203\pi$
$810$	0	0
$811$	3768.40	0.163165	0.0815823	−	0.996667i	$-0.474003\pi$
$811$	3768.40	0.163165	0.0815823	+	0.996667i	$0.474003\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17238.6	−0.740910
$816$	0	0
$817$	21849.0	0.935619
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29287.6	−1.24500	−0.622500	−	0.782620i	$-0.713883\pi$
$821$	−29287.6	−1.24500	−0.622500	+	0.782620i	$0.713883\pi$
$822$	0	0
$823$	−1061.18	−0.0449459	−0.0224729	−	0.999747i	$-0.507154\pi$
$823$	−1061.18	−0.0449459	−0.0224729	+	0.999747i	$0.507154\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43016.7	1.80875	0.904375	−	0.426739i	$-0.140338\pi$
$827$	43016.7	1.80875	0.904375	+	0.426739i	$0.140338\pi$
$828$	0	0
$829$	−27947.5	−1.17088	−0.585439	−	0.810716i	$-0.699078\pi$
$829$	−27947.5	−1.17088	−0.585439	+	0.810716i	$0.699078\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−35376.8	−1.47147
$834$	0	0
$835$	5274.45	0.218599
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−13095.6	−0.538866	−0.269433	−	0.963019i	$-0.586836\pi$
$839$	−13095.6	−0.538866	−0.269433	+	0.963019i	$0.586836\pi$
$840$	0	0
$841$	−21301.8	−0.873419
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10743.5	0.437382
$846$	0	0
$847$	2463.86	0.0999517
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1629.01	0.0656189
$852$	0	0
$853$	32225.5	1.29353	0.646764	−	0.762690i	$-0.276122\pi$
$853$	32225.5	1.29353	0.646764	+	0.762690i	$0.276122\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5251.95	0.209339	0.104669	−	0.994507i	$-0.466622\pi$
$857$	5251.95	0.209339	0.104669	+	0.994507i	$0.466622\pi$
$858$	0	0
$859$	−35129.6	−1.39535	−0.697675	−	0.716414i	$-0.745782\pi$
$859$	−35129.6	−1.39535	−0.697675	+	0.716414i	$0.745782\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47402.3	1.86975	0.934874	−	0.354979i	$-0.115512\pi$
$863$	47402.3	1.86975	0.934874	+	0.354979i	$0.115512\pi$
$864$	0	0
$865$	3160.72	0.124240
$866$	0	0
$867$	0	0
$868$	0	0
$869$	41724.0	1.62876
$870$	0	0
$871$	4389.08	0.170744
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−118.698	−0.00458597
$876$	0	0
$877$	23727.3	0.913584	0.456792	−	0.889573i	$-0.348998\pi$
$877$	23727.3	0.913584	0.456792	+	0.889573i	$0.348998\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−43690.9	−1.67081	−0.835406	−	0.549634i	$-0.814767\pi$
$881$	−43690.9	−1.67081	−0.835406	+	0.549634i	$0.814767\pi$
$882$	0	0
$883$	7852.57	0.299275	0.149638	−	0.988741i	$-0.452189\pi$
$883$	7852.57	0.299275	0.149638	+	0.988741i	$0.452189\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16014.3	−0.606209	−0.303105	−	0.952957i	$-0.598023\pi$
$887$	−16014.3	−0.606209	−0.303105	+	0.952957i	$0.598023\pi$
$888$	0	0
$889$	1105.02	0.0416886
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4246.71	−0.159138
$894$	0	0
$895$	6646.66	0.248238
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11105.0	0.411981
$900$	0	0
$901$	−69492.8	−2.56952
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7342.60	−0.269698
$906$	0	0
$907$	25873.3	0.947200	0.473600	−	0.880740i	$-0.342954\pi$
$907$	25873.3	0.947200	0.473600	+	0.880740i	$0.342954\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23293.1	0.847128	0.423564	−	0.905866i	$-0.360779\pi$
$911$	23293.1	0.847128	0.423564	+	0.905866i	$0.360779\pi$
$912$	0	0
$913$	84421.9	3.06020
$914$	0	0
$915$	0	0
$916$	0	0
$917$	94.0038	0.00338525
$918$	0	0
$919$	42422.5	1.52273	0.761365	−	0.648323i	$-0.224529\pi$
$919$	42422.5	1.52273	0.761365	+	0.648323i	$0.224529\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−715.401	−0.0255121
$924$	0	0
$925$	−472.674	−0.0168016
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34596.0	1.22181	0.610903	−	0.791705i	$-0.290807\pi$
$929$	34596.0	1.22181	0.610903	+	0.791705i	$0.290807\pi$
$930$	0	0
$931$	−25249.0	−0.888833
$932$	0	0
$933$	0	0
$934$	0	0
$935$	32396.2	1.13312
$936$	0	0
$937$	17173.8	0.598767	0.299383	−	0.954133i	$-0.403219\pi$
$937$	17173.8	0.598767	0.299383	+	0.954133i	$0.403219\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−33493.5	−1.16032	−0.580158	−	0.814504i	$-0.697009\pi$
$941$	−33493.5	−1.16032	−0.580158	+	0.814504i	$0.697009\pi$
$942$	0	0
$943$	−4937.13	−0.170493
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21477.0	0.736968	0.368484	−	0.929634i	$-0.379877\pi$
$947$	21477.0	0.736968	0.368484	+	0.929634i	$0.379877\pi$
$948$	0	0
$949$	−1394.45	−0.0476984
$950$	0	0
$951$	0	0
$952$	0	0
$953$	19415.1	0.659934	0.329967	−	0.943992i	$-0.392962\pi$
$953$	19415.1	0.659934	0.329967	+	0.943992i	$0.392962\pi$
$954$	0	0
$955$	2411.95	0.0817266
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−166.089	−0.00559260
$960$	0	0
$961$	10154.9	0.340870
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1716.77	0.0572691
$966$	0	0
$967$	−54931.8	−1.82677	−0.913386	−	0.407094i	$-0.866542\pi$
$967$	−54931.8	−1.82677	−0.913386	+	0.407094i	$0.866542\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.202715	6.69972e−6 0	3.34986e−6	−	1.00000i	$-0.499999\pi$
$971$	0.202715	6.69972e−6 0	3.34986e−6		1.00000i	$0.499999\pi$
$972$	0	0
$973$	932.030	0.0307086
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−22873.5	−0.749015	−0.374508	−	0.927224i	$-0.622188\pi$
$977$	−22873.5	−0.749015	−0.374508	+	0.927224i	$0.622188\pi$
$978$	0	0
$979$	−794.884	−0.0259495
$980$	0	0
$981$	0	0
$982$	0	0
$983$	26259.5	0.852032	0.426016	−	0.904716i	$-0.359917\pi$
$983$	26259.5	0.852032	0.426016	+	0.904716i	$0.359917\pi$
$984$	0	0
$985$	22537.7	0.729047
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−25505.8	−0.820058
$990$	0	0
$991$	−40250.1	−1.29020	−0.645099	−	0.764099i	$-0.723184\pi$
$991$	−40250.1	−1.29020	−0.645099	+	0.764099i	$0.723184\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8750.13	0.278792
$996$	0	0
$997$	−59087.1	−1.87694	−0.938469	−	0.345362i	$-0.887756\pi$
$997$	−59087.1	−1.87694	−0.938469	+	0.345362i	$0.887756\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.d.1.2	✓	3
3.2	odd	2		1620.4.a.f.1.2	yes	3
9.2	odd	6		1620.4.i.s.1081.2		6
9.4	even	3		1620.4.i.u.541.2		6
9.5	odd	6		1620.4.i.s.541.2		6
9.7	even	3		1620.4.i.u.1081.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.d.1.2	✓	3	1.1	even	1	trivial
1620.4.a.f.1.2	yes	3	3.2	odd	2
1620.4.i.s.541.2		6	9.5	odd	6
1620.4.i.s.1081.2		6	9.2	odd	6
1620.4.i.u.541.2		6	9.4	even	3
1620.4.i.u.1081.2		6	9.7	even	3

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +0.949584 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +0.949584 q^{7} -62.6552 q^{11} +6.94958 q^{13} +103.411 q^{17} +73.8064 q^{19} -86.1591 q^{23} +25.0000 q^{25} +55.5624 q^{29} +199.865 q^{31} -4.74792 q^{35} -18.9070 q^{37} +57.3025 q^{41} +296.032 q^{43} -57.5384 q^{47} -342.098 q^{49} -672.005 q^{53} +313.276 q^{55} +546.644 q^{59} -791.559 q^{61} -34.7479 q^{65} +631.560 q^{67} -102.942 q^{71} -200.652 q^{73} -59.4963 q^{77} -665.931 q^{79} -1347.41 q^{83} -517.056 q^{85} +12.6866 q^{89} +6.59922 q^{91} -369.032 q^{95} -637.379 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} + 15 q^{7}+O(q^{10}) \) 3 * q - 15 * q^5 + 15 * q^7	\( 3 q - 15 q^{5} + 15 q^{7} - 24 q^{11} + 33 q^{13} - 42 q^{17} + 21 q^{19} + 33 q^{23} + 75 q^{25} - 222 q^{29} + 132 q^{31} - 75 q^{35} + 174 q^{37} + 99 q^{41} - 120 q^{43} - 537 q^{47} + 492 q^{49} - 267 q^{53} + 120 q^{55} + 225 q^{59} - 480 q^{61} - 165 q^{65} + 12 q^{67} - 570 q^{71} + 1062 q^{73} - 312 q^{77} - 1026 q^{79} - 702 q^{83} + 210 q^{85} - 1140 q^{89} + 1611 q^{91} - 105 q^{95} - 2556 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 + 15 * q^7 - 24 * q^11 + 33 * q^13 - 42 * q^17 + 21 * q^19 + 33 * q^23 + 75 * q^25 - 222 * q^29 + 132 * q^31 - 75 * q^35 + 174 * q^37 + 99 * q^41 - 120 * q^43 - 537 * q^47 + 492 * q^49 - 267 * q^53 + 120 * q^55 + 225 * q^59 - 480 * q^61 - 165 * q^65 + 12 * q^67 - 570 * q^71 + 1062 * q^73 - 312 * q^77 - 1026 * q^79 - 702 * q^83 + 210 * q^85 - 1140 * q^89 + 1611 * q^91 - 105 * q^95 - 2556 * q^97

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