LMFDB - Embedded newform 6004.2.a.e.1.9

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:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-1.93761 q^{3} +0.795901 q^{5} +1.65858 q^{7} +0.754317 q^{9} +O(q^{10})$	$q-1.93761 q^{3} +0.795901 q^{5} +1.65858 q^{7} +0.754317 q^{9} +4.98552 q^{11} -5.14293 q^{13} -1.54214 q^{15} -7.30716 q^{17} +1.00000 q^{19} -3.21368 q^{21} -0.482448 q^{23} -4.36654 q^{25} +4.35125 q^{27} -1.64880 q^{29} -2.09426 q^{31} -9.65997 q^{33} +1.32007 q^{35} +7.37275 q^{37} +9.96498 q^{39} +8.34657 q^{41} +6.41319 q^{43} +0.600362 q^{45} -9.11366 q^{47} -4.24910 q^{49} +14.1584 q^{51} +5.62276 q^{53} +3.96798 q^{55} -1.93761 q^{57} +1.75283 q^{59} -0.496017 q^{61} +1.25110 q^{63} -4.09327 q^{65} +4.01880 q^{67} +0.934795 q^{69} +8.17837 q^{71} +8.24116 q^{73} +8.46064 q^{75} +8.26890 q^{77} +1.00000 q^{79} -10.6940 q^{81} -7.67303 q^{83} -5.81578 q^{85} +3.19472 q^{87} -5.33996 q^{89} -8.52999 q^{91} +4.05785 q^{93} +0.795901 q^{95} +3.47972 q^{97} +3.76066 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * 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
$3$	−1.93761	−1.11868	−0.559339	−	0.828939i	$-0.688945\pi$
$3$	−1.93761	−1.11868	−0.559339	+	0.828939i	$0.688945\pi$
$4$	0	0
$5$	0.795901	0.355938	0.177969	−	0.984036i	$-0.443047\pi$
$5$	0.795901	0.355938	0.177969	+	0.984036i	$0.443047\pi$
$6$	0	0
$7$	1.65858	0.626886	0.313443	−	0.949607i	$-0.398518\pi$
$7$	1.65858	0.626886	0.313443	+	0.949607i	$0.398518\pi$
$8$	0	0
$9$	0.754317	0.251439
$10$	0	0
$11$	4.98552	1.50319	0.751595	−	0.659625i	$-0.229285\pi$
$11$	4.98552	1.50319	0.751595	+	0.659625i	$0.229285\pi$
$12$	0	0
$13$	−5.14293	−1.42639	−0.713196	−	0.700964i	$-0.752753\pi$
$13$	−5.14293	−1.42639	−0.713196	+	0.700964i	$0.752753\pi$
$14$	0	0
$15$	−1.54214	−0.398180
$16$	0	0
$17$	−7.30716	−1.77225	−0.886123	−	0.463450i	$-0.846611\pi$
$17$	−7.30716	−1.77225	−0.886123	+	0.463450i	$0.846611\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.21368	−0.701283
$22$	0	0
$23$	−0.482448	−0.100597	−0.0502987	−	0.998734i	$-0.516017\pi$
$23$	−0.482448	−0.100597	−0.0502987	+	0.998734i	$0.516017\pi$
$24$	0	0
$25$	−4.36654	−0.873308
$26$	0	0
$27$	4.35125	0.837398
$28$	0	0
$29$	−1.64880	−0.306174	−0.153087	−	0.988213i	$-0.548921\pi$
$29$	−1.64880	−0.306174	−0.153087	+	0.988213i	$0.548921\pi$
$30$	0	0
$31$	−2.09426	−0.376140	−0.188070	−	0.982156i	$-0.560223\pi$
$31$	−2.09426	−0.376140	−0.188070	+	0.982156i	$0.560223\pi$
$32$	0	0
$33$	−9.65997	−1.68158
$34$	0	0
$35$	1.32007	0.223132
$36$	0	0
$37$	7.37275	1.21207	0.606036	−	0.795437i	$-0.292759\pi$
$37$	7.37275	1.21207	0.606036	+	0.795437i	$0.292759\pi$
$38$	0	0
$39$	9.96498	1.59567
$40$	0	0
$41$	8.34657	1.30351	0.651757	−	0.758428i	$-0.274032\pi$
$41$	8.34657	1.30351	0.651757	+	0.758428i	$0.274032\pi$
$42$	0	0
$43$	6.41319	0.978002	0.489001	−	0.872283i	$-0.337361\pi$
$43$	6.41319	0.978002	0.489001	+	0.872283i	$0.337361\pi$
$44$	0	0
$45$	0.600362	0.0894966
$46$	0	0
$47$	−9.11366	−1.32936	−0.664682	−	0.747126i	$-0.731433\pi$
$47$	−9.11366	−1.32936	−0.664682	+	0.747126i	$0.731433\pi$
$48$	0	0
$49$	−4.24910	−0.607014
$50$	0	0
$51$	14.1584	1.98257
$52$	0	0
$53$	5.62276	0.772345	0.386173	−	0.922426i	$-0.373797\pi$
$53$	5.62276	0.772345	0.386173	+	0.922426i	$0.373797\pi$
$54$	0	0
$55$	3.96798	0.535042
$56$	0	0
$57$	−1.93761	−0.256642
$58$	0	0
$59$	1.75283	0.228199	0.114099	−	0.993469i	$-0.463602\pi$
$59$	1.75283	0.228199	0.114099	+	0.993469i	$0.463602\pi$
$60$	0	0
$61$	−0.496017	−0.0635084	−0.0317542	−	0.999496i	$-0.510109\pi$
$61$	−0.496017	−0.0635084	−0.0317542	+	0.999496i	$0.510109\pi$
$62$	0	0
$63$	1.25110	0.157624
$64$	0	0
$65$	−4.09327	−0.507707
$66$	0	0
$67$	4.01880	0.490974	0.245487	−	0.969400i	$-0.421052\pi$
$67$	4.01880	0.490974	0.245487	+	0.969400i	$0.421052\pi$
$68$	0	0
$69$	0.934795	0.112536
$70$	0	0
$71$	8.17837	0.970594	0.485297	−	0.874349i	$-0.338712\pi$
$71$	8.17837	0.970594	0.485297	+	0.874349i	$0.338712\pi$
$72$	0	0
$73$	8.24116	0.964555	0.482277	−	0.876019i	$-0.339810\pi$
$73$	8.24116	0.964555	0.482277	+	0.876019i	$0.339810\pi$
$74$	0	0
$75$	8.46064	0.976950
$76$	0	0
$77$	8.26890	0.942329
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−10.6940	−1.18822
$82$	0	0
$83$	−7.67303	−0.842224	−0.421112	−	0.907009i	$-0.638360\pi$
$83$	−7.67303	−0.842224	−0.421112	+	0.907009i	$0.638360\pi$
$84$	0	0
$85$	−5.81578	−0.630809
$86$	0	0
$87$	3.19472	0.342510
$88$	0	0
$89$	−5.33996	−0.566035	−0.283017	−	0.959115i	$-0.591335\pi$
$89$	−5.33996	−0.566035	−0.283017	+	0.959115i	$0.591335\pi$
$90$	0	0
$91$	−8.52999	−0.894186
$92$	0	0
$93$	4.05785	0.420780
$94$	0	0
$95$	0.795901	0.0816577
$96$	0	0
$97$	3.47972	0.353312	0.176656	−	0.984273i	$-0.443472\pi$
$97$	3.47972	0.353312	0.176656	+	0.984273i	$0.443472\pi$
$98$	0	0
$99$	3.76066	0.377961
$100$	0	0
$101$	17.9397	1.78507	0.892533	−	0.450981i	$-0.148926\pi$
$101$	17.9397	1.78507	0.892533	+	0.450981i	$0.148926\pi$
$102$	0	0
$103$	−5.36069	−0.528205	−0.264102	−	0.964495i	$-0.585076\pi$
$103$	−5.36069	−0.528205	−0.264102	+	0.964495i	$0.585076\pi$
$104$	0	0
$105$	−2.55777	−0.249613
$106$	0	0
$107$	5.33437	0.515693	0.257847	−	0.966186i	$-0.416987\pi$
$107$	5.33437	0.515693	0.257847	+	0.966186i	$0.416987\pi$
$108$	0	0
$109$	7.02612	0.672980	0.336490	−	0.941687i	$-0.390760\pi$
$109$	7.02612	0.672980	0.336490	+	0.941687i	$0.390760\pi$
$110$	0	0
$111$	−14.2855	−1.35592
$112$	0	0
$113$	−4.24725	−0.399548	−0.199774	−	0.979842i	$-0.564021\pi$
$113$	−4.24725	−0.399548	−0.199774	+	0.979842i	$0.564021\pi$
$114$	0	0
$115$	−0.383981	−0.0358064
$116$	0	0
$117$	−3.87940	−0.358651
$118$	0	0
$119$	−12.1195	−1.11100
$120$	0	0
$121$	13.8554	1.25958
$122$	0	0
$123$	−16.1724	−1.45821
$124$	0	0
$125$	−7.45484	−0.666781
$126$	0	0
$127$	−13.1565	−1.16745	−0.583725	−	0.811951i	$-0.698405\pi$
$127$	−13.1565	−1.16745	−0.583725	+	0.811951i	$0.698405\pi$
$128$	0	0
$129$	−12.4262	−1.09407
$130$	0	0
$131$	11.0156	0.962441	0.481221	−	0.876600i	$-0.340194\pi$
$131$	11.0156	0.962441	0.481221	+	0.876600i	$0.340194\pi$
$132$	0	0
$133$	1.65858	0.143818
$134$	0	0
$135$	3.46316	0.298062
$136$	0	0
$137$	−5.00449	−0.427563	−0.213781	−	0.976882i	$-0.568578\pi$
$137$	−5.00449	−0.427563	−0.213781	+	0.976882i	$0.568578\pi$
$138$	0	0
$139$	−0.563582	−0.0478024	−0.0239012	−	0.999714i	$-0.507609\pi$
$139$	−0.563582	−0.0478024	−0.0239012	+	0.999714i	$0.507609\pi$
$140$	0	0
$141$	17.6587	1.48713
$142$	0	0
$143$	−25.6402	−2.14414
$144$	0	0
$145$	−1.31228	−0.108979
$146$	0	0
$147$	8.23308	0.679053
$148$	0	0
$149$	−14.3883	−1.17874	−0.589369	−	0.807864i	$-0.700624\pi$
$149$	−14.3883	−1.17874	−0.589369	+	0.807864i	$0.700624\pi$
$150$	0	0
$151$	−2.13270	−0.173556	−0.0867781	−	0.996228i	$-0.527657\pi$
$151$	−2.13270	−0.173556	−0.0867781	+	0.996228i	$0.527657\pi$
$152$	0	0
$153$	−5.51191	−0.445612
$154$	0	0
$155$	−1.66682	−0.133883
$156$	0	0
$157$	0.300794	0.0240060	0.0120030	−	0.999928i	$-0.496179\pi$
$157$	0.300794	0.0240060	0.0120030	+	0.999928i	$0.496179\pi$
$158$	0	0
$159$	−10.8947	−0.864005
$160$	0	0
$161$	−0.800182	−0.0630631
$162$	0	0
$163$	16.1377	1.26400	0.632002	−	0.774967i	$-0.282233\pi$
$163$	16.1377	1.26400	0.632002	+	0.774967i	$0.282233\pi$
$164$	0	0
$165$	−7.68838	−0.598539
$166$	0	0
$167$	14.7497	1.14137	0.570684	−	0.821169i	$-0.306678\pi$
$167$	14.7497	1.14137	0.570684	+	0.821169i	$0.306678\pi$
$168$	0	0
$169$	13.4498	1.03460
$170$	0	0
$171$	0.754317	0.0576841
$172$	0	0
$173$	4.80030	0.364960	0.182480	−	0.983210i	$-0.441587\pi$
$173$	4.80030	0.364960	0.182480	+	0.983210i	$0.441587\pi$
$174$	0	0
$175$	−7.24228	−0.547465
$176$	0	0
$177$	−3.39629	−0.255281
$178$	0	0
$179$	−12.7770	−0.954998	−0.477499	−	0.878632i	$-0.658457\pi$
$179$	−12.7770	−0.954998	−0.477499	+	0.878632i	$0.658457\pi$
$180$	0	0
$181$	−14.7494	−1.09631	−0.548157	−	0.836376i	$-0.684670\pi$
$181$	−14.7494	−1.09631	−0.548157	+	0.836376i	$0.684670\pi$
$182$	0	0
$183$	0.961085	0.0710454
$184$	0	0
$185$	5.86798	0.431422
$186$	0	0
$187$	−36.4300	−2.66402
$188$	0	0
$189$	7.21691	0.524953
$190$	0	0
$191$	23.8412	1.72509	0.862544	−	0.505982i	$-0.168870\pi$
$191$	23.8412	1.72509	0.862544	+	0.505982i	$0.168870\pi$
$192$	0	0
$193$	14.8857	1.07150	0.535748	−	0.844378i	$-0.320030\pi$
$193$	14.8857	1.07150	0.535748	+	0.844378i	$0.320030\pi$
$194$	0	0
$195$	7.93114	0.567960
$196$	0	0
$197$	−8.12451	−0.578848	−0.289424	−	0.957201i	$-0.593464\pi$
$197$	−8.12451	−0.578848	−0.289424	+	0.957201i	$0.593464\pi$
$198$	0	0
$199$	15.7745	1.11823	0.559114	−	0.829091i	$-0.311141\pi$
$199$	15.7745	1.11823	0.559114	+	0.829091i	$0.311141\pi$
$200$	0	0
$201$	−7.78685	−0.549242
$202$	0	0
$203$	−2.73467	−0.191936
$204$	0	0
$205$	6.64304	0.463970
$206$	0	0
$207$	−0.363919	−0.0252941
$208$	0	0
$209$	4.98552	0.344855
$210$	0	0
$211$	0.213221	0.0146787	0.00733936	−	0.999973i	$-0.497664\pi$
$211$	0.213221	0.0146787	0.00733936	+	0.999973i	$0.497664\pi$
$212$	0	0
$213$	−15.8465	−1.08578
$214$	0	0
$215$	5.10426	0.348108
$216$	0	0
$217$	−3.47351	−0.235797
$218$	0	0
$219$	−15.9681	−1.07903
$220$	0	0
$221$	37.5802	2.52792
$222$	0	0
$223$	15.6800	1.05001	0.525007	−	0.851098i	$-0.324063\pi$
$223$	15.6800	1.05001	0.525007	+	0.851098i	$0.324063\pi$
$224$	0	0
$225$	−3.29376	−0.219584
$226$	0	0
$227$	−18.3589	−1.21852	−0.609260	−	0.792970i	$-0.708534\pi$
$227$	−18.3589	−1.21852	−0.609260	+	0.792970i	$0.708534\pi$
$228$	0	0
$229$	5.20573	0.344004	0.172002	−	0.985097i	$-0.444976\pi$
$229$	5.20573	0.344004	0.172002	+	0.985097i	$0.444976\pi$
$230$	0	0
$231$	−16.0219	−1.05416
$232$	0	0
$233$	18.6489	1.22173	0.610866	−	0.791734i	$-0.290821\pi$
$233$	18.6489	1.22173	0.610866	+	0.791734i	$0.290821\pi$
$234$	0	0
$235$	−7.25357	−0.473171
$236$	0	0
$237$	−1.93761	−0.125861
$238$	0	0
$239$	9.75777	0.631178	0.315589	−	0.948896i	$-0.397798\pi$
$239$	9.75777	0.631178	0.315589	+	0.948896i	$0.397798\pi$
$240$	0	0
$241$	14.0228	0.903288	0.451644	−	0.892198i	$-0.350838\pi$
$241$	14.0228	0.903288	0.451644	+	0.892198i	$0.350838\pi$
$242$	0	0
$243$	7.66693	0.491834
$244$	0	0
$245$	−3.38186	−0.216059
$246$	0	0
$247$	−5.14293	−0.327237
$248$	0	0
$249$	14.8673	0.942177
$250$	0	0
$251$	−20.6530	−1.30361	−0.651804	−	0.758388i	$-0.725987\pi$
$251$	−20.6530	−1.30361	−0.651804	+	0.758388i	$0.725987\pi$
$252$	0	0
$253$	−2.40525	−0.151217
$254$	0	0
$255$	11.2687	0.705672
$256$	0	0
$257$	−5.43088	−0.338769	−0.169385	−	0.985550i	$-0.554178\pi$
$257$	−5.43088	−0.338769	−0.169385	+	0.985550i	$0.554178\pi$
$258$	0	0
$259$	12.2283	0.759831
$260$	0	0
$261$	−1.24371	−0.0769840
$262$	0	0
$263$	19.1652	1.18177	0.590887	−	0.806754i	$-0.298778\pi$
$263$	19.1652	1.18177	0.590887	+	0.806754i	$0.298778\pi$
$264$	0	0
$265$	4.47516	0.274907
$266$	0	0
$267$	10.3467	0.633210
$268$	0	0
$269$	8.10231	0.494006	0.247003	−	0.969015i	$-0.420554\pi$
$269$	8.10231	0.494006	0.247003	+	0.969015i	$0.420554\pi$
$270$	0	0
$271$	6.75001	0.410034	0.205017	−	0.978758i	$-0.434275\pi$
$271$	6.75001	0.410034	0.205017	+	0.978758i	$0.434275\pi$
$272$	0	0
$273$	16.5278	1.00031
$274$	0	0
$275$	−21.7695	−1.31275
$276$	0	0
$277$	−9.22750	−0.554427	−0.277213	−	0.960808i	$-0.589411\pi$
$277$	−9.22750	−0.554427	−0.277213	+	0.960808i	$0.589411\pi$
$278$	0	0
$279$	−1.57974	−0.0945763
$280$	0	0
$281$	19.2624	1.14910	0.574551	−	0.818469i	$-0.305177\pi$
$281$	19.2624	1.14910	0.574551	+	0.818469i	$0.305177\pi$
$282$	0	0
$283$	−17.9552	−1.06733	−0.533664	−	0.845697i	$-0.679185\pi$
$283$	−17.9552	−1.06733	−0.533664	+	0.845697i	$0.679185\pi$
$284$	0	0
$285$	−1.54214	−0.0913486
$286$	0	0
$287$	13.8435	0.817155
$288$	0	0
$289$	36.3946	2.14086
$290$	0	0
$291$	−6.74233	−0.395243
$292$	0	0
$293$	12.2961	0.718347	0.359174	−	0.933271i	$-0.383059\pi$
$293$	12.2961	0.718347	0.359174	+	0.933271i	$0.383059\pi$
$294$	0	0
$295$	1.39508	0.0812245
$296$	0	0
$297$	21.6932	1.25877
$298$	0	0
$299$	2.48120	0.143491
$300$	0	0
$301$	10.6368	0.613096
$302$	0	0
$303$	−34.7601	−1.99691
$304$	0	0
$305$	−0.394780	−0.0226050
$306$	0	0
$307$	−2.27019	−0.129567	−0.0647834	−	0.997899i	$-0.520636\pi$
$307$	−2.27019	−0.129567	−0.0647834	+	0.997899i	$0.520636\pi$
$308$	0	0
$309$	10.3869	0.590891
$310$	0	0
$311$	3.41575	0.193690	0.0968448	−	0.995299i	$-0.469125\pi$
$311$	3.41575	0.193690	0.0968448	+	0.995299i	$0.469125\pi$
$312$	0	0
$313$	9.23719	0.522117	0.261059	−	0.965323i	$-0.415928\pi$
$313$	9.23719	0.522117	0.261059	+	0.965323i	$0.415928\pi$
$314$	0	0
$315$	0.995751	0.0561042
$316$	0	0
$317$	7.21355	0.405153	0.202577	−	0.979266i	$-0.435069\pi$
$317$	7.21355	0.405153	0.202577	+	0.979266i	$0.435069\pi$
$318$	0	0
$319$	−8.22010	−0.460237
$320$	0	0
$321$	−10.3359	−0.576894
$322$	0	0
$323$	−7.30716	−0.406581
$324$	0	0
$325$	22.4568	1.24568
$326$	0	0
$327$	−13.6139	−0.752848
$328$	0	0
$329$	−15.1158	−0.833360
$330$	0	0
$331$	35.1160	1.93015	0.965075	−	0.261973i	$-0.0843732\pi$
$331$	35.1160	1.93015	0.965075	+	0.261973i	$0.0843732\pi$
$332$	0	0
$333$	5.56139	0.304762
$334$	0	0
$335$	3.19857	0.174756
$336$	0	0
$337$	0.809793	0.0441122	0.0220561	−	0.999757i	$-0.492979\pi$
$337$	0.809793	0.0441122	0.0220561	+	0.999757i	$0.492979\pi$
$338$	0	0
$339$	8.22951	0.446965
$340$	0	0
$341$	−10.4410	−0.565410
$342$	0	0
$343$	−18.6576	−1.00741
$344$	0	0
$345$	0.744004	0.0400558
$346$	0	0
$347$	−8.42653	−0.452360	−0.226180	−	0.974086i	$-0.572624\pi$
$347$	−8.42653	−0.452360	−0.226180	+	0.974086i	$0.572624\pi$
$348$	0	0
$349$	31.8367	1.70418	0.852091	−	0.523394i	$-0.175334\pi$
$349$	31.8367	1.70418	0.852091	+	0.523394i	$0.175334\pi$
$350$	0	0
$351$	−22.3782	−1.19446
$352$	0	0
$353$	−0.931213	−0.0495635	−0.0247817	−	0.999693i	$-0.507889\pi$
$353$	−0.931213	−0.0495635	−0.0247817	+	0.999693i	$0.507889\pi$
$354$	0	0
$355$	6.50917	0.345471
$356$	0	0
$357$	23.4829	1.24285
$358$	0	0
$359$	4.19191	0.221241	0.110620	−	0.993863i	$-0.464716\pi$
$359$	4.19191	0.221241	0.110620	+	0.993863i	$0.464716\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−26.8463	−1.40906
$364$	0	0
$365$	6.55915	0.343321
$366$	0	0
$367$	−34.0454	−1.77716	−0.888578	−	0.458725i	$-0.848306\pi$
$367$	−34.0454	−1.77716	−0.888578	+	0.458725i	$0.848306\pi$
$368$	0	0
$369$	6.29596	0.327754
$370$	0	0
$371$	9.32582	0.484173
$372$	0	0
$373$	26.3002	1.36177	0.680885	−	0.732390i	$-0.261595\pi$
$373$	26.3002	1.36177	0.680885	+	0.732390i	$0.261595\pi$
$374$	0	0
$375$	14.4445	0.745913
$376$	0	0
$377$	8.47965	0.436724
$378$	0	0
$379$	35.9883	1.84860	0.924298	−	0.381671i	$-0.124651\pi$
$379$	35.9883	1.84860	0.924298	+	0.381671i	$0.124651\pi$
$380$	0	0
$381$	25.4921	1.30600
$382$	0	0
$383$	4.61510	0.235821	0.117910	−	0.993024i	$-0.462380\pi$
$383$	4.61510	0.235821	0.117910	+	0.993024i	$0.462380\pi$
$384$	0	0
$385$	6.58123	0.335410
$386$	0	0
$387$	4.83758	0.245908
$388$	0	0
$389$	−16.1756	−0.820136	−0.410068	−	0.912055i	$-0.634495\pi$
$389$	−16.1756	−0.820136	−0.410068	+	0.912055i	$0.634495\pi$
$390$	0	0
$391$	3.52533	0.178283
$392$	0	0
$393$	−21.3440	−1.07666
$394$	0	0
$395$	0.795901	0.0400461
$396$	0	0
$397$	18.2538	0.916134	0.458067	−	0.888918i	$-0.348542\pi$
$397$	18.2538	0.916134	0.458067	+	0.888918i	$0.348542\pi$
$398$	0	0
$399$	−3.21368	−0.160885
$400$	0	0
$401$	17.8325	0.890512	0.445256	−	0.895403i	$-0.353113\pi$
$401$	17.8325	0.890512	0.445256	+	0.895403i	$0.353113\pi$
$402$	0	0
$403$	10.7706	0.536524
$404$	0	0
$405$	−8.51133	−0.422931
$406$	0	0
$407$	36.7570	1.82198
$408$	0	0
$409$	−11.5036	−0.568818	−0.284409	−	0.958703i	$-0.591797\pi$
$409$	−11.5036	−0.568818	−0.284409	+	0.958703i	$0.591797\pi$
$410$	0	0
$411$	9.69673	0.478305
$412$	0	0
$413$	2.90721	0.143055
$414$	0	0
$415$	−6.10697	−0.299779
$416$	0	0
$417$	1.09200	0.0534755
$418$	0	0
$419$	1.64932	0.0805746	0.0402873	−	0.999188i	$-0.487173\pi$
$419$	1.64932	0.0805746	0.0402873	+	0.999188i	$0.487173\pi$
$420$	0	0
$421$	29.8550	1.45504	0.727521	−	0.686085i	$-0.240672\pi$
$421$	29.8550	1.45504	0.727521	+	0.686085i	$0.240672\pi$
$422$	0	0
$423$	−6.87459	−0.334254
$424$	0	0
$425$	31.9070	1.54772
$426$	0	0
$427$	−0.822685	−0.0398125
$428$	0	0
$429$	49.6806	2.39860
$430$	0	0
$431$	33.0446	1.59170	0.795852	−	0.605491i	$-0.207023\pi$
$431$	33.0446	1.59170	0.795852	+	0.605491i	$0.207023\pi$
$432$	0	0
$433$	7.66181	0.368203	0.184102	−	0.982907i	$-0.441062\pi$
$433$	7.66181	0.368203	0.184102	+	0.982907i	$0.441062\pi$
$434$	0	0
$435$	2.54268	0.121912
$436$	0	0
$437$	−0.482448	−0.0230786
$438$	0	0
$439$	−19.3967	−0.925756	−0.462878	−	0.886422i	$-0.653183\pi$
$439$	−19.3967	−0.925756	−0.462878	+	0.886422i	$0.653183\pi$
$440$	0	0
$441$	−3.20517	−0.152627
$442$	0	0
$443$	−36.9444	−1.75528	−0.877640	−	0.479320i	$-0.840883\pi$
$443$	−36.9444	−1.75528	−0.877640	+	0.479320i	$0.840883\pi$
$444$	0	0
$445$	−4.25008	−0.201473
$446$	0	0
$447$	27.8789	1.31863
$448$	0	0
$449$	10.1429	0.478671	0.239335	−	0.970937i	$-0.423070\pi$
$449$	10.1429	0.478671	0.239335	+	0.970937i	$0.423070\pi$
$450$	0	0
$451$	41.6119	1.95943
$452$	0	0
$453$	4.13232	0.194153
$454$	0	0
$455$	−6.78903	−0.318274
$456$	0	0
$457$	−33.2646	−1.55605	−0.778026	−	0.628233i	$-0.783779\pi$
$457$	−33.2646	−1.55605	−0.778026	+	0.628233i	$0.783779\pi$
$458$	0	0
$459$	−31.7953	−1.48408
$460$	0	0
$461$	−2.22835	−0.103784	−0.0518922	−	0.998653i	$-0.516525\pi$
$461$	−2.22835	−0.103784	−0.0518922	+	0.998653i	$0.516525\pi$
$462$	0	0
$463$	16.5060	0.767098	0.383549	−	0.923521i	$-0.374702\pi$
$463$	16.5060	0.767098	0.383549	+	0.923521i	$0.374702\pi$
$464$	0	0
$465$	3.22965	0.149771
$466$	0	0
$467$	2.14106	0.0990762	0.0495381	−	0.998772i	$-0.484225\pi$
$467$	2.14106	0.0990762	0.0495381	+	0.998772i	$0.484225\pi$
$468$	0	0
$469$	6.66552	0.307785
$470$	0	0
$471$	−0.582820	−0.0268549
$472$	0	0
$473$	31.9730	1.47012
$474$	0	0
$475$	−4.36654	−0.200351
$476$	0	0
$477$	4.24134	0.194198
$478$	0	0
$479$	20.5907	0.940812	0.470406	−	0.882450i	$-0.344107\pi$
$479$	20.5907	0.940812	0.470406	+	0.882450i	$0.344107\pi$
$480$	0	0
$481$	−37.9176	−1.72889
$482$	0	0
$483$	1.55044	0.0705473
$484$	0	0
$485$	2.76952	0.125757
$486$	0	0
$487$	12.8738	0.583366	0.291683	−	0.956515i	$-0.405785\pi$
$487$	12.8738	0.583366	0.291683	+	0.956515i	$0.405785\pi$
$488$	0	0
$489$	−31.2685	−1.41401
$490$	0	0
$491$	−6.60347	−0.298010	−0.149005	−	0.988836i	$-0.547607\pi$
$491$	−6.60347	−0.298010	−0.149005	+	0.988836i	$0.547607\pi$
$492$	0	0
$493$	12.0480	0.542615
$494$	0	0
$495$	2.99311	0.134530
$496$	0	0
$497$	13.5645	0.608452
$498$	0	0
$499$	−5.60903	−0.251095	−0.125547	−	0.992088i	$-0.540069\pi$
$499$	−5.60903	−0.251095	−0.125547	+	0.992088i	$0.540069\pi$
$500$	0	0
$501$	−28.5792	−1.27682
$502$	0	0
$503$	−30.7464	−1.37091	−0.685457	−	0.728113i	$-0.740398\pi$
$503$	−30.7464	−1.37091	−0.685457	+	0.728113i	$0.740398\pi$
$504$	0	0
$505$	14.2782	0.635373
$506$	0	0
$507$	−26.0603	−1.15738
$508$	0	0
$509$	22.0999	0.979560	0.489780	−	0.871846i	$-0.337077\pi$
$509$	22.0999	0.979560	0.489780	+	0.871846i	$0.337077\pi$
$510$	0	0
$511$	13.6687	0.604666
$512$	0	0
$513$	4.35125	0.192112
$514$	0	0
$515$	−4.26658	−0.188008
$516$	0	0
$517$	−45.4363	−1.99829
$518$	0	0
$519$	−9.30110	−0.408273
$520$	0	0
$521$	−9.33462	−0.408957	−0.204478	−	0.978871i	$-0.565550\pi$
$521$	−9.33462	−0.408957	−0.204478	+	0.978871i	$0.565550\pi$
$522$	0	0
$523$	−13.9555	−0.610230	−0.305115	−	0.952316i	$-0.598695\pi$
$523$	−13.9555	−0.610230	−0.305115	+	0.952316i	$0.598695\pi$
$524$	0	0
$525$	14.0327	0.612436
$526$	0	0
$527$	15.3031	0.666613
$528$	0	0
$529$	−22.7672	−0.989880
$530$	0	0
$531$	1.32219	0.0573780
$532$	0	0
$533$	−42.9258	−1.85932
$534$	0	0
$535$	4.24563	0.183555
$536$	0	0
$537$	24.7568	1.06833
$538$	0	0
$539$	−21.1839	−0.912457
$540$	0	0
$541$	−10.2736	−0.441695	−0.220848	−	0.975308i	$-0.570882\pi$
$541$	−10.2736	−0.441695	−0.220848	+	0.975308i	$0.570882\pi$
$542$	0	0
$543$	28.5785	1.22642
$544$	0	0
$545$	5.59210	0.239539
$546$	0	0
$547$	−40.2164	−1.71953	−0.859765	−	0.510691i	$-0.829390\pi$
$547$	−40.2164	−1.71953	−0.859765	+	0.510691i	$0.829390\pi$
$548$	0	0
$549$	−0.374154	−0.0159685
$550$	0	0
$551$	−1.64880	−0.0702411
$552$	0	0
$553$	1.65858	0.0705302
$554$	0	0
$555$	−11.3698	−0.482622
$556$	0	0
$557$	−5.77384	−0.244646	−0.122323	−	0.992490i	$-0.539034\pi$
$557$	−5.77384	−0.244646	−0.122323	+	0.992490i	$0.539034\pi$
$558$	0	0
$559$	−32.9826	−1.39501
$560$	0	0
$561$	70.5869	2.98018
$562$	0	0
$563$	25.8148	1.08797	0.543983	−	0.839096i	$-0.316916\pi$
$563$	25.8148	1.08797	0.543983	+	0.839096i	$0.316916\pi$
$564$	0	0
$565$	−3.38039	−0.142214
$566$	0	0
$567$	−17.7368	−0.744877
$568$	0	0
$569$	18.7869	0.787587	0.393794	−	0.919199i	$-0.371162\pi$
$569$	18.7869	0.787587	0.393794	+	0.919199i	$0.371162\pi$
$570$	0	0
$571$	−27.1901	−1.13787	−0.568936	−	0.822382i	$-0.692645\pi$
$571$	−27.1901	−1.13787	−0.568936	+	0.822382i	$0.692645\pi$
$572$	0	0
$573$	−46.1948	−1.92982
$574$	0	0
$575$	2.10663	0.0878526
$576$	0	0
$577$	5.94569	0.247522	0.123761	−	0.992312i	$-0.460504\pi$
$577$	5.94569	0.247522	0.123761	+	0.992312i	$0.460504\pi$
$578$	0	0
$579$	−28.8426	−1.19866
$580$	0	0
$581$	−12.7264	−0.527978
$582$	0	0
$583$	28.0324	1.16098
$584$	0	0
$585$	−3.08762	−0.127657
$586$	0	0
$587$	−9.70806	−0.400695	−0.200347	−	0.979725i	$-0.564207\pi$
$587$	−9.70806	−0.400695	−0.200347	+	0.979725i	$0.564207\pi$
$588$	0	0
$589$	−2.09426	−0.0862925
$590$	0	0
$591$	15.7421	0.647544
$592$	0	0
$593$	10.4067	0.427351	0.213675	−	0.976905i	$-0.431457\pi$
$593$	10.4067	0.427351	0.213675	+	0.976905i	$0.431457\pi$
$594$	0	0
$595$	−9.64596	−0.395446
$596$	0	0
$597$	−30.5648	−1.25094
$598$	0	0
$599$	−7.47585	−0.305455	−0.152727	−	0.988268i	$-0.548806\pi$
$599$	−7.47585	−0.305455	−0.152727	+	0.988268i	$0.548806\pi$
$600$	0	0
$601$	31.9917	1.30497	0.652484	−	0.757803i	$-0.273727\pi$
$601$	31.9917	1.30497	0.652484	+	0.757803i	$0.273727\pi$
$602$	0	0
$603$	3.03145	0.123450
$604$	0	0
$605$	11.0275	0.448332
$606$	0	0
$607$	−44.6386	−1.81182	−0.905912	−	0.423467i	$-0.860813\pi$
$607$	−44.6386	−1.81182	−0.905912	+	0.423467i	$0.860813\pi$
$608$	0	0
$609$	5.29871	0.214714
$610$	0	0
$611$	46.8710	1.89620
$612$	0	0
$613$	−14.4892	−0.585212	−0.292606	−	0.956233i	$-0.594522\pi$
$613$	−14.4892	−0.585212	−0.292606	+	0.956233i	$0.594522\pi$
$614$	0	0
$615$	−12.8716	−0.519033
$616$	0	0
$617$	40.1877	1.61790	0.808949	−	0.587879i	$-0.200037\pi$
$617$	40.1877	1.61790	0.808949	+	0.587879i	$0.200037\pi$
$618$	0	0
$619$	15.1152	0.607531	0.303766	−	0.952747i	$-0.401756\pi$
$619$	15.1152	0.607531	0.303766	+	0.952747i	$0.401756\pi$
$620$	0	0
$621$	−2.09925	−0.0842401
$622$	0	0
$623$	−8.85678	−0.354839
$624$	0	0
$625$	15.8994	0.635976
$626$	0	0
$627$	−9.65997	−0.385782
$628$	0	0
$629$	−53.8739	−2.14809
$630$	0	0
$631$	18.0826	0.719855	0.359928	−	0.932980i	$-0.382801\pi$
$631$	18.0826	0.719855	0.359928	+	0.932980i	$0.382801\pi$
$632$	0	0
$633$	−0.413138	−0.0164208
$634$	0	0
$635$	−10.4713	−0.415540
$636$	0	0
$637$	21.8528	0.865840
$638$	0	0
$639$	6.16908	0.244045
$640$	0	0
$641$	48.6167	1.92025	0.960123	−	0.279579i	$-0.0901947\pi$
$641$	48.6167	1.92025	0.960123	+	0.279579i	$0.0901947\pi$
$642$	0	0
$643$	10.9680	0.432536	0.216268	−	0.976334i	$-0.430611\pi$
$643$	10.9680	0.432536	0.216268	+	0.976334i	$0.430611\pi$
$644$	0	0
$645$	−9.89005	−0.389420
$646$	0	0
$647$	2.65573	0.104408	0.0522038	−	0.998636i	$-0.483375\pi$
$647$	2.65573	0.104408	0.0522038	+	0.998636i	$0.483375\pi$
$648$	0	0
$649$	8.73875	0.343026
$650$	0	0
$651$	6.73029	0.263781
$652$	0	0
$653$	−43.4498	−1.70032	−0.850162	−	0.526521i	$-0.823496\pi$
$653$	−43.4498	−1.70032	−0.850162	+	0.526521i	$0.823496\pi$
$654$	0	0
$655$	8.76736	0.342569
$656$	0	0
$657$	6.21645	0.242527
$658$	0	0
$659$	31.4504	1.22513	0.612567	−	0.790419i	$-0.290137\pi$
$659$	31.4504	1.22513	0.612567	+	0.790419i	$0.290137\pi$
$660$	0	0
$661$	17.5512	0.682662	0.341331	−	0.939943i	$-0.389122\pi$
$661$	17.5512	0.682662	0.341331	+	0.939943i	$0.389122\pi$
$662$	0	0
$663$	−72.8157	−2.82793
$664$	0	0
$665$	1.32007	0.0511901
$666$	0	0
$667$	0.795459	0.0308003
$668$	0	0
$669$	−30.3817	−1.17463
$670$	0	0
$671$	−2.47290	−0.0954652
$672$	0	0
$673$	−33.5199	−1.29210	−0.646048	−	0.763297i	$-0.723579\pi$
$673$	−33.5199	−1.29210	−0.646048	+	0.763297i	$0.723579\pi$
$674$	0	0
$675$	−18.9999	−0.731307
$676$	0	0
$677$	27.0116	1.03814	0.519070	−	0.854731i	$-0.326278\pi$
$677$	27.0116	1.03814	0.519070	+	0.854731i	$0.326278\pi$
$678$	0	0
$679$	5.77142	0.221487
$680$	0	0
$681$	35.5723	1.36313
$682$	0	0
$683$	20.4166	0.781219	0.390609	−	0.920557i	$-0.372264\pi$
$683$	20.4166	0.781219	0.390609	+	0.920557i	$0.372264\pi$
$684$	0	0
$685$	−3.98308	−0.152186
$686$	0	0
$687$	−10.0867	−0.384830
$688$	0	0
$689$	−28.9175	−1.10167
$690$	0	0
$691$	25.5171	0.970715	0.485357	−	0.874316i	$-0.338689\pi$
$691$	25.5171	0.970715	0.485357	+	0.874316i	$0.338689\pi$
$692$	0	0
$693$	6.23737	0.236938
$694$	0	0
$695$	−0.448555	−0.0170147
$696$	0	0
$697$	−60.9897	−2.31015
$698$	0	0
$699$	−36.1343	−1.36672
$700$	0	0
$701$	9.93099	0.375088	0.187544	−	0.982256i	$-0.439947\pi$
$701$	9.93099	0.375088	0.187544	+	0.982256i	$0.439947\pi$
$702$	0	0
$703$	7.37275	0.278069
$704$	0	0
$705$	14.0546	0.529326
$706$	0	0
$707$	29.7545	1.11903
$708$	0	0
$709$	−28.9104	−1.08575	−0.542877	−	0.839812i	$-0.682665\pi$
$709$	−28.9104	−1.08575	−0.542877	+	0.839812i	$0.682665\pi$
$710$	0	0
$711$	0.754317	0.0282891
$712$	0	0
$713$	1.01037	0.0378387
$714$	0	0
$715$	−20.4070	−0.763180
$716$	0	0
$717$	−18.9067	−0.706084
$718$	0	0
$719$	7.04371	0.262686	0.131343	−	0.991337i	$-0.458071\pi$
$719$	7.04371	0.262686	0.131343	+	0.991337i	$0.458071\pi$
$720$	0	0
$721$	−8.89116	−0.331124
$722$	0	0
$723$	−27.1707	−1.01049
$724$	0	0
$725$	7.19954	0.267384
$726$	0	0
$727$	−14.5274	−0.538793	−0.269396	−	0.963029i	$-0.586824\pi$
$727$	−14.5274	−0.538793	−0.269396	+	0.963029i	$0.586824\pi$
$728$	0	0
$729$	17.2264	0.638014
$730$	0	0
$731$	−46.8622	−1.73326
$732$	0	0
$733$	28.9482	1.06923	0.534613	−	0.845097i	$-0.320457\pi$
$733$	28.9482	1.06923	0.534613	+	0.845097i	$0.320457\pi$
$734$	0	0
$735$	6.55271	0.241700
$736$	0	0
$737$	20.0358	0.738028
$738$	0	0
$739$	−23.1293	−0.850824	−0.425412	−	0.905000i	$-0.639871\pi$
$739$	−23.1293	−0.850824	−0.425412	+	0.905000i	$0.639871\pi$
$740$	0	0
$741$	9.96498	0.366073
$742$	0	0
$743$	9.69475	0.355666	0.177833	−	0.984061i	$-0.443091\pi$
$743$	9.69475	0.355666	0.177833	+	0.984061i	$0.443091\pi$
$744$	0	0
$745$	−11.4517	−0.419558
$746$	0	0
$747$	−5.78789	−0.211768
$748$	0	0
$749$	8.84751	0.323281
$750$	0	0
$751$	8.94000	0.326225	0.163113	−	0.986607i	$-0.447847\pi$
$751$	8.94000	0.326225	0.163113	+	0.986607i	$0.447847\pi$
$752$	0	0
$753$	40.0174	1.45832
$754$	0	0
$755$	−1.69741	−0.0617752
$756$	0	0
$757$	27.9131	1.01452	0.507260	−	0.861793i	$-0.330659\pi$
$757$	27.9131	1.01452	0.507260	+	0.861793i	$0.330659\pi$
$758$	0	0
$759$	4.66044	0.169163
$760$	0	0
$761$	−39.8107	−1.44314	−0.721569	−	0.692343i	$-0.756579\pi$
$761$	−39.8107	−1.44314	−0.721569	+	0.692343i	$0.756579\pi$
$762$	0	0
$763$	11.6534	0.421882
$764$	0	0
$765$	−4.38694	−0.158610
$766$	0	0
$767$	−9.01467	−0.325501
$768$	0	0
$769$	−2.73212	−0.0985228	−0.0492614	−	0.998786i	$-0.515687\pi$
$769$	−2.73212	−0.0985228	−0.0492614	+	0.998786i	$0.515687\pi$
$770$	0	0
$771$	10.5229	0.378974
$772$	0	0
$773$	31.7370	1.14150	0.570750	−	0.821124i	$-0.306652\pi$
$773$	31.7370	1.14150	0.570750	+	0.821124i	$0.306652\pi$
$774$	0	0
$775$	9.14467	0.328486
$776$	0	0
$777$	−23.6937	−0.850006
$778$	0	0
$779$	8.34657	0.299047
$780$	0	0
$781$	40.7734	1.45899
$782$	0	0
$783$	−7.17432	−0.256389
$784$	0	0
$785$	0.239402	0.00854463
$786$	0	0
$787$	−41.5640	−1.48160	−0.740798	−	0.671728i	$-0.765552\pi$
$787$	−41.5640	−1.48160	−0.740798	+	0.671728i	$0.765552\pi$
$788$	0	0
$789$	−37.1345	−1.32202
$790$	0	0
$791$	−7.04443	−0.250471
$792$	0	0
$793$	2.55098	0.0905879
$794$	0	0
$795$	−8.67110	−0.307532
$796$	0	0
$797$	5.68571	0.201398	0.100699	−	0.994917i	$-0.467892\pi$
$797$	5.68571	0.201398	0.100699	+	0.994917i	$0.467892\pi$
$798$	0	0
$799$	66.5950	2.35596
$800$	0	0
$801$	−4.02802	−0.142323
$802$	0	0
$803$	41.0864	1.44991
$804$	0	0
$805$	−0.636865	−0.0224466
$806$	0	0
$807$	−15.6991	−0.552634
$808$	0	0
$809$	−11.5321	−0.405448	−0.202724	−	0.979236i	$-0.564979\pi$
$809$	−11.5321	−0.405448	−0.202724	+	0.979236i	$0.564979\pi$
$810$	0	0
$811$	37.0153	1.29978	0.649891	−	0.760027i	$-0.274814\pi$
$811$	37.0153	1.29978	0.649891	+	0.760027i	$0.274814\pi$
$812$	0	0
$813$	−13.0789	−0.458696
$814$	0	0
$815$	12.8440	0.449907
$816$	0	0
$817$	6.41319	0.224369
$818$	0	0
$819$	−6.43432	−0.224833
$820$	0	0
$821$	20.2426	0.706471	0.353235	−	0.935534i	$-0.385081\pi$
$821$	20.2426	0.706471	0.353235	+	0.935534i	$0.385081\pi$
$822$	0	0
$823$	53.9914	1.88202	0.941010	−	0.338378i	$-0.109878\pi$
$823$	53.9914	1.88202	0.941010	+	0.338378i	$0.109878\pi$
$824$	0	0
$825$	42.1806	1.46854
$826$	0	0
$827$	4.57896	0.159226	0.0796130	−	0.996826i	$-0.474632\pi$
$827$	4.57896	0.159226	0.0796130	+	0.996826i	$0.474632\pi$
$828$	0	0
$829$	20.5636	0.714203	0.357102	−	0.934066i	$-0.383765\pi$
$829$	20.5636	0.714203	0.357102	+	0.934066i	$0.383765\pi$
$830$	0	0
$831$	17.8793	0.620225
$832$	0	0
$833$	31.0488	1.07578
$834$	0	0
$835$	11.7393	0.406256
$836$	0	0
$837$	−9.11265	−0.314979
$838$	0	0
$839$	−5.94367	−0.205198	−0.102599	−	0.994723i	$-0.532716\pi$
$839$	−5.94367	−0.205198	−0.102599	+	0.994723i	$0.532716\pi$
$840$	0	0
$841$	−26.2815	−0.906258
$842$	0	0
$843$	−37.3230	−1.28547
$844$	0	0
$845$	10.7047	0.368252
$846$	0	0
$847$	22.9803	0.789613
$848$	0	0
$849$	34.7901	1.19399
$850$	0	0
$851$	−3.55697	−0.121931
$852$	0	0
$853$	−37.0920	−1.27001	−0.635003	−	0.772510i	$-0.719001\pi$
$853$	−37.0920	−1.27001	−0.635003	+	0.772510i	$0.719001\pi$
$854$	0	0
$855$	0.600362	0.0205319
$856$	0	0
$857$	4.25912	0.145489	0.0727443	−	0.997351i	$-0.476824\pi$
$857$	4.25912	0.145489	0.0727443	+	0.997351i	$0.476824\pi$
$858$	0	0
$859$	−4.44182	−0.151553	−0.0757764	−	0.997125i	$-0.524144\pi$
$859$	−4.44182	−0.151553	−0.0757764	+	0.997125i	$0.524144\pi$
$860$	0	0
$861$	−26.8232	−0.914133
$862$	0	0
$863$	11.4119	0.388465	0.194232	−	0.980956i	$-0.437778\pi$
$863$	11.4119	0.388465	0.194232	+	0.980956i	$0.437778\pi$
$864$	0	0
$865$	3.82057	0.129903
$866$	0	0
$867$	−70.5184	−2.39493
$868$	0	0
$869$	4.98552	0.169122
$870$	0	0
$871$	−20.6684	−0.700322
$872$	0	0
$873$	2.62482	0.0888365
$874$	0	0
$875$	−12.3645	−0.417996
$876$	0	0
$877$	−22.8962	−0.773149	−0.386574	−	0.922258i	$-0.626342\pi$
$877$	−22.8962	−0.773149	−0.386574	+	0.922258i	$0.626342\pi$
$878$	0	0
$879$	−23.8250	−0.803599
$880$	0	0
$881$	9.30880	0.313622	0.156811	−	0.987629i	$-0.449879\pi$
$881$	9.30880	0.313622	0.156811	+	0.987629i	$0.449879\pi$
$882$	0	0
$883$	0.325188	0.0109435	0.00547173	−	0.999985i	$-0.498258\pi$
$883$	0.325188	0.0109435	0.00547173	+	0.999985i	$0.498258\pi$
$884$	0	0
$885$	−2.70311	−0.0908640
$886$	0	0
$887$	6.57363	0.220721	0.110361	−	0.993892i	$-0.464799\pi$
$887$	6.57363	0.220721	0.110361	+	0.993892i	$0.464799\pi$
$888$	0	0
$889$	−21.8212	−0.731858
$890$	0	0
$891$	−53.3149	−1.78612
$892$	0	0
$893$	−9.11366	−0.304977
$894$	0	0
$895$	−10.1692	−0.339920
$896$	0	0
$897$	−4.80759	−0.160521
$898$	0	0
$899$	3.45301	0.115164
$900$	0	0
$901$	−41.0864	−1.36879
$902$	0	0
$903$	−20.6100	−0.685856
$904$	0	0
$905$	−11.7391	−0.390219
$906$	0	0
$907$	42.1680	1.40017	0.700083	−	0.714062i	$-0.253147\pi$
$907$	42.1680	1.40017	0.700083	+	0.714062i	$0.253147\pi$
$908$	0	0
$909$	13.5322	0.448835
$910$	0	0
$911$	−29.7888	−0.986948	−0.493474	−	0.869761i	$-0.664273\pi$
$911$	−29.7888	−0.986948	−0.493474	+	0.869761i	$0.664273\pi$
$912$	0	0
$913$	−38.2540	−1.26602
$914$	0	0
$915$	0.764928	0.0252877
$916$	0	0
$917$	18.2704	0.603341
$918$	0	0
$919$	27.5367	0.908351	0.454176	−	0.890912i	$-0.349934\pi$
$919$	27.5367	0.908351	0.454176	+	0.890912i	$0.349934\pi$
$920$	0	0
$921$	4.39874	0.144943
$922$	0	0
$923$	−42.0608	−1.38445
$924$	0	0
$925$	−32.1934	−1.05851
$926$	0	0
$927$	−4.04366	−0.132811
$928$	0	0
$929$	−46.8292	−1.53642	−0.768208	−	0.640201i	$-0.778851\pi$
$929$	−46.8292	−1.53642	−0.768208	+	0.640201i	$0.778851\pi$
$930$	0	0
$931$	−4.24910	−0.139259
$932$	0	0
$933$	−6.61839	−0.216676
$934$	0	0
$935$	−28.9946	−0.948226
$936$	0	0
$937$	−0.600963	−0.0196326	−0.00981631	−	0.999952i	$-0.503125\pi$
$937$	−0.600963	−0.0196326	−0.00981631	+	0.999952i	$0.503125\pi$
$938$	0	0
$939$	−17.8980	−0.584081
$940$	0	0
$941$	−21.9147	−0.714399	−0.357199	−	0.934028i	$-0.616268\pi$
$941$	−21.9147	−0.714399	−0.357199	+	0.934028i	$0.616268\pi$
$942$	0	0
$943$	−4.02679	−0.131130
$944$	0	0
$945$	5.74395	0.186851
$946$	0	0
$947$	13.1409	0.427021	0.213510	−	0.976941i	$-0.431510\pi$
$947$	13.1409	0.427021	0.213510	+	0.976941i	$0.431510\pi$
$948$	0	0
$949$	−42.3837	−1.37583
$950$	0	0
$951$	−13.9770	−0.453236
$952$	0	0
$953$	41.6990	1.35076	0.675382	−	0.737468i	$-0.263979\pi$
$953$	41.6990	1.35076	0.675382	+	0.737468i	$0.263979\pi$
$954$	0	0
$955$	18.9752	0.614024
$956$	0	0
$957$	15.9273	0.514857
$958$	0	0
$959$	−8.30037	−0.268033
$960$	0	0
$961$	−26.6141	−0.858519
$962$	0	0
$963$	4.02381	0.129665
$964$	0	0
$965$	11.8476	0.381386
$966$	0	0
$967$	1.20042	0.0386028	0.0193014	−	0.999814i	$-0.493856\pi$
$967$	1.20042	0.0386028	0.0193014	+	0.999814i	$0.493856\pi$
$968$	0	0
$969$	14.1584	0.454833
$970$	0	0
$971$	−9.61639	−0.308605	−0.154302	−	0.988024i	$-0.549313\pi$
$971$	−9.61639	−0.308605	−0.154302	+	0.988024i	$0.549313\pi$
$972$	0	0
$973$	−0.934748	−0.0299667
$974$	0	0
$975$	−43.5125	−1.39351
$976$	0	0
$977$	−35.4975	−1.13567	−0.567833	−	0.823144i	$-0.692218\pi$
$977$	−35.4975	−1.13567	−0.567833	+	0.823144i	$0.692218\pi$
$978$	0	0
$979$	−26.6225	−0.850858
$980$	0	0
$981$	5.29992	0.169213
$982$	0	0
$983$	−47.5823	−1.51764	−0.758819	−	0.651301i	$-0.774223\pi$
$983$	−47.5823	−1.51764	−0.758819	+	0.651301i	$0.774223\pi$
$984$	0	0
$985$	−6.46631	−0.206034
$986$	0	0
$987$	29.2884	0.932261
$988$	0	0
$989$	−3.09403	−0.0983845
$990$	0	0
$991$	11.4851	0.364836	0.182418	−	0.983221i	$-0.441608\pi$
$991$	11.4851	0.364836	0.182418	+	0.983221i	$0.441608\pi$
$992$	0	0
$993$	−68.0410	−2.15922
$994$	0	0
$995$	12.5550	0.398019
$996$	0	0
$997$	−12.6590	−0.400916	−0.200458	−	0.979702i	$-0.564243\pi$
$997$	−12.6590	−0.400916	−0.200458	+	0.979702i	$0.564243\pi$
$998$	0	0
$999$	32.0807	1.01499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.e.1.9	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.e.1.9	✓	24	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)

\(f(q)\)	\(=\)	\(q-1.93761 q^{3} +0.795901 q^{5} +1.65858 q^{7} +0.754317 q^{9} +O(q^{10})\)	\(q-1.93761 q^{3} +0.795901 q^{5} +1.65858 q^{7} +0.754317 q^{9} +4.98552 q^{11} -5.14293 q^{13} -1.54214 q^{15} -7.30716 q^{17} +1.00000 q^{19} -3.21368 q^{21} -0.482448 q^{23} -4.36654 q^{25} +4.35125 q^{27} -1.64880 q^{29} -2.09426 q^{31} -9.65997 q^{33} +1.32007 q^{35} +7.37275 q^{37} +9.96498 q^{39} +8.34657 q^{41} +6.41319 q^{43} +0.600362 q^{45} -9.11366 q^{47} -4.24910 q^{49} +14.1584 q^{51} +5.62276 q^{53} +3.96798 q^{55} -1.93761 q^{57} +1.75283 q^{59} -0.496017 q^{61} +1.25110 q^{63} -4.09327 q^{65} +4.01880 q^{67} +0.934795 q^{69} +8.17837 q^{71} +8.24116 q^{73} +8.46064 q^{75} +8.26890 q^{77} +1.00000 q^{79} -10.6940 q^{81} -7.67303 q^{83} -5.81578 q^{85} +3.19472 q^{87} -5.33996 q^{89} -8.52999 q^{91} +4.05785 q^{93} +0.795901 q^{95} +3.47972 q^{97} +3.76066 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

Introduction

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