LMFDB - Embedded newform 3040.2.a.u.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3040,2,Mod(1,3040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3040.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3040 = 2^{5} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3040.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-4,0,-5,0,4,0,7,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$24.2745222145$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.387268.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2 $ x^5 - x^4 - 7x^3 + 4x^2 + 12*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.50372$ of defining polynomial
Character	$\chi$	$=$	3040.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.76491 q^{3} -1.00000 q^{5} -3.89255 q^{7} +0.114895 q^{9} +4.80767 q^{11} +0.0353128 q^{13} +1.76491 q^{15} -4.39275 q^{17} +1.00000 q^{19} +6.86999 q^{21} +9.12195 q^{23} +1.00000 q^{25} +5.09194 q^{27} -6.21514 q^{29} +9.71448 q^{31} -8.48508 q^{33} +3.89255 q^{35} -0.749158 q^{37} -0.0623238 q^{39} -3.52981 q^{41} -11.4625 q^{43} -0.114895 q^{45} +5.57827 q^{47} +8.15196 q^{49} +7.75279 q^{51} +8.67877 q^{53} -4.80767 q^{55} -1.76491 q^{57} -2.56252 q^{59} +6.50725 q^{61} -0.447233 q^{63} -0.0353128 q^{65} -13.2496 q^{67} -16.0994 q^{69} +1.67106 q^{71} -11.3217 q^{73} -1.76491 q^{75} -18.7141 q^{77} +11.4141 q^{79} -9.33148 q^{81} -5.64889 q^{83} +4.39275 q^{85} +10.9691 q^{87} -1.24079 q^{89} -0.137457 q^{91} -17.1451 q^{93} -1.00000 q^{95} +9.31976 q^{97} +0.552375 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} - 5 q^{5} + 4 q^{7} + 7 q^{9} - 2 q^{11} - 4 q^{13} + 4 q^{15} - 12 q^{17} + 5 q^{19} - 10 q^{21} + 8 q^{23} + 5 q^{25} - 16 q^{27} - 6 q^{29} + 10 q^{31} - 18 q^{33} - 4 q^{35} - 6 q^{37}+ \cdots + 14 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 - 5 * q^5 + 4 * q^7 + 7 * q^9 - 2 * q^11 - 4 * q^13 + 4 * q^15 - 12 * q^17 + 5 * q^19 - 10 * q^21 + 8 * q^23 + 5 * q^25 - 16 * q^27 - 6 * q^29 + 10 * q^31 - 18 * q^33 - 4 * q^35 - 6 * q^37 + 18 * q^39 - 8 * q^41 - 12 * q^43 - 7 * q^45 + 16 * q^47 + 7 * q^49 + 14 * q^51 - 18 * q^53 + 2 * q^55 - 4 * q^57 - 8 * q^59 + 2 * q^61 + 36 * q^63 + 4 * q^65 - 10 * q^67 - 22 * q^69 - 18 * q^71 - 28 * q^73 - 4 * q^75 - 28 * q^77 + 14 * q^79 + 25 * q^81 - 8 * q^83 + 12 * q^85 + 24 * q^87 - 30 * q^89 - 28 * q^91 - 24 * q^93 - 5 * q^95 - 18 * q^97 + 14 * q^99

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
$3$	−1.76491	−1.01897	−0.509485	−	0.860480i	$-0.670164\pi$
$3$	−1.76491	−1.01897	−0.509485	+	0.860480i	$0.670164\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.89255	−1.47125	−0.735623	−	0.677391i	$-0.763111\pi$
$7$	−3.89255	−1.47125	−0.735623	+	0.677391i	$0.763111\pi$
$8$	0	0
$9$	0.114895	0.0382982
$10$	0	0
$11$	4.80767	1.44957	0.724783	−	0.688977i	$-0.241940\pi$
$11$	4.80767	1.44957	0.724783	+	0.688977i	$0.241940\pi$
$12$	0	0
$13$	0.0353128	0.00979401	0.00489701	−	0.999988i	$-0.498441\pi$
$13$	0.0353128	0.00979401	0.00489701	+	0.999988i	$0.498441\pi$
$14$	0	0
$15$	1.76491	0.455697
$16$	0	0
$17$	−4.39275	−1.06540	−0.532699	−	0.846305i	$-0.678822\pi$
$17$	−4.39275	−1.06540	−0.532699	+	0.846305i	$0.678822\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	6.86999	1.49915
$22$	0	0
$23$	9.12195	1.90206	0.951029	−	0.309102i	$-0.100028\pi$
$23$	9.12195	1.90206	0.951029	+	0.309102i	$0.100028\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.09194	0.979944
$28$	0	0
$29$	−6.21514	−1.15412	−0.577061	−	0.816701i	$-0.695801\pi$
$29$	−6.21514	−1.15412	−0.577061	+	0.816701i	$0.695801\pi$
$30$	0	0
$31$	9.71448	1.74477	0.872386	−	0.488818i	$-0.162572\pi$
$31$	9.71448	1.74477	0.872386	+	0.488818i	$0.162572\pi$
$32$	0	0
$33$	−8.48508	−1.47706
$34$	0	0
$35$	3.89255	0.657961
$36$	0	0
$37$	−0.749158	−0.123161	−0.0615804	−	0.998102i	$-0.519614\pi$
$37$	−0.749158	−0.123161	−0.0615804	+	0.998102i	$0.519614\pi$
$38$	0	0
$39$	−0.0623238	−0.00997980
$40$	0	0
$41$	−3.52981	−0.551264	−0.275632	−	0.961263i	$-0.588887\pi$
$41$	−3.52981	−0.551264	−0.275632	+	0.961263i	$0.588887\pi$
$42$	0	0
$43$	−11.4625	−1.74802	−0.874009	−	0.485910i	$-0.838488\pi$
$43$	−11.4625	−1.74802	−0.874009	+	0.485910i	$0.838488\pi$
$44$	0	0
$45$	−0.114895	−0.0171275
$46$	0	0
$47$	5.57827	0.813674	0.406837	−	0.913501i	$-0.366632\pi$
$47$	5.57827	0.813674	0.406837	+	0.913501i	$0.366632\pi$
$48$	0	0
$49$	8.15196	1.16457
$50$	0	0
$51$	7.75279	1.08561
$52$	0	0
$53$	8.67877	1.19212	0.596061	−	0.802939i	$-0.296732\pi$
$53$	8.67877	1.19212	0.596061	+	0.802939i	$0.296732\pi$
$54$	0	0
$55$	−4.80767	−0.648265
$56$	0	0
$57$	−1.76491	−0.233768
$58$	0	0
$59$	−2.56252	−0.333612	−0.166806	−	0.985990i	$-0.553345\pi$
$59$	−2.56252	−0.333612	−0.166806	+	0.985990i	$0.553345\pi$
$60$	0	0
$61$	6.50725	0.833168	0.416584	−	0.909097i	$-0.363227\pi$
$61$	6.50725	0.833168	0.416584	+	0.909097i	$0.363227\pi$
$62$	0	0
$63$	−0.447233	−0.0563461
$64$	0	0
$65$	−0.0353128	−0.00438002
$66$	0	0
$67$	−13.2496	−1.61870	−0.809348	−	0.587330i	$-0.800179\pi$
$67$	−13.2496	−1.61870	−0.809348	+	0.587330i	$0.800179\pi$
$68$	0	0
$69$	−16.0994	−1.93814
$70$	0	0
$71$	1.67106	0.198319	0.0991594	−	0.995072i	$-0.468385\pi$
$71$	1.67106	0.198319	0.0991594	+	0.995072i	$0.468385\pi$
$72$	0	0
$73$	−11.3217	−1.32511	−0.662554	−	0.749014i	$-0.730528\pi$
$73$	−11.3217	−1.32511	−0.662554	+	0.749014i	$0.730528\pi$
$74$	0	0
$75$	−1.76491	−0.203794
$76$	0	0
$77$	−18.7141	−2.13267
$78$	0	0
$79$	11.4141	1.28418	0.642091	−	0.766628i	$-0.278067\pi$
$79$	11.4141	1.28418	0.642091	+	0.766628i	$0.278067\pi$
$80$	0	0
$81$	−9.33148	−1.03683
$82$	0	0
$83$	−5.64889	−0.620047	−0.310023	−	0.950729i	$-0.600337\pi$
$83$	−5.64889	−0.620047	−0.310023	+	0.950729i	$0.600337\pi$
$84$	0	0
$85$	4.39275	0.476460
$86$	0	0
$87$	10.9691	1.17601
$88$	0	0
$89$	−1.24079	−0.131523	−0.0657617	−	0.997835i	$-0.520948\pi$
$89$	−1.24079	−0.131523	−0.0657617	+	0.997835i	$0.520948\pi$
$90$	0	0
$91$	−0.137457	−0.0144094
$92$	0	0
$93$	−17.1451	−1.77787
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	9.31976	0.946278	0.473139	−	0.880988i	$-0.343121\pi$
$97$	9.31976	0.946278	0.473139	+	0.880988i	$0.343121\pi$
$98$	0	0
$99$	0.552375	0.0555158
$100$	0	0
$101$	0.807666	0.0803657	0.0401829	−	0.999192i	$-0.487206\pi$
$101$	0.807666	0.0803657	0.0401829	+	0.999192i	$0.487206\pi$
$102$	0	0
$103$	−17.0564	−1.68062	−0.840309	−	0.542108i	$-0.817626\pi$
$103$	−17.0564	−1.68062	−0.840309	+	0.542108i	$0.817626\pi$
$104$	0	0
$105$	−6.86999	−0.670442
$106$	0	0
$107$	−8.41008	−0.813033	−0.406516	−	0.913643i	$-0.633257\pi$
$107$	−8.41008	−0.813033	−0.406516	+	0.913643i	$0.633257\pi$
$108$	0	0
$109$	−16.1309	−1.54506	−0.772529	−	0.634979i	$-0.781009\pi$
$109$	−16.1309	−1.54506	−0.772529	+	0.634979i	$0.781009\pi$
$110$	0	0
$111$	1.32219	0.125497
$112$	0	0
$113$	13.0794	1.23040	0.615201	−	0.788370i	$-0.289075\pi$
$113$	13.0794	1.23040	0.615201	+	0.788370i	$0.289075\pi$
$114$	0	0
$115$	−9.12195	−0.850626
$116$	0	0
$117$	0.00405725	0.000375093 0
$118$	0	0
$119$	17.0990	1.56746
$120$	0	0
$121$	12.1136	1.10124
$122$	0	0
$123$	6.22979	0.561721
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.0024	−1.33125	−0.665623	−	0.746288i	$-0.731834\pi$
$127$	−15.0024	−1.33125	−0.665623	+	0.746288i	$0.731834\pi$
$128$	0	0
$129$	20.2303	1.78118
$130$	0	0
$131$	4.11404	0.359445	0.179723	−	0.983717i	$-0.442480\pi$
$131$	4.11404	0.359445	0.179723	+	0.983717i	$0.442480\pi$
$132$	0	0
$133$	−3.89255	−0.337527
$134$	0	0
$135$	−5.09194	−0.438244
$136$	0	0
$137$	−2.77742	−0.237291	−0.118645	−	0.992937i	$-0.537855\pi$
$137$	−2.77742	−0.237291	−0.118645	+	0.992937i	$0.537855\pi$
$138$	0	0
$139$	−5.40850	−0.458743	−0.229371	−	0.973339i	$-0.573667\pi$
$139$	−5.40850	−0.458743	−0.229371	+	0.973339i	$0.573667\pi$
$140$	0	0
$141$	−9.84512	−0.829109
$142$	0	0
$143$	0.169772	0.0141971
$144$	0	0
$145$	6.21514	0.516139
$146$	0	0
$147$	−14.3874	−1.18666
$148$	0	0
$149$	−0.291090	−0.0238470	−0.0119235	−	0.999929i	$-0.503795\pi$
$149$	−0.291090	−0.0238470	−0.0119235	+	0.999929i	$0.503795\pi$
$150$	0	0
$151$	15.0259	1.22279	0.611395	−	0.791326i	$-0.290609\pi$
$151$	15.0259	1.22279	0.611395	+	0.791326i	$0.290609\pi$
$152$	0	0
$153$	−0.504703	−0.0408028
$154$	0	0
$155$	−9.71448	−0.780286
$156$	0	0
$157$	−9.60044	−0.766198	−0.383099	−	0.923707i	$-0.625143\pi$
$157$	−9.60044	−0.766198	−0.383099	+	0.923707i	$0.625143\pi$
$158$	0	0
$159$	−15.3172	−1.21473
$160$	0	0
$161$	−35.5077	−2.79840
$162$	0	0
$163$	−4.90681	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$163$	−4.90681	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$164$	0	0
$165$	8.48508	0.660563
$166$	0	0
$167$	−6.54582	−0.506531	−0.253266	−	0.967397i	$-0.581505\pi$
$167$	−6.54582	−0.506531	−0.253266	+	0.967397i	$0.581505\pi$
$168$	0	0
$169$	−12.9988	−0.999904
$170$	0	0
$171$	0.114895	0.00878621
$172$	0	0
$173$	−23.9483	−1.82076	−0.910379	−	0.413776i	$-0.864210\pi$
$173$	−23.9483	−1.82076	−0.910379	+	0.413776i	$0.864210\pi$
$174$	0	0
$175$	−3.89255	−0.294249
$176$	0	0
$177$	4.52261	0.339940
$178$	0	0
$179$	9.98637	0.746416	0.373208	−	0.927748i	$-0.378258\pi$
$179$	9.98637	0.746416	0.373208	+	0.927748i	$0.378258\pi$
$180$	0	0
$181$	−7.41406	−0.551083	−0.275541	−	0.961289i	$-0.588857\pi$
$181$	−7.41406	−0.551083	−0.275541	+	0.961289i	$0.588857\pi$
$182$	0	0
$183$	−11.4847	−0.848973
$184$	0	0
$185$	0.749158	0.0550792
$186$	0	0
$187$	−21.1189	−1.54436
$188$	0	0
$189$	−19.8206	−1.44174
$190$	0	0
$191$	−22.3007	−1.61362	−0.806809	−	0.590812i	$-0.798808\pi$
$191$	−22.3007	−1.61362	−0.806809	+	0.590812i	$0.798808\pi$
$192$	0	0
$193$	−3.15393	−0.227025	−0.113512	−	0.993537i	$-0.536210\pi$
$193$	−3.15393	−0.227025	−0.113512	+	0.993537i	$0.536210\pi$
$194$	0	0
$195$	0.0623238	0.00446310
$196$	0	0
$197$	5.21801	0.371768	0.185884	−	0.982572i	$-0.440485\pi$
$197$	5.21801	0.371768	0.185884	+	0.982572i	$0.440485\pi$
$198$	0	0
$199$	22.2841	1.57967	0.789837	−	0.613316i	$-0.210165\pi$
$199$	22.2841	1.57967	0.789837	+	0.613316i	$0.210165\pi$
$200$	0	0
$201$	23.3843	1.64940
$202$	0	0
$203$	24.1927	1.69800
$204$	0	0
$205$	3.52981	0.246533
$206$	0	0
$207$	1.04806	0.0728454
$208$	0	0
$209$	4.80767	0.332553
$210$	0	0
$211$	−2.27871	−0.156873	−0.0784364	−	0.996919i	$-0.524993\pi$
$211$	−2.27871	−0.156873	−0.0784364	+	0.996919i	$0.524993\pi$
$212$	0	0
$213$	−2.94927	−0.202081
$214$	0	0
$215$	11.4625	0.781737
$216$	0	0
$217$	−37.8141	−2.56699
$218$	0	0
$219$	19.9818	1.35024
$220$	0	0
$221$	−0.155120	−0.0104345
$222$	0	0
$223$	20.1108	1.34672	0.673360	−	0.739315i	$-0.264850\pi$
$223$	20.1108	1.34672	0.673360	+	0.739315i	$0.264850\pi$
$224$	0	0
$225$	0.114895	0.00765964
$226$	0	0
$227$	1.16184	0.0771140	0.0385570	−	0.999256i	$-0.487724\pi$
$227$	1.16184	0.0771140	0.0385570	+	0.999256i	$0.487724\pi$
$228$	0	0
$229$	2.77617	0.183454	0.0917272	−	0.995784i	$-0.470761\pi$
$229$	2.77617	0.183454	0.0917272	+	0.995784i	$0.470761\pi$
$230$	0	0
$231$	33.0286	2.17312
$232$	0	0
$233$	25.2847	1.65645	0.828227	−	0.560392i	$-0.189350\pi$
$233$	25.2847	1.65645	0.828227	+	0.560392i	$0.189350\pi$
$234$	0	0
$235$	−5.57827	−0.363886
$236$	0	0
$237$	−20.1448	−1.30854
$238$	0	0
$239$	−0.228766	−0.0147976	−0.00739881	−	0.999973i	$-0.502355\pi$
$239$	−0.228766	−0.0147976	−0.00739881	+	0.999973i	$0.502355\pi$
$240$	0	0
$241$	−3.21056	−0.206810	−0.103405	−	0.994639i	$-0.532974\pi$
$241$	−3.21056	−0.206810	−0.103405	+	0.994639i	$0.532974\pi$
$242$	0	0
$243$	1.19337	0.0765548
$244$	0	0
$245$	−8.15196	−0.520809
$246$	0	0
$247$	0.0353128	0.00224690
$248$	0	0
$249$	9.96977	0.631808
$250$	0	0
$251$	−26.5329	−1.67474	−0.837372	−	0.546634i	$-0.815909\pi$
$251$	−26.5329	−1.67474	−0.837372	+	0.546634i	$0.815909\pi$
$252$	0	0
$253$	43.8553	2.75716
$254$	0	0
$255$	−7.75279	−0.485498
$256$	0	0
$257$	−28.5807	−1.78281	−0.891406	−	0.453206i	$-0.850280\pi$
$257$	−28.5807	−1.78281	−0.891406	+	0.453206i	$0.850280\pi$
$258$	0	0
$259$	2.91614	0.181200
$260$	0	0
$261$	−0.714086	−0.0442008
$262$	0	0
$263$	18.4959	1.14050	0.570252	−	0.821470i	$-0.306846\pi$
$263$	18.4959	1.14050	0.570252	+	0.821470i	$0.306846\pi$
$264$	0	0
$265$	−8.67877	−0.533133
$266$	0	0
$267$	2.18988	0.134018
$268$	0	0
$269$	−15.6882	−0.956526	−0.478263	−	0.878217i	$-0.658733\pi$
$269$	−15.6882	−0.956526	−0.478263	+	0.878217i	$0.658733\pi$
$270$	0	0
$271$	−27.7803	−1.68753	−0.843767	−	0.536710i	$-0.819667\pi$
$271$	−27.7803	−1.68753	−0.843767	+	0.536710i	$0.819667\pi$
$272$	0	0
$273$	0.242599	0.0146827
$274$	0	0
$275$	4.80767	0.289913
$276$	0	0
$277$	19.9044	1.19594	0.597968	−	0.801520i	$-0.295975\pi$
$277$	19.9044	1.19594	0.597968	+	0.801520i	$0.295975\pi$
$278$	0	0
$279$	1.11614	0.0668217
$280$	0	0
$281$	10.5220	0.627688	0.313844	−	0.949475i	$-0.398383\pi$
$281$	10.5220	0.627688	0.313844	+	0.949475i	$0.398383\pi$
$282$	0	0
$283$	−4.26164	−0.253328	−0.126664	−	0.991946i	$-0.540427\pi$
$283$	−4.26164	−0.253328	−0.126664	+	0.991946i	$0.540427\pi$
$284$	0	0
$285$	1.76491	0.104544
$286$	0	0
$287$	13.7400	0.811045
$288$	0	0
$289$	2.29623	0.135072
$290$	0	0
$291$	−16.4485	−0.964228
$292$	0	0
$293$	−2.16596	−0.126536	−0.0632682	−	0.997997i	$-0.520152\pi$
$293$	−2.16596	−0.126536	−0.0632682	+	0.997997i	$0.520152\pi$
$294$	0	0
$295$	2.56252	0.149196
$296$	0	0
$297$	24.4803	1.42049
$298$	0	0
$299$	0.322122	0.0186288
$300$	0	0
$301$	44.6184	2.57176
$302$	0	0
$303$	−1.42545	−0.0818902
$304$	0	0
$305$	−6.50725	−0.372604
$306$	0	0
$307$	−15.1287	−0.863443	−0.431721	−	0.902007i	$-0.642094\pi$
$307$	−15.1287	−0.863443	−0.431721	+	0.902007i	$0.642094\pi$
$308$	0	0
$309$	30.1030	1.71250
$310$	0	0
$311$	12.0180	0.681477	0.340739	−	0.940158i	$-0.389323\pi$
$311$	12.0180	0.681477	0.340739	+	0.940158i	$0.389323\pi$
$312$	0	0
$313$	−23.1532	−1.30870	−0.654349	−	0.756193i	$-0.727057\pi$
$313$	−23.1532	−1.30870	−0.654349	+	0.756193i	$0.727057\pi$
$314$	0	0
$315$	0.447233	0.0251987
$316$	0	0
$317$	6.88004	0.386422	0.193211	−	0.981157i	$-0.438110\pi$
$317$	6.88004	0.386422	0.193211	+	0.981157i	$0.438110\pi$
$318$	0	0
$319$	−29.8803	−1.67298
$320$	0	0
$321$	14.8430	0.828456
$322$	0	0
$323$	−4.39275	−0.244419
$324$	0	0
$325$	0.0353128	0.00195880
$326$	0	0
$327$	28.4695	1.57437
$328$	0	0
$329$	−21.7137	−1.19711
$330$	0	0
$331$	10.6924	0.587706	0.293853	−	0.955851i	$-0.405062\pi$
$331$	10.6924	0.587706	0.293853	+	0.955851i	$0.405062\pi$
$332$	0	0
$333$	−0.0860743	−0.00471684
$334$	0	0
$335$	13.2496	0.723903
$336$	0	0
$337$	−29.7049	−1.61813	−0.809065	−	0.587720i	$-0.800026\pi$
$337$	−29.7049	−1.61813	−0.809065	+	0.587720i	$0.800026\pi$
$338$	0	0
$339$	−23.0838	−1.25374
$340$	0	0
$341$	46.7040	2.52916
$342$	0	0
$343$	−4.48406	−0.242116
$344$	0	0
$345$	16.0994	0.866762
$346$	0	0
$347$	−3.25157	−0.174553	−0.0872767	−	0.996184i	$-0.527816\pi$
$347$	−3.25157	−0.174553	−0.0872767	+	0.996184i	$0.527816\pi$
$348$	0	0
$349$	−31.4979	−1.68604	−0.843021	−	0.537880i	$-0.819225\pi$
$349$	−31.4979	−1.68604	−0.843021	+	0.537880i	$0.819225\pi$
$350$	0	0
$351$	0.179811	0.00959759
$352$	0	0
$353$	−23.0362	−1.22609	−0.613047	−	0.790047i	$-0.710056\pi$
$353$	−23.0362	−1.22609	−0.613047	+	0.790047i	$0.710056\pi$
$354$	0	0
$355$	−1.67106	−0.0886909
$356$	0	0
$357$	−30.1781	−1.59720
$358$	0	0
$359$	−13.0768	−0.690168	−0.345084	−	0.938572i	$-0.612150\pi$
$359$	−13.0768	−0.690168	−0.345084	+	0.938572i	$0.612150\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−21.3795	−1.12213
$364$	0	0
$365$	11.3217	0.592606
$366$	0	0
$367$	32.4936	1.69615	0.848077	−	0.529874i	$-0.177761\pi$
$367$	32.4936	1.69615	0.848077	+	0.529874i	$0.177761\pi$
$368$	0	0
$369$	−0.405557	−0.0211124
$370$	0	0
$371$	−33.7826	−1.75390
$372$	0	0
$373$	6.75164	0.349586	0.174793	−	0.984605i	$-0.444074\pi$
$373$	6.75164	0.349586	0.174793	+	0.984605i	$0.444074\pi$
$374$	0	0
$375$	1.76491	0.0911394
$376$	0	0
$377$	−0.219474	−0.0113035
$378$	0	0
$379$	34.7328	1.78410	0.892051	−	0.451934i	$-0.149266\pi$
$379$	34.7328	1.78410	0.892051	+	0.451934i	$0.149266\pi$
$380$	0	0
$381$	26.4778	1.35650
$382$	0	0
$383$	−0.230516	−0.0117788	−0.00588942	−	0.999983i	$-0.501875\pi$
$383$	−0.230516	−0.0117788	−0.00588942	+	0.999983i	$0.501875\pi$
$384$	0	0
$385$	18.7141	0.953758
$386$	0	0
$387$	−1.31698	−0.0669460
$388$	0	0
$389$	−26.1973	−1.32826	−0.664128	−	0.747619i	$-0.731197\pi$
$389$	−26.1973	−1.32826	−0.664128	+	0.747619i	$0.731197\pi$
$390$	0	0
$391$	−40.0704	−2.02645
$392$	0	0
$393$	−7.26089	−0.366264
$394$	0	0
$395$	−11.4141	−0.574304
$396$	0	0
$397$	−24.7005	−1.23968	−0.619840	−	0.784728i	$-0.712803\pi$
$397$	−24.7005	−1.23968	−0.619840	+	0.784728i	$0.712803\pi$
$398$	0	0
$399$	6.86999	0.343930
$400$	0	0
$401$	2.43645	0.121671	0.0608354	−	0.998148i	$-0.480624\pi$
$401$	2.43645	0.121671	0.0608354	+	0.998148i	$0.480624\pi$
$402$	0	0
$403$	0.343046	0.0170883
$404$	0	0
$405$	9.33148	0.463685
$406$	0	0
$407$	−3.60170	−0.178530
$408$	0	0
$409$	−22.2293	−1.09917	−0.549585	−	0.835438i	$-0.685214\pi$
$409$	−22.2293	−1.09917	−0.549585	+	0.835438i	$0.685214\pi$
$410$	0	0
$411$	4.90188	0.241792
$412$	0	0
$413$	9.97474	0.490825
$414$	0	0
$415$	5.64889	0.277293
$416$	0	0
$417$	9.54549	0.467445
$418$	0	0
$419$	4.34305	0.212172	0.106086	−	0.994357i	$-0.466168\pi$
$419$	4.34305	0.212172	0.106086	+	0.994357i	$0.466168\pi$
$420$	0	0
$421$	−0.903722	−0.0440447	−0.0220224	−	0.999757i	$-0.507011\pi$
$421$	−0.903722	−0.0440447	−0.0220224	+	0.999757i	$0.507011\pi$
$422$	0	0
$423$	0.640913	0.0311623
$424$	0	0
$425$	−4.39275	−0.213080
$426$	0	0
$427$	−25.3298	−1.22580
$428$	0	0
$429$	−0.299632	−0.0144664
$430$	0	0
$431$	−27.6872	−1.33365	−0.666823	−	0.745216i	$-0.732346\pi$
$431$	−27.6872	−1.33365	−0.666823	+	0.745216i	$0.732346\pi$
$432$	0	0
$433$	15.7540	0.757088	0.378544	−	0.925583i	$-0.376425\pi$
$433$	15.7540	0.757088	0.378544	+	0.925583i	$0.376425\pi$
$434$	0	0
$435$	−10.9691	−0.525930
$436$	0	0
$437$	9.12195	0.436362
$438$	0	0
$439$	−2.54081	−0.121266	−0.0606332	−	0.998160i	$-0.519312\pi$
$439$	−2.54081	−0.121266	−0.0606332	+	0.998160i	$0.519312\pi$
$440$	0	0
$441$	0.936616	0.0446008
$442$	0	0
$443$	20.3166	0.965270	0.482635	−	0.875821i	$-0.339680\pi$
$443$	20.3166	0.965270	0.482635	+	0.875821i	$0.339680\pi$
$444$	0	0
$445$	1.24079	0.0588191
$446$	0	0
$447$	0.513746	0.0242993
$448$	0	0
$449$	−20.0132	−0.944480	−0.472240	−	0.881470i	$-0.656555\pi$
$449$	−20.0132	−0.944480	−0.472240	+	0.881470i	$0.656555\pi$
$450$	0	0
$451$	−16.9702	−0.799094
$452$	0	0
$453$	−26.5193	−1.24599
$454$	0	0
$455$	0.137457	0.00644408
$456$	0	0
$457$	28.9331	1.35343	0.676717	−	0.736243i	$-0.263402\pi$
$457$	28.9331	1.35343	0.676717	+	0.736243i	$0.263402\pi$
$458$	0	0
$459$	−22.3676	−1.04403
$460$	0	0
$461$	9.04395	0.421219	0.210609	−	0.977570i	$-0.432455\pi$
$461$	9.04395	0.421219	0.210609	+	0.977570i	$0.432455\pi$
$462$	0	0
$463$	33.2226	1.54399	0.771993	−	0.635632i	$-0.219260\pi$
$463$	33.2226	1.54399	0.771993	+	0.635632i	$0.219260\pi$
$464$	0	0
$465$	17.1451	0.795087
$466$	0	0
$467$	−34.0888	−1.57744	−0.788722	−	0.614750i	$-0.789257\pi$
$467$	−34.0888	−1.57744	−0.788722	+	0.614750i	$0.789257\pi$
$468$	0	0
$469$	51.5747	2.38150
$470$	0	0
$471$	16.9439	0.780733
$472$	0	0
$473$	−55.1079	−2.53387
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0.997145	0.0456561
$478$	0	0
$479$	−28.2735	−1.29185	−0.645925	−	0.763401i	$-0.723528\pi$
$479$	−28.2735	−1.29185	−0.645925	+	0.763401i	$0.723528\pi$
$480$	0	0
$481$	−0.0264549	−0.00120624
$482$	0	0
$483$	62.6677	2.85148
$484$	0	0
$485$	−9.31976	−0.423188
$486$	0	0
$487$	−25.5836	−1.15930	−0.579652	−	0.814864i	$-0.696811\pi$
$487$	−25.5836	−1.15930	−0.579652	+	0.814864i	$0.696811\pi$
$488$	0	0
$489$	8.66006	0.391622
$490$	0	0
$491$	−3.53949	−0.159735	−0.0798675	−	0.996805i	$-0.525450\pi$
$491$	−3.53949	−0.159735	−0.0798675	+	0.996805i	$0.525450\pi$
$492$	0	0
$493$	27.3015	1.22960
$494$	0	0
$495$	−0.552375	−0.0248274
$496$	0	0
$497$	−6.50470	−0.291776
$498$	0	0
$499$	26.7046	1.19546	0.597730	−	0.801697i	$-0.296069\pi$
$499$	26.7046	1.19546	0.597730	+	0.801697i	$0.296069\pi$
$500$	0	0
$501$	11.5528	0.516140
$502$	0	0
$503$	−24.7956	−1.10558	−0.552790	−	0.833320i	$-0.686437\pi$
$503$	−24.7956	−1.10558	−0.552790	+	0.833320i	$0.686437\pi$
$504$	0	0
$505$	−0.807666	−0.0359406
$506$	0	0
$507$	22.9416	1.01887
$508$	0	0
$509$	17.4860	0.775051	0.387526	−	0.921859i	$-0.373330\pi$
$509$	17.4860	0.775051	0.387526	+	0.921859i	$0.373330\pi$
$510$	0	0
$511$	44.0704	1.94956
$512$	0	0
$513$	5.09194	0.224815
$514$	0	0
$515$	17.0564	0.751595
$516$	0	0
$517$	26.8184	1.17947
$518$	0	0
$519$	42.2666	1.85530
$520$	0	0
$521$	−1.37888	−0.0604099	−0.0302049	−	0.999544i	$-0.509616\pi$
$521$	−1.37888	−0.0604099	−0.0302049	+	0.999544i	$0.509616\pi$
$522$	0	0
$523$	38.5666	1.68640	0.843199	−	0.537602i	$-0.180670\pi$
$523$	38.5666	1.68640	0.843199	+	0.537602i	$0.180670\pi$
$524$	0	0
$525$	6.86999	0.299831
$526$	0	0
$527$	−42.6732	−1.85888
$528$	0	0
$529$	60.2100	2.61782
$530$	0	0
$531$	−0.294420	−0.0127767
$532$	0	0
$533$	−0.124648	−0.00539909
$534$	0	0
$535$	8.41008	0.363599
$536$	0	0
$537$	−17.6250	−0.760575
$538$	0	0
$539$	39.1919	1.68811
$540$	0	0
$541$	8.73354	0.375484	0.187742	−	0.982218i	$-0.439883\pi$
$541$	8.73354	0.375484	0.187742	+	0.982218i	$0.439883\pi$
$542$	0	0
$543$	13.0851	0.561537
$544$	0	0
$545$	16.1309	0.690971
$546$	0	0
$547$	31.8117	1.36017	0.680084	−	0.733134i	$-0.261943\pi$
$547$	31.8117	1.36017	0.680084	+	0.733134i	$0.261943\pi$
$548$	0	0
$549$	0.747648	0.0319089
$550$	0	0
$551$	−6.21514	−0.264774
$552$	0	0
$553$	−44.4298	−1.88935
$554$	0	0
$555$	−1.32219	−0.0561240
$556$	0	0
$557$	−31.9044	−1.35183	−0.675915	−	0.736979i	$-0.736252\pi$
$557$	−31.9044	−1.35183	−0.675915	+	0.736979i	$0.736252\pi$
$558$	0	0
$559$	−0.404774	−0.0171201
$560$	0	0
$561$	37.2728	1.57366
$562$	0	0
$563$	−40.7223	−1.71624	−0.858119	−	0.513450i	$-0.828367\pi$
$563$	−40.7223	−1.71624	−0.858119	+	0.513450i	$0.828367\pi$
$564$	0	0
$565$	−13.0794	−0.550253
$566$	0	0
$567$	36.3233	1.52543
$568$	0	0
$569$	12.9176	0.541534	0.270767	−	0.962645i	$-0.412723\pi$
$569$	12.9176	0.541534	0.270767	+	0.962645i	$0.412723\pi$
$570$	0	0
$571$	−14.3129	−0.598976	−0.299488	−	0.954100i	$-0.596816\pi$
$571$	−14.3129	−0.598976	−0.299488	+	0.954100i	$0.596816\pi$
$572$	0	0
$573$	39.3586	1.64423
$574$	0	0
$575$	9.12195	0.380412
$576$	0	0
$577$	36.0033	1.49884	0.749419	−	0.662095i	$-0.230333\pi$
$577$	36.0033	1.49884	0.749419	+	0.662095i	$0.230333\pi$
$578$	0	0
$579$	5.56639	0.231331
$580$	0	0
$581$	21.9886	0.912241
$582$	0	0
$583$	41.7246	1.72806
$584$	0	0
$585$	−0.00405725	−0.000167747 0
$586$	0	0
$587$	−6.30974	−0.260431	−0.130215	−	0.991486i	$-0.541567\pi$
$587$	−6.30974	−0.260431	−0.130215	+	0.991486i	$0.541567\pi$
$588$	0	0
$589$	9.71448	0.400278
$590$	0	0
$591$	−9.20929	−0.378820
$592$	0	0
$593$	6.17104	0.253414	0.126707	−	0.991940i	$-0.459559\pi$
$593$	6.17104	0.253414	0.126707	+	0.991940i	$0.459559\pi$
$594$	0	0
$595$	−17.0990	−0.700990
$596$	0	0
$597$	−39.3293	−1.60964
$598$	0	0
$599$	23.3127	0.952530	0.476265	−	0.879302i	$-0.341990\pi$
$599$	23.3127	0.952530	0.476265	+	0.879302i	$0.341990\pi$
$600$	0	0
$601$	−20.0776	−0.818984	−0.409492	−	0.912314i	$-0.634294\pi$
$601$	−20.0776	−0.818984	−0.409492	+	0.912314i	$0.634294\pi$
$602$	0	0
$603$	−1.52231	−0.0619932
$604$	0	0
$605$	−12.1136	−0.492490
$606$	0	0
$607$	5.12748	0.208118	0.104059	−	0.994571i	$-0.466817\pi$
$607$	5.12748	0.208118	0.104059	+	0.994571i	$0.466817\pi$
$608$	0	0
$609$	−42.6979	−1.73021
$610$	0	0
$611$	0.196984	0.00796913
$612$	0	0
$613$	24.2342	0.978811	0.489405	−	0.872056i	$-0.337214\pi$
$613$	24.2342	0.978811	0.489405	+	0.872056i	$0.337214\pi$
$614$	0	0
$615$	−6.22979	−0.251209
$616$	0	0
$617$	10.8591	0.437171	0.218585	−	0.975818i	$-0.429856\pi$
$617$	10.8591	0.437171	0.218585	+	0.975818i	$0.429856\pi$
$618$	0	0
$619$	19.2362	0.773169	0.386585	−	0.922254i	$-0.373655\pi$
$619$	19.2362	0.773169	0.386585	+	0.922254i	$0.373655\pi$
$620$	0	0
$621$	46.4484	1.86391
$622$	0	0
$623$	4.82984	0.193503
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.48508	−0.338861
$628$	0	0
$629$	3.29086	0.131215
$630$	0	0
$631$	46.0007	1.83126	0.915629	−	0.402025i	$-0.131694\pi$
$631$	46.0007	1.83126	0.915629	+	0.402025i	$0.131694\pi$
$632$	0	0
$633$	4.02171	0.159848
$634$	0	0
$635$	15.0024	0.595351
$636$	0	0
$637$	0.287869	0.0114058
$638$	0	0
$639$	0.191996	0.00759526
$640$	0	0
$641$	7.97279	0.314906	0.157453	−	0.987526i	$-0.449672\pi$
$641$	7.97279	0.314906	0.157453	+	0.987526i	$0.449672\pi$
$642$	0	0
$643$	−42.2951	−1.66795	−0.833977	−	0.551799i	$-0.813942\pi$
$643$	−42.2951	−1.66795	−0.833977	+	0.551799i	$0.813942\pi$
$644$	0	0
$645$	−20.2303	−0.796566
$646$	0	0
$647$	0.422756	0.0166202	0.00831012	−	0.999965i	$-0.497355\pi$
$647$	0.422756	0.0166202	0.00831012	+	0.999965i	$0.497355\pi$
$648$	0	0
$649$	−12.3197	−0.483592
$650$	0	0
$651$	66.7384	2.61568
$652$	0	0
$653$	9.74222	0.381242	0.190621	−	0.981664i	$-0.438950\pi$
$653$	9.74222	0.381242	0.190621	+	0.981664i	$0.438950\pi$
$654$	0	0
$655$	−4.11404	−0.160749
$656$	0	0
$657$	−1.30081	−0.0507493
$658$	0	0
$659$	−4.05689	−0.158034	−0.0790170	−	0.996873i	$-0.525178\pi$
$659$	−4.05689	−0.158034	−0.0790170	+	0.996873i	$0.525178\pi$
$660$	0	0
$661$	−21.5278	−0.837335	−0.418668	−	0.908140i	$-0.637503\pi$
$661$	−21.5278	−0.837335	−0.418668	+	0.908140i	$0.637503\pi$
$662$	0	0
$663$	0.273773	0.0106325
$664$	0	0
$665$	3.89255	0.150947
$666$	0	0
$667$	−56.6942	−2.19521
$668$	0	0
$669$	−35.4937	−1.37227
$670$	0	0
$671$	31.2847	1.20773
$672$	0	0
$673$	−30.3026	−1.16808	−0.584039	−	0.811725i	$-0.698529\pi$
$673$	−30.3026	−1.16808	−0.584039	+	0.811725i	$0.698529\pi$
$674$	0	0
$675$	5.09194	0.195989
$676$	0	0
$677$	−18.3642	−0.705795	−0.352898	−	0.935662i	$-0.614804\pi$
$677$	−18.3642	−0.705795	−0.352898	+	0.935662i	$0.614804\pi$
$678$	0	0
$679$	−36.2776	−1.39221
$680$	0	0
$681$	−2.05054	−0.0785768
$682$	0	0
$683$	−9.35629	−0.358009	−0.179004	−	0.983848i	$-0.557288\pi$
$683$	−9.35629	−0.358009	−0.179004	+	0.983848i	$0.557288\pi$
$684$	0	0
$685$	2.77742	0.106120
$686$	0	0
$687$	−4.89968	−0.186934
$688$	0	0
$689$	0.306472	0.0116757
$690$	0	0
$691$	−24.6498	−0.937723	−0.468861	−	0.883272i	$-0.655336\pi$
$691$	−24.6498	−0.937723	−0.468861	+	0.883272i	$0.655336\pi$
$692$	0	0
$693$	−2.15015	−0.0816774
$694$	0	0
$695$	5.40850	0.205156
$696$	0	0
$697$	15.5056	0.587316
$698$	0	0
$699$	−44.6251	−1.68788
$700$	0	0
$701$	−13.2467	−0.500321	−0.250160	−	0.968204i	$-0.580483\pi$
$701$	−13.2467	−0.500321	−0.250160	+	0.968204i	$0.580483\pi$
$702$	0	0
$703$	−0.749158	−0.0282550
$704$	0	0
$705$	9.84512	0.370789
$706$	0	0
$707$	−3.14388	−0.118238
$708$	0	0
$709$	−30.0979	−1.13035	−0.565175	−	0.824971i	$-0.691192\pi$
$709$	−30.0979	−1.13035	−0.565175	+	0.824971i	$0.691192\pi$
$710$	0	0
$711$	1.31141	0.0491819
$712$	0	0
$713$	88.6150	3.31866
$714$	0	0
$715$	−0.169772	−0.00634912
$716$	0	0
$717$	0.403750	0.0150783
$718$	0	0
$719$	25.5662	0.953458	0.476729	−	0.879050i	$-0.341822\pi$
$719$	25.5662	0.953458	0.476729	+	0.879050i	$0.341822\pi$
$720$	0	0
$721$	66.3929	2.47260
$722$	0	0
$723$	5.66633	0.210733
$724$	0	0
$725$	−6.21514	−0.230824
$726$	0	0
$727$	46.2807	1.71646	0.858228	−	0.513268i	$-0.171565\pi$
$727$	46.2807	1.71646	0.858228	+	0.513268i	$0.171565\pi$
$728$	0	0
$729$	25.8883	0.958824
$730$	0	0
$731$	50.3519	1.86233
$732$	0	0
$733$	33.1328	1.22379	0.611894	−	0.790940i	$-0.290408\pi$
$733$	33.1328	1.22379	0.611894	+	0.790940i	$0.290408\pi$
$734$	0	0
$735$	14.3874	0.530689
$736$	0	0
$737$	−63.6996	−2.34641
$738$	0	0
$739$	2.75493	0.101342	0.0506708	−	0.998715i	$-0.483864\pi$
$739$	2.75493	0.101342	0.0506708	+	0.998715i	$0.483864\pi$
$740$	0	0
$741$	−0.0623238	−0.00228952
$742$	0	0
$743$	−9.01027	−0.330555	−0.165277	−	0.986247i	$-0.552852\pi$
$743$	−9.01027	−0.330555	−0.165277	+	0.986247i	$0.552852\pi$
$744$	0	0
$745$	0.291090	0.0106647
$746$	0	0
$747$	−0.649028	−0.0237467
$748$	0	0
$749$	32.7367	1.19617
$750$	0	0
$751$	−10.7150	−0.390996	−0.195498	−	0.980704i	$-0.562632\pi$
$751$	−10.7150	−0.390996	−0.195498	+	0.980704i	$0.562632\pi$
$752$	0	0
$753$	46.8281	1.70651
$754$	0	0
$755$	−15.0259	−0.546848
$756$	0	0
$757$	22.4197	0.814857	0.407428	−	0.913237i	$-0.366426\pi$
$757$	22.4197	0.814857	0.407428	+	0.913237i	$0.366426\pi$
$758$	0	0
$759$	−77.4005	−2.80946
$760$	0	0
$761$	−7.21311	−0.261475	−0.130738	−	0.991417i	$-0.541735\pi$
$761$	−7.21311	−0.261475	−0.130738	+	0.991417i	$0.541735\pi$
$762$	0	0
$763$	62.7903	2.27316
$764$	0	0
$765$	0.504703	0.0182476
$766$	0	0
$767$	−0.0904898	−0.00326740
$768$	0	0
$769$	6.14157	0.221471	0.110735	−	0.993850i	$-0.464679\pi$
$769$	6.14157	0.221471	0.110735	+	0.993850i	$0.464679\pi$
$770$	0	0
$771$	50.4422	1.81663
$772$	0	0
$773$	−51.0612	−1.83655	−0.918273	−	0.395949i	$-0.870416\pi$
$773$	−51.0612	−1.83655	−0.918273	+	0.395949i	$0.870416\pi$
$774$	0	0
$775$	9.71448	0.348954
$776$	0	0
$777$	−5.14671	−0.184637
$778$	0	0
$779$	−3.52981	−0.126469
$780$	0	0
$781$	8.03392	0.287476
$782$	0	0
$783$	−31.6471	−1.13098
$784$	0	0
$785$	9.60044	0.342654
$786$	0	0
$787$	−38.6448	−1.37754	−0.688769	−	0.724980i	$-0.741849\pi$
$787$	−38.6448	−1.37754	−0.688769	+	0.724980i	$0.741849\pi$
$788$	0	0
$789$	−32.6435	−1.16214
$790$	0	0
$791$	−50.9121	−1.81023
$792$	0	0
$793$	0.229789	0.00816006
$794$	0	0
$795$	15.3172	0.543246
$796$	0	0
$797$	−8.38007	−0.296837	−0.148419	−	0.988925i	$-0.547418\pi$
$797$	−8.38007	−0.296837	−0.148419	+	0.988925i	$0.547418\pi$
$798$	0	0
$799$	−24.5039	−0.866886
$800$	0	0
$801$	−0.142560	−0.00503711
$802$	0	0
$803$	−54.4311	−1.92083
$804$	0	0
$805$	35.5077	1.25148
$806$	0	0
$807$	27.6882	0.974671
$808$	0	0
$809$	−20.2642	−0.712450	−0.356225	−	0.934400i	$-0.615936\pi$
$809$	−20.2642	−0.712450	−0.356225	+	0.934400i	$0.615936\pi$
$810$	0	0
$811$	15.1391	0.531606	0.265803	−	0.964027i	$-0.414363\pi$
$811$	15.1391	0.531606	0.265803	+	0.964027i	$0.414363\pi$
$812$	0	0
$813$	49.0296	1.71954
$814$	0	0
$815$	4.90681	0.171878
$816$	0	0
$817$	−11.4625	−0.401023
$818$	0	0
$819$	−0.0157931	−0.000551855 0
$820$	0	0
$821$	11.6562	0.406803	0.203402	−	0.979095i	$-0.434800\pi$
$821$	11.6562	0.406803	0.203402	+	0.979095i	$0.434800\pi$
$822$	0	0
$823$	37.6062	1.31087	0.655436	−	0.755251i	$-0.272485\pi$
$823$	37.6062	1.31087	0.655436	+	0.755251i	$0.272485\pi$
$824$	0	0
$825$	−8.48508	−0.295413
$826$	0	0
$827$	50.0978	1.74207	0.871036	−	0.491219i	$-0.163449\pi$
$827$	50.0978	1.74207	0.871036	+	0.491219i	$0.163449\pi$
$828$	0	0
$829$	−18.7309	−0.650552	−0.325276	−	0.945619i	$-0.605457\pi$
$829$	−18.7309	−0.650552	−0.325276	+	0.945619i	$0.605457\pi$
$830$	0	0
$831$	−35.1293	−1.21862
$832$	0	0
$833$	−35.8095	−1.24073
$834$	0	0
$835$	6.54582	0.226528
$836$	0	0
$837$	49.4655	1.70978
$838$	0	0
$839$	−23.7615	−0.820339	−0.410170	−	0.912009i	$-0.634531\pi$
$839$	−23.7615	−0.820339	−0.410170	+	0.912009i	$0.634531\pi$
$840$	0	0
$841$	9.62793	0.331998
$842$	0	0
$843$	−18.5703	−0.639595
$844$	0	0
$845$	12.9988	0.447171
$846$	0	0
$847$	−47.1530	−1.62020
$848$	0	0
$849$	7.52140	0.258134
$850$	0	0
$851$	−6.83378	−0.234259
$852$	0	0
$853$	−3.61980	−0.123940	−0.0619699	−	0.998078i	$-0.519738\pi$
$853$	−3.61980	−0.123940	−0.0619699	+	0.998078i	$0.519738\pi$
$854$	0	0
$855$	−0.114895	−0.00392931
$856$	0	0
$857$	−36.1820	−1.23595	−0.617977	−	0.786196i	$-0.712048\pi$
$857$	−36.1820	−1.23595	−0.617977	+	0.786196i	$0.712048\pi$
$858$	0	0
$859$	−52.3890	−1.78749	−0.893745	−	0.448576i	$-0.851931\pi$
$859$	−52.3890	−1.78749	−0.893745	+	0.448576i	$0.851931\pi$
$860$	0	0
$861$	−24.2498	−0.826430
$862$	0	0
$863$	−48.7594	−1.65979	−0.829895	−	0.557919i	$-0.811600\pi$
$863$	−48.7594	−1.65979	−0.829895	+	0.557919i	$0.811600\pi$
$864$	0	0
$865$	23.9483	0.814267
$866$	0	0
$867$	−4.05263	−0.137635
$868$	0	0
$869$	54.8750	1.86151
$870$	0	0
$871$	−0.467880	−0.0158535
$872$	0	0
$873$	1.07079	0.0362408
$874$	0	0
$875$	3.89255	0.131592
$876$	0	0
$877$	−9.17301	−0.309751	−0.154875	−	0.987934i	$-0.549498\pi$
$877$	−9.17301	−0.309751	−0.154875	+	0.987934i	$0.549498\pi$
$878$	0	0
$879$	3.82271	0.128937
$880$	0	0
$881$	44.7985	1.50930	0.754649	−	0.656128i	$-0.227807\pi$
$881$	44.7985	1.50930	0.754649	+	0.656128i	$0.227807\pi$
$882$	0	0
$883$	40.4371	1.36082	0.680408	−	0.732833i	$-0.261802\pi$
$883$	40.4371	1.36082	0.680408	+	0.732833i	$0.261802\pi$
$884$	0	0
$885$	−4.52261	−0.152026
$886$	0	0
$887$	11.6569	0.391400	0.195700	−	0.980664i	$-0.437302\pi$
$887$	11.6569	0.391400	0.195700	+	0.980664i	$0.437302\pi$
$888$	0	0
$889$	58.3975	1.95859
$890$	0	0
$891$	−44.8627	−1.50296
$892$	0	0
$893$	5.57827	0.186670
$894$	0	0
$895$	−9.98637	−0.333808
$896$	0	0
$897$	−0.568515	−0.0189822
$898$	0	0
$899$	−60.3768	−2.01368
$900$	0	0
$901$	−38.1237	−1.27008
$902$	0	0
$903$	−78.7474	−2.62055
$904$	0	0
$905$	7.41406	0.246452
$906$	0	0
$907$	−29.7723	−0.988575	−0.494287	−	0.869299i	$-0.664571\pi$
$907$	−29.7723	−0.988575	−0.494287	+	0.869299i	$0.664571\pi$
$908$	0	0
$909$	0.0927965	0.00307786
$910$	0	0
$911$	−7.86284	−0.260507	−0.130254	−	0.991481i	$-0.541579\pi$
$911$	−7.86284	−0.260507	−0.130254	+	0.991481i	$0.541579\pi$
$912$	0	0
$913$	−27.1580	−0.898798
$914$	0	0
$915$	11.4847	0.379672
$916$	0	0
$917$	−16.0141	−0.528832
$918$	0	0
$919$	−11.2607	−0.371455	−0.185727	−	0.982601i	$-0.559464\pi$
$919$	−11.2607	−0.371455	−0.185727	+	0.982601i	$0.559464\pi$
$920$	0	0
$921$	26.7008	0.879822
$922$	0	0
$923$	0.0590100	0.00194234
$924$	0	0
$925$	−0.749158	−0.0246322
$926$	0	0
$927$	−1.95969	−0.0643647
$928$	0	0
$929$	−6.12699	−0.201020	−0.100510	−	0.994936i	$-0.532047\pi$
$929$	−6.12699	−0.201020	−0.100510	+	0.994936i	$0.532047\pi$
$930$	0	0
$931$	8.15196	0.267170
$932$	0	0
$933$	−21.2106	−0.694404
$934$	0	0
$935$	21.1189	0.690661
$936$	0	0
$937$	20.2530	0.661635	0.330818	−	0.943695i	$-0.392676\pi$
$937$	20.2530	0.661635	0.330818	+	0.943695i	$0.392676\pi$
$938$	0	0
$939$	40.8633	1.33352
$940$	0	0
$941$	0.565626	0.0184389	0.00921945	−	0.999958i	$-0.497065\pi$
$941$	0.565626	0.0184389	0.00921945	+	0.999958i	$0.497065\pi$
$942$	0	0
$943$	−32.1988	−1.04854
$944$	0	0
$945$	19.8206	0.644766
$946$	0	0
$947$	6.16157	0.200224	0.100112	−	0.994976i	$-0.468080\pi$
$947$	6.16157	0.200224	0.100112	+	0.994976i	$0.468080\pi$
$948$	0	0
$949$	−0.399802	−0.0129781
$950$	0	0
$951$	−12.1426	−0.393752
$952$	0	0
$953$	15.1913	0.492095	0.246047	−	0.969258i	$-0.420868\pi$
$953$	15.1913	0.492095	0.246047	+	0.969258i	$0.420868\pi$
$954$	0	0
$955$	22.3007	0.721632
$956$	0	0
$957$	52.7359	1.70471
$958$	0	0
$959$	10.8112	0.349113
$960$	0	0
$961$	63.3711	2.04423
$962$	0	0
$963$	−0.966273	−0.0311377
$964$	0	0
$965$	3.15393	0.101529
$966$	0	0
$967$	26.5387	0.853429	0.426714	−	0.904386i	$-0.359671\pi$
$967$	26.5387	0.853429	0.426714	+	0.904386i	$0.359671\pi$
$968$	0	0
$969$	7.75279	0.249055
$970$	0	0
$971$	−59.8610	−1.92103	−0.960516	−	0.278226i	$-0.910254\pi$
$971$	−59.8610	−1.92103	−0.960516	+	0.278226i	$0.910254\pi$
$972$	0	0
$973$	21.0528	0.674923
$974$	0	0
$975$	−0.0623238	−0.00199596
$976$	0	0
$977$	22.5670	0.721983	0.360991	−	0.932569i	$-0.382438\pi$
$977$	22.5670	0.721983	0.360991	+	0.932569i	$0.382438\pi$
$978$	0	0
$979$	−5.96530	−0.190652
$980$	0	0
$981$	−1.85335	−0.0591730
$982$	0	0
$983$	−5.97094	−0.190443	−0.0952217	−	0.995456i	$-0.530356\pi$
$983$	−5.97094	−0.190443	−0.0952217	+	0.995456i	$0.530356\pi$
$984$	0	0
$985$	−5.21801	−0.166260
$986$	0	0
$987$	38.3226	1.21982
$988$	0	0
$989$	−104.561	−3.32483
$990$	0	0
$991$	12.8794	0.409127	0.204564	−	0.978853i	$-0.434422\pi$
$991$	12.8794	0.409127	0.204564	+	0.978853i	$0.434422\pi$
$992$	0	0
$993$	−18.8710	−0.598855
$994$	0	0
$995$	−22.2841	−0.706452
$996$	0	0
$997$	0.948160	0.0300285	0.0150143	−	0.999887i	$-0.495221\pi$
$997$	0.948160	0.0300285	0.0150143	+	0.999887i	$0.495221\pi$
$998$	0	0
$999$	−3.81467	−0.120691

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.a.u.1.2	✓	5
4.3	odd	2		3040.2.a.x.1.4	yes	5
8.3	odd	2		6080.2.a.ci.1.2		5
8.5	even	2		6080.2.a.cl.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.u.1.2	✓	5	1.1	even	1	trivial
3040.2.a.x.1.4	yes	5	4.3	odd	2
6080.2.a.ci.1.2		5	8.3	odd	2
6080.2.a.cl.1.4		5	8.5	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.76491 q^{3} -1.00000 q^{5} -3.89255 q^{7} +0.114895 q^{9} +4.80767 q^{11} +0.0353128 q^{13} +1.76491 q^{15} -4.39275 q^{17} +1.00000 q^{19} +6.86999 q^{21} +9.12195 q^{23} +1.00000 q^{25} +5.09194 q^{27} -6.21514 q^{29} +9.71448 q^{31} -8.48508 q^{33} +3.89255 q^{35} -0.749158 q^{37} -0.0623238 q^{39} -3.52981 q^{41} -11.4625 q^{43} -0.114895 q^{45} +5.57827 q^{47} +8.15196 q^{49} +7.75279 q^{51} +8.67877 q^{53} -4.80767 q^{55} -1.76491 q^{57} -2.56252 q^{59} +6.50725 q^{61} -0.447233 q^{63} -0.0353128 q^{65} -13.2496 q^{67} -16.0994 q^{69} +1.67106 q^{71} -11.3217 q^{73} -1.76491 q^{75} -18.7141 q^{77} +11.4141 q^{79} -9.33148 q^{81} -5.64889 q^{83} +4.39275 q^{85} +10.9691 q^{87} -1.24079 q^{89} -0.137457 q^{91} -17.1451 q^{93} -1.00000 q^{95} +9.31976 q^{97} +0.552375 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} - 5 q^{5} + 4 q^{7} + 7 q^{9} - 2 q^{11} - 4 q^{13} + 4 q^{15} - 12 q^{17} + 5 q^{19} - 10 q^{21} + 8 q^{23} + 5 q^{25} - 16 q^{27} - 6 q^{29} + 10 q^{31} - 18 q^{33} - 4 q^{35} - 6 q^{37}+ \cdots + 14 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 - 5 * q^5 + 4 * q^7 + 7 * q^9 - 2 * q^11 - 4 * q^13 + 4 * q^15 - 12 * q^17 + 5 * q^19 - 10 * q^21 + 8 * q^23 + 5 * q^25 - 16 * q^27 - 6 * q^29 + 10 * q^31 - 18 * q^33 - 4 * q^35 - 6 * q^37 + 18 * q^39 - 8 * q^41 - 12 * q^43 - 7 * q^45 + 16 * q^47 + 7 * q^49 + 14 * q^51 - 18 * q^53 + 2 * q^55 - 4 * q^57 - 8 * q^59 + 2 * q^61 + 36 * q^63 + 4 * q^65 - 10 * q^67 - 22 * q^69 - 18 * q^71 - 28 * q^73 - 4 * q^75 - 28 * q^77 + 14 * q^79 + 25 * q^81 - 8 * q^83 + 12 * q^85 + 24 * q^87 - 30 * q^89 - 28 * q^91 - 24 * q^93 - 5 * q^95 - 18 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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