LMFDB - Embedded newform 8024.2.a.w.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 29 x^{18} + 51 x^{17} + 341 x^{16} - 514 x^{15} - 2114 x^{14} + 2629 x^{13} + \cdots + 4 $ x^20 - 2x^19 - 29x^18 + 51x^17 + 341x^16 - 514x^15 - 2114x^14 + 2629x^13 + 7532x^12 - 7342x^11 - 15668x^10 + 11260x^9 + 18269x^8 - 9059x^7 - 10680x^6 + 3386x^5 + 2473x^4 - 427x^3 - 193x^2 + 18*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-1.36564$ of defining polynomial
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.36564 q^{3} +2.75901 q^{5} -0.969675 q^{7} -1.13502 q^{9} +O(q^{10})$	$q+1.36564 q^{3} +2.75901 q^{5} -0.969675 q^{7} -1.13502 q^{9} +0.196912 q^{11} -4.14111 q^{13} +3.76783 q^{15} +1.00000 q^{17} +0.254948 q^{19} -1.32423 q^{21} -1.49251 q^{23} +2.61216 q^{25} -5.64696 q^{27} -1.54069 q^{29} +0.552038 q^{31} +0.268912 q^{33} -2.67535 q^{35} -6.54827 q^{37} -5.65528 q^{39} -0.607102 q^{41} -4.06298 q^{43} -3.13153 q^{45} -5.48901 q^{47} -6.05973 q^{49} +1.36564 q^{51} +7.97438 q^{53} +0.543283 q^{55} +0.348168 q^{57} -1.00000 q^{59} +0.656892 q^{61} +1.10060 q^{63} -11.4254 q^{65} +8.80482 q^{67} -2.03823 q^{69} -1.04562 q^{71} -14.4390 q^{73} +3.56728 q^{75} -0.190941 q^{77} -7.84467 q^{79} -4.30669 q^{81} -18.0860 q^{83} +2.75901 q^{85} -2.10404 q^{87} +8.88258 q^{89} +4.01553 q^{91} +0.753887 q^{93} +0.703404 q^{95} +17.2469 q^{97} -0.223498 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) $ 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9	$ 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9} + 8 q^{11} - 3 q^{13} - 2 q^{15} + 20 q^{17} - 9 q^{19} - 31 q^{21} - 8 q^{23} - 4 q^{25} - 17 q^{27} - 37 q^{29} - 9 q^{31} - 8 q^{33} - 8 q^{35} - 11 q^{37} - 12 q^{39} - 27 q^{41} + 17 q^{43} - 4 q^{45} + 7 q^{47} - 9 q^{49} - 2 q^{51} - q^{53} - 21 q^{55} - 57 q^{57} - 20 q^{59} - 12 q^{61} - 14 q^{63} - 59 q^{65} - 26 q^{67} + 14 q^{69} - 3 q^{71} - 45 q^{73} + 15 q^{75} + 6 q^{77} + 5 q^{79} + 12 q^{81} - 17 q^{83} - 2 q^{85} - 32 q^{87} - 40 q^{89} - 36 q^{91} - 4 q^{93} + 5 q^{95} - 77 q^{97} + 54 q^{99}+O(q^{100}) $ 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9 + 8 * q^11 - 3 * q^13 - 2 * q^15 + 20 * q^17 - 9 * q^19 - 31 * q^21 - 8 * q^23 - 4 * q^25 - 17 * q^27 - 37 * q^29 - 9 * q^31 - 8 * q^33 - 8 * q^35 - 11 * q^37 - 12 * q^39 - 27 * q^41 + 17 * q^43 - 4 * q^45 + 7 * q^47 - 9 * q^49 - 2 * q^51 - q^53 - 21 * q^55 - 57 * q^57 - 20 * q^59 - 12 * q^61 - 14 * q^63 - 59 * q^65 - 26 * q^67 + 14 * q^69 - 3 * q^71 - 45 * q^73 + 15 * q^75 + 6 * q^77 + 5 * q^79 + 12 * q^81 - 17 * q^83 - 2 * q^85 - 32 * q^87 - 40 * q^89 - 36 * q^91 - 4 * q^93 + 5 * q^95 - 77 * q^97 + 54 * 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
$3$	1.36564	0.788455	0.394227	−	0.919013i	$-0.371012\pi$
$3$	1.36564	0.788455	0.394227	+	0.919013i	$0.371012\pi$
$4$	0	0
$5$	2.75901	1.23387	0.616934	−	0.787015i	$-0.288374\pi$
$5$	2.75901	1.23387	0.616934	+	0.787015i	$0.288374\pi$
$6$	0	0
$7$	−0.969675	−0.366503	−0.183251	−	0.983066i	$-0.558662\pi$
$7$	−0.969675	−0.366503	−0.183251	+	0.983066i	$0.558662\pi$
$8$	0	0
$9$	−1.13502	−0.378339
$10$	0	0
$11$	0.196912	0.0593712	0.0296856	−	0.999559i	$-0.490549\pi$
$11$	0.196912	0.0593712	0.0296856	+	0.999559i	$0.490549\pi$
$12$	0	0
$13$	−4.14111	−1.14854	−0.574268	−	0.818667i	$-0.694713\pi$
$13$	−4.14111	−1.14854	−0.574268	+	0.818667i	$0.694713\pi$
$14$	0	0
$15$	3.76783	0.972850
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.254948	0.0584890	0.0292445	−	0.999572i	$-0.490690\pi$
$19$	0.254948	0.0584890	0.0292445	+	0.999572i	$0.490690\pi$
$20$	0	0
$21$	−1.32423	−0.288971
$22$	0	0
$23$	−1.49251	−0.311209	−0.155604	−	0.987819i	$-0.549733\pi$
$23$	−1.49251	−0.311209	−0.155604	+	0.987819i	$0.549733\pi$
$24$	0	0
$25$	2.61216	0.522433
$26$	0	0
$27$	−5.64696	−1.08676
$28$	0	0
$29$	−1.54069	−0.286099	−0.143050	−	0.989716i	$-0.545691\pi$
$29$	−1.54069	−0.286099	−0.143050	+	0.989716i	$0.545691\pi$
$30$	0	0
$31$	0.552038	0.0991489	0.0495744	−	0.998770i	$-0.484213\pi$
$31$	0.552038	0.0991489	0.0495744	+	0.998770i	$0.484213\pi$
$32$	0	0
$33$	0.268912	0.0468115
$34$	0	0
$35$	−2.67535	−0.452216
$36$	0	0
$37$	−6.54827	−1.07653	−0.538265	−	0.842776i	$-0.680920\pi$
$37$	−6.54827	−1.07653	−0.538265	+	0.842776i	$0.680920\pi$
$38$	0	0
$39$	−5.65528	−0.905569
$40$	0	0
$41$	−0.607102	−0.0948135	−0.0474067	−	0.998876i	$-0.515096\pi$
$41$	−0.607102	−0.0948135	−0.0474067	+	0.998876i	$0.515096\pi$
$42$	0	0
$43$	−4.06298	−0.619599	−0.309799	−	0.950802i	$-0.600262\pi$
$43$	−4.06298	−0.619599	−0.309799	+	0.950802i	$0.600262\pi$
$44$	0	0
$45$	−3.13153	−0.466821
$46$	0	0
$47$	−5.48901	−0.800654	−0.400327	−	0.916372i	$-0.631103\pi$
$47$	−5.48901	−0.800654	−0.400327	+	0.916372i	$0.631103\pi$
$48$	0	0
$49$	−6.05973	−0.865676
$50$	0	0
$51$	1.36564	0.191228
$52$	0	0
$53$	7.97438	1.09537	0.547683	−	0.836686i	$-0.315510\pi$
$53$	7.97438	1.09537	0.547683	+	0.836686i	$0.315510\pi$
$54$	0	0
$55$	0.543283	0.0732563
$56$	0	0
$57$	0.348168	0.0461159
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	0.656892	0.0841064	0.0420532	−	0.999115i	$-0.486610\pi$
$61$	0.656892	0.0841064	0.0420532	+	0.999115i	$0.486610\pi$
$62$	0	0
$63$	1.10060	0.138662
$64$	0	0
$65$	−11.4254	−1.41714
$66$	0	0
$67$	8.80482	1.07568	0.537840	−	0.843047i	$-0.319241\pi$
$67$	8.80482	1.07568	0.537840	+	0.843047i	$0.319241\pi$
$68$	0	0
$69$	−2.03823	−0.245374
$70$	0	0
$71$	−1.04562	−0.124092	−0.0620459	−	0.998073i	$-0.519763\pi$
$71$	−1.04562	−0.124092	−0.0620459	+	0.998073i	$0.519763\pi$
$72$	0	0
$73$	−14.4390	−1.68996	−0.844980	−	0.534798i	$-0.820388\pi$
$73$	−14.4390	−1.68996	−0.844980	+	0.534798i	$0.820388\pi$
$74$	0	0
$75$	3.56728	0.411915
$76$	0	0
$77$	−0.190941	−0.0217597
$78$	0	0
$79$	−7.84467	−0.882594	−0.441297	−	0.897361i	$-0.645481\pi$
$79$	−7.84467	−0.882594	−0.441297	+	0.897361i	$0.645481\pi$
$80$	0	0
$81$	−4.30669	−0.478521
$82$	0	0
$83$	−18.0860	−1.98520	−0.992599	−	0.121436i	$-0.961250\pi$
$83$	−18.0860	−1.98520	−0.992599	+	0.121436i	$0.961250\pi$
$84$	0	0
$85$	2.75901	0.299257
$86$	0	0
$87$	−2.10404	−0.225576
$88$	0	0
$89$	8.88258	0.941552	0.470776	−	0.882253i	$-0.343974\pi$
$89$	8.88258	0.941552	0.470776	+	0.882253i	$0.343974\pi$
$90$	0	0
$91$	4.01553	0.420942
$92$	0	0
$93$	0.753887	0.0781744
$94$	0	0
$95$	0.703404	0.0721678
$96$	0	0
$97$	17.2469	1.75115	0.875576	−	0.483080i	$-0.160482\pi$
$97$	17.2469	1.75115	0.875576	+	0.483080i	$0.160482\pi$
$98$	0	0
$99$	−0.223498	−0.0224624
$100$	0	0
$101$	−10.0818	−1.00317	−0.501587	−	0.865107i	$-0.667250\pi$
$101$	−10.0818	−1.00317	−0.501587	+	0.865107i	$0.667250\pi$
$102$	0	0
$103$	7.99299	0.787572	0.393786	−	0.919202i	$-0.371165\pi$
$103$	7.99299	0.787572	0.393786	+	0.919202i	$0.371165\pi$
$104$	0	0
$105$	−3.65357	−0.356552
$106$	0	0
$107$	6.24976	0.604187	0.302094	−	0.953278i	$-0.402314\pi$
$107$	6.24976	0.604187	0.302094	+	0.953278i	$0.402314\pi$
$108$	0	0
$109$	3.81234	0.365156	0.182578	−	0.983191i	$-0.441556\pi$
$109$	3.81234	0.365156	0.182578	+	0.983191i	$0.441556\pi$
$110$	0	0
$111$	−8.94261	−0.848795
$112$	0	0
$113$	1.27724	0.120153	0.0600765	−	0.998194i	$-0.480866\pi$
$113$	1.27724	0.120153	0.0600765	+	0.998194i	$0.480866\pi$
$114$	0	0
$115$	−4.11785	−0.383991
$116$	0	0
$117$	4.70023	0.434536
$118$	0	0
$119$	−0.969675	−0.0888899
$120$	0	0
$121$	−10.9612	−0.996475
$122$	0	0
$123$	−0.829086	−0.0747561
$124$	0	0
$125$	−6.58808	−0.589256
$126$	0	0
$127$	5.64995	0.501352	0.250676	−	0.968071i	$-0.419347\pi$
$127$	5.64995	0.501352	0.250676	+	0.968071i	$0.419347\pi$
$128$	0	0
$129$	−5.54859	−0.488526
$130$	0	0
$131$	−17.7206	−1.54826	−0.774128	−	0.633029i	$-0.781812\pi$
$131$	−17.7206	−1.54826	−0.774128	+	0.633029i	$0.781812\pi$
$132$	0	0
$133$	−0.247216	−0.0214364
$134$	0	0
$135$	−15.5800	−1.34092
$136$	0	0
$137$	−13.7267	−1.17275	−0.586375	−	0.810040i	$-0.699445\pi$
$137$	−13.7267	−1.17275	−0.586375	+	0.810040i	$0.699445\pi$
$138$	0	0
$139$	−6.30475	−0.534762	−0.267381	−	0.963591i	$-0.586158\pi$
$139$	−6.30475	−0.534762	−0.267381	+	0.963591i	$0.586158\pi$
$140$	0	0
$141$	−7.49603	−0.631280
$142$	0	0
$143$	−0.815434	−0.0681900
$144$	0	0
$145$	−4.25079	−0.353009
$146$	0	0
$147$	−8.27544	−0.682546
$148$	0	0
$149$	10.2966	0.843526	0.421763	−	0.906706i	$-0.361411\pi$
$149$	10.2966	0.843526	0.421763	+	0.906706i	$0.361411\pi$
$150$	0	0
$151$	16.9177	1.37675	0.688373	−	0.725357i	$-0.258325\pi$
$151$	16.9177	1.37675	0.688373	+	0.725357i	$0.258325\pi$
$152$	0	0
$153$	−1.13502	−0.0917607
$154$	0	0
$155$	1.52308	0.122337
$156$	0	0
$157$	−2.77629	−0.221572	−0.110786	−	0.993844i	$-0.535337\pi$
$157$	−2.77629	−0.221572	−0.110786	+	0.993844i	$0.535337\pi$
$158$	0	0
$159$	10.8902	0.863646
$160$	0	0
$161$	1.44725	0.114059
$162$	0	0
$163$	13.7493	1.07693	0.538463	−	0.842649i	$-0.319005\pi$
$163$	13.7493	1.07693	0.538463	+	0.842649i	$0.319005\pi$
$164$	0	0
$165$	0.741931	0.0577593
$166$	0	0
$167$	−19.7223	−1.52616	−0.763078	−	0.646307i	$-0.776313\pi$
$167$	−19.7223	−1.52616	−0.763078	+	0.646307i	$0.776313\pi$
$168$	0	0
$169$	4.14877	0.319136
$170$	0	0
$171$	−0.289370	−0.0221287
$172$	0	0
$173$	10.7583	0.817936	0.408968	−	0.912549i	$-0.365889\pi$
$173$	10.7583	0.817936	0.408968	+	0.912549i	$0.365889\pi$
$174$	0	0
$175$	−2.53295	−0.191473
$176$	0	0
$177$	−1.36564	−0.102648
$178$	0	0
$179$	19.2735	1.44057	0.720286	−	0.693677i	$-0.244011\pi$
$179$	19.2735	1.44057	0.720286	+	0.693677i	$0.244011\pi$
$180$	0	0
$181$	−19.1268	−1.42169	−0.710844	−	0.703350i	$-0.751687\pi$
$181$	−19.1268	−1.42169	−0.710844	+	0.703350i	$0.751687\pi$
$182$	0	0
$183$	0.897081	0.0663141
$184$	0	0
$185$	−18.0668	−1.32830
$186$	0	0
$187$	0.196912	0.0143996
$188$	0	0
$189$	5.47571	0.398300
$190$	0	0
$191$	−6.98275	−0.505254	−0.252627	−	0.967564i	$-0.581295\pi$
$191$	−6.98275	−0.505254	−0.252627	+	0.967564i	$0.581295\pi$
$192$	0	0
$193$	−22.8300	−1.64334	−0.821670	−	0.569964i	$-0.806957\pi$
$193$	−22.8300	−1.64334	−0.821670	+	0.569964i	$0.806957\pi$
$194$	0	0
$195$	−15.6030	−1.11735
$196$	0	0
$197$	−13.7727	−0.981261	−0.490631	−	0.871368i	$-0.663234\pi$
$197$	−13.7727	−0.981261	−0.490631	+	0.871368i	$0.663234\pi$
$198$	0	0
$199$	2.79181	0.197906	0.0989531	−	0.995092i	$-0.468451\pi$
$199$	2.79181	0.197906	0.0989531	+	0.995092i	$0.468451\pi$
$200$	0	0
$201$	12.0242	0.848125
$202$	0	0
$203$	1.49397	0.104856
$204$	0	0
$205$	−1.67500	−0.116987
$206$	0	0
$207$	1.69402	0.117742
$208$	0	0
$209$	0.0502022	0.00347256
$210$	0	0
$211$	18.0105	1.23989	0.619947	−	0.784644i	$-0.287154\pi$
$211$	18.0105	1.23989	0.619947	+	0.784644i	$0.287154\pi$
$212$	0	0
$213$	−1.42794	−0.0978408
$214$	0	0
$215$	−11.2098	−0.764504
$216$	0	0
$217$	−0.535297	−0.0363383
$218$	0	0
$219$	−19.7186	−1.33246
$220$	0	0
$221$	−4.14111	−0.278561
$222$	0	0
$223$	−16.8363	−1.12744	−0.563721	−	0.825966i	$-0.690631\pi$
$223$	−16.8363	−1.12744	−0.563721	+	0.825966i	$0.690631\pi$
$224$	0	0
$225$	−2.96485	−0.197657
$226$	0	0
$227$	7.68056	0.509777	0.254888	−	0.966970i	$-0.417961\pi$
$227$	7.68056	0.509777	0.254888	+	0.966970i	$0.417961\pi$
$228$	0	0
$229$	−8.52953	−0.563647	−0.281824	−	0.959466i	$-0.590939\pi$
$229$	−8.52953	−0.563647	−0.281824	+	0.959466i	$0.590939\pi$
$230$	0	0
$231$	−0.260757	−0.0171565
$232$	0	0
$233$	−24.0487	−1.57548	−0.787740	−	0.616008i	$-0.788749\pi$
$233$	−24.0487	−1.57548	−0.787740	+	0.616008i	$0.788749\pi$
$234$	0	0
$235$	−15.1443	−0.987902
$236$	0	0
$237$	−10.7130	−0.695885
$238$	0	0
$239$	18.3875	1.18939	0.594693	−	0.803953i	$-0.297274\pi$
$239$	18.3875	1.18939	0.594693	+	0.803953i	$0.297274\pi$
$240$	0	0
$241$	−9.53197	−0.614008	−0.307004	−	0.951708i	$-0.599327\pi$
$241$	−9.53197	−0.614008	−0.307004	+	0.951708i	$0.599327\pi$
$242$	0	0
$243$	11.0595	0.709466
$244$	0	0
$245$	−16.7189	−1.06813
$246$	0	0
$247$	−1.05577	−0.0671768
$248$	0	0
$249$	−24.6991	−1.56524
$250$	0	0
$251$	−11.1801	−0.705681	−0.352841	−	0.935683i	$-0.614784\pi$
$251$	−11.1801	−0.705681	−0.352841	+	0.935683i	$0.614784\pi$
$252$	0	0
$253$	−0.293892	−0.0184768
$254$	0	0
$255$	3.76783	0.235951
$256$	0	0
$257$	−25.9172	−1.61667	−0.808336	−	0.588722i	$-0.799631\pi$
$257$	−25.9172	−1.61667	−0.808336	+	0.588722i	$0.799631\pi$
$258$	0	0
$259$	6.34969	0.394551
$260$	0	0
$261$	1.74871	0.108242
$262$	0	0
$263$	21.6733	1.33643	0.668216	−	0.743967i	$-0.267058\pi$
$263$	21.6733	1.33643	0.668216	+	0.743967i	$0.267058\pi$
$264$	0	0
$265$	22.0014	1.35154
$266$	0	0
$267$	12.1304	0.742371
$268$	0	0
$269$	−10.6429	−0.648908	−0.324454	−	0.945901i	$-0.605181\pi$
$269$	−10.6429	−0.648908	−0.324454	+	0.945901i	$0.605181\pi$
$270$	0	0
$271$	15.0666	0.915228	0.457614	−	0.889151i	$-0.348704\pi$
$271$	15.0666	0.915228	0.457614	+	0.889151i	$0.348704\pi$
$272$	0	0
$273$	5.48378	0.331893
$274$	0	0
$275$	0.514366	0.0310174
$276$	0	0
$277$	−1.17069	−0.0703398	−0.0351699	−	0.999381i	$-0.511197\pi$
$277$	−1.17069	−0.0703398	−0.0351699	+	0.999381i	$0.511197\pi$
$278$	0	0
$279$	−0.626572	−0.0375119
$280$	0	0
$281$	−0.580856	−0.0346510	−0.0173255	−	0.999850i	$-0.505515\pi$
$281$	−0.580856	−0.0346510	−0.0173255	+	0.999850i	$0.505515\pi$
$282$	0	0
$283$	14.1652	0.842036	0.421018	−	0.907052i	$-0.361673\pi$
$283$	14.1652	0.842036	0.421018	+	0.907052i	$0.361673\pi$
$284$	0	0
$285$	0.960600	0.0569010
$286$	0	0
$287$	0.588692	0.0347494
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	23.5531	1.38070
$292$	0	0
$293$	26.6034	1.55419	0.777095	−	0.629383i	$-0.216692\pi$
$293$	26.6034	1.55419	0.777095	+	0.629383i	$0.216692\pi$
$294$	0	0
$295$	−2.75901	−0.160636
$296$	0	0
$297$	−1.11195	−0.0645221
$298$	0	0
$299$	6.18063	0.357435
$300$	0	0
$301$	3.93977	0.227085
$302$	0	0
$303$	−13.7681	−0.790957
$304$	0	0
$305$	1.81237	0.103776
$306$	0	0
$307$	−17.6615	−1.00800	−0.503998	−	0.863705i	$-0.668138\pi$
$307$	−17.6615	−1.00800	−0.503998	+	0.863705i	$0.668138\pi$
$308$	0	0
$309$	10.9156	0.620965
$310$	0	0
$311$	−21.6520	−1.22777	−0.613885	−	0.789396i	$-0.710394\pi$
$311$	−21.6520	−1.22777	−0.613885	+	0.789396i	$0.710394\pi$
$312$	0	0
$313$	24.1856	1.36705	0.683525	−	0.729927i	$-0.260446\pi$
$313$	24.1856	1.36705	0.683525	+	0.729927i	$0.260446\pi$
$314$	0	0
$315$	3.03656	0.171091
$316$	0	0
$317$	22.3222	1.25374	0.626870	−	0.779124i	$-0.284336\pi$
$317$	22.3222	1.25374	0.626870	+	0.779124i	$0.284336\pi$
$318$	0	0
$319$	−0.303381	−0.0169861
$320$	0	0
$321$	8.53495	0.476374
$322$	0	0
$323$	0.254948	0.0141857
$324$	0	0
$325$	−10.8172	−0.600033
$326$	0	0
$327$	5.20630	0.287909
$328$	0	0
$329$	5.32255	0.293442
$330$	0	0
$331$	22.7229	1.24897	0.624483	−	0.781038i	$-0.285310\pi$
$331$	22.7229	1.24897	0.624483	+	0.781038i	$0.285310\pi$
$332$	0	0
$333$	7.43240	0.407293
$334$	0	0
$335$	24.2926	1.32725
$336$	0	0
$337$	−8.28632	−0.451385	−0.225692	−	0.974199i	$-0.572464\pi$
$337$	−8.28632	−0.451385	−0.225692	+	0.974199i	$0.572464\pi$
$338$	0	0
$339$	1.74426	0.0947353
$340$	0	0
$341$	0.108703	0.00588659
$342$	0	0
$343$	12.6637	0.683775
$344$	0	0
$345$	−5.62351	−0.302760
$346$	0	0
$347$	−11.4651	−0.615478	−0.307739	−	0.951471i	$-0.599572\pi$
$347$	−11.4651	−0.615478	−0.307739	+	0.951471i	$0.599572\pi$
$348$	0	0
$349$	−30.8608	−1.65194	−0.825971	−	0.563712i	$-0.809373\pi$
$349$	−30.8608	−1.65194	−0.825971	+	0.563712i	$0.809373\pi$
$350$	0	0
$351$	23.3847	1.24818
$352$	0	0
$353$	29.5523	1.57291	0.786454	−	0.617648i	$-0.211915\pi$
$353$	29.5523	1.57291	0.786454	+	0.617648i	$0.211915\pi$
$354$	0	0
$355$	−2.88487	−0.153113
$356$	0	0
$357$	−1.32423	−0.0700857
$358$	0	0
$359$	−25.3792	−1.33946	−0.669732	−	0.742603i	$-0.733591\pi$
$359$	−25.3792	−1.33946	−0.669732	+	0.742603i	$0.733591\pi$
$360$	0	0
$361$	−18.9350	−0.996579
$362$	0	0
$363$	−14.9691	−0.785676
$364$	0	0
$365$	−39.8375	−2.08519
$366$	0	0
$367$	13.8842	0.724749	0.362374	−	0.932033i	$-0.381966\pi$
$367$	13.8842	0.724749	0.362374	+	0.932033i	$0.381966\pi$
$368$	0	0
$369$	0.689071	0.0358716
$370$	0	0
$371$	−7.73255	−0.401454
$372$	0	0
$373$	31.1168	1.61117	0.805583	−	0.592483i	$-0.201852\pi$
$373$	31.1168	1.61117	0.805583	+	0.592483i	$0.201852\pi$
$374$	0	0
$375$	−8.99697	−0.464601
$376$	0	0
$377$	6.38017	0.328596
$378$	0	0
$379$	−19.8688	−1.02059	−0.510297	−	0.859998i	$-0.670464\pi$
$379$	−19.8688	−1.02059	−0.510297	+	0.859998i	$0.670464\pi$
$380$	0	0
$381$	7.71582	0.395294
$382$	0	0
$383$	26.8512	1.37203	0.686017	−	0.727585i	$-0.259357\pi$
$383$	26.8512	1.37203	0.686017	+	0.727585i	$0.259357\pi$
$384$	0	0
$385$	−0.526808	−0.0268486
$386$	0	0
$387$	4.61155	0.234418
$388$	0	0
$389$	3.21937	0.163229	0.0816143	−	0.996664i	$-0.473992\pi$
$389$	3.21937	0.163229	0.0816143	+	0.996664i	$0.473992\pi$
$390$	0	0
$391$	−1.49251	−0.0754793
$392$	0	0
$393$	−24.2000	−1.22073
$394$	0	0
$395$	−21.6436	−1.08901
$396$	0	0
$397$	−32.1291	−1.61252	−0.806258	−	0.591565i	$-0.798510\pi$
$397$	−32.1291	−1.61252	−0.806258	+	0.591565i	$0.798510\pi$
$398$	0	0
$399$	−0.337609	−0.0169016
$400$	0	0
$401$	−11.4694	−0.572755	−0.286378	−	0.958117i	$-0.592451\pi$
$401$	−11.4694	−0.572755	−0.286378	+	0.958117i	$0.592451\pi$
$402$	0	0
$403$	−2.28605	−0.113876
$404$	0	0
$405$	−11.8822	−0.590432
$406$	0	0
$407$	−1.28943	−0.0639148
$408$	0	0
$409$	17.6643	0.873445	0.436723	−	0.899596i	$-0.356139\pi$
$409$	17.6643	0.873445	0.436723	+	0.899596i	$0.356139\pi$
$410$	0	0
$411$	−18.7458	−0.924660
$412$	0	0
$413$	0.969675	0.0477146
$414$	0	0
$415$	−49.8996	−2.44947
$416$	0	0
$417$	−8.61005	−0.421636
$418$	0	0
$419$	30.7205	1.50079	0.750396	−	0.660988i	$-0.229863\pi$
$419$	30.7205	1.50079	0.750396	+	0.660988i	$0.229863\pi$
$420$	0	0
$421$	11.0319	0.537660	0.268830	−	0.963188i	$-0.413363\pi$
$421$	11.0319	0.537660	0.268830	+	0.963188i	$0.413363\pi$
$422$	0	0
$423$	6.23011	0.302919
$424$	0	0
$425$	2.61216	0.126709
$426$	0	0
$427$	−0.636971	−0.0308252
$428$	0	0
$429$	−1.11359	−0.0537647
$430$	0	0
$431$	−2.02545	−0.0975626	−0.0487813	−	0.998809i	$-0.515534\pi$
$431$	−2.02545	−0.0975626	−0.0487813	+	0.998809i	$0.515534\pi$
$432$	0	0
$433$	−12.5712	−0.604133	−0.302067	−	0.953287i	$-0.597677\pi$
$433$	−12.5712	−0.604133	−0.302067	+	0.953287i	$0.597677\pi$
$434$	0	0
$435$	−5.80507	−0.278332
$436$	0	0
$437$	−0.380511	−0.0182023
$438$	0	0
$439$	11.8092	0.563625	0.281812	−	0.959470i	$-0.409064\pi$
$439$	11.8092	0.563625	0.281812	+	0.959470i	$0.409064\pi$
$440$	0	0
$441$	6.87790	0.327519
$442$	0	0
$443$	25.7203	1.22201	0.611003	−	0.791628i	$-0.290766\pi$
$443$	25.7203	1.22201	0.611003	+	0.791628i	$0.290766\pi$
$444$	0	0
$445$	24.5072	1.16175
$446$	0	0
$447$	14.0614	0.665082
$448$	0	0
$449$	41.8367	1.97440	0.987198	−	0.159500i	$-0.0509883\pi$
$449$	41.8367	1.97440	0.987198	+	0.159500i	$0.0509883\pi$
$450$	0	0
$451$	−0.119546	−0.00562919
$452$	0	0
$453$	23.1036	1.08550
$454$	0	0
$455$	11.0789	0.519387
$456$	0	0
$457$	32.3075	1.51128	0.755640	−	0.654987i	$-0.227326\pi$
$457$	32.3075	1.51128	0.755640	+	0.654987i	$0.227326\pi$
$458$	0	0
$459$	−5.64696	−0.263578
$460$	0	0
$461$	−36.7805	−1.71304	−0.856519	−	0.516115i	$-0.827378\pi$
$461$	−36.7805	−1.71304	−0.856519	+	0.516115i	$0.827378\pi$
$462$	0	0
$463$	38.1538	1.77316	0.886579	−	0.462577i	$-0.153075\pi$
$463$	38.1538	1.77316	0.886579	+	0.462577i	$0.153075\pi$
$464$	0	0
$465$	2.07998	0.0964570
$466$	0	0
$467$	−26.0339	−1.20471	−0.602353	−	0.798230i	$-0.705770\pi$
$467$	−26.0339	−1.20471	−0.602353	+	0.798230i	$0.705770\pi$
$468$	0	0
$469$	−8.53781	−0.394239
$470$	0	0
$471$	−3.79142	−0.174700
$472$	0	0
$473$	−0.800050	−0.0367863
$474$	0	0
$475$	0.665965	0.0305566
$476$	0	0
$477$	−9.05105	−0.414419
$478$	0	0
$479$	−18.5051	−0.845521	−0.422760	−	0.906242i	$-0.638939\pi$
$479$	−18.5051	−0.845521	−0.422760	+	0.906242i	$0.638939\pi$
$480$	0	0
$481$	27.1171	1.23643
$482$	0	0
$483$	1.97642	0.0899303
$484$	0	0
$485$	47.5843	2.16069
$486$	0	0
$487$	−1.48754	−0.0674067	−0.0337034	−	0.999432i	$-0.510730\pi$
$487$	−1.48754	−0.0674067	−0.0337034	+	0.999432i	$0.510730\pi$
$488$	0	0
$489$	18.7766	0.849107
$490$	0	0
$491$	−39.4850	−1.78193	−0.890966	−	0.454070i	$-0.849972\pi$
$491$	−39.4850	−1.78193	−0.890966	+	0.454070i	$0.849972\pi$
$492$	0	0
$493$	−1.54069	−0.0693893
$494$	0	0
$495$	−0.616635	−0.0277157
$496$	0	0
$497$	1.01391	0.0454800
$498$	0	0
$499$	−19.5049	−0.873161	−0.436581	−	0.899665i	$-0.643811\pi$
$499$	−19.5049	−0.873161	−0.436581	+	0.899665i	$0.643811\pi$
$500$	0	0
$501$	−26.9336	−1.20330
$502$	0	0
$503$	31.9496	1.42456	0.712281	−	0.701894i	$-0.247662\pi$
$503$	31.9496	1.42456	0.712281	+	0.701894i	$0.247662\pi$
$504$	0	0
$505$	−27.8158	−1.23779
$506$	0	0
$507$	5.66574	0.251624
$508$	0	0
$509$	−32.7655	−1.45231	−0.726153	−	0.687534i	$-0.758693\pi$
$509$	−32.7655	−1.45231	−0.726153	+	0.687534i	$0.758693\pi$
$510$	0	0
$511$	14.0012	0.619375
$512$	0	0
$513$	−1.43968	−0.0635634
$514$	0	0
$515$	22.0528	0.971761
$516$	0	0
$517$	−1.08085	−0.0475358
$518$	0	0
$519$	14.6920	0.644906
$520$	0	0
$521$	−12.2924	−0.538540	−0.269270	−	0.963065i	$-0.586782\pi$
$521$	−12.2924	−0.538540	−0.269270	+	0.963065i	$0.586782\pi$
$522$	0	0
$523$	15.9573	0.697763	0.348882	−	0.937167i	$-0.386561\pi$
$523$	15.9573	0.697763	0.348882	+	0.937167i	$0.386561\pi$
$524$	0	0
$525$	−3.45911	−0.150968
$526$	0	0
$527$	0.552038	0.0240471
$528$	0	0
$529$	−20.7724	−0.903149
$530$	0	0
$531$	1.13502	0.0492555
$532$	0	0
$533$	2.51408	0.108897
$534$	0	0
$535$	17.2432	0.745488
$536$	0	0
$537$	26.3208	1.13583
$538$	0	0
$539$	−1.19323	−0.0513962
$540$	0	0
$541$	−18.1400	−0.779901	−0.389950	−	0.920836i	$-0.627508\pi$
$541$	−18.1400	−0.779901	−0.389950	+	0.920836i	$0.627508\pi$
$542$	0	0
$543$	−26.1205	−1.12094
$544$	0	0
$545$	10.5183	0.450555
$546$	0	0
$547$	34.5738	1.47827	0.739134	−	0.673558i	$-0.235235\pi$
$547$	34.5738	1.47827	0.739134	+	0.673558i	$0.235235\pi$
$548$	0	0
$549$	−0.745583	−0.0318207
$550$	0	0
$551$	−0.392796	−0.0167337
$552$	0	0
$553$	7.60677	0.323473
$554$	0	0
$555$	−24.6728	−1.04730
$556$	0	0
$557$	−7.05840	−0.299074	−0.149537	−	0.988756i	$-0.547778\pi$
$557$	−7.05840	−0.299074	−0.149537	+	0.988756i	$0.547778\pi$
$558$	0	0
$559$	16.8252	0.711632
$560$	0	0
$561$	0.268912	0.0113535
$562$	0	0
$563$	−23.7042	−0.999011	−0.499506	−	0.866311i	$-0.666485\pi$
$563$	−23.7042	−0.999011	−0.499506	+	0.866311i	$0.666485\pi$
$564$	0	0
$565$	3.52394	0.148253
$566$	0	0
$567$	4.17609	0.175379
$568$	0	0
$569$	−31.3187	−1.31295	−0.656475	−	0.754348i	$-0.727953\pi$
$569$	−31.3187	−1.31295	−0.656475	+	0.754348i	$0.727953\pi$
$570$	0	0
$571$	37.7402	1.57938	0.789689	−	0.613507i	$-0.210242\pi$
$571$	37.7402	1.57938	0.789689	+	0.613507i	$0.210242\pi$
$572$	0	0
$573$	−9.53595	−0.398370
$574$	0	0
$575$	−3.89867	−0.162586
$576$	0	0
$577$	10.6662	0.444042	0.222021	−	0.975042i	$-0.428735\pi$
$577$	10.6662	0.444042	0.222021	+	0.975042i	$0.428735\pi$
$578$	0	0
$579$	−31.1776	−1.29570
$580$	0	0
$581$	17.5376	0.727580
$582$	0	0
$583$	1.57025	0.0650331
$584$	0	0
$585$	12.9680	0.536160
$586$	0	0
$587$	−17.5696	−0.725175	−0.362587	−	0.931950i	$-0.618107\pi$
$587$	−17.5696	−0.725175	−0.362587	+	0.931950i	$0.618107\pi$
$588$	0	0
$589$	0.140741	0.00579912
$590$	0	0
$591$	−18.8085	−0.773680
$592$	0	0
$593$	16.8568	0.692227	0.346114	−	0.938193i	$-0.387501\pi$
$593$	16.8568	0.692227	0.346114	+	0.938193i	$0.387501\pi$
$594$	0	0
$595$	−2.67535	−0.109679
$596$	0	0
$597$	3.81262	0.156040
$598$	0	0
$599$	40.4901	1.65438	0.827191	−	0.561921i	$-0.189937\pi$
$599$	40.4901	1.65438	0.827191	+	0.561921i	$0.189937\pi$
$600$	0	0
$601$	44.5790	1.81842	0.909209	−	0.416341i	$-0.136688\pi$
$601$	44.5790	1.81842	0.909209	+	0.416341i	$0.136688\pi$
$602$	0	0
$603$	−9.99361	−0.406971
$604$	0	0
$605$	−30.2422	−1.22952
$606$	0	0
$607$	35.0045	1.42079	0.710393	−	0.703805i	$-0.248517\pi$
$607$	35.0045	1.42079	0.710393	+	0.703805i	$0.248517\pi$
$608$	0	0
$609$	2.04023	0.0826743
$610$	0	0
$611$	22.7306	0.919580
$612$	0	0
$613$	−17.9483	−0.724926	−0.362463	−	0.931998i	$-0.618064\pi$
$613$	−17.9483	−0.724926	−0.362463	+	0.931998i	$0.618064\pi$
$614$	0	0
$615$	−2.28746	−0.0922393
$616$	0	0
$617$	−1.48117	−0.0596295	−0.0298148	−	0.999555i	$-0.509492\pi$
$617$	−1.48117	−0.0596295	−0.0298148	+	0.999555i	$0.509492\pi$
$618$	0	0
$619$	−33.5914	−1.35015	−0.675076	−	0.737748i	$-0.735889\pi$
$619$	−33.5914	−1.35015	−0.675076	+	0.737748i	$0.735889\pi$
$620$	0	0
$621$	8.42812	0.338209
$622$	0	0
$623$	−8.61321	−0.345081
$624$	0	0
$625$	−31.2374	−1.24950
$626$	0	0
$627$	0.0685584	0.00273796
$628$	0	0
$629$	−6.54827	−0.261097
$630$	0	0
$631$	−3.64082	−0.144939	−0.0724694	−	0.997371i	$-0.523088\pi$
$631$	−3.64082	−0.144939	−0.0724694	+	0.997371i	$0.523088\pi$
$632$	0	0
$633$	24.5959	0.977600
$634$	0	0
$635$	15.5883	0.618603
$636$	0	0
$637$	25.0940	0.994260
$638$	0	0
$639$	1.18679	0.0469488
$640$	0	0
$641$	29.1882	1.15286	0.576432	−	0.817145i	$-0.304444\pi$
$641$	29.1882	1.15286	0.576432	+	0.817145i	$0.304444\pi$
$642$	0	0
$643$	−29.9449	−1.18091	−0.590456	−	0.807070i	$-0.701052\pi$
$643$	−29.9449	−1.18091	−0.590456	+	0.807070i	$0.701052\pi$
$644$	0	0
$645$	−15.3086	−0.602777
$646$	0	0
$647$	−20.4547	−0.804158	−0.402079	−	0.915605i	$-0.631712\pi$
$647$	−20.4547	−0.804158	−0.402079	+	0.915605i	$0.631712\pi$
$648$	0	0
$649$	−0.196912	−0.00772947
$650$	0	0
$651$	−0.731025	−0.0286511
$652$	0	0
$653$	12.3178	0.482031	0.241016	−	0.970521i	$-0.422520\pi$
$653$	12.3178	0.482031	0.241016	+	0.970521i	$0.422520\pi$
$654$	0	0
$655$	−48.8914	−1.91035
$656$	0	0
$657$	16.3885	0.639377
$658$	0	0
$659$	−43.2773	−1.68584	−0.842922	−	0.538035i	$-0.819167\pi$
$659$	−43.2773	−1.68584	−0.842922	+	0.538035i	$0.819167\pi$
$660$	0	0
$661$	24.1989	0.941229	0.470615	−	0.882339i	$-0.344032\pi$
$661$	24.1989	0.941229	0.470615	+	0.882339i	$0.344032\pi$
$662$	0	0
$663$	−5.65528	−0.219633
$664$	0	0
$665$	−0.682073	−0.0264497
$666$	0	0
$667$	2.29949	0.0890367
$668$	0	0
$669$	−22.9924	−0.888936
$670$	0	0
$671$	0.129350	0.00499350
$672$	0	0
$673$	13.1692	0.507633	0.253817	−	0.967252i	$-0.418314\pi$
$673$	13.1692	0.507633	0.253817	+	0.967252i	$0.418314\pi$
$674$	0	0
$675$	−14.7508	−0.567758
$676$	0	0
$677$	−20.9880	−0.806635	−0.403317	−	0.915060i	$-0.632143\pi$
$677$	−20.9880	−0.806635	−0.403317	+	0.915060i	$0.632143\pi$
$678$	0	0
$679$	−16.7238	−0.641802
$680$	0	0
$681$	10.4889	0.401936
$682$	0	0
$683$	−37.6702	−1.44141	−0.720705	−	0.693242i	$-0.756182\pi$
$683$	−37.6702	−1.44141	−0.720705	+	0.693242i	$0.756182\pi$
$684$	0	0
$685$	−37.8721	−1.44702
$686$	0	0
$687$	−11.6483	−0.444411
$688$	0	0
$689$	−33.0228	−1.25807
$690$	0	0
$691$	−11.1240	−0.423175	−0.211588	−	0.977359i	$-0.567863\pi$
$691$	−11.1240	−0.423175	−0.211588	+	0.977359i	$0.567863\pi$
$692$	0	0
$693$	0.216721	0.00823254
$694$	0	0
$695$	−17.3949	−0.659826
$696$	0	0
$697$	−0.607102	−0.0229956
$698$	0	0
$699$	−32.8419	−1.24220
$700$	0	0
$701$	7.47993	0.282513	0.141257	−	0.989973i	$-0.454886\pi$
$701$	7.47993	0.282513	0.141257	+	0.989973i	$0.454886\pi$
$702$	0	0
$703$	−1.66947	−0.0629651
$704$	0	0
$705$	−20.6817	−0.778916
$706$	0	0
$707$	9.77604	0.367666
$708$	0	0
$709$	30.7021	1.15304	0.576521	−	0.817082i	$-0.304410\pi$
$709$	30.7021	1.15304	0.576521	+	0.817082i	$0.304410\pi$
$710$	0	0
$711$	8.90383	0.333920
$712$	0	0
$713$	−0.823919	−0.0308560
$714$	0	0
$715$	−2.24979	−0.0841375
$716$	0	0
$717$	25.1107	0.937777
$718$	0	0
$719$	36.9604	1.37839	0.689195	−	0.724576i	$-0.257964\pi$
$719$	36.9604	1.37839	0.689195	+	0.724576i	$0.257964\pi$
$720$	0	0
$721$	−7.75060	−0.288647
$722$	0	0
$723$	−13.0173	−0.484118
$724$	0	0
$725$	−4.02454	−0.149468
$726$	0	0
$727$	−0.877257	−0.0325357	−0.0162678	−	0.999868i	$-0.505178\pi$
$727$	−0.877257	−0.0325357	−0.0162678	+	0.999868i	$0.505178\pi$
$728$	0	0
$729$	28.0234	1.03790
$730$	0	0
$731$	−4.06298	−0.150275
$732$	0	0
$733$	−21.7373	−0.802885	−0.401442	−	0.915884i	$-0.631491\pi$
$733$	−21.7373	−0.802885	−0.401442	+	0.915884i	$0.631491\pi$
$734$	0	0
$735$	−22.8320	−0.842173
$736$	0	0
$737$	1.73377	0.0638644
$738$	0	0
$739$	49.2824	1.81288	0.906441	−	0.422332i	$-0.138788\pi$
$739$	49.2824	1.81288	0.906441	+	0.422332i	$0.138788\pi$
$740$	0	0
$741$	−1.44180	−0.0529658
$742$	0	0
$743$	−19.0992	−0.700680	−0.350340	−	0.936623i	$-0.613934\pi$
$743$	−19.0992	−0.700680	−0.350340	+	0.936623i	$0.613934\pi$
$744$	0	0
$745$	28.4083	1.04080
$746$	0	0
$747$	20.5279	0.751078
$748$	0	0
$749$	−6.06023	−0.221436
$750$	0	0
$751$	28.8215	1.05171	0.525856	−	0.850573i	$-0.323745\pi$
$751$	28.8215	1.05171	0.525856	+	0.850573i	$0.323745\pi$
$752$	0	0
$753$	−15.2680	−0.556398
$754$	0	0
$755$	46.6763	1.69873
$756$	0	0
$757$	−36.3977	−1.32290	−0.661449	−	0.749990i	$-0.730058\pi$
$757$	−36.3977	−1.32290	−0.661449	+	0.749990i	$0.730058\pi$
$758$	0	0
$759$	−0.401352	−0.0145682
$760$	0	0
$761$	−8.49368	−0.307896	−0.153948	−	0.988079i	$-0.549199\pi$
$761$	−8.49368	−0.307896	−0.153948	+	0.988079i	$0.549199\pi$
$762$	0	0
$763$	−3.69673	−0.133831
$764$	0	0
$765$	−3.13153	−0.113221
$766$	0	0
$767$	4.14111	0.149527
$768$	0	0
$769$	−18.9470	−0.683246	−0.341623	−	0.939837i	$-0.610977\pi$
$769$	−18.9470	−0.683246	−0.341623	+	0.939837i	$0.610977\pi$
$770$	0	0
$771$	−35.3937	−1.27467
$772$	0	0
$773$	−30.3046	−1.08998	−0.544991	−	0.838442i	$-0.683467\pi$
$773$	−30.3046	−1.08998	−0.544991	+	0.838442i	$0.683467\pi$
$774$	0	0
$775$	1.44201	0.0517986
$776$	0	0
$777$	8.67142	0.311085
$778$	0	0
$779$	−0.154779	−0.00554555
$780$	0	0
$781$	−0.205894	−0.00736748
$782$	0	0
$783$	8.70023	0.310921
$784$	0	0
$785$	−7.65983	−0.273391
$786$	0	0
$787$	−14.2835	−0.509152	−0.254576	−	0.967053i	$-0.581936\pi$
$787$	−14.2835	−0.509152	−0.254576	+	0.967053i	$0.581936\pi$
$788$	0	0
$789$	29.5980	1.05372
$790$	0	0
$791$	−1.23851	−0.0440364
$792$	0	0
$793$	−2.72026	−0.0965993
$794$	0	0
$795$	30.0461	1.06563
$796$	0	0
$797$	3.83984	0.136014	0.0680070	−	0.997685i	$-0.478336\pi$
$797$	3.83984	0.136014	0.0680070	+	0.997685i	$0.478336\pi$
$798$	0	0
$799$	−5.48901	−0.194187
$800$	0	0
$801$	−10.0819	−0.356226
$802$	0	0
$803$	−2.84322	−0.100335
$804$	0	0
$805$	3.99297	0.140734
$806$	0	0
$807$	−14.5344	−0.511635
$808$	0	0
$809$	−28.1485	−0.989650	−0.494825	−	0.868993i	$-0.664768\pi$
$809$	−28.1485	−0.989650	−0.494825	+	0.868993i	$0.664768\pi$
$810$	0	0
$811$	25.6767	0.901632	0.450816	−	0.892617i	$-0.351133\pi$
$811$	25.6767	0.901632	0.450816	+	0.892617i	$0.351133\pi$
$812$	0	0
$813$	20.5756	0.721616
$814$	0	0
$815$	37.9344	1.32878
$816$	0	0
$817$	−1.03585	−0.0362397
$818$	0	0
$819$	−4.55769	−0.159259
$820$	0	0
$821$	21.9788	0.767066	0.383533	−	0.923527i	$-0.374707\pi$
$821$	21.9788	0.767066	0.383533	+	0.923527i	$0.374707\pi$
$822$	0	0
$823$	−35.4701	−1.23641	−0.618205	−	0.786017i	$-0.712140\pi$
$823$	−35.4701	−1.23641	−0.618205	+	0.786017i	$0.712140\pi$
$824$	0	0
$825$	0.702441	0.0244559
$826$	0	0
$827$	−13.3801	−0.465272	−0.232636	−	0.972564i	$-0.574735\pi$
$827$	−13.3801	−0.465272	−0.232636	+	0.972564i	$0.574735\pi$
$828$	0	0
$829$	−48.9352	−1.69959	−0.849794	−	0.527114i	$-0.823274\pi$
$829$	−48.9352	−1.69959	−0.849794	+	0.527114i	$0.823274\pi$
$830$	0	0
$831$	−1.59874	−0.0554598
$832$	0	0
$833$	−6.05973	−0.209957
$834$	0	0
$835$	−54.4140	−1.88308
$836$	0	0
$837$	−3.11733	−0.107751
$838$	0	0
$839$	−55.5276	−1.91703	−0.958513	−	0.285047i	$-0.907991\pi$
$839$	−55.5276	−1.91703	−0.958513	+	0.285047i	$0.907991\pi$
$840$	0	0
$841$	−26.6263	−0.918147
$842$	0	0
$843$	−0.793243	−0.0273207
$844$	0	0
$845$	11.4465	0.393772
$846$	0	0
$847$	10.6288	0.365211
$848$	0	0
$849$	19.3447	0.663907
$850$	0	0
$851$	9.77333	0.335026
$852$	0	0
$853$	41.7235	1.42859	0.714293	−	0.699847i	$-0.246748\pi$
$853$	41.7235	1.42859	0.714293	+	0.699847i	$0.246748\pi$
$854$	0	0
$855$	−0.798376	−0.0273039
$856$	0	0
$857$	5.88301	0.200960	0.100480	−	0.994939i	$-0.467962\pi$
$857$	5.88301	0.200960	0.100480	+	0.994939i	$0.467962\pi$
$858$	0	0
$859$	27.1530	0.926449	0.463224	−	0.886241i	$-0.346692\pi$
$859$	27.1530	0.926449	0.463224	+	0.886241i	$0.346692\pi$
$860$	0	0
$861$	0.803943	0.0273983
$862$	0	0
$863$	36.9823	1.25889	0.629446	−	0.777045i	$-0.283282\pi$
$863$	36.9823	1.25889	0.629446	+	0.777045i	$0.283282\pi$
$864$	0	0
$865$	29.6822	1.00923
$866$	0	0
$867$	1.36564	0.0463797
$868$	0	0
$869$	−1.54471	−0.0524006
$870$	0	0
$871$	−36.4617	−1.23546
$872$	0	0
$873$	−19.5755	−0.662529
$874$	0	0
$875$	6.38829	0.215964
$876$	0	0
$877$	−50.0569	−1.69030	−0.845151	−	0.534527i	$-0.820490\pi$
$877$	−50.0569	−1.69030	−0.845151	+	0.534527i	$0.820490\pi$
$878$	0	0
$879$	36.3308	1.22541
$880$	0	0
$881$	−32.1718	−1.08390	−0.541948	−	0.840412i	$-0.682313\pi$
$881$	−32.1718	−1.08390	−0.541948	+	0.840412i	$0.682313\pi$
$882$	0	0
$883$	31.2409	1.05134	0.525670	−	0.850688i	$-0.323815\pi$
$883$	31.2409	1.05134	0.525670	+	0.850688i	$0.323815\pi$
$884$	0	0
$885$	−3.76783	−0.126654
$886$	0	0
$887$	35.6143	1.19581	0.597906	−	0.801566i	$-0.296000\pi$
$887$	35.6143	1.19581	0.597906	+	0.801566i	$0.296000\pi$
$888$	0	0
$889$	−5.47862	−0.183747
$890$	0	0
$891$	−0.848038	−0.0284104
$892$	0	0
$893$	−1.39941	−0.0468295
$894$	0	0
$895$	53.1760	1.77748
$896$	0	0
$897$	8.44054	0.281821
$898$	0	0
$899$	−0.850520	−0.0283664
$900$	0	0
$901$	7.97438	0.265665
$902$	0	0
$903$	5.38032	0.179046
$904$	0	0
$905$	−52.7712	−1.75418
$906$	0	0
$907$	−26.3446	−0.874758	−0.437379	−	0.899277i	$-0.644093\pi$
$907$	−26.3446	−0.874758	−0.437379	+	0.899277i	$0.644093\pi$
$908$	0	0
$909$	11.4430	0.379540
$910$	0	0
$911$	−16.2048	−0.536890	−0.268445	−	0.963295i	$-0.586510\pi$
$911$	−16.2048	−0.536890	−0.268445	+	0.963295i	$0.586510\pi$
$912$	0	0
$913$	−3.56135	−0.117864
$914$	0	0
$915$	2.47506	0.0818229
$916$	0	0
$917$	17.1832	0.567440
$918$	0	0
$919$	2.72307	0.0898259	0.0449129	−	0.998991i	$-0.485699\pi$
$919$	2.72307	0.0898259	0.0449129	+	0.998991i	$0.485699\pi$
$920$	0	0
$921$	−24.1194	−0.794760
$922$	0	0
$923$	4.33001	0.142524
$924$	0	0
$925$	−17.1052	−0.562414
$926$	0	0
$927$	−9.07217	−0.297969
$928$	0	0
$929$	−54.5280	−1.78901	−0.894503	−	0.447062i	$-0.852470\pi$
$929$	−54.5280	−1.78901	−0.894503	+	0.447062i	$0.852470\pi$
$930$	0	0
$931$	−1.54491	−0.0506325
$932$	0	0
$933$	−29.5689	−0.968041
$934$	0	0
$935$	0.543283	0.0177673
$936$	0	0
$937$	−36.0499	−1.17770	−0.588850	−	0.808242i	$-0.700419\pi$
$937$	−36.0499	−1.17770	−0.588850	+	0.808242i	$0.700419\pi$
$938$	0	0
$939$	33.0289	1.07786
$940$	0	0
$941$	7.08389	0.230928	0.115464	−	0.993312i	$-0.463164\pi$
$941$	7.08389	0.230928	0.115464	+	0.993312i	$0.463164\pi$
$942$	0	0
$943$	0.906104	0.0295068
$944$	0	0
$945$	15.1076	0.491449
$946$	0	0
$947$	44.2876	1.43915	0.719577	−	0.694413i	$-0.244336\pi$
$947$	44.2876	1.43915	0.719577	+	0.694413i	$0.244336\pi$
$948$	0	0
$949$	59.7935	1.94098
$950$	0	0
$951$	30.4842	0.988517
$952$	0	0
$953$	−50.9249	−1.64962	−0.824809	−	0.565411i	$-0.808717\pi$
$953$	−50.9249	−1.64962	−0.824809	+	0.565411i	$0.808717\pi$
$954$	0	0
$955$	−19.2655	−0.623417
$956$	0	0
$957$	−0.414310	−0.0133927
$958$	0	0
$959$	13.3104	0.429816
$960$	0	0
$961$	−30.6953	−0.990170
$962$	0	0
$963$	−7.09358	−0.228588
$964$	0	0
$965$	−62.9883	−2.02767
$966$	0	0
$967$	2.88918	0.0929099	0.0464550	−	0.998920i	$-0.485208\pi$
$967$	2.88918	0.0929099	0.0464550	+	0.998920i	$0.485208\pi$
$968$	0	0
$969$	0.348168	0.0111848
$970$	0	0
$971$	14.6344	0.469642	0.234821	−	0.972039i	$-0.424550\pi$
$971$	14.6344	0.469642	0.234821	+	0.972039i	$0.424550\pi$
$972$	0	0
$973$	6.11356	0.195992
$974$	0	0
$975$	−14.7725	−0.473099
$976$	0	0
$977$	35.6710	1.14122	0.570608	−	0.821222i	$-0.306708\pi$
$977$	35.6710	1.14122	0.570608	+	0.821222i	$0.306708\pi$
$978$	0	0
$979$	1.74909	0.0559010
$980$	0	0
$981$	−4.32707	−0.138153
$982$	0	0
$983$	24.7782	0.790303	0.395151	−	0.918616i	$-0.370692\pi$
$983$	24.7782	0.790303	0.395151	+	0.918616i	$0.370692\pi$
$984$	0	0
$985$	−37.9990	−1.21075
$986$	0	0
$987$	7.26871	0.231366
$988$	0	0
$989$	6.06402	0.192825
$990$	0	0
$991$	55.0083	1.74740	0.873698	−	0.486469i	$-0.161715\pi$
$991$	55.0083	1.74740	0.873698	+	0.486469i	$0.161715\pi$
$992$	0	0
$993$	31.0315	0.984753
$994$	0	0
$995$	7.70265	0.244190
$996$	0	0
$997$	−22.3185	−0.706833	−0.353417	−	0.935466i	$-0.614980\pi$
$997$	−22.3185	−0.706833	−0.353417	+	0.935466i	$0.614980\pi$
$998$	0	0
$999$	36.9778	1.16993

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.w.1.15	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.w.1.15	✓	20	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.36564 q^{3} +2.75901 q^{5} -0.969675 q^{7} -1.13502 q^{9} +O(q^{10})\)	\(q+1.36564 q^{3} +2.75901 q^{5} -0.969675 q^{7} -1.13502 q^{9} +0.196912 q^{11} -4.14111 q^{13} +3.76783 q^{15} +1.00000 q^{17} +0.254948 q^{19} -1.32423 q^{21} -1.49251 q^{23} +2.61216 q^{25} -5.64696 q^{27} -1.54069 q^{29} +0.552038 q^{31} +0.268912 q^{33} -2.67535 q^{35} -6.54827 q^{37} -5.65528 q^{39} -0.607102 q^{41} -4.06298 q^{43} -3.13153 q^{45} -5.48901 q^{47} -6.05973 q^{49} +1.36564 q^{51} +7.97438 q^{53} +0.543283 q^{55} +0.348168 q^{57} -1.00000 q^{59} +0.656892 q^{61} +1.10060 q^{63} -11.4254 q^{65} +8.80482 q^{67} -2.03823 q^{69} -1.04562 q^{71} -14.4390 q^{73} +3.56728 q^{75} -0.190941 q^{77} -7.84467 q^{79} -4.30669 q^{81} -18.0860 q^{83} +2.75901 q^{85} -2.10404 q^{87} +8.88258 q^{89} +4.01553 q^{91} +0.753887 q^{93} +0.703404 q^{95} +17.2469 q^{97} -0.223498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) \) 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9	\( 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9} + 8 q^{11} - 3 q^{13} - 2 q^{15} + 20 q^{17} - 9 q^{19} - 31 q^{21} - 8 q^{23} - 4 q^{25} - 17 q^{27} - 37 q^{29} - 9 q^{31} - 8 q^{33} - 8 q^{35} - 11 q^{37} - 12 q^{39} - 27 q^{41} + 17 q^{43} - 4 q^{45} + 7 q^{47} - 9 q^{49} - 2 q^{51} - q^{53} - 21 q^{55} - 57 q^{57} - 20 q^{59} - 12 q^{61} - 14 q^{63} - 59 q^{65} - 26 q^{67} + 14 q^{69} - 3 q^{71} - 45 q^{73} + 15 q^{75} + 6 q^{77} + 5 q^{79} + 12 q^{81} - 17 q^{83} - 2 q^{85} - 32 q^{87} - 40 q^{89} - 36 q^{91} - 4 q^{93} + 5 q^{95} - 77 q^{97} + 54 q^{99}+O(q^{100}) \) 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9 + 8 * q^11 - 3 * q^13 - 2 * q^15 + 20 * q^17 - 9 * q^19 - 31 * q^21 - 8 * q^23 - 4 * q^25 - 17 * q^27 - 37 * q^29 - 9 * q^31 - 8 * q^33 - 8 * q^35 - 11 * q^37 - 12 * q^39 - 27 * q^41 + 17 * q^43 - 4 * q^45 + 7 * q^47 - 9 * q^49 - 2 * q^51 - q^53 - 21 * q^55 - 57 * q^57 - 20 * q^59 - 12 * q^61 - 14 * q^63 - 59 * q^65 - 26 * q^67 + 14 * q^69 - 3 * q^71 - 45 * q^73 + 15 * q^75 + 6 * q^77 + 5 * q^79 + 12 * q^81 - 17 * q^83 - 2 * q^85 - 32 * q^87 - 40 * q^89 - 36 * q^91 - 4 * q^93 + 5 * q^95 - 77 * q^97 + 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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