LMFDB - Embedded newform 567.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,4,Mod(1,567)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("567.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	567.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:	$33.4540829733$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 49x^{6} + 138x^{5} + 708x^{4} - 1941x^{3} - 2506x^{2} + 8592x - 4616 $ x^8 - 3x^7 - 49x^6 + 138x^5 + 708x^4 - 1941x^3 - 2506x^2 + 8592*x - 4616
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.95181$ of defining polynomial
Character	$\chi$	$=$	567.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.95181 q^{2} +16.5205 q^{4} +16.1400 q^{5} -7.00000 q^{7} -42.1917 q^{8} +O(q^{10})$	\(q-4.95181 q^{2} +16.5205 q^{4} +16.1400 q^{5} -7.00000 q^{7} -42.1917 q^{8} -79.9221 q^{10} -30.6371 q^{11} -78.0119 q^{13} +34.6627 q^{14} +76.7617 q^{16} +106.992 q^{17} -49.5081 q^{19} +266.639 q^{20} +151.709 q^{22} +39.0142 q^{23} +135.498 q^{25} +386.300 q^{26} -115.643 q^{28} +5.74776 q^{29} +184.493 q^{31} -42.5762 q^{32} -529.803 q^{34} -112.980 q^{35} +91.0901 q^{37} +245.155 q^{38} -680.972 q^{40} -82.8692 q^{41} -91.2742 q^{43} -506.139 q^{44} -193.191 q^{46} +237.964 q^{47} +49.0000 q^{49} -670.962 q^{50} -1288.79 q^{52} +497.669 q^{53} -494.481 q^{55} +295.342 q^{56} -28.4618 q^{58} -4.20594 q^{59} +626.391 q^{61} -913.577 q^{62} -403.264 q^{64} -1259.11 q^{65} -678.620 q^{67} +1767.55 q^{68} +559.454 q^{70} +747.071 q^{71} -23.2628 q^{73} -451.061 q^{74} -817.897 q^{76} +214.460 q^{77} +154.041 q^{79} +1238.93 q^{80} +410.353 q^{82} +282.196 q^{83} +1726.84 q^{85} +451.973 q^{86} +1292.63 q^{88} +111.091 q^{89} +546.083 q^{91} +644.532 q^{92} -1178.35 q^{94} -799.059 q^{95} +1110.09 q^{97} -242.639 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8}+O(q^{10}) $ 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8	$ 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8} - 14 q^{10} + 24 q^{11} + 68 q^{13} - 21 q^{14} + 103 q^{16} + 168 q^{17} + 176 q^{19} + 330 q^{20} + 151 q^{22} + 228 q^{23} + 244 q^{25} + 795 q^{26} - 301 q^{28} + 618 q^{29} + 72 q^{31} + 786 q^{32} - 261 q^{34} - 210 q^{35} + 210 q^{37} + 1032 q^{38} - 375 q^{40} + 420 q^{41} - 2 q^{43} + 387 q^{44} + 402 q^{46} + 570 q^{47} + 392 q^{49} + 1110 q^{50} - 431 q^{52} + 528 q^{53} - 838 q^{55} - 42 q^{56} + 37 q^{58} - 150 q^{59} + 578 q^{61} + 1170 q^{62} - 112 q^{64} - 366 q^{65} - 898 q^{67} + 2526 q^{68} + 98 q^{70} + 882 q^{71} + 972 q^{73} - 222 q^{74} + 1423 q^{76} - 168 q^{77} - 158 q^{79} + 2475 q^{80} - 211 q^{82} + 2958 q^{83} - 774 q^{85} - 114 q^{86} + 1317 q^{88} + 4380 q^{89} - 476 q^{91} + 4629 q^{92} - 3234 q^{94} + 930 q^{95} - 60 q^{97} + 147 q^{98}+O(q^{100}) $ 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8 - 14 * q^10 + 24 * q^11 + 68 * q^13 - 21 * q^14 + 103 * q^16 + 168 * q^17 + 176 * q^19 + 330 * q^20 + 151 * q^22 + 228 * q^23 + 244 * q^25 + 795 * q^26 - 301 * q^28 + 618 * q^29 + 72 * q^31 + 786 * q^32 - 261 * q^34 - 210 * q^35 + 210 * q^37 + 1032 * q^38 - 375 * q^40 + 420 * q^41 - 2 * q^43 + 387 * q^44 + 402 * q^46 + 570 * q^47 + 392 * q^49 + 1110 * q^50 - 431 * q^52 + 528 * q^53 - 838 * q^55 - 42 * q^56 + 37 * q^58 - 150 * q^59 + 578 * q^61 + 1170 * q^62 - 112 * q^64 - 366 * q^65 - 898 * q^67 + 2526 * q^68 + 98 * q^70 + 882 * q^71 + 972 * q^73 - 222 * q^74 + 1423 * q^76 - 168 * q^77 - 158 * q^79 + 2475 * q^80 - 211 * q^82 + 2958 * q^83 - 774 * q^85 - 114 * q^86 + 1317 * q^88 + 4380 * q^89 - 476 * q^91 + 4629 * q^92 - 3234 * q^94 + 930 * q^95 - 60 * q^97 + 147 * 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 $

$2$	−4.95181	−1.75073	−0.875365	−	0.483462i	$-0.839379\pi$
$2$	−4.95181	−1.75073	−0.875365	+	0.483462i	$0.839379\pi$
$3$	0	0
$3$	0	0
$4$	16.5205	2.06506
$5$	16.1400	1.44360	0.721801	−	0.692101i	$-0.243315\pi$
$5$	16.1400	1.44360	0.721801	+	0.692101i	$0.243315\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−42.1917	−1.86463
$9$	0	0
$10$	−79.9221	−2.52736
$11$	−30.6371	−0.839766	−0.419883	−	0.907578i	$-0.637929\pi$
$11$	−30.6371	−0.839766	−0.419883	+	0.907578i	$0.637929\pi$
$12$	0	0
$13$	−78.0119	−1.66435	−0.832177	−	0.554510i	$-0.812906\pi$
$13$	−78.0119	−1.66435	−0.832177	+	0.554510i	$0.812906\pi$
$14$	34.6627	0.661714
$15$	0	0
$16$	76.7617	1.19940
$17$	106.992	1.52643	0.763214	−	0.646146i	$-0.223620\pi$
$17$	106.992	1.52643	0.763214	+	0.646146i	$0.223620\pi$
$18$	0	0
$19$	−49.5081	−0.597787	−0.298893	−	0.954287i	$-0.596617\pi$
$19$	−49.5081	−0.597787	−0.298893	+	0.954287i	$0.596617\pi$
$20$	266.639	2.98112
$21$	0	0
$22$	151.709	1.47020
$23$	39.0142	0.353696	0.176848	−	0.984238i	$-0.443410\pi$
$23$	39.0142	0.353696	0.176848	+	0.984238i	$0.443410\pi$
$24$	0	0
$25$	135.498	1.08399
$26$	386.300	2.91383
$27$	0	0
$28$	−115.643	−0.780518
$29$	5.74776	0.0368046	0.0184023	−	0.999831i	$-0.494142\pi$
$29$	5.74776	0.0368046	0.0184023	+	0.999831i	$0.494142\pi$
$30$	0	0
$31$	184.493	1.06890	0.534451	−	0.845199i	$-0.320518\pi$
$31$	184.493	1.06890	0.534451	+	0.845199i	$0.320518\pi$
$32$	−42.5762	−0.235202
$33$	0	0
$34$	−529.803	−2.67236
$35$	−112.980	−0.545630
$36$	0	0
$37$	91.0901	0.404733	0.202366	−	0.979310i	$-0.435137\pi$
$37$	91.0901	0.404733	0.202366	+	0.979310i	$0.435137\pi$
$38$	245.155	1.04656
$39$	0	0
$40$	−680.972	−2.69178
$41$	−82.8692	−0.315658	−0.157829	−	0.987466i	$-0.550450\pi$
$41$	−82.8692	−0.315658	−0.157829	+	0.987466i	$0.550450\pi$
$42$	0	0
$43$	−91.2742	−0.323702	−0.161851	−	0.986815i	$-0.551746\pi$
$43$	−91.2742	−0.323702	−0.161851	+	0.986815i	$0.551746\pi$
$44$	−506.139	−1.73417
$45$	0	0
$46$	−193.191	−0.619227
$47$	237.964	0.738524	0.369262	−	0.929325i	$-0.379611\pi$
$47$	237.964	0.738524	0.369262	+	0.929325i	$0.379611\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−670.962	−1.89777
$51$	0	0
$52$	−1288.79	−3.43698
$53$	497.669	1.28981	0.644906	−	0.764262i	$-0.276896\pi$
$53$	497.669	1.28981	0.644906	+	0.764262i	$0.276896\pi$
$54$	0	0
$55$	−494.481	−1.21229
$56$	295.342	0.704763
$57$	0	0
$58$	−28.4618	−0.0644349
$59$	−4.20594	−0.00928079	−0.00464039	−	0.999989i	$-0.501477\pi$
$59$	−4.20594	−0.00928079	−0.00464039	+	0.999989i	$0.501477\pi$
$60$	0	0
$61$	626.391	1.31477	0.657387	−	0.753553i	$-0.271662\pi$
$61$	626.391	1.31477	0.657387	+	0.753553i	$0.271662\pi$
$62$	−913.577	−1.87136
$63$	0	0
$64$	−403.264	−0.787626
$65$	−1259.11	−2.40266
$66$	0	0
$67$	−678.620	−1.23741	−0.618706	−	0.785622i	$-0.712343\pi$
$67$	−678.620	−1.23741	−0.618706	+	0.785622i	$0.712343\pi$
$68$	1767.55	3.15216
$69$	0	0
$70$	559.454	0.955251
$71$	747.071	1.24875	0.624374	−	0.781126i	$-0.285354\pi$
$71$	747.071	1.24875	0.624374	+	0.781126i	$0.285354\pi$
$72$	0	0
$73$	−23.2628	−0.0372974	−0.0186487	−	0.999826i	$-0.505936\pi$
$73$	−23.2628	−0.0372974	−0.0186487	+	0.999826i	$0.505936\pi$
$74$	−451.061	−0.708578
$75$	0	0
$76$	−817.897	−1.23446
$77$	214.460	0.317402
$78$	0	0
$79$	154.041	0.219379	0.109689	−	0.993966i	$-0.465014\pi$
$79$	154.041	0.219379	0.109689	+	0.993966i	$0.465014\pi$
$80$	1238.93	1.73146
$81$	0	0
$82$	410.353	0.552633
$83$	282.196	0.373193	0.186597	−	0.982437i	$-0.440254\pi$
$83$	282.196	0.373193	0.186597	+	0.982437i	$0.440254\pi$
$84$	0	0
$85$	1726.84	2.20355
$86$	451.973	0.566715
$87$	0	0
$88$	1292.63	1.56585
$89$	111.091	0.132311	0.0661554	−	0.997809i	$-0.478927\pi$
$89$	111.091	0.132311	0.0661554	+	0.997809i	$0.478927\pi$
$90$	0	0
$91$	546.083	0.629067
$92$	644.532	0.730403
$93$	0	0
$94$	−1178.35	−1.29296
$95$	−799.059	−0.862966
$96$	0	0
$97$	1110.09	1.16199	0.580993	−	0.813909i	$-0.302664\pi$
$97$	1110.09	1.16199	0.580993	+	0.813909i	$0.302664\pi$
$98$	−242.639	−0.250104
$99$	0	0
$100$	2238.49	2.23849
$101$	−1398.33	−1.37761	−0.688805	−	0.724946i	$-0.741865\pi$
$101$	−1398.33	−1.37761	−0.688805	+	0.724946i	$0.741865\pi$
$102$	0	0
$103$	1447.75	1.38497	0.692483	−	0.721434i	$-0.256517\pi$
$103$	1447.75	1.38497	0.692483	+	0.721434i	$0.256517\pi$
$104$	3291.45	3.10340
$105$	0	0
$106$	−2464.36	−2.25811
$107$	−1063.28	−0.960664	−0.480332	−	0.877087i	$-0.659484\pi$
$107$	−1063.28	−0.960664	−0.480332	+	0.877087i	$0.659484\pi$
$108$	0	0
$109$	1933.34	1.69890	0.849450	−	0.527669i	$-0.176934\pi$
$109$	1933.34	1.69890	0.849450	+	0.527669i	$0.176934\pi$
$110$	2448.58	2.12239
$111$	0	0
$112$	−537.332	−0.453331
$113$	1540.63	1.28257	0.641284	−	0.767304i	$-0.278402\pi$
$113$	1540.63	1.28257	0.641284	+	0.767304i	$0.278402\pi$
$114$	0	0
$115$	629.687	0.510597
$116$	94.9556	0.0760035
$117$	0	0
$118$	20.8270	0.0162482
$119$	−748.941	−0.576936
$120$	0	0
$121$	−392.369	−0.294792
$122$	−3101.77	−2.30181
$123$	0	0
$124$	3047.91	2.20734
$125$	169.442	0.121243
$126$	0	0
$127$	1471.23	1.02796	0.513980	−	0.857802i	$-0.328171\pi$
$127$	1471.23	1.02796	0.513980	+	0.857802i	$0.328171\pi$
$128$	2337.50	1.61412
$129$	0	0
$130$	6234.87	4.20642
$131$	678.752	0.452693	0.226347	−	0.974047i	$-0.427322\pi$
$131$	678.752	0.452693	0.226347	+	0.974047i	$0.427322\pi$
$132$	0	0
$133$	346.557	0.225942
$134$	3360.40	2.16638
$135$	0	0
$136$	−4514.16	−2.84622
$137$	1819.96	1.13496	0.567480	−	0.823387i	$-0.307918\pi$
$137$	1819.96	1.13496	0.567480	+	0.823387i	$0.307918\pi$
$138$	0	0
$139$	−2918.67	−1.78100	−0.890499	−	0.454986i	$-0.849644\pi$
$139$	−2918.67	−1.78100	−0.890499	+	0.454986i	$0.849644\pi$
$140$	−1866.48	−1.12676
$141$	0	0
$142$	−3699.36	−2.18622
$143$	2390.06	1.39767
$144$	0	0
$145$	92.7686	0.0531311
$146$	115.193	0.0652976
$147$	0	0
$148$	1504.85	0.835796
$149$	−984.486	−0.541290	−0.270645	−	0.962679i	$-0.587237\pi$
$149$	−984.486	−0.541290	−0.270645	+	0.962679i	$0.587237\pi$
$150$	0	0
$151$	−129.865	−0.0699887	−0.0349943	−	0.999388i	$-0.511141\pi$
$151$	−129.865	−0.0699887	−0.0349943	+	0.999388i	$0.511141\pi$
$152$	2088.83	1.11465
$153$	0	0
$154$	−1061.96	−0.555685
$155$	2977.72	1.54307
$156$	0	0
$157$	−217.030	−0.110324	−0.0551621	−	0.998477i	$-0.517568\pi$
$157$	−217.030	−0.110324	−0.0551621	+	0.998477i	$0.517568\pi$
$158$	−762.780	−0.384073
$159$	0	0
$160$	−687.177	−0.339538
$161$	−273.099	−0.133685
$162$	0	0
$163$	40.7178	0.0195660	0.00978302	−	0.999952i	$-0.496886\pi$
$163$	40.7178	0.0195660	0.00978302	+	0.999952i	$0.496886\pi$
$164$	−1369.04	−0.651853
$165$	0	0
$166$	−1397.38	−0.653361
$167$	1725.99	0.799766	0.399883	−	0.916566i	$-0.369051\pi$
$167$	1725.99	0.799766	0.399883	+	0.916566i	$0.369051\pi$
$168$	0	0
$169$	3888.85	1.77007
$170$	−8550.99	−3.85783
$171$	0	0
$172$	−1507.89	−0.668463
$173$	−582.877	−0.256158	−0.128079	−	0.991764i	$-0.540881\pi$
$173$	−582.877	−0.256158	−0.128079	+	0.991764i	$0.540881\pi$
$174$	0	0
$175$	−948.488	−0.409708
$176$	−2351.76	−1.00722
$177$	0	0
$178$	−550.104	−0.231641
$179$	−3699.88	−1.54493	−0.772465	−	0.635058i	$-0.780976\pi$
$179$	−3699.88	−1.54493	−0.772465	+	0.635058i	$0.780976\pi$
$180$	0	0
$181$	2872.98	1.17982	0.589908	−	0.807470i	$-0.299164\pi$
$181$	2872.98	1.17982	0.589908	+	0.807470i	$0.299164\pi$
$182$	−2704.10	−1.10133
$183$	0	0
$184$	−1646.07	−0.659512
$185$	1470.19	0.584273
$186$	0	0
$187$	−3277.91	−1.28184
$188$	3931.27	1.52509
$189$	0	0
$190$	3956.79	1.51082
$191$	−273.292	−0.103533	−0.0517663	−	0.998659i	$-0.516485\pi$
$191$	−273.292	−0.103533	−0.0517663	+	0.998659i	$0.516485\pi$
$192$	0	0
$193$	844.278	0.314883	0.157442	−	0.987528i	$-0.449675\pi$
$193$	844.278	0.314883	0.157442	+	0.987528i	$0.449675\pi$
$194$	−5496.96	−2.03432
$195$	0	0
$196$	809.502	0.295008
$197$	2506.77	0.906600	0.453300	−	0.891358i	$-0.350247\pi$
$197$	2506.77	0.906600	0.453300	+	0.891358i	$0.350247\pi$
$198$	0	0
$199$	−3938.12	−1.40284	−0.701421	−	0.712747i	$-0.747451\pi$
$199$	−3938.12	−1.40284	−0.701421	+	0.712747i	$0.747451\pi$
$200$	−5716.90	−2.02123
$201$	0	0
$202$	6924.25	2.41182
$203$	−40.2343	−0.0139108
$204$	0	0
$205$	−1337.51	−0.455685
$206$	−7169.01	−2.42470
$207$	0	0
$208$	−5988.32	−1.99623
$209$	1516.79	0.502001
$210$	0	0
$211$	−66.8506	−0.0218113	−0.0109057	−	0.999941i	$-0.503471\pi$
$211$	−66.8506	−0.0218113	−0.0109057	+	0.999941i	$0.503471\pi$
$212$	8221.71	2.66354
$213$	0	0
$214$	5265.16	1.68186
$215$	−1473.16	−0.467297
$216$	0	0
$217$	−1291.45	−0.404007
$218$	−9573.53	−2.97432
$219$	0	0
$220$	−8169.06	−2.50344
$221$	−8346.62	−2.54052
$222$	0	0
$223$	2487.60	0.747005	0.373502	−	0.927629i	$-0.378157\pi$
$223$	2487.60	0.747005	0.373502	+	0.927629i	$0.378157\pi$
$224$	298.033	0.0888981
$225$	0	0
$226$	−7628.90	−2.24543
$227$	3846.73	1.12474	0.562370	−	0.826885i	$-0.309890\pi$
$227$	3846.73	1.12474	0.562370	+	0.826885i	$0.309890\pi$
$228$	0	0
$229$	−4011.89	−1.15770	−0.578849	−	0.815435i	$-0.696498\pi$
$229$	−4011.89	−1.15770	−0.578849	+	0.815435i	$0.696498\pi$
$230$	−3118.09	−0.893917
$231$	0	0
$232$	−242.508	−0.0686268
$233$	5724.20	1.60946	0.804732	−	0.593639i	$-0.202309\pi$
$233$	5724.20	1.60946	0.804732	+	0.593639i	$0.202309\pi$
$234$	0	0
$235$	3840.73	1.06613
$236$	−69.4840	−0.0191653
$237$	0	0
$238$	3708.62	1.01006
$239$	6119.57	1.65624	0.828122	−	0.560548i	$-0.189409\pi$
$239$	6119.57	1.65624	0.828122	+	0.560548i	$0.189409\pi$
$240$	0	0
$241$	492.803	0.131719	0.0658594	−	0.997829i	$-0.479021\pi$
$241$	492.803	0.131719	0.0658594	+	0.997829i	$0.479021\pi$
$242$	1942.94	0.516102
$243$	0	0
$244$	10348.3	2.71508
$245$	790.858	0.206229
$246$	0	0
$247$	3862.22	0.994928
$248$	−7784.09	−1.99310
$249$	0	0
$250$	−839.046	−0.212264
$251$	−1859.25	−0.467548	−0.233774	−	0.972291i	$-0.575108\pi$
$251$	−1859.25	−0.467548	−0.233774	+	0.972291i	$0.575108\pi$
$252$	0	0
$253$	−1195.28	−0.297022
$254$	−7285.27	−1.79968
$255$	0	0
$256$	−8348.74	−2.03827
$257$	3771.91	0.915507	0.457753	−	0.889079i	$-0.348654\pi$
$257$	3771.91	0.915507	0.457753	+	0.889079i	$0.348654\pi$
$258$	0	0
$259$	−637.631	−0.152975
$260$	−20801.0	−4.96164
$261$	0	0
$262$	−3361.05	−0.792544
$263$	40.7379	0.00955135	0.00477568	−	0.999989i	$-0.498480\pi$
$263$	40.7379	0.00955135	0.00477568	+	0.999989i	$0.498480\pi$
$264$	0	0
$265$	8032.35	1.86198
$266$	−1716.09	−0.395564
$267$	0	0
$268$	−11211.1	−2.55533
$269$	1404.08	0.318247	0.159124	−	0.987259i	$-0.449133\pi$
$269$	1404.08	0.318247	0.159124	+	0.987259i	$0.449133\pi$
$270$	0	0
$271$	−222.681	−0.0499148	−0.0249574	−	0.999689i	$-0.507945\pi$
$271$	−222.681	−0.0499148	−0.0249574	+	0.999689i	$0.507945\pi$
$272$	8212.86	1.83080
$273$	0	0
$274$	−9012.10	−1.98701
$275$	−4151.27	−0.910295
$276$	0	0
$277$	−5779.25	−1.25358	−0.626789	−	0.779189i	$-0.715631\pi$
$277$	−5779.25	−1.25358	−0.626789	+	0.779189i	$0.715631\pi$
$278$	14452.7	3.11805
$279$	0	0
$280$	4766.80	1.01740
$281$	3770.20	0.800395	0.400198	−	0.916429i	$-0.368942\pi$
$281$	3770.20	0.800395	0.400198	+	0.916429i	$0.368942\pi$
$282$	0	0
$283$	−4804.11	−1.00910	−0.504549	−	0.863383i	$-0.668341\pi$
$283$	−4804.11	−1.00910	−0.504549	+	0.863383i	$0.668341\pi$
$284$	12342.0	2.57873
$285$	0	0
$286$	−11835.1	−2.44694
$287$	580.084	0.119308
$288$	0	0
$289$	6534.21	1.32998
$290$	−459.373	−0.0930183
$291$	0	0
$292$	−384.312	−0.0770211
$293$	2986.91	0.595553	0.297777	−	0.954636i	$-0.403755\pi$
$293$	2986.91	0.595553	0.297777	+	0.954636i	$0.403755\pi$
$294$	0	0
$295$	−67.8837	−0.0133978
$296$	−3843.24	−0.754676
$297$	0	0
$298$	4874.99	0.947653
$299$	−3043.57	−0.588676
$300$	0	0
$301$	638.920	0.122348
$302$	643.069	0.122531
$303$	0	0
$304$	−3800.33	−0.716986
$305$	10109.9	1.89801
$306$	0	0
$307$	−2881.52	−0.535691	−0.267845	−	0.963462i	$-0.586312\pi$
$307$	−2881.52	−0.535691	−0.267845	+	0.963462i	$0.586312\pi$
$308$	3542.97	0.655453
$309$	0	0
$310$	−14745.1	−2.70150
$311$	6797.68	1.23942	0.619712	−	0.784829i	$-0.287249\pi$
$311$	6797.68	1.23942	0.619712	+	0.784829i	$0.287249\pi$
$312$	0	0
$313$	941.742	0.170065	0.0850326	−	0.996378i	$-0.472901\pi$
$313$	941.742	0.170065	0.0850326	+	0.996378i	$0.472901\pi$
$314$	1074.69	0.193148
$315$	0	0
$316$	2544.82	0.453029
$317$	3633.00	0.643690	0.321845	−	0.946792i	$-0.395697\pi$
$317$	3633.00	0.643690	0.321845	+	0.946792i	$0.395697\pi$
$318$	0	0
$319$	−176.095	−0.0309072
$320$	−6508.67	−1.13702
$321$	0	0
$322$	1352.34	0.234046
$323$	−5296.96	−0.912478
$324$	0	0
$325$	−10570.5	−1.80414
$326$	−201.627	−0.0342549
$327$	0	0
$328$	3496.39	0.588585
$329$	−1665.75	−0.279136
$330$	0	0
$331$	2750.33	0.456713	0.228357	−	0.973578i	$-0.426665\pi$
$331$	2750.33	0.456713	0.228357	+	0.973578i	$0.426665\pi$
$332$	4662.01	0.770665
$333$	0	0
$334$	−8546.77	−1.40017
$335$	−10952.9	−1.78633
$336$	0	0
$337$	5832.28	0.942743	0.471372	−	0.881935i	$-0.343759\pi$
$337$	5832.28	0.942743	0.471372	+	0.881935i	$0.343759\pi$
$338$	−19256.9	−3.09892
$339$	0	0
$340$	28528.2	4.55047
$341$	−5652.34	−0.897629
$342$	0	0
$343$	−343.000	−0.0539949
$344$	3851.01	0.603583
$345$	0	0
$346$	2886.30	0.448463
$347$	3409.96	0.527540	0.263770	−	0.964586i	$-0.415034\pi$
$347$	3409.96	0.527540	0.263770	+	0.964586i	$0.415034\pi$
$348$	0	0
$349$	5947.47	0.912208	0.456104	−	0.889926i	$-0.349244\pi$
$349$	5947.47	0.912208	0.456104	+	0.889926i	$0.349244\pi$
$350$	4696.74	0.717289
$351$	0	0
$352$	1304.41	0.197515
$353$	−531.694	−0.0801677	−0.0400838	−	0.999196i	$-0.512763\pi$
$353$	−531.694	−0.0801677	−0.0400838	+	0.999196i	$0.512763\pi$
$354$	0	0
$355$	12057.7	1.80269
$356$	1835.28	0.273229
$357$	0	0
$358$	18321.1	2.70475
$359$	−7555.95	−1.11083	−0.555415	−	0.831573i	$-0.687441\pi$
$359$	−7555.95	−1.11083	−0.555415	+	0.831573i	$0.687441\pi$
$360$	0	0
$361$	−4407.94	−0.642651
$362$	−14226.4	−2.06554
$363$	0	0
$364$	9021.54	1.29906
$365$	−375.461	−0.0538425
$366$	0	0
$367$	−9703.22	−1.38012	−0.690060	−	0.723752i	$-0.742416\pi$
$367$	−9703.22	−1.38012	−0.690060	+	0.723752i	$0.742416\pi$
$368$	2994.79	0.424224
$369$	0	0
$370$	−7280.11	−1.02290
$371$	−3483.68	−0.487503
$372$	0	0
$373$	14.1946	0.00197042	0.000985210	−	1.00000i	$-0.499686\pi$
$373$	14.1946	0.00197042	0.000985210		1.00000i	$0.499686\pi$
$374$	16231.6	2.24416
$375$	0	0
$376$	−10040.1	−1.37707
$377$	−448.393	−0.0612558
$378$	0	0
$379$	−2618.92	−0.354948	−0.177474	−	0.984126i	$-0.556792\pi$
$379$	−2618.92	−0.354948	−0.177474	+	0.984126i	$0.556792\pi$
$380$	−13200.8	−1.78207
$381$	0	0
$382$	1353.29	0.181258
$383$	−8689.24	−1.15927	−0.579633	−	0.814877i	$-0.696804\pi$
$383$	−8689.24	−1.15927	−0.579633	+	0.814877i	$0.696804\pi$
$384$	0	0
$385$	3461.37	0.458202
$386$	−4180.71	−0.551276
$387$	0	0
$388$	18339.2	2.39957
$389$	10604.5	1.38219	0.691094	−	0.722764i	$-0.257129\pi$
$389$	10604.5	1.38219	0.691094	+	0.722764i	$0.257129\pi$
$390$	0	0
$391$	4174.19	0.539892
$392$	−2067.39	−0.266375
$393$	0	0
$394$	−12413.1	−1.58721
$395$	2486.21	0.316695
$396$	0	0
$397$	2918.97	0.369015	0.184508	−	0.982831i	$-0.440931\pi$
$397$	2918.97	0.369015	0.184508	+	0.982831i	$0.440931\pi$
$398$	19500.8	2.45600
$399$	0	0
$400$	10401.1	1.30014
$401$	4258.03	0.530264	0.265132	−	0.964212i	$-0.414585\pi$
$401$	4258.03	0.530264	0.265132	+	0.964212i	$0.414585\pi$
$402$	0	0
$403$	−14392.7	−1.77903
$404$	−23101.0	−2.84484
$405$	0	0
$406$	199.233	0.0243541
$407$	−2790.74	−0.339881
$408$	0	0
$409$	−4616.66	−0.558139	−0.279070	−	0.960271i	$-0.590026\pi$
$409$	−4616.66	−0.558139	−0.279070	+	0.960271i	$0.590026\pi$
$410$	6623.08	0.797782
$411$	0	0
$412$	23917.6	2.86003
$413$	29.4416	0.00350781
$414$	0	0
$415$	4554.63	0.538742
$416$	3321.45	0.391460
$417$	0	0
$418$	−7510.84	−0.878869
$419$	1995.13	0.232622	0.116311	−	0.993213i	$-0.462893\pi$
$419$	1995.13	0.232622	0.116311	+	0.993213i	$0.462893\pi$
$420$	0	0
$421$	16563.5	1.91747	0.958736	−	0.284297i	$-0.0917602\pi$
$421$	16563.5	1.91747	0.958736	+	0.284297i	$0.0917602\pi$
$422$	331.032	0.0381857
$423$	0	0
$424$	−20997.5	−2.40502
$425$	14497.2	1.65463
$426$	0	0
$427$	−4384.74	−0.496938
$428$	−17565.8	−1.98382
$429$	0	0
$430$	7294.82	0.818111
$431$	1439.27	0.160852	0.0804258	−	0.996761i	$-0.474372\pi$
$431$	1439.27	0.160852	0.0804258	+	0.996761i	$0.474372\pi$
$432$	0	0
$433$	−6467.98	−0.717856	−0.358928	−	0.933365i	$-0.616858\pi$
$433$	−6467.98	−0.717856	−0.358928	+	0.933365i	$0.616858\pi$
$434$	6395.04	0.707308
$435$	0	0
$436$	31939.6	3.50833
$437$	−1931.52	−0.211435
$438$	0	0
$439$	−12510.0	−1.36007	−0.680035	−	0.733180i	$-0.738035\pi$
$439$	−12510.0	−1.36007	−0.680035	+	0.733180i	$0.738035\pi$
$440$	20863.0	2.26047
$441$	0	0
$442$	41330.9	4.44776
$443$	7021.16	0.753015	0.376507	−	0.926414i	$-0.377125\pi$
$443$	7021.16	0.753015	0.376507	+	0.926414i	$0.377125\pi$
$444$	0	0
$445$	1793.01	0.191004
$446$	−12318.1	−1.30780
$447$	0	0
$448$	2822.85	0.297695
$449$	5184.19	0.544894	0.272447	−	0.962171i	$-0.412167\pi$
$449$	5184.19	0.544894	0.272447	+	0.962171i	$0.412167\pi$
$450$	0	0
$451$	2538.87	0.265079
$452$	25451.9	2.64857
$453$	0	0
$454$	−19048.3	−1.96912
$455$	8813.76	0.908122
$456$	0	0
$457$	2576.28	0.263705	0.131853	−	0.991269i	$-0.457907\pi$
$457$	2576.28	0.263705	0.131853	+	0.991269i	$0.457907\pi$
$458$	19866.1	2.02682
$459$	0	0
$460$	10402.7	1.05441
$461$	−17904.4	−1.80888	−0.904438	−	0.426606i	$-0.859709\pi$
$461$	−17904.4	−1.80888	−0.904438	+	0.426606i	$0.859709\pi$
$462$	0	0
$463$	11263.9	1.13062	0.565309	−	0.824879i	$-0.308757\pi$
$463$	11263.9	1.13062	0.565309	+	0.824879i	$0.308757\pi$
$464$	441.208	0.0441434
$465$	0	0
$466$	−28345.2	−2.81774
$467$	9817.00	0.972755	0.486377	−	0.873749i	$-0.338318\pi$
$467$	9817.00	0.972755	0.486377	+	0.873749i	$0.338318\pi$
$468$	0	0
$469$	4750.34	0.467698
$470$	−19018.6	−1.86651
$471$	0	0
$472$	177.456	0.0173052
$473$	2796.38	0.271834
$474$	0	0
$475$	−6708.27	−0.647993
$476$	−12372.9	−1.19140
$477$	0	0
$478$	−30303.0	−2.89964
$479$	7351.73	0.701271	0.350636	−	0.936512i	$-0.385966\pi$
$479$	7351.73	0.701271	0.350636	+	0.936512i	$0.385966\pi$
$480$	0	0
$481$	−7106.11	−0.673619
$482$	−2440.27	−0.230604
$483$	0	0
$484$	−6482.11	−0.608763
$485$	17916.8	1.67745
$486$	0	0
$487$	15312.1	1.42476	0.712378	−	0.701796i	$-0.247618\pi$
$487$	15312.1	1.42476	0.712378	+	0.701796i	$0.247618\pi$
$488$	−26428.5	−2.45156
$489$	0	0
$490$	−3916.18	−0.361051
$491$	−17978.9	−1.65250	−0.826251	−	0.563303i	$-0.809531\pi$
$491$	−17978.9	−1.65250	−0.826251	+	0.563303i	$0.809531\pi$
$492$	0	0
$493$	614.962	0.0561795
$494$	−19125.0	−1.74185
$495$	0	0
$496$	14162.0	1.28204
$497$	−5229.50	−0.471982
$498$	0	0
$499$	−18815.0	−1.68793	−0.843965	−	0.536398i	$-0.819784\pi$
$499$	−18815.0	−1.68793	−0.843965	+	0.536398i	$0.819784\pi$
$500$	2799.26	0.250374
$501$	0	0
$502$	9206.64	0.818551
$503$	−6228.19	−0.552090	−0.276045	−	0.961145i	$-0.589024\pi$
$503$	−6228.19	−0.552090	−0.276045	+	0.961145i	$0.589024\pi$
$504$	0	0
$505$	−22568.9	−1.98872
$506$	5918.81	0.520006
$507$	0	0
$508$	24305.4	2.12279
$509$	−11803.4	−1.02785	−0.513925	−	0.857835i	$-0.671809\pi$
$509$	−11803.4	−1.02785	−0.513925	+	0.857835i	$0.671809\pi$
$510$	0	0
$511$	162.840	0.0140971
$512$	22641.4	1.95433
$513$	0	0
$514$	−18677.8	−1.60281
$515$	23366.7	1.99934
$516$	0	0
$517$	−7290.53	−0.620188
$518$	3157.43	0.267817
$519$	0	0
$520$	53123.9	4.48007
$521$	−7277.82	−0.611990	−0.305995	−	0.952033i	$-0.598989\pi$
$521$	−7277.82	−0.611990	−0.305995	+	0.952033i	$0.598989\pi$
$522$	0	0
$523$	18929.2	1.58263	0.791316	−	0.611408i	$-0.209397\pi$
$523$	18929.2	1.58263	0.791316	+	0.611408i	$0.209397\pi$
$524$	11213.3	0.934837
$525$	0	0
$526$	−201.726	−0.0167218
$527$	19739.2	1.63160
$528$	0	0
$529$	−10644.9	−0.874899
$530$	−39774.7	−3.25982
$531$	0	0
$532$	5725.28	0.466583
$533$	6464.78	0.525367
$534$	0	0
$535$	−17161.3	−1.38682
$536$	28632.1	2.30731
$537$	0	0
$538$	−6952.76	−0.557165
$539$	−1501.22	−0.119967
$540$	0	0
$541$	3018.24	0.239860	0.119930	−	0.992782i	$-0.461733\pi$
$541$	3018.24	0.239860	0.119930	+	0.992782i	$0.461733\pi$
$542$	1102.67	0.0873873
$543$	0	0
$544$	−4555.29	−0.359019
$545$	31204.0	2.45254
$546$	0	0
$547$	−9453.37	−0.738934	−0.369467	−	0.929244i	$-0.620460\pi$
$547$	−9453.37	−0.738934	−0.369467	+	0.929244i	$0.620460\pi$
$548$	30066.6	2.34376
$549$	0	0
$550$	20556.3	1.59368
$551$	−284.561	−0.0220013
$552$	0	0
$553$	−1078.28	−0.0829173
$554$	28617.8	2.19468
$555$	0	0
$556$	−48217.8	−3.67786
$557$	6175.74	0.469793	0.234896	−	0.972020i	$-0.424525\pi$
$557$	6175.74	0.469793	0.234896	+	0.972020i	$0.424525\pi$
$558$	0	0
$559$	7120.47	0.538755
$560$	−8672.52	−0.654430
$561$	0	0
$562$	−18669.3	−1.40128
$563$	12659.1	0.947634	0.473817	−	0.880623i	$-0.342876\pi$
$563$	12659.1	0.947634	0.473817	+	0.880623i	$0.342876\pi$
$564$	0	0
$565$	24865.7	1.85152
$566$	23789.1	1.76666
$567$	0	0
$568$	−31520.2	−2.32845
$569$	19211.0	1.41541	0.707704	−	0.706509i	$-0.249731\pi$
$569$	19211.0	1.41541	0.707704	+	0.706509i	$0.249731\pi$
$570$	0	0
$571$	−7560.25	−0.554092	−0.277046	−	0.960857i	$-0.589355\pi$
$571$	−7560.25	−0.554092	−0.277046	+	0.960857i	$0.589355\pi$
$572$	39484.8	2.88626
$573$	0	0
$574$	−2872.47	−0.208876
$575$	5286.36	0.383402
$576$	0	0
$577$	−14126.1	−1.01920	−0.509598	−	0.860413i	$-0.670206\pi$
$577$	−14126.1	−1.01920	−0.509598	+	0.860413i	$0.670206\pi$
$578$	−32356.2	−2.32844
$579$	0	0
$580$	1532.58	0.109719
$581$	−1975.37	−0.141054
$582$	0	0
$583$	−15247.1	−1.08314
$584$	981.497	0.0695456
$585$	0	0
$586$	−14790.6	−1.04265
$587$	11622.3	0.817212	0.408606	−	0.912711i	$-0.366015\pi$
$587$	11622.3	0.817212	0.408606	+	0.912711i	$0.366015\pi$
$588$	0	0
$589$	−9133.92	−0.638976
$590$	336.147	0.0234559
$591$	0	0
$592$	6992.23	0.485437
$593$	−22182.5	−1.53613	−0.768064	−	0.640373i	$-0.778780\pi$
$593$	−22182.5	−1.53613	−0.768064	+	0.640373i	$0.778780\pi$
$594$	0	0
$595$	−12087.9	−0.832865
$596$	−16264.1	−1.11779
$597$	0	0
$598$	15071.2	1.03061
$599$	−22406.4	−1.52838	−0.764189	−	0.644992i	$-0.776861\pi$
$599$	−22406.4	−1.52838	−0.764189	+	0.644992i	$0.776861\pi$
$600$	0	0
$601$	11255.9	0.763955	0.381977	−	0.924172i	$-0.375243\pi$
$601$	11255.9	0.763955	0.381977	+	0.924172i	$0.375243\pi$
$602$	−3163.81	−0.214198
$603$	0	0
$604$	−2145.43	−0.144531
$605$	−6332.81	−0.425563
$606$	0	0
$607$	−12312.6	−0.823316	−0.411658	−	0.911339i	$-0.635050\pi$
$607$	−12312.6	−0.823316	−0.411658	+	0.911339i	$0.635050\pi$
$608$	2107.87	0.140601
$609$	0	0
$610$	−50062.5	−3.32290
$611$	−18564.0	−1.22916
$612$	0	0
$613$	−26835.4	−1.76814	−0.884070	−	0.467354i	$-0.845207\pi$
$613$	−26835.4	−1.76814	−0.884070	+	0.467354i	$0.845207\pi$
$614$	14268.7	0.937850
$615$	0	0
$616$	−9048.41	−0.591836
$617$	−21765.5	−1.42017	−0.710084	−	0.704117i	$-0.751343\pi$
$617$	−21765.5	−1.42017	−0.710084	+	0.704117i	$0.751343\pi$
$618$	0	0
$619$	7115.28	0.462015	0.231007	−	0.972952i	$-0.425798\pi$
$619$	7115.28	0.462015	0.231007	+	0.972952i	$0.425798\pi$
$620$	49193.2	3.18653
$621$	0	0
$622$	−33660.8	−2.16990
$623$	−777.640	−0.0500088
$624$	0	0
$625$	−14202.5	−0.908960
$626$	−4663.33	−0.297738
$627$	0	0
$628$	−3585.44	−0.227826
$629$	9745.88	0.617796
$630$	0	0
$631$	−11639.8	−0.734345	−0.367173	−	0.930153i	$-0.619674\pi$
$631$	−11639.8	−0.734345	−0.367173	+	0.930153i	$0.619674\pi$
$632$	−6499.23	−0.409059
$633$	0	0
$634$	−17990.0	−1.12693
$635$	23745.6	1.48396
$636$	0	0
$637$	−3822.58	−0.237765
$638$	871.988	0.0541102
$639$	0	0
$640$	37727.1	2.33015
$641$	−25675.6	−1.58210	−0.791048	−	0.611754i	$-0.790464\pi$
$641$	−25675.6	−1.58210	−0.791048	+	0.611754i	$0.790464\pi$
$642$	0	0
$643$	−17146.8	−1.05164	−0.525821	−	0.850595i	$-0.676242\pi$
$643$	−17146.8	−1.05164	−0.525821	+	0.850595i	$0.676242\pi$
$644$	−4511.72	−0.276066
$645$	0	0
$646$	26229.5	1.59750
$647$	−13104.3	−0.796263	−0.398132	−	0.917328i	$-0.630341\pi$
$647$	−13104.3	−0.796263	−0.398132	+	0.917328i	$0.630341\pi$
$648$	0	0
$649$	128.858	0.00779369
$650$	52343.0	3.15856
$651$	0	0
$652$	672.677	0.0404050
$653$	−16746.1	−1.00356	−0.501780	−	0.864995i	$-0.667321\pi$
$653$	−16746.1	−1.00356	−0.501780	+	0.864995i	$0.667321\pi$
$654$	0	0
$655$	10955.0	0.653509
$656$	−6361.18	−0.378601
$657$	0	0
$658$	8248.47	0.488691
$659$	−24267.1	−1.43446	−0.717232	−	0.696834i	$-0.754591\pi$
$659$	−24267.1	−1.43446	−0.717232	+	0.696834i	$0.754591\pi$
$660$	0	0
$661$	14467.9	0.851340	0.425670	−	0.904878i	$-0.360038\pi$
$661$	14467.9	0.851340	0.425670	+	0.904878i	$0.360038\pi$
$662$	−13619.1	−0.799582
$663$	0	0
$664$	−11906.3	−0.695866
$665$	5593.42	0.326170
$666$	0	0
$667$	224.244	0.0130176
$668$	28514.1	1.65156
$669$	0	0
$670$	54236.7	3.12738
$671$	−19190.8	−1.10410
$672$	0	0
$673$	−14958.3	−0.856759	−0.428379	−	0.903599i	$-0.640915\pi$
$673$	−14958.3	−0.856759	−0.428379	+	0.903599i	$0.640915\pi$
$674$	−28880.4	−1.65049
$675$	0	0
$676$	64245.6	3.65530
$677$	−25723.7	−1.46033	−0.730165	−	0.683271i	$-0.760557\pi$
$677$	−25723.7	−1.46033	−0.730165	+	0.683271i	$0.760557\pi$
$678$	0	0
$679$	−7770.63	−0.439189
$680$	−72858.3	−4.10881
$681$	0	0
$682$	27989.3	1.57151
$683$	−2949.64	−0.165248	−0.0826242	−	0.996581i	$-0.526330\pi$
$683$	−2949.64	−0.165248	−0.0826242	+	0.996581i	$0.526330\pi$
$684$	0	0
$685$	29374.1	1.63843
$686$	1698.47	0.0945305
$687$	0	0
$688$	−7006.36	−0.388249
$689$	−38824.1	−2.14670
$690$	0	0
$691$	−8119.91	−0.447028	−0.223514	−	0.974701i	$-0.571753\pi$
$691$	−8119.91	−0.447028	−0.223514	+	0.974701i	$0.571753\pi$
$692$	−9629.38	−0.528980
$693$	0	0
$694$	−16885.5	−0.923579
$695$	−47107.3	−2.57105
$696$	0	0
$697$	−8866.31	−0.481830
$698$	−29450.8	−1.59703
$699$	0	0
$700$	−15669.5	−0.846071
$701$	−15870.9	−0.855115	−0.427558	−	0.903988i	$-0.640626\pi$
$701$	−15870.9	−0.855115	−0.427558	+	0.903988i	$0.640626\pi$
$702$	0	0
$703$	−4509.70	−0.241944
$704$	12354.9	0.661422
$705$	0	0
$706$	2632.85	0.140352
$707$	9788.28	0.520688
$708$	0	0
$709$	−14873.6	−0.787855	−0.393928	−	0.919141i	$-0.628884\pi$
$709$	−14873.6	−0.787855	−0.393928	+	0.919141i	$0.628884\pi$
$710$	−59707.5	−3.15603
$711$	0	0
$712$	−4687.13	−0.246710
$713$	7197.86	0.378067
$714$	0	0
$715$	38575.4	2.01768
$716$	−61123.8	−3.19037
$717$	0	0
$718$	37415.7	1.94476
$719$	31809.4	1.64992	0.824959	−	0.565192i	$-0.191198\pi$
$719$	31809.4	1.64992	0.824959	+	0.565192i	$0.191198\pi$
$720$	0	0
$721$	−10134.3	−0.523468
$722$	21827.3	1.12511
$723$	0	0
$724$	47462.9	2.43639
$725$	778.812	0.0398956
$726$	0	0
$727$	16771.8	0.855615	0.427808	−	0.903870i	$-0.359286\pi$
$727$	16771.8	0.855615	0.427808	+	0.903870i	$0.359286\pi$
$728$	−23040.2	−1.17297
$729$	0	0
$730$	1859.21	0.0942638
$731$	−9765.58	−0.494108
$732$	0	0
$733$	−31788.3	−1.60181	−0.800906	−	0.598790i	$-0.795648\pi$
$733$	−31788.3	−1.60181	−0.800906	+	0.598790i	$0.795648\pi$
$734$	48048.5	2.41622
$735$	0	0
$736$	−1661.07	−0.0831902
$737$	20791.0	1.03914
$738$	0	0
$739$	−17098.2	−0.851107	−0.425553	−	0.904933i	$-0.639921\pi$
$739$	−17098.2	−0.851107	−0.425553	+	0.904933i	$0.639921\pi$
$740$	24288.2	1.20656
$741$	0	0
$742$	17250.5	0.853487
$743$	−617.901	−0.0305095	−0.0152548	−	0.999884i	$-0.504856\pi$
$743$	−617.901	−0.0305095	−0.0152548	+	0.999884i	$0.504856\pi$
$744$	0	0
$745$	−15889.6	−0.781407
$746$	−70.2888	−0.00344967
$747$	0	0
$748$	−54152.6	−2.64708
$749$	7442.95	0.363097
$750$	0	0
$751$	26469.0	1.28611	0.643053	−	0.765821i	$-0.277667\pi$
$751$	26469.0	1.28611	0.643053	+	0.765821i	$0.277667\pi$
$752$	18266.5	0.885787
$753$	0	0
$754$	2220.36	0.107242
$755$	−2096.02	−0.101036
$756$	0	0
$757$	40262.5	1.93311	0.966557	−	0.256453i	$-0.0825538\pi$
$757$	40262.5	1.93311	0.966557	+	0.256453i	$0.0825538\pi$
$758$	12968.4	0.621417
$759$	0	0
$760$	33713.7	1.60911
$761$	38891.3	1.85257	0.926286	−	0.376820i	$-0.122983\pi$
$761$	38891.3	1.85257	0.926286	+	0.376820i	$0.122983\pi$
$762$	0	0
$763$	−13533.4	−0.642124
$764$	−4514.91	−0.213801
$765$	0	0
$766$	43027.5	2.02956
$767$	328.113	0.0154465
$768$	0	0
$769$	27820.5	1.30459	0.652296	−	0.757964i	$-0.273806\pi$
$769$	27820.5	1.30459	0.652296	+	0.757964i	$0.273806\pi$
$770$	−17140.1	−0.802188
$771$	0	0
$772$	13947.9	0.650252
$773$	25952.0	1.20754	0.603770	−	0.797159i	$-0.293665\pi$
$773$	25952.0	1.20754	0.603770	+	0.797159i	$0.293665\pi$
$774$	0	0
$775$	24998.5	1.15868
$776$	−46836.6	−2.16667
$777$	0	0
$778$	−52511.7	−2.41984
$779$	4102.70	0.188696
$780$	0	0
$781$	−22888.1	−1.04866
$782$	−20669.8	−0.945206
$783$	0	0
$784$	3761.32	0.171343
$785$	−3502.86	−0.159264
$786$	0	0
$787$	42116.3	1.90760	0.953802	−	0.300437i	$-0.0971324\pi$
$787$	42116.3	1.90760	0.953802	+	0.300437i	$0.0971324\pi$
$788$	41413.0	1.87218
$789$	0	0
$790$	−12311.2	−0.554448
$791$	−10784.4	−0.484765
$792$	0	0
$793$	−48866.0	−2.18825
$794$	−14454.2	−0.646046
$795$	0	0
$796$	−65059.5	−2.89695
$797$	12971.4	0.576501	0.288250	−	0.957555i	$-0.406926\pi$
$797$	12971.4	0.576501	0.288250	+	0.957555i	$0.406926\pi$
$798$	0	0
$799$	25460.2	1.12730
$800$	−5769.00	−0.254956
$801$	0	0
$802$	−21085.0	−0.928349
$803$	712.705	0.0313211
$804$	0	0
$805$	−4407.81	−0.192987
$806$	71269.8	3.11461
$807$	0	0
$808$	58997.7	2.56873
$809$	−9053.03	−0.393433	−0.196717	−	0.980460i	$-0.563028\pi$
$809$	−9053.03	−0.393433	−0.196717	+	0.980460i	$0.563028\pi$
$810$	0	0
$811$	−2374.82	−0.102825	−0.0514127	−	0.998677i	$-0.516372\pi$
$811$	−2374.82	−0.102825	−0.0514127	+	0.998677i	$0.516372\pi$
$812$	−664.689	−0.0287266
$813$	0	0
$814$	13819.2	0.595040
$815$	657.184	0.0282456
$816$	0	0
$817$	4518.82	0.193505
$818$	22860.8	0.977152
$819$	0	0
$820$	−22096.2	−0.941016
$821$	5149.86	0.218918	0.109459	−	0.993991i	$-0.465088\pi$
$821$	5149.86	0.218918	0.109459	+	0.993991i	$0.465088\pi$
$822$	0	0
$823$	2874.96	0.121768	0.0608839	−	0.998145i	$-0.480608\pi$
$823$	2874.96	0.121768	0.0608839	+	0.998145i	$0.480608\pi$
$824$	−61083.2	−2.58244
$825$	0	0
$826$	−145.789	−0.00614122
$827$	515.502	0.0216757	0.0108378	−	0.999941i	$-0.496550\pi$
$827$	515.502	0.0216757	0.0108378	+	0.999941i	$0.496550\pi$
$828$	0	0
$829$	−14698.3	−0.615793	−0.307897	−	0.951420i	$-0.599625\pi$
$829$	−14698.3	−0.615793	−0.307897	+	0.951420i	$0.599625\pi$
$830$	−22553.7	−0.943193
$831$	0	0
$832$	31459.4	1.31089
$833$	5242.59	0.218061
$834$	0	0
$835$	27857.4	1.15454
$836$	25058.0	1.03666
$837$	0	0
$838$	−9879.53	−0.407259
$839$	−4411.10	−0.181511	−0.0907557	−	0.995873i	$-0.528928\pi$
$839$	−4411.10	−0.181511	−0.0907557	+	0.995873i	$0.528928\pi$
$840$	0	0
$841$	−24356.0	−0.998645
$842$	−82019.4	−3.35698
$843$	0	0
$844$	−1104.40	−0.0450416
$845$	62765.9	2.55528
$846$	0	0
$847$	2746.58	0.111421
$848$	38201.9	1.54700
$849$	0	0
$850$	−71787.4	−2.89681
$851$	3553.80	0.143153
$852$	0	0
$853$	21900.4	0.879080	0.439540	−	0.898223i	$-0.355141\pi$
$853$	21900.4	0.879080	0.439540	+	0.898223i	$0.355141\pi$
$854$	21712.4	0.870004
$855$	0	0
$856$	44861.5	1.79128
$857$	23803.2	0.948775	0.474388	−	0.880316i	$-0.342670\pi$
$857$	23803.2	0.948775	0.474388	+	0.880316i	$0.342670\pi$
$858$	0	0
$859$	44488.1	1.76707	0.883536	−	0.468363i	$-0.155156\pi$
$859$	44488.1	1.76707	0.883536	+	0.468363i	$0.155156\pi$
$860$	−24337.3	−0.964994
$861$	0	0
$862$	−7126.98	−0.281608
$863$	−34544.3	−1.36258	−0.681288	−	0.732016i	$-0.738580\pi$
$863$	−34544.3	−1.36258	−0.681288	+	0.732016i	$0.738580\pi$
$864$	0	0
$865$	−9407.60	−0.369790
$866$	32028.2	1.25677
$867$	0	0
$868$	−21335.4	−0.834298
$869$	−4719.35	−0.184227
$870$	0	0
$871$	52940.4	2.05949
$872$	−81570.8	−3.16782
$873$	0	0
$874$	9564.52	0.370166
$875$	−1186.10	−0.0458256
$876$	0	0
$877$	33204.5	1.27849	0.639246	−	0.769003i	$-0.279247\pi$
$877$	33204.5	1.27849	0.639246	+	0.769003i	$0.279247\pi$
$878$	61947.2	2.38111
$879$	0	0
$880$	−37957.2	−1.45402
$881$	23411.1	0.895278	0.447639	−	0.894214i	$-0.352265\pi$
$881$	23411.1	0.895278	0.447639	+	0.894214i	$0.352265\pi$
$882$	0	0
$883$	−12881.8	−0.490947	−0.245473	−	0.969403i	$-0.578943\pi$
$883$	−12881.8	−0.490947	−0.245473	+	0.969403i	$0.578943\pi$
$884$	−137890.	−5.24631
$885$	0	0
$886$	−34767.5	−1.31833
$887$	5623.57	0.212876	0.106438	−	0.994319i	$-0.466055\pi$
$887$	5623.57	0.212876	0.106438	+	0.994319i	$0.466055\pi$
$888$	0	0
$889$	−10298.6	−0.388532
$890$	−8878.66	−0.334397
$891$	0	0
$892$	41096.3	1.54261
$893$	−11781.2	−0.441480
$894$	0	0
$895$	−59716.0	−2.23026
$896$	−16362.5	−0.610081
$897$	0	0
$898$	−25671.2	−0.953962
$899$	1060.42	0.0393405
$900$	0	0
$901$	53246.4	1.96881
$902$	−12572.0	−0.464082
$903$	0	0
$904$	−65001.7	−2.39151
$905$	46369.7	1.70318
$906$	0	0
$907$	36563.0	1.33854	0.669269	−	0.743020i	$-0.266608\pi$
$907$	36563.0	1.33854	0.669269	+	0.743020i	$0.266608\pi$
$908$	63549.6	2.32265
$909$	0	0
$910$	−43644.1	−1.58988
$911$	19926.9	0.724707	0.362354	−	0.932041i	$-0.381973\pi$
$911$	19926.9	0.724707	0.362354	+	0.932041i	$0.381973\pi$
$912$	0	0
$913$	−8645.67	−0.313395
$914$	−12757.3	−0.461677
$915$	0	0
$916$	−66278.2	−2.39071
$917$	−4751.26	−0.171102
$918$	0	0
$919$	48298.1	1.73363	0.866816	−	0.498628i	$-0.166163\pi$
$919$	48298.1	1.73363	0.866816	+	0.498628i	$0.166163\pi$
$920$	−26567.6	−0.952073
$921$	0	0
$922$	88659.3	3.16685
$923$	−58280.4	−2.07836
$924$	0	0
$925$	12342.6	0.438725
$926$	−55776.5	−1.97941
$927$	0	0
$928$	−244.718	−0.00865652
$929$	15616.0	0.551501	0.275751	−	0.961229i	$-0.411074\pi$
$929$	15616.0	0.551501	0.275751	+	0.961229i	$0.411074\pi$
$930$	0	0
$931$	−2425.90	−0.0853981
$932$	94566.4	3.32363
$933$	0	0
$934$	−48611.9	−1.70303
$935$	−52905.4	−1.85047
$936$	0	0
$937$	25030.1	0.872678	0.436339	−	0.899782i	$-0.356275\pi$
$937$	25030.1	0.872678	0.436339	+	0.899782i	$0.356275\pi$
$938$	−23522.8	−0.818813
$939$	0	0
$940$	63450.6	2.20163
$941$	16012.9	0.554734	0.277367	−	0.960764i	$-0.410538\pi$
$941$	16012.9	0.554734	0.277367	+	0.960764i	$0.410538\pi$
$942$	0	0
$943$	−3233.07	−0.111647
$944$	−322.855	−0.0111314
$945$	0	0
$946$	−13847.1	−0.475908
$947$	−1213.23	−0.0416310	−0.0208155	−	0.999783i	$-0.506626\pi$
$947$	−1213.23	−0.0416310	−0.0208155	+	0.999783i	$0.506626\pi$
$948$	0	0
$949$	1814.78	0.0620760
$950$	33218.1	1.13446
$951$	0	0
$952$	31599.1	1.07577
$953$	−20143.9	−0.684705	−0.342352	−	0.939572i	$-0.611224\pi$
$953$	−20143.9	−0.684705	−0.342352	+	0.939572i	$0.611224\pi$
$954$	0	0
$955$	−4410.92	−0.149460
$956$	101098.	3.42024
$957$	0	0
$958$	−36404.4	−1.22774
$959$	−12739.7	−0.428975
$960$	0	0
$961$	4246.81	0.142553
$962$	35188.1	1.17932
$963$	0	0
$964$	8141.32	0.272007
$965$	13626.6	0.454566
$966$	0	0
$967$	27402.7	0.911285	0.455642	−	0.890163i	$-0.349410\pi$
$967$	27402.7	0.911285	0.455642	+	0.890163i	$0.349410\pi$
$968$	16554.7	0.549678
$969$	0	0
$970$	−88720.7	−2.93675
$971$	−25837.8	−0.853938	−0.426969	−	0.904266i	$-0.640419\pi$
$971$	−25837.8	−0.853938	−0.426969	+	0.904266i	$0.640419\pi$
$972$	0	0
$973$	20430.7	0.673154
$974$	−75822.5	−2.49436
$975$	0	0
$976$	48082.9	1.57694
$977$	−22062.0	−0.722441	−0.361221	−	0.932480i	$-0.617640\pi$
$977$	−22062.0	−0.722441	−0.361221	+	0.932480i	$0.617640\pi$
$978$	0	0
$979$	−3403.52	−0.111110
$980$	13065.3	0.425874
$981$	0	0
$982$	89028.3	2.89308
$983$	−11801.8	−0.382928	−0.191464	−	0.981500i	$-0.561324\pi$
$983$	−11801.8	−0.382928	−0.191464	+	0.981500i	$0.561324\pi$
$984$	0	0
$985$	40459.2	1.30877
$986$	−3045.18	−0.0983552
$987$	0	0
$988$	63805.7	2.05458
$989$	−3560.99	−0.114492
$990$	0	0
$991$	−46761.9	−1.49893	−0.749465	−	0.662044i	$-0.769689\pi$
$991$	−46761.9	−1.49893	−0.749465	+	0.662044i	$0.769689\pi$
$992$	−7855.02	−0.251408
$993$	0	0
$994$	25895.5	0.826313
$995$	−63561.1	−2.02515
$996$	0	0
$997$	−38261.2	−1.21539	−0.607695	−	0.794171i	$-0.707906\pi$
$997$	−38261.2	−1.21539	−0.607695	+	0.794171i	$0.707906\pi$
$998$	93168.6	2.95511
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.4.a.i.1.1		8
3.2	odd	2		567.4.a.g.1.8		8
9.2	odd	6		189.4.f.b.64.1		16
9.4	even	3		63.4.f.b.43.8	yes	16
9.5	odd	6		189.4.f.b.127.1		16
9.7	even	3		63.4.f.b.22.8	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.f.b.22.8	✓	16	9.7	even	3
63.4.f.b.43.8	yes	16	9.4	even	3
189.4.f.b.64.1		16	9.2	odd	6
189.4.f.b.127.1		16	9.5	odd	6
567.4.a.g.1.8		8	3.2	odd	2
567.4.a.i.1.1		8	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:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	567.a (trivial)

\(f(q)\)	\(=\)	\(q-4.95181 q^{2} +16.5205 q^{4} +16.1400 q^{5} -7.00000 q^{7} -42.1917 q^{8} +O(q^{10})\)	\(q-4.95181 q^{2} +16.5205 q^{4} +16.1400 q^{5} -7.00000 q^{7} -42.1917 q^{8} -79.9221 q^{10} -30.6371 q^{11} -78.0119 q^{13} +34.6627 q^{14} +76.7617 q^{16} +106.992 q^{17} -49.5081 q^{19} +266.639 q^{20} +151.709 q^{22} +39.0142 q^{23} +135.498 q^{25} +386.300 q^{26} -115.643 q^{28} +5.74776 q^{29} +184.493 q^{31} -42.5762 q^{32} -529.803 q^{34} -112.980 q^{35} +91.0901 q^{37} +245.155 q^{38} -680.972 q^{40} -82.8692 q^{41} -91.2742 q^{43} -506.139 q^{44} -193.191 q^{46} +237.964 q^{47} +49.0000 q^{49} -670.962 q^{50} -1288.79 q^{52} +497.669 q^{53} -494.481 q^{55} +295.342 q^{56} -28.4618 q^{58} -4.20594 q^{59} +626.391 q^{61} -913.577 q^{62} -403.264 q^{64} -1259.11 q^{65} -678.620 q^{67} +1767.55 q^{68} +559.454 q^{70} +747.071 q^{71} -23.2628 q^{73} -451.061 q^{74} -817.897 q^{76} +214.460 q^{77} +154.041 q^{79} +1238.93 q^{80} +410.353 q^{82} +282.196 q^{83} +1726.84 q^{85} +451.973 q^{86} +1292.63 q^{88} +111.091 q^{89} +546.083 q^{91} +644.532 q^{92} -1178.35 q^{94} -799.059 q^{95} +1110.09 q^{97} -242.639 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8}+O(q^{10}) \) 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8	\( 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8} - 14 q^{10} + 24 q^{11} + 68 q^{13} - 21 q^{14} + 103 q^{16} + 168 q^{17} + 176 q^{19} + 330 q^{20} + 151 q^{22} + 228 q^{23} + 244 q^{25} + 795 q^{26} - 301 q^{28} + 618 q^{29} + 72 q^{31} + 786 q^{32} - 261 q^{34} - 210 q^{35} + 210 q^{37} + 1032 q^{38} - 375 q^{40} + 420 q^{41} - 2 q^{43} + 387 q^{44} + 402 q^{46} + 570 q^{47} + 392 q^{49} + 1110 q^{50} - 431 q^{52} + 528 q^{53} - 838 q^{55} - 42 q^{56} + 37 q^{58} - 150 q^{59} + 578 q^{61} + 1170 q^{62} - 112 q^{64} - 366 q^{65} - 898 q^{67} + 2526 q^{68} + 98 q^{70} + 882 q^{71} + 972 q^{73} - 222 q^{74} + 1423 q^{76} - 168 q^{77} - 158 q^{79} + 2475 q^{80} - 211 q^{82} + 2958 q^{83} - 774 q^{85} - 114 q^{86} + 1317 q^{88} + 4380 q^{89} - 476 q^{91} + 4629 q^{92} - 3234 q^{94} + 930 q^{95} - 60 q^{97} + 147 q^{98}+O(q^{100}) \) 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8 - 14 * q^10 + 24 * q^11 + 68 * q^13 - 21 * q^14 + 103 * q^16 + 168 * q^17 + 176 * q^19 + 330 * q^20 + 151 * q^22 + 228 * q^23 + 244 * q^25 + 795 * q^26 - 301 * q^28 + 618 * q^29 + 72 * q^31 + 786 * q^32 - 261 * q^34 - 210 * q^35 + 210 * q^37 + 1032 * q^38 - 375 * q^40 + 420 * q^41 - 2 * q^43 + 387 * q^44 + 402 * q^46 + 570 * q^47 + 392 * q^49 + 1110 * q^50 - 431 * q^52 + 528 * q^53 - 838 * q^55 - 42 * q^56 + 37 * q^58 - 150 * q^59 + 578 * q^61 + 1170 * q^62 - 112 * q^64 - 366 * q^65 - 898 * q^67 + 2526 * q^68 + 98 * q^70 + 882 * q^71 + 972 * q^73 - 222 * q^74 + 1423 * q^76 - 168 * q^77 - 158 * q^79 + 2475 * q^80 - 211 * q^82 + 2958 * q^83 - 774 * q^85 - 114 * q^86 + 1317 * q^88 + 4380 * q^89 - 476 * q^91 + 4629 * q^92 - 3234 * q^94 + 930 * q^95 - 60 * q^97 + 147 * q^98

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