LMFDB - Embedded newform 6004.2.a.f.1.3

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			1.3
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.45149 q^{3} +1.53010 q^{5} +1.05276 q^{7} +3.00979 q^{9} +O(q^{10})$	$q-2.45149 q^{3} +1.53010 q^{5} +1.05276 q^{7} +3.00979 q^{9} +2.99221 q^{11} -0.170649 q^{13} -3.75101 q^{15} +3.06585 q^{17} -1.00000 q^{19} -2.58083 q^{21} -8.77490 q^{23} -2.65880 q^{25} -0.0239932 q^{27} -5.85129 q^{29} +9.59685 q^{31} -7.33536 q^{33} +1.61082 q^{35} +2.36380 q^{37} +0.418344 q^{39} -6.12106 q^{41} +4.52915 q^{43} +4.60527 q^{45} -9.15759 q^{47} -5.89170 q^{49} -7.51589 q^{51} -11.9474 q^{53} +4.57837 q^{55} +2.45149 q^{57} -4.41013 q^{59} -9.85190 q^{61} +3.16858 q^{63} -0.261110 q^{65} +5.05110 q^{67} +21.5116 q^{69} +2.90184 q^{71} +2.32834 q^{73} +6.51802 q^{75} +3.15008 q^{77} +1.00000 q^{79} -8.97054 q^{81} -7.65477 q^{83} +4.69105 q^{85} +14.3444 q^{87} -2.18396 q^{89} -0.179652 q^{91} -23.5265 q^{93} -1.53010 q^{95} +10.0959 q^{97} +9.00591 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.45149	−1.41537	−0.707683	−	0.706530i	$-0.750260\pi$
$3$	−2.45149	−1.41537	−0.707683	+	0.706530i	$0.750260\pi$
$4$	0	0
$5$	1.53010	0.684280	0.342140	−	0.939649i	$-0.388848\pi$
$5$	1.53010	0.684280	0.342140	+	0.939649i	$0.388848\pi$
$6$	0	0
$7$	1.05276	0.397906	0.198953	−	0.980009i	$-0.436246\pi$
$7$	1.05276	0.397906	0.198953	+	0.980009i	$0.436246\pi$
$8$	0	0
$9$	3.00979	1.00326
$10$	0	0
$11$	2.99221	0.902184	0.451092	−	0.892477i	$-0.351035\pi$
$11$	2.99221	0.902184	0.451092	+	0.892477i	$0.351035\pi$
$12$	0	0
$13$	−0.170649	−0.0473295	−0.0236648	−	0.999720i	$-0.507533\pi$
$13$	−0.170649	−0.0473295	−0.0236648	+	0.999720i	$0.507533\pi$
$14$	0	0
$15$	−3.75101	−0.968507
$16$	0	0
$17$	3.06585	0.743578	0.371789	−	0.928317i	$-0.378744\pi$
$17$	3.06585	0.743578	0.371789	+	0.928317i	$0.378744\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.58083	−0.563183
$22$	0	0
$23$	−8.77490	−1.82969	−0.914847	−	0.403801i	$-0.867689\pi$
$23$	−8.77490	−1.82969	−0.914847	+	0.403801i	$0.867689\pi$
$24$	0	0
$25$	−2.65880	−0.531761
$26$	0	0
$27$	−0.0239932	−0.00461750
$28$	0	0
$29$	−5.85129	−1.08656	−0.543279	−	0.839552i	$-0.682817\pi$
$29$	−5.85129	−1.08656	−0.543279	+	0.839552i	$0.682817\pi$
$30$	0	0
$31$	9.59685	1.72364	0.861822	−	0.507210i	$-0.169323\pi$
$31$	9.59685	1.72364	0.861822	+	0.507210i	$0.169323\pi$
$32$	0	0
$33$	−7.33536	−1.27692
$34$	0	0
$35$	1.61082	0.272279
$36$	0	0
$37$	2.36380	0.388606	0.194303	−	0.980942i	$-0.437756\pi$
$37$	2.36380	0.388606	0.194303	+	0.980942i	$0.437756\pi$
$38$	0	0
$39$	0.418344	0.0669886
$40$	0	0
$41$	−6.12106	−0.955949	−0.477974	−	0.878374i	$-0.658629\pi$
$41$	−6.12106	−0.955949	−0.477974	+	0.878374i	$0.658629\pi$
$42$	0	0
$43$	4.52915	0.690688	0.345344	−	0.938476i	$-0.387762\pi$
$43$	4.52915	0.690688	0.345344	+	0.938476i	$0.387762\pi$
$44$	0	0
$45$	4.60527	0.686513
$46$	0	0
$47$	−9.15759	−1.33577	−0.667886	−	0.744264i	$-0.732800\pi$
$47$	−9.15759	−1.33577	−0.667886	+	0.744264i	$0.732800\pi$
$48$	0	0
$49$	−5.89170	−0.841671
$50$	0	0
$51$	−7.51589	−1.05244
$52$	0	0
$53$	−11.9474	−1.64111	−0.820553	−	0.571571i	$-0.806334\pi$
$53$	−11.9474	−1.64111	−0.820553	+	0.571571i	$0.806334\pi$
$54$	0	0
$55$	4.57837	0.617347
$56$	0	0
$57$	2.45149	0.324707
$58$	0	0
$59$	−4.41013	−0.574150	−0.287075	−	0.957908i	$-0.592683\pi$
$59$	−4.41013	−0.574150	−0.287075	+	0.957908i	$0.592683\pi$
$60$	0	0
$61$	−9.85190	−1.26141	−0.630704	−	0.776024i	$-0.717234\pi$
$61$	−9.85190	−1.26141	−0.630704	+	0.776024i	$0.717234\pi$
$62$	0	0
$63$	3.16858	0.399204
$64$	0	0
$65$	−0.261110	−0.0323866
$66$	0	0
$67$	5.05110	0.617091	0.308545	−	0.951210i	$-0.400158\pi$
$67$	5.05110	0.617091	0.308545	+	0.951210i	$0.400158\pi$
$68$	0	0
$69$	21.5116	2.58969
$70$	0	0
$71$	2.90184	0.344385	0.172192	−	0.985063i	$-0.444915\pi$
$71$	2.90184	0.344385	0.172192	+	0.985063i	$0.444915\pi$
$72$	0	0
$73$	2.32834	0.272511	0.136256	−	0.990674i	$-0.456493\pi$
$73$	2.32834	0.272511	0.136256	+	0.990674i	$0.456493\pi$
$74$	0	0
$75$	6.51802	0.752636
$76$	0	0
$77$	3.15008	0.358984
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−8.97054	−0.996727
$82$	0	0
$83$	−7.65477	−0.840221	−0.420110	−	0.907473i	$-0.638009\pi$
$83$	−7.65477	−0.840221	−0.420110	+	0.907473i	$0.638009\pi$
$84$	0	0
$85$	4.69105	0.508816
$86$	0	0
$87$	14.3444	1.53788
$88$	0	0
$89$	−2.18396	−0.231499	−0.115749	−	0.993278i	$-0.536927\pi$
$89$	−2.18396	−0.231499	−0.115749	+	0.993278i	$0.536927\pi$
$90$	0	0
$91$	−0.179652	−0.0188327
$92$	0	0
$93$	−23.5265	−2.43959
$94$	0	0
$95$	−1.53010	−0.156985
$96$	0	0
$97$	10.0959	1.02508	0.512541	−	0.858663i	$-0.328704\pi$
$97$	10.0959	1.02508	0.512541	+	0.858663i	$0.328704\pi$
$98$	0	0
$99$	9.00591	0.905128
$100$	0	0
$101$	−10.0697	−1.00197	−0.500987	−	0.865455i	$-0.667030\pi$
$101$	−10.0697	−1.00197	−0.500987	+	0.865455i	$0.667030\pi$
$102$	0	0
$103$	5.18306	0.510703	0.255351	−	0.966848i	$-0.417809\pi$
$103$	5.18306	0.510703	0.255351	+	0.966848i	$0.417809\pi$
$104$	0	0
$105$	−3.94892	−0.385375
$106$	0	0
$107$	8.85245	0.855798	0.427899	−	0.903826i	$-0.359254\pi$
$107$	8.85245	0.855798	0.427899	+	0.903826i	$0.359254\pi$
$108$	0	0
$109$	−4.10296	−0.392992	−0.196496	−	0.980505i	$-0.562956\pi$
$109$	−4.10296	−0.392992	−0.196496	+	0.980505i	$0.562956\pi$
$110$	0	0
$111$	−5.79481	−0.550019
$112$	0	0
$113$	17.3308	1.63035	0.815173	−	0.579218i	$-0.196642\pi$
$113$	17.3308	1.63035	0.815173	+	0.579218i	$0.196642\pi$
$114$	0	0
$115$	−13.4265	−1.25202
$116$	0	0
$117$	−0.513617	−0.0474839
$118$	0	0
$119$	3.22760	0.295874
$120$	0	0
$121$	−2.04670	−0.186064
$122$	0	0
$123$	15.0057	1.35302
$124$	0	0
$125$	−11.7187	−1.04815
$126$	0	0
$127$	7.44593	0.660719	0.330360	−	0.943855i	$-0.392830\pi$
$127$	7.44593	0.660719	0.330360	+	0.943855i	$0.392830\pi$
$128$	0	0
$129$	−11.1031	−0.977577
$130$	0	0
$131$	−6.03882	−0.527614	−0.263807	−	0.964575i	$-0.584978\pi$
$131$	−6.03882	−0.527614	−0.263807	+	0.964575i	$0.584978\pi$
$132$	0	0
$133$	−1.05276	−0.0912859
$134$	0	0
$135$	−0.0367120	−0.00315966
$136$	0	0
$137$	−1.85942	−0.158861	−0.0794304	−	0.996840i	$-0.525310\pi$
$137$	−1.85942	−0.158861	−0.0794304	+	0.996840i	$0.525310\pi$
$138$	0	0
$139$	11.4789	0.973624	0.486812	−	0.873507i	$-0.338160\pi$
$139$	11.4789	0.973624	0.486812	+	0.873507i	$0.338160\pi$
$140$	0	0
$141$	22.4497	1.89061
$142$	0	0
$143$	−0.510617	−0.0426999
$144$	0	0
$145$	−8.95305	−0.743510
$146$	0	0
$147$	14.4434	1.19127
$148$	0	0
$149$	20.4743	1.67732	0.838662	−	0.544652i	$-0.183338\pi$
$149$	20.4743	1.67732	0.838662	+	0.544652i	$0.183338\pi$
$150$	0	0
$151$	−11.0024	−0.895366	−0.447683	−	0.894192i	$-0.647751\pi$
$151$	−11.0024	−0.895366	−0.447683	+	0.894192i	$0.647751\pi$
$152$	0	0
$153$	9.22756	0.746004
$154$	0	0
$155$	14.6841	1.17946
$156$	0	0
$157$	−21.8489	−1.74373	−0.871866	−	0.489744i	$-0.837090\pi$
$157$	−21.8489	−1.74373	−0.871866	+	0.489744i	$0.837090\pi$
$158$	0	0
$159$	29.2890	2.32277
$160$	0	0
$161$	−9.23787	−0.728046
$162$	0	0
$163$	0.686244	0.0537508	0.0268754	−	0.999639i	$-0.491444\pi$
$163$	0.686244	0.0537508	0.0268754	+	0.999639i	$0.491444\pi$
$164$	0	0
$165$	−11.2238	−0.873772
$166$	0	0
$167$	−8.55726	−0.662181	−0.331090	−	0.943599i	$-0.607416\pi$
$167$	−8.55726	−0.662181	−0.331090	+	0.943599i	$0.607416\pi$
$168$	0	0
$169$	−12.9709	−0.997760
$170$	0	0
$171$	−3.00979	−0.230164
$172$	0	0
$173$	7.91923	0.602088	0.301044	−	0.953610i	$-0.402665\pi$
$173$	7.91923	0.602088	0.301044	+	0.953610i	$0.402665\pi$
$174$	0	0
$175$	−2.79908	−0.211591
$176$	0	0
$177$	10.8114	0.812633
$178$	0	0
$179$	16.0665	1.20087	0.600433	−	0.799675i	$-0.294995\pi$
$179$	16.0665	1.20087	0.600433	+	0.799675i	$0.294995\pi$
$180$	0	0
$181$	8.20164	0.609623	0.304812	−	0.952413i	$-0.401406\pi$
$181$	8.20164	0.609623	0.304812	+	0.952413i	$0.401406\pi$
$182$	0	0
$183$	24.1518	1.78535
$184$	0	0
$185$	3.61684	0.265915
$186$	0	0
$187$	9.17366	0.670844
$188$	0	0
$189$	−0.0252591	−0.00183733
$190$	0	0
$191$	13.0572	0.944787	0.472393	−	0.881388i	$-0.343390\pi$
$191$	13.0572	0.944787	0.472393	+	0.881388i	$0.343390\pi$
$192$	0	0
$193$	−5.71937	−0.411689	−0.205845	−	0.978585i	$-0.565994\pi$
$193$	−5.71937	−0.411689	−0.205845	+	0.978585i	$0.565994\pi$
$194$	0	0
$195$	0.640106	0.0458390
$196$	0	0
$197$	−3.46615	−0.246953	−0.123477	−	0.992347i	$-0.539404\pi$
$197$	−3.46615	−0.246953	−0.123477	+	0.992347i	$0.539404\pi$
$198$	0	0
$199$	7.95079	0.563616	0.281808	−	0.959471i	$-0.409066\pi$
$199$	7.95079	0.563616	0.281808	+	0.959471i	$0.409066\pi$
$200$	0	0
$201$	−12.3827	−0.873409
$202$	0	0
$203$	−6.16001	−0.432348
$204$	0	0
$205$	−9.36582	−0.654137
$206$	0	0
$207$	−26.4106	−1.83566
$208$	0	0
$209$	−2.99221	−0.206975
$210$	0	0
$211$	−9.48727	−0.653131	−0.326565	−	0.945175i	$-0.605891\pi$
$211$	−9.48727	−0.653131	−0.326565	+	0.945175i	$0.605891\pi$
$212$	0	0
$213$	−7.11381	−0.487430
$214$	0	0
$215$	6.93003	0.472624
$216$	0	0
$217$	10.1032	0.685848
$218$	0	0
$219$	−5.70788	−0.385703
$220$	0	0
$221$	−0.523184	−0.0351932
$222$	0	0
$223$	−13.0468	−0.873676	−0.436838	−	0.899540i	$-0.643902\pi$
$223$	−13.0468	−0.873676	−0.436838	+	0.899540i	$0.643902\pi$
$224$	0	0
$225$	−8.00243	−0.533495
$226$	0	0
$227$	−5.58018	−0.370370	−0.185185	−	0.982704i	$-0.559288\pi$
$227$	−5.58018	−0.370370	−0.185185	+	0.982704i	$0.559288\pi$
$228$	0	0
$229$	−21.5083	−1.42131	−0.710655	−	0.703541i	$-0.751601\pi$
$229$	−21.5083	−1.42131	−0.710655	+	0.703541i	$0.751601\pi$
$230$	0	0
$231$	−7.72237	−0.508095
$232$	0	0
$233$	6.55691	0.429558	0.214779	−	0.976663i	$-0.431097\pi$
$233$	6.55691	0.429558	0.214779	+	0.976663i	$0.431097\pi$
$234$	0	0
$235$	−14.0120	−0.914042
$236$	0	0
$237$	−2.45149	−0.159241
$238$	0	0
$239$	−4.28033	−0.276871	−0.138436	−	0.990371i	$-0.544207\pi$
$239$	−4.28033	−0.276871	−0.138436	+	0.990371i	$0.544207\pi$
$240$	0	0
$241$	−10.2523	−0.660409	−0.330205	−	0.943909i	$-0.607118\pi$
$241$	−10.2523	−0.660409	−0.330205	+	0.943909i	$0.607118\pi$
$242$	0	0
$243$	22.0631	1.41535
$244$	0	0
$245$	−9.01487	−0.575939
$246$	0	0
$247$	0.170649	0.0108581
$248$	0	0
$249$	18.7656	1.18922
$250$	0	0
$251$	−19.4433	−1.22725	−0.613627	−	0.789596i	$-0.710290\pi$
$251$	−19.4433	−1.22725	−0.613627	+	0.789596i	$0.710290\pi$
$252$	0	0
$253$	−26.2563	−1.65072
$254$	0	0
$255$	−11.5000	−0.720161
$256$	0	0
$257$	−13.8906	−0.866469	−0.433234	−	0.901281i	$-0.642628\pi$
$257$	−13.8906	−0.866469	−0.433234	+	0.901281i	$0.642628\pi$
$258$	0	0
$259$	2.48851	0.154628
$260$	0	0
$261$	−17.6111	−1.09010
$262$	0	0
$263$	−26.4582	−1.63148	−0.815742	−	0.578415i	$-0.803671\pi$
$263$	−26.4582	−1.63148	−0.815742	+	0.578415i	$0.803671\pi$
$264$	0	0
$265$	−18.2807	−1.12298
$266$	0	0
$267$	5.35394	0.327656
$268$	0	0
$269$	9.16047	0.558524	0.279262	−	0.960215i	$-0.409910\pi$
$269$	9.16047	0.558524	0.279262	+	0.960215i	$0.409910\pi$
$270$	0	0
$271$	−28.7062	−1.74377	−0.871887	−	0.489707i	$-0.837104\pi$
$271$	−28.7062	−1.74377	−0.871887	+	0.489707i	$0.837104\pi$
$272$	0	0
$273$	0.440416	0.0266552
$274$	0	0
$275$	−7.95569	−0.479746
$276$	0	0
$277$	−12.2488	−0.735957	−0.367979	−	0.929834i	$-0.619950\pi$
$277$	−12.2488	−0.735957	−0.367979	+	0.929834i	$0.619950\pi$
$278$	0	0
$279$	28.8845	1.72927
$280$	0	0
$281$	1.98494	0.118412	0.0592058	−	0.998246i	$-0.481143\pi$
$281$	1.98494	0.118412	0.0592058	+	0.998246i	$0.481143\pi$
$282$	0	0
$283$	9.18340	0.545896	0.272948	−	0.962029i	$-0.412001\pi$
$283$	9.18340	0.545896	0.272948	+	0.962029i	$0.412001\pi$
$284$	0	0
$285$	3.75101	0.222191
$286$	0	0
$287$	−6.44401	−0.380378
$288$	0	0
$289$	−7.60056	−0.447092
$290$	0	0
$291$	−24.7499	−1.45087
$292$	0	0
$293$	−8.67335	−0.506703	−0.253351	−	0.967374i	$-0.581533\pi$
$293$	−8.67335	−0.506703	−0.253351	+	0.967374i	$0.581533\pi$
$294$	0	0
$295$	−6.74793	−0.392880
$296$	0	0
$297$	−0.0717927	−0.00416583
$298$	0	0
$299$	1.49743	0.0865985
$300$	0	0
$301$	4.76810	0.274829
$302$	0	0
$303$	24.6858	1.41816
$304$	0	0
$305$	−15.0744	−0.863156
$306$	0	0
$307$	31.7741	1.81344	0.906722	−	0.421730i	$-0.138577\pi$
$307$	31.7741	1.81344	0.906722	+	0.421730i	$0.138577\pi$
$308$	0	0
$309$	−12.7062	−0.722831
$310$	0	0
$311$	10.0872	0.571994	0.285997	−	0.958230i	$-0.407675\pi$
$311$	10.0872	0.571994	0.285997	+	0.958230i	$0.407675\pi$
$312$	0	0
$313$	21.0295	1.18866	0.594329	−	0.804222i	$-0.297418\pi$
$313$	21.0295	1.18866	0.594329	+	0.804222i	$0.297418\pi$
$314$	0	0
$315$	4.84824	0.273167
$316$	0	0
$317$	−14.6293	−0.821664	−0.410832	−	0.911711i	$-0.634762\pi$
$317$	−14.6293	−0.821664	−0.410832	+	0.911711i	$0.634762\pi$
$318$	0	0
$319$	−17.5083	−0.980275
$320$	0	0
$321$	−21.7017	−1.21127
$322$	0	0
$323$	−3.06585	−0.170588
$324$	0	0
$325$	0.453722	0.0251680
$326$	0	0
$327$	10.0584	0.556228
$328$	0	0
$329$	−9.64074	−0.531511
$330$	0	0
$331$	−15.1451	−0.832450	−0.416225	−	0.909262i	$-0.636647\pi$
$331$	−15.1451	−0.832450	−0.416225	+	0.909262i	$0.636647\pi$
$332$	0	0
$333$	7.11452	0.389873
$334$	0	0
$335$	7.72868	0.422263
$336$	0	0
$337$	3.88927	0.211862	0.105931	−	0.994373i	$-0.466218\pi$
$337$	3.88927	0.211862	0.105931	+	0.994373i	$0.466218\pi$
$338$	0	0
$339$	−42.4863	−2.30754
$340$	0	0
$341$	28.7158	1.55505
$342$	0	0
$343$	−13.5719	−0.732812
$344$	0	0
$345$	32.9148	1.77207
$346$	0	0
$347$	23.1817	1.24446	0.622230	−	0.782835i	$-0.286227\pi$
$347$	23.1817	1.24446	0.622230	+	0.782835i	$0.286227\pi$
$348$	0	0
$349$	−10.0579	−0.538385	−0.269193	−	0.963086i	$-0.586757\pi$
$349$	−10.0579	−0.538385	−0.269193	+	0.963086i	$0.586757\pi$
$350$	0	0
$351$	0.00409442	0.000218544 0
$352$	0	0
$353$	3.25825	0.173419	0.0867096	−	0.996234i	$-0.472365\pi$
$353$	3.25825	0.173419	0.0867096	+	0.996234i	$0.472365\pi$
$354$	0	0
$355$	4.44009	0.235656
$356$	0	0
$357$	−7.91243	−0.418770
$358$	0	0
$359$	−10.0836	−0.532190	−0.266095	−	0.963947i	$-0.585733\pi$
$359$	−10.0836	−0.532190	−0.266095	+	0.963947i	$0.585733\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.01746	0.263348
$364$	0	0
$365$	3.56258	0.186474
$366$	0	0
$367$	−10.8894	−0.568421	−0.284211	−	0.958762i	$-0.591732\pi$
$367$	−10.8894	−0.568421	−0.284211	+	0.958762i	$0.591732\pi$
$368$	0	0
$369$	−18.4231	−0.959068
$370$	0	0
$371$	−12.5778	−0.653006
$372$	0	0
$373$	6.22064	0.322093	0.161046	−	0.986947i	$-0.448513\pi$
$373$	6.22064	0.322093	0.161046	+	0.986947i	$0.448513\pi$
$374$	0	0
$375$	28.7283	1.48352
$376$	0	0
$377$	0.998517	0.0514263
$378$	0	0
$379$	0.0558057	0.00286655	0.00143327	−	0.999999i	$-0.499544\pi$
$379$	0.0558057	0.00286655	0.00143327	+	0.999999i	$0.499544\pi$
$380$	0	0
$381$	−18.2536	−0.935160
$382$	0	0
$383$	−19.8946	−1.01657	−0.508284	−	0.861189i	$-0.669720\pi$
$383$	−19.8946	−1.01657	−0.508284	+	0.861189i	$0.669720\pi$
$384$	0	0
$385$	4.81992	0.245646
$386$	0	0
$387$	13.6318	0.692941
$388$	0	0
$389$	12.1457	0.615809	0.307905	−	0.951417i	$-0.400372\pi$
$389$	12.1457	0.615809	0.307905	+	0.951417i	$0.400372\pi$
$390$	0	0
$391$	−26.9025	−1.36052
$392$	0	0
$393$	14.8041	0.746767
$394$	0	0
$395$	1.53010	0.0769875
$396$	0	0
$397$	7.37561	0.370171	0.185086	−	0.982722i	$-0.440744\pi$
$397$	7.37561	0.370171	0.185086	+	0.982722i	$0.440744\pi$
$398$	0	0
$399$	2.58083	0.129203
$400$	0	0
$401$	6.88458	0.343800	0.171900	−	0.985114i	$-0.445009\pi$
$401$	6.88458	0.343800	0.171900	+	0.985114i	$0.445009\pi$
$402$	0	0
$403$	−1.63769	−0.0815793
$404$	0	0
$405$	−13.7258	−0.682040
$406$	0	0
$407$	7.07297	0.350594
$408$	0	0
$409$	−8.68156	−0.429275	−0.214638	−	0.976694i	$-0.568857\pi$
$409$	−8.68156	−0.429275	−0.214638	+	0.976694i	$0.568857\pi$
$410$	0	0
$411$	4.55834	0.224846
$412$	0	0
$413$	−4.64281	−0.228458
$414$	0	0
$415$	−11.7125	−0.574946
$416$	0	0
$417$	−28.1403	−1.37803
$418$	0	0
$419$	−9.96019	−0.486587	−0.243294	−	0.969953i	$-0.578228\pi$
$419$	−9.96019	−0.486587	−0.243294	+	0.969953i	$0.578228\pi$
$420$	0	0
$421$	−32.9277	−1.60480	−0.802399	−	0.596788i	$-0.796443\pi$
$421$	−32.9277	−1.60480	−0.802399	+	0.596788i	$0.796443\pi$
$422$	0	0
$423$	−27.5624	−1.34013
$424$	0	0
$425$	−8.15149	−0.395405
$426$	0	0
$427$	−10.3717	−0.501921
$428$	0	0
$429$	1.25177	0.0604361
$430$	0	0
$431$	−25.5868	−1.23247	−0.616236	−	0.787561i	$-0.711343\pi$
$431$	−25.5868	−1.23247	−0.616236	+	0.787561i	$0.711343\pi$
$432$	0	0
$433$	13.7364	0.660132	0.330066	−	0.943958i	$-0.392929\pi$
$433$	13.7364	0.660132	0.330066	+	0.943958i	$0.392929\pi$
$434$	0	0
$435$	21.9483	1.05234
$436$	0	0
$437$	8.77490	0.419761
$438$	0	0
$439$	−15.2116	−0.726008	−0.363004	−	0.931787i	$-0.618249\pi$
$439$	−15.2116	−0.726008	−0.363004	+	0.931787i	$0.618249\pi$
$440$	0	0
$441$	−17.7328	−0.844417
$442$	0	0
$443$	−9.05741	−0.430331	−0.215165	−	0.976578i	$-0.569029\pi$
$443$	−9.05741	−0.430331	−0.215165	+	0.976578i	$0.569029\pi$
$444$	0	0
$445$	−3.34166	−0.158410
$446$	0	0
$447$	−50.1926	−2.37403
$448$	0	0
$449$	−14.0375	−0.662468	−0.331234	−	0.943549i	$-0.607465\pi$
$449$	−14.0375	−0.662468	−0.331234	+	0.943549i	$0.607465\pi$
$450$	0	0
$451$	−18.3155	−0.862442
$452$	0	0
$453$	26.9723	1.26727
$454$	0	0
$455$	−0.274886	−0.0128868
$456$	0	0
$457$	−19.1673	−0.896608	−0.448304	−	0.893881i	$-0.647972\pi$
$457$	−19.1673	−0.896608	−0.448304	+	0.893881i	$0.647972\pi$
$458$	0	0
$459$	−0.0735596	−0.00343347
$460$	0	0
$461$	−29.1555	−1.35791	−0.678953	−	0.734182i	$-0.737566\pi$
$461$	−29.1555	−1.35791	−0.678953	+	0.734182i	$0.737566\pi$
$462$	0	0
$463$	18.4511	0.857497	0.428748	−	0.903424i	$-0.358955\pi$
$463$	18.4511	0.857497	0.428748	+	0.903424i	$0.358955\pi$
$464$	0	0
$465$	−35.9979	−1.66936
$466$	0	0
$467$	−10.4736	−0.484661	−0.242331	−	0.970194i	$-0.577912\pi$
$467$	−10.4736	−0.484661	−0.242331	+	0.970194i	$0.577912\pi$
$468$	0	0
$469$	5.31760	0.245544
$470$	0	0
$471$	53.5623	2.46802
$472$	0	0
$473$	13.5521	0.623128
$474$	0	0
$475$	2.65880	0.121994
$476$	0	0
$477$	−35.9592	−1.64646
$478$	0	0
$479$	−23.9382	−1.09376	−0.546882	−	0.837210i	$-0.684185\pi$
$479$	−23.9382	−1.09376	−0.546882	+	0.837210i	$0.684185\pi$
$480$	0	0
$481$	−0.403379	−0.0183925
$482$	0	0
$483$	22.6465	1.03045
$484$	0	0
$485$	15.4477	0.701444
$486$	0	0
$487$	5.00357	0.226734	0.113367	−	0.993553i	$-0.463836\pi$
$487$	5.00357	0.226734	0.113367	+	0.993553i	$0.463836\pi$
$488$	0	0
$489$	−1.68232	−0.0760770
$490$	0	0
$491$	23.3396	1.05330	0.526650	−	0.850082i	$-0.323448\pi$
$491$	23.3396	1.05330	0.526650	+	0.850082i	$0.323448\pi$
$492$	0	0
$493$	−17.9392	−0.807940
$494$	0	0
$495$	13.7799	0.619361
$496$	0	0
$497$	3.05494	0.137033
$498$	0	0
$499$	2.27823	0.101988	0.0509938	−	0.998699i	$-0.483761\pi$
$499$	2.27823	0.101988	0.0509938	+	0.998699i	$0.483761\pi$
$500$	0	0
$501$	20.9780	0.937228
$502$	0	0
$503$	−12.5695	−0.560448	−0.280224	−	0.959935i	$-0.590409\pi$
$503$	−12.5695	−0.560448	−0.280224	+	0.959935i	$0.590409\pi$
$504$	0	0
$505$	−15.4076	−0.685631
$506$	0	0
$507$	31.7979	1.41220
$508$	0	0
$509$	21.2526	0.942005	0.471003	−	0.882132i	$-0.343892\pi$
$509$	21.2526	0.942005	0.471003	+	0.882132i	$0.343892\pi$
$510$	0	0
$511$	2.45118	0.108434
$512$	0	0
$513$	0.0239932	0.00105933
$514$	0	0
$515$	7.93059	0.349464
$516$	0	0
$517$	−27.4014	−1.20511
$518$	0	0
$519$	−19.4139	−0.852175
$520$	0	0
$521$	−17.1614	−0.751856	−0.375928	−	0.926649i	$-0.622676\pi$
$521$	−17.1614	−0.751856	−0.375928	+	0.926649i	$0.622676\pi$
$522$	0	0
$523$	44.0605	1.92663	0.963316	−	0.268369i	$-0.0864846\pi$
$523$	44.0605	1.92663	0.963316	+	0.268369i	$0.0864846\pi$
$524$	0	0
$525$	6.86191	0.299478
$526$	0	0
$527$	29.4225	1.28166
$528$	0	0
$529$	53.9989	2.34778
$530$	0	0
$531$	−13.2736	−0.576023
$532$	0	0
$533$	1.04455	0.0452446
$534$	0	0
$535$	13.5451	0.585606
$536$	0	0
$537$	−39.3868	−1.69967
$538$	0	0
$539$	−17.6292	−0.759342
$540$	0	0
$541$	−23.5101	−1.01078	−0.505388	−	0.862892i	$-0.668651\pi$
$541$	−23.5101	−1.01078	−0.505388	+	0.862892i	$0.668651\pi$
$542$	0	0
$543$	−20.1062	−0.862840
$544$	0	0
$545$	−6.27793	−0.268917
$546$	0	0
$547$	34.3536	1.46885	0.734427	−	0.678688i	$-0.237451\pi$
$547$	34.3536	1.46885	0.734427	+	0.678688i	$0.237451\pi$
$548$	0	0
$549$	−29.6521	−1.26552
$550$	0	0
$551$	5.85129	0.249273
$552$	0	0
$553$	1.05276	0.0447679
$554$	0	0
$555$	−8.86663	−0.376367
$556$	0	0
$557$	21.9411	0.929675	0.464838	−	0.885396i	$-0.346113\pi$
$557$	21.9411	0.929675	0.464838	+	0.885396i	$0.346113\pi$
$558$	0	0
$559$	−0.772894	−0.0326899
$560$	0	0
$561$	−22.4891	−0.949491
$562$	0	0
$563$	−10.0644	−0.424166	−0.212083	−	0.977252i	$-0.568025\pi$
$563$	−10.0644	−0.424166	−0.212083	+	0.977252i	$0.568025\pi$
$564$	0	0
$565$	26.5178	1.11561
$566$	0	0
$567$	−9.44383	−0.396604
$568$	0	0
$569$	−44.7230	−1.87489	−0.937443	−	0.348140i	$-0.886813\pi$
$569$	−44.7230	−1.87489	−0.937443	+	0.348140i	$0.886813\pi$
$570$	0	0
$571$	−14.3586	−0.600888	−0.300444	−	0.953799i	$-0.597135\pi$
$571$	−14.3586	−0.600888	−0.300444	+	0.953799i	$0.597135\pi$
$572$	0	0
$573$	−32.0096	−1.33722
$574$	0	0
$575$	23.3307	0.972959
$576$	0	0
$577$	−20.8255	−0.866979	−0.433489	−	0.901159i	$-0.642718\pi$
$577$	−20.8255	−0.866979	−0.433489	+	0.901159i	$0.642718\pi$
$578$	0	0
$579$	14.0210	0.582691
$580$	0	0
$581$	−8.05864	−0.334329
$582$	0	0
$583$	−35.7492	−1.48058
$584$	0	0
$585$	−0.785884	−0.0324923
$586$	0	0
$587$	11.5042	0.474829	0.237414	−	0.971408i	$-0.423700\pi$
$587$	11.5042	0.474829	0.237414	+	0.971408i	$0.423700\pi$
$588$	0	0
$589$	−9.59685	−0.395431
$590$	0	0
$591$	8.49723	0.349529
$592$	0	0
$593$	14.9113	0.612333	0.306166	−	0.951978i	$-0.400954\pi$
$593$	14.9113	0.612333	0.306166	+	0.951978i	$0.400954\pi$
$594$	0	0
$595$	4.93855	0.202461
$596$	0	0
$597$	−19.4913	−0.797724
$598$	0	0
$599$	−44.4656	−1.81682	−0.908408	−	0.418085i	$-0.862701\pi$
$599$	−44.4656	−1.81682	−0.908408	+	0.418085i	$0.862701\pi$
$600$	0	0
$601$	−6.94269	−0.283198	−0.141599	−	0.989924i	$-0.545224\pi$
$601$	−6.94269	−0.283198	−0.141599	+	0.989924i	$0.545224\pi$
$602$	0	0
$603$	15.2027	0.619104
$604$	0	0
$605$	−3.13165	−0.127320
$606$	0	0
$607$	5.56369	0.225823	0.112912	−	0.993605i	$-0.463982\pi$
$607$	5.56369	0.225823	0.112912	+	0.993605i	$0.463982\pi$
$608$	0	0
$609$	15.1012	0.611931
$610$	0	0
$611$	1.56273	0.0632214
$612$	0	0
$613$	24.8679	1.00441	0.502203	−	0.864750i	$-0.332523\pi$
$613$	24.8679	1.00441	0.502203	+	0.864750i	$0.332523\pi$
$614$	0	0
$615$	22.9602	0.925844
$616$	0	0
$617$	−16.2435	−0.653939	−0.326969	−	0.945035i	$-0.606027\pi$
$617$	−16.2435	−0.653939	−0.326969	+	0.945035i	$0.606027\pi$
$618$	0	0
$619$	40.6132	1.63238	0.816191	−	0.577782i	$-0.196082\pi$
$619$	40.6132	1.63238	0.816191	+	0.577782i	$0.196082\pi$
$620$	0	0
$621$	0.210538	0.00844861
$622$	0	0
$623$	−2.29918	−0.0921147
$624$	0	0
$625$	−4.63675	−0.185470
$626$	0	0
$627$	7.33536	0.292946
$628$	0	0
$629$	7.24705	0.288959
$630$	0	0
$631$	13.8944	0.553128	0.276564	−	0.960996i	$-0.410804\pi$
$631$	13.8944	0.553128	0.276564	+	0.960996i	$0.410804\pi$
$632$	0	0
$633$	23.2579	0.924419
$634$	0	0
$635$	11.3930	0.452117
$636$	0	0
$637$	1.00541	0.0398359
$638$	0	0
$639$	8.73391	0.345508
$640$	0	0
$641$	−14.1486	−0.558836	−0.279418	−	0.960170i	$-0.590142\pi$
$641$	−14.1486	−0.558836	−0.279418	+	0.960170i	$0.590142\pi$
$642$	0	0
$643$	11.0210	0.434628	0.217314	−	0.976102i	$-0.430270\pi$
$643$	11.0210	0.434628	0.217314	+	0.976102i	$0.430270\pi$
$644$	0	0
$645$	−16.9889	−0.668936
$646$	0	0
$647$	−9.98052	−0.392375	−0.196187	−	0.980566i	$-0.562856\pi$
$647$	−9.98052	−0.392375	−0.196187	+	0.980566i	$0.562856\pi$
$648$	0	0
$649$	−13.1960	−0.517989
$650$	0	0
$651$	−24.7678	−0.970727
$652$	0	0
$653$	25.3847	0.993381	0.496690	−	0.867928i	$-0.334548\pi$
$653$	25.3847	0.993381	0.496690	+	0.867928i	$0.334548\pi$
$654$	0	0
$655$	−9.23998	−0.361036
$656$	0	0
$657$	7.00780	0.273400
$658$	0	0
$659$	31.9915	1.24621	0.623106	−	0.782137i	$-0.285870\pi$
$659$	31.9915	1.24621	0.623106	+	0.782137i	$0.285870\pi$
$660$	0	0
$661$	2.73242	0.106279	0.0531394	−	0.998587i	$-0.483077\pi$
$661$	2.73242	0.106279	0.0531394	+	0.998587i	$0.483077\pi$
$662$	0	0
$663$	1.28258	0.0498113
$664$	0	0
$665$	−1.61082	−0.0624651
$666$	0	0
$667$	51.3445	1.98807
$668$	0	0
$669$	31.9840	1.23657
$670$	0	0
$671$	−29.4789	−1.13802
$672$	0	0
$673$	−43.3739	−1.67194	−0.835970	−	0.548775i	$-0.815094\pi$
$673$	−43.3739	−1.67194	−0.835970	+	0.548775i	$0.815094\pi$
$674$	0	0
$675$	0.0637933	0.00245540
$676$	0	0
$677$	−2.01124	−0.0772984	−0.0386492	−	0.999253i	$-0.512305\pi$
$677$	−2.01124	−0.0772984	−0.0386492	+	0.999253i	$0.512305\pi$
$678$	0	0
$679$	10.6286	0.407886
$680$	0	0
$681$	13.6797	0.524209
$682$	0	0
$683$	−46.6718	−1.78585	−0.892923	−	0.450210i	$-0.851349\pi$
$683$	−46.6718	−1.78585	−0.892923	+	0.450210i	$0.851349\pi$
$684$	0	0
$685$	−2.84509	−0.108705
$686$	0	0
$687$	52.7274	2.01167
$688$	0	0
$689$	2.03882	0.0776727
$690$	0	0
$691$	−27.3380	−1.03999	−0.519993	−	0.854170i	$-0.674066\pi$
$691$	−27.3380	−1.03999	−0.519993	+	0.854170i	$0.674066\pi$
$692$	0	0
$693$	9.48106	0.360156
$694$	0	0
$695$	17.5638	0.666231
$696$	0	0
$697$	−18.7663	−0.710823
$698$	0	0
$699$	−16.0742	−0.607982
$700$	0	0
$701$	−11.3543	−0.428848	−0.214424	−	0.976741i	$-0.568787\pi$
$701$	−11.3543	−0.428848	−0.214424	+	0.976741i	$0.568787\pi$
$702$	0	0
$703$	−2.36380	−0.0891523
$704$	0	0
$705$	34.3502	1.29370
$706$	0	0
$707$	−10.6010	−0.398692
$708$	0	0
$709$	12.4659	0.468168	0.234084	−	0.972216i	$-0.424791\pi$
$709$	12.4659	0.468168	0.234084	+	0.972216i	$0.424791\pi$
$710$	0	0
$711$	3.00979	0.112876
$712$	0	0
$713$	−84.2114	−3.15374
$714$	0	0
$715$	−0.781294	−0.0292187
$716$	0	0
$717$	10.4932	0.391874
$718$	0	0
$719$	22.0003	0.820474	0.410237	−	0.911979i	$-0.365446\pi$
$719$	22.0003	0.820474	0.410237	+	0.911979i	$0.365446\pi$
$720$	0	0
$721$	5.45652	0.203212
$722$	0	0
$723$	25.1334	0.934721
$724$	0	0
$725$	15.5574	0.577789
$726$	0	0
$727$	5.04635	0.187159	0.0935794	−	0.995612i	$-0.470169\pi$
$727$	5.04635	0.187159	0.0935794	+	0.995612i	$0.470169\pi$
$728$	0	0
$729$	−27.1759	−1.00651
$730$	0	0
$731$	13.8857	0.513580
$732$	0	0
$733$	−18.1001	−0.668543	−0.334271	−	0.942477i	$-0.608490\pi$
$733$	−18.1001	−0.668543	−0.334271	+	0.942477i	$0.608490\pi$
$734$	0	0
$735$	22.0998	0.815164
$736$	0	0
$737$	15.1139	0.556729
$738$	0	0
$739$	−3.26351	−0.120050	−0.0600250	−	0.998197i	$-0.519118\pi$
$739$	−3.26351	−0.120050	−0.0600250	+	0.998197i	$0.519118\pi$
$740$	0	0
$741$	−0.418344	−0.0153682
$742$	0	0
$743$	−16.5828	−0.608365	−0.304182	−	0.952614i	$-0.598383\pi$
$743$	−16.5828	−0.608365	−0.304182	+	0.952614i	$0.598383\pi$
$744$	0	0
$745$	31.3277	1.14776
$746$	0	0
$747$	−23.0392	−0.842962
$748$	0	0
$749$	9.31950	0.340527
$750$	0	0
$751$	5.92910	0.216356	0.108178	−	0.994132i	$-0.465498\pi$
$751$	5.92910	0.216356	0.108178	+	0.994132i	$0.465498\pi$
$752$	0	0
$753$	47.6651	1.73701
$754$	0	0
$755$	−16.8348	−0.612681
$756$	0	0
$757$	−10.0960	−0.366946	−0.183473	−	0.983025i	$-0.558734\pi$
$757$	−10.0960	−0.366946	−0.183473	+	0.983025i	$0.558734\pi$
$758$	0	0
$759$	64.3670	2.33638
$760$	0	0
$761$	−46.3578	−1.68047	−0.840234	−	0.542224i	$-0.817582\pi$
$761$	−46.3578	−1.68047	−0.840234	+	0.542224i	$0.817582\pi$
$762$	0	0
$763$	−4.31943	−0.156374
$764$	0	0
$765$	14.1191	0.510476
$766$	0	0
$767$	0.752585	0.0271743
$768$	0	0
$769$	25.6714	0.925735	0.462867	−	0.886428i	$-0.346821\pi$
$769$	25.6714	0.925735	0.462867	+	0.886428i	$0.346821\pi$
$770$	0	0
$771$	34.0525	1.22637
$772$	0	0
$773$	−52.8294	−1.90014	−0.950071	−	0.312033i	$-0.898990\pi$
$773$	−52.8294	−1.90014	−0.950071	+	0.312033i	$0.898990\pi$
$774$	0	0
$775$	−25.5161	−0.916566
$776$	0	0
$777$	−6.10055	−0.218856
$778$	0	0
$779$	6.12106	0.219310
$780$	0	0
$781$	8.68289	0.310698
$782$	0	0
$783$	0.140391	0.00501718
$784$	0	0
$785$	−33.4309	−1.19320
$786$	0	0
$787$	40.1417	1.43090	0.715448	−	0.698666i	$-0.246223\pi$
$787$	40.1417	1.43090	0.715448	+	0.698666i	$0.246223\pi$
$788$	0	0
$789$	64.8620	2.30915
$790$	0	0
$791$	18.2452	0.648724
$792$	0	0
$793$	1.68122	0.0597018
$794$	0	0
$795$	44.8150	1.58942
$796$	0	0
$797$	−3.59829	−0.127458	−0.0637289	−	0.997967i	$-0.520299\pi$
$797$	−3.59829	−0.127458	−0.0637289	+	0.997967i	$0.520299\pi$
$798$	0	0
$799$	−28.0758	−0.993250
$800$	0	0
$801$	−6.57324	−0.232254
$802$	0	0
$803$	6.96686	0.245855
$804$	0	0
$805$	−14.1348	−0.498187
$806$	0	0
$807$	−22.4568	−0.790516
$808$	0	0
$809$	12.4249	0.436835	0.218417	−	0.975855i	$-0.429911\pi$
$809$	12.4249	0.436835	0.218417	+	0.975855i	$0.429911\pi$
$810$	0	0
$811$	27.8743	0.978801	0.489400	−	0.872059i	$-0.337216\pi$
$811$	27.8743	0.978801	0.489400	+	0.872059i	$0.337216\pi$
$812$	0	0
$813$	70.3727	2.46808
$814$	0	0
$815$	1.05002	0.0367806
$816$	0	0
$817$	−4.52915	−0.158455
$818$	0	0
$819$	−0.540716	−0.0188941
$820$	0	0
$821$	−15.3921	−0.537188	−0.268594	−	0.963253i	$-0.586559\pi$
$821$	−15.3921	−0.537188	−0.268594	+	0.963253i	$0.586559\pi$
$822$	0	0
$823$	33.7827	1.17759	0.588795	−	0.808282i	$-0.299602\pi$
$823$	33.7827	1.17759	0.588795	+	0.808282i	$0.299602\pi$
$824$	0	0
$825$	19.5033	0.679017
$826$	0	0
$827$	−5.93645	−0.206431	−0.103215	−	0.994659i	$-0.532913\pi$
$827$	−5.93645	−0.206431	−0.103215	+	0.994659i	$0.532913\pi$
$828$	0	0
$829$	33.8632	1.17612	0.588059	−	0.808818i	$-0.299892\pi$
$829$	33.8632	1.17612	0.588059	+	0.808818i	$0.299892\pi$
$830$	0	0
$831$	30.0277	1.04165
$832$	0	0
$833$	−18.0631	−0.625848
$834$	0	0
$835$	−13.0934	−0.453117
$836$	0	0
$837$	−0.230259	−0.00795893
$838$	0	0
$839$	51.6439	1.78295	0.891473	−	0.453075i	$-0.149673\pi$
$839$	51.6439	1.78295	0.891473	+	0.453075i	$0.149673\pi$
$840$	0	0
$841$	5.23763	0.180608
$842$	0	0
$843$	−4.86605	−0.167596
$844$	0	0
$845$	−19.8467	−0.682747
$846$	0	0
$847$	−2.15468	−0.0740358
$848$	0	0
$849$	−22.5130	−0.772643
$850$	0	0
$851$	−20.7421	−0.711029
$852$	0	0
$853$	4.54858	0.155741	0.0778703	−	0.996964i	$-0.475188\pi$
$853$	4.54858	0.155741	0.0778703	+	0.996964i	$0.475188\pi$
$854$	0	0
$855$	−4.60527	−0.157497
$856$	0	0
$857$	−33.2044	−1.13424	−0.567121	−	0.823635i	$-0.691943\pi$
$857$	−33.2044	−1.13424	−0.567121	+	0.823635i	$0.691943\pi$
$858$	0	0
$859$	−38.6071	−1.31726	−0.658629	−	0.752468i	$-0.728863\pi$
$859$	−38.6071	−1.31726	−0.658629	+	0.752468i	$0.728863\pi$
$860$	0	0
$861$	15.7974	0.538374
$862$	0	0
$863$	−41.7167	−1.42005	−0.710026	−	0.704175i	$-0.751317\pi$
$863$	−41.7167	−1.42005	−0.710026	+	0.704175i	$0.751317\pi$
$864$	0	0
$865$	12.1172	0.411997
$866$	0	0
$867$	18.6327	0.632799
$868$	0	0
$869$	2.99221	0.101504
$870$	0	0
$871$	−0.861966	−0.0292066
$872$	0	0
$873$	30.3865	1.02843
$874$	0	0
$875$	−12.3370	−0.417066
$876$	0	0
$877$	28.9800	0.978584	0.489292	−	0.872120i	$-0.337255\pi$
$877$	28.9800	0.978584	0.489292	+	0.872120i	$0.337255\pi$
$878$	0	0
$879$	21.2626	0.717170
$880$	0	0
$881$	0.745168	0.0251054	0.0125527	−	0.999921i	$-0.496004\pi$
$881$	0.745168	0.0251054	0.0125527	+	0.999921i	$0.496004\pi$
$882$	0	0
$883$	−37.5981	−1.26528	−0.632638	−	0.774447i	$-0.718028\pi$
$883$	−37.5981	−1.26528	−0.632638	+	0.774447i	$0.718028\pi$
$884$	0	0
$885$	16.5425	0.556069
$886$	0	0
$887$	50.9578	1.71100	0.855498	−	0.517806i	$-0.173251\pi$
$887$	50.9578	1.71100	0.855498	+	0.517806i	$0.173251\pi$
$888$	0	0
$889$	7.83877	0.262904
$890$	0	0
$891$	−26.8417	−0.899231
$892$	0	0
$893$	9.15759	0.306447
$894$	0	0
$895$	24.5833	0.821729
$896$	0	0
$897$	−3.67093	−0.122569
$898$	0	0
$899$	−56.1540	−1.87284
$900$	0	0
$901$	−36.6290	−1.22029
$902$	0	0
$903$	−11.6889	−0.388984
$904$	0	0
$905$	12.5493	0.417153
$906$	0	0
$907$	36.0506	1.19704	0.598520	−	0.801108i	$-0.295756\pi$
$907$	36.0506	1.19704	0.598520	+	0.801108i	$0.295756\pi$
$908$	0	0
$909$	−30.3077	−1.00524
$910$	0	0
$911$	−36.2427	−1.20077	−0.600387	−	0.799710i	$-0.704987\pi$
$911$	−36.2427	−1.20077	−0.600387	+	0.799710i	$0.704987\pi$
$912$	0	0
$913$	−22.9047	−0.758034
$914$	0	0
$915$	36.9546	1.22168
$916$	0	0
$917$	−6.35743	−0.209941
$918$	0	0
$919$	−14.4787	−0.477610	−0.238805	−	0.971068i	$-0.576756\pi$
$919$	−14.4787	−0.477610	−0.238805	+	0.971068i	$0.576756\pi$
$920$	0	0
$921$	−77.8938	−2.56669
$922$	0	0
$923$	−0.495195	−0.0162996
$924$	0	0
$925$	−6.28487	−0.206645
$926$	0	0
$927$	15.5999	0.512369
$928$	0	0
$929$	−32.0622	−1.05193	−0.525964	−	0.850507i	$-0.676295\pi$
$929$	−32.0622	−1.05193	−0.525964	+	0.850507i	$0.676295\pi$
$930$	0	0
$931$	5.89170	0.193093
$932$	0	0
$933$	−24.7287	−0.809582
$934$	0	0
$935$	14.0366	0.459045
$936$	0	0
$937$	11.9704	0.391056	0.195528	−	0.980698i	$-0.437358\pi$
$937$	11.9704	0.391056	0.195528	+	0.980698i	$0.437358\pi$
$938$	0	0
$939$	−51.5536	−1.68239
$940$	0	0
$941$	50.6611	1.65151	0.825753	−	0.564032i	$-0.190750\pi$
$941$	50.6611	1.65151	0.825753	+	0.564032i	$0.190750\pi$
$942$	0	0
$943$	53.7117	1.74909
$944$	0	0
$945$	−0.0386489	−0.00125725
$946$	0	0
$947$	35.6161	1.15737	0.578684	−	0.815552i	$-0.303566\pi$
$947$	35.6161	1.15737	0.578684	+	0.815552i	$0.303566\pi$
$948$	0	0
$949$	−0.397328	−0.0128978
$950$	0	0
$951$	35.8635	1.16296
$952$	0	0
$953$	−42.0325	−1.36156	−0.680782	−	0.732486i	$-0.738360\pi$
$953$	−42.0325	−1.36156	−0.680782	+	0.732486i	$0.738360\pi$
$954$	0	0
$955$	19.9788	0.646499
$956$	0	0
$957$	42.9213	1.38745
$958$	0	0
$959$	−1.95752	−0.0632116
$960$	0	0
$961$	61.0995	1.97095
$962$	0	0
$963$	26.6440	0.858590
$964$	0	0
$965$	−8.75119	−0.281711
$966$	0	0
$967$	−24.6132	−0.791507	−0.395753	−	0.918357i	$-0.629517\pi$
$967$	−24.6132	−0.791507	−0.395753	+	0.918357i	$0.629517\pi$
$968$	0	0
$969$	7.51589	0.241445
$970$	0	0
$971$	−49.7410	−1.59626	−0.798132	−	0.602482i	$-0.794178\pi$
$971$	−49.7410	−1.59626	−0.798132	+	0.602482i	$0.794178\pi$
$972$	0	0
$973$	12.0845	0.387411
$974$	0	0
$975$	−1.11229	−0.0356219
$976$	0	0
$977$	23.0288	0.736757	0.368378	−	0.929676i	$-0.379913\pi$
$977$	23.0288	0.736757	0.368378	+	0.929676i	$0.379913\pi$
$978$	0	0
$979$	−6.53485	−0.208855
$980$	0	0
$981$	−12.3490	−0.394274
$982$	0	0
$983$	12.2107	0.389461	0.194730	−	0.980857i	$-0.437617\pi$
$983$	12.2107	0.389461	0.194730	+	0.980857i	$0.437617\pi$
$984$	0	0
$985$	−5.30355	−0.168985
$986$	0	0
$987$	23.6341	0.752283
$988$	0	0
$989$	−39.7428	−1.26375
$990$	0	0
$991$	28.4066	0.902365	0.451183	−	0.892432i	$-0.351002\pi$
$991$	28.4066	0.902365	0.451183	+	0.892432i	$0.351002\pi$
$992$	0	0
$993$	37.1280	1.17822
$994$	0	0
$995$	12.1655	0.385672
$996$	0	0
$997$	59.2545	1.87661	0.938305	−	0.345809i	$-0.112396\pi$
$997$	59.2545	1.87661	0.938305	+	0.345809i	$0.112396\pi$
$998$	0	0
$999$	−0.0567151	−0.00179439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.f.1.3	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.f.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-2.45149 q^{3} +1.53010 q^{5} +1.05276 q^{7} +3.00979 q^{9} +O(q^{10})\)	\(q-2.45149 q^{3} +1.53010 q^{5} +1.05276 q^{7} +3.00979 q^{9} +2.99221 q^{11} -0.170649 q^{13} -3.75101 q^{15} +3.06585 q^{17} -1.00000 q^{19} -2.58083 q^{21} -8.77490 q^{23} -2.65880 q^{25} -0.0239932 q^{27} -5.85129 q^{29} +9.59685 q^{31} -7.33536 q^{33} +1.61082 q^{35} +2.36380 q^{37} +0.418344 q^{39} -6.12106 q^{41} +4.52915 q^{43} +4.60527 q^{45} -9.15759 q^{47} -5.89170 q^{49} -7.51589 q^{51} -11.9474 q^{53} +4.57837 q^{55} +2.45149 q^{57} -4.41013 q^{59} -9.85190 q^{61} +3.16858 q^{63} -0.261110 q^{65} +5.05110 q^{67} +21.5116 q^{69} +2.90184 q^{71} +2.32834 q^{73} +6.51802 q^{75} +3.15008 q^{77} +1.00000 q^{79} -8.97054 q^{81} -7.65477 q^{83} +4.69105 q^{85} +14.3444 q^{87} -2.18396 q^{89} -0.179652 q^{91} -23.5265 q^{93} -1.53010 q^{95} +10.0959 q^{97} +9.00591 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 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

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