LMFDB - Embedded newform 6004.2.a.e.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:	$0$
Dimension:	$24$
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)

$f(q)$	$=$	$q-2.58747 q^{3} -3.96262 q^{5} +2.03599 q^{7} +3.69501 q^{9} +O(q^{10})$	$q-2.58747 q^{3} -3.96262 q^{5} +2.03599 q^{7} +3.69501 q^{9} +3.34861 q^{11} +3.37440 q^{13} +10.2532 q^{15} +6.55328 q^{17} +1.00000 q^{19} -5.26808 q^{21} +6.84096 q^{23} +10.7023 q^{25} -1.79832 q^{27} +3.95273 q^{29} -1.10884 q^{31} -8.66443 q^{33} -8.06786 q^{35} +1.25679 q^{37} -8.73117 q^{39} +4.56766 q^{41} +11.0112 q^{43} -14.6419 q^{45} +1.48761 q^{47} -2.85473 q^{49} -16.9564 q^{51} +0.297131 q^{53} -13.2693 q^{55} -2.58747 q^{57} +10.0490 q^{59} -0.742975 q^{61} +7.52302 q^{63} -13.3714 q^{65} -5.24839 q^{67} -17.7008 q^{69} +8.78090 q^{71} +4.34190 q^{73} -27.6920 q^{75} +6.81775 q^{77} +1.00000 q^{79} -6.43193 q^{81} -16.2445 q^{83} -25.9681 q^{85} -10.2276 q^{87} +8.65090 q^{89} +6.87026 q^{91} +2.86910 q^{93} -3.96262 q^{95} -3.42719 q^{97} +12.3732 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.58747	−1.49388	−0.746939	−	0.664893i	$-0.768477\pi$
$3$	−2.58747	−1.49388	−0.746939	+	0.664893i	$0.768477\pi$
$4$	0	0
$5$	−3.96262	−1.77214	−0.886068	−	0.463556i	$-0.846573\pi$
$5$	−3.96262	−1.77214	−0.886068	+	0.463556i	$0.846573\pi$
$6$	0	0
$7$	2.03599	0.769533	0.384767	−	0.923014i	$-0.374282\pi$
$7$	2.03599	0.769533	0.384767	+	0.923014i	$0.374282\pi$
$8$	0	0
$9$	3.69501	1.23167
$10$	0	0
$11$	3.34861	1.00964	0.504822	−	0.863224i	$-0.331558\pi$
$11$	3.34861	1.00964	0.504822	+	0.863224i	$0.331558\pi$
$12$	0	0
$13$	3.37440	0.935890	0.467945	−	0.883758i	$-0.344994\pi$
$13$	3.37440	0.935890	0.467945	+	0.883758i	$0.344994\pi$
$14$	0	0
$15$	10.2532	2.64735
$16$	0	0
$17$	6.55328	1.58940	0.794701	−	0.607001i	$-0.207627\pi$
$17$	6.55328	1.58940	0.794701	+	0.607001i	$0.207627\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−5.26808	−1.14959
$22$	0	0
$23$	6.84096	1.42644	0.713219	−	0.700941i	$-0.247236\pi$
$23$	6.84096	1.42644	0.713219	+	0.700941i	$0.247236\pi$
$24$	0	0
$25$	10.7023	2.14046
$26$	0	0
$27$	−1.79832	−0.346087
$28$	0	0
$29$	3.95273	0.734003	0.367001	−	0.930220i	$-0.380384\pi$
$29$	3.95273	0.734003	0.367001	+	0.930220i	$0.380384\pi$
$30$	0	0
$31$	−1.10884	−0.199154	−0.0995772	−	0.995030i	$-0.531749\pi$
$31$	−1.10884	−0.199154	−0.0995772	+	0.995030i	$0.531749\pi$
$32$	0	0
$33$	−8.66443	−1.50828
$34$	0	0
$35$	−8.06786	−1.36372
$36$	0	0
$37$	1.25679	0.206614	0.103307	−	0.994650i	$-0.467058\pi$
$37$	1.25679	0.206614	0.103307	+	0.994650i	$0.467058\pi$
$38$	0	0
$39$	−8.73117	−1.39811
$40$	0	0
$41$	4.56766	0.713349	0.356675	−	0.934229i	$-0.383910\pi$
$41$	4.56766	0.713349	0.356675	+	0.934229i	$0.383910\pi$
$42$	0	0
$43$	11.0112	1.67919	0.839594	−	0.543215i	$-0.182793\pi$
$43$	11.0112	1.67919	0.839594	+	0.543215i	$0.182793\pi$
$44$	0	0
$45$	−14.6419	−2.18269
$46$	0	0
$47$	1.48761	0.216991	0.108495	−	0.994097i	$-0.465397\pi$
$47$	1.48761	0.216991	0.108495	+	0.994097i	$0.465397\pi$
$48$	0	0
$49$	−2.85473	−0.407819
$50$	0	0
$51$	−16.9564	−2.37437
$52$	0	0
$53$	0.297131	0.0408140	0.0204070	−	0.999792i	$-0.493504\pi$
$53$	0.297131	0.0408140	0.0204070	+	0.999792i	$0.493504\pi$
$54$	0	0
$55$	−13.2693	−1.78923
$56$	0	0
$57$	−2.58747	−0.342719
$58$	0	0
$59$	10.0490	1.30827	0.654133	−	0.756379i	$-0.273033\pi$
$59$	10.0490	1.30827	0.654133	+	0.756379i	$0.273033\pi$
$60$	0	0
$61$	−0.742975	−0.0951283	−0.0475641	−	0.998868i	$-0.515146\pi$
$61$	−0.742975	−0.0951283	−0.0475641	+	0.998868i	$0.515146\pi$
$62$	0	0
$63$	7.52302	0.947811
$64$	0	0
$65$	−13.3714	−1.65852
$66$	0	0
$67$	−5.24839	−0.641193	−0.320597	−	0.947216i	$-0.603883\pi$
$67$	−5.24839	−0.641193	−0.320597	+	0.947216i	$0.603883\pi$
$68$	0	0
$69$	−17.7008	−2.13093
$70$	0	0
$71$	8.78090	1.04210	0.521050	−	0.853526i	$-0.325540\pi$
$71$	8.78090	1.04210	0.521050	+	0.853526i	$0.325540\pi$
$72$	0	0
$73$	4.34190	0.508180	0.254090	−	0.967181i	$-0.418224\pi$
$73$	4.34190	0.508180	0.254090	+	0.967181i	$0.418224\pi$
$74$	0	0
$75$	−27.6920	−3.19759
$76$	0	0
$77$	6.81775	0.776954
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−6.43193	−0.714658
$82$	0	0
$83$	−16.2445	−1.78307	−0.891535	−	0.452951i	$-0.850371\pi$
$83$	−16.2445	−1.78307	−0.891535	+	0.452951i	$0.850371\pi$
$84$	0	0
$85$	−25.9681	−2.81664
$86$	0	0
$87$	−10.2276	−1.09651
$88$	0	0
$89$	8.65090	0.916993	0.458497	−	0.888696i	$-0.348388\pi$
$89$	8.65090	0.916993	0.458497	+	0.888696i	$0.348388\pi$
$90$	0	0
$91$	6.87026	0.720198
$92$	0	0
$93$	2.86910	0.297512
$94$	0	0
$95$	−3.96262	−0.406556
$96$	0	0
$97$	−3.42719	−0.347978	−0.173989	−	0.984748i	$-0.555666\pi$
$97$	−3.42719	−0.347978	−0.173989	+	0.984748i	$0.555666\pi$
$98$	0	0
$99$	12.3732	1.24355
$100$	0	0
$101$	−19.8588	−1.97603	−0.988014	−	0.154363i	$-0.950668\pi$
$101$	−19.8588	−1.97603	−0.988014	+	0.154363i	$0.950668\pi$
$102$	0	0
$103$	12.6115	1.24265	0.621323	−	0.783554i	$-0.286595\pi$
$103$	12.6115	1.24265	0.621323	+	0.783554i	$0.286595\pi$
$104$	0	0
$105$	20.8754	2.03723
$106$	0	0
$107$	−1.06089	−0.102560	−0.0512800	−	0.998684i	$-0.516330\pi$
$107$	−1.06089	−0.102560	−0.0512800	+	0.998684i	$0.516330\pi$
$108$	0	0
$109$	6.08579	0.582913	0.291456	−	0.956584i	$-0.405860\pi$
$109$	6.08579	0.582913	0.291456	+	0.956584i	$0.405860\pi$
$110$	0	0
$111$	−3.25190	−0.308656
$112$	0	0
$113$	−0.156619	−0.0147335	−0.00736675	−	0.999973i	$-0.502345\pi$
$113$	−0.156619	−0.0147335	−0.00736675	+	0.999973i	$0.502345\pi$
$114$	0	0
$115$	−27.1081	−2.52784
$116$	0	0
$117$	12.4684	1.15271
$118$	0	0
$119$	13.3424	1.22310
$120$	0	0
$121$	0.213186	0.0193806
$122$	0	0
$123$	−11.8187	−1.06566
$124$	0	0
$125$	−22.5961	−2.02106
$126$	0	0
$127$	1.75739	0.155943	0.0779714	−	0.996956i	$-0.475156\pi$
$127$	1.75739	0.155943	0.0779714	+	0.996956i	$0.475156\pi$
$128$	0	0
$129$	−28.4911	−2.50850
$130$	0	0
$131$	−1.12475	−0.0982701	−0.0491351	−	0.998792i	$-0.515646\pi$
$131$	−1.12475	−0.0982701	−0.0491351	+	0.998792i	$0.515646\pi$
$132$	0	0
$133$	2.03599	0.176543
$134$	0	0
$135$	7.12606	0.613314
$136$	0	0
$137$	19.0691	1.62918	0.814592	−	0.580034i	$-0.196961\pi$
$137$	19.0691	1.62918	0.814592	+	0.580034i	$0.196961\pi$
$138$	0	0
$139$	−20.8202	−1.76595	−0.882974	−	0.469423i	$-0.844462\pi$
$139$	−20.8202	−1.76595	−0.882974	+	0.469423i	$0.844462\pi$
$140$	0	0
$141$	−3.84916	−0.324158
$142$	0	0
$143$	11.2995	0.944916
$144$	0	0
$145$	−15.6631	−1.30075
$146$	0	0
$147$	7.38654	0.609231
$148$	0	0
$149$	12.1945	0.999014	0.499507	−	0.866310i	$-0.333514\pi$
$149$	12.1945	0.999014	0.499507	+	0.866310i	$0.333514\pi$
$150$	0	0
$151$	9.81670	0.798872	0.399436	−	0.916761i	$-0.369206\pi$
$151$	9.81670	0.798872	0.399436	+	0.916761i	$0.369206\pi$
$152$	0	0
$153$	24.2144	1.95762
$154$	0	0
$155$	4.39392	0.352929
$156$	0	0
$157$	−0.387118	−0.0308954	−0.0154477	−	0.999881i	$-0.504917\pi$
$157$	−0.387118	−0.0308954	−0.0154477	+	0.999881i	$0.504917\pi$
$158$	0	0
$159$	−0.768817	−0.0609712
$160$	0	0
$161$	13.9281	1.09769
$162$	0	0
$163$	0.337910	0.0264671	0.0132336	−	0.999912i	$-0.495787\pi$
$163$	0.337910	0.0264671	0.0132336	+	0.999912i	$0.495787\pi$
$164$	0	0
$165$	34.3338	2.67288
$166$	0	0
$167$	−2.86618	−0.221792	−0.110896	−	0.993832i	$-0.535372\pi$
$167$	−2.86618	−0.221792	−0.110896	+	0.993832i	$0.535372\pi$
$168$	0	0
$169$	−1.61342	−0.124110
$170$	0	0
$171$	3.69501	0.282565
$172$	0	0
$173$	24.5519	1.86665	0.933325	−	0.359033i	$-0.116894\pi$
$173$	24.5519	1.86665	0.933325	+	0.359033i	$0.116894\pi$
$174$	0	0
$175$	21.7899	1.64716
$176$	0	0
$177$	−26.0015	−1.95439
$178$	0	0
$179$	−2.42694	−0.181398	−0.0906991	−	0.995878i	$-0.528910\pi$
$179$	−2.42694	−0.181398	−0.0906991	+	0.995878i	$0.528910\pi$
$180$	0	0
$181$	−11.2102	−0.833250	−0.416625	−	0.909078i	$-0.636787\pi$
$181$	−11.2102	−0.833250	−0.416625	+	0.909078i	$0.636787\pi$
$182$	0	0
$183$	1.92243	0.142110
$184$	0	0
$185$	−4.98016	−0.366149
$186$	0	0
$187$	21.9444	1.60473
$188$	0	0
$189$	−3.66137	−0.266326
$190$	0	0
$191$	10.1667	0.735636	0.367818	−	0.929898i	$-0.380105\pi$
$191$	10.1667	0.735636	0.367818	+	0.929898i	$0.380105\pi$
$192$	0	0
$193$	3.88507	0.279654	0.139827	−	0.990176i	$-0.455345\pi$
$193$	3.88507	0.279654	0.139827	+	0.990176i	$0.455345\pi$
$194$	0	0
$195$	34.5983	2.47763
$196$	0	0
$197$	−10.4329	−0.743313	−0.371657	−	0.928370i	$-0.621210\pi$
$197$	−10.4329	−0.743313	−0.371657	+	0.928370i	$0.621210\pi$
$198$	0	0
$199$	11.2272	0.795877	0.397938	−	0.917412i	$-0.369726\pi$
$199$	11.2272	0.795877	0.397938	+	0.917412i	$0.369726\pi$
$200$	0	0
$201$	13.5801	0.957864
$202$	0	0
$203$	8.04772	0.564840
$204$	0	0
$205$	−18.0999	−1.26415
$206$	0	0
$207$	25.2774	1.75690
$208$	0	0
$209$	3.34861	0.231628
$210$	0	0
$211$	14.8698	1.02368	0.511841	−	0.859080i	$-0.328964\pi$
$211$	14.8698	1.02368	0.511841	+	0.859080i	$0.328964\pi$
$212$	0	0
$213$	−22.7203	−1.55677
$214$	0	0
$215$	−43.6330	−2.97575
$216$	0	0
$217$	−2.25760	−0.153256
$218$	0	0
$219$	−11.2345	−0.759159
$220$	0	0
$221$	22.1134	1.48751
$222$	0	0
$223$	−19.0799	−1.27769	−0.638844	−	0.769336i	$-0.720587\pi$
$223$	−19.0799	−1.27769	−0.638844	+	0.769336i	$0.720587\pi$
$224$	0	0
$225$	39.5452	2.63635
$226$	0	0
$227$	−16.5748	−1.10011	−0.550054	−	0.835129i	$-0.685393\pi$
$227$	−16.5748	−1.10011	−0.550054	+	0.835129i	$0.685393\pi$
$228$	0	0
$229$	−0.762938	−0.0504164	−0.0252082	−	0.999682i	$-0.508025\pi$
$229$	−0.762938	−0.0504164	−0.0252082	+	0.999682i	$0.508025\pi$
$230$	0	0
$231$	−17.6407	−1.16067
$232$	0	0
$233$	−5.10198	−0.334241	−0.167121	−	0.985936i	$-0.553447\pi$
$233$	−5.10198	−0.334241	−0.167121	+	0.985936i	$0.553447\pi$
$234$	0	0
$235$	−5.89484	−0.384537
$236$	0	0
$237$	−2.58747	−0.168074
$238$	0	0
$239$	−5.59682	−0.362028	−0.181014	−	0.983480i	$-0.557938\pi$
$239$	−5.59682	−0.362028	−0.181014	+	0.983480i	$0.557938\pi$
$240$	0	0
$241$	−14.4748	−0.932401	−0.466201	−	0.884679i	$-0.654378\pi$
$241$	−14.4748	−0.932401	−0.466201	+	0.884679i	$0.654378\pi$
$242$	0	0
$243$	22.0374	1.41370
$244$	0	0
$245$	11.3122	0.722710
$246$	0	0
$247$	3.37440	0.214708
$248$	0	0
$249$	42.0323	2.66369
$250$	0	0
$251$	−24.3345	−1.53598	−0.767991	−	0.640461i	$-0.778743\pi$
$251$	−24.3345	−1.53598	−0.767991	+	0.640461i	$0.778743\pi$
$252$	0	0
$253$	22.9077	1.44020
$254$	0	0
$255$	67.1918	4.20771
$256$	0	0
$257$	4.93986	0.308140	0.154070	−	0.988060i	$-0.450762\pi$
$257$	4.93986	0.308140	0.154070	+	0.988060i	$0.450762\pi$
$258$	0	0
$259$	2.55881	0.158997
$260$	0	0
$261$	14.6054	0.904050
$262$	0	0
$263$	−3.22276	−0.198724	−0.0993619	−	0.995051i	$-0.531680\pi$
$263$	−3.22276	−0.198724	−0.0993619	+	0.995051i	$0.531680\pi$
$264$	0	0
$265$	−1.17741	−0.0723280
$266$	0	0
$267$	−22.3840	−1.36988
$268$	0	0
$269$	21.3685	1.30286	0.651431	−	0.758708i	$-0.274169\pi$
$269$	21.3685	1.30286	0.651431	+	0.758708i	$0.274169\pi$
$270$	0	0
$271$	6.44222	0.391337	0.195669	−	0.980670i	$-0.437312\pi$
$271$	6.44222	0.391337	0.195669	+	0.980670i	$0.437312\pi$
$272$	0	0
$273$	−17.7766	−1.07589
$274$	0	0
$275$	35.8379	2.16111
$276$	0	0
$277$	3.26075	0.195919	0.0979597	−	0.995190i	$-0.468768\pi$
$277$	3.26075	0.195919	0.0979597	+	0.995190i	$0.468768\pi$
$278$	0	0
$279$	−4.09719	−0.245293
$280$	0	0
$281$	2.98688	0.178182	0.0890912	−	0.996023i	$-0.471604\pi$
$281$	2.98688	0.178182	0.0890912	+	0.996023i	$0.471604\pi$
$282$	0	0
$283$	−25.7368	−1.52989	−0.764946	−	0.644095i	$-0.777234\pi$
$283$	−25.7368	−1.52989	−0.764946	+	0.644095i	$0.777234\pi$
$284$	0	0
$285$	10.2532	0.607345
$286$	0	0
$287$	9.29973	0.548946
$288$	0	0
$289$	25.9454	1.52620
$290$	0	0
$291$	8.86775	0.519837
$292$	0	0
$293$	25.7729	1.50567	0.752834	−	0.658210i	$-0.228686\pi$
$293$	25.7729	1.50567	0.752834	+	0.658210i	$0.228686\pi$
$294$	0	0
$295$	−39.8203	−2.31843
$296$	0	0
$297$	−6.02188	−0.349425
$298$	0	0
$299$	23.0841	1.33499
$300$	0	0
$301$	22.4187	1.29219
$302$	0	0
$303$	51.3842	2.95194
$304$	0	0
$305$	2.94413	0.168580
$306$	0	0
$307$	22.3387	1.27494	0.637469	−	0.770476i	$-0.279981\pi$
$307$	22.3387	1.27494	0.637469	+	0.770476i	$0.279981\pi$
$308$	0	0
$309$	−32.6319	−1.85636
$310$	0	0
$311$	23.1528	1.31287	0.656437	−	0.754381i	$-0.272063\pi$
$311$	23.1528	1.31287	0.656437	+	0.754381i	$0.272063\pi$
$312$	0	0
$313$	−6.42191	−0.362988	−0.181494	−	0.983392i	$-0.558093\pi$
$313$	−6.42191	−0.362988	−0.181494	+	0.983392i	$0.558093\pi$
$314$	0	0
$315$	−29.8108	−1.67965
$316$	0	0
$317$	5.14721	0.289096	0.144548	−	0.989498i	$-0.453827\pi$
$317$	5.14721	0.289096	0.144548	+	0.989498i	$0.453827\pi$
$318$	0	0
$319$	13.2361	0.741082
$320$	0	0
$321$	2.74502	0.153212
$322$	0	0
$323$	6.55328	0.364634
$324$	0	0
$325$	36.1139	2.00324
$326$	0	0
$327$	−15.7468	−0.870800
$328$	0	0
$329$	3.02877	0.166982
$330$	0	0
$331$	−16.5029	−0.907083	−0.453542	−	0.891235i	$-0.649840\pi$
$331$	−16.5029	−0.907083	−0.453542	+	0.891235i	$0.649840\pi$
$332$	0	0
$333$	4.64384	0.254481
$334$	0	0
$335$	20.7974	1.13628
$336$	0	0
$337$	22.7245	1.23788	0.618942	−	0.785437i	$-0.287562\pi$
$337$	22.7245	1.23788	0.618942	+	0.785437i	$0.287562\pi$
$338$	0	0
$339$	0.405248	0.0220100
$340$	0	0
$341$	−3.71309	−0.201075
$342$	0	0
$343$	−20.0642	−1.08336
$344$	0	0
$345$	70.1414	3.77629
$346$	0	0
$347$	−28.9602	−1.55467	−0.777333	−	0.629089i	$-0.783428\pi$
$347$	−28.9602	−1.55467	−0.777333	+	0.629089i	$0.783428\pi$
$348$	0	0
$349$	−23.1395	−1.23863	−0.619315	−	0.785142i	$-0.712590\pi$
$349$	−23.1395	−1.23863	−0.619315	+	0.785142i	$0.712590\pi$
$350$	0	0
$351$	−6.06826	−0.323900
$352$	0	0
$353$	11.6506	0.620096	0.310048	−	0.950721i	$-0.399655\pi$
$353$	11.6506	0.620096	0.310048	+	0.950721i	$0.399655\pi$
$354$	0	0
$355$	−34.7953	−1.84674
$356$	0	0
$357$	−34.5232	−1.82716
$358$	0	0
$359$	−14.6745	−0.774490	−0.387245	−	0.921977i	$-0.626573\pi$
$359$	−14.6745	−0.774490	−0.387245	+	0.921977i	$0.626573\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.551613	−0.0289522
$364$	0	0
$365$	−17.2053	−0.900565
$366$	0	0
$367$	−30.0314	−1.56763	−0.783813	−	0.620997i	$-0.786728\pi$
$367$	−30.0314	−1.56763	−0.783813	+	0.620997i	$0.786728\pi$
$368$	0	0
$369$	16.8776	0.878611
$370$	0	0
$371$	0.604956	0.0314078
$372$	0	0
$373$	−23.7960	−1.23211	−0.616055	−	0.787703i	$-0.711270\pi$
$373$	−23.7960	−1.23211	−0.616055	+	0.787703i	$0.711270\pi$
$374$	0	0
$375$	58.4668	3.01921
$376$	0	0
$377$	13.3381	0.686946
$378$	0	0
$379$	−16.0836	−0.826159	−0.413080	−	0.910695i	$-0.635547\pi$
$379$	−16.0836	−0.826159	−0.413080	+	0.910695i	$0.635547\pi$
$380$	0	0
$381$	−4.54719	−0.232959
$382$	0	0
$383$	−21.3130	−1.08904	−0.544522	−	0.838746i	$-0.683289\pi$
$383$	−21.3130	−1.08904	−0.544522	+	0.838746i	$0.683289\pi$
$384$	0	0
$385$	−27.0161	−1.37687
$386$	0	0
$387$	40.6864	2.06821
$388$	0	0
$389$	17.1013	0.867069	0.433534	−	0.901137i	$-0.357266\pi$
$389$	17.1013	0.867069	0.433534	+	0.901137i	$0.357266\pi$
$390$	0	0
$391$	44.8307	2.26719
$392$	0	0
$393$	2.91027	0.146804
$394$	0	0
$395$	−3.96262	−0.199381
$396$	0	0
$397$	−6.21519	−0.311932	−0.155966	−	0.987762i	$-0.549849\pi$
$397$	−6.21519	−0.311932	−0.155966	+	0.987762i	$0.549849\pi$
$398$	0	0
$399$	−5.26808	−0.263734
$400$	0	0
$401$	0.525405	0.0262375	0.0131187	−	0.999914i	$-0.495824\pi$
$401$	0.525405	0.0262375	0.0131187	+	0.999914i	$0.495824\pi$
$402$	0	0
$403$	−3.74169	−0.186387
$404$	0	0
$405$	25.4872	1.26647
$406$	0	0
$407$	4.20848	0.208607
$408$	0	0
$409$	22.4970	1.11241	0.556203	−	0.831047i	$-0.312258\pi$
$409$	22.4970	1.11241	0.556203	+	0.831047i	$0.312258\pi$
$410$	0	0
$411$	−49.3408	−2.43380
$412$	0	0
$413$	20.4597	1.00675
$414$	0	0
$415$	64.3709	3.15984
$416$	0	0
$417$	53.8717	2.63811
$418$	0	0
$419$	−31.0558	−1.51717	−0.758587	−	0.651571i	$-0.774110\pi$
$419$	−31.0558	−1.51717	−0.758587	+	0.651571i	$0.774110\pi$
$420$	0	0
$421$	−7.42079	−0.361667	−0.180833	−	0.983514i	$-0.557880\pi$
$421$	−7.42079	−0.361667	−0.180833	+	0.983514i	$0.557880\pi$
$422$	0	0
$423$	5.49675	0.267261
$424$	0	0
$425$	70.1353	3.40206
$426$	0	0
$427$	−1.51269	−0.0732043
$428$	0	0
$429$	−29.2373	−1.41159
$430$	0	0
$431$	33.9388	1.63477	0.817387	−	0.576089i	$-0.195422\pi$
$431$	33.9388	1.63477	0.817387	+	0.576089i	$0.195422\pi$
$432$	0	0
$433$	−16.1110	−0.774243	−0.387122	−	0.922029i	$-0.626531\pi$
$433$	−16.1110	−0.774243	−0.387122	+	0.922029i	$0.626531\pi$
$434$	0	0
$435$	40.5279	1.94317
$436$	0	0
$437$	6.84096	0.327248
$438$	0	0
$439$	27.1603	1.29629	0.648146	−	0.761516i	$-0.275545\pi$
$439$	27.1603	1.29629	0.648146	+	0.761516i	$0.275545\pi$
$440$	0	0
$441$	−10.5483	−0.502298
$442$	0	0
$443$	11.6600	0.553984	0.276992	−	0.960872i	$-0.410662\pi$
$443$	11.6600	0.553984	0.276992	+	0.960872i	$0.410662\pi$
$444$	0	0
$445$	−34.2802	−1.62504
$446$	0	0
$447$	−31.5530	−1.49241
$448$	0	0
$449$	−15.5070	−0.731820	−0.365910	−	0.930650i	$-0.619242\pi$
$449$	−15.5070	−0.731820	−0.365910	+	0.930650i	$0.619242\pi$
$450$	0	0
$451$	15.2953	0.720229
$452$	0	0
$453$	−25.4004	−1.19342
$454$	0	0
$455$	−27.2242	−1.27629
$456$	0	0
$457$	3.71929	0.173981	0.0869905	−	0.996209i	$-0.472275\pi$
$457$	3.71929	0.173981	0.0869905	+	0.996209i	$0.472275\pi$
$458$	0	0
$459$	−11.7849	−0.550072
$460$	0	0
$461$	16.6900	0.777331	0.388666	−	0.921379i	$-0.372936\pi$
$461$	16.6900	0.777331	0.388666	+	0.921379i	$0.372936\pi$
$462$	0	0
$463$	−24.8221	−1.15358	−0.576790	−	0.816893i	$-0.695695\pi$
$463$	−24.8221	−1.15358	−0.576790	+	0.816893i	$0.695695\pi$
$464$	0	0
$465$	−11.3692	−0.527232
$466$	0	0
$467$	5.11482	0.236686	0.118343	−	0.992973i	$-0.462242\pi$
$467$	5.11482	0.236686	0.118343	+	0.992973i	$0.462242\pi$
$468$	0	0
$469$	−10.6857	−0.493419
$470$	0	0
$471$	1.00166	0.0461539
$472$	0	0
$473$	36.8721	1.69538
$474$	0	0
$475$	10.7023	0.491056
$476$	0	0
$477$	1.09790	0.0502694
$478$	0	0
$479$	−25.5111	−1.16563	−0.582815	−	0.812605i	$-0.698049\pi$
$479$	−25.5111	−1.16563	−0.582815	+	0.812605i	$0.698049\pi$
$480$	0	0
$481$	4.24090	0.193368
$482$	0	0
$483$	−36.0387	−1.63982
$484$	0	0
$485$	13.5806	0.616665
$486$	0	0
$487$	−28.6662	−1.29899	−0.649496	−	0.760365i	$-0.725020\pi$
$487$	−28.6662	−1.29899	−0.649496	+	0.760365i	$0.725020\pi$
$488$	0	0
$489$	−0.874332	−0.0395387
$490$	0	0
$491$	−15.9985	−0.722001	−0.361001	−	0.932566i	$-0.617565\pi$
$491$	−15.9985	−0.722001	−0.361001	+	0.932566i	$0.617565\pi$
$492$	0	0
$493$	25.9033	1.16663
$494$	0	0
$495$	−49.0300	−2.20374
$496$	0	0
$497$	17.8778	0.801931
$498$	0	0
$499$	−37.6014	−1.68327	−0.841635	−	0.540046i	$-0.818407\pi$
$499$	−37.6014	−1.68327	−0.841635	+	0.540046i	$0.818407\pi$
$500$	0	0
$501$	7.41617	0.331330
$502$	0	0
$503$	6.15608	0.274486	0.137243	−	0.990537i	$-0.456176\pi$
$503$	6.15608	0.274486	0.137243	+	0.990537i	$0.456176\pi$
$504$	0	0
$505$	78.6929	3.50179
$506$	0	0
$507$	4.17469	0.185405
$508$	0	0
$509$	−0.0285970	−0.00126754	−0.000633771	−	1.00000i	$-0.500202\pi$
$509$	−0.0285970	−0.00126754	−0.000633771		1.00000i	$0.500202\pi$
$510$	0	0
$511$	8.84007	0.391062
$512$	0	0
$513$	−1.79832	−0.0793979
$514$	0	0
$515$	−49.9745	−2.20214
$516$	0	0
$517$	4.98144	0.219083
$518$	0	0
$519$	−63.5275	−2.78855
$520$	0	0
$521$	2.49196	0.109175	0.0545875	−	0.998509i	$-0.482616\pi$
$521$	2.49196	0.109175	0.0545875	+	0.998509i	$0.482616\pi$
$522$	0	0
$523$	−3.47083	−0.151769	−0.0758845	−	0.997117i	$-0.524178\pi$
$523$	−3.47083	−0.151769	−0.0758845	+	0.997117i	$0.524178\pi$
$524$	0	0
$525$	−56.3806	−2.46065
$526$	0	0
$527$	−7.26656	−0.316537
$528$	0	0
$529$	23.7987	1.03473
$530$	0	0
$531$	37.1311	1.61135
$532$	0	0
$533$	15.4131	0.667617
$534$	0	0
$535$	4.20390	0.181750
$536$	0	0
$537$	6.27965	0.270987
$538$	0	0
$539$	−9.55938	−0.411752
$540$	0	0
$541$	5.40907	0.232554	0.116277	−	0.993217i	$-0.462904\pi$
$541$	5.40907	0.232554	0.116277	+	0.993217i	$0.462904\pi$
$542$	0	0
$543$	29.0062	1.24477
$544$	0	0
$545$	−24.1156	−1.03300
$546$	0	0
$547$	−30.4651	−1.30259	−0.651297	−	0.758823i	$-0.725775\pi$
$547$	−30.4651	−1.30259	−0.651297	+	0.758823i	$0.725775\pi$
$548$	0	0
$549$	−2.74530	−0.117167
$550$	0	0
$551$	3.95273	0.168392
$552$	0	0
$553$	2.03599	0.0865792
$554$	0	0
$555$	12.8860	0.546981
$556$	0	0
$557$	30.4028	1.28821	0.644103	−	0.764939i	$-0.277231\pi$
$557$	30.4028	1.28821	0.644103	+	0.764939i	$0.277231\pi$
$558$	0	0
$559$	37.1561	1.57153
$560$	0	0
$561$	−56.7804	−2.39727
$562$	0	0
$563$	18.7411	0.789843	0.394922	−	0.918715i	$-0.370772\pi$
$563$	18.7411	0.789843	0.394922	+	0.918715i	$0.370772\pi$
$564$	0	0
$565$	0.620622	0.0261098
$566$	0	0
$567$	−13.0954	−0.549953
$568$	0	0
$569$	−35.7485	−1.49866	−0.749328	−	0.662199i	$-0.769623\pi$
$569$	−35.7485	−1.49866	−0.749328	+	0.662199i	$0.769623\pi$
$570$	0	0
$571$	15.2180	0.636852	0.318426	−	0.947948i	$-0.396846\pi$
$571$	15.2180	0.636852	0.318426	+	0.947948i	$0.396846\pi$
$572$	0	0
$573$	−26.3060	−1.09895
$574$	0	0
$575$	73.2142	3.05324
$576$	0	0
$577$	7.18475	0.299105	0.149553	−	0.988754i	$-0.452217\pi$
$577$	7.18475	0.299105	0.149553	+	0.988754i	$0.452217\pi$
$578$	0	0
$579$	−10.0525	−0.417768
$580$	0	0
$581$	−33.0738	−1.37213
$582$	0	0
$583$	0.994975	0.0412076
$584$	0	0
$585$	−49.4077	−2.04276
$586$	0	0
$587$	5.15014	0.212569	0.106285	−	0.994336i	$-0.466105\pi$
$587$	5.15014	0.212569	0.106285	+	0.994336i	$0.466105\pi$
$588$	0	0
$589$	−1.10884	−0.0456891
$590$	0	0
$591$	26.9948	1.11042
$592$	0	0
$593$	4.38274	0.179978	0.0899888	−	0.995943i	$-0.471317\pi$
$593$	4.38274	0.179978	0.0899888	+	0.995943i	$0.471317\pi$
$594$	0	0
$595$	−52.8709	−2.16750
$596$	0	0
$597$	−29.0501	−1.18894
$598$	0	0
$599$	17.8024	0.727387	0.363693	−	0.931519i	$-0.381516\pi$
$599$	17.8024	0.727387	0.363693	+	0.931519i	$0.381516\pi$
$600$	0	0
$601$	−38.7225	−1.57952	−0.789762	−	0.613413i	$-0.789796\pi$
$601$	−38.7225	−1.57952	−0.789762	+	0.613413i	$0.789796\pi$
$602$	0	0
$603$	−19.3929	−0.789739
$604$	0	0
$605$	−0.844775	−0.0343450
$606$	0	0
$607$	−36.2312	−1.47058	−0.735289	−	0.677754i	$-0.762953\pi$
$607$	−36.2312	−1.47058	−0.735289	+	0.677754i	$0.762953\pi$
$608$	0	0
$609$	−20.8233	−0.843801
$610$	0	0
$611$	5.01980	0.203080
$612$	0	0
$613$	−18.0115	−0.727479	−0.363740	−	0.931501i	$-0.618500\pi$
$613$	−18.0115	−0.727479	−0.363740	+	0.931501i	$0.618500\pi$
$614$	0	0
$615$	46.8330	1.88849
$616$	0	0
$617$	22.4562	0.904051	0.452025	−	0.892005i	$-0.350702\pi$
$617$	22.4562	0.904051	0.452025	+	0.892005i	$0.350702\pi$
$618$	0	0
$619$	10.4302	0.419224	0.209612	−	0.977785i	$-0.432780\pi$
$619$	10.4302	0.419224	0.209612	+	0.977785i	$0.432780\pi$
$620$	0	0
$621$	−12.3023	−0.493672
$622$	0	0
$623$	17.6132	0.705657
$624$	0	0
$625$	36.0281	1.44112
$626$	0	0
$627$	−8.66443	−0.346024
$628$	0	0
$629$	8.23606	0.328393
$630$	0	0
$631$	14.3758	0.572291	0.286146	−	0.958186i	$-0.407626\pi$
$631$	14.3758	0.572291	0.286146	+	0.958186i	$0.407626\pi$
$632$	0	0
$633$	−38.4753	−1.52925
$634$	0	0
$635$	−6.96384	−0.276352
$636$	0	0
$637$	−9.63301	−0.381674
$638$	0	0
$639$	32.4455	1.28352
$640$	0	0
$641$	13.7191	0.541873	0.270937	−	0.962597i	$-0.412667\pi$
$641$	13.7191	0.541873	0.270937	+	0.962597i	$0.412667\pi$
$642$	0	0
$643$	−15.2730	−0.602308	−0.301154	−	0.953576i	$-0.597372\pi$
$643$	−15.2730	−0.602308	−0.301154	+	0.953576i	$0.597372\pi$
$644$	0	0
$645$	112.899	4.44540
$646$	0	0
$647$	−22.0062	−0.865154	−0.432577	−	0.901597i	$-0.642396\pi$
$647$	−22.0062	−0.865154	−0.432577	+	0.901597i	$0.642396\pi$
$648$	0	0
$649$	33.6501	1.32088
$650$	0	0
$651$	5.84148	0.228946
$652$	0	0
$653$	15.3448	0.600490	0.300245	−	0.953862i	$-0.402932\pi$
$653$	15.3448	0.600490	0.300245	+	0.953862i	$0.402932\pi$
$654$	0	0
$655$	4.45696	0.174148
$656$	0	0
$657$	16.0434	0.625911
$658$	0	0
$659$	−23.0331	−0.897243	−0.448622	−	0.893722i	$-0.648085\pi$
$659$	−23.0331	−0.897243	−0.448622	+	0.893722i	$0.648085\pi$
$660$	0	0
$661$	34.6887	1.34923	0.674617	−	0.738168i	$-0.264309\pi$
$661$	34.6887	1.34923	0.674617	+	0.738168i	$0.264309\pi$
$662$	0	0
$663$	−57.2177	−2.22215
$664$	0	0
$665$	−8.06786	−0.312858
$666$	0	0
$667$	27.0404	1.04701
$668$	0	0
$669$	49.3688	1.90871
$670$	0	0
$671$	−2.48793	−0.0960456
$672$	0	0
$673$	−44.1223	−1.70079	−0.850395	−	0.526146i	$-0.823637\pi$
$673$	−44.1223	−1.70079	−0.850395	+	0.526146i	$0.823637\pi$
$674$	0	0
$675$	−19.2462	−0.740788
$676$	0	0
$677$	−18.6991	−0.718665	−0.359332	−	0.933210i	$-0.616996\pi$
$677$	−18.6991	−0.718665	−0.359332	+	0.933210i	$0.616996\pi$
$678$	0	0
$679$	−6.97773	−0.267781
$680$	0	0
$681$	42.8868	1.64343
$682$	0	0
$683$	1.84076	0.0704346	0.0352173	−	0.999380i	$-0.488788\pi$
$683$	1.84076	0.0704346	0.0352173	+	0.999380i	$0.488788\pi$
$684$	0	0
$685$	−75.5636	−2.88714
$686$	0	0
$687$	1.97408	0.0753159
$688$	0	0
$689$	1.00264	0.0381975
$690$	0	0
$691$	−25.9059	−0.985507	−0.492754	−	0.870169i	$-0.664010\pi$
$691$	−25.9059	−0.985507	−0.492754	+	0.870169i	$0.664010\pi$
$692$	0	0
$693$	25.1916	0.956952
$694$	0	0
$695$	82.5025	3.12950
$696$	0	0
$697$	29.9332	1.13380
$698$	0	0
$699$	13.2012	0.499316
$700$	0	0
$701$	−25.6782	−0.969850	−0.484925	−	0.874556i	$-0.661153\pi$
$701$	−25.6782	−0.969850	−0.484925	+	0.874556i	$0.661153\pi$
$702$	0	0
$703$	1.25679	0.0474006
$704$	0	0
$705$	15.2527	0.574451
$706$	0	0
$707$	−40.4325	−1.52062
$708$	0	0
$709$	−27.5203	−1.03355	−0.516773	−	0.856122i	$-0.672867\pi$
$709$	−27.5203	−1.03355	−0.516773	+	0.856122i	$0.672867\pi$
$710$	0	0
$711$	3.69501	0.138574
$712$	0	0
$713$	−7.58556	−0.284082
$714$	0	0
$715$	−44.7758	−1.67452
$716$	0	0
$717$	14.4816	0.540826
$718$	0	0
$719$	10.5725	0.394288	0.197144	−	0.980375i	$-0.436833\pi$
$719$	10.5725	0.394288	0.197144	+	0.980375i	$0.436833\pi$
$720$	0	0
$721$	25.6769	0.956258
$722$	0	0
$723$	37.4530	1.39289
$724$	0	0
$725$	42.3034	1.57111
$726$	0	0
$727$	−38.9131	−1.44321	−0.721603	−	0.692307i	$-0.756594\pi$
$727$	−38.9131	−1.44321	−0.721603	+	0.692307i	$0.756594\pi$
$728$	0	0
$729$	−37.7254	−1.39724
$730$	0	0
$731$	72.1592	2.66890
$732$	0	0
$733$	14.0420	0.518654	0.259327	−	0.965790i	$-0.416499\pi$
$733$	14.0420	0.518654	0.259327	+	0.965790i	$0.416499\pi$
$734$	0	0
$735$	−29.2700	−1.07964
$736$	0	0
$737$	−17.5748	−0.647377
$738$	0	0
$739$	−42.3552	−1.55806	−0.779031	−	0.626985i	$-0.784289\pi$
$739$	−42.3552	−1.55806	−0.779031	+	0.626985i	$0.784289\pi$
$740$	0	0
$741$	−8.73117	−0.320747
$742$	0	0
$743$	47.1334	1.72916	0.864579	−	0.502497i	$-0.167585\pi$
$743$	47.1334	1.72916	0.864579	+	0.502497i	$0.167585\pi$
$744$	0	0
$745$	−48.3222	−1.77039
$746$	0	0
$747$	−60.0238	−2.19616
$748$	0	0
$749$	−2.15996	−0.0789234
$750$	0	0
$751$	−16.4139	−0.598953	−0.299476	−	0.954104i	$-0.596812\pi$
$751$	−16.4139	−0.598953	−0.299476	+	0.954104i	$0.596812\pi$
$752$	0	0
$753$	62.9649	2.29457
$754$	0	0
$755$	−38.8998	−1.41571
$756$	0	0
$757$	−20.8323	−0.757162	−0.378581	−	0.925568i	$-0.623588\pi$
$757$	−20.8323	−0.757162	−0.378581	+	0.925568i	$0.623588\pi$
$758$	0	0
$759$	−59.2731	−2.15148
$760$	0	0
$761$	21.0652	0.763612	0.381806	−	0.924243i	$-0.375302\pi$
$761$	21.0652	0.763612	0.381806	+	0.924243i	$0.375302\pi$
$762$	0	0
$763$	12.3906	0.448571
$764$	0	0
$765$	−95.9525	−3.46917
$766$	0	0
$767$	33.9093	1.22439
$768$	0	0
$769$	22.7153	0.819135	0.409567	−	0.912280i	$-0.365680\pi$
$769$	22.7153	0.819135	0.409567	+	0.912280i	$0.365680\pi$
$770$	0	0
$771$	−12.7818	−0.460324
$772$	0	0
$773$	50.7784	1.82637	0.913187	−	0.407542i	$-0.133614\pi$
$773$	50.7784	1.82637	0.913187	+	0.407542i	$0.133614\pi$
$774$	0	0
$775$	−11.8672	−0.426283
$776$	0	0
$777$	−6.62084	−0.237521
$778$	0	0
$779$	4.56766	0.163654
$780$	0	0
$781$	29.4038	1.05215
$782$	0	0
$783$	−7.10828	−0.254029
$784$	0	0
$785$	1.53400	0.0547508
$786$	0	0
$787$	−34.2706	−1.22162	−0.610808	−	0.791779i	$-0.709155\pi$
$787$	−34.2706	−1.22162	−0.610808	+	0.791779i	$0.709155\pi$
$788$	0	0
$789$	8.33880	0.296869
$790$	0	0
$791$	−0.318876	−0.0113379
$792$	0	0
$793$	−2.50710	−0.0890296
$794$	0	0
$795$	3.04653	0.108049
$796$	0	0
$797$	−5.54749	−0.196502	−0.0982511	−	0.995162i	$-0.531325\pi$
$797$	−5.54749	−0.196502	−0.0982511	+	0.995162i	$0.531325\pi$
$798$	0	0
$799$	9.74875	0.344886
$800$	0	0
$801$	31.9652	1.12943
$802$	0	0
$803$	14.5393	0.513081
$804$	0	0
$805$	−55.1919	−1.94526
$806$	0	0
$807$	−55.2904	−1.94632
$808$	0	0
$809$	40.9100	1.43832	0.719159	−	0.694846i	$-0.244527\pi$
$809$	40.9100	1.43832	0.719159	+	0.694846i	$0.244527\pi$
$810$	0	0
$811$	−28.7377	−1.00912	−0.504558	−	0.863378i	$-0.668345\pi$
$811$	−28.7377	−1.00912	−0.504558	+	0.863378i	$0.668345\pi$
$812$	0	0
$813$	−16.6691	−0.584610
$814$	0	0
$815$	−1.33901	−0.0469034
$816$	0	0
$817$	11.0112	0.385232
$818$	0	0
$819$	25.3857	0.887047
$820$	0	0
$821$	14.4231	0.503369	0.251684	−	0.967809i	$-0.419016\pi$
$821$	14.4231	0.503369	0.251684	+	0.967809i	$0.419016\pi$
$822$	0	0
$823$	−24.5855	−0.856995	−0.428498	−	0.903543i	$-0.640957\pi$
$823$	−24.5855	−0.856995	−0.428498	+	0.903543i	$0.640957\pi$
$824$	0	0
$825$	−92.7296	−3.22843
$826$	0	0
$827$	−7.34455	−0.255395	−0.127698	−	0.991813i	$-0.540759\pi$
$827$	−7.34455	−0.255395	−0.127698	+	0.991813i	$0.540759\pi$
$828$	0	0
$829$	36.0865	1.25334	0.626669	−	0.779286i	$-0.284418\pi$
$829$	36.0865	1.25334	0.626669	+	0.779286i	$0.284418\pi$
$830$	0	0
$831$	−8.43710	−0.292680
$832$	0	0
$833$	−18.7078	−0.648188
$834$	0	0
$835$	11.3576	0.393045
$836$	0	0
$837$	1.99406	0.0689248
$838$	0	0
$839$	−44.1971	−1.52585	−0.762927	−	0.646484i	$-0.776239\pi$
$839$	−44.1971	−1.52585	−0.762927	+	0.646484i	$0.776239\pi$
$840$	0	0
$841$	−13.3760	−0.461240
$842$	0	0
$843$	−7.72847	−0.266183
$844$	0	0
$845$	6.39338	0.219939
$846$	0	0
$847$	0.434046	0.0149140
$848$	0	0
$849$	66.5931	2.28547
$850$	0	0
$851$	8.59762	0.294723
$852$	0	0
$853$	−53.7136	−1.83912	−0.919559	−	0.392953i	$-0.871454\pi$
$853$	−53.7136	−1.83912	−0.919559	+	0.392953i	$0.871454\pi$
$854$	0	0
$855$	−14.6419	−0.500743
$856$	0	0
$857$	−45.4340	−1.55200	−0.775998	−	0.630735i	$-0.782753\pi$
$857$	−45.4340	−1.55200	−0.775998	+	0.630735i	$0.782753\pi$
$858$	0	0
$859$	−1.53144	−0.0522522	−0.0261261	−	0.999659i	$-0.508317\pi$
$859$	−1.53144	−0.0522522	−0.0261261	+	0.999659i	$0.508317\pi$
$860$	0	0
$861$	−24.0628	−0.820058
$862$	0	0
$863$	−17.0118	−0.579088	−0.289544	−	0.957165i	$-0.593504\pi$
$863$	−17.0118	−0.579088	−0.289544	+	0.957165i	$0.593504\pi$
$864$	0	0
$865$	−97.2899	−3.30796
$866$	0	0
$867$	−67.1331	−2.27996
$868$	0	0
$869$	3.34861	0.113594
$870$	0	0
$871$	−17.7102	−0.600086
$872$	0	0
$873$	−12.6635	−0.428594
$874$	0	0
$875$	−46.0055	−1.55527
$876$	0	0
$877$	−55.5390	−1.87542	−0.937709	−	0.347423i	$-0.887057\pi$
$877$	−55.5390	−1.87542	−0.937709	+	0.347423i	$0.887057\pi$
$878$	0	0
$879$	−66.6866	−2.24928
$880$	0	0
$881$	31.4180	1.05850	0.529250	−	0.848466i	$-0.322473\pi$
$881$	31.4180	1.05850	0.529250	+	0.848466i	$0.322473\pi$
$882$	0	0
$883$	21.9198	0.737660	0.368830	−	0.929497i	$-0.379758\pi$
$883$	21.9198	0.737660	0.368830	+	0.929497i	$0.379758\pi$
$884$	0	0
$885$	103.034	3.46345
$886$	0	0
$887$	25.7177	0.863514	0.431757	−	0.901990i	$-0.357894\pi$
$887$	25.7177	0.863514	0.431757	+	0.901990i	$0.357894\pi$
$888$	0	0
$889$	3.57802	0.120003
$890$	0	0
$891$	−21.5380	−0.721550
$892$	0	0
$893$	1.48761	0.0497811
$894$	0	0
$895$	9.61704	0.321462
$896$	0	0
$897$	−59.7296	−1.99431
$898$	0	0
$899$	−4.38296	−0.146180
$900$	0	0
$901$	1.94718	0.0648700
$902$	0	0
$903$	−58.0077	−1.93037
$904$	0	0
$905$	44.4218	1.47663
$906$	0	0
$907$	40.3151	1.33864	0.669320	−	0.742974i	$-0.266585\pi$
$907$	40.3151	1.33864	0.669320	+	0.742974i	$0.266585\pi$
$908$	0	0
$909$	−73.3786	−2.43382
$910$	0	0
$911$	−4.01257	−0.132942	−0.0664712	−	0.997788i	$-0.521174\pi$
$911$	−4.01257	−0.132942	−0.0664712	+	0.997788i	$0.521174\pi$
$912$	0	0
$913$	−54.3967	−1.80027
$914$	0	0
$915$	−7.61784	−0.251838
$916$	0	0
$917$	−2.28999	−0.0756221
$918$	0	0
$919$	40.5456	1.33748	0.668738	−	0.743498i	$-0.266835\pi$
$919$	40.5456	1.33748	0.668738	+	0.743498i	$0.266835\pi$
$920$	0	0
$921$	−57.8008	−1.90460
$922$	0	0
$923$	29.6303	0.975292
$924$	0	0
$925$	13.4505	0.442251
$926$	0	0
$927$	46.5996	1.53053
$928$	0	0
$929$	−37.5645	−1.23245	−0.616225	−	0.787570i	$-0.711339\pi$
$929$	−37.5645	−1.23245	−0.616225	+	0.787570i	$0.711339\pi$
$930$	0	0
$931$	−2.85473	−0.0935601
$932$	0	0
$933$	−59.9072	−1.96127
$934$	0	0
$935$	−86.9571	−2.84380
$936$	0	0
$937$	25.6424	0.837700	0.418850	−	0.908056i	$-0.362433\pi$
$937$	25.6424	0.837700	0.418850	+	0.908056i	$0.362433\pi$
$938$	0	0
$939$	16.6165	0.542260
$940$	0	0
$941$	53.3462	1.73904	0.869518	−	0.493902i	$-0.164430\pi$
$941$	53.3462	1.73904	0.869518	+	0.493902i	$0.164430\pi$
$942$	0	0
$943$	31.2472	1.01755
$944$	0	0
$945$	14.5086	0.471965
$946$	0	0
$947$	49.2747	1.60121	0.800606	−	0.599191i	$-0.204511\pi$
$947$	49.2747	1.60121	0.800606	+	0.599191i	$0.204511\pi$
$948$	0	0
$949$	14.6513	0.475601
$950$	0	0
$951$	−13.3183	−0.431874
$952$	0	0
$953$	−9.91994	−0.321338	−0.160669	−	0.987008i	$-0.551365\pi$
$953$	−9.91994	−0.321338	−0.160669	+	0.987008i	$0.551365\pi$
$954$	0	0
$955$	−40.2867	−1.30365
$956$	0	0
$957$	−34.2481	−1.10709
$958$	0	0
$959$	38.8246	1.25371
$960$	0	0
$961$	−29.7705	−0.960338
$962$	0	0
$963$	−3.92000	−0.126320
$964$	0	0
$965$	−15.3950	−0.495584
$966$	0	0
$967$	42.6242	1.37070	0.685351	−	0.728213i	$-0.259649\pi$
$967$	42.6242	1.37070	0.685351	+	0.728213i	$0.259649\pi$
$968$	0	0
$969$	−16.9564	−0.544719
$970$	0	0
$971$	32.2181	1.03393	0.516963	−	0.856007i	$-0.327062\pi$
$971$	32.2181	1.03393	0.516963	+	0.856007i	$0.327062\pi$
$972$	0	0
$973$	−42.3898	−1.35895
$974$	0	0
$975$	−93.4437	−2.99259
$976$	0	0
$977$	−21.4160	−0.685159	−0.342580	−	0.939489i	$-0.611301\pi$
$977$	−21.4160	−0.685159	−0.342580	+	0.939489i	$0.611301\pi$
$978$	0	0
$979$	28.9685	0.925837
$980$	0	0
$981$	22.4871	0.717956
$982$	0	0
$983$	−40.3051	−1.28553	−0.642766	−	0.766063i	$-0.722213\pi$
$983$	−40.3051	−1.28553	−0.642766	+	0.766063i	$0.722213\pi$
$984$	0	0
$985$	41.3416	1.31725
$986$	0	0
$987$	−7.83686	−0.249450
$988$	0	0
$989$	75.3270	2.39526
$990$	0	0
$991$	50.4213	1.60168	0.800842	−	0.598876i	$-0.204386\pi$
$991$	50.4213	1.60168	0.800842	+	0.598876i	$0.204386\pi$
$992$	0	0
$993$	42.7009	1.35507
$994$	0	0
$995$	−44.4892	−1.41040
$996$	0	0
$997$	−13.5479	−0.429067	−0.214533	−	0.976717i	$-0.568823\pi$
$997$	−13.5479	−0.429067	−0.214533	+	0.976717i	$0.568823\pi$
$998$	0	0
$999$	−2.26011	−0.0715066

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.e.1.5	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.e.1.5	✓	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58747 q^{3} -3.96262 q^{5} +2.03599 q^{7} +3.69501 q^{9} +O(q^{10})\)	\(q-2.58747 q^{3} -3.96262 q^{5} +2.03599 q^{7} +3.69501 q^{9} +3.34861 q^{11} +3.37440 q^{13} +10.2532 q^{15} +6.55328 q^{17} +1.00000 q^{19} -5.26808 q^{21} +6.84096 q^{23} +10.7023 q^{25} -1.79832 q^{27} +3.95273 q^{29} -1.10884 q^{31} -8.66443 q^{33} -8.06786 q^{35} +1.25679 q^{37} -8.73117 q^{39} +4.56766 q^{41} +11.0112 q^{43} -14.6419 q^{45} +1.48761 q^{47} -2.85473 q^{49} -16.9564 q^{51} +0.297131 q^{53} -13.2693 q^{55} -2.58747 q^{57} +10.0490 q^{59} -0.742975 q^{61} +7.52302 q^{63} -13.3714 q^{65} -5.24839 q^{67} -17.7008 q^{69} +8.78090 q^{71} +4.34190 q^{73} -27.6920 q^{75} +6.81775 q^{77} +1.00000 q^{79} -6.43193 q^{81} -16.2445 q^{83} -25.9681 q^{85} -10.2276 q^{87} +8.65090 q^{89} +6.87026 q^{91} +2.86910 q^{93} -3.96262 q^{95} -3.42719 q^{97} +12.3732 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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