LMFDB - Embedded newform 8037.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 28 x^{16} + 25 x^{15} + 322 x^{14} - 247 x^{13} - 1971 x^{12} + 1231 x^{11} + 6953 x^{10} + \cdots + 9 $ x^18 - x^17 - 28x^16 + 25x^15 + 322x^14 - 247x^13 - 1971x^12 + 1231x^11 + 6953x^10 - 3283x^9 - 14235x^8 + 4562x^7 + 15962x^6 - 2882x^5 - 8159x^4 + 606x^3 + 890x^2 - 179x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.33471$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.33471 q^{2} +3.45089 q^{4} -4.12904 q^{5} -0.743390 q^{7} -3.38741 q^{8} +O(q^{10})$	\(q-2.33471 q^{2} +3.45089 q^{4} -4.12904 q^{5} -0.743390 q^{7} -3.38741 q^{8} +9.64013 q^{10} -2.71900 q^{11} -0.574568 q^{13} +1.73560 q^{14} +1.00686 q^{16} +2.51629 q^{17} -1.00000 q^{19} -14.2489 q^{20} +6.34810 q^{22} +1.58514 q^{23} +12.0490 q^{25} +1.34145 q^{26} -2.56536 q^{28} +5.39073 q^{29} +0.300880 q^{31} +4.42409 q^{32} -5.87481 q^{34} +3.06949 q^{35} +2.44994 q^{37} +2.33471 q^{38} +13.9868 q^{40} +6.94009 q^{41} -1.25528 q^{43} -9.38299 q^{44} -3.70085 q^{46} -1.00000 q^{47} -6.44737 q^{49} -28.1310 q^{50} -1.98277 q^{52} -4.86144 q^{53} +11.2269 q^{55} +2.51817 q^{56} -12.5858 q^{58} -14.3106 q^{59} +6.57508 q^{61} -0.702468 q^{62} -12.3427 q^{64} +2.37241 q^{65} +3.89333 q^{67} +8.68343 q^{68} -7.16638 q^{70} -5.74257 q^{71} -14.7469 q^{73} -5.71991 q^{74} -3.45089 q^{76} +2.02128 q^{77} -2.74290 q^{79} -4.15737 q^{80} -16.2031 q^{82} +1.82125 q^{83} -10.3899 q^{85} +2.93071 q^{86} +9.21039 q^{88} -8.53153 q^{89} +0.427128 q^{91} +5.47014 q^{92} +2.33471 q^{94} +4.12904 q^{95} +7.00765 q^{97} +15.0528 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * q^98

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 $

$2$	−2.33471	−1.65089	−0.825446	−	0.564481i	$-0.809076\pi$
$2$	−2.33471	−1.65089	−0.825446	+	0.564481i	$0.809076\pi$
$3$	0	0
$3$	0	0
$4$	3.45089	1.72544
$5$	−4.12904	−1.84656	−0.923282	−	0.384123i	$-0.874504\pi$
$5$	−4.12904	−1.84656	−0.923282	+	0.384123i	$0.874504\pi$
$6$	0	0
$7$	−0.743390	−0.280975	−0.140488	−	0.990082i	$-0.544867\pi$
$7$	−0.743390	−0.280975	−0.140488	+	0.990082i	$0.544867\pi$
$8$	−3.38741	−1.19763
$9$	0	0
$10$	9.64013	3.04848
$11$	−2.71900	−0.819811	−0.409905	−	0.912128i	$-0.634438\pi$
$11$	−2.71900	−0.819811	−0.409905	+	0.912128i	$0.634438\pi$
$12$	0	0
$13$	−0.574568	−0.159356	−0.0796782	−	0.996821i	$-0.525389\pi$
$13$	−0.574568	−0.159356	−0.0796782	+	0.996821i	$0.525389\pi$
$14$	1.73560	0.463860
$15$	0	0
$16$	1.00686	0.251715
$17$	2.51629	0.610289	0.305145	−	0.952306i	$-0.401295\pi$
$17$	2.51629	0.610289	0.305145	+	0.952306i	$0.401295\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−14.2489	−3.18614
$21$	0	0
$22$	6.34810	1.35342
$23$	1.58514	0.330524	0.165262	−	0.986250i	$-0.447153\pi$
$23$	1.58514	0.330524	0.165262	+	0.986250i	$0.447153\pi$
$24$	0	0
$25$	12.0490	2.40980
$26$	1.34145	0.263080
$27$	0	0
$28$	−2.56536	−0.484807
$29$	5.39073	1.00103	0.500517	−	0.865727i	$-0.333143\pi$
$29$	5.39073	1.00103	0.500517	+	0.865727i	$0.333143\pi$
$30$	0	0
$31$	0.300880	0.0540396	0.0270198	−	0.999635i	$-0.491398\pi$
$31$	0.300880	0.0540396	0.0270198	+	0.999635i	$0.491398\pi$
$32$	4.42409	0.782077
$33$	0	0
$34$	−5.87481	−1.00752
$35$	3.06949	0.518839
$36$	0	0
$37$	2.44994	0.402768	0.201384	−	0.979512i	$-0.435456\pi$
$37$	2.44994	0.402768	0.201384	+	0.979512i	$0.435456\pi$
$38$	2.33471	0.378741
$39$	0	0
$40$	13.9868	2.21150
$41$	6.94009	1.08386	0.541930	−	0.840424i	$-0.317694\pi$
$41$	6.94009	1.08386	0.541930	+	0.840424i	$0.317694\pi$
$42$	0	0
$43$	−1.25528	−0.191428	−0.0957139	−	0.995409i	$-0.530513\pi$
$43$	−1.25528	−0.191428	−0.0957139	+	0.995409i	$0.530513\pi$
$44$	−9.38299	−1.41454
$45$	0	0
$46$	−3.70085	−0.545660
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−6.44737	−0.921053
$50$	−28.1310	−3.97832
$51$	0	0
$52$	−1.98277	−0.274961
$53$	−4.86144	−0.667771	−0.333885	−	0.942614i	$-0.608360\pi$
$53$	−4.86144	−0.667771	−0.333885	+	0.942614i	$0.608360\pi$
$54$	0	0
$55$	11.2269	1.51383
$56$	2.51817	0.336505
$57$	0	0
$58$	−12.5858	−1.65260
$59$	−14.3106	−1.86308	−0.931541	−	0.363636i	$-0.881535\pi$
$59$	−14.3106	−1.86308	−0.931541	+	0.363636i	$0.881535\pi$
$60$	0	0
$61$	6.57508	0.841853	0.420927	−	0.907095i	$-0.361705\pi$
$61$	6.57508	0.841853	0.420927	+	0.907095i	$0.361705\pi$
$62$	−0.702468	−0.0892135
$63$	0	0
$64$	−12.3427	−1.54284
$65$	2.37241	0.294262
$66$	0	0
$67$	3.89333	0.475645	0.237823	−	0.971309i	$-0.423566\pi$
$67$	3.89333	0.475645	0.237823	+	0.971309i	$0.423566\pi$
$68$	8.68343	1.05302
$69$	0	0
$70$	−7.16638	−0.856547
$71$	−5.74257	−0.681517	−0.340759	−	0.940151i	$-0.610684\pi$
$71$	−5.74257	−0.681517	−0.340759	+	0.940151i	$0.610684\pi$
$72$	0	0
$73$	−14.7469	−1.72600	−0.862999	−	0.505206i	$-0.831417\pi$
$73$	−14.7469	−1.72600	−0.862999	+	0.505206i	$0.831417\pi$
$74$	−5.71991	−0.664926
$75$	0	0
$76$	−3.45089	−0.395844
$77$	2.02128	0.230347
$78$	0	0
$79$	−2.74290	−0.308600	−0.154300	−	0.988024i	$-0.549312\pi$
$79$	−2.74290	−0.308600	−0.154300	+	0.988024i	$0.549312\pi$
$80$	−4.15737	−0.464808
$81$	0	0
$82$	−16.2031	−1.78933
$83$	1.82125	0.199908	0.0999538	−	0.994992i	$-0.468130\pi$
$83$	1.82125	0.199908	0.0999538	+	0.994992i	$0.468130\pi$
$84$	0	0
$85$	−10.3899	−1.12694
$86$	2.93071	0.316027
$87$	0	0
$88$	9.21039	0.981831
$89$	−8.53153	−0.904340	−0.452170	−	0.891932i	$-0.649350\pi$
$89$	−8.53153	−0.904340	−0.452170	+	0.891932i	$0.649350\pi$
$90$	0	0
$91$	0.427128	0.0447752
$92$	5.47014	0.570301
$93$	0	0
$94$	2.33471	0.240807
$95$	4.12904	0.423631
$96$	0	0
$97$	7.00765	0.711519	0.355760	−	0.934577i	$-0.384222\pi$
$97$	7.00765	0.711519	0.355760	+	0.934577i	$0.384222\pi$
$98$	15.0528	1.52056
$99$	0	0
$100$	41.5797	4.15797
$101$	10.4595	1.04076	0.520379	−	0.853935i	$-0.325791\pi$
$101$	10.4595	1.04076	0.520379	+	0.853935i	$0.325791\pi$
$102$	0	0
$103$	−7.95141	−0.783476	−0.391738	−	0.920077i	$-0.628126\pi$
$103$	−7.95141	−0.783476	−0.391738	+	0.920077i	$0.628126\pi$
$104$	1.94630	0.190850
$105$	0	0
$106$	11.3501	1.10242
$107$	−6.55339	−0.633540	−0.316770	−	0.948502i	$-0.602598\pi$
$107$	−6.55339	−0.633540	−0.316770	+	0.948502i	$0.602598\pi$
$108$	0	0
$109$	−5.30570	−0.508194	−0.254097	−	0.967179i	$-0.581778\pi$
$109$	−5.30570	−0.508194	−0.254097	+	0.967179i	$0.581778\pi$
$110$	−26.2116	−2.49918
$111$	0	0
$112$	−0.748491	−0.0707257
$113$	−12.5108	−1.17692	−0.588459	−	0.808527i	$-0.700265\pi$
$113$	−12.5108	−1.17692	−0.588459	+	0.808527i	$0.700265\pi$
$114$	0	0
$115$	−6.54511	−0.610334
$116$	18.6028	1.72723
$117$	0	0
$118$	33.4112	3.07575
$119$	−1.87058	−0.171476
$120$	0	0
$121$	−3.60701	−0.327910
$122$	−15.3509	−1.38981
$123$	0	0
$124$	1.03830	0.0932423
$125$	−29.1056	−2.60328
$126$	0	0
$127$	−6.36190	−0.564527	−0.282264	−	0.959337i	$-0.591085\pi$
$127$	−6.36190	−0.564527	−0.282264	+	0.959337i	$0.591085\pi$
$128$	19.9685	1.76498
$129$	0	0
$130$	−5.53891	−0.485795
$131$	18.1598	1.58663	0.793316	−	0.608810i	$-0.208353\pi$
$131$	18.1598	1.58663	0.793316	+	0.608810i	$0.208353\pi$
$132$	0	0
$133$	0.743390	0.0644601
$134$	−9.08980	−0.785239
$135$	0	0
$136$	−8.52371	−0.730902
$137$	−2.21717	−0.189426	−0.0947130	−	0.995505i	$-0.530193\pi$
$137$	−2.21717	−0.189426	−0.0947130	+	0.995505i	$0.530193\pi$
$138$	0	0
$139$	8.21453	0.696748	0.348374	−	0.937356i	$-0.386734\pi$
$139$	8.21453	0.696748	0.348374	+	0.937356i	$0.386734\pi$
$140$	10.5925	0.895228
$141$	0	0
$142$	13.4072	1.12511
$143$	1.56225	0.130642
$144$	0	0
$145$	−22.2585	−1.84847
$146$	34.4299	2.84944
$147$	0	0
$148$	8.45447	0.694953
$149$	9.95273	0.815359	0.407680	−	0.913125i	$-0.366338\pi$
$149$	9.95273	0.815359	0.407680	+	0.913125i	$0.366338\pi$
$150$	0	0
$151$	13.0196	1.05952	0.529759	−	0.848148i	$-0.322282\pi$
$151$	13.0196	1.05952	0.529759	+	0.848148i	$0.322282\pi$
$152$	3.38741	0.274755
$153$	0	0
$154$	−4.71912	−0.380277
$155$	−1.24234	−0.0997875
$156$	0	0
$157$	−17.5077	−1.39726	−0.698631	−	0.715482i	$-0.746207\pi$
$157$	−17.5077	−1.39726	−0.698631	+	0.715482i	$0.746207\pi$
$158$	6.40389	0.509466
$159$	0	0
$160$	−18.2673	−1.44415
$161$	−1.17838	−0.0928691
$162$	0	0
$163$	10.9428	0.857104	0.428552	−	0.903517i	$-0.359024\pi$
$163$	10.9428	0.857104	0.428552	+	0.903517i	$0.359024\pi$
$164$	23.9495	1.87014
$165$	0	0
$166$	−4.25209	−0.330026
$167$	−13.5355	−1.04741	−0.523706	−	0.851899i	$-0.675451\pi$
$167$	−13.5355	−1.04741	−0.523706	+	0.851899i	$0.675451\pi$
$168$	0	0
$169$	−12.6699	−0.974606
$170$	24.2574	1.86045
$171$	0	0
$172$	−4.33182	−0.330298
$173$	−2.09102	−0.158977	−0.0794885	−	0.996836i	$-0.525329\pi$
$173$	−2.09102	−0.158977	−0.0794885	+	0.996836i	$0.525329\pi$
$174$	0	0
$175$	−8.95711	−0.677094
$176$	−2.73766	−0.206359
$177$	0	0
$178$	19.9187	1.49297
$179$	15.1142	1.12969	0.564845	−	0.825197i	$-0.308936\pi$
$179$	15.1142	1.12969	0.564845	+	0.825197i	$0.308936\pi$
$180$	0	0
$181$	5.81211	0.432011	0.216005	−	0.976392i	$-0.430697\pi$
$181$	5.81211	0.432011	0.216005	+	0.976392i	$0.430697\pi$
$182$	−0.997222	−0.0739190
$183$	0	0
$184$	−5.36952	−0.395846
$185$	−10.1159	−0.743736
$186$	0	0
$187$	−6.84180	−0.500322
$188$	−3.45089	−0.251682
$189$	0	0
$190$	−9.64013	−0.699369
$191$	−17.9984	−1.30232	−0.651159	−	0.758942i	$-0.725717\pi$
$191$	−17.9984	−1.30232	−0.651159	+	0.758942i	$0.725717\pi$
$192$	0	0
$193$	−26.6245	−1.91648	−0.958238	−	0.285973i	$-0.907683\pi$
$193$	−26.6245	−1.91648	−0.958238	+	0.285973i	$0.907683\pi$
$194$	−16.3609	−1.17464
$195$	0	0
$196$	−22.2492	−1.58923
$197$	−4.37501	−0.311707	−0.155853	−	0.987780i	$-0.549813\pi$
$197$	−4.37501	−0.311707	−0.155853	+	0.987780i	$0.549813\pi$
$198$	0	0
$199$	15.3632	1.08907	0.544533	−	0.838739i	$-0.316707\pi$
$199$	15.3632	1.08907	0.544533	+	0.838739i	$0.316707\pi$
$200$	−40.8149	−2.88605
$201$	0	0
$202$	−24.4199	−1.71818
$203$	−4.00742	−0.281265
$204$	0	0
$205$	−28.6559	−2.00142
$206$	18.5643	1.29343
$207$	0	0
$208$	−0.578510	−0.0401124
$209$	2.71900	0.188077
$210$	0	0
$211$	0.846053	0.0582447	0.0291224	−	0.999576i	$-0.490729\pi$
$211$	0.846053	0.0582447	0.0291224	+	0.999576i	$0.490729\pi$
$212$	−16.7763	−1.15220
$213$	0	0
$214$	15.3003	1.04591
$215$	5.18309	0.353484
$216$	0	0
$217$	−0.223671	−0.0151838
$218$	12.3873	0.838973
$219$	0	0
$220$	38.7427	2.61204
$221$	−1.44578	−0.0972535
$222$	0	0
$223$	11.7579	0.787367	0.393684	−	0.919246i	$-0.371201\pi$
$223$	11.7579	0.787367	0.393684	+	0.919246i	$0.371201\pi$
$224$	−3.28883	−0.219744
$225$	0	0
$226$	29.2092	1.94297
$227$	5.74551	0.381343	0.190671	−	0.981654i	$-0.438933\pi$
$227$	5.74551	0.381343	0.190671	+	0.981654i	$0.438933\pi$
$228$	0	0
$229$	−25.7369	−1.70074	−0.850372	−	0.526182i	$-0.823623\pi$
$229$	−25.7369	−1.70074	−0.850372	+	0.526182i	$0.823623\pi$
$230$	15.2810	1.00760
$231$	0	0
$232$	−18.2606	−1.19887
$233$	−5.27786	−0.345764	−0.172882	−	0.984943i	$-0.555308\pi$
$233$	−5.27786	−0.345764	−0.172882	+	0.984943i	$0.555308\pi$
$234$	0	0
$235$	4.12904	0.269349
$236$	−49.3843	−3.21465
$237$	0	0
$238$	4.36728	0.283089
$239$	−15.1099	−0.977376	−0.488688	−	0.872459i	$-0.662524\pi$
$239$	−15.1099	−0.977376	−0.488688	+	0.872459i	$0.662524\pi$
$240$	0	0
$241$	10.8358	0.697994	0.348997	−	0.937124i	$-0.386522\pi$
$241$	10.8358	0.697994	0.348997	+	0.937124i	$0.386522\pi$
$242$	8.42134	0.541344
$243$	0	0
$244$	22.6899	1.45257
$245$	26.6215	1.70078
$246$	0	0
$247$	0.574568	0.0365589
$248$	−1.01920	−0.0647195
$249$	0	0
$250$	67.9532	4.29774
$251$	−22.8781	−1.44406	−0.722028	−	0.691864i	$-0.756790\pi$
$251$	−22.8781	−1.44406	−0.722028	+	0.691864i	$0.756790\pi$
$252$	0	0
$253$	−4.31000	−0.270967
$254$	14.8532	0.931974
$255$	0	0
$256$	−21.9354	−1.37096
$257$	−4.66667	−0.291099	−0.145549	−	0.989351i	$-0.546495\pi$
$257$	−4.66667	−0.291099	−0.145549	+	0.989351i	$0.546495\pi$
$258$	0	0
$259$	−1.82126	−0.113168
$260$	8.18694	0.507733
$261$	0	0
$262$	−42.3980	−2.61936
$263$	24.8025	1.52939	0.764693	−	0.644395i	$-0.222891\pi$
$263$	24.8025	1.52939	0.764693	+	0.644395i	$0.222891\pi$
$264$	0	0
$265$	20.0731	1.23308
$266$	−1.73560	−0.106417
$267$	0	0
$268$	13.4354	0.820700
$269$	29.4133	1.79336	0.896680	−	0.442680i	$-0.145972\pi$
$269$	29.4133	1.79336	0.896680	+	0.442680i	$0.145972\pi$
$270$	0	0
$271$	4.18461	0.254197	0.127099	−	0.991890i	$-0.459434\pi$
$271$	4.18461	0.254197	0.127099	+	0.991890i	$0.459434\pi$
$272$	2.53355	0.153619
$273$	0	0
$274$	5.17647	0.312722
$275$	−32.7613	−1.97558
$276$	0	0
$277$	19.9829	1.20065	0.600327	−	0.799755i	$-0.295037\pi$
$277$	19.9829	1.20065	0.600327	+	0.799755i	$0.295037\pi$
$278$	−19.1786	−1.15026
$279$	0	0
$280$	−10.3976	−0.621377
$281$	−5.97269	−0.356301	−0.178151	−	0.984003i	$-0.557011\pi$
$281$	−5.97269	−0.356301	−0.178151	+	0.984003i	$0.557011\pi$
$282$	0	0
$283$	11.3591	0.675231	0.337616	−	0.941284i	$-0.390380\pi$
$283$	11.3591	0.675231	0.337616	+	0.941284i	$0.390380\pi$
$284$	−19.8170	−1.17592
$285$	0	0
$286$	−3.64741	−0.215676
$287$	−5.15919	−0.304538
$288$	0	0
$289$	−10.6683	−0.627547
$290$	51.9673	3.05163
$291$	0	0
$292$	−50.8900	−2.97811
$293$	−28.5527	−1.66807	−0.834035	−	0.551712i	$-0.813975\pi$
$293$	−28.5527	−1.66807	−0.834035	+	0.551712i	$0.813975\pi$
$294$	0	0
$295$	59.0891	3.44030
$296$	−8.29896	−0.482367
$297$	0	0
$298$	−23.2368	−1.34607
$299$	−0.910770	−0.0526712
$300$	0	0
$301$	0.933160	0.0537865
$302$	−30.3970	−1.74915
$303$	0	0
$304$	−1.00686	−0.0577474
$305$	−27.1488	−1.55454
$306$	0	0
$307$	26.4732	1.51091	0.755454	−	0.655202i	$-0.227416\pi$
$307$	26.4732	1.51091	0.755454	+	0.655202i	$0.227416\pi$
$308$	6.97522	0.397450
$309$	0	0
$310$	2.90052	0.164738
$311$	−26.5702	−1.50666	−0.753328	−	0.657645i	$-0.771553\pi$
$311$	−26.5702	−1.50666	−0.753328	+	0.657645i	$0.771553\pi$
$312$	0	0
$313$	25.3681	1.43389	0.716944	−	0.697131i	$-0.245540\pi$
$313$	25.3681	1.43389	0.716944	+	0.697131i	$0.245540\pi$
$314$	40.8754	2.30673
$315$	0	0
$316$	−9.46545	−0.532473
$317$	−9.97972	−0.560517	−0.280259	−	0.959925i	$-0.590420\pi$
$317$	−9.97972	−0.560517	−0.280259	+	0.959925i	$0.590420\pi$
$318$	0	0
$319$	−14.6574	−0.820658
$320$	50.9636	2.84895
$321$	0	0
$322$	2.75117	0.153317
$323$	−2.51629	−0.140010
$324$	0	0
$325$	−6.92296	−0.384017
$326$	−25.5482	−1.41499
$327$	0	0
$328$	−23.5089	−1.29806
$329$	0.743390	0.0409844
$330$	0	0
$331$	−9.13508	−0.502110	−0.251055	−	0.967973i	$-0.580777\pi$
$331$	−9.13508	−0.502110	−0.251055	+	0.967973i	$0.580777\pi$
$332$	6.28492	0.344930
$333$	0	0
$334$	31.6016	1.72916
$335$	−16.0757	−0.878310
$336$	0	0
$337$	14.7683	0.804478	0.402239	−	0.915535i	$-0.368232\pi$
$337$	14.7683	0.804478	0.402239	+	0.915535i	$0.368232\pi$
$338$	29.5805	1.60897
$339$	0	0
$340$	−35.8543	−1.94447
$341$	−0.818093	−0.0443022
$342$	0	0
$343$	9.99665	0.539768
$344$	4.25214	0.229260
$345$	0	0
$346$	4.88193	0.262454
$347$	33.8156	1.81532	0.907659	−	0.419708i	$-0.137867\pi$
$347$	33.8156	1.81532	0.907659	+	0.419708i	$0.137867\pi$
$348$	0	0
$349$	−9.07557	−0.485804	−0.242902	−	0.970051i	$-0.578099\pi$
$349$	−9.07557	−0.485804	−0.242902	+	0.970051i	$0.578099\pi$
$350$	20.9123	1.11781
$351$	0	0
$352$	−12.0291	−0.641155
$353$	36.1587	1.92453	0.962267	−	0.272107i	$-0.0877204\pi$
$353$	36.1587	1.92453	0.962267	+	0.272107i	$0.0877204\pi$
$354$	0	0
$355$	23.7113	1.25847
$356$	−29.4414	−1.56039
$357$	0	0
$358$	−35.2874	−1.86500
$359$	18.4570	0.974122	0.487061	−	0.873368i	$-0.338069\pi$
$359$	18.4570	0.974122	0.487061	+	0.873368i	$0.338069\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−13.5696	−0.713204
$363$	0	0
$364$	1.47397	0.0772571
$365$	60.8907	3.18717
$366$	0	0
$367$	24.0980	1.25791	0.628953	−	0.777443i	$-0.283484\pi$
$367$	24.0980	1.25791	0.628953	+	0.777443i	$0.283484\pi$
$368$	1.59601	0.0831980
$369$	0	0
$370$	23.6177	1.22783
$371$	3.61395	0.187627
$372$	0	0
$373$	1.50882	0.0781240	0.0390620	−	0.999237i	$-0.487563\pi$
$373$	1.50882	0.0781240	0.0390620	+	0.999237i	$0.487563\pi$
$374$	15.9736	0.825977
$375$	0	0
$376$	3.38741	0.174692
$377$	−3.09734	−0.159521
$378$	0	0
$379$	−13.6588	−0.701604	−0.350802	−	0.936450i	$-0.614091\pi$
$379$	−13.6588	−0.701604	−0.350802	+	0.936450i	$0.614091\pi$
$380$	14.2489	0.730952
$381$	0	0
$382$	42.0211	2.14999
$383$	−34.5612	−1.76600	−0.882998	−	0.469377i	$-0.844479\pi$
$383$	−34.5612	−1.76600	−0.882998	+	0.469377i	$0.844479\pi$
$384$	0	0
$385$	−8.34596	−0.425350
$386$	62.1607	3.16389
$387$	0	0
$388$	24.1826	1.22769
$389$	−31.4235	−1.59323	−0.796617	−	0.604484i	$-0.793379\pi$
$389$	−31.4235	−1.59323	−0.796617	+	0.604484i	$0.793379\pi$
$390$	0	0
$391$	3.98867	0.201715
$392$	21.8399	1.10308
$393$	0	0
$394$	10.2144	0.514594
$395$	11.3256	0.569850
$396$	0	0
$397$	2.98717	0.149922	0.0749608	−	0.997186i	$-0.476117\pi$
$397$	2.98717	0.149922	0.0749608	+	0.997186i	$0.476117\pi$
$398$	−35.8686	−1.79793
$399$	0	0
$400$	12.1317	0.606583
$401$	−11.9038	−0.594449	−0.297225	−	0.954808i	$-0.596061\pi$
$401$	−11.9038	−0.594449	−0.297225	+	0.954808i	$0.596061\pi$
$402$	0	0
$403$	−0.172876	−0.00861155
$404$	36.0945	1.79577
$405$	0	0
$406$	9.35617	0.464339
$407$	−6.66140	−0.330193
$408$	0	0
$409$	15.4244	0.762688	0.381344	−	0.924433i	$-0.375461\pi$
$409$	15.4244	0.762688	0.381344	+	0.924433i	$0.375461\pi$
$410$	66.9033	3.30412
$411$	0	0
$412$	−27.4395	−1.35184
$413$	10.6384	0.523480
$414$	0	0
$415$	−7.52000	−0.369142
$416$	−2.54194	−0.124629
$417$	0	0
$418$	−6.34810	−0.310496
$419$	−17.4888	−0.854385	−0.427192	−	0.904161i	$-0.640497\pi$
$419$	−17.4888	−0.854385	−0.427192	+	0.904161i	$0.640497\pi$
$420$	0	0
$421$	−10.9807	−0.535165	−0.267582	−	0.963535i	$-0.586225\pi$
$421$	−10.9807	−0.535165	−0.267582	+	0.963535i	$0.586225\pi$
$422$	−1.97529	−0.0961557
$423$	0	0
$424$	16.4677	0.799743
$425$	30.3187	1.47067
$426$	0	0
$427$	−4.88786	−0.236540
$428$	−22.6150	−1.09314
$429$	0	0
$430$	−12.1010	−0.583563
$431$	−24.0404	−1.15799	−0.578994	−	0.815332i	$-0.696554\pi$
$431$	−24.0404	−1.15799	−0.578994	+	0.815332i	$0.696554\pi$
$432$	0	0
$433$	−7.68310	−0.369226	−0.184613	−	0.982811i	$-0.559103\pi$
$433$	−7.68310	−0.369226	−0.184613	+	0.982811i	$0.559103\pi$
$434$	0.522208	0.0250668
$435$	0	0
$436$	−18.3094	−0.876860
$437$	−1.58514	−0.0758275
$438$	0	0
$439$	−8.58399	−0.409692	−0.204846	−	0.978794i	$-0.565669\pi$
$439$	−8.58399	−0.409692	−0.204846	+	0.978794i	$0.565669\pi$
$440$	−38.0301	−1.81301
$441$	0	0
$442$	3.37548	0.160555
$443$	36.6098	1.73938	0.869692	−	0.493595i	$-0.164318\pi$
$443$	36.6098	1.73938	0.869692	+	0.493595i	$0.164318\pi$
$444$	0	0
$445$	35.2270	1.66992
$446$	−27.4513	−1.29986
$447$	0	0
$448$	9.17546	0.433500
$449$	7.67532	0.362221	0.181110	−	0.983463i	$-0.442031\pi$
$449$	7.67532	0.362221	0.181110	+	0.983463i	$0.442031\pi$
$450$	0	0
$451$	−18.8701	−0.888560
$452$	−43.1734	−2.03071
$453$	0	0
$454$	−13.4141	−0.629556
$455$	−1.76363	−0.0826803
$456$	0	0
$457$	33.7633	1.57938	0.789689	−	0.613507i	$-0.210242\pi$
$457$	33.7633	1.57938	0.789689	+	0.613507i	$0.210242\pi$
$458$	60.0884	2.80774
$459$	0	0
$460$	−22.5864	−1.05310
$461$	−4.25040	−0.197961	−0.0989803	−	0.995089i	$-0.531558\pi$
$461$	−4.25040	−0.197961	−0.0989803	+	0.995089i	$0.531558\pi$
$462$	0	0
$463$	−16.1164	−0.748993	−0.374496	−	0.927228i	$-0.622184\pi$
$463$	−16.1164	−0.748993	−0.374496	+	0.927228i	$0.622184\pi$
$464$	5.42771	0.251975
$465$	0	0
$466$	12.3223	0.570819
$467$	−4.53688	−0.209942	−0.104971	−	0.994475i	$-0.533475\pi$
$467$	−4.53688	−0.209942	−0.104971	+	0.994475i	$0.533475\pi$
$468$	0	0
$469$	−2.89426	−0.133645
$470$	−9.64013	−0.444666
$471$	0	0
$472$	48.4759	2.23129
$473$	3.41310	0.156935
$474$	0	0
$475$	−12.0490	−0.552846
$476$	−6.45518	−0.295873
$477$	0	0
$478$	35.2772	1.61354
$479$	−11.6229	−0.531062	−0.265531	−	0.964102i	$-0.585547\pi$
$479$	−11.6229	−0.531062	−0.265531	+	0.964102i	$0.585547\pi$
$480$	0	0
$481$	−1.40766	−0.0641836
$482$	−25.2984	−1.15231
$483$	0	0
$484$	−12.4474	−0.565791
$485$	−28.9349	−1.31387
$486$	0	0
$487$	25.2698	1.14508	0.572542	−	0.819876i	$-0.305957\pi$
$487$	25.2698	1.14508	0.572542	+	0.819876i	$0.305957\pi$
$488$	−22.2725	−1.00823
$489$	0	0
$490$	−62.1535	−2.80781
$491$	6.98577	0.315263	0.157632	−	0.987498i	$-0.449614\pi$
$491$	6.98577	0.315263	0.157632	+	0.987498i	$0.449614\pi$
$492$	0	0
$493$	13.5646	0.610920
$494$	−1.34145	−0.0603548
$495$	0	0
$496$	0.302944	0.0136026
$497$	4.26897	0.191489
$498$	0	0
$499$	−36.3335	−1.62651	−0.813256	−	0.581907i	$-0.802307\pi$
$499$	−36.3335	−1.62651	−0.813256	+	0.581907i	$0.802307\pi$
$500$	−100.440	−4.49182
$501$	0	0
$502$	53.4139	2.38398
$503$	10.4922	0.467823	0.233911	−	0.972258i	$-0.424847\pi$
$503$	10.4922	0.467823	0.233911	+	0.972258i	$0.424847\pi$
$504$	0	0
$505$	−43.1877	−1.92183
$506$	10.0626	0.447338
$507$	0	0
$508$	−21.9542	−0.974061
$509$	31.8342	1.41103	0.705514	−	0.708696i	$-0.250716\pi$
$509$	31.8342	1.41103	0.705514	+	0.708696i	$0.250716\pi$
$510$	0	0
$511$	10.9627	0.484963
$512$	11.2758	0.498323
$513$	0	0
$514$	10.8953	0.480572
$515$	32.8317	1.44674
$516$	0	0
$517$	2.71900	0.119582
$518$	4.25213	0.186828
$519$	0	0
$520$	−8.03635	−0.352417
$521$	−31.6574	−1.38694	−0.693468	−	0.720487i	$-0.743918\pi$
$521$	−31.6574	−1.38694	−0.693468	+	0.720487i	$0.743918\pi$
$522$	0	0
$523$	−2.42607	−0.106085	−0.0530423	−	0.998592i	$-0.516892\pi$
$523$	−2.42607	−0.106085	−0.0530423	+	0.998592i	$0.516892\pi$
$524$	62.6676	2.73765
$525$	0	0
$526$	−57.9066	−2.52485
$527$	0.757100	0.0329798
$528$	0	0
$529$	−20.4873	−0.890754
$530$	−46.8650	−2.03568
$531$	0	0
$532$	2.56536	0.111222
$533$	−3.98755	−0.172720
$534$	0	0
$535$	27.0592	1.16987
$536$	−13.1883	−0.569648
$537$	0	0
$538$	−68.6716	−2.96064
$539$	17.5304	0.755089
$540$	0	0
$541$	−26.8374	−1.15383	−0.576915	−	0.816804i	$-0.695744\pi$
$541$	−26.8374	−1.15383	−0.576915	+	0.816804i	$0.695744\pi$
$542$	−9.76987	−0.419652
$543$	0	0
$544$	11.1323	0.477293
$545$	21.9074	0.938412
$546$	0	0
$547$	−25.9674	−1.11029	−0.555144	−	0.831755i	$-0.687337\pi$
$547$	−25.9674	−1.11029	−0.555144	+	0.831755i	$0.687337\pi$
$548$	−7.65122	−0.326844
$549$	0	0
$550$	76.4882	3.26147
$551$	−5.39073	−0.229653
$552$	0	0
$553$	2.03905	0.0867091
$554$	−46.6543	−1.98215
$555$	0	0
$556$	28.3474	1.20220
$557$	19.7935	0.838678	0.419339	−	0.907830i	$-0.362262\pi$
$557$	19.7935	0.838678	0.419339	+	0.907830i	$0.362262\pi$
$558$	0	0
$559$	0.721241	0.0305052
$560$	3.09055	0.130600
$561$	0	0
$562$	13.9445	0.588215
$563$	22.4496	0.946138	0.473069	−	0.881025i	$-0.343146\pi$
$563$	22.4496	0.946138	0.473069	+	0.881025i	$0.343146\pi$
$564$	0	0
$565$	51.6577	2.17326
$566$	−26.5204	−1.11473
$567$	0	0
$568$	19.4524	0.816206
$569$	8.33059	0.349236	0.174618	−	0.984636i	$-0.444131\pi$
$569$	8.33059	0.349236	0.174618	+	0.984636i	$0.444131\pi$
$570$	0	0
$571$	18.6921	0.782238	0.391119	−	0.920340i	$-0.372088\pi$
$571$	18.6921	0.782238	0.391119	+	0.920340i	$0.372088\pi$
$572$	5.39116	0.225416
$573$	0	0
$574$	12.0452	0.502759
$575$	19.0993	0.796497
$576$	0	0
$577$	−20.2860	−0.844516	−0.422258	−	0.906476i	$-0.638762\pi$
$577$	−20.2860	−0.844516	−0.422258	+	0.906476i	$0.638762\pi$
$578$	24.9074	1.03601
$579$	0	0
$580$	−76.8118	−3.18944
$581$	−1.35390	−0.0561691
$582$	0	0
$583$	13.2183	0.547445
$584$	49.9539	2.06711
$585$	0	0
$586$	66.6625	2.75380
$587$	−8.52647	−0.351925	−0.175963	−	0.984397i	$-0.556304\pi$
$587$	−8.52647	−0.351925	−0.175963	+	0.984397i	$0.556304\pi$
$588$	0	0
$589$	−0.300880	−0.0123975
$590$	−137.956	−5.67957
$591$	0	0
$592$	2.46675	0.101383
$593$	−31.0172	−1.27372	−0.636862	−	0.770978i	$-0.719768\pi$
$593$	−31.0172	−1.27372	−0.636862	+	0.770978i	$0.719768\pi$
$594$	0	0
$595$	7.72372	0.316642
$596$	34.3458	1.40686
$597$	0	0
$598$	2.12639	0.0869544
$599$	−43.1902	−1.76470	−0.882352	−	0.470590i	$-0.844041\pi$
$599$	−43.1902	−1.76470	−0.882352	+	0.470590i	$0.844041\pi$
$600$	0	0
$601$	−36.5385	−1.49044	−0.745219	−	0.666820i	$-0.767655\pi$
$601$	−36.5385	−1.49044	−0.745219	+	0.666820i	$0.767655\pi$
$602$	−2.17866	−0.0887956
$603$	0	0
$604$	44.9291	1.82814
$605$	14.8935	0.605507
$606$	0	0
$607$	43.6832	1.77305	0.886524	−	0.462682i	$-0.153113\pi$
$607$	43.6832	1.77305	0.886524	+	0.462682i	$0.153113\pi$
$608$	−4.42409	−0.179421
$609$	0	0
$610$	63.3847	2.56637
$611$	0.574568	0.0232445
$612$	0	0
$613$	7.66039	0.309400	0.154700	−	0.987961i	$-0.450559\pi$
$613$	7.66039	0.309400	0.154700	+	0.987961i	$0.450559\pi$
$614$	−61.8074	−2.49435
$615$	0	0
$616$	−6.84692	−0.275870
$617$	6.04372	0.243311	0.121656	−	0.992572i	$-0.461180\pi$
$617$	6.04372	0.243311	0.121656	+	0.992572i	$0.461180\pi$
$618$	0	0
$619$	15.6729	0.629948	0.314974	−	0.949100i	$-0.398004\pi$
$619$	15.6729	0.629948	0.314974	+	0.949100i	$0.398004\pi$
$620$	−4.28719	−0.172178
$621$	0	0
$622$	62.0337	2.48733
$623$	6.34226	0.254097
$624$	0	0
$625$	59.9332	2.39733
$626$	−59.2272	−2.36719
$627$	0	0
$628$	−60.4170	−2.41090
$629$	6.16475	0.245805
$630$	0	0
$631$	−1.75541	−0.0698819	−0.0349410	−	0.999389i	$-0.511124\pi$
$631$	−1.75541	−0.0698819	−0.0349410	+	0.999389i	$0.511124\pi$
$632$	9.29134	0.369590
$633$	0	0
$634$	23.2998	0.925353
$635$	26.2686	1.04244
$636$	0	0
$637$	3.70445	0.146776
$638$	34.2209	1.35482
$639$	0	0
$640$	−82.4509	−3.25916
$641$	18.2331	0.720165	0.360083	−	0.932920i	$-0.382749\pi$
$641$	18.2331	0.720165	0.360083	+	0.932920i	$0.382749\pi$
$642$	0	0
$643$	26.6045	1.04918	0.524590	−	0.851355i	$-0.324219\pi$
$643$	26.6045	1.04918	0.524590	+	0.851355i	$0.324219\pi$
$644$	−4.06645	−0.160241
$645$	0	0
$646$	5.87481	0.231141
$647$	12.7846	0.502615	0.251307	−	0.967907i	$-0.419140\pi$
$647$	12.7846	0.502615	0.251307	+	0.967907i	$0.419140\pi$
$648$	0	0
$649$	38.9106	1.52738
$650$	16.1631	0.633970
$651$	0	0
$652$	37.7623	1.47889
$653$	19.2761	0.754332	0.377166	−	0.926146i	$-0.376899\pi$
$653$	19.2761	0.754332	0.377166	+	0.926146i	$0.376899\pi$
$654$	0	0
$655$	−74.9828	−2.92982
$656$	6.98770	0.272824
$657$	0	0
$658$	−1.73560	−0.0676609
$659$	35.2927	1.37481	0.687404	−	0.726275i	$-0.258750\pi$
$659$	35.2927	1.37481	0.687404	+	0.726275i	$0.258750\pi$
$660$	0	0
$661$	21.4224	0.833235	0.416617	−	0.909082i	$-0.363215\pi$
$661$	21.4224	0.833235	0.416617	+	0.909082i	$0.363215\pi$
$662$	21.3278	0.828929
$663$	0	0
$664$	−6.16931	−0.239416
$665$	−3.06949	−0.119030
$666$	0	0
$667$	8.54505	0.330866
$668$	−46.7097	−1.80725
$669$	0	0
$670$	37.5322	1.44999
$671$	−17.8777	−0.690160
$672$	0	0
$673$	−16.0994	−0.620587	−0.310294	−	0.950641i	$-0.600427\pi$
$673$	−16.0994	−0.620587	−0.310294	+	0.950641i	$0.600427\pi$
$674$	−34.4797	−1.32811
$675$	0	0
$676$	−43.7223	−1.68163
$677$	−18.6442	−0.716555	−0.358278	−	0.933615i	$-0.616636\pi$
$677$	−18.6442	−0.716555	−0.358278	+	0.933615i	$0.616636\pi$
$678$	0	0
$679$	−5.20942	−0.199919
$680$	35.1947	1.34966
$681$	0	0
$682$	1.91001	0.0731382
$683$	38.1510	1.45981	0.729903	−	0.683551i	$-0.239565\pi$
$683$	38.1510	1.45981	0.729903	+	0.683551i	$0.239565\pi$
$684$	0	0
$685$	9.15481	0.349787
$686$	−23.3393	−0.891099
$687$	0	0
$688$	−1.26389	−0.0481853
$689$	2.79323	0.106414
$690$	0	0
$691$	−12.1250	−0.461255	−0.230628	−	0.973042i	$-0.574078\pi$
$691$	−12.1250	−0.461255	−0.230628	+	0.973042i	$0.574078\pi$
$692$	−7.21587	−0.274306
$693$	0	0
$694$	−78.9498	−2.99690
$695$	−33.9182	−1.28659
$696$	0	0
$697$	17.4633	0.661468
$698$	21.1889	0.802010
$699$	0	0
$700$	−30.9100	−1.16829
$701$	17.5045	0.661135	0.330568	−	0.943782i	$-0.392760\pi$
$701$	17.5045	0.661135	0.330568	+	0.943782i	$0.392760\pi$
$702$	0	0
$703$	−2.44994	−0.0924012
$704$	33.5599	1.26484
$705$	0	0
$706$	−84.4203	−3.17720
$707$	−7.77548	−0.292427
$708$	0	0
$709$	−8.19382	−0.307725	−0.153863	−	0.988092i	$-0.549171\pi$
$709$	−8.19382	−0.307725	−0.153863	+	0.988092i	$0.549171\pi$
$710$	−55.3591	−2.07759
$711$	0	0
$712$	28.8998	1.08307
$713$	0.476936	0.0178614
$714$	0	0
$715$	−6.45061	−0.241239
$716$	52.1575	1.94922
$717$	0	0
$718$	−43.0918	−1.60817
$719$	0.201561	0.00751697	0.00375849	−	0.999993i	$-0.498804\pi$
$719$	0.201561	0.00751697	0.00375849	+	0.999993i	$0.498804\pi$
$720$	0	0
$721$	5.91101	0.220137
$722$	−2.33471	−0.0868891
$723$	0	0
$724$	20.0570	0.745411
$725$	64.9528	2.41229
$726$	0	0
$727$	−42.8009	−1.58740	−0.793700	−	0.608310i	$-0.791848\pi$
$727$	−42.8009	−1.58740	−0.793700	+	0.608310i	$0.791848\pi$
$728$	−1.44686	−0.0536242
$729$	0	0
$730$	−142.162	−5.26167
$731$	−3.15864	−0.116826
$732$	0	0
$733$	23.7969	0.878959	0.439480	−	0.898253i	$-0.355163\pi$
$733$	23.7969	0.878959	0.439480	+	0.898253i	$0.355163\pi$
$734$	−56.2620	−2.07667
$735$	0	0
$736$	7.01280	0.258495
$737$	−10.5860	−0.389939
$738$	0	0
$739$	−9.49947	−0.349444	−0.174722	−	0.984618i	$-0.555903\pi$
$739$	−9.49947	−0.349444	−0.174722	+	0.984618i	$0.555903\pi$
$740$	−34.9089	−1.28328
$741$	0	0
$742$	−8.43754	−0.309752
$743$	22.5532	0.827398	0.413699	−	0.910414i	$-0.364237\pi$
$743$	22.5532	0.827398	0.413699	+	0.910414i	$0.364237\pi$
$744$	0	0
$745$	−41.0952	−1.50561
$746$	−3.52267	−0.128974
$747$	0	0
$748$	−23.6103	−0.863278
$749$	4.87172	0.178009
$750$	0	0
$751$	−21.9980	−0.802720	−0.401360	−	0.915920i	$-0.631462\pi$
$751$	−21.9980	−0.802720	−0.401360	+	0.915920i	$0.631462\pi$
$752$	−1.00686	−0.0367164
$753$	0	0
$754$	7.23140	0.263352
$755$	−53.7584	−1.95647
$756$	0	0
$757$	28.6814	1.04244	0.521222	−	0.853421i	$-0.325476\pi$
$757$	28.6814	1.04244	0.521222	+	0.853421i	$0.325476\pi$
$758$	31.8893	1.15827
$759$	0	0
$760$	−13.9868	−0.507354
$761$	−10.1405	−0.367593	−0.183796	−	0.982964i	$-0.558839\pi$
$761$	−10.1405	−0.367593	−0.183796	+	0.982964i	$0.558839\pi$
$762$	0	0
$763$	3.94420	0.142790
$764$	−62.1104	−2.24708
$765$	0	0
$766$	80.6906	2.91547
$767$	8.22241	0.296894
$768$	0	0
$769$	−21.0989	−0.760845	−0.380422	−	0.924813i	$-0.624221\pi$
$769$	−21.0989	−0.760845	−0.380422	+	0.924813i	$0.624221\pi$
$770$	19.4854	0.702206
$771$	0	0
$772$	−91.8783	−3.30677
$773$	−24.0539	−0.865159	−0.432579	−	0.901596i	$-0.642396\pi$
$773$	−24.0539	−0.865159	−0.432579	+	0.901596i	$0.642396\pi$
$774$	0	0
$775$	3.62530	0.130224
$776$	−23.7378	−0.852138
$777$	0	0
$778$	73.3649	2.63026
$779$	−6.94009	−0.248654
$780$	0	0
$781$	15.6141	0.558715
$782$	−9.31239	−0.333011
$783$	0	0
$784$	−6.49160	−0.231843
$785$	72.2898	2.58013
$786$	0	0
$787$	34.0784	1.21476	0.607382	−	0.794410i	$-0.292220\pi$
$787$	34.0784	1.21476	0.607382	+	0.794410i	$0.292220\pi$
$788$	−15.0977	−0.537833
$789$	0	0
$790$	−26.4419	−0.940762
$791$	9.30042	0.330685
$792$	0	0
$793$	−3.77783	−0.134155
$794$	−6.97418	−0.247504
$795$	0	0
$796$	53.0166	1.87912
$797$	16.4371	0.582231	0.291115	−	0.956688i	$-0.405974\pi$
$797$	16.4371	0.582231	0.291115	+	0.956688i	$0.405974\pi$
$798$	0	0
$799$	−2.51629	−0.0890199
$800$	53.3059	1.88465
$801$	0	0
$802$	27.7921	0.981372
$803$	40.0970	1.41499
$804$	0	0
$805$	4.86557	0.171489
$806$	0.403615	0.0142167
$807$	0	0
$808$	−35.4306	−1.24644
$809$	40.7997	1.43444	0.717221	−	0.696846i	$-0.245414\pi$
$809$	40.7997	1.43444	0.717221	+	0.696846i	$0.245414\pi$
$810$	0	0
$811$	35.0274	1.22998	0.614990	−	0.788535i	$-0.289160\pi$
$811$	35.0274	1.22998	0.614990	+	0.788535i	$0.289160\pi$
$812$	−13.8292	−0.485308
$813$	0	0
$814$	15.5525	0.545113
$815$	−45.1832	−1.58270
$816$	0	0
$817$	1.25528	0.0439165
$818$	−36.0116	−1.25911
$819$	0	0
$820$	−98.8884	−3.45333
$821$	−11.9610	−0.417443	−0.208722	−	0.977975i	$-0.566930\pi$
$821$	−11.9610	−0.417443	−0.208722	+	0.977975i	$0.566930\pi$
$822$	0	0
$823$	17.1752	0.598689	0.299345	−	0.954145i	$-0.403232\pi$
$823$	17.1752	0.598689	0.299345	+	0.954145i	$0.403232\pi$
$824$	26.9347	0.938315
$825$	0	0
$826$	−24.8376	−0.864209
$827$	35.2584	1.22605	0.613027	−	0.790062i	$-0.289952\pi$
$827$	35.2584	1.22605	0.613027	+	0.790062i	$0.289952\pi$
$828$	0	0
$829$	−41.6234	−1.44564	−0.722821	−	0.691036i	$-0.757155\pi$
$829$	−41.6234	−1.44564	−0.722821	+	0.691036i	$0.757155\pi$
$830$	17.5571	0.609414
$831$	0	0
$832$	7.09173	0.245861
$833$	−16.2234	−0.562109
$834$	0	0
$835$	55.8888	1.93411
$836$	9.38299	0.324517
$837$	0	0
$838$	40.8314	1.41050
$839$	41.0969	1.41882	0.709412	−	0.704794i	$-0.248961\pi$
$839$	41.0969	1.41882	0.709412	+	0.704794i	$0.248961\pi$
$840$	0	0
$841$	0.0599466	0.00206712
$842$	25.6367	0.883499
$843$	0	0
$844$	2.91964	0.100498
$845$	52.3144	1.79967
$846$	0	0
$847$	2.68142	0.0921346
$848$	−4.89480	−0.168088
$849$	0	0
$850$	−70.7856	−2.42792
$851$	3.88349	0.133124
$852$	0	0
$853$	−46.7710	−1.60141	−0.800705	−	0.599059i	$-0.795541\pi$
$853$	−46.7710	−1.60141	−0.800705	+	0.599059i	$0.795541\pi$
$854$	11.4117	0.390502
$855$	0	0
$856$	22.1990	0.758747
$857$	22.0231	0.752295	0.376147	−	0.926560i	$-0.377249\pi$
$857$	22.0231	0.752295	0.376147	+	0.926560i	$0.377249\pi$
$858$	0	0
$859$	47.1413	1.60844	0.804220	−	0.594332i	$-0.202583\pi$
$859$	47.1413	1.60844	0.804220	+	0.594332i	$0.202583\pi$
$860$	17.8863	0.609917
$861$	0	0
$862$	56.1276	1.91171
$863$	48.2690	1.64310	0.821548	−	0.570139i	$-0.193111\pi$
$863$	48.2690	1.64310	0.821548	+	0.570139i	$0.193111\pi$
$864$	0	0
$865$	8.63390	0.293561
$866$	17.9378	0.609553
$867$	0	0
$868$	−0.771864	−0.0261988
$869$	7.45796	0.252994
$870$	0	0
$871$	−2.23698	−0.0757972
$872$	17.9726	0.608628
$873$	0	0
$874$	3.70085	0.125183
$875$	21.6368	0.731458
$876$	0	0
$877$	33.2601	1.12311	0.561557	−	0.827438i	$-0.310203\pi$
$877$	33.2601	1.12311	0.561557	+	0.827438i	$0.310203\pi$
$878$	20.0412	0.676357
$879$	0	0
$880$	11.3039	0.381055
$881$	−5.04910	−0.170109	−0.0850543	−	0.996376i	$-0.527106\pi$
$881$	−5.04910	−0.170109	−0.0850543	+	0.996376i	$0.527106\pi$
$882$	0	0
$883$	9.17644	0.308812	0.154406	−	0.988008i	$-0.450654\pi$
$883$	9.17644	0.308812	0.154406	+	0.988008i	$0.450654\pi$
$884$	−4.98922	−0.167806
$885$	0	0
$886$	−85.4734	−2.87153
$887$	25.5824	0.858973	0.429486	−	0.903073i	$-0.358695\pi$
$887$	25.5824	0.858973	0.429486	+	0.903073i	$0.358695\pi$
$888$	0	0
$889$	4.72938	0.158618
$890$	−82.2451	−2.75686
$891$	0	0
$892$	40.5752	1.35856
$893$	1.00000	0.0334637
$894$	0	0
$895$	−62.4073	−2.08605
$896$	−14.8444	−0.495917
$897$	0	0
$898$	−17.9197	−0.597987
$899$	1.62196	0.0540954
$900$	0	0
$901$	−12.2328	−0.407533
$902$	44.0563	1.46692
$903$	0	0
$904$	42.3793	1.40951
$905$	−23.9985	−0.797736
$906$	0	0
$907$	7.65589	0.254210	0.127105	−	0.991889i	$-0.459432\pi$
$907$	7.65589	0.254210	0.127105	+	0.991889i	$0.459432\pi$
$908$	19.8271	0.657986
$909$	0	0
$910$	4.11757	0.136496
$911$	59.6374	1.97588	0.987938	−	0.154852i	$-0.0494901\pi$
$911$	59.6374	1.97588	0.987938	+	0.154852i	$0.0494901\pi$
$912$	0	0
$913$	−4.95198	−0.163886
$914$	−78.8275	−2.60738
$915$	0	0
$916$	−88.8153	−2.93454
$917$	−13.4999	−0.445804
$918$	0	0
$919$	−18.0017	−0.593822	−0.296911	−	0.954905i	$-0.595957\pi$
$919$	−18.0017	−0.593822	−0.296911	+	0.954905i	$0.595957\pi$
$920$	22.1710	0.730955
$921$	0	0
$922$	9.92346	0.326812
$923$	3.29949	0.108604
$924$	0	0
$925$	29.5193	0.970589
$926$	37.6272	1.23651
$927$	0	0
$928$	23.8491	0.782885
$929$	42.2366	1.38574	0.692869	−	0.721063i	$-0.256346\pi$
$929$	42.2366	1.38574	0.692869	+	0.721063i	$0.256346\pi$
$930$	0	0
$931$	6.44737	0.211304
$932$	−18.2133	−0.596597
$933$	0	0
$934$	10.5923	0.346591
$935$	28.2501	0.923876
$936$	0	0
$937$	−24.2165	−0.791119	−0.395559	−	0.918440i	$-0.629449\pi$
$937$	−24.2165	−0.791119	−0.395559	+	0.918440i	$0.629449\pi$
$938$	6.75727	0.220633
$939$	0	0
$940$	14.2489	0.464747
$941$	−5.79127	−0.188790	−0.0943950	−	0.995535i	$-0.530092\pi$
$941$	−5.79127	−0.188790	−0.0943950	+	0.995535i	$0.530092\pi$
$942$	0	0
$943$	11.0010	0.358242
$944$	−14.4088	−0.468966
$945$	0	0
$946$	−7.96862	−0.259082
$947$	2.64008	0.0857911	0.0428956	−	0.999080i	$-0.486342\pi$
$947$	2.64008	0.0857911	0.0428956	+	0.999080i	$0.486342\pi$
$948$	0	0
$949$	8.47311	0.275049
$950$	28.1310	0.912689
$951$	0	0
$952$	6.33644	0.205365
$953$	−51.9899	−1.68412	−0.842060	−	0.539385i	$-0.818657\pi$
$953$	−51.9899	−1.68412	−0.842060	+	0.539385i	$0.818657\pi$
$954$	0	0
$955$	74.3161	2.40481
$956$	−52.1425	−1.68641
$957$	0	0
$958$	27.1360	0.876726
$959$	1.64823	0.0532240
$960$	0	0
$961$	−30.9095	−0.997080
$962$	3.28648	0.105960
$963$	0	0
$964$	37.3931	1.20435
$965$	109.934	3.53889
$966$	0	0
$967$	−12.3052	−0.395707	−0.197854	−	0.980232i	$-0.563397\pi$
$967$	−12.3052	−0.395707	−0.197854	+	0.980232i	$0.563397\pi$
$968$	12.2184	0.392716
$969$	0	0
$970$	67.5547	2.16905
$971$	47.4186	1.52174	0.760868	−	0.648907i	$-0.224774\pi$
$971$	47.4186	1.52174	0.760868	+	0.648907i	$0.224774\pi$
$972$	0	0
$973$	−6.10661	−0.195769
$974$	−58.9977	−1.89041
$975$	0	0
$976$	6.62019	0.211907
$977$	18.9038	0.604786	0.302393	−	0.953183i	$-0.402215\pi$
$977$	18.9038	0.604786	0.302393	+	0.953183i	$0.402215\pi$
$978$	0	0
$979$	23.1973	0.741388
$980$	91.8677	2.93461
$981$	0	0
$982$	−16.3098	−0.520466
$983$	9.00449	0.287199	0.143599	−	0.989636i	$-0.454132\pi$
$983$	9.00449	0.287199	0.143599	+	0.989636i	$0.454132\pi$
$984$	0	0
$985$	18.0646	0.575587
$986$	−31.6695	−1.00856
$987$	0	0
$988$	1.98277	0.0630803
$989$	−1.98979	−0.0632715
$990$	0	0
$991$	−38.2094	−1.21376	−0.606881	−	0.794793i	$-0.707580\pi$
$991$	−38.2094	−1.21376	−0.606881	+	0.794793i	$0.707580\pi$
$992$	1.33112	0.0422631
$993$	0	0
$994$	−9.96682	−0.316128
$995$	−63.4352	−2.01103
$996$	0	0
$997$	45.8241	1.45126	0.725632	−	0.688083i	$-0.241548\pi$
$997$	45.8241	1.45126	0.725632	+	0.688083i	$0.241548\pi$
$998$	84.8283	2.68519
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.o.1.3		18
3.2	odd	2		893.2.a.c.1.16	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.c.1.16	✓	18	3.2	odd	2
8037.2.a.o.1.3		18	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-2.33471 q^{2} +3.45089 q^{4} -4.12904 q^{5} -0.743390 q^{7} -3.38741 q^{8} +O(q^{10})\)	\(q-2.33471 q^{2} +3.45089 q^{4} -4.12904 q^{5} -0.743390 q^{7} -3.38741 q^{8} +9.64013 q^{10} -2.71900 q^{11} -0.574568 q^{13} +1.73560 q^{14} +1.00686 q^{16} +2.51629 q^{17} -1.00000 q^{19} -14.2489 q^{20} +6.34810 q^{22} +1.58514 q^{23} +12.0490 q^{25} +1.34145 q^{26} -2.56536 q^{28} +5.39073 q^{29} +0.300880 q^{31} +4.42409 q^{32} -5.87481 q^{34} +3.06949 q^{35} +2.44994 q^{37} +2.33471 q^{38} +13.9868 q^{40} +6.94009 q^{41} -1.25528 q^{43} -9.38299 q^{44} -3.70085 q^{46} -1.00000 q^{47} -6.44737 q^{49} -28.1310 q^{50} -1.98277 q^{52} -4.86144 q^{53} +11.2269 q^{55} +2.51817 q^{56} -12.5858 q^{58} -14.3106 q^{59} +6.57508 q^{61} -0.702468 q^{62} -12.3427 q^{64} +2.37241 q^{65} +3.89333 q^{67} +8.68343 q^{68} -7.16638 q^{70} -5.74257 q^{71} -14.7469 q^{73} -5.71991 q^{74} -3.45089 q^{76} +2.02128 q^{77} -2.74290 q^{79} -4.15737 q^{80} -16.2031 q^{82} +1.82125 q^{83} -10.3899 q^{85} +2.93071 q^{86} +9.21039 q^{88} -8.53153 q^{89} +0.427128 q^{91} +5.47014 q^{92} +2.33471 q^{94} +4.12904 q^{95} +7.00765 q^{97} +15.0528 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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