LMFDB - Embedded newform 320.4.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,4,Mod(1,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("320.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	320.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:	$18.8806112018$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.21110 q^{3} +5.00000 q^{5} +7.21110 q^{7} +25.0000 q^{9} +O(q^{10})$	$q-7.21110 q^{3} +5.00000 q^{5} +7.21110 q^{7} +25.0000 q^{9} -43.2666 q^{11} -34.0000 q^{13} -36.0555 q^{15} +114.000 q^{17} -52.0000 q^{21} -209.122 q^{23} +25.0000 q^{25} +14.4222 q^{27} +26.0000 q^{29} +100.955 q^{31} +312.000 q^{33} +36.0555 q^{35} +150.000 q^{37} +245.177 q^{39} +342.000 q^{41} +454.299 q^{43} +125.000 q^{45} +584.099 q^{47} -291.000 q^{49} -822.066 q^{51} +262.000 q^{53} -216.333 q^{55} +490.355 q^{59} +262.000 q^{61} +180.278 q^{63} -170.000 q^{65} +497.566 q^{67} +1508.00 q^{69} -1052.82 q^{71} +682.000 q^{73} -180.278 q^{75} -312.000 q^{77} -201.911 q^{79} -779.000 q^{81} -151.433 q^{83} +570.000 q^{85} -187.489 q^{87} -630.000 q^{89} -245.177 q^{91} -728.000 q^{93} -966.000 q^{97} -1081.67 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{5} + 50 q^{9}+O(q^{10}) $ 2 * q + 10 * q^5 + 50 * q^9	$ 2 q + 10 q^{5} + 50 q^{9} - 68 q^{13} + 228 q^{17} - 104 q^{21} + 50 q^{25} + 52 q^{29} + 624 q^{33} + 300 q^{37} + 684 q^{41} + 250 q^{45} - 582 q^{49} + 524 q^{53} + 524 q^{61} - 340 q^{65} + 3016 q^{69} + 1364 q^{73} - 624 q^{77} - 1558 q^{81} + 1140 q^{85} - 1260 q^{89} - 1456 q^{93} - 1932 q^{97}+O(q^{100}) $ 2 * q + 10 * q^5 + 50 * q^9 - 68 * q^13 + 228 * q^17 - 104 * q^21 + 50 * q^25 + 52 * q^29 + 624 * q^33 + 300 * q^37 + 684 * q^41 + 250 * q^45 - 582 * q^49 + 524 * q^53 + 524 * q^61 - 340 * q^65 + 3016 * q^69 + 1364 * q^73 - 624 * q^77 - 1558 * q^81 + 1140 * q^85 - 1260 * q^89 - 1456 * q^93 - 1932 * q^97

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
$3$	−7.21110	−1.38778	−0.693889	−	0.720082i	$-0.744104\pi$
$3$	−7.21110	−1.38778	−0.693889	+	0.720082i	$0.744104\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	7.21110	0.389363	0.194681	−	0.980867i	$-0.437633\pi$
$7$	7.21110	0.389363	0.194681	+	0.980867i	$0.437633\pi$
$8$	0	0
$9$	25.0000	0.925926
$10$	0	0
$11$	−43.2666	−1.18594	−0.592972	−	0.805223i	$-0.702045\pi$
$11$	−43.2666	−1.18594	−0.592972	+	0.805223i	$0.702045\pi$
$12$	0	0
$13$	−34.0000	−0.725377	−0.362689	−	0.931910i	$-0.618141\pi$
$13$	−34.0000	−0.725377	−0.362689	+	0.931910i	$0.618141\pi$
$14$	0	0
$15$	−36.0555	−0.620633
$16$	0	0
$17$	114.000	1.62642	0.813208	−	0.581974i	$-0.197719\pi$
$17$	114.000	1.62642	0.813208	+	0.581974i	$0.197719\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−52.0000	−0.540349
$22$	0	0
$23$	−209.122	−1.89587	−0.947934	−	0.318468i	$-0.896832\pi$
$23$	−209.122	−1.89587	−0.947934	+	0.318468i	$0.896832\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	14.4222	0.102798
$28$	0	0
$29$	26.0000	0.166485	0.0832427	−	0.996529i	$-0.473472\pi$
$29$	26.0000	0.166485	0.0832427	+	0.996529i	$0.473472\pi$
$30$	0	0
$31$	100.955	0.584907	0.292454	−	0.956280i	$-0.405528\pi$
$31$	100.955	0.584907	0.292454	+	0.956280i	$0.405528\pi$
$32$	0	0
$33$	312.000	1.64583
$34$	0	0
$35$	36.0555	0.174128
$36$	0	0
$37$	150.000	0.666482	0.333241	−	0.942842i	$-0.391858\pi$
$37$	150.000	0.666482	0.333241	+	0.942842i	$0.391858\pi$
$38$	0	0
$39$	245.177	1.00666
$40$	0	0
$41$	342.000	1.30272	0.651359	−	0.758770i	$-0.274199\pi$
$41$	342.000	1.30272	0.651359	+	0.758770i	$0.274199\pi$
$42$	0	0
$43$	454.299	1.61116	0.805582	−	0.592485i	$-0.201853\pi$
$43$	454.299	1.61116	0.805582	+	0.592485i	$0.201853\pi$
$44$	0	0
$45$	125.000	0.414087
$46$	0	0
$47$	584.099	1.81276	0.906379	−	0.422465i	$-0.138835\pi$
$47$	584.099	1.81276	0.906379	+	0.422465i	$0.138835\pi$
$48$	0	0
$49$	−291.000	−0.848397
$50$	0	0
$51$	−822.066	−2.25710
$52$	0	0
$53$	262.000	0.679028	0.339514	−	0.940601i	$-0.389737\pi$
$53$	262.000	0.679028	0.339514	+	0.940601i	$0.389737\pi$
$54$	0	0
$55$	−216.333	−0.530370
$56$	0	0
$57$	0	0
$58$	0	0
$59$	490.355	1.08201	0.541007	−	0.841018i	$-0.318043\pi$
$59$	490.355	1.08201	0.541007	+	0.841018i	$0.318043\pi$
$60$	0	0
$61$	262.000	0.549929	0.274964	−	0.961454i	$-0.411334\pi$
$61$	262.000	0.549929	0.274964	+	0.961454i	$0.411334\pi$
$62$	0	0
$63$	180.278	0.360521
$64$	0	0
$65$	−170.000	−0.324399
$66$	0	0
$67$	497.566	0.907274	0.453637	−	0.891187i	$-0.350126\pi$
$67$	497.566	0.907274	0.453637	+	0.891187i	$0.350126\pi$
$68$	0	0
$69$	1508.00	2.63104
$70$	0	0
$71$	−1052.82	−1.75981	−0.879907	−	0.475145i	$-0.842396\pi$
$71$	−1052.82	−1.75981	−0.879907	+	0.475145i	$0.842396\pi$
$72$	0	0
$73$	682.000	1.09345	0.546726	−	0.837311i	$-0.315874\pi$
$73$	682.000	1.09345	0.546726	+	0.837311i	$0.315874\pi$
$74$	0	0
$75$	−180.278	−0.277555
$76$	0	0
$77$	−312.000	−0.461762
$78$	0	0
$79$	−201.911	−0.287554	−0.143777	−	0.989610i	$-0.545925\pi$
$79$	−201.911	−0.287554	−0.143777	+	0.989610i	$0.545925\pi$
$80$	0	0
$81$	−779.000	−1.06859
$82$	0	0
$83$	−151.433	−0.200264	−0.100132	−	0.994974i	$-0.531927\pi$
$83$	−151.433	−0.200264	−0.100132	+	0.994974i	$0.531927\pi$
$84$	0	0
$85$	570.000	0.727355
$86$	0	0
$87$	−187.489	−0.231045
$88$	0	0
$89$	−630.000	−0.750336	−0.375168	−	0.926957i	$-0.622415\pi$
$89$	−630.000	−0.750336	−0.375168	+	0.926957i	$0.622415\pi$
$90$	0	0
$91$	−245.177	−0.282435
$92$	0	0
$93$	−728.000	−0.811721
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−966.000	−1.01116	−0.505580	−	0.862780i	$-0.668721\pi$
$97$	−966.000	−1.01116	−0.505580	+	0.862780i	$0.668721\pi$
$98$	0	0
$99$	−1081.67	−1.09810
$100$	0	0
$101$	−1638.00	−1.61373	−0.806867	−	0.590733i	$-0.798838\pi$
$101$	−1638.00	−1.61373	−0.806867	+	0.590733i	$0.798838\pi$
$102$	0	0
$103$	685.055	0.655344	0.327672	−	0.944792i	$-0.393736\pi$
$103$	685.055	0.655344	0.327672	+	0.944792i	$0.393736\pi$
$104$	0	0
$105$	−260.000	−0.241651
$106$	0	0
$107$	612.944	0.553790	0.276895	−	0.960900i	$-0.410695\pi$
$107$	612.944	0.553790	0.276895	+	0.960900i	$0.410695\pi$
$108$	0	0
$109$	342.000	0.300529	0.150264	−	0.988646i	$-0.451987\pi$
$109$	342.000	0.300529	0.150264	+	0.988646i	$0.451987\pi$
$110$	0	0
$111$	−1081.67	−0.924929
$112$	0	0
$113$	2106.00	1.75324	0.876619	−	0.481186i	$-0.159794\pi$
$113$	2106.00	1.75324	0.876619	+	0.481186i	$0.159794\pi$
$114$	0	0
$115$	−1045.61	−0.847858
$116$	0	0
$117$	−850.000	−0.671646
$118$	0	0
$119$	822.066	0.633266
$120$	0	0
$121$	541.000	0.406461
$122$	0	0
$123$	−2466.20	−1.80788
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	439.877	0.307345	0.153672	−	0.988122i	$-0.450890\pi$
$127$	439.877	0.307345	0.153672	+	0.988122i	$0.450890\pi$
$128$	0	0
$129$	−3276.00	−2.23594
$130$	0	0
$131$	1773.93	1.18312	0.591561	−	0.806260i	$-0.298512\pi$
$131$	1773.93	1.18312	0.591561	+	0.806260i	$0.298512\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	72.1110	0.0459728
$136$	0	0
$137$	1618.00	1.00902	0.504508	−	0.863407i	$-0.331674\pi$
$137$	1618.00	1.00902	0.504508	+	0.863407i	$0.331674\pi$
$138$	0	0
$139$	2509.46	1.53129	0.765647	−	0.643261i	$-0.222419\pi$
$139$	2509.46	1.53129	0.765647	+	0.643261i	$0.222419\pi$
$140$	0	0
$141$	−4212.00	−2.51570
$142$	0	0
$143$	1471.06	0.860256
$144$	0	0
$145$	130.000	0.0744546
$146$	0	0
$147$	2098.43	1.17739
$148$	0	0
$149$	−1010.00	−0.555318	−0.277659	−	0.960680i	$-0.589559\pi$
$149$	−1010.00	−0.555318	−0.277659	+	0.960680i	$0.589559\pi$
$150$	0	0
$151$	−14.4222	−0.00777260	−0.00388630	−	0.999992i	$-0.501237\pi$
$151$	−14.4222	−0.00777260	−0.00388630	+	0.999992i	$0.501237\pi$
$152$	0	0
$153$	2850.00	1.50594
$154$	0	0
$155$	504.777	0.261579
$156$	0	0
$157$	−1794.00	−0.911954	−0.455977	−	0.889992i	$-0.650710\pi$
$157$	−1794.00	−0.911954	−0.455977	+	0.889992i	$0.650710\pi$
$158$	0	0
$159$	−1889.31	−0.942339
$160$	0	0
$161$	−1508.00	−0.738180
$162$	0	0
$163$	1983.05	0.952912	0.476456	−	0.879198i	$-0.341921\pi$
$163$	1983.05	0.952912	0.476456	+	0.879198i	$0.341921\pi$
$164$	0	0
$165$	1560.00	0.736035
$166$	0	0
$167$	757.166	0.350846	0.175423	−	0.984493i	$-0.443871\pi$
$167$	757.166	0.350846	0.175423	+	0.984493i	$0.443871\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2834.00	−1.24546	−0.622731	−	0.782436i	$-0.713977\pi$
$173$	−2834.00	−1.24546	−0.622731	+	0.782436i	$0.713977\pi$
$174$	0	0
$175$	180.278	0.0778726
$176$	0	0
$177$	−3536.00	−1.50159
$178$	0	0
$179$	−951.866	−0.397462	−0.198731	−	0.980054i	$-0.563682\pi$
$179$	−951.866	−0.397462	−0.198731	+	0.980054i	$0.563682\pi$
$180$	0	0
$181$	1466.00	0.602027	0.301014	−	0.953620i	$-0.402675\pi$
$181$	1466.00	0.602027	0.301014	+	0.953620i	$0.402675\pi$
$182$	0	0
$183$	−1889.31	−0.763179
$184$	0	0
$185$	750.000	0.298060
$186$	0	0
$187$	−4932.39	−1.92884
$188$	0	0
$189$	104.000	0.0400259
$190$	0	0
$191$	3475.75	1.31674	0.658368	−	0.752696i	$-0.271247\pi$
$191$	3475.75	1.31674	0.658368	+	0.752696i	$0.271247\pi$
$192$	0	0
$193$	−46.0000	−0.0171562	−0.00857812	−	0.999963i	$-0.502731\pi$
$193$	−46.0000	−0.0171562	−0.00857812	+	0.999963i	$0.502731\pi$
$194$	0	0
$195$	1225.89	0.450193
$196$	0	0
$197$	−1122.00	−0.405783	−0.202891	−	0.979201i	$-0.565034\pi$
$197$	−1122.00	−0.405783	−0.202891	+	0.979201i	$0.565034\pi$
$198$	0	0
$199$	2999.82	1.06860	0.534300	−	0.845295i	$-0.320575\pi$
$199$	2999.82	1.06860	0.534300	+	0.845295i	$0.320575\pi$
$200$	0	0
$201$	−3588.00	−1.25909
$202$	0	0
$203$	187.489	0.0648233
$204$	0	0
$205$	1710.00	0.582593
$206$	0	0
$207$	−5228.05	−1.75543
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−418.244	−0.136460	−0.0682301	−	0.997670i	$-0.521735\pi$
$211$	−418.244	−0.136460	−0.0682301	+	0.997670i	$0.521735\pi$
$212$	0	0
$213$	7592.00	2.44223
$214$	0	0
$215$	2271.50	0.720534
$216$	0	0
$217$	728.000	0.227741
$218$	0	0
$219$	−4917.97	−1.51747
$220$	0	0
$221$	−3876.00	−1.17976
$222$	0	0
$223$	−2545.52	−0.764397	−0.382199	−	0.924080i	$-0.624833\pi$
$223$	−2545.52	−0.764397	−0.382199	+	0.924080i	$0.624833\pi$
$224$	0	0
$225$	625.000	0.185185
$226$	0	0
$227$	−5069.41	−1.48224	−0.741119	−	0.671373i	$-0.765705\pi$
$227$	−5069.41	−1.48224	−0.741119	+	0.671373i	$0.765705\pi$
$228$	0	0
$229$	6194.00	1.78738	0.893692	−	0.448681i	$-0.148106\pi$
$229$	6194.00	1.78738	0.893692	+	0.448681i	$0.148106\pi$
$230$	0	0
$231$	2249.86	0.640823
$232$	0	0
$233$	4290.00	1.20621	0.603106	−	0.797661i	$-0.293930\pi$
$233$	4290.00	1.20621	0.603106	+	0.797661i	$0.293930\pi$
$234$	0	0
$235$	2920.50	0.810690
$236$	0	0
$237$	1456.00	0.399061
$238$	0	0
$239$	−5278.53	−1.42862	−0.714309	−	0.699831i	$-0.753259\pi$
$239$	−5278.53	−1.42862	−0.714309	+	0.699831i	$0.753259\pi$
$240$	0	0
$241$	−3074.00	−0.821634	−0.410817	−	0.911718i	$-0.634756\pi$
$241$	−3074.00	−0.821634	−0.410817	+	0.911718i	$0.634756\pi$
$242$	0	0
$243$	5228.05	1.38016
$244$	0	0
$245$	−1455.00	−0.379414
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1092.00	0.277922
$250$	0	0
$251$	−2062.38	−0.518629	−0.259315	−	0.965793i	$-0.583497\pi$
$251$	−2062.38	−0.518629	−0.259315	+	0.965793i	$0.583497\pi$
$252$	0	0
$253$	9048.00	2.24839
$254$	0	0
$255$	−4110.33	−1.00941
$256$	0	0
$257$	−3718.00	−0.902422	−0.451211	−	0.892417i	$-0.649008\pi$
$257$	−3718.00	−0.902422	−0.451211	+	0.892417i	$0.649008\pi$
$258$	0	0
$259$	1081.67	0.259504
$260$	0	0
$261$	650.000	0.154153
$262$	0	0
$263$	−7045.25	−1.65182	−0.825910	−	0.563802i	$-0.809338\pi$
$263$	−7045.25	−1.65182	−0.825910	+	0.563802i	$0.809338\pi$
$264$	0	0
$265$	1310.00	0.303670
$266$	0	0
$267$	4542.99	1.04130
$268$	0	0
$269$	−6058.00	−1.37310	−0.686548	−	0.727085i	$-0.740875\pi$
$269$	−6058.00	−1.37310	−0.686548	+	0.727085i	$0.740875\pi$
$270$	0	0
$271$	5206.42	1.16704	0.583519	−	0.812100i	$-0.301675\pi$
$271$	5206.42	1.16704	0.583519	+	0.812100i	$0.301675\pi$
$272$	0	0
$273$	1768.00	0.391957
$274$	0	0
$275$	−1081.67	−0.237189
$276$	0	0
$277$	2990.00	0.648562	0.324281	−	0.945961i	$-0.394878\pi$
$277$	2990.00	0.648562	0.324281	+	0.945961i	$0.394878\pi$
$278$	0	0
$279$	2523.89	0.541581
$280$	0	0
$281$	2710.00	0.575320	0.287660	−	0.957733i	$-0.407123\pi$
$281$	2710.00	0.575320	0.287660	+	0.957733i	$0.407123\pi$
$282$	0	0
$283$	−4593.47	−0.964854	−0.482427	−	0.875936i	$-0.660245\pi$
$283$	−4593.47	−0.964854	−0.482427	+	0.875936i	$0.660245\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2466.20	0.507230
$288$	0	0
$289$	8083.00	1.64523
$290$	0	0
$291$	6965.93	1.40326
$292$	0	0
$293$	3750.00	0.747704	0.373852	−	0.927488i	$-0.378037\pi$
$293$	3750.00	0.747704	0.373852	+	0.927488i	$0.378037\pi$
$294$	0	0
$295$	2451.77	0.483891
$296$	0	0
$297$	−624.000	−0.121913
$298$	0	0
$299$	7110.15	1.37522
$300$	0	0
$301$	3276.00	0.627327
$302$	0	0
$303$	11811.8	2.23950
$304$	0	0
$305$	1310.00	0.245936
$306$	0	0
$307$	−4405.98	−0.819097	−0.409548	−	0.912288i	$-0.634314\pi$
$307$	−4405.98	−0.819097	−0.409548	+	0.912288i	$0.634314\pi$
$308$	0	0
$309$	−4940.00	−0.909472
$310$	0	0
$311$	2956.55	0.539070	0.269535	−	0.962991i	$-0.413130\pi$
$311$	2956.55	0.539070	0.269535	+	0.962991i	$0.413130\pi$
$312$	0	0
$313$	5642.00	1.01886	0.509432	−	0.860511i	$-0.329855\pi$
$313$	5642.00	1.01886	0.509432	+	0.860511i	$0.329855\pi$
$314$	0	0
$315$	901.388	0.161230
$316$	0	0
$317$	−4650.00	−0.823880	−0.411940	−	0.911211i	$-0.635149\pi$
$317$	−4650.00	−0.823880	−0.411940	+	0.911211i	$0.635149\pi$
$318$	0	0
$319$	−1124.93	−0.197442
$320$	0	0
$321$	−4420.00	−0.768537
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−850.000	−0.145075
$326$	0	0
$327$	−2466.20	−0.417067
$328$	0	0
$329$	4212.00	0.705821
$330$	0	0
$331$	5523.70	0.917252	0.458626	−	0.888630i	$-0.348342\pi$
$331$	5523.70	0.917252	0.458626	+	0.888630i	$0.348342\pi$
$332$	0	0
$333$	3750.00	0.617113
$334$	0	0
$335$	2487.83	0.405745
$336$	0	0
$337$	9266.00	1.49778	0.748889	−	0.662695i	$-0.230588\pi$
$337$	9266.00	1.49778	0.748889	+	0.662695i	$0.230588\pi$
$338$	0	0
$339$	−15186.6	−2.43310
$340$	0	0
$341$	−4368.00	−0.693667
$342$	0	0
$343$	−4571.84	−0.719697
$344$	0	0
$345$	7540.00	1.17664
$346$	0	0
$347$	814.855	0.126062	0.0630312	−	0.998012i	$-0.479923\pi$
$347$	814.855	0.126062	0.0630312	+	0.998012i	$0.479923\pi$
$348$	0	0
$349$	−7494.00	−1.14941	−0.574706	−	0.818360i	$-0.694883\pi$
$349$	−7494.00	−1.14941	−0.574706	+	0.818360i	$0.694883\pi$
$350$	0	0
$351$	−490.355	−0.0745676
$352$	0	0
$353$	−6270.00	−0.945378	−0.472689	−	0.881229i	$-0.656717\pi$
$353$	−6270.00	−0.945378	−0.472689	+	0.881229i	$0.656717\pi$
$354$	0	0
$355$	−5264.10	−0.787013
$356$	0	0
$357$	−5928.00	−0.878832
$358$	0	0
$359$	−692.266	−0.101773	−0.0508863	−	0.998704i	$-0.516205\pi$
$359$	−692.266	−0.101773	−0.0508863	+	0.998704i	$0.516205\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	0	0
$363$	−3901.21	−0.564078
$364$	0	0
$365$	3410.00	0.489007
$366$	0	0
$367$	2141.70	0.304620	0.152310	−	0.988333i	$-0.451329\pi$
$367$	2141.70	0.304620	0.152310	+	0.988333i	$0.451329\pi$
$368$	0	0
$369$	8550.00	1.20622
$370$	0	0
$371$	1889.31	0.264388
$372$	0	0
$373$	2574.00	0.357310	0.178655	−	0.983912i	$-0.442825\pi$
$373$	2574.00	0.357310	0.178655	+	0.983912i	$0.442825\pi$
$374$	0	0
$375$	−901.388	−0.124127
$376$	0	0
$377$	−884.000	−0.120765
$378$	0	0
$379$	13729.9	1.86084	0.930422	−	0.366491i	$-0.119441\pi$
$379$	13729.9	1.86084	0.930422	+	0.366491i	$0.119441\pi$
$380$	0	0
$381$	−3172.00	−0.426526
$382$	0	0
$383$	4204.07	0.560883	0.280441	−	0.959871i	$-0.409519\pi$
$383$	4204.07	0.560883	0.280441	+	0.959871i	$0.409519\pi$
$384$	0	0
$385$	−1560.00	−0.206506
$386$	0	0
$387$	11357.5	1.49182
$388$	0	0
$389$	−5314.00	−0.692623	−0.346312	−	0.938120i	$-0.612566\pi$
$389$	−5314.00	−0.692623	−0.346312	+	0.938120i	$0.612566\pi$
$390$	0	0
$391$	−23839.9	−3.08347
$392$	0	0
$393$	−12792.0	−1.64191
$394$	0	0
$395$	−1009.55	−0.128598
$396$	0	0
$397$	8638.00	1.09201	0.546006	−	0.837781i	$-0.316148\pi$
$397$	8638.00	1.09201	0.546006	+	0.837781i	$0.316148\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2802.00	0.348941	0.174470	−	0.984662i	$-0.444179\pi$
$401$	2802.00	0.348941	0.174470	+	0.984662i	$0.444179\pi$
$402$	0	0
$403$	−3432.48	−0.424279
$404$	0	0
$405$	−3895.00	−0.477887
$406$	0	0
$407$	−6489.99	−0.790410
$408$	0	0
$409$	−82.0000	−0.00991354	−0.00495677	−	0.999988i	$-0.501578\pi$
$409$	−82.0000	−0.00991354	−0.00495677	+	0.999988i	$0.501578\pi$
$410$	0	0
$411$	−11667.6	−1.40029
$412$	0	0
$413$	3536.00	0.421296
$414$	0	0
$415$	−757.166	−0.0895610
$416$	0	0
$417$	−18096.0	−2.12510
$418$	0	0
$419$	−1067.24	−0.124435	−0.0622175	−	0.998063i	$-0.519817\pi$
$419$	−1067.24	−0.124435	−0.0622175	+	0.998063i	$0.519817\pi$
$420$	0	0
$421$	5742.00	0.664722	0.332361	−	0.943152i	$-0.392155\pi$
$421$	5742.00	0.664722	0.332361	+	0.943152i	$0.392155\pi$
$422$	0	0
$423$	14602.5	1.67848
$424$	0	0
$425$	2850.00	0.325283
$426$	0	0
$427$	1889.31	0.214122
$428$	0	0
$429$	−10608.0	−1.19384
$430$	0	0
$431$	−14234.7	−1.59086	−0.795432	−	0.606043i	$-0.792756\pi$
$431$	−14234.7	−1.59086	−0.795432	+	0.606043i	$0.792756\pi$
$432$	0	0
$433$	7098.00	0.787779	0.393889	−	0.919158i	$-0.371129\pi$
$433$	7098.00	0.787779	0.393889	+	0.919158i	$0.371129\pi$
$434$	0	0
$435$	−937.443	−0.103326
$436$	0	0
$437$	0	0
$438$	0	0
$439$	461.511	0.0501747	0.0250874	−	0.999685i	$-0.492014\pi$
$439$	461.511	0.0501747	0.0250874	+	0.999685i	$0.492014\pi$
$440$	0	0
$441$	−7275.00	−0.785552
$442$	0	0
$443$	−6064.54	−0.650417	−0.325209	−	0.945642i	$-0.605435\pi$
$443$	−6064.54	−0.650417	−0.325209	+	0.945642i	$0.605435\pi$
$444$	0	0
$445$	−3150.00	−0.335560
$446$	0	0
$447$	7283.21	0.770658
$448$	0	0
$449$	−11706.0	−1.23038	−0.615190	−	0.788379i	$-0.710921\pi$
$449$	−11706.0	−1.23038	−0.615190	+	0.788379i	$0.710921\pi$
$450$	0	0
$451$	−14797.2	−1.54495
$452$	0	0
$453$	104.000	0.0107866
$454$	0	0
$455$	−1225.89	−0.126309
$456$	0	0
$457$	5066.00	0.518550	0.259275	−	0.965803i	$-0.416516\pi$
$457$	5066.00	0.518550	0.259275	+	0.965803i	$0.416516\pi$
$458$	0	0
$459$	1644.13	0.167193
$460$	0	0
$461$	594.000	0.0600116	0.0300058	−	0.999550i	$-0.490447\pi$
$461$	594.000	0.0600116	0.0300058	+	0.999550i	$0.490447\pi$
$462$	0	0
$463$	483.144	0.0484959	0.0242479	−	0.999706i	$-0.492281\pi$
$463$	483.144	0.0484959	0.0242479	+	0.999706i	$0.492281\pi$
$464$	0	0
$465$	−3640.00	−0.363013
$466$	0	0
$467$	−2271.50	−0.225080	−0.112540	−	0.993647i	$-0.535899\pi$
$467$	−2271.50	−0.225080	−0.112540	+	0.993647i	$0.535899\pi$
$468$	0	0
$469$	3588.00	0.353259
$470$	0	0
$471$	12936.7	1.26559
$472$	0	0
$473$	−19656.0	−1.91075
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6550.00	0.628729
$478$	0	0
$479$	4067.06	0.387952	0.193976	−	0.981006i	$-0.437862\pi$
$479$	4067.06	0.387952	0.193976	+	0.981006i	$0.437862\pi$
$480$	0	0
$481$	−5100.00	−0.483451
$482$	0	0
$483$	10874.3	1.02443
$484$	0	0
$485$	−4830.00	−0.452204
$486$	0	0
$487$	11170.0	1.03934	0.519672	−	0.854366i	$-0.326054\pi$
$487$	11170.0	1.03934	0.519672	+	0.854366i	$0.326054\pi$
$488$	0	0
$489$	−14300.0	−1.32243
$490$	0	0
$491$	−4600.68	−0.422863	−0.211432	−	0.977393i	$-0.567813\pi$
$491$	−4600.68	−0.422863	−0.211432	+	0.977393i	$0.567813\pi$
$492$	0	0
$493$	2964.00	0.270775
$494$	0	0
$495$	−5408.33	−0.491083
$496$	0	0
$497$	−7592.00	−0.685207
$498$	0	0
$499$	−288.444	−0.0258768	−0.0129384	−	0.999916i	$-0.504119\pi$
$499$	−288.444	−0.0258768	−0.0129384	+	0.999916i	$0.504119\pi$
$500$	0	0
$501$	−5460.00	−0.486896
$502$	0	0
$503$	−18409.9	−1.63193	−0.815963	−	0.578104i	$-0.803793\pi$
$503$	−18409.9	−1.63193	−0.815963	+	0.578104i	$0.803793\pi$
$504$	0	0
$505$	−8190.00	−0.721684
$506$	0	0
$507$	7506.76	0.657568
$508$	0	0
$509$	8714.00	0.758824	0.379412	−	0.925228i	$-0.376126\pi$
$509$	8714.00	0.758824	0.379412	+	0.925228i	$0.376126\pi$
$510$	0	0
$511$	4917.97	0.425750
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3425.27	0.293079
$516$	0	0
$517$	−25272.0	−2.14983
$518$	0	0
$519$	20436.3	1.72842
$520$	0	0
$521$	−11830.0	−0.994783	−0.497391	−	0.867526i	$-0.665709\pi$
$521$	−11830.0	−0.994783	−0.497391	+	0.867526i	$0.665709\pi$
$522$	0	0
$523$	8963.40	0.749411	0.374706	−	0.927144i	$-0.377744\pi$
$523$	8963.40	0.749411	0.374706	+	0.927144i	$0.377744\pi$
$524$	0	0
$525$	−1300.00	−0.108070
$526$	0	0
$527$	11508.9	0.951302
$528$	0	0
$529$	31565.0	2.59431
$530$	0	0
$531$	12258.9	1.00186
$532$	0	0
$533$	−11628.0	−0.944962
$534$	0	0
$535$	3064.72	0.247662
$536$	0	0
$537$	6864.00	0.551589
$538$	0	0
$539$	12590.6	1.00615
$540$	0	0
$541$	15490.0	1.23099	0.615496	−	0.788140i	$-0.288956\pi$
$541$	15490.0	1.23099	0.615496	+	0.788140i	$0.288956\pi$
$542$	0	0
$543$	−10571.5	−0.835480
$544$	0	0
$545$	1710.00	0.134401
$546$	0	0
$547$	11429.6	0.893408	0.446704	−	0.894682i	$-0.352598\pi$
$547$	11429.6	0.893408	0.446704	+	0.894682i	$0.352598\pi$
$548$	0	0
$549$	6550.00	0.509193
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−1456.00	−0.111963
$554$	0	0
$555$	−5408.33	−0.413641
$556$	0	0
$557$	23862.0	1.81520	0.907599	−	0.419838i	$-0.137913\pi$
$557$	23862.0	1.81520	0.907599	+	0.419838i	$0.137913\pi$
$558$	0	0
$559$	−15446.2	−1.16870
$560$	0	0
$561$	35568.0	2.67680
$562$	0	0
$563$	7261.58	0.543586	0.271793	−	0.962356i	$-0.412383\pi$
$563$	7261.58	0.543586	0.271793	+	0.962356i	$0.412383\pi$
$564$	0	0
$565$	10530.0	0.784072
$566$	0	0
$567$	−5617.45	−0.416068
$568$	0	0
$569$	−9074.00	−0.668545	−0.334272	−	0.942477i	$-0.608491\pi$
$569$	−9074.00	−0.668545	−0.334272	+	0.942477i	$0.608491\pi$
$570$	0	0
$571$	−7860.10	−0.576068	−0.288034	−	0.957620i	$-0.593002\pi$
$571$	−7860.10	−0.576068	−0.288034	+	0.957620i	$0.593002\pi$
$572$	0	0
$573$	−25064.0	−1.82734
$574$	0	0
$575$	−5228.05	−0.379173
$576$	0	0
$577$	4762.00	0.343578	0.171789	−	0.985134i	$-0.445045\pi$
$577$	4762.00	0.343578	0.171789	+	0.985134i	$0.445045\pi$
$578$	0	0
$579$	331.711	0.0238090
$580$	0	0
$581$	−1092.00	−0.0779755
$582$	0	0
$583$	−11335.9	−0.805288
$584$	0	0
$585$	−4250.00	−0.300369
$586$	0	0
$587$	−21481.9	−1.51048	−0.755240	−	0.655448i	$-0.772480\pi$
$587$	−21481.9	−1.51048	−0.755240	+	0.655448i	$0.772480\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	8090.86	0.563136
$592$	0	0
$593$	11954.0	0.827811	0.413906	−	0.910320i	$-0.364164\pi$
$593$	11954.0	0.827811	0.413906	+	0.910320i	$0.364164\pi$
$594$	0	0
$595$	4110.33	0.283205
$596$	0	0
$597$	−21632.0	−1.48298
$598$	0	0
$599$	10759.0	0.733889	0.366944	−	0.930243i	$-0.380404\pi$
$599$	10759.0	0.733889	0.366944	+	0.930243i	$0.380404\pi$
$600$	0	0
$601$	17862.0	1.21232	0.606162	−	0.795342i	$-0.292708\pi$
$601$	17862.0	1.21232	0.606162	+	0.795342i	$0.292708\pi$
$602$	0	0
$603$	12439.2	0.840069
$604$	0	0
$605$	2705.00	0.181775
$606$	0	0
$607$	7506.76	0.501960	0.250980	−	0.967992i	$-0.419247\pi$
$607$	7506.76	0.501960	0.250980	+	0.967992i	$0.419247\pi$
$608$	0	0
$609$	−1352.00	−0.0899603
$610$	0	0
$611$	−19859.4	−1.31493
$612$	0	0
$613$	−11522.0	−0.759167	−0.379583	−	0.925158i	$-0.623933\pi$
$613$	−11522.0	−0.759167	−0.379583	+	0.925158i	$0.623933\pi$
$614$	0	0
$615$	−12331.0	−0.808509
$616$	0	0
$617$	8290.00	0.540912	0.270456	−	0.962732i	$-0.412826\pi$
$617$	8290.00	0.540912	0.270456	+	0.962732i	$0.412826\pi$
$618$	0	0
$619$	−24171.6	−1.56953	−0.784765	−	0.619793i	$-0.787216\pi$
$619$	−24171.6	−1.56953	−0.784765	+	0.619793i	$0.787216\pi$
$620$	0	0
$621$	−3016.00	−0.194892
$622$	0	0
$623$	−4542.99	−0.292153
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17100.0	1.08398
$630$	0	0
$631$	12388.7	0.781593	0.390797	−	0.920477i	$-0.372200\pi$
$631$	12388.7	0.781593	0.390797	+	0.920477i	$0.372200\pi$
$632$	0	0
$633$	3016.00	0.189376
$634$	0	0
$635$	2199.39	0.137449
$636$	0	0
$637$	9894.00	0.615407
$638$	0	0
$639$	−26320.5	−1.62946
$640$	0	0
$641$	6750.00	0.415927	0.207963	−	0.978137i	$-0.433317\pi$
$641$	6750.00	0.415927	0.207963	+	0.978137i	$0.433317\pi$
$642$	0	0
$643$	−23428.9	−1.43693	−0.718464	−	0.695564i	$-0.755154\pi$
$643$	−23428.9	−1.43693	−0.718464	+	0.695564i	$0.755154\pi$
$644$	0	0
$645$	−16380.0	−0.999941
$646$	0	0
$647$	8689.38	0.527998	0.263999	−	0.964523i	$-0.414958\pi$
$647$	8689.38	0.527998	0.263999	+	0.964523i	$0.414958\pi$
$648$	0	0
$649$	−21216.0	−1.28321
$650$	0	0
$651$	−5249.68	−0.316054
$652$	0	0
$653$	−22282.0	−1.33532	−0.667659	−	0.744468i	$-0.732703\pi$
$653$	−22282.0	−1.33532	−0.667659	+	0.744468i	$0.732703\pi$
$654$	0	0
$655$	8869.66	0.529109
$656$	0	0
$657$	17050.0	1.01246
$658$	0	0
$659$	15835.6	0.936065	0.468032	−	0.883711i	$-0.344963\pi$
$659$	15835.6	0.936065	0.468032	+	0.883711i	$0.344963\pi$
$660$	0	0
$661$	11758.0	0.691881	0.345940	−	0.938256i	$-0.387560\pi$
$661$	11758.0	0.691881	0.345940	+	0.938256i	$0.387560\pi$
$662$	0	0
$663$	27950.2	1.63725
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5437.17	−0.315634
$668$	0	0
$669$	18356.0	1.06081
$670$	0	0
$671$	−11335.9	−0.652184
$672$	0	0
$673$	11866.0	0.679644	0.339822	−	0.940490i	$-0.389633\pi$
$673$	11866.0	0.679644	0.339822	+	0.940490i	$0.389633\pi$
$674$	0	0
$675$	360.555	0.0205597
$676$	0	0
$677$	26574.0	1.50860	0.754300	−	0.656530i	$-0.227977\pi$
$677$	26574.0	1.50860	0.754300	+	0.656530i	$0.227977\pi$
$678$	0	0
$679$	−6965.93	−0.393708
$680$	0	0
$681$	36556.0	2.05702
$682$	0	0
$683$	−13737.2	−0.769601	−0.384800	−	0.923000i	$-0.625730\pi$
$683$	−13737.2	−0.769601	−0.384800	+	0.923000i	$0.625730\pi$
$684$	0	0
$685$	8090.00	0.451245
$686$	0	0
$687$	−44665.6	−2.48049
$688$	0	0
$689$	−8908.00	−0.492551
$690$	0	0
$691$	22570.8	1.24259	0.621297	−	0.783576i	$-0.286606\pi$
$691$	22570.8	1.24259	0.621297	+	0.783576i	$0.286606\pi$
$692$	0	0
$693$	−7800.00	−0.427558
$694$	0	0
$695$	12547.3	0.684816
$696$	0	0
$697$	38988.0	2.11876
$698$	0	0
$699$	−30935.6	−1.67395
$700$	0	0
$701$	7062.00	0.380497	0.190248	−	0.981736i	$-0.439071\pi$
$701$	7062.00	0.380497	0.190248	+	0.981736i	$0.439071\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−21060.0	−1.12506
$706$	0	0
$707$	−11811.8	−0.628328
$708$	0	0
$709$	1554.00	0.0823155	0.0411578	−	0.999153i	$-0.486895\pi$
$709$	1554.00	0.0823155	0.0411578	+	0.999153i	$0.486895\pi$
$710$	0	0
$711$	−5047.77	−0.266253
$712$	0	0
$713$	−21112.0	−1.10891
$714$	0	0
$715$	7355.32	0.384718
$716$	0	0
$717$	38064.0	1.98260
$718$	0	0
$719$	31065.4	1.61133	0.805664	−	0.592373i	$-0.201809\pi$
$719$	31065.4	1.61133	0.805664	+	0.592373i	$0.201809\pi$
$720$	0	0
$721$	4940.00	0.255167
$722$	0	0
$723$	22166.9	1.14024
$724$	0	0
$725$	650.000	0.0332971
$726$	0	0
$727$	−24452.8	−1.24746	−0.623732	−	0.781638i	$-0.714384\pi$
$727$	−24452.8	−1.24746	−0.623732	+	0.781638i	$0.714384\pi$
$728$	0	0
$729$	−16667.0	−0.846771
$730$	0	0
$731$	51790.1	2.62042
$732$	0	0
$733$	−15058.0	−0.758772	−0.379386	−	0.925238i	$-0.623865\pi$
$733$	−15058.0	−0.758772	−0.379386	+	0.925238i	$0.623865\pi$
$734$	0	0
$735$	10492.2	0.526543
$736$	0	0
$737$	−21528.0	−1.07598
$738$	0	0
$739$	11912.7	0.592987	0.296493	−	0.955035i	$-0.404183\pi$
$739$	11912.7	0.592987	0.296493	+	0.955035i	$0.404183\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17270.6	0.852754	0.426377	−	0.904545i	$-0.359790\pi$
$743$	17270.6	0.852754	0.426377	+	0.904545i	$0.359790\pi$
$744$	0	0
$745$	−5050.00	−0.248346
$746$	0	0
$747$	−3785.83	−0.185430
$748$	0	0
$749$	4420.00	0.215625
$750$	0	0
$751$	17869.1	0.868247	0.434123	−	0.900853i	$-0.357058\pi$
$751$	17869.1	0.868247	0.434123	+	0.900853i	$0.357058\pi$
$752$	0	0
$753$	14872.0	0.719742
$754$	0	0
$755$	−72.1110	−0.00347601
$756$	0	0
$757$	−35410.0	−1.70013	−0.850065	−	0.526678i	$-0.823437\pi$
$757$	−35410.0	−1.70013	−0.850065	+	0.526678i	$0.823437\pi$
$758$	0	0
$759$	−65246.1	−3.12027
$760$	0	0
$761$	3386.00	0.161291	0.0806455	−	0.996743i	$-0.474302\pi$
$761$	3386.00	0.161291	0.0806455	+	0.996743i	$0.474302\pi$
$762$	0	0
$763$	2466.20	0.117015
$764$	0	0
$765$	14250.0	0.673477
$766$	0	0
$767$	−16672.1	−0.784868
$768$	0	0
$769$	11522.0	0.540304	0.270152	−	0.962818i	$-0.412926\pi$
$769$	11522.0	0.540304	0.270152	+	0.962818i	$0.412926\pi$
$770$	0	0
$771$	26810.9	1.25236
$772$	0	0
$773$	14382.0	0.669191	0.334595	−	0.942362i	$-0.391400\pi$
$773$	14382.0	0.669191	0.334595	+	0.942362i	$0.391400\pi$
$774$	0	0
$775$	2523.89	0.116981
$776$	0	0
$777$	−7800.00	−0.360133
$778$	0	0
$779$	0	0
$780$	0	0
$781$	45552.0	2.08704
$782$	0	0
$783$	374.977	0.0171144
$784$	0	0
$785$	−8970.00	−0.407838
$786$	0	0
$787$	814.855	0.0369078	0.0184539	−	0.999830i	$-0.494126\pi$
$787$	814.855	0.0369078	0.0184539	+	0.999830i	$0.494126\pi$
$788$	0	0
$789$	50804.0	2.29236
$790$	0	0
$791$	15186.6	0.682646
$792$	0	0
$793$	−8908.00	−0.398906
$794$	0	0
$795$	−9446.54	−0.421427
$796$	0	0
$797$	14550.0	0.646659	0.323330	−	0.946286i	$-0.395198\pi$
$797$	14550.0	0.646659	0.323330	+	0.946286i	$0.395198\pi$
$798$	0	0
$799$	66587.3	2.94830
$800$	0	0
$801$	−15750.0	−0.694755
$802$	0	0
$803$	−29507.8	−1.29677
$804$	0	0
$805$	−7540.00	−0.330124
$806$	0	0
$807$	43684.9	1.90555
$808$	0	0
$809$	−37622.0	−1.63501	−0.817503	−	0.575925i	$-0.804642\pi$
$809$	−37622.0	−1.63501	−0.817503	+	0.575925i	$0.804642\pi$
$810$	0	0
$811$	−331.711	−0.0143624	−0.00718122	−	0.999974i	$-0.502286\pi$
$811$	−331.711	−0.0143624	−0.00718122	+	0.999974i	$0.502286\pi$
$812$	0	0
$813$	−37544.0	−1.61959
$814$	0	0
$815$	9915.27	0.426155
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−6129.44	−0.261514
$820$	0	0
$821$	−18690.0	−0.794501	−0.397251	−	0.917710i	$-0.630036\pi$
$821$	−18690.0	−0.794501	−0.397251	+	0.917710i	$0.630036\pi$
$822$	0	0
$823$	16866.8	0.714385	0.357192	−	0.934031i	$-0.383734\pi$
$823$	16866.8	0.714385	0.357192	+	0.934031i	$0.383734\pi$
$824$	0	0
$825$	7800.00	0.329165
$826$	0	0
$827$	39235.6	1.64977	0.824883	−	0.565304i	$-0.191241\pi$
$827$	39235.6	1.64977	0.824883	+	0.565304i	$0.191241\pi$
$828$	0	0
$829$	3718.00	0.155768	0.0778839	−	0.996962i	$-0.475184\pi$
$829$	3718.00	0.155768	0.0778839	+	0.996962i	$0.475184\pi$
$830$	0	0
$831$	−21561.2	−0.900060
$832$	0	0
$833$	−33174.0	−1.37985
$834$	0	0
$835$	3785.83	0.156903
$836$	0	0
$837$	1456.00	0.0601275
$838$	0	0
$839$	−14335.7	−0.589896	−0.294948	−	0.955513i	$-0.595302\pi$
$839$	−14335.7	−0.589896	−0.294948	+	0.955513i	$0.595302\pi$
$840$	0	0
$841$	−23713.0	−0.972283
$842$	0	0
$843$	−19542.1	−0.798417
$844$	0	0
$845$	−5205.00	−0.211902
$846$	0	0
$847$	3901.21	0.158261
$848$	0	0
$849$	33124.0	1.33900
$850$	0	0
$851$	−31368.3	−1.26356
$852$	0	0
$853$	−26786.0	−1.07519	−0.537594	−	0.843204i	$-0.680667\pi$
$853$	−26786.0	−1.07519	−0.537594	+	0.843204i	$0.680667\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19682.0	0.784509	0.392255	−	0.919857i	$-0.371695\pi$
$857$	19682.0	0.784509	0.392255	+	0.919857i	$0.371695\pi$
$858$	0	0
$859$	33199.9	1.31870	0.659352	−	0.751834i	$-0.270831\pi$
$859$	33199.9	1.31870	0.659352	+	0.751834i	$0.270831\pi$
$860$	0	0
$861$	−17784.0	−0.703922
$862$	0	0
$863$	10203.7	0.402478	0.201239	−	0.979542i	$-0.435503\pi$
$863$	10203.7	0.402478	0.201239	+	0.979542i	$0.435503\pi$
$864$	0	0
$865$	−14170.0	−0.556988
$866$	0	0
$867$	−58287.3	−2.28321
$868$	0	0
$869$	8736.00	0.341022
$870$	0	0
$871$	−16917.2	−0.658116
$872$	0	0
$873$	−24150.0	−0.936258
$874$	0	0
$875$	901.388	0.0348257
$876$	0	0
$877$	−41986.0	−1.61661	−0.808305	−	0.588764i	$-0.799615\pi$
$877$	−41986.0	−1.61661	−0.808305	+	0.588764i	$0.799615\pi$
$878$	0	0
$879$	−27041.6	−1.03765
$880$	0	0
$881$	38142.0	1.45861	0.729306	−	0.684188i	$-0.239843\pi$
$881$	38142.0	1.45861	0.729306	+	0.684188i	$0.239843\pi$
$882$	0	0
$883$	−39235.6	−1.49534	−0.747669	−	0.664072i	$-0.768827\pi$
$883$	−39235.6	−1.49534	−0.747669	+	0.664072i	$0.768827\pi$
$884$	0	0
$885$	−17680.0	−0.671533
$886$	0	0
$887$	−41528.7	−1.57204	−0.786020	−	0.618202i	$-0.787861\pi$
$887$	−41528.7	−1.57204	−0.786020	+	0.618202i	$0.787861\pi$
$888$	0	0
$889$	3172.00	0.119669
$890$	0	0
$891$	33704.7	1.26728
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−4759.33	−0.177751
$896$	0	0
$897$	−51272.0	−1.90850
$898$	0	0
$899$	2624.84	0.0973786
$900$	0	0
$901$	29868.0	1.10438
$902$	0	0
$903$	−23623.6	−0.870591
$904$	0	0
$905$	7330.00	0.269235
$906$	0	0
$907$	−4319.45	−0.158131	−0.0790656	−	0.996869i	$-0.525194\pi$
$907$	−4319.45	−0.158131	−0.0790656	+	0.996869i	$0.525194\pi$
$908$	0	0
$909$	−40950.0	−1.49420
$910$	0	0
$911$	49713.3	1.80799	0.903994	−	0.427546i	$-0.140622\pi$
$911$	49713.3	1.80799	0.903994	+	0.427546i	$0.140622\pi$
$912$	0	0
$913$	6552.00	0.237502
$914$	0	0
$915$	−9446.54	−0.341304
$916$	0	0
$917$	12792.0	0.460664
$918$	0	0
$919$	−5220.84	−0.187399	−0.0936994	−	0.995601i	$-0.529869\pi$
$919$	−5220.84	−0.187399	−0.0936994	+	0.995601i	$0.529869\pi$
$920$	0	0
$921$	31772.0	1.13672
$922$	0	0
$923$	35795.9	1.27653
$924$	0	0
$925$	3750.00	0.133296
$926$	0	0
$927$	17126.4	0.606800
$928$	0	0
$929$	−17546.0	−0.619662	−0.309831	−	0.950792i	$-0.600272\pi$
$929$	−17546.0	−0.619662	−0.309831	+	0.950792i	$0.600272\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−21320.0	−0.748109
$934$	0	0
$935$	−24662.0	−0.862602
$936$	0	0
$937$	−11390.0	−0.397113	−0.198557	−	0.980089i	$-0.563625\pi$
$937$	−11390.0	−0.397113	−0.198557	+	0.980089i	$0.563625\pi$
$938$	0	0
$939$	−40685.0	−1.41396
$940$	0	0
$941$	−41838.0	−1.44939	−0.724697	−	0.689068i	$-0.758020\pi$
$941$	−41838.0	−1.44939	−0.724697	+	0.689068i	$0.758020\pi$
$942$	0	0
$943$	−71519.7	−2.46978
$944$	0	0
$945$	520.000	0.0179001
$946$	0	0
$947$	35226.2	1.20876	0.604382	−	0.796695i	$-0.293420\pi$
$947$	35226.2	1.20876	0.604382	+	0.796695i	$0.293420\pi$
$948$	0	0
$949$	−23188.0	−0.793166
$950$	0	0
$951$	33531.6	1.14336
$952$	0	0
$953$	28522.0	0.969484	0.484742	−	0.874657i	$-0.338914\pi$
$953$	28522.0	0.969484	0.484742	+	0.874657i	$0.338914\pi$
$954$	0	0
$955$	17378.8	0.588862
$956$	0	0
$957$	8112.00	0.274006
$958$	0	0
$959$	11667.6	0.392873
$960$	0	0
$961$	−19599.0	−0.657883
$962$	0	0
$963$	15323.6	0.512768
$964$	0	0
$965$	−230.000	−0.00767250
$966$	0	0
$967$	−4579.05	−0.152277	−0.0761387	−	0.997097i	$-0.524259\pi$
$967$	−4579.05	−0.152277	−0.0761387	+	0.997097i	$0.524259\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22282.3	−0.736430	−0.368215	−	0.929741i	$-0.620031\pi$
$971$	−22282.3	−0.736430	−0.368215	+	0.929741i	$0.620031\pi$
$972$	0	0
$973$	18096.0	0.596229
$974$	0	0
$975$	6129.44	0.201332
$976$	0	0
$977$	−2214.00	−0.0724996	−0.0362498	−	0.999343i	$-0.511541\pi$
$977$	−2214.00	−0.0724996	−0.0362498	+	0.999343i	$0.511541\pi$
$978$	0	0
$979$	27258.0	0.889855
$980$	0	0
$981$	8550.00	0.278268
$982$	0	0
$983$	−24871.1	−0.806983	−0.403492	−	0.914983i	$-0.632204\pi$
$983$	−24871.1	−0.806983	−0.403492	+	0.914983i	$0.632204\pi$
$984$	0	0
$985$	−5610.00	−0.181472
$986$	0	0
$987$	−30373.2	−0.979522
$988$	0	0
$989$	−95004.0	−3.05455
$990$	0	0
$991$	−46453.9	−1.48906	−0.744530	−	0.667590i	$-0.767326\pi$
$991$	−46453.9	−1.48906	−0.744530	+	0.667590i	$0.767326\pi$
$992$	0	0
$993$	−39832.0	−1.27294
$994$	0	0
$995$	14999.1	0.477893
$996$	0	0
$997$	−57930.0	−1.84018	−0.920091	−	0.391705i	$-0.871886\pi$
$997$	−57930.0	−1.84018	−0.920091	+	0.391705i	$0.871886\pi$
$998$	0	0
$999$	2163.33	0.0685133

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.4.a.r.1.1		2
4.3	odd	2	inner	320.4.a.r.1.2		2
5.4	even	2		1600.4.a.ci.1.2		2
8.3	odd	2		160.4.a.e.1.1	✓	2
8.5	even	2		160.4.a.e.1.2	yes	2
16.3	odd	4		1280.4.d.t.641.3		4
16.5	even	4		1280.4.d.t.641.4		4
16.11	odd	4		1280.4.d.t.641.2		4
16.13	even	4		1280.4.d.t.641.1		4
20.19	odd	2		1600.4.a.ci.1.1		2
24.5	odd	2		1440.4.a.bd.1.2		2
24.11	even	2		1440.4.a.bd.1.1		2
40.3	even	4		800.4.c.h.449.1		4
40.13	odd	4		800.4.c.h.449.4		4
40.19	odd	2		800.4.a.q.1.2		2
40.27	even	4		800.4.c.h.449.3		4
40.29	even	2		800.4.a.q.1.1		2
40.37	odd	4		800.4.c.h.449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.e.1.1	✓	2	8.3	odd	2
160.4.a.e.1.2	yes	2	8.5	even	2
320.4.a.r.1.1		2	1.1	even	1	trivial
320.4.a.r.1.2		2	4.3	odd	2	inner
800.4.a.q.1.1		2	40.29	even	2
800.4.a.q.1.2		2	40.19	odd	2
800.4.c.h.449.1		4	40.3	even	4
800.4.c.h.449.2		4	40.37	odd	4
800.4.c.h.449.3		4	40.27	even	4
800.4.c.h.449.4		4	40.13	odd	4
1280.4.d.t.641.1		4	16.13	even	4
1280.4.d.t.641.2		4	16.11	odd	4
1280.4.d.t.641.3		4	16.3	odd	4
1280.4.d.t.641.4		4	16.5	even	4
1440.4.a.bd.1.1		2	24.11	even	2
1440.4.a.bd.1.2		2	24.5	odd	2
1600.4.a.ci.1.1		2	20.19	odd	2
1600.4.a.ci.1.2		2	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

\(f(q)\)	\(=\)	\(q-7.21110 q^{3} +5.00000 q^{5} +7.21110 q^{7} +25.0000 q^{9} +O(q^{10})\)	\(q-7.21110 q^{3} +5.00000 q^{5} +7.21110 q^{7} +25.0000 q^{9} -43.2666 q^{11} -34.0000 q^{13} -36.0555 q^{15} +114.000 q^{17} -52.0000 q^{21} -209.122 q^{23} +25.0000 q^{25} +14.4222 q^{27} +26.0000 q^{29} +100.955 q^{31} +312.000 q^{33} +36.0555 q^{35} +150.000 q^{37} +245.177 q^{39} +342.000 q^{41} +454.299 q^{43} +125.000 q^{45} +584.099 q^{47} -291.000 q^{49} -822.066 q^{51} +262.000 q^{53} -216.333 q^{55} +490.355 q^{59} +262.000 q^{61} +180.278 q^{63} -170.000 q^{65} +497.566 q^{67} +1508.00 q^{69} -1052.82 q^{71} +682.000 q^{73} -180.278 q^{75} -312.000 q^{77} -201.911 q^{79} -779.000 q^{81} -151.433 q^{83} +570.000 q^{85} -187.489 q^{87} -630.000 q^{89} -245.177 q^{91} -728.000 q^{93} -966.000 q^{97} -1081.67 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{5} + 50 q^{9}+O(q^{10}) \) 2 * q + 10 * q^5 + 50 * q^9	\( 2 q + 10 q^{5} + 50 q^{9} - 68 q^{13} + 228 q^{17} - 104 q^{21} + 50 q^{25} + 52 q^{29} + 624 q^{33} + 300 q^{37} + 684 q^{41} + 250 q^{45} - 582 q^{49} + 524 q^{53} + 524 q^{61} - 340 q^{65} + 3016 q^{69} + 1364 q^{73} - 624 q^{77} - 1558 q^{81} + 1140 q^{85} - 1260 q^{89} - 1456 q^{93} - 1932 q^{97}+O(q^{100}) \) 2 * q + 10 * q^5 + 50 * q^9 - 68 * q^13 + 228 * q^17 - 104 * q^21 + 50 * q^25 + 52 * q^29 + 624 * q^33 + 300 * q^37 + 684 * q^41 + 250 * q^45 - 582 * q^49 + 524 * q^53 + 524 * q^61 - 340 * q^65 + 3016 * q^69 + 1364 * q^73 - 624 * q^77 - 1558 * q^81 + 1140 * q^85 - 1260 * q^89 - 1456 * q^93 - 1932 * q^97

Introduction

L-functions

Modular forms

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