LMFDB - Embedded newform 6004.2.a.f.1.16

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			1.16
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)

$f(q)$	$=$	$q+1.01059 q^{3} +3.45227 q^{5} -1.97653 q^{7} -1.97870 q^{9} +O(q^{10})$	$q+1.01059 q^{3} +3.45227 q^{5} -1.97653 q^{7} -1.97870 q^{9} -1.83783 q^{11} +0.528598 q^{13} +3.48884 q^{15} -4.05909 q^{17} -1.00000 q^{19} -1.99746 q^{21} -0.152313 q^{23} +6.91814 q^{25} -5.03144 q^{27} -5.34295 q^{29} +5.65026 q^{31} -1.85729 q^{33} -6.82350 q^{35} +3.09995 q^{37} +0.534198 q^{39} -8.94010 q^{41} +4.33951 q^{43} -6.83100 q^{45} -7.23851 q^{47} -3.09334 q^{49} -4.10209 q^{51} -8.53322 q^{53} -6.34466 q^{55} -1.01059 q^{57} -2.00076 q^{59} +12.4779 q^{61} +3.91096 q^{63} +1.82486 q^{65} +5.00387 q^{67} -0.153926 q^{69} +3.70905 q^{71} +2.60334 q^{73} +6.99142 q^{75} +3.63251 q^{77} +1.00000 q^{79} +0.851365 q^{81} -13.2107 q^{83} -14.0131 q^{85} -5.39955 q^{87} -11.6619 q^{89} -1.04479 q^{91} +5.71011 q^{93} -3.45227 q^{95} -6.39030 q^{97} +3.63651 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.01059	0.583466	0.291733	−	0.956500i	$-0.405768\pi$
$3$	1.01059	0.583466	0.291733	+	0.956500i	$0.405768\pi$
$4$	0	0
$5$	3.45227	1.54390	0.771950	−	0.635683i	$-0.219282\pi$
$5$	3.45227	1.54390	0.771950	+	0.635683i	$0.219282\pi$
$6$	0	0
$7$	−1.97653	−0.747057	−0.373528	−	0.927619i	$-0.621852\pi$
$7$	−1.97653	−0.747057	−0.373528	+	0.927619i	$0.621852\pi$
$8$	0	0
$9$	−1.97870	−0.659567
$10$	0	0
$11$	−1.83783	−0.554125	−0.277063	−	0.960852i	$-0.589361\pi$
$11$	−1.83783	−0.554125	−0.277063	+	0.960852i	$0.589361\pi$
$12$	0	0
$13$	0.528598	0.146607	0.0733034	−	0.997310i	$-0.476646\pi$
$13$	0.528598	0.146607	0.0733034	+	0.997310i	$0.476646\pi$
$14$	0	0
$15$	3.48884	0.900814
$16$	0	0
$17$	−4.05909	−0.984474	−0.492237	−	0.870461i	$-0.663821\pi$
$17$	−4.05909	−0.984474	−0.492237	+	0.870461i	$0.663821\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.99746	−0.435882
$22$	0	0
$23$	−0.152313	−0.0317594	−0.0158797	−	0.999874i	$-0.505055\pi$
$23$	−0.152313	−0.0317594	−0.0158797	+	0.999874i	$0.505055\pi$
$24$	0	0
$25$	6.91814	1.38363
$26$	0	0
$27$	−5.03144	−0.968301
$28$	0	0
$29$	−5.34295	−0.992162	−0.496081	−	0.868276i	$-0.665228\pi$
$29$	−5.34295	−0.992162	−0.496081	+	0.868276i	$0.665228\pi$
$30$	0	0
$31$	5.65026	1.01482	0.507408	−	0.861706i	$-0.330604\pi$
$31$	5.65026	1.01482	0.507408	+	0.861706i	$0.330604\pi$
$32$	0	0
$33$	−1.85729	−0.323313
$34$	0	0
$35$	−6.82350	−1.15338
$36$	0	0
$37$	3.09995	0.509629	0.254814	−	0.966990i	$-0.417986\pi$
$37$	3.09995	0.509629	0.254814	+	0.966990i	$0.417986\pi$
$38$	0	0
$39$	0.534198	0.0855401
$40$	0	0
$41$	−8.94010	−1.39621	−0.698105	−	0.715996i	$-0.745973\pi$
$41$	−8.94010	−1.39621	−0.698105	+	0.715996i	$0.745973\pi$
$42$	0	0
$43$	4.33951	0.661769	0.330885	−	0.943671i	$-0.392653\pi$
$43$	4.33951	0.661769	0.330885	+	0.943671i	$0.392653\pi$
$44$	0	0
$45$	−6.83100	−1.01831
$46$	0	0
$47$	−7.23851	−1.05584	−0.527922	−	0.849293i	$-0.677029\pi$
$47$	−7.23851	−1.05584	−0.527922	+	0.849293i	$0.677029\pi$
$48$	0	0
$49$	−3.09334	−0.441906
$50$	0	0
$51$	−4.10209	−0.574407
$52$	0	0
$53$	−8.53322	−1.17213	−0.586064	−	0.810265i	$-0.699323\pi$
$53$	−8.53322	−1.17213	−0.586064	+	0.810265i	$0.699323\pi$
$54$	0	0
$55$	−6.34466	−0.855514
$56$	0	0
$57$	−1.01059	−0.133856
$58$	0	0
$59$	−2.00076	−0.260476	−0.130238	−	0.991483i	$-0.541574\pi$
$59$	−2.00076	−0.260476	−0.130238	+	0.991483i	$0.541574\pi$
$60$	0	0
$61$	12.4779	1.59763	0.798816	−	0.601576i	$-0.205460\pi$
$61$	12.4779	1.59763	0.798816	+	0.601576i	$0.205460\pi$
$62$	0	0
$63$	3.91096	0.492734
$64$	0	0
$65$	1.82486	0.226346
$66$	0	0
$67$	5.00387	0.611320	0.305660	−	0.952141i	$-0.401123\pi$
$67$	5.00387	0.611320	0.305660	+	0.952141i	$0.401123\pi$
$68$	0	0
$69$	−0.153926	−0.0185305
$70$	0	0
$71$	3.70905	0.440183	0.220092	−	0.975479i	$-0.429364\pi$
$71$	3.70905	0.440183	0.220092	+	0.975479i	$0.429364\pi$
$72$	0	0
$73$	2.60334	0.304698	0.152349	−	0.988327i	$-0.451316\pi$
$73$	2.60334	0.304698	0.152349	+	0.988327i	$0.451316\pi$
$74$	0	0
$75$	6.99142	0.807300
$76$	0	0
$77$	3.63251	0.413963
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	0.851365	0.0945961
$82$	0	0
$83$	−13.2107	−1.45007	−0.725033	−	0.688714i	$-0.758176\pi$
$83$	−13.2107	−1.45007	−0.725033	+	0.688714i	$0.758176\pi$
$84$	0	0
$85$	−14.0131	−1.51993
$86$	0	0
$87$	−5.39955	−0.578893
$88$	0	0
$89$	−11.6619	−1.23616	−0.618079	−	0.786116i	$-0.712089\pi$
$89$	−11.6619	−1.23616	−0.618079	+	0.786116i	$0.712089\pi$
$90$	0	0
$91$	−1.04479	−0.109524
$92$	0	0
$93$	5.71011	0.592111
$94$	0	0
$95$	−3.45227	−0.354195
$96$	0	0
$97$	−6.39030	−0.648837	−0.324418	−	0.945914i	$-0.605169\pi$
$97$	−6.39030	−0.648837	−0.324418	+	0.945914i	$0.605169\pi$
$98$	0	0
$99$	3.63651	0.365483
$100$	0	0
$101$	−8.42643	−0.838461	−0.419231	−	0.907880i	$-0.637700\pi$
$101$	−8.42643	−0.838461	−0.419231	+	0.907880i	$0.637700\pi$
$102$	0	0
$103$	3.73397	0.367919	0.183959	−	0.982934i	$-0.441108\pi$
$103$	3.73397	0.367919	0.183959	+	0.982934i	$0.441108\pi$
$104$	0	0
$105$	−6.89578	−0.672959
$106$	0	0
$107$	1.67418	0.161850	0.0809248	−	0.996720i	$-0.474213\pi$
$107$	1.67418	0.161850	0.0809248	+	0.996720i	$0.474213\pi$
$108$	0	0
$109$	−7.09957	−0.680015	−0.340008	−	0.940423i	$-0.610430\pi$
$109$	−7.09957	−0.680015	−0.340008	+	0.940423i	$0.610430\pi$
$110$	0	0
$111$	3.13279	0.297351
$112$	0	0
$113$	−10.0157	−0.942202	−0.471101	−	0.882079i	$-0.656143\pi$
$113$	−10.0157	−0.942202	−0.471101	+	0.882079i	$0.656143\pi$
$114$	0	0
$115$	−0.525824	−0.0490333
$116$	0	0
$117$	−1.04594	−0.0966970
$118$	0	0
$119$	8.02290	0.735458
$120$	0	0
$121$	−7.62240	−0.692945
$122$	0	0
$123$	−9.03480	−0.814641
$124$	0	0
$125$	6.62192	0.592283
$126$	0	0
$127$	−0.873984	−0.0775535	−0.0387768	−	0.999248i	$-0.512346\pi$
$127$	−0.873984	−0.0775535	−0.0387768	+	0.999248i	$0.512346\pi$
$128$	0	0
$129$	4.38548	0.386120
$130$	0	0
$131$	−15.7138	−1.37292	−0.686461	−	0.727167i	$-0.740837\pi$
$131$	−15.7138	−1.37292	−0.686461	+	0.727167i	$0.740837\pi$
$132$	0	0
$133$	1.97653	0.171387
$134$	0	0
$135$	−17.3699	−1.49496
$136$	0	0
$137$	−19.4871	−1.66489	−0.832446	−	0.554106i	$-0.813060\pi$
$137$	−19.4871	−1.66489	−0.832446	+	0.554106i	$0.813060\pi$
$138$	0	0
$139$	−9.82769	−0.833574	−0.416787	−	0.909004i	$-0.636844\pi$
$139$	−9.82769	−0.833574	−0.416787	+	0.909004i	$0.636844\pi$
$140$	0	0
$141$	−7.31518	−0.616050
$142$	0	0
$143$	−0.971471	−0.0812385
$144$	0	0
$145$	−18.4453	−1.53180
$146$	0	0
$147$	−3.12611	−0.257837
$148$	0	0
$149$	11.4641	0.939180	0.469590	−	0.882885i	$-0.344402\pi$
$149$	11.4641	0.939180	0.469590	+	0.882885i	$0.344402\pi$
$150$	0	0
$151$	−8.73681	−0.710992	−0.355496	−	0.934678i	$-0.615688\pi$
$151$	−8.73681	−0.710992	−0.355496	+	0.934678i	$0.615688\pi$
$152$	0	0
$153$	8.03173	0.649327
$154$	0	0
$155$	19.5062	1.56678
$156$	0	0
$157$	15.1935	1.21257	0.606286	−	0.795246i	$-0.292659\pi$
$157$	15.1935	1.21257	0.606286	+	0.795246i	$0.292659\pi$
$158$	0	0
$159$	−8.62361	−0.683897
$160$	0	0
$161$	0.301050	0.0237261
$162$	0	0
$163$	1.58839	0.124412	0.0622060	−	0.998063i	$-0.480186\pi$
$163$	1.58839	0.124412	0.0622060	+	0.998063i	$0.480186\pi$
$164$	0	0
$165$	−6.41187	−0.499163
$166$	0	0
$167$	3.30167	0.255491	0.127745	−	0.991807i	$-0.459226\pi$
$167$	3.30167	0.255491	0.127745	+	0.991807i	$0.459226\pi$
$168$	0	0
$169$	−12.7206	−0.978506
$170$	0	0
$171$	1.97870	0.151315
$172$	0	0
$173$	−13.9575	−1.06117	−0.530584	−	0.847633i	$-0.678027\pi$
$173$	−13.9575	−1.06117	−0.530584	+	0.847633i	$0.678027\pi$
$174$	0	0
$175$	−13.6739	−1.03365
$176$	0	0
$177$	−2.02195	−0.151979
$178$	0	0
$179$	6.14532	0.459322	0.229661	−	0.973271i	$-0.426238\pi$
$179$	6.14532	0.459322	0.229661	+	0.973271i	$0.426238\pi$
$180$	0	0
$181$	13.9420	1.03630	0.518150	−	0.855290i	$-0.326621\pi$
$181$	13.9420	1.03630	0.518150	+	0.855290i	$0.326621\pi$
$182$	0	0
$183$	12.6101	0.932164
$184$	0	0
$185$	10.7019	0.786816
$186$	0	0
$187$	7.45990	0.545522
$188$	0	0
$189$	9.94478	0.723376
$190$	0	0
$191$	−2.93339	−0.212253	−0.106126	−	0.994353i	$-0.533845\pi$
$191$	−2.93339	−0.212253	−0.106126	+	0.994353i	$0.533845\pi$
$192$	0	0
$193$	8.51936	0.613237	0.306618	−	0.951833i	$-0.400802\pi$
$193$	8.51936	0.613237	0.306618	+	0.951833i	$0.400802\pi$
$194$	0	0
$195$	1.84419	0.132065
$196$	0	0
$197$	−13.0545	−0.930096	−0.465048	−	0.885285i	$-0.653963\pi$
$197$	−13.0545	−0.930096	−0.465048	+	0.885285i	$0.653963\pi$
$198$	0	0
$199$	22.7350	1.61164	0.805819	−	0.592161i	$-0.201725\pi$
$199$	22.7350	1.61164	0.805819	+	0.592161i	$0.201725\pi$
$200$	0	0
$201$	5.05688	0.356685
$202$	0	0
$203$	10.5605	0.741201
$204$	0	0
$205$	−30.8636	−2.15561
$206$	0	0
$207$	0.301381	0.0209474
$208$	0	0
$209$	1.83783	0.127125
$210$	0	0
$211$	8.26886	0.569252	0.284626	−	0.958639i	$-0.408131\pi$
$211$	8.26886	0.569252	0.284626	+	0.958639i	$0.408131\pi$
$212$	0	0
$213$	3.74834	0.256832
$214$	0	0
$215$	14.9811	1.02171
$216$	0	0
$217$	−11.1679	−0.758126
$218$	0	0
$219$	2.63092	0.177781
$220$	0	0
$221$	−2.14563	−0.144331
$222$	0	0
$223$	−1.78366	−0.119443	−0.0597213	−	0.998215i	$-0.519021\pi$
$223$	−1.78366	−0.119443	−0.0597213	+	0.998215i	$0.519021\pi$
$224$	0	0
$225$	−13.6889	−0.912595
$226$	0	0
$227$	16.5492	1.09841	0.549204	−	0.835688i	$-0.314931\pi$
$227$	16.5492	1.09841	0.549204	+	0.835688i	$0.314931\pi$
$228$	0	0
$229$	0.465288	0.0307471	0.0153735	−	0.999882i	$-0.495106\pi$
$229$	0.465288	0.0307471	0.0153735	+	0.999882i	$0.495106\pi$
$230$	0	0
$231$	3.67099	0.241533
$232$	0	0
$233$	19.0082	1.24527	0.622633	−	0.782514i	$-0.286063\pi$
$233$	19.0082	1.24527	0.622633	+	0.782514i	$0.286063\pi$
$234$	0	0
$235$	−24.9892	−1.63012
$236$	0	0
$237$	1.01059	0.0656451
$238$	0	0
$239$	−27.0861	−1.75206	−0.876029	−	0.482259i	$-0.839816\pi$
$239$	−27.0861	−1.75206	−0.876029	+	0.482259i	$0.839816\pi$
$240$	0	0
$241$	4.06140	0.261618	0.130809	−	0.991408i	$-0.458243\pi$
$241$	4.06140	0.261618	0.130809	+	0.991408i	$0.458243\pi$
$242$	0	0
$243$	15.9547	1.02349
$244$	0	0
$245$	−10.6790	−0.682259
$246$	0	0
$247$	−0.528598	−0.0336339
$248$	0	0
$249$	−13.3507	−0.846065
$250$	0	0
$251$	14.0040	0.883927	0.441963	−	0.897033i	$-0.354282\pi$
$251$	14.0040	0.883927	0.441963	+	0.897033i	$0.354282\pi$
$252$	0	0
$253$	0.279924	0.0175987
$254$	0	0
$255$	−14.1615	−0.886827
$256$	0	0
$257$	−16.9979	−1.06030	−0.530151	−	0.847903i	$-0.677865\pi$
$257$	−16.9979	−1.06030	−0.530151	+	0.847903i	$0.677865\pi$
$258$	0	0
$259$	−6.12714	−0.380722
$260$	0	0
$261$	10.5721	0.654397
$262$	0	0
$263$	28.6740	1.76812	0.884059	−	0.467376i	$-0.154800\pi$
$263$	28.6740	1.76812	0.884059	+	0.467376i	$0.154800\pi$
$264$	0	0
$265$	−29.4589	−1.80965
$266$	0	0
$267$	−11.7854	−0.721257
$268$	0	0
$269$	−16.8668	−1.02839	−0.514194	−	0.857674i	$-0.671909\pi$
$269$	−16.8668	−1.02839	−0.514194	+	0.857674i	$0.671909\pi$
$270$	0	0
$271$	7.17803	0.436034	0.218017	−	0.975945i	$-0.430041\pi$
$271$	7.17803	0.436034	0.218017	+	0.975945i	$0.430041\pi$
$272$	0	0
$273$	−1.05586	−0.0639033
$274$	0	0
$275$	−12.7143	−0.766703
$276$	0	0
$277$	−24.2496	−1.45702	−0.728510	−	0.685035i	$-0.759787\pi$
$277$	−24.2496	−1.45702	−0.728510	+	0.685035i	$0.759787\pi$
$278$	0	0
$279$	−11.1802	−0.669340
$280$	0	0
$281$	−0.481544	−0.0287265	−0.0143633	−	0.999897i	$-0.504572\pi$
$281$	−0.481544	−0.0287265	−0.0143633	+	0.999897i	$0.504572\pi$
$282$	0	0
$283$	−1.47511	−0.0876864	−0.0438432	−	0.999038i	$-0.513960\pi$
$283$	−1.47511	−0.0876864	−0.0438432	+	0.999038i	$0.513960\pi$
$284$	0	0
$285$	−3.48884	−0.206661
$286$	0	0
$287$	17.6703	1.04305
$288$	0	0
$289$	−0.523794	−0.0308114
$290$	0	0
$291$	−6.45799	−0.378574
$292$	0	0
$293$	4.24765	0.248151	0.124075	−	0.992273i	$-0.460404\pi$
$293$	4.24765	0.248151	0.124075	+	0.992273i	$0.460404\pi$
$294$	0	0
$295$	−6.90715	−0.402150
$296$	0	0
$297$	9.24691	0.536560
$298$	0	0
$299$	−0.0805122	−0.00465614
$300$	0	0
$301$	−8.57716	−0.494379
$302$	0	0
$303$	−8.51569	−0.489214
$304$	0	0
$305$	43.0770	2.46658
$306$	0	0
$307$	−21.2523	−1.21293	−0.606467	−	0.795109i	$-0.707414\pi$
$307$	−21.2523	−1.21293	−0.606467	+	0.795109i	$0.707414\pi$
$308$	0	0
$309$	3.77352	0.214668
$310$	0	0
$311$	28.6984	1.62734	0.813669	−	0.581329i	$-0.197467\pi$
$311$	28.6984	1.62734	0.813669	+	0.581329i	$0.197467\pi$
$312$	0	0
$313$	17.0251	0.962315	0.481158	−	0.876634i	$-0.340216\pi$
$313$	17.0251	0.962315	0.481158	+	0.876634i	$0.340216\pi$
$314$	0	0
$315$	13.5017	0.760732
$316$	0	0
$317$	31.8557	1.78919	0.894597	−	0.446874i	$-0.147463\pi$
$317$	31.8557	1.78919	0.894597	+	0.446874i	$0.147463\pi$
$318$	0	0
$319$	9.81942	0.549782
$320$	0	0
$321$	1.69192	0.0944337
$322$	0	0
$323$	4.05909	0.225854
$324$	0	0
$325$	3.65692	0.202849
$326$	0	0
$327$	−7.17477	−0.396766
$328$	0	0
$329$	14.3071	0.788776
$330$	0	0
$331$	7.91627	0.435117	0.217559	−	0.976047i	$-0.430191\pi$
$331$	7.91627	0.435117	0.217559	+	0.976047i	$0.430191\pi$
$332$	0	0
$333$	−6.13388	−0.336134
$334$	0	0
$335$	17.2747	0.943817
$336$	0	0
$337$	−16.0956	−0.876783	−0.438391	−	0.898784i	$-0.644452\pi$
$337$	−16.0956	−0.876783	−0.438391	+	0.898784i	$0.644452\pi$
$338$	0	0
$339$	−10.1218	−0.549743
$340$	0	0
$341$	−10.3842	−0.562335
$342$	0	0
$343$	19.9498	1.07719
$344$	0	0
$345$	−0.531394	−0.0286093
$346$	0	0
$347$	−17.4893	−0.938873	−0.469437	−	0.882966i	$-0.655543\pi$
$347$	−17.4893	−0.938873	−0.469437	+	0.882966i	$0.655543\pi$
$348$	0	0
$349$	28.5080	1.52600	0.762999	−	0.646400i	$-0.223726\pi$
$349$	28.5080	1.52600	0.762999	+	0.646400i	$0.223726\pi$
$350$	0	0
$351$	−2.65961	−0.141960
$352$	0	0
$353$	−35.7508	−1.90282	−0.951412	−	0.307922i	$-0.900366\pi$
$353$	−35.7508	−1.90282	−0.951412	+	0.307922i	$0.900366\pi$
$354$	0	0
$355$	12.8046	0.679599
$356$	0	0
$357$	8.10789	0.429115
$358$	0	0
$359$	−5.03101	−0.265526	−0.132763	−	0.991148i	$-0.542385\pi$
$359$	−5.03101	−0.265526	−0.132763	+	0.991148i	$0.542385\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−7.70314	−0.404310
$364$	0	0
$365$	8.98742	0.470423
$366$	0	0
$367$	26.0344	1.35898	0.679492	−	0.733683i	$-0.262200\pi$
$367$	26.0344	1.35898	0.679492	+	0.733683i	$0.262200\pi$
$368$	0	0
$369$	17.6898	0.920894
$370$	0	0
$371$	16.8661	0.875646
$372$	0	0
$373$	10.9587	0.567420	0.283710	−	0.958910i	$-0.408435\pi$
$373$	10.9587	0.567420	0.283710	+	0.958910i	$0.408435\pi$
$374$	0	0
$375$	6.69207	0.345577
$376$	0	0
$377$	−2.82428	−0.145458
$378$	0	0
$379$	−11.3234	−0.581645	−0.290823	−	0.956777i	$-0.593929\pi$
$379$	−11.3234	−0.581645	−0.290823	+	0.956777i	$0.593929\pi$
$380$	0	0
$381$	−0.883242	−0.0452499
$382$	0	0
$383$	8.00552	0.409063	0.204532	−	0.978860i	$-0.434433\pi$
$383$	8.00552	0.409063	0.204532	+	0.978860i	$0.434433\pi$
$384$	0	0
$385$	12.5404	0.639118
$386$	0	0
$387$	−8.58660	−0.436481
$388$	0	0
$389$	6.06589	0.307553	0.153776	−	0.988106i	$-0.450856\pi$
$389$	6.06589	0.307553	0.153776	+	0.988106i	$0.450856\pi$
$390$	0	0
$391$	0.618251	0.0312663
$392$	0	0
$393$	−15.8803	−0.801054
$394$	0	0
$395$	3.45227	0.173702
$396$	0	0
$397$	−17.2735	−0.866931	−0.433466	−	0.901170i	$-0.642709\pi$
$397$	−17.2735	−0.866931	−0.433466	+	0.901170i	$0.642709\pi$
$398$	0	0
$399$	1.99746	0.0999983
$400$	0	0
$401$	34.5655	1.72612	0.863061	−	0.505100i	$-0.168545\pi$
$401$	34.5655	1.72612	0.863061	+	0.505100i	$0.168545\pi$
$402$	0	0
$403$	2.98672	0.148779
$404$	0	0
$405$	2.93914	0.146047
$406$	0	0
$407$	−5.69717	−0.282398
$408$	0	0
$409$	−30.1147	−1.48908	−0.744539	−	0.667579i	$-0.767331\pi$
$409$	−30.1147	−1.48908	−0.744539	+	0.667579i	$0.767331\pi$
$410$	0	0
$411$	−19.6935	−0.971409
$412$	0	0
$413$	3.95455	0.194591
$414$	0	0
$415$	−45.6070	−2.23876
$416$	0	0
$417$	−9.93179	−0.486362
$418$	0	0
$419$	25.1073	1.22657	0.613287	−	0.789860i	$-0.289847\pi$
$419$	25.1073	1.22657	0.613287	+	0.789860i	$0.289847\pi$
$420$	0	0
$421$	−3.69341	−0.180006	−0.0900028	−	0.995942i	$-0.528688\pi$
$421$	−3.69341	−0.180006	−0.0900028	+	0.995942i	$0.528688\pi$
$422$	0	0
$423$	14.3228	0.696400
$424$	0	0
$425$	−28.0813	−1.36214
$426$	0	0
$427$	−24.6629	−1.19352
$428$	0	0
$429$	−0.981762	−0.0473999
$430$	0	0
$431$	15.5925	0.751065	0.375532	−	0.926809i	$-0.377460\pi$
$431$	15.5925	0.751065	0.375532	+	0.926809i	$0.377460\pi$
$432$	0	0
$433$	−4.52983	−0.217690	−0.108845	−	0.994059i	$-0.534715\pi$
$433$	−4.52983	−0.217690	−0.108845	+	0.994059i	$0.534715\pi$
$434$	0	0
$435$	−18.6407	−0.893753
$436$	0	0
$437$	0.152313	0.00728610
$438$	0	0
$439$	1.87167	0.0893297	0.0446649	−	0.999002i	$-0.485778\pi$
$439$	1.87167	0.0893297	0.0446649	+	0.999002i	$0.485778\pi$
$440$	0	0
$441$	6.12080	0.291467
$442$	0	0
$443$	−13.7722	−0.654337	−0.327168	−	0.944966i	$-0.606094\pi$
$443$	−13.7722	−0.654337	−0.327168	+	0.944966i	$0.606094\pi$
$444$	0	0
$445$	−40.2600	−1.90850
$446$	0	0
$447$	11.5856	0.547980
$448$	0	0
$449$	−25.0746	−1.18334	−0.591672	−	0.806179i	$-0.701532\pi$
$449$	−25.0746	−1.18334	−0.591672	+	0.806179i	$0.701532\pi$
$450$	0	0
$451$	16.4303	0.773675
$452$	0	0
$453$	−8.82936	−0.414840
$454$	0	0
$455$	−3.60689	−0.169093
$456$	0	0
$457$	17.3129	0.809863	0.404932	−	0.914347i	$-0.367295\pi$
$457$	17.3129	0.809863	0.404932	+	0.914347i	$0.367295\pi$
$458$	0	0
$459$	20.4231	0.953267
$460$	0	0
$461$	4.42277	0.205989	0.102994	−	0.994682i	$-0.467158\pi$
$461$	4.42277	0.205989	0.102994	+	0.994682i	$0.467158\pi$
$462$	0	0
$463$	19.8950	0.924600	0.462300	−	0.886723i	$-0.347024\pi$
$463$	19.8950	0.924600	0.462300	+	0.886723i	$0.347024\pi$
$464$	0	0
$465$	19.7128	0.914161
$466$	0	0
$467$	27.3073	1.26363	0.631815	−	0.775119i	$-0.282310\pi$
$467$	27.3073	1.26363	0.631815	+	0.775119i	$0.282310\pi$
$468$	0	0
$469$	−9.89029	−0.456691
$470$	0	0
$471$	15.3544	0.707495
$472$	0	0
$473$	−7.97526	−0.366703
$474$	0	0
$475$	−6.91814	−0.317426
$476$	0	0
$477$	16.8847	0.773097
$478$	0	0
$479$	9.79817	0.447690	0.223845	−	0.974625i	$-0.428139\pi$
$479$	9.79817	0.447690	0.223845	+	0.974625i	$0.428139\pi$
$480$	0	0
$481$	1.63863	0.0747150
$482$	0	0
$483$	0.304239	0.0138434
$484$	0	0
$485$	−22.0610	−1.00174
$486$	0	0
$487$	3.76457	0.170589	0.0852945	−	0.996356i	$-0.472817\pi$
$487$	3.76457	0.170589	0.0852945	+	0.996356i	$0.472817\pi$
$488$	0	0
$489$	1.60521	0.0725902
$490$	0	0
$491$	−34.9983	−1.57945	−0.789725	−	0.613461i	$-0.789777\pi$
$491$	−34.9983	−1.57945	−0.789725	+	0.613461i	$0.789777\pi$
$492$	0	0
$493$	21.6875	0.976757
$494$	0	0
$495$	12.5542	0.564269
$496$	0	0
$497$	−7.33104	−0.328842
$498$	0	0
$499$	11.9705	0.535875	0.267938	−	0.963436i	$-0.413658\pi$
$499$	11.9705	0.535875	0.267938	+	0.963436i	$0.413658\pi$
$500$	0	0
$501$	3.33664	0.149070
$502$	0	0
$503$	21.5278	0.959879	0.479940	−	0.877302i	$-0.340659\pi$
$503$	21.5278	0.959879	0.479940	+	0.877302i	$0.340659\pi$
$504$	0	0
$505$	−29.0903	−1.29450
$506$	0	0
$507$	−12.8553	−0.570925
$508$	0	0
$509$	−32.8843	−1.45757	−0.728785	−	0.684743i	$-0.759915\pi$
$509$	−32.8843	−1.45757	−0.728785	+	0.684743i	$0.759915\pi$
$510$	0	0
$511$	−5.14557	−0.227627
$512$	0	0
$513$	5.03144	0.222144
$514$	0	0
$515$	12.8907	0.568030
$516$	0	0
$517$	13.3031	0.585070
$518$	0	0
$519$	−14.1053	−0.619155
$520$	0	0
$521$	22.2671	0.975539	0.487770	−	0.872972i	$-0.337811\pi$
$521$	22.2671	0.975539	0.487770	+	0.872972i	$0.337811\pi$
$522$	0	0
$523$	−23.7031	−1.03647	−0.518233	−	0.855240i	$-0.673410\pi$
$523$	−23.7031	−1.03647	−0.518233	+	0.855240i	$0.673410\pi$
$524$	0	0
$525$	−13.8187	−0.603099
$526$	0	0
$527$	−22.9349	−0.999060
$528$	0	0
$529$	−22.9768	−0.998991
$530$	0	0
$531$	3.95890	0.171802
$532$	0	0
$533$	−4.72572	−0.204694
$534$	0	0
$535$	5.77973	0.249879
$536$	0	0
$537$	6.21041	0.267999
$538$	0	0
$539$	5.68502	0.244871
$540$	0	0
$541$	26.8359	1.15376	0.576882	−	0.816827i	$-0.304269\pi$
$541$	26.8359	1.15376	0.576882	+	0.816827i	$0.304269\pi$
$542$	0	0
$543$	14.0897	0.604646
$544$	0	0
$545$	−24.5096	−1.04988
$546$	0	0
$547$	13.7398	0.587472	0.293736	−	0.955887i	$-0.405101\pi$
$547$	13.7398	0.587472	0.293736	+	0.955887i	$0.405101\pi$
$548$	0	0
$549$	−24.6900	−1.05375
$550$	0	0
$551$	5.34295	0.227617
$552$	0	0
$553$	−1.97653	−0.0840505
$554$	0	0
$555$	10.8152	0.459080
$556$	0	0
$557$	12.3114	0.521650	0.260825	−	0.965386i	$-0.416005\pi$
$557$	12.3114	0.521650	0.260825	+	0.965386i	$0.416005\pi$
$558$	0	0
$559$	2.29386	0.0970199
$560$	0	0
$561$	7.53892	0.318293
$562$	0	0
$563$	−6.02384	−0.253875	−0.126937	−	0.991911i	$-0.540515\pi$
$563$	−6.02384	−0.253875	−0.126937	+	0.991911i	$0.540515\pi$
$564$	0	0
$565$	−34.5770	−1.45466
$566$	0	0
$567$	−1.68275	−0.0706686
$568$	0	0
$569$	8.83789	0.370504	0.185252	−	0.982691i	$-0.440690\pi$
$569$	8.83789	0.370504	0.185252	+	0.982691i	$0.440690\pi$
$570$	0	0
$571$	−44.0454	−1.84324	−0.921621	−	0.388092i	$-0.873134\pi$
$571$	−44.0454	−1.84324	−0.921621	+	0.388092i	$0.873134\pi$
$572$	0	0
$573$	−2.96447	−0.123842
$574$	0	0
$575$	−1.05372	−0.0439432
$576$	0	0
$577$	−3.85027	−0.160289	−0.0801445	−	0.996783i	$-0.525538\pi$
$577$	−3.85027	−0.160289	−0.0801445	+	0.996783i	$0.525538\pi$
$578$	0	0
$579$	8.60960	0.357803
$580$	0	0
$581$	26.1114	1.08328
$582$	0	0
$583$	15.6826	0.649505
$584$	0	0
$585$	−3.61086	−0.149291
$586$	0	0
$587$	24.6559	1.01766	0.508829	−	0.860868i	$-0.330079\pi$
$587$	24.6559	1.01766	0.508829	+	0.860868i	$0.330079\pi$
$588$	0	0
$589$	−5.65026	−0.232815
$590$	0	0
$591$	−13.1928	−0.542680
$592$	0	0
$593$	−7.41162	−0.304359	−0.152179	−	0.988353i	$-0.548629\pi$
$593$	−7.41162	−0.304359	−0.152179	+	0.988353i	$0.548629\pi$
$594$	0	0
$595$	27.6972	1.13547
$596$	0	0
$597$	22.9758	0.940337
$598$	0	0
$599$	0.265626	0.0108532	0.00542659	−	0.999985i	$-0.498273\pi$
$599$	0.265626	0.0108532	0.00542659	+	0.999985i	$0.498273\pi$
$600$	0	0
$601$	9.72457	0.396674	0.198337	−	0.980134i	$-0.436446\pi$
$601$	9.72457	0.396674	0.198337	+	0.980134i	$0.436446\pi$
$602$	0	0
$603$	−9.90117	−0.403207
$604$	0	0
$605$	−26.3145	−1.06984
$606$	0	0
$607$	−17.1913	−0.697773	−0.348887	−	0.937165i	$-0.613440\pi$
$607$	−17.1913	−0.697773	−0.348887	+	0.937165i	$0.613440\pi$
$608$	0	0
$609$	10.6724	0.432466
$610$	0	0
$611$	−3.82626	−0.154794
$612$	0	0
$613$	−24.2604	−0.979868	−0.489934	−	0.871760i	$-0.662979\pi$
$613$	−24.2604	−0.979868	−0.489934	+	0.871760i	$0.662979\pi$
$614$	0	0
$615$	−31.1905	−1.25772
$616$	0	0
$617$	−0.335074	−0.0134896	−0.00674479	−	0.999977i	$-0.502147\pi$
$617$	−0.335074	−0.0134896	−0.00674479	+	0.999977i	$0.502147\pi$
$618$	0	0
$619$	−2.11127	−0.0848589	−0.0424295	−	0.999099i	$-0.513510\pi$
$619$	−2.11127	−0.0848589	−0.0424295	+	0.999099i	$0.513510\pi$
$620$	0	0
$621$	0.766352	0.0307527
$622$	0	0
$623$	23.0500	0.923481
$624$	0	0
$625$	−11.7301	−0.469202
$626$	0	0
$627$	1.85729	0.0741732
$628$	0	0
$629$	−12.5830	−0.501716
$630$	0	0
$631$	−11.6476	−0.463683	−0.231841	−	0.972754i	$-0.574475\pi$
$631$	−11.6476	−0.463683	−0.231841	+	0.972754i	$0.574475\pi$
$632$	0	0
$633$	8.35645	0.332139
$634$	0	0
$635$	−3.01722	−0.119735
$636$	0	0
$637$	−1.63514	−0.0647864
$638$	0	0
$639$	−7.33910	−0.290331
$640$	0	0
$641$	20.6000	0.813653	0.406826	−	0.913505i	$-0.366635\pi$
$641$	20.6000	0.813653	0.406826	+	0.913505i	$0.366635\pi$
$642$	0	0
$643$	−25.1778	−0.992916	−0.496458	−	0.868061i	$-0.665366\pi$
$643$	−25.1778	−0.992916	−0.496458	+	0.868061i	$0.665366\pi$
$644$	0	0
$645$	15.1398	0.596131
$646$	0	0
$647$	36.7489	1.44475	0.722374	−	0.691503i	$-0.243051\pi$
$647$	36.7489	1.44475	0.722374	+	0.691503i	$0.243051\pi$
$648$	0	0
$649$	3.67704	0.144337
$650$	0	0
$651$	−11.2862	−0.442341
$652$	0	0
$653$	−29.8978	−1.16999	−0.584996	−	0.811036i	$-0.698904\pi$
$653$	−29.8978	−1.16999	−0.584996	+	0.811036i	$0.698904\pi$
$654$	0	0
$655$	−54.2483	−2.11965
$656$	0	0
$657$	−5.15123	−0.200969
$658$	0	0
$659$	−34.1247	−1.32931	−0.664655	−	0.747150i	$-0.731421\pi$
$659$	−34.1247	−1.32931	−0.664655	+	0.747150i	$0.731421\pi$
$660$	0	0
$661$	21.9632	0.854271	0.427136	−	0.904188i	$-0.359523\pi$
$661$	21.9632	0.854271	0.427136	+	0.904188i	$0.359523\pi$
$662$	0	0
$663$	−2.16836	−0.0842120
$664$	0	0
$665$	6.82350	0.264604
$666$	0	0
$667$	0.813799	0.0315104
$668$	0	0
$669$	−1.80255	−0.0696907
$670$	0	0
$671$	−22.9322	−0.885288
$672$	0	0
$673$	13.8779	0.534952	0.267476	−	0.963565i	$-0.413810\pi$
$673$	13.8779	0.534952	0.267476	+	0.963565i	$0.413810\pi$
$674$	0	0
$675$	−34.8082	−1.33977
$676$	0	0
$677$	−19.7815	−0.760265	−0.380133	−	0.924932i	$-0.624122\pi$
$677$	−19.7815	−0.760265	−0.380133	+	0.924932i	$0.624122\pi$
$678$	0	0
$679$	12.6306	0.484718
$680$	0	0
$681$	16.7245	0.640884
$682$	0	0
$683$	3.17855	0.121624	0.0608119	−	0.998149i	$-0.480631\pi$
$683$	3.17855	0.121624	0.0608119	+	0.998149i	$0.480631\pi$
$684$	0	0
$685$	−67.2745	−2.57043
$686$	0	0
$687$	0.470216	0.0179399
$688$	0	0
$689$	−4.51064	−0.171842
$690$	0	0
$691$	−16.7634	−0.637709	−0.318855	−	0.947804i	$-0.603298\pi$
$691$	−16.7634	−0.637709	−0.318855	+	0.947804i	$0.603298\pi$
$692$	0	0
$693$	−7.18765	−0.273036
$694$	0	0
$695$	−33.9278	−1.28695
$696$	0	0
$697$	36.2887	1.37453
$698$	0	0
$699$	19.2095	0.726571
$700$	0	0
$701$	−12.5386	−0.473575	−0.236787	−	0.971562i	$-0.576094\pi$
$701$	−12.5386	−0.473575	−0.236787	+	0.971562i	$0.576094\pi$
$702$	0	0
$703$	−3.09995	−0.116917
$704$	0	0
$705$	−25.2540	−0.951119
$706$	0	0
$707$	16.6551	0.626378
$708$	0	0
$709$	38.4248	1.44307	0.721536	−	0.692377i	$-0.243436\pi$
$709$	38.4248	1.44307	0.721536	+	0.692377i	$0.243436\pi$
$710$	0	0
$711$	−1.97870	−0.0742071
$712$	0	0
$713$	−0.860606	−0.0322299
$714$	0	0
$715$	−3.35378	−0.125424
$716$	0	0
$717$	−27.3731	−1.02227
$718$	0	0
$719$	−8.00929	−0.298696	−0.149348	−	0.988785i	$-0.547717\pi$
$719$	−8.00929	−0.298696	−0.149348	+	0.988785i	$0.547717\pi$
$720$	0	0
$721$	−7.38029	−0.274856
$722$	0	0
$723$	4.10443	0.152645
$724$	0	0
$725$	−36.9633	−1.37278
$726$	0	0
$727$	−6.99092	−0.259279	−0.129640	−	0.991561i	$-0.541382\pi$
$727$	−6.99092	−0.259279	−0.129640	+	0.991561i	$0.541382\pi$
$728$	0	0
$729$	13.5696	0.502579
$730$	0	0
$731$	−17.6145	−0.651495
$732$	0	0
$733$	10.7958	0.398754	0.199377	−	0.979923i	$-0.436108\pi$
$733$	10.7958	0.398754	0.199377	+	0.979923i	$0.436108\pi$
$734$	0	0
$735$	−10.7922	−0.398075
$736$	0	0
$737$	−9.19624	−0.338748
$738$	0	0
$739$	−14.9112	−0.548518	−0.274259	−	0.961656i	$-0.588433\pi$
$739$	−14.9112	−0.548518	−0.274259	+	0.961656i	$0.588433\pi$
$740$	0	0
$741$	−0.534198	−0.0196242
$742$	0	0
$743$	−49.7910	−1.82666	−0.913328	−	0.407225i	$-0.866496\pi$
$743$	−49.7910	−1.82666	−0.913328	+	0.407225i	$0.866496\pi$
$744$	0	0
$745$	39.5773	1.45000
$746$	0	0
$747$	26.1401	0.956416
$748$	0	0
$749$	−3.30907	−0.120911
$750$	0	0
$751$	24.4689	0.892882	0.446441	−	0.894813i	$-0.352691\pi$
$751$	24.4689	0.892882	0.446441	+	0.894813i	$0.352691\pi$
$752$	0	0
$753$	14.1524	0.515741
$754$	0	0
$755$	−30.1618	−1.09770
$756$	0	0
$757$	12.6157	0.458526	0.229263	−	0.973364i	$-0.426368\pi$
$757$	12.6157	0.458526	0.229263	+	0.973364i	$0.426368\pi$
$758$	0	0
$759$	0.282889	0.0102682
$760$	0	0
$761$	5.58532	0.202468	0.101234	−	0.994863i	$-0.467721\pi$
$761$	5.58532	0.202468	0.101234	+	0.994863i	$0.467721\pi$
$762$	0	0
$763$	14.0325	0.508010
$764$	0	0
$765$	27.7277	1.00250
$766$	0	0
$767$	−1.05760	−0.0381876
$768$	0	0
$769$	−9.10585	−0.328366	−0.164183	−	0.986430i	$-0.552499\pi$
$769$	−9.10585	−0.328366	−0.164183	+	0.986430i	$0.552499\pi$
$770$	0	0
$771$	−17.1780	−0.618650
$772$	0	0
$773$	−4.63293	−0.166635	−0.0833174	−	0.996523i	$-0.526552\pi$
$773$	−4.63293	−0.166635	−0.0833174	+	0.996523i	$0.526552\pi$
$774$	0	0
$775$	39.0893	1.40413
$776$	0	0
$777$	−6.19204	−0.222138
$778$	0	0
$779$	8.94010	0.320312
$780$	0	0
$781$	−6.81659	−0.243917
$782$	0	0
$783$	26.8828	0.960711
$784$	0	0
$785$	52.4520	1.87209
$786$	0	0
$787$	−29.9574	−1.06787	−0.533934	−	0.845526i	$-0.679287\pi$
$787$	−29.9574	−1.06787	−0.533934	+	0.845526i	$0.679287\pi$
$788$	0	0
$789$	28.9778	1.03164
$790$	0	0
$791$	19.7964	0.703878
$792$	0	0
$793$	6.59580	0.234224
$794$	0	0
$795$	−29.7710	−1.05587
$796$	0	0
$797$	−35.5118	−1.25789	−0.628947	−	0.777448i	$-0.716514\pi$
$797$	−35.5118	−1.25789	−0.628947	+	0.777448i	$0.716514\pi$
$798$	0	0
$799$	29.3817	1.03945
$800$	0	0
$801$	23.0754	0.815329
$802$	0	0
$803$	−4.78449	−0.168841
$804$	0	0
$805$	1.03930	0.0366307
$806$	0	0
$807$	−17.0455	−0.600030
$808$	0	0
$809$	13.5820	0.477519	0.238760	−	0.971079i	$-0.423259\pi$
$809$	13.5820	0.477519	0.238760	+	0.971079i	$0.423259\pi$
$810$	0	0
$811$	11.1380	0.391107	0.195554	−	0.980693i	$-0.437350\pi$
$811$	11.1380	0.391107	0.195554	+	0.980693i	$0.437350\pi$
$812$	0	0
$813$	7.25406	0.254411
$814$	0	0
$815$	5.48353	0.192080
$816$	0	0
$817$	−4.33951	−0.151820
$818$	0	0
$819$	2.06732	0.0722382
$820$	0	0
$821$	−3.21443	−0.112184	−0.0560922	−	0.998426i	$-0.517864\pi$
$821$	−3.21443	−0.112184	−0.0560922	+	0.998426i	$0.517864\pi$
$822$	0	0
$823$	53.3683	1.86030	0.930151	−	0.367177i	$-0.119676\pi$
$823$	53.3683	1.86030	0.930151	+	0.367177i	$0.119676\pi$
$824$	0	0
$825$	−12.8490	−0.447345
$826$	0	0
$827$	30.8107	1.07139	0.535696	−	0.844411i	$-0.320049\pi$
$827$	30.8107	1.07139	0.535696	+	0.844411i	$0.320049\pi$
$828$	0	0
$829$	−34.9886	−1.21520	−0.607602	−	0.794242i	$-0.707868\pi$
$829$	−34.9886	−1.21520	−0.607602	+	0.794242i	$0.707868\pi$
$830$	0	0
$831$	−24.5065	−0.850122
$832$	0	0
$833$	12.5562	0.435045
$834$	0	0
$835$	11.3982	0.394452
$836$	0	0
$837$	−28.4290	−0.982648
$838$	0	0
$839$	−30.1999	−1.04262	−0.521308	−	0.853369i	$-0.674556\pi$
$839$	−30.1999	−1.04262	−0.521308	+	0.853369i	$0.674556\pi$
$840$	0	0
$841$	−0.452841	−0.0156152
$842$	0	0
$843$	−0.486645	−0.0167610
$844$	0	0
$845$	−43.9148	−1.51072
$846$	0	0
$847$	15.0659	0.517670
$848$	0	0
$849$	−1.49074	−0.0511621
$850$	0	0
$851$	−0.472162	−0.0161855
$852$	0	0
$853$	52.9445	1.81279	0.906393	−	0.422435i	$-0.138825\pi$
$853$	52.9445	1.81279	0.906393	+	0.422435i	$0.138825\pi$
$854$	0	0
$855$	6.83100	0.233615
$856$	0	0
$857$	−30.8736	−1.05462	−0.527312	−	0.849672i	$-0.676800\pi$
$857$	−30.8736	−1.05462	−0.527312	+	0.849672i	$0.676800\pi$
$858$	0	0
$859$	−8.12476	−0.277213	−0.138607	−	0.990348i	$-0.544262\pi$
$859$	−8.12476	−0.277213	−0.138607	+	0.990348i	$0.544262\pi$
$860$	0	0
$861$	17.8575	0.608583
$862$	0	0
$863$	−33.1112	−1.12712	−0.563559	−	0.826076i	$-0.690568\pi$
$863$	−33.1112	−1.12712	−0.563559	+	0.826076i	$0.690568\pi$
$864$	0	0
$865$	−48.1849	−1.63834
$866$	0	0
$867$	−0.529343	−0.0179774
$868$	0	0
$869$	−1.83783	−0.0623439
$870$	0	0
$871$	2.64504	0.0896237
$872$	0	0
$873$	12.6445	0.427951
$874$	0	0
$875$	−13.0884	−0.442469
$876$	0	0
$877$	−58.8832	−1.98834	−0.994172	−	0.107805i	$-0.965618\pi$
$877$	−58.8832	−1.98834	−0.994172	+	0.107805i	$0.965618\pi$
$878$	0	0
$879$	4.29265	0.144788
$880$	0	0
$881$	51.3051	1.72851	0.864257	−	0.503051i	$-0.167789\pi$
$881$	51.3051	1.72851	0.864257	+	0.503051i	$0.167789\pi$
$882$	0	0
$883$	10.5694	0.355688	0.177844	−	0.984059i	$-0.443088\pi$
$883$	10.5694	0.355688	0.177844	+	0.984059i	$0.443088\pi$
$884$	0	0
$885$	−6.98032	−0.234641
$886$	0	0
$887$	−30.6163	−1.02799	−0.513997	−	0.857792i	$-0.671836\pi$
$887$	−30.6163	−1.02799	−0.513997	+	0.857792i	$0.671836\pi$
$888$	0	0
$889$	1.72745	0.0579369
$890$	0	0
$891$	−1.56466	−0.0524181
$892$	0	0
$893$	7.23851	0.242227
$894$	0	0
$895$	21.2153	0.709148
$896$	0	0
$897$	−0.0813651	−0.00271670
$898$	0	0
$899$	−30.1891	−1.00686
$900$	0	0
$901$	34.6371	1.15393
$902$	0	0
$903$	−8.66802	−0.288454
$904$	0	0
$905$	48.1315	1.59994
$906$	0	0
$907$	−53.9255	−1.79057	−0.895284	−	0.445496i	$-0.853027\pi$
$907$	−53.9255	−1.79057	−0.895284	+	0.445496i	$0.853027\pi$
$908$	0	0
$909$	16.6734	0.553021
$910$	0	0
$911$	55.1503	1.82721	0.913605	−	0.406602i	$-0.133286\pi$
$911$	55.1503	1.82721	0.913605	+	0.406602i	$0.133286\pi$
$912$	0	0
$913$	24.2790	0.803518
$914$	0	0
$915$	43.5334	1.43917
$916$	0	0
$917$	31.0588	1.02565
$918$	0	0
$919$	−22.3378	−0.736856	−0.368428	−	0.929656i	$-0.620104\pi$
$919$	−22.3378	−0.736856	−0.368428	+	0.929656i	$0.620104\pi$
$920$	0	0
$921$	−21.4775	−0.707706
$922$	0	0
$923$	1.96060	0.0645339
$924$	0	0
$925$	21.4459	0.705136
$926$	0	0
$927$	−7.38841	−0.242667
$928$	0	0
$929$	12.6401	0.414707	0.207353	−	0.978266i	$-0.433515\pi$
$929$	12.6401	0.414707	0.207353	+	0.978266i	$0.433515\pi$
$930$	0	0
$931$	3.09334	0.101380
$932$	0	0
$933$	29.0024	0.949496
$934$	0	0
$935$	25.7535	0.842231
$936$	0	0
$937$	52.5211	1.71579	0.857894	−	0.513827i	$-0.171773\pi$
$937$	52.5211	1.71579	0.857894	+	0.513827i	$0.171773\pi$
$938$	0	0
$939$	17.2054	0.561478
$940$	0	0
$941$	−26.5127	−0.864289	−0.432145	−	0.901804i	$-0.642243\pi$
$941$	−26.5127	−0.864289	−0.432145	+	0.901804i	$0.642243\pi$
$942$	0	0
$943$	1.36169	0.0443427
$944$	0	0
$945$	34.3320	1.11682
$946$	0	0
$947$	6.21818	0.202064	0.101032	−	0.994883i	$-0.467786\pi$
$947$	6.21818	0.202064	0.101032	+	0.994883i	$0.467786\pi$
$948$	0	0
$949$	1.37612	0.0446708
$950$	0	0
$951$	32.1931	1.04393
$952$	0	0
$953$	58.0286	1.87973	0.939865	−	0.341547i	$-0.110951\pi$
$953$	58.0286	1.87973	0.939865	+	0.341547i	$0.110951\pi$
$954$	0	0
$955$	−10.1268	−0.327697
$956$	0	0
$957$	9.92343	0.320779
$958$	0	0
$959$	38.5167	1.24377
$960$	0	0
$961$	0.925438	0.0298528
$962$	0	0
$963$	−3.31271	−0.106751
$964$	0	0
$965$	29.4111	0.946776
$966$	0	0
$967$	−5.83713	−0.187709	−0.0938546	−	0.995586i	$-0.529919\pi$
$967$	−5.83713	−0.187709	−0.0938546	+	0.995586i	$0.529919\pi$
$968$	0	0
$969$	4.10209	0.131778
$970$	0	0
$971$	−50.0127	−1.60498	−0.802491	−	0.596664i	$-0.796493\pi$
$971$	−50.0127	−1.60498	−0.802491	+	0.596664i	$0.796493\pi$
$972$	0	0
$973$	19.4247	0.622727
$974$	0	0
$975$	3.69565	0.118356
$976$	0	0
$977$	−16.5839	−0.530566	−0.265283	−	0.964171i	$-0.585465\pi$
$977$	−16.5839	−0.530566	−0.265283	+	0.964171i	$0.585465\pi$
$978$	0	0
$979$	21.4325	0.684986
$980$	0	0
$981$	14.0479	0.448516
$982$	0	0
$983$	−5.43221	−0.173260	−0.0866302	−	0.996241i	$-0.527610\pi$
$983$	−5.43221	−0.173260	−0.0866302	+	0.996241i	$0.527610\pi$
$984$	0	0
$985$	−45.0677	−1.43598
$986$	0	0
$987$	14.4587	0.460224
$988$	0	0
$989$	−0.660962	−0.0210174
$990$	0	0
$991$	10.4490	0.331922	0.165961	−	0.986132i	$-0.446927\pi$
$991$	10.4490	0.331922	0.165961	+	0.986132i	$0.446927\pi$
$992$	0	0
$993$	8.00013	0.253876
$994$	0	0
$995$	78.4871	2.48821
$996$	0	0
$997$	−33.2044	−1.05159	−0.525797	−	0.850610i	$-0.676233\pi$
$997$	−33.2044	−1.05159	−0.525797	+	0.850610i	$0.676233\pi$
$998$	0	0
$999$	−15.5972	−0.493474

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.f.1.16	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.f.1.16	✓	25	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)

\(f(q)\)	\(=\)	\(q+1.01059 q^{3} +3.45227 q^{5} -1.97653 q^{7} -1.97870 q^{9} +O(q^{10})\)	\(q+1.01059 q^{3} +3.45227 q^{5} -1.97653 q^{7} -1.97870 q^{9} -1.83783 q^{11} +0.528598 q^{13} +3.48884 q^{15} -4.05909 q^{17} -1.00000 q^{19} -1.99746 q^{21} -0.152313 q^{23} +6.91814 q^{25} -5.03144 q^{27} -5.34295 q^{29} +5.65026 q^{31} -1.85729 q^{33} -6.82350 q^{35} +3.09995 q^{37} +0.534198 q^{39} -8.94010 q^{41} +4.33951 q^{43} -6.83100 q^{45} -7.23851 q^{47} -3.09334 q^{49} -4.10209 q^{51} -8.53322 q^{53} -6.34466 q^{55} -1.01059 q^{57} -2.00076 q^{59} +12.4779 q^{61} +3.91096 q^{63} +1.82486 q^{65} +5.00387 q^{67} -0.153926 q^{69} +3.70905 q^{71} +2.60334 q^{73} +6.99142 q^{75} +3.63251 q^{77} +1.00000 q^{79} +0.851365 q^{81} -13.2107 q^{83} -14.0131 q^{85} -5.39955 q^{87} -11.6619 q^{89} -1.04479 q^{91} +5.71011 q^{93} -3.45227 q^{95} -6.39030 q^{97} +3.63651 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more