LMFDB - Embedded newform 6004.2.a.f.1.5

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:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-1.61532 q^{3} +1.18505 q^{5} +3.44544 q^{7} -0.390754 q^{9} +O(q^{10})$	$q-1.61532 q^{3} +1.18505 q^{5} +3.44544 q^{7} -0.390754 q^{9} -5.43776 q^{11} +4.64341 q^{13} -1.91424 q^{15} +1.71018 q^{17} -1.00000 q^{19} -5.56548 q^{21} -2.61889 q^{23} -3.59565 q^{25} +5.47714 q^{27} -0.806736 q^{29} +1.06235 q^{31} +8.78370 q^{33} +4.08303 q^{35} -5.55276 q^{37} -7.50058 q^{39} -8.63256 q^{41} +2.18814 q^{43} -0.463065 q^{45} +9.20135 q^{47} +4.87108 q^{49} -2.76248 q^{51} -12.3427 q^{53} -6.44403 q^{55} +1.61532 q^{57} +5.65530 q^{59} -5.83852 q^{61} -1.34632 q^{63} +5.50269 q^{65} +4.38626 q^{67} +4.23034 q^{69} -3.01663 q^{71} -4.48547 q^{73} +5.80811 q^{75} -18.7355 q^{77} +1.00000 q^{79} -7.67505 q^{81} -6.91424 q^{83} +2.02666 q^{85} +1.30313 q^{87} -5.30790 q^{89} +15.9986 q^{91} -1.71602 q^{93} -1.18505 q^{95} -17.2074 q^{97} +2.12483 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * 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
$3$	−1.61532	−0.932603	−0.466302	−	0.884626i	$-0.654414\pi$
$3$	−1.61532	−0.932603	−0.466302	+	0.884626i	$0.654414\pi$
$4$	0	0
$5$	1.18505	0.529972	0.264986	−	0.964252i	$-0.414633\pi$
$5$	1.18505	0.529972	0.264986	+	0.964252i	$0.414633\pi$
$6$	0	0
$7$	3.44544	1.30225	0.651127	−	0.758968i	$-0.274296\pi$
$7$	3.44544	1.30225	0.651127	+	0.758968i	$0.274296\pi$
$8$	0	0
$9$	−0.390754	−0.130251
$10$	0	0
$11$	−5.43776	−1.63955	−0.819773	−	0.572688i	$-0.805900\pi$
$11$	−5.43776	−1.63955	−0.819773	+	0.572688i	$0.805900\pi$
$12$	0	0
$13$	4.64341	1.28785	0.643926	−	0.765088i	$-0.277305\pi$
$13$	4.64341	1.28785	0.643926	+	0.765088i	$0.277305\pi$
$14$	0	0
$15$	−1.91424	−0.494253
$16$	0	0
$17$	1.71018	0.414780	0.207390	−	0.978258i	$-0.433503\pi$
$17$	1.71018	0.414780	0.207390	+	0.978258i	$0.433503\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−5.56548	−1.21449
$22$	0	0
$23$	−2.61889	−0.546076	−0.273038	−	0.962003i	$-0.588029\pi$
$23$	−2.61889	−0.546076	−0.273038	+	0.962003i	$0.588029\pi$
$24$	0	0
$25$	−3.59565	−0.719130
$26$	0	0
$27$	5.47714	1.05408
$28$	0	0
$29$	−0.806736	−0.149807	−0.0749036	−	0.997191i	$-0.523865\pi$
$29$	−0.806736	−0.149807	−0.0749036	+	0.997191i	$0.523865\pi$
$30$	0	0
$31$	1.06235	0.190803	0.0954014	−	0.995439i	$-0.469587\pi$
$31$	1.06235	0.190803	0.0954014	+	0.995439i	$0.469587\pi$
$32$	0	0
$33$	8.78370	1.52905
$34$	0	0
$35$	4.08303	0.690159
$36$	0	0
$37$	−5.55276	−0.912868	−0.456434	−	0.889757i	$-0.650874\pi$
$37$	−5.55276	−0.912868	−0.456434	+	0.889757i	$0.650874\pi$
$38$	0	0
$39$	−7.50058	−1.20105
$40$	0	0
$41$	−8.63256	−1.34818	−0.674090	−	0.738649i	$-0.735464\pi$
$41$	−8.63256	−1.34818	−0.674090	+	0.738649i	$0.735464\pi$
$42$	0	0
$43$	2.18814	0.333688	0.166844	−	0.985983i	$-0.446642\pi$
$43$	2.18814	0.333688	0.166844	+	0.985983i	$0.446642\pi$
$44$	0	0
$45$	−0.463065	−0.0690296
$46$	0	0
$47$	9.20135	1.34216	0.671078	−	0.741387i	$-0.265832\pi$
$47$	9.20135	1.34216	0.671078	+	0.741387i	$0.265832\pi$
$48$	0	0
$49$	4.87108	0.695868
$50$	0	0
$51$	−2.76248	−0.386825
$52$	0	0
$53$	−12.3427	−1.69539	−0.847697	−	0.530480i	$-0.822012\pi$
$53$	−12.3427	−1.69539	−0.847697	+	0.530480i	$0.822012\pi$
$54$	0	0
$55$	−6.44403	−0.868914
$56$	0	0
$57$	1.61532	0.213954
$58$	0	0
$59$	5.65530	0.736257	0.368128	−	0.929775i	$-0.379999\pi$
$59$	5.65530	0.736257	0.368128	+	0.929775i	$0.379999\pi$
$60$	0	0
$61$	−5.83852	−0.747546	−0.373773	−	0.927520i	$-0.621936\pi$
$61$	−5.83852	−0.747546	−0.373773	+	0.927520i	$0.621936\pi$
$62$	0	0
$63$	−1.34632	−0.169621
$64$	0	0
$65$	5.50269	0.682525
$66$	0	0
$67$	4.38626	0.535867	0.267934	−	0.963437i	$-0.413659\pi$
$67$	4.38626	0.535867	0.267934	+	0.963437i	$0.413659\pi$
$68$	0	0
$69$	4.23034	0.509273
$70$	0	0
$71$	−3.01663	−0.358008	−0.179004	−	0.983848i	$-0.557287\pi$
$71$	−3.01663	−0.358008	−0.179004	+	0.983848i	$0.557287\pi$
$72$	0	0
$73$	−4.48547	−0.524985	−0.262492	−	0.964934i	$-0.584544\pi$
$73$	−4.48547	−0.524985	−0.262492	+	0.964934i	$0.584544\pi$
$74$	0	0
$75$	5.80811	0.670663
$76$	0	0
$77$	−18.7355	−2.13511
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−7.67505	−0.852783
$82$	0	0
$83$	−6.91424	−0.758936	−0.379468	−	0.925205i	$-0.623893\pi$
$83$	−6.91424	−0.758936	−0.379468	+	0.925205i	$0.623893\pi$
$84$	0	0
$85$	2.02666	0.219822
$86$	0	0
$87$	1.30313	0.139711
$88$	0	0
$89$	−5.30790	−0.562636	−0.281318	−	0.959615i	$-0.590772\pi$
$89$	−5.30790	−0.562636	−0.281318	+	0.959615i	$0.590772\pi$
$90$	0	0
$91$	15.9986	1.67711
$92$	0	0
$93$	−1.71602	−0.177943
$94$	0	0
$95$	−1.18505	−0.121584
$96$	0	0
$97$	−17.2074	−1.74715	−0.873575	−	0.486689i	$-0.838204\pi$
$97$	−17.2074	−1.74715	−0.873575	+	0.486689i	$0.838204\pi$
$98$	0	0
$99$	2.12483	0.213553
$100$	0	0
$101$	6.60632	0.657353	0.328677	−	0.944443i	$-0.393397\pi$
$101$	6.60632	0.657353	0.328677	+	0.944443i	$0.393397\pi$
$102$	0	0
$103$	−16.1905	−1.59530	−0.797650	−	0.603121i	$-0.793924\pi$
$103$	−16.1905	−1.59530	−0.797650	+	0.603121i	$0.793924\pi$
$104$	0	0
$105$	−6.59539	−0.643644
$106$	0	0
$107$	12.6560	1.22350	0.611752	−	0.791050i	$-0.290465\pi$
$107$	12.6560	1.22350	0.611752	+	0.791050i	$0.290465\pi$
$108$	0	0
$109$	13.9436	1.33555	0.667776	−	0.744362i	$-0.267246\pi$
$109$	13.9436	1.33555	0.667776	+	0.744362i	$0.267246\pi$
$110$	0	0
$111$	8.96946	0.851344
$112$	0	0
$113$	−7.02361	−0.660726	−0.330363	−	0.943854i	$-0.607171\pi$
$113$	−7.02361	−0.660726	−0.330363	+	0.943854i	$0.607171\pi$
$114$	0	0
$115$	−3.10352	−0.289405
$116$	0	0
$117$	−1.81443	−0.167745
$118$	0	0
$119$	5.89233	0.540149
$120$	0	0
$121$	18.5692	1.68811
$122$	0	0
$123$	13.9443	1.25732
$124$	0	0
$125$	−10.1863	−0.911091
$126$	0	0
$127$	−14.7459	−1.30848	−0.654242	−	0.756285i	$-0.727012\pi$
$127$	−14.7459	−1.30848	−0.654242	+	0.756285i	$0.727012\pi$
$128$	0	0
$129$	−3.53454	−0.311199
$130$	0	0
$131$	4.39829	0.384280	0.192140	−	0.981368i	$-0.438457\pi$
$131$	4.39829	0.384280	0.192140	+	0.981368i	$0.438457\pi$
$132$	0	0
$133$	−3.44544	−0.298758
$134$	0	0
$135$	6.49070	0.558631
$136$	0	0
$137$	14.9832	1.28010	0.640052	−	0.768332i	$-0.278913\pi$
$137$	14.9832	1.28010	0.640052	+	0.768332i	$0.278913\pi$
$138$	0	0
$139$	−10.5566	−0.895395	−0.447698	−	0.894185i	$-0.647756\pi$
$139$	−10.5566	−0.895395	−0.447698	+	0.894185i	$0.647756\pi$
$140$	0	0
$141$	−14.8631	−1.25170
$142$	0	0
$143$	−25.2498	−2.11149
$144$	0	0
$145$	−0.956025	−0.0793936
$146$	0	0
$147$	−7.86833	−0.648969
$148$	0	0
$149$	−18.7617	−1.53702	−0.768508	−	0.639840i	$-0.779001\pi$
$149$	−18.7617	−1.53702	−0.768508	+	0.639840i	$0.779001\pi$
$150$	0	0
$151$	20.4683	1.66569	0.832844	−	0.553508i	$-0.186711\pi$
$151$	20.4683	1.66569	0.832844	+	0.553508i	$0.186711\pi$
$152$	0	0
$153$	−0.668261	−0.0540257
$154$	0	0
$155$	1.25894	0.101120
$156$	0	0
$157$	19.3142	1.54144	0.770721	−	0.637173i	$-0.219896\pi$
$157$	19.3142	1.54144	0.770721	+	0.637173i	$0.219896\pi$
$158$	0	0
$159$	19.9373	1.58113
$160$	0	0
$161$	−9.02324	−0.711131
$162$	0	0
$163$	8.00618	0.627093	0.313546	−	0.949573i	$-0.398483\pi$
$163$	8.00618	0.627093	0.313546	+	0.949573i	$0.398483\pi$
$164$	0	0
$165$	10.4092	0.810351
$166$	0	0
$167$	14.4783	1.12037	0.560183	−	0.828369i	$-0.310731\pi$
$167$	14.4783	1.12037	0.560183	+	0.828369i	$0.310731\pi$
$168$	0	0
$169$	8.56130	0.658562
$170$	0	0
$171$	0.390754	0.0298817
$172$	0	0
$173$	−22.0694	−1.67791	−0.838954	−	0.544203i	$-0.816832\pi$
$173$	−22.0694	−1.67791	−0.838954	+	0.544203i	$0.816832\pi$
$174$	0	0
$175$	−12.3886	−0.936490
$176$	0	0
$177$	−9.13509	−0.686635
$178$	0	0
$179$	−20.7105	−1.54797	−0.773986	−	0.633202i	$-0.781740\pi$
$179$	−20.7105	−1.54797	−0.773986	+	0.633202i	$0.781740\pi$
$180$	0	0
$181$	14.3283	1.06502	0.532509	−	0.846425i	$-0.321249\pi$
$181$	14.3283	1.06502	0.532509	+	0.846425i	$0.321249\pi$
$182$	0	0
$183$	9.43105	0.697164
$184$	0	0
$185$	−6.58032	−0.483795
$186$	0	0
$187$	−9.29955	−0.680051
$188$	0	0
$189$	18.8712	1.37268
$190$	0	0
$191$	2.05380	0.148608	0.0743040	−	0.997236i	$-0.476326\pi$
$191$	2.05380	0.148608	0.0743040	+	0.997236i	$0.476326\pi$
$192$	0	0
$193$	−3.81409	−0.274544	−0.137272	−	0.990533i	$-0.543833\pi$
$193$	−3.81409	−0.274544	−0.137272	+	0.990533i	$0.543833\pi$
$194$	0	0
$195$	−8.88859	−0.636525
$196$	0	0
$197$	7.30212	0.520255	0.260127	−	0.965574i	$-0.416235\pi$
$197$	7.30212	0.520255	0.260127	+	0.965574i	$0.416235\pi$
$198$	0	0
$199$	23.0024	1.63060	0.815299	−	0.579040i	$-0.196573\pi$
$199$	23.0024	1.63060	0.815299	+	0.579040i	$0.196573\pi$
$200$	0	0
$201$	−7.08520	−0.499752
$202$	0	0
$203$	−2.77956	−0.195087
$204$	0	0
$205$	−10.2300	−0.714497
$206$	0	0
$207$	1.02334	0.0711272
$208$	0	0
$209$	5.43776	0.376138
$210$	0	0
$211$	13.3860	0.921531	0.460766	−	0.887522i	$-0.347575\pi$
$211$	13.3860	0.921531	0.460766	+	0.887522i	$0.347575\pi$
$212$	0	0
$213$	4.87281	0.333879
$214$	0	0
$215$	2.59306	0.176845
$216$	0	0
$217$	3.66025	0.248474
$218$	0	0
$219$	7.24545	0.489602
$220$	0	0
$221$	7.94108	0.534175
$222$	0	0
$223$	−7.23905	−0.484762	−0.242381	−	0.970181i	$-0.577928\pi$
$223$	−7.23905	−0.484762	−0.242381	+	0.970181i	$0.577928\pi$
$224$	0	0
$225$	1.40502	0.0936677
$226$	0	0
$227$	13.9688	0.927144	0.463572	−	0.886059i	$-0.346568\pi$
$227$	13.9688	0.927144	0.463572	+	0.886059i	$0.346568\pi$
$228$	0	0
$229$	−26.7269	−1.76616	−0.883081	−	0.469220i	$-0.844535\pi$
$229$	−26.7269	−1.76616	−0.883081	+	0.469220i	$0.844535\pi$
$230$	0	0
$231$	30.2637	1.99121
$232$	0	0
$233$	−24.8449	−1.62764	−0.813822	−	0.581114i	$-0.802617\pi$
$233$	−24.8449	−1.62764	−0.813822	+	0.581114i	$0.802617\pi$
$234$	0	0
$235$	10.9041	0.711305
$236$	0	0
$237$	−1.61532	−0.104926
$238$	0	0
$239$	−25.2227	−1.63152	−0.815759	−	0.578392i	$-0.803681\pi$
$239$	−25.2227	−1.63152	−0.815759	+	0.578392i	$0.803681\pi$
$240$	0	0
$241$	−8.81067	−0.567545	−0.283773	−	0.958892i	$-0.591586\pi$
$241$	−8.81067	−0.567545	−0.283773	+	0.958892i	$0.591586\pi$
$242$	0	0
$243$	−4.03379	−0.258768
$244$	0	0
$245$	5.77249	0.368791
$246$	0	0
$247$	−4.64341	−0.295453
$248$	0	0
$249$	11.1687	0.707786
$250$	0	0
$251$	−26.4408	−1.66893	−0.834466	−	0.551060i	$-0.814224\pi$
$251$	−26.4408	−1.66893	−0.834466	+	0.551060i	$0.814224\pi$
$252$	0	0
$253$	14.2409	0.895318
$254$	0	0
$255$	−3.27369	−0.205006
$256$	0	0
$257$	−20.4631	−1.27645	−0.638227	−	0.769848i	$-0.720332\pi$
$257$	−20.4631	−1.27645	−0.638227	+	0.769848i	$0.720332\pi$
$258$	0	0
$259$	−19.1317	−1.18879
$260$	0	0
$261$	0.315236	0.0195126
$262$	0	0
$263$	−19.8588	−1.22455	−0.612274	−	0.790645i	$-0.709745\pi$
$263$	−19.8588	−1.22455	−0.612274	+	0.790645i	$0.709745\pi$
$264$	0	0
$265$	−14.6267	−0.898512
$266$	0	0
$267$	8.57394	0.524716
$268$	0	0
$269$	31.9660	1.94900	0.974502	−	0.224379i	$-0.0720354\pi$
$269$	31.9660	1.94900	0.974502	+	0.224379i	$0.0720354\pi$
$270$	0	0
$271$	4.80839	0.292089	0.146044	−	0.989278i	$-0.453346\pi$
$271$	4.80839	0.292089	0.146044	+	0.989278i	$0.453346\pi$
$272$	0	0
$273$	−25.8428	−1.56408
$274$	0	0
$275$	19.5523	1.17905
$276$	0	0
$277$	1.28103	0.0769696	0.0384848	−	0.999259i	$-0.487747\pi$
$277$	1.28103	0.0769696	0.0384848	+	0.999259i	$0.487747\pi$
$278$	0	0
$279$	−0.415116	−0.0248524
$280$	0	0
$281$	−25.4228	−1.51660	−0.758298	−	0.651908i	$-0.773969\pi$
$281$	−25.4228	−1.51660	−0.758298	+	0.651908i	$0.773969\pi$
$282$	0	0
$283$	−15.2196	−0.904711	−0.452356	−	0.891838i	$-0.649416\pi$
$283$	−15.2196	−0.904711	−0.452356	+	0.891838i	$0.649416\pi$
$284$	0	0
$285$	1.91424	0.113390
$286$	0	0
$287$	−29.7430	−1.75567
$288$	0	0
$289$	−14.0753	−0.827958
$290$	0	0
$291$	27.7954	1.62940
$292$	0	0
$293$	3.04597	0.177948	0.0889738	−	0.996034i	$-0.471641\pi$
$293$	3.04597	0.177948	0.0889738	+	0.996034i	$0.471641\pi$
$294$	0	0
$295$	6.70183	0.390196
$296$	0	0
$297$	−29.7834	−1.72821
$298$	0	0
$299$	−12.1606	−0.703265
$300$	0	0
$301$	7.53911	0.434547
$302$	0	0
$303$	−10.6713	−0.613050
$304$	0	0
$305$	−6.91896	−0.396178
$306$	0	0
$307$	−2.53950	−0.144937	−0.0724685	−	0.997371i	$-0.523088\pi$
$307$	−2.53950	−0.144937	−0.0724685	+	0.997371i	$0.523088\pi$
$308$	0	0
$309$	26.1528	1.48778
$310$	0	0
$311$	10.6382	0.603236	0.301618	−	0.953429i	$-0.402473\pi$
$311$	10.6382	0.603236	0.301618	+	0.953429i	$0.402473\pi$
$312$	0	0
$313$	−15.8186	−0.894121	−0.447061	−	0.894504i	$-0.647529\pi$
$313$	−15.8186	−0.894121	−0.447061	+	0.894504i	$0.647529\pi$
$314$	0	0
$315$	−1.59546	−0.0898942
$316$	0	0
$317$	−0.0648254	−0.00364096	−0.00182048	−	0.999998i	$-0.500579\pi$
$317$	−0.0648254	−0.00364096	−0.00182048	+	0.999998i	$0.500579\pi$
$318$	0	0
$319$	4.38684	0.245616
$320$	0	0
$321$	−20.4435	−1.14104
$322$	0	0
$323$	−1.71018	−0.0951570
$324$	0	0
$325$	−16.6961	−0.926132
$326$	0	0
$327$	−22.5233	−1.24554
$328$	0	0
$329$	31.7027	1.74783
$330$	0	0
$331$	31.9946	1.75858	0.879292	−	0.476282i	$-0.158016\pi$
$331$	31.9946	1.75858	0.879292	+	0.476282i	$0.158016\pi$
$332$	0	0
$333$	2.16977	0.118902
$334$	0	0
$335$	5.19796	0.283995
$336$	0	0
$337$	27.7643	1.51242	0.756208	−	0.654331i	$-0.227050\pi$
$337$	27.7643	1.51242	0.756208	+	0.654331i	$0.227050\pi$
$338$	0	0
$339$	11.3454	0.616195
$340$	0	0
$341$	−5.77678	−0.312830
$342$	0	0
$343$	−7.33508	−0.396057
$344$	0	0
$345$	5.01317	0.269900
$346$	0	0
$347$	−20.5438	−1.10285	−0.551425	−	0.834225i	$-0.685916\pi$
$347$	−20.5438	−1.10285	−0.551425	+	0.834225i	$0.685916\pi$
$348$	0	0
$349$	14.7986	0.792149	0.396075	−	0.918218i	$-0.370372\pi$
$349$	14.7986	0.792149	0.396075	+	0.918218i	$0.370372\pi$
$350$	0	0
$351$	25.4326	1.35749
$352$	0	0
$353$	10.9461	0.582600	0.291300	−	0.956632i	$-0.405912\pi$
$353$	10.9461	0.582600	0.291300	+	0.956632i	$0.405912\pi$
$354$	0	0
$355$	−3.57486	−0.189734
$356$	0	0
$357$	−9.51797	−0.503745
$358$	0	0
$359$	−22.3680	−1.18054	−0.590269	−	0.807207i	$-0.700978\pi$
$359$	−22.3680	−1.18054	−0.590269	+	0.807207i	$0.700978\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−29.9952	−1.57434
$364$	0	0
$365$	−5.31552	−0.278227
$366$	0	0
$367$	−9.32448	−0.486734	−0.243367	−	0.969934i	$-0.578252\pi$
$367$	−9.32448	−0.486734	−0.243367	+	0.969934i	$0.578252\pi$
$368$	0	0
$369$	3.37321	0.175602
$370$	0	0
$371$	−42.5259	−2.20784
$372$	0	0
$373$	−13.4335	−0.695559	−0.347779	−	0.937576i	$-0.613064\pi$
$373$	−13.4335	−0.695559	−0.347779	+	0.937576i	$0.613064\pi$
$374$	0	0
$375$	16.4541	0.849686
$376$	0	0
$377$	−3.74601	−0.192929
$378$	0	0
$379$	30.8773	1.58606	0.793029	−	0.609184i	$-0.208503\pi$
$379$	30.8773	1.58606	0.793029	+	0.609184i	$0.208503\pi$
$380$	0	0
$381$	23.8192	1.22030
$382$	0	0
$383$	−3.44439	−0.176000	−0.0880001	−	0.996120i	$-0.528048\pi$
$383$	−3.44439	−0.176000	−0.0880001	+	0.996120i	$0.528048\pi$
$384$	0	0
$385$	−22.2026	−1.13155
$386$	0	0
$387$	−0.855025	−0.0434634
$388$	0	0
$389$	−12.9939	−0.658815	−0.329408	−	0.944188i	$-0.606849\pi$
$389$	−12.9939	−0.658815	−0.329408	+	0.944188i	$0.606849\pi$
$390$	0	0
$391$	−4.47878	−0.226501
$392$	0	0
$393$	−7.10462	−0.358381
$394$	0	0
$395$	1.18505	0.0596265
$396$	0	0
$397$	−33.0856	−1.66052	−0.830260	−	0.557377i	$-0.811808\pi$
$397$	−33.0856	−1.66052	−0.830260	+	0.557377i	$0.811808\pi$
$398$	0	0
$399$	5.56548	0.278622
$400$	0	0
$401$	−13.5689	−0.677598	−0.338799	−	0.940859i	$-0.610021\pi$
$401$	−13.5689	−0.677598	−0.338799	+	0.940859i	$0.610021\pi$
$402$	0	0
$403$	4.93291	0.245726
$404$	0	0
$405$	−9.09534	−0.451951
$406$	0	0
$407$	30.1946	1.49669
$408$	0	0
$409$	−18.6595	−0.922655	−0.461327	−	0.887230i	$-0.652627\pi$
$409$	−18.6595	−0.922655	−0.461327	+	0.887230i	$0.652627\pi$
$410$	0	0
$411$	−24.2026	−1.19383
$412$	0	0
$413$	19.4850	0.958794
$414$	0	0
$415$	−8.19374	−0.402215
$416$	0	0
$417$	17.0522	0.835049
$418$	0	0
$419$	−34.5635	−1.68854	−0.844269	−	0.535920i	$-0.819965\pi$
$419$	−34.5635	−1.68854	−0.844269	+	0.535920i	$0.819965\pi$
$420$	0	0
$421$	34.7325	1.69276	0.846379	−	0.532581i	$-0.178778\pi$
$421$	34.7325	1.69276	0.846379	+	0.532581i	$0.178778\pi$
$422$	0	0
$423$	−3.59547	−0.174818
$424$	0	0
$425$	−6.14921	−0.298280
$426$	0	0
$427$	−20.1163	−0.973495
$428$	0	0
$429$	40.7864	1.96918
$430$	0	0
$431$	−7.49379	−0.360963	−0.180482	−	0.983578i	$-0.557766\pi$
$431$	−7.49379	−0.360963	−0.180482	+	0.983578i	$0.557766\pi$
$432$	0	0
$433$	13.2912	0.638733	0.319366	−	0.947631i	$-0.396530\pi$
$433$	13.2912	0.638733	0.319366	+	0.947631i	$0.396530\pi$
$434$	0	0
$435$	1.54428	0.0740427
$436$	0	0
$437$	2.61889	0.125279
$438$	0	0
$439$	15.4598	0.737855	0.368928	−	0.929458i	$-0.379725\pi$
$439$	15.4598	0.737855	0.368928	+	0.929458i	$0.379725\pi$
$440$	0	0
$441$	−1.90339	−0.0906378
$442$	0	0
$443$	−26.2671	−1.24799	−0.623995	−	0.781428i	$-0.714491\pi$
$443$	−26.2671	−1.24799	−0.623995	+	0.781428i	$0.714491\pi$
$444$	0	0
$445$	−6.29014	−0.298181
$446$	0	0
$447$	30.3060	1.43343
$448$	0	0
$449$	−25.7091	−1.21329	−0.606643	−	0.794975i	$-0.707484\pi$
$449$	−25.7091	−1.21329	−0.606643	+	0.794975i	$0.707484\pi$
$450$	0	0
$451$	46.9418	2.21040
$452$	0	0
$453$	−33.0628	−1.55343
$454$	0	0
$455$	18.9592	0.888822
$456$	0	0
$457$	22.3717	1.04650	0.523251	−	0.852179i	$-0.324719\pi$
$457$	22.3717	1.04650	0.523251	+	0.852179i	$0.324719\pi$
$458$	0	0
$459$	9.36690	0.437209
$460$	0	0
$461$	20.0230	0.932566	0.466283	−	0.884636i	$-0.345593\pi$
$461$	20.0230	0.932566	0.466283	+	0.884636i	$0.345593\pi$
$462$	0	0
$463$	−3.76680	−0.175058	−0.0875291	−	0.996162i	$-0.527897\pi$
$463$	−3.76680	−0.175058	−0.0875291	+	0.996162i	$0.527897\pi$
$464$	0	0
$465$	−2.03358	−0.0943050
$466$	0	0
$467$	−20.4190	−0.944880	−0.472440	−	0.881363i	$-0.656627\pi$
$467$	−20.4190	−0.944880	−0.472440	+	0.881363i	$0.656627\pi$
$468$	0	0
$469$	15.1126	0.697836
$470$	0	0
$471$	−31.1986	−1.43755
$472$	0	0
$473$	−11.8986	−0.547097
$474$	0	0
$475$	3.59565	0.164980
$476$	0	0
$477$	4.82295	0.220828
$478$	0	0
$479$	−10.2513	−0.468393	−0.234197	−	0.972189i	$-0.575246\pi$
$479$	−10.2513	−0.468393	−0.234197	+	0.972189i	$0.575246\pi$
$480$	0	0
$481$	−25.7838	−1.17564
$482$	0	0
$483$	14.5754	0.663203
$484$	0	0
$485$	−20.3917	−0.925941
$486$	0	0
$487$	−33.0510	−1.49769	−0.748843	−	0.662747i	$-0.769390\pi$
$487$	−33.0510	−1.49769	−0.748843	+	0.662747i	$0.769390\pi$
$488$	0	0
$489$	−12.9325	−0.584829
$490$	0	0
$491$	−18.6519	−0.841750	−0.420875	−	0.907119i	$-0.638277\pi$
$491$	−18.6519	−0.841750	−0.420875	+	0.907119i	$0.638277\pi$
$492$	0	0
$493$	−1.37966	−0.0621370
$494$	0	0
$495$	2.51803	0.113177
$496$	0	0
$497$	−10.3936	−0.466217
$498$	0	0
$499$	13.5250	0.605462	0.302731	−	0.953076i	$-0.402102\pi$
$499$	13.5250	0.605462	0.302731	+	0.953076i	$0.402102\pi$
$500$	0	0
$501$	−23.3871	−1.04486
$502$	0	0
$503$	8.73467	0.389459	0.194730	−	0.980857i	$-0.437617\pi$
$503$	8.73467	0.389459	0.194730	+	0.980857i	$0.437617\pi$
$504$	0	0
$505$	7.82884	0.348379
$506$	0	0
$507$	−13.8292	−0.614177
$508$	0	0
$509$	−0.560197	−0.0248303	−0.0124151	−	0.999923i	$-0.503952\pi$
$509$	−0.560197	−0.0248303	−0.0124151	+	0.999923i	$0.503952\pi$
$510$	0	0
$511$	−15.4544	−0.683664
$512$	0	0
$513$	−5.47714	−0.241822
$514$	0	0
$515$	−19.1866	−0.845464
$516$	0	0
$517$	−50.0348	−2.20053
$518$	0	0
$519$	35.6491	1.56482
$520$	0	0
$521$	−20.4651	−0.896591	−0.448295	−	0.893886i	$-0.647969\pi$
$521$	−20.4651	−0.896591	−0.448295	+	0.893886i	$0.647969\pi$
$522$	0	0
$523$	−29.3411	−1.28300	−0.641499	−	0.767124i	$-0.721687\pi$
$523$	−29.3411	−1.28300	−0.641499	+	0.767124i	$0.721687\pi$
$524$	0	0
$525$	20.0115	0.873374
$526$	0	0
$527$	1.81680	0.0791412
$528$	0	0
$529$	−16.1414	−0.701801
$530$	0	0
$531$	−2.20983	−0.0958985
$532$	0	0
$533$	−40.0846	−1.73626
$534$	0	0
$535$	14.9981	0.648423
$536$	0	0
$537$	33.4539	1.44364
$538$	0	0
$539$	−26.4877	−1.14091
$540$	0	0
$541$	−8.77071	−0.377082	−0.188541	−	0.982065i	$-0.560376\pi$
$541$	−8.77071	−0.377082	−0.188541	+	0.982065i	$0.560376\pi$
$542$	0	0
$543$	−23.1448	−0.993238
$544$	0	0
$545$	16.5239	0.707805
$546$	0	0
$547$	−23.9124	−1.02242	−0.511210	−	0.859456i	$-0.670802\pi$
$547$	−23.9124	−1.02242	−0.511210	+	0.859456i	$0.670802\pi$
$548$	0	0
$549$	2.28143	0.0973689
$550$	0	0
$551$	0.806736	0.0343681
$552$	0	0
$553$	3.44544	0.146515
$554$	0	0
$555$	10.6293	0.451188
$556$	0	0
$557$	9.45034	0.400424	0.200212	−	0.979753i	$-0.435837\pi$
$557$	9.45034	0.400424	0.200212	+	0.979753i	$0.435837\pi$
$558$	0	0
$559$	10.1604	0.429741
$560$	0	0
$561$	15.0217	0.634217
$562$	0	0
$563$	5.80189	0.244521	0.122260	−	0.992498i	$-0.460986\pi$
$563$	5.80189	0.244521	0.122260	+	0.992498i	$0.460986\pi$
$564$	0	0
$565$	−8.32336	−0.350166
$566$	0	0
$567$	−26.4439	−1.11054
$568$	0	0
$569$	16.7152	0.700738	0.350369	−	0.936612i	$-0.386056\pi$
$569$	16.7152	0.700738	0.350369	+	0.936612i	$0.386056\pi$
$570$	0	0
$571$	−3.51700	−0.147182	−0.0735908	−	0.997289i	$-0.523446\pi$
$571$	−3.51700	−0.147182	−0.0735908	+	0.997289i	$0.523446\pi$
$572$	0	0
$573$	−3.31754	−0.138592
$574$	0	0
$575$	9.41661	0.392700
$576$	0	0
$577$	−31.6755	−1.31867	−0.659335	−	0.751850i	$-0.729162\pi$
$577$	−31.6755	−1.31867	−0.659335	+	0.751850i	$0.729162\pi$
$578$	0	0
$579$	6.16095	0.256041
$580$	0	0
$581$	−23.8226	−0.988328
$582$	0	0
$583$	67.1164	2.77968
$584$	0	0
$585$	−2.15020	−0.0888999
$586$	0	0
$587$	−27.4349	−1.13236	−0.566181	−	0.824281i	$-0.691579\pi$
$587$	−27.4349	−1.13236	−0.566181	+	0.824281i	$0.691579\pi$
$588$	0	0
$589$	−1.06235	−0.0437732
$590$	0	0
$591$	−11.7952	−0.485191
$592$	0	0
$593$	−46.6882	−1.91725	−0.958627	−	0.284664i	$-0.908118\pi$
$593$	−46.6882	−1.91725	−0.958627	+	0.284664i	$0.908118\pi$
$594$	0	0
$595$	6.98272	0.286264
$596$	0	0
$597$	−37.1562	−1.52070
$598$	0	0
$599$	29.5787	1.20855	0.604277	−	0.796774i	$-0.293462\pi$
$599$	29.5787	1.20855	0.604277	+	0.796774i	$0.293462\pi$
$600$	0	0
$601$	−34.7745	−1.41848	−0.709240	−	0.704967i	$-0.750962\pi$
$601$	−34.7745	−1.41848	−0.709240	+	0.704967i	$0.750962\pi$
$602$	0	0
$603$	−1.71395	−0.0697975
$604$	0	0
$605$	22.0055	0.894652
$606$	0	0
$607$	−32.9130	−1.33590	−0.667949	−	0.744207i	$-0.732828\pi$
$607$	−32.9130	−1.33590	−0.667949	+	0.744207i	$0.732828\pi$
$608$	0	0
$609$	4.48987	0.181939
$610$	0	0
$611$	42.7257	1.72850
$612$	0	0
$613$	19.5859	0.791068	0.395534	−	0.918451i	$-0.370560\pi$
$613$	19.5859	0.791068	0.395534	+	0.918451i	$0.370560\pi$
$614$	0	0
$615$	16.5248	0.666342
$616$	0	0
$617$	24.3887	0.981854	0.490927	−	0.871201i	$-0.336658\pi$
$617$	24.3887	0.981854	0.490927	+	0.871201i	$0.336658\pi$
$618$	0	0
$619$	−44.8558	−1.80291	−0.901453	−	0.432877i	$-0.857499\pi$
$619$	−44.8558	−1.80291	−0.901453	+	0.432877i	$0.857499\pi$
$620$	0	0
$621$	−14.3440	−0.575606
$622$	0	0
$623$	−18.2881	−0.732696
$624$	0	0
$625$	5.90693	0.236277
$626$	0	0
$627$	−8.78370	−0.350787
$628$	0	0
$629$	−9.49622	−0.378639
$630$	0	0
$631$	39.0615	1.55501	0.777507	−	0.628874i	$-0.216484\pi$
$631$	39.0615	1.55501	0.777507	+	0.628874i	$0.216484\pi$
$632$	0	0
$633$	−21.6226	−0.859423
$634$	0	0
$635$	−17.4746	−0.693460
$636$	0	0
$637$	22.6184	0.896175
$638$	0	0
$639$	1.17876	0.0466310
$640$	0	0
$641$	−24.4718	−0.966579	−0.483290	−	0.875461i	$-0.660558\pi$
$641$	−24.4718	−0.966579	−0.483290	+	0.875461i	$0.660558\pi$
$642$	0	0
$643$	9.37313	0.369640	0.184820	−	0.982772i	$-0.440830\pi$
$643$	9.37313	0.369640	0.184820	+	0.982772i	$0.440830\pi$
$644$	0	0
$645$	−4.18861	−0.164927
$646$	0	0
$647$	9.81761	0.385970	0.192985	−	0.981202i	$-0.438183\pi$
$647$	9.81761	0.385970	0.192985	+	0.981202i	$0.438183\pi$
$648$	0	0
$649$	−30.7521	−1.20713
$650$	0	0
$651$	−5.91246	−0.231728
$652$	0	0
$653$	−13.0034	−0.508863	−0.254431	−	0.967091i	$-0.581888\pi$
$653$	−13.0034	−0.508863	−0.254431	+	0.967091i	$0.581888\pi$
$654$	0	0
$655$	5.21220	0.203658
$656$	0	0
$657$	1.75272	0.0683800
$658$	0	0
$659$	−35.7289	−1.39180	−0.695901	−	0.718138i	$-0.744995\pi$
$659$	−35.7289	−1.39180	−0.695901	+	0.718138i	$0.744995\pi$
$660$	0	0
$661$	−24.2807	−0.944410	−0.472205	−	0.881489i	$-0.656542\pi$
$661$	−24.2807	−0.944410	−0.472205	+	0.881489i	$0.656542\pi$
$662$	0	0
$663$	−12.8274	−0.498173
$664$	0	0
$665$	−4.08303	−0.158333
$666$	0	0
$667$	2.11275	0.0818062
$668$	0	0
$669$	11.6933	0.452091
$670$	0	0
$671$	31.7485	1.22564
$672$	0	0
$673$	−1.92837	−0.0743333	−0.0371666	−	0.999309i	$-0.511833\pi$
$673$	−1.92837	−0.0743333	−0.0371666	+	0.999309i	$0.511833\pi$
$674$	0	0
$675$	−19.6939	−0.758017
$676$	0	0
$677$	15.3731	0.590836	0.295418	−	0.955368i	$-0.404541\pi$
$677$	15.3731	0.590836	0.295418	+	0.955368i	$0.404541\pi$
$678$	0	0
$679$	−59.2872	−2.27524
$680$	0	0
$681$	−22.5641	−0.864657
$682$	0	0
$683$	47.0393	1.79991	0.899955	−	0.435984i	$-0.143599\pi$
$683$	47.0393	1.79991	0.899955	+	0.435984i	$0.143599\pi$
$684$	0	0
$685$	17.7559	0.678419
$686$	0	0
$687$	43.1724	1.64713
$688$	0	0
$689$	−57.3121	−2.18342
$690$	0	0
$691$	−20.4209	−0.776848	−0.388424	−	0.921481i	$-0.626980\pi$
$691$	−20.4209	−0.776848	−0.388424	+	0.921481i	$0.626980\pi$
$692$	0	0
$693$	7.32097	0.278101
$694$	0	0
$695$	−12.5101	−0.474534
$696$	0	0
$697$	−14.7632	−0.559198
$698$	0	0
$699$	40.1324	1.51795
$700$	0	0
$701$	24.2673	0.916563	0.458281	−	0.888807i	$-0.348465\pi$
$701$	24.2673	0.916563	0.458281	+	0.888807i	$0.348465\pi$
$702$	0	0
$703$	5.55276	0.209426
$704$	0	0
$705$	−17.6136	−0.663365
$706$	0	0
$707$	22.7617	0.856041
$708$	0	0
$709$	−6.40771	−0.240647	−0.120323	−	0.992735i	$-0.538393\pi$
$709$	−6.40771	−0.240647	−0.120323	+	0.992735i	$0.538393\pi$
$710$	0	0
$711$	−0.390754	−0.0146544
$712$	0	0
$713$	−2.78217	−0.104193
$714$	0	0
$715$	−29.9223	−1.11903
$716$	0	0
$717$	40.7426	1.52156
$718$	0	0
$719$	34.9142	1.30208	0.651040	−	0.759044i	$-0.274333\pi$
$719$	34.9142	1.30208	0.651040	+	0.759044i	$0.274333\pi$
$720$	0	0
$721$	−55.7835	−2.07749
$722$	0	0
$723$	14.2320	0.529294
$724$	0	0
$725$	2.90074	0.107731
$726$	0	0
$727$	29.3193	1.08739	0.543697	−	0.839282i	$-0.317024\pi$
$727$	29.3193	1.08739	0.543697	+	0.839282i	$0.317024\pi$
$728$	0	0
$729$	29.5410	1.09411
$730$	0	0
$731$	3.74211	0.138407
$732$	0	0
$733$	37.6082	1.38909	0.694546	−	0.719449i	$-0.255605\pi$
$733$	37.6082	1.38909	0.694546	+	0.719449i	$0.255605\pi$
$734$	0	0
$735$	−9.32439	−0.343935
$736$	0	0
$737$	−23.8514	−0.878579
$738$	0	0
$739$	32.9972	1.21382	0.606912	−	0.794769i	$-0.292408\pi$
$739$	32.9972	1.21382	0.606912	+	0.794769i	$0.292408\pi$
$740$	0	0
$741$	7.50058	0.275541
$742$	0	0
$743$	41.6773	1.52899	0.764496	−	0.644628i	$-0.222988\pi$
$743$	41.6773	1.52899	0.764496	+	0.644628i	$0.222988\pi$
$744$	0	0
$745$	−22.2336	−0.814576
$746$	0	0
$747$	2.70177	0.0988525
$748$	0	0
$749$	43.6056	1.59331
$750$	0	0
$751$	0.711656	0.0259687	0.0129843	−	0.999916i	$-0.495867\pi$
$751$	0.711656	0.0259687	0.0129843	+	0.999916i	$0.495867\pi$
$752$	0	0
$753$	42.7103	1.55645
$754$	0	0
$755$	24.2560	0.882768
$756$	0	0
$757$	21.3888	0.777390	0.388695	−	0.921367i	$-0.372926\pi$
$757$	21.3888	0.777390	0.388695	+	0.921367i	$0.372926\pi$
$758$	0	0
$759$	−23.0035	−0.834976
$760$	0	0
$761$	22.2328	0.805938	0.402969	−	0.915214i	$-0.367978\pi$
$761$	22.2328	0.805938	0.402969	+	0.915214i	$0.367978\pi$
$762$	0	0
$763$	48.0418	1.73923
$764$	0	0
$765$	−0.791924	−0.0286321
$766$	0	0
$767$	26.2599	0.948190
$768$	0	0
$769$	7.49512	0.270281	0.135140	−	0.990826i	$-0.456851\pi$
$769$	7.49512	0.270281	0.135140	+	0.990826i	$0.456851\pi$
$770$	0	0
$771$	33.0544	1.19042
$772$	0	0
$773$	24.8860	0.895087	0.447543	−	0.894262i	$-0.352299\pi$
$773$	24.8860	0.895087	0.447543	+	0.894262i	$0.352299\pi$
$774$	0	0
$775$	−3.81982	−0.137212
$776$	0	0
$777$	30.9038	1.10867
$778$	0	0
$779$	8.63256	0.309294
$780$	0	0
$781$	16.4037	0.586970
$782$	0	0
$783$	−4.41861	−0.157908
$784$	0	0
$785$	22.8884	0.816921
$786$	0	0
$787$	38.5288	1.37340	0.686702	−	0.726939i	$-0.259058\pi$
$787$	38.5288	1.37340	0.686702	+	0.726939i	$0.259058\pi$
$788$	0	0
$789$	32.0783	1.14202
$790$	0	0
$791$	−24.1995	−0.860434
$792$	0	0
$793$	−27.1107	−0.962728
$794$	0	0
$795$	23.6268	0.837955
$796$	0	0
$797$	18.5646	0.657593	0.328796	−	0.944401i	$-0.393357\pi$
$797$	18.5646	0.657593	0.328796	+	0.944401i	$0.393357\pi$
$798$	0	0
$799$	15.7360	0.556699
$800$	0	0
$801$	2.07409	0.0732842
$802$	0	0
$803$	24.3909	0.860737
$804$	0	0
$805$	−10.6930	−0.376879
$806$	0	0
$807$	−51.6353	−1.81765
$808$	0	0
$809$	54.8809	1.92951	0.964755	−	0.263151i	$-0.0847619\pi$
$809$	54.8809	1.92951	0.964755	+	0.263151i	$0.0847619\pi$
$810$	0	0
$811$	31.8437	1.11818	0.559092	−	0.829106i	$-0.311150\pi$
$811$	31.8437	1.11818	0.559092	+	0.829106i	$0.311150\pi$
$812$	0	0
$813$	−7.76707	−0.272403
$814$	0	0
$815$	9.48775	0.332342
$816$	0	0
$817$	−2.18814	−0.0765533
$818$	0	0
$819$	−6.25153	−0.218446
$820$	0	0
$821$	−0.357647	−0.0124820	−0.00624099	−	0.999981i	$-0.501987\pi$
$821$	−0.357647	−0.0124820	−0.00624099	+	0.999981i	$0.501987\pi$
$822$	0	0
$823$	−7.87566	−0.274528	−0.137264	−	0.990534i	$-0.543831\pi$
$823$	−7.87566	−0.274528	−0.137264	+	0.990534i	$0.543831\pi$
$824$	0	0
$825$	−31.5831	−1.09958
$826$	0	0
$827$	−2.44374	−0.0849771	−0.0424885	−	0.999097i	$-0.513529\pi$
$827$	−2.44374	−0.0849771	−0.0424885	+	0.999097i	$0.513529\pi$
$828$	0	0
$829$	−5.57494	−0.193626	−0.0968128	−	0.995303i	$-0.530865\pi$
$829$	−5.57494	−0.193626	−0.0968128	+	0.995303i	$0.530865\pi$
$830$	0	0
$831$	−2.06927	−0.0717821
$832$	0	0
$833$	8.33042	0.288632
$834$	0	0
$835$	17.1576	0.593763
$836$	0	0
$837$	5.81862	0.201121
$838$	0	0
$839$	−20.8433	−0.719590	−0.359795	−	0.933031i	$-0.617153\pi$
$839$	−20.8433	−0.719590	−0.359795	+	0.933031i	$0.617153\pi$
$840$	0	0
$841$	−28.3492	−0.977558
$842$	0	0
$843$	41.0658	1.41438
$844$	0	0
$845$	10.1456	0.349019
$846$	0	0
$847$	63.9792	2.19835
$848$	0	0
$849$	24.5845	0.843737
$850$	0	0
$851$	14.5421	0.498496
$852$	0	0
$853$	39.4904	1.35212	0.676062	−	0.736844i	$-0.263685\pi$
$853$	39.4904	1.35212	0.676062	+	0.736844i	$0.263685\pi$
$854$	0	0
$855$	0.463065	0.0158365
$856$	0	0
$857$	−23.5217	−0.803484	−0.401742	−	0.915753i	$-0.631595\pi$
$857$	−23.5217	−0.803484	−0.401742	+	0.915753i	$0.631595\pi$
$858$	0	0
$859$	−21.2488	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$859$	−21.2488	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$860$	0	0
$861$	48.0443	1.63735
$862$	0	0
$863$	25.4284	0.865591	0.432796	−	0.901492i	$-0.357527\pi$
$863$	25.4284	0.865591	0.432796	+	0.901492i	$0.357527\pi$
$864$	0	0
$865$	−26.1535	−0.889244
$866$	0	0
$867$	22.7360	0.772156
$868$	0	0
$869$	−5.43776	−0.184463
$870$	0	0
$871$	20.3672	0.690118
$872$	0	0
$873$	6.72388	0.227569
$874$	0	0
$875$	−35.0963	−1.18647
$876$	0	0
$877$	34.1523	1.15324	0.576621	−	0.817012i	$-0.304371\pi$
$877$	34.1523	1.15324	0.576621	+	0.817012i	$0.304371\pi$
$878$	0	0
$879$	−4.92021	−0.165954
$880$	0	0
$881$	−46.9438	−1.58158	−0.790788	−	0.612090i	$-0.790329\pi$
$881$	−46.9438	−1.58158	−0.790788	+	0.612090i	$0.790329\pi$
$882$	0	0
$883$	−3.84520	−0.129401	−0.0647007	−	0.997905i	$-0.520609\pi$
$883$	−3.84520	−0.129401	−0.0647007	+	0.997905i	$0.520609\pi$
$884$	0	0
$885$	−10.8256	−0.363898
$886$	0	0
$887$	−34.4085	−1.15532	−0.577661	−	0.816276i	$-0.696035\pi$
$887$	−34.4085	−1.15532	−0.577661	+	0.816276i	$0.696035\pi$
$888$	0	0
$889$	−50.8060	−1.70398
$890$	0	0
$891$	41.7351	1.39818
$892$	0	0
$893$	−9.20135	−0.307912
$894$	0	0
$895$	−24.5430	−0.820382
$896$	0	0
$897$	19.6432	0.655867
$898$	0	0
$899$	−0.857033	−0.0285836
$900$	0	0
$901$	−21.1082	−0.703215
$902$	0	0
$903$	−12.1780	−0.405260
$904$	0	0
$905$	16.9798	0.564429
$906$	0	0
$907$	51.2956	1.70324	0.851621	−	0.524158i	$-0.175620\pi$
$907$	51.2956	1.70324	0.851621	+	0.524158i	$0.175620\pi$
$908$	0	0
$909$	−2.58145	−0.0856212
$910$	0	0
$911$	−4.03426	−0.133661	−0.0668305	−	0.997764i	$-0.521289\pi$
$911$	−4.03426	−0.133661	−0.0668305	+	0.997764i	$0.521289\pi$
$912$	0	0
$913$	37.5980	1.24431
$914$	0	0
$915$	11.1763	0.369477
$916$	0	0
$917$	15.1540	0.500431
$918$	0	0
$919$	−16.1394	−0.532389	−0.266195	−	0.963919i	$-0.585766\pi$
$919$	−16.1394	−0.532389	−0.266195	+	0.963919i	$0.585766\pi$
$920$	0	0
$921$	4.10210	0.135169
$922$	0	0
$923$	−14.0075	−0.461061
$924$	0	0
$925$	19.9658	0.656471
$926$	0	0
$927$	6.32652	0.207790
$928$	0	0
$929$	−28.5862	−0.937884	−0.468942	−	0.883229i	$-0.655365\pi$
$929$	−28.5862	−0.937884	−0.468942	+	0.883229i	$0.655365\pi$
$930$	0	0
$931$	−4.87108	−0.159643
$932$	0	0
$933$	−17.1840	−0.562580
$934$	0	0
$935$	−11.0205	−0.360408
$936$	0	0
$937$	−38.7196	−1.26491	−0.632457	−	0.774595i	$-0.717954\pi$
$937$	−38.7196	−1.26491	−0.632457	+	0.774595i	$0.717954\pi$
$938$	0	0
$939$	25.5521	0.833860
$940$	0	0
$941$	−57.3448	−1.86939	−0.934694	−	0.355453i	$-0.884326\pi$
$941$	−57.3448	−1.86939	−0.934694	+	0.355453i	$0.884326\pi$
$942$	0	0
$943$	22.6077	0.736209
$944$	0	0
$945$	22.3633	0.727480
$946$	0	0
$947$	20.7156	0.673165	0.336582	−	0.941654i	$-0.390729\pi$
$947$	20.7156	0.673165	0.336582	+	0.941654i	$0.390729\pi$
$948$	0	0
$949$	−20.8279	−0.676102
$950$	0	0
$951$	0.104714	0.00339557
$952$	0	0
$953$	20.8695	0.676029	0.338014	−	0.941141i	$-0.390245\pi$
$953$	20.8695	0.676029	0.338014	+	0.941141i	$0.390245\pi$
$954$	0	0
$955$	2.43387	0.0787581
$956$	0	0
$957$	−7.08613	−0.229062
$958$	0	0
$959$	51.6238	1.66702
$960$	0	0
$961$	−29.8714	−0.963594
$962$	0	0
$963$	−4.94539	−0.159363
$964$	0	0
$965$	−4.51990	−0.145501
$966$	0	0
$967$	6.36701	0.204749	0.102375	−	0.994746i	$-0.467356\pi$
$967$	6.36701	0.204749	0.102375	+	0.994746i	$0.467356\pi$
$968$	0	0
$969$	2.76248	0.0887437
$970$	0	0
$971$	53.5899	1.71978	0.859890	−	0.510479i	$-0.170532\pi$
$971$	53.5899	1.71978	0.859890	+	0.510479i	$0.170532\pi$
$972$	0	0
$973$	−36.3720	−1.16603
$974$	0	0
$975$	26.9695	0.863714
$976$	0	0
$977$	5.45811	0.174620	0.0873102	−	0.996181i	$-0.472173\pi$
$977$	5.45811	0.174620	0.0873102	+	0.996181i	$0.472173\pi$
$978$	0	0
$979$	28.8631	0.922468
$980$	0	0
$981$	−5.44851	−0.173958
$982$	0	0
$983$	4.20849	0.134230	0.0671150	−	0.997745i	$-0.478621\pi$
$983$	4.20849	0.134230	0.0671150	+	0.997745i	$0.478621\pi$
$984$	0	0
$985$	8.65340	0.275720
$986$	0	0
$987$	−51.2099	−1.63003
$988$	0	0
$989$	−5.73050	−0.182219
$990$	0	0
$991$	57.6760	1.83214	0.916069	−	0.401021i	$-0.131344\pi$
$991$	57.6760	1.83214	0.916069	+	0.401021i	$0.131344\pi$
$992$	0	0
$993$	−51.6815	−1.64006
$994$	0	0
$995$	27.2591	0.864171
$996$	0	0
$997$	−39.8984	−1.26359	−0.631797	−	0.775134i	$-0.717682\pi$
$997$	−39.8984	−1.26359	−0.631797	+	0.775134i	$0.717682\pi$
$998$	0	0
$999$	−30.4132	−0.962233

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.f.1.5	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.f.1.5	✓	25	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.61532 q^{3} +1.18505 q^{5} +3.44544 q^{7} -0.390754 q^{9} +O(q^{10})\)	\(q-1.61532 q^{3} +1.18505 q^{5} +3.44544 q^{7} -0.390754 q^{9} -5.43776 q^{11} +4.64341 q^{13} -1.91424 q^{15} +1.71018 q^{17} -1.00000 q^{19} -5.56548 q^{21} -2.61889 q^{23} -3.59565 q^{25} +5.47714 q^{27} -0.806736 q^{29} +1.06235 q^{31} +8.78370 q^{33} +4.08303 q^{35} -5.55276 q^{37} -7.50058 q^{39} -8.63256 q^{41} +2.18814 q^{43} -0.463065 q^{45} +9.20135 q^{47} +4.87108 q^{49} -2.76248 q^{51} -12.3427 q^{53} -6.44403 q^{55} +1.61532 q^{57} +5.65530 q^{59} -5.83852 q^{61} -1.34632 q^{63} +5.50269 q^{65} +4.38626 q^{67} +4.23034 q^{69} -3.01663 q^{71} -4.48547 q^{73} +5.80811 q^{75} -18.7355 q^{77} +1.00000 q^{79} -7.67505 q^{81} -6.91424 q^{83} +2.02666 q^{85} +1.30313 q^{87} -5.30790 q^{89} +15.9986 q^{91} -1.71602 q^{93} -1.18505 q^{95} -17.2074 q^{97} +2.12483 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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