LMFDB - Embedded newform 539.4.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,4,Mod(1,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, 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("539.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	539.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:	$31.8020294931$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.522072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 5x + 1 $ x^4 - x^3 - 12x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.148103$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.60395 q^{2} -2.77399 q^{3} +13.1964 q^{4} -1.84418 q^{5} -12.7713 q^{6} +23.9238 q^{8} -19.3050 q^{9} -8.49053 q^{10} -11.0000 q^{11} -36.6066 q^{12} -24.6401 q^{13} +5.11574 q^{15} +4.57310 q^{16} -17.8800 q^{17} -88.8792 q^{18} -32.1459 q^{19} -24.3365 q^{20} -50.6435 q^{22} +14.1248 q^{23} -66.3643 q^{24} -121.599 q^{25} -113.442 q^{26} +128.450 q^{27} -41.5471 q^{29} +23.5526 q^{30} -175.766 q^{31} -170.336 q^{32} +30.5139 q^{33} -82.3187 q^{34} -254.756 q^{36} +292.877 q^{37} -147.998 q^{38} +68.3513 q^{39} -44.1199 q^{40} -154.296 q^{41} -277.144 q^{43} -145.160 q^{44} +35.6019 q^{45} +65.0300 q^{46} +52.1450 q^{47} -12.6857 q^{48} -559.836 q^{50} +49.5989 q^{51} -325.160 q^{52} +82.3907 q^{53} +591.375 q^{54} +20.2860 q^{55} +89.1723 q^{57} -191.281 q^{58} -712.816 q^{59} +67.5092 q^{60} +647.078 q^{61} -809.217 q^{62} -820.804 q^{64} +45.4408 q^{65} +140.484 q^{66} +260.867 q^{67} -235.951 q^{68} -39.1821 q^{69} +369.025 q^{71} -461.849 q^{72} -1145.77 q^{73} +1348.39 q^{74} +337.314 q^{75} -424.209 q^{76} +314.686 q^{78} +488.885 q^{79} -8.43362 q^{80} +164.917 q^{81} -710.372 q^{82} -548.982 q^{83} +32.9740 q^{85} -1275.96 q^{86} +115.251 q^{87} -263.162 q^{88} -105.039 q^{89} +163.910 q^{90} +186.396 q^{92} +487.572 q^{93} +240.073 q^{94} +59.2829 q^{95} +472.510 q^{96} +1361.91 q^{97} +212.355 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 14 q^{3} + 26 q^{4} - 10 q^{5} - 14 q^{6} - 18 q^{8} + 76 q^{9} + 2 q^{10} - 44 q^{11} - 70 q^{12} - 58 q^{13} + 284 q^{15} + 2 q^{16} - 4 q^{17} - 62 q^{18} - 258 q^{19} - 182 q^{20} + 22 q^{22}+ \cdots - 836 q^{99}+O(q^{100}) $ 4 * q - 2 * q^2 - 14 * q^3 + 26 * q^4 - 10 * q^5 - 14 * q^6 - 18 * q^8 + 76 * q^9 + 2 * q^10 - 44 * q^11 - 70 * q^12 - 58 * q^13 + 284 * q^15 + 2 * q^16 - 4 * q^17 - 62 * q^18 - 258 * q^19 - 182 * q^20 + 22 * q^22 + 8 * q^23 + 498 * q^24 + 80 * q^25 + 482 * q^26 - 428 * q^27 - 396 * q^29 - 628 * q^30 + 56 * q^31 + 134 * q^32 + 154 * q^33 - 472 * q^34 - 418 * q^36 + 84 * q^37 + 942 * q^38 - 412 * q^39 + 1026 * q^40 - 52 * q^41 + 408 * q^43 - 286 * q^44 - 826 * q^45 + 368 * q^46 - 8 * q^47 - 982 * q^48 - 1642 * q^50 - 388 * q^51 - 2030 * q^52 + 624 * q^53 - 92 * q^54 + 110 * q^55 + 48 * q^57 + 864 * q^58 + 238 * q^59 + 1420 * q^60 + 162 * q^61 - 688 * q^62 - 902 * q^64 - 32 * q^65 + 154 * q^66 + 1340 * q^67 + 1384 * q^68 - 2416 * q^69 + 1788 * q^71 - 2622 * q^72 - 1456 * q^73 + 996 * q^74 + 806 * q^75 - 3042 * q^76 - 2632 * q^78 - 1324 * q^79 - 2342 * q^80 + 1444 * q^81 - 1984 * q^82 - 450 * q^83 - 1736 * q^85 - 4380 * q^86 - 588 * q^87 + 198 * q^88 + 3072 * q^89 + 218 * q^90 + 544 * q^92 - 1264 * q^93 + 1696 * q^94 + 24 * q^95 - 862 * q^96 + 652 * q^97 - 836 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.60395	1.62774	0.813871	−	0.581045i	$-0.197356\pi$
$2$	4.60395	1.62774	0.813871	+	0.581045i	$0.197356\pi$
$3$	−2.77399	−0.533854	−0.266927	−	0.963717i	$-0.586008\pi$
$3$	−2.77399	−0.533854	−0.266927	+	0.963717i	$0.586008\pi$
$4$	13.1964	1.64955
$5$	−1.84418	−0.164949	−0.0824744	−	0.996593i	$-0.526282\pi$
$5$	−1.84418	−0.164949	−0.0824744	+	0.996593i	$0.526282\pi$
$6$	−12.7713	−0.868977
$7$	0	0
$7$	0	0
$8$	23.9238	1.05729
$9$	−19.3050	−0.715000
$10$	−8.49053	−0.268494
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−36.6066	−0.880617
$13$	−24.6401	−0.525687	−0.262844	−	0.964838i	$-0.584660\pi$
$13$	−24.6401	−0.525687	−0.262844	+	0.964838i	$0.584660\pi$
$14$	0	0
$15$	5.11574	0.0880586
$16$	4.57310	0.0714546
$17$	−17.8800	−0.255090	−0.127545	−	0.991833i	$-0.540710\pi$
$17$	−17.8800	−0.255090	−0.127545	+	0.991833i	$0.540710\pi$
$18$	−88.8792	−1.16384
$19$	−32.1459	−0.388146	−0.194073	−	0.980987i	$-0.562170\pi$
$19$	−32.1459	−0.388146	−0.194073	+	0.980987i	$0.562170\pi$
$20$	−24.3365	−0.272090
$21$	0	0
$22$	−50.6435	−0.490783
$23$	14.1248	0.128054	0.0640268	−	0.997948i	$-0.479606\pi$
$23$	14.1248	0.128054	0.0640268	+	0.997948i	$0.479606\pi$
$24$	−66.3643	−0.564440
$25$	−121.599	−0.972792
$26$	−113.442	−0.855683
$27$	128.450	0.915560
$28$	0	0
$29$	−41.5471	−0.266038	−0.133019	−	0.991113i	$-0.542467\pi$
$29$	−41.5471	−0.266038	−0.133019	+	0.991113i	$0.542467\pi$
$30$	23.5526	0.143337
$31$	−175.766	−1.01834	−0.509169	−	0.860667i	$-0.670047\pi$
$31$	−175.766	−1.01834	−0.509169	+	0.860667i	$0.670047\pi$
$32$	−170.336	−0.940983
$33$	30.5139	0.160963
$34$	−82.3187	−0.415222
$35$	0	0
$36$	−254.756	−1.17942
$37$	292.877	1.30131	0.650657	−	0.759372i	$-0.274494\pi$
$37$	292.877	1.30131	0.650657	+	0.759372i	$0.274494\pi$
$38$	−147.998	−0.631801
$39$	68.3513	0.280640
$40$	−44.1199	−0.174399
$41$	−154.296	−0.587732	−0.293866	−	0.955847i	$-0.594942\pi$
$41$	−154.296	−0.587732	−0.293866	+	0.955847i	$0.594942\pi$
$42$	0	0
$43$	−277.144	−0.982887	−0.491443	−	0.870910i	$-0.663530\pi$
$43$	−277.144	−0.982887	−0.491443	+	0.870910i	$0.663530\pi$
$44$	−145.160	−0.497357
$45$	35.6019	0.117938
$46$	65.0300	0.208438
$47$	52.1450	0.161832	0.0809162	−	0.996721i	$-0.474215\pi$
$47$	52.1450	0.161832	0.0809162	+	0.996721i	$0.474215\pi$
$48$	−12.6857	−0.0381464
$49$	0	0
$50$	−559.836	−1.58345
$51$	49.5989	0.136181
$52$	−325.160	−0.867145
$53$	82.3907	0.213533	0.106766	−	0.994284i	$-0.465950\pi$
$53$	82.3907	0.213533	0.106766	+	0.994284i	$0.465950\pi$
$54$	591.375	1.49030
$55$	20.2860	0.0497339
$56$	0	0
$57$	89.1723	0.207213
$58$	−191.281	−0.433041
$59$	−712.816	−1.57289	−0.786447	−	0.617658i	$-0.788081\pi$
$59$	−712.816	−1.57289	−0.786447	+	0.617658i	$0.788081\pi$
$60$	67.5092	0.145257
$61$	647.078	1.35819	0.679097	−	0.734048i	$-0.262371\pi$
$61$	647.078	1.35819	0.679097	+	0.734048i	$0.262371\pi$
$62$	−809.217	−1.65759
$63$	0	0
$64$	−820.804	−1.60313
$65$	45.4408	0.0867114
$66$	140.484	0.262007
$67$	260.867	0.475672	0.237836	−	0.971305i	$-0.423562\pi$
$67$	260.867	0.475672	0.237836	+	0.971305i	$0.423562\pi$
$68$	−235.951	−0.420783
$69$	−39.1821	−0.0683619
$70$	0	0
$71$	369.025	0.616833	0.308417	−	0.951251i	$-0.400201\pi$
$71$	369.025	0.616833	0.308417	+	0.951251i	$0.400201\pi$
$72$	−461.849	−0.755964
$73$	−1145.77	−1.83702	−0.918509	−	0.395401i	$-0.870606\pi$
$73$	−1145.77	−1.83702	−0.918509	+	0.395401i	$0.870606\pi$
$74$	1348.39	2.11820
$75$	337.314	0.519329
$76$	−424.209	−0.640264
$77$	0	0
$78$	314.686	0.456810
$79$	488.885	0.696251	0.348125	−	0.937448i	$-0.386818\pi$
$79$	488.885	0.696251	0.348125	+	0.937448i	$0.386818\pi$
$80$	−8.43362	−0.0117863
$81$	164.917	0.226224
$82$	−710.372	−0.956676
$83$	−548.982	−0.726008	−0.363004	−	0.931788i	$-0.618249\pi$
$83$	−548.982	−0.726008	−0.363004	+	0.931788i	$0.618249\pi$
$84$	0	0
$85$	32.9740	0.0420769
$86$	−1275.96	−1.59989
$87$	115.251	0.142026
$88$	−263.162	−0.318786
$89$	−105.039	−0.125102	−0.0625510	−	0.998042i	$-0.519924\pi$
$89$	−105.039	−0.125102	−0.0625510	+	0.998042i	$0.519924\pi$
$90$	163.910	0.191973
$91$	0	0
$92$	186.396	0.211230
$93$	487.572	0.543644
$94$	240.073	0.263422
$95$	59.2829	0.0640242
$96$	472.510	0.502348
$97$	1361.91	1.42557	0.712787	−	0.701381i	$-0.247433\pi$
$97$	1361.91	1.42557	0.712787	+	0.701381i	$0.247433\pi$
$98$	0	0
$99$	212.355	0.215580
$100$	−1604.66	−1.60466
$101$	1610.32	1.58647	0.793234	−	0.608917i	$-0.208396\pi$
$101$	1610.32	1.58647	0.793234	+	0.608917i	$0.208396\pi$
$102$	228.351	0.221668
$103$	123.044	0.117708	0.0588540	−	0.998267i	$-0.481255\pi$
$103$	123.044	0.117708	0.0588540	+	0.998267i	$0.481255\pi$
$104$	−589.485	−0.555805
$105$	0	0
$106$	379.323	0.347576
$107$	−1740.90	−1.57289	−0.786446	−	0.617660i	$-0.788081\pi$
$107$	−1740.90	−1.57289	−0.786446	+	0.617660i	$0.788081\pi$
$108$	1695.07	1.51026
$109$	248.938	0.218752	0.109376	−	0.994000i	$-0.465115\pi$
$109$	248.938	0.218752	0.109376	+	0.994000i	$0.465115\pi$
$110$	93.3958	0.0809540
$111$	−812.436	−0.694712
$112$	0	0
$113$	−494.465	−0.411641	−0.205820	−	0.978590i	$-0.565986\pi$
$113$	−494.465	−0.411641	−0.205820	+	0.978590i	$0.565986\pi$
$114$	410.545	0.337290
$115$	−26.0488	−0.0211223
$116$	−548.271	−0.438842
$117$	475.677	0.375866
$118$	−3281.77	−2.56026
$119$	0	0
$120$	122.388	0.0931037
$121$	121.000	0.0909091
$122$	2979.12	2.21079
$123$	428.016	0.313763
$124$	−2319.47	−1.67979
$125$	454.774	0.325410
$126$	0	0
$127$	−979.104	−0.684106	−0.342053	−	0.939681i	$-0.611122\pi$
$127$	−979.104	−0.684106	−0.342053	+	0.939681i	$0.611122\pi$
$128$	−2416.25	−1.66850
$129$	768.795	0.524718
$130$	209.207	0.141144
$131$	1660.85	1.10770	0.553851	−	0.832616i	$-0.313158\pi$
$131$	1660.85	1.10770	0.553851	+	0.832616i	$0.313158\pi$
$132$	402.672	0.265516
$133$	0	0
$134$	1201.02	0.774271
$135$	−236.884	−0.151020
$136$	−427.758	−0.269705
$137$	−1618.17	−1.00912	−0.504559	−	0.863377i	$-0.668345\pi$
$137$	−1618.17	−1.00912	−0.504559	+	0.863377i	$0.668345\pi$
$138$	−180.393	−0.111276
$139$	−695.736	−0.424544	−0.212272	−	0.977211i	$-0.568086\pi$
$139$	−695.736	−0.424544	−0.212272	+	0.977211i	$0.568086\pi$
$140$	0	0
$141$	−144.650	−0.0863950
$142$	1698.97	1.00405
$143$	271.041	0.158501
$144$	−88.2835	−0.0510900
$145$	76.6205	0.0438826
$146$	−5275.07	−2.99019
$147$	0	0
$148$	3864.90	2.14658
$149$	−2081.84	−1.14464	−0.572318	−	0.820032i	$-0.693956\pi$
$149$	−2081.84	−1.14464	−0.572318	+	0.820032i	$0.693956\pi$
$150$	1552.98	0.845334
$151$	2679.28	1.44395	0.721975	−	0.691919i	$-0.243234\pi$
$151$	2679.28	1.44395	0.721975	+	0.691919i	$0.243234\pi$
$152$	−769.052	−0.410384
$153$	345.173	0.182390
$154$	0	0
$155$	324.144	0.167973
$156$	901.989	0.462929
$157$	−2410.32	−1.22525	−0.612627	−	0.790372i	$-0.709887\pi$
$157$	−2410.32	−1.22525	−0.612627	+	0.790372i	$0.709887\pi$
$158$	2250.80	1.13332
$159$	−228.551	−0.113995
$160$	314.131	0.155214
$161$	0	0
$162$	759.271	0.368234
$163$	3629.08	1.74388	0.871938	−	0.489616i	$-0.162863\pi$
$163$	3629.08	1.74388	0.871938	+	0.489616i	$0.162863\pi$
$164$	−2036.15	−0.969491
$165$	−56.2732	−0.0265507
$166$	−2527.49	−1.18175
$167$	3826.04	1.77286	0.886432	−	0.462859i	$-0.153176\pi$
$167$	3826.04	1.77286	0.886432	+	0.462859i	$0.153176\pi$
$168$	0	0
$169$	−1589.87	−0.723653
$170$	151.811	0.0684903
$171$	620.576	0.277524
$172$	−3657.30	−1.62132
$173$	4310.67	1.89442	0.947209	−	0.320617i	$-0.103890\pi$
$173$	4310.67	1.89442	0.947209	+	0.320617i	$0.103890\pi$
$174$	530.611	0.231181
$175$	0	0
$176$	−50.3040	−0.0215444
$177$	1977.34	0.839696
$178$	−483.593	−0.203634
$179$	2491.12	1.04019	0.520097	−	0.854107i	$-0.325896\pi$
$179$	2491.12	1.04019	0.520097	+	0.854107i	$0.325896\pi$
$180$	469.816	0.194545
$181$	4315.49	1.77220	0.886098	−	0.463498i	$-0.153406\pi$
$181$	4315.49	1.77220	0.886098	+	0.463498i	$0.153406\pi$
$182$	0	0
$183$	−1794.99	−0.725078
$184$	337.920	0.135390
$185$	−540.118	−0.214650
$186$	2244.76	0.884912
$187$	196.680	0.0769127
$188$	688.124	0.266950
$189$	0	0
$190$	272.936	0.104215
$191$	2840.41	1.07605	0.538024	−	0.842930i	$-0.319171\pi$
$191$	2840.41	1.07605	0.538024	+	0.842930i	$0.319171\pi$
$192$	2276.90	0.855839
$193$	−1734.68	−0.646969	−0.323485	−	0.946233i	$-0.604854\pi$
$193$	−1734.68	−0.646969	−0.323485	+	0.946233i	$0.604854\pi$
$194$	6270.15	2.32047
$195$	−126.052	−0.0462913
$196$	0	0
$197$	−3098.42	−1.12057	−0.560287	−	0.828298i	$-0.689309\pi$
$197$	−3098.42	−1.12057	−0.560287	+	0.828298i	$0.689309\pi$
$198$	977.671	0.350909
$199$	−4497.38	−1.60206	−0.801032	−	0.598622i	$-0.795715\pi$
$199$	−4497.38	−1.60206	−0.801032	+	0.598622i	$0.795715\pi$
$200$	−2909.11	−1.02853
$201$	−723.643	−0.253940
$202$	7413.85	2.58236
$203$	0	0
$204$	654.525	0.224637
$205$	284.550	0.0969456
$206$	566.491	0.191598
$207$	−272.680	−0.0915582
$208$	−112.682	−0.0375628
$209$	353.605	0.117030
$210$	0	0
$211$	1262.32	0.411857	0.205929	−	0.978567i	$-0.433979\pi$
$211$	1262.32	0.411857	0.205929	+	0.978567i	$0.433979\pi$
$212$	1087.26	0.352232
$213$	−1023.67	−0.329299
$214$	−8015.03	−2.56026
$215$	511.105	0.162126
$216$	3073.00	0.968015
$217$	0	0
$218$	1146.10	0.356071
$219$	3178.35	0.980700
$220$	267.702	0.0820384
$221$	440.565	0.134098
$222$	−3740.42	−1.13081
$223$	−2931.38	−0.880268	−0.440134	−	0.897932i	$-0.645069\pi$
$223$	−2931.38	−0.880268	−0.440134	+	0.897932i	$0.645069\pi$
$224$	0	0
$225$	2347.47	0.695546
$226$	−2276.49	−0.670045
$227$	−4298.37	−1.25680	−0.628399	−	0.777891i	$-0.716289\pi$
$227$	−4298.37	−1.25680	−0.628399	+	0.777891i	$0.716289\pi$
$228$	1176.75	0.341808
$229$	−698.500	−0.201564	−0.100782	−	0.994909i	$-0.532134\pi$
$229$	−698.500	−0.201564	−0.100782	+	0.994909i	$0.532134\pi$
$230$	−119.927	−0.0343816
$231$	0	0
$232$	−993.965	−0.281280
$233$	1887.78	0.530783	0.265391	−	0.964141i	$-0.414499\pi$
$233$	1887.78	0.530783	0.265391	+	0.964141i	$0.414499\pi$
$234$	2189.99	0.611813
$235$	−96.1649	−0.0266941
$236$	−9406.57	−2.59456
$237$	−1356.16	−0.371697
$238$	0	0
$239$	−6449.93	−1.74565	−0.872827	−	0.488030i	$-0.837716\pi$
$239$	−6449.93	−1.74565	−0.872827	+	0.488030i	$0.837716\pi$
$240$	23.3948	0.00629219
$241$	−4636.98	−1.23939	−0.619697	−	0.784841i	$-0.712744\pi$
$241$	−4636.98	−1.23939	−0.619697	+	0.784841i	$0.712744\pi$
$242$	557.078	0.147977
$243$	−3925.62	−1.03633
$244$	8539.08	2.24040
$245$	0	0
$246$	1970.56	0.510726
$247$	792.078	0.204043
$248$	−4204.98	−1.07668
$249$	1522.87	0.387582
$250$	2093.76	0.529683
$251$	−2194.11	−0.551758	−0.275879	−	0.961192i	$-0.588969\pi$
$251$	−2194.11	−0.551758	−0.275879	+	0.961192i	$0.588969\pi$
$252$	0	0
$253$	−155.373	−0.0386096
$254$	−4507.75	−1.11355
$255$	−91.4695	−0.0224629
$256$	−4557.87	−1.11276
$257$	2966.97	0.720133	0.360067	−	0.932927i	$-0.382754\pi$
$257$	2966.97	0.720133	0.360067	+	0.932927i	$0.382754\pi$
$258$	3539.50	0.854106
$259$	0	0
$260$	599.654	0.143034
$261$	802.066	0.190217
$262$	7646.47	1.80305
$263$	915.810	0.214720	0.107360	−	0.994220i	$-0.465760\pi$
$263$	915.810	0.214720	0.107360	+	0.994220i	$0.465760\pi$
$264$	730.008	0.170185
$265$	−151.944	−0.0352220
$266$	0	0
$267$	291.376	0.0667863
$268$	3442.50	0.784642
$269$	164.462	0.0372768	0.0186384	−	0.999826i	$-0.494067\pi$
$269$	164.462	0.0372768	0.0186384	+	0.999826i	$0.494067\pi$
$270$	−1090.60	−0.245822
$271$	−1502.60	−0.336815	−0.168407	−	0.985718i	$-0.553862\pi$
$271$	−1502.60	−0.336815	−0.168407	+	0.985718i	$0.553862\pi$
$272$	−81.7670	−0.0182274
$273$	0	0
$274$	−7449.96	−1.64259
$275$	1337.59	0.293308
$276$	−517.061	−0.112766
$277$	−7500.11	−1.62685	−0.813426	−	0.581669i	$-0.802400\pi$
$277$	−7500.11	−1.62685	−0.813426	+	0.581669i	$0.802400\pi$
$278$	−3203.14	−0.691048
$279$	3393.15	0.728111
$280$	0	0
$281$	−2945.12	−0.625235	−0.312618	−	0.949879i	$-0.601206\pi$
$281$	−2945.12	−0.625235	−0.312618	+	0.949879i	$0.601206\pi$
$282$	−665.959	−0.140629
$283$	−5215.29	−1.09547	−0.547733	−	0.836653i	$-0.684509\pi$
$283$	−5215.29	−1.09547	−0.547733	+	0.836653i	$0.684509\pi$
$284$	4869.78	1.01749
$285$	−164.450	−0.0341796
$286$	1247.86	0.257998
$287$	0	0
$288$	3288.34	0.672802
$289$	−4593.31	−0.934929
$290$	352.757	0.0714296
$291$	−3777.91	−0.761049
$292$	−15120.0	−3.03024
$293$	−7407.99	−1.47706	−0.738531	−	0.674219i	$-0.764480\pi$
$293$	−7407.99	−1.47706	−0.738531	+	0.674219i	$0.764480\pi$
$294$	0	0
$295$	1314.56	0.259447
$296$	7006.72	1.37587
$297$	−1412.94	−0.276052
$298$	−9584.68	−1.86317
$299$	−348.037	−0.0673161
$300$	4451.32	0.856657
$301$	0	0
$302$	12335.3	2.35038
$303$	−4467.02	−0.846942
$304$	−147.006	−0.0277348
$305$	−1193.33	−0.224033
$306$	1589.16	0.296883
$307$	6850.01	1.27345	0.636727	−	0.771089i	$-0.280288\pi$
$307$	6850.01	1.27345	0.636727	+	0.771089i	$0.280288\pi$
$308$	0	0
$309$	−341.324	−0.0628390
$310$	1492.34	0.273417
$311$	5538.62	1.00986	0.504929	−	0.863161i	$-0.331519\pi$
$311$	5538.62	1.00986	0.504929	+	0.863161i	$0.331519\pi$
$312$	1635.22	0.296719
$313$	−9361.13	−1.69049	−0.845243	−	0.534382i	$-0.820544\pi$
$313$	−9361.13	−1.69049	−0.845243	+	0.534382i	$0.820544\pi$
$314$	−11097.0	−1.99440
$315$	0	0
$316$	6451.50	1.14850
$317$	219.221	0.0388413	0.0194206	−	0.999811i	$-0.493818\pi$
$317$	219.221	0.0388413	0.0194206	+	0.999811i	$0.493818\pi$
$318$	−1052.24	−0.185555
$319$	457.018	0.0802135
$320$	1513.71	0.264435
$321$	4829.24	0.839695
$322$	0	0
$323$	574.769	0.0990123
$324$	2176.31	0.373167
$325$	2996.21	0.511384
$326$	16708.1	2.83858
$327$	−690.551	−0.116781
$328$	−3691.35	−0.621405
$329$	0	0
$330$	−259.079	−0.0432176
$331$	3377.63	0.560880	0.280440	−	0.959872i	$-0.409520\pi$
$331$	3377.63	0.560880	0.280440	+	0.959872i	$0.409520\pi$
$332$	−7244.57	−1.19758
$333$	−5653.98	−0.930439
$334$	17614.9	2.88577
$335$	−481.087	−0.0784615
$336$	0	0
$337$	7755.93	1.25369	0.626843	−	0.779145i	$-0.284347\pi$
$337$	7755.93	1.25369	0.626843	+	0.779145i	$0.284347\pi$
$338$	−7319.66	−1.17792
$339$	1371.64	0.219756
$340$	435.137	0.0694077
$341$	1933.42	0.307040
$342$	2857.10	0.451738
$343$	0	0
$344$	−6630.35	−1.03920
$345$	72.2590	0.0112762
$346$	19846.1	3.08362
$347$	−831.044	−0.128567	−0.0642835	−	0.997932i	$-0.520476\pi$
$347$	−831.044	−0.128567	−0.0642835	+	0.997932i	$0.520476\pi$
$348$	1520.90	0.234278
$349$	−1840.94	−0.282359	−0.141180	−	0.989984i	$-0.545089\pi$
$349$	−1840.94	−0.282359	−0.141180	+	0.989984i	$0.545089\pi$
$350$	0	0
$351$	−3165.01	−0.481298
$352$	1873.70	0.283717
$353$	−3409.11	−0.514019	−0.257009	−	0.966409i	$-0.582737\pi$
$353$	−3409.11	−0.514019	−0.257009	+	0.966409i	$0.582737\pi$
$354$	9103.59	1.36681
$355$	−680.549	−0.101746
$356$	−1386.13	−0.206361
$357$	0	0
$358$	11469.0	1.69317
$359$	2199.75	0.323394	0.161697	−	0.986840i	$-0.448303\pi$
$359$	2199.75	0.323394	0.161697	+	0.986840i	$0.448303\pi$
$360$	851.734	0.124695
$361$	−5825.64	−0.849343
$362$	19868.3	2.88468
$363$	−335.653	−0.0485322
$364$	0	0
$365$	2113.01	0.303014
$366$	−8264.04	−1.18024
$367$	855.008	0.121611	0.0608053	−	0.998150i	$-0.480633\pi$
$367$	855.008	0.121611	0.0608053	+	0.998150i	$0.480633\pi$
$368$	64.5942	0.00915002
$369$	2978.69	0.420228
$370$	−2486.68	−0.349395
$371$	0	0
$372$	6434.18	0.896765
$373$	3193.40	0.443293	0.221646	−	0.975127i	$-0.428857\pi$
$373$	3193.40	0.443293	0.221646	+	0.975127i	$0.428857\pi$
$374$	905.505	0.125194
$375$	−1261.54	−0.173721
$376$	1247.51	0.171104
$377$	1023.72	0.139853
$378$	0	0
$379$	−5614.48	−0.760940	−0.380470	−	0.924793i	$-0.624238\pi$
$379$	−5614.48	−0.760940	−0.380470	+	0.924793i	$0.624238\pi$
$380$	782.319	0.105611
$381$	2716.02	0.365213
$382$	13077.1	1.75153
$383$	1736.86	0.231721	0.115861	−	0.993265i	$-0.463037\pi$
$383$	1736.86	0.231721	0.115861	+	0.993265i	$0.463037\pi$
$384$	6702.65	0.890738
$385$	0	0
$386$	−7986.38	−1.05310
$387$	5350.27	0.702763
$388$	17972.2	2.35155
$389$	8710.78	1.13536	0.567679	−	0.823250i	$-0.307841\pi$
$389$	8710.78	1.13536	0.567679	+	0.823250i	$0.307841\pi$
$390$	−580.339	−0.0753503
$391$	−252.552	−0.0326652
$392$	0	0
$393$	−4607.17	−0.591352
$394$	−14265.0	−1.82401
$395$	−901.593	−0.114846
$396$	2802.31	0.355610
$397$	−11731.6	−1.48311	−0.741553	−	0.670894i	$-0.765911\pi$
$397$	−11731.6	−1.48311	−0.741553	+	0.670894i	$0.765911\pi$
$398$	−20705.7	−2.60775
$399$	0	0
$400$	−556.084	−0.0695105
$401$	−14408.8	−1.79437	−0.897183	−	0.441659i	$-0.854390\pi$
$401$	−14408.8	−1.79437	−0.897183	+	0.441659i	$0.854390\pi$
$402$	−3331.62	−0.413348
$403$	4330.88	0.535327
$404$	21250.4	2.61695
$405$	−304.138	−0.0373154
$406$	0	0
$407$	−3221.64	−0.392361
$408$	1186.59	0.143983
$409$	2155.54	0.260598	0.130299	−	0.991475i	$-0.458406\pi$
$409$	2155.54	0.260598	0.130299	+	0.991475i	$0.458406\pi$
$410$	1310.06	0.157803
$411$	4488.77	0.538722
$412$	1623.74	0.194165
$413$	0	0
$414$	−1255.40	−0.149033
$415$	1012.42	0.119754
$416$	4197.10	0.494663
$417$	1929.96	0.226644
$418$	1627.98	0.190495
$419$	−8443.41	−0.984458	−0.492229	−	0.870466i	$-0.663818\pi$
$419$	−8443.41	−0.984458	−0.492229	+	0.870466i	$0.663818\pi$
$420$	0	0
$421$	−3070.28	−0.355430	−0.177715	−	0.984082i	$-0.556871\pi$
$421$	−3070.28	−0.355430	−0.177715	+	0.984082i	$0.556871\pi$
$422$	5811.67	0.670398
$423$	−1006.66	−0.115710
$424$	1971.10	0.225767
$425$	2174.19	0.248150
$426$	−4712.93	−0.536014
$427$	0	0
$428$	−22973.6	−2.59456
$429$	−751.865	−0.0846163
$430$	2353.10	0.263899
$431$	−7004.40	−0.782808	−0.391404	−	0.920219i	$-0.628010\pi$
$431$	−7004.40	−0.782808	−0.391404	+	0.920219i	$0.628010\pi$
$432$	587.412	0.0654210
$433$	−7486.00	−0.830841	−0.415421	−	0.909629i	$-0.636366\pi$
$433$	−7486.00	−0.830841	−0.415421	+	0.909629i	$0.636366\pi$
$434$	0	0
$435$	−212.544	−0.0234269
$436$	3285.07	0.360841
$437$	−454.055	−0.0497034
$438$	14633.0	1.59633
$439$	13046.2	1.41836	0.709179	−	0.705029i	$-0.249066\pi$
$439$	13046.2	1.41836	0.709179	+	0.705029i	$0.249066\pi$
$440$	485.319	0.0525833
$441$	0	0
$442$	2028.34	0.218277
$443$	−11781.3	−1.26354	−0.631768	−	0.775157i	$-0.717671\pi$
$443$	−11781.3	−1.26354	−0.631768	+	0.775157i	$0.717671\pi$
$444$	−10721.2	−1.14596
$445$	193.711	0.0206354
$446$	−13495.9	−1.43285
$447$	5774.99	0.611069
$448$	0	0
$449$	7576.58	0.796349	0.398175	−	0.917310i	$-0.369644\pi$
$449$	7576.58	0.796349	0.398175	+	0.917310i	$0.369644\pi$
$450$	10807.6	1.13217
$451$	1697.26	0.177208
$452$	−6525.14	−0.679020
$453$	−7432.28	−0.770859
$454$	−19789.5	−2.04574
$455$	0	0
$456$	2133.34	0.219085
$457$	11793.0	1.20712	0.603560	−	0.797318i	$-0.293748\pi$
$457$	11793.0	1.20712	0.603560	+	0.797318i	$0.293748\pi$
$458$	−3215.86	−0.328095
$459$	−2296.68	−0.233551
$460$	−343.749	−0.0348421
$461$	−4228.32	−0.427185	−0.213592	−	0.976923i	$-0.568516\pi$
$461$	−4228.32	−0.427185	−0.213592	+	0.976923i	$0.568516\pi$
$462$	0	0
$463$	14448.8	1.45031	0.725154	−	0.688586i	$-0.241768\pi$
$463$	14448.8	1.45031	0.725154	+	0.688586i	$0.241768\pi$
$464$	−189.999	−0.0190096
$465$	−899.172	−0.0896733
$466$	8691.23	0.863977
$467$	−16547.5	−1.63967	−0.819836	−	0.572599i	$-0.805935\pi$
$467$	−16547.5	−1.63967	−0.819836	+	0.572599i	$0.805935\pi$
$468$	6277.20	0.620008
$469$	0	0
$470$	−442.738	−0.0434511
$471$	6686.21	0.654107
$472$	−17053.3	−1.66301
$473$	3048.59	0.296351
$474$	−6243.69	−0.605026
$475$	3908.91	0.377585
$476$	0	0
$477$	−1590.55	−0.152676
$478$	−29695.2	−2.84148
$479$	2989.34	0.285149	0.142575	−	0.989784i	$-0.454462\pi$
$479$	2989.34	0.285149	0.142575	+	0.989784i	$0.454462\pi$
$480$	−871.396	−0.0828616
$481$	−7216.51	−0.684084
$482$	−21348.4	−2.01741
$483$	0	0
$484$	1596.76	0.149959
$485$	−2511.60	−0.235147
$486$	−18073.3	−1.68688
$487$	−5549.61	−0.516379	−0.258190	−	0.966094i	$-0.583126\pi$
$487$	−5549.61	−0.516379	−0.258190	+	0.966094i	$0.583126\pi$
$488$	15480.6	1.43601
$489$	−10067.0	−0.930976
$490$	0	0
$491$	−7751.20	−0.712438	−0.356219	−	0.934403i	$-0.615934\pi$
$491$	−7751.20	−0.712438	−0.356219	+	0.934403i	$0.615934\pi$
$492$	5648.25	0.517567
$493$	742.862	0.0678638
$494$	3646.69	0.332130
$495$	−391.621	−0.0355597
$496$	−803.793	−0.0727649
$497$	0	0
$498$	7011.22	0.630884
$499$	−6841.83	−0.613792	−0.306896	−	0.951743i	$-0.599290\pi$
$499$	−6841.83	−0.613792	−0.306896	+	0.951743i	$0.599290\pi$
$500$	6001.36	0.536778
$501$	−10613.4	−0.946451
$502$	−10101.6	−0.898120
$503$	11518.5	1.02105	0.510523	−	0.859864i	$-0.329452\pi$
$503$	11518.5	1.02105	0.510523	+	0.859864i	$0.329452\pi$
$504$	0	0
$505$	−2969.73	−0.261686
$506$	−715.330	−0.0628465
$507$	4410.27	0.386325
$508$	−12920.6	−1.12846
$509$	−8292.40	−0.722110	−0.361055	−	0.932545i	$-0.617583\pi$
$509$	−8292.40	−0.722110	−0.361055	+	0.932545i	$0.617583\pi$
$510$	−421.121	−0.0365638
$511$	0	0
$512$	−1654.21	−0.142786
$513$	−4129.12	−0.355371
$514$	13659.8	1.17219
$515$	−226.917	−0.0194158
$516$	10145.3	0.865547
$517$	−573.595	−0.0487943
$518$	0	0
$519$	−11957.8	−1.01134
$520$	1087.12	0.0916794
$521$	−10891.6	−0.915868	−0.457934	−	0.888986i	$-0.651410\pi$
$521$	−10891.6	−0.915868	−0.457934	+	0.888986i	$0.651410\pi$
$522$	3692.67	0.309624
$523$	−14662.0	−1.22586	−0.612928	−	0.790139i	$-0.710008\pi$
$523$	−14662.0	−1.22586	−0.612928	+	0.790139i	$0.710008\pi$
$524$	21917.2	1.82721
$525$	0	0
$526$	4216.34	0.349508
$527$	3142.69	0.259768
$528$	139.543	0.0115016
$529$	−11967.5	−0.983602
$530$	−699.541	−0.0573323
$531$	13760.9	1.12462
$532$	0	0
$533$	3801.87	0.308963
$534$	1341.48	0.108711
$535$	3210.54	0.259446
$536$	6240.94	0.502924
$537$	−6910.33	−0.555313
$538$	757.177	0.0606770
$539$	0	0
$540$	−3126.01	−0.249115
$541$	−19825.8	−1.57556	−0.787778	−	0.615960i	$-0.788768\pi$
$541$	−19825.8	−1.57556	−0.787778	+	0.615960i	$0.788768\pi$
$542$	−6917.92	−0.548247
$543$	−11971.1	−0.946094
$544$	3045.61	0.240036
$545$	−459.087	−0.0360828
$546$	0	0
$547$	12706.1	0.993187	0.496593	−	0.867983i	$-0.334584\pi$
$547$	12706.1	0.993187	0.496593	+	0.867983i	$0.334584\pi$
$548$	−21353.9	−1.66459
$549$	−12491.8	−0.971109
$550$	6158.19	0.477430
$551$	1335.57	0.103262
$552$	−937.385	−0.0722786
$553$	0	0
$554$	−34530.1	−2.64809
$555$	1498.28	0.114592
$556$	−9181.19	−0.700304
$557$	−12599.1	−0.958421	−0.479211	−	0.877700i	$-0.659077\pi$
$557$	−12599.1	−0.958421	−0.479211	+	0.877700i	$0.659077\pi$
$558$	15621.9	1.18518
$559$	6828.86	0.516691
$560$	0	0
$561$	−545.588	−0.0410602
$562$	−13559.2	−1.01772
$563$	−6004.47	−0.449482	−0.224741	−	0.974419i	$-0.572154\pi$
$563$	−6004.47	−0.449482	−0.224741	+	0.974419i	$0.572154\pi$
$564$	−1908.85	−0.142512
$565$	911.884	0.0678996
$566$	−24010.9	−1.78314
$567$	0	0
$568$	8828.47	0.652173
$569$	−3145.89	−0.231779	−0.115890	−	0.993262i	$-0.536972\pi$
$569$	−3145.89	−0.231779	−0.115890	+	0.993262i	$0.536972\pi$
$570$	−757.120	−0.0556356
$571$	23549.1	1.72592	0.862960	−	0.505273i	$-0.168608\pi$
$571$	23549.1	1.72592	0.862960	+	0.505273i	$0.168608\pi$
$572$	3576.76	0.261454
$573$	−7879.27	−0.574453
$574$	0	0
$575$	−1717.57	−0.124569
$576$	15845.6	1.14624
$577$	−327.335	−0.0236172	−0.0118086	−	0.999930i	$-0.503759\pi$
$577$	−327.335	−0.0236172	−0.0118086	+	0.999930i	$0.503759\pi$
$578$	−21147.4	−1.52182
$579$	4811.98	0.345387
$580$	1011.11	0.0723864
$581$	0	0
$582$	−17393.3	−1.23879
$583$	−906.298	−0.0643825
$584$	−27411.2	−1.94226
$585$	−877.235	−0.0619986
$586$	−34106.0	−2.40428
$587$	13270.8	0.933123	0.466562	−	0.884489i	$-0.345493\pi$
$587$	13270.8	0.933123	0.466562	+	0.884489i	$0.345493\pi$
$588$	0	0
$589$	5650.14	0.395263
$590$	6052.18	0.422312
$591$	8594.98	0.598224
$592$	1339.35	0.0929849
$593$	−5098.92	−0.353099	−0.176549	−	0.984292i	$-0.556494\pi$
$593$	−5098.92	−0.353099	−0.176549	+	0.984292i	$0.556494\pi$
$594$	−6505.13	−0.449341
$595$	0	0
$596$	−27472.7	−1.88813
$597$	12475.7	0.855268
$598$	−1602.35	−0.109573
$599$	19358.2	1.32046	0.660230	−	0.751064i	$-0.270459\pi$
$599$	19358.2	1.32046	0.660230	+	0.751064i	$0.270459\pi$
$600$	8069.84	0.549083
$601$	−1238.87	−0.0840841	−0.0420420	−	0.999116i	$-0.513386\pi$
$601$	−1238.87	−0.0840841	−0.0420420	+	0.999116i	$0.513386\pi$
$602$	0	0
$603$	−5036.04	−0.340105
$604$	35356.7	2.38186
$605$	−223.146	−0.0149953
$606$	−20565.9	−1.37860
$607$	14175.6	0.947888	0.473944	−	0.880555i	$-0.342830\pi$
$607$	14175.6	0.947888	0.473944	+	0.880555i	$0.342830\pi$
$608$	5475.60	0.365239
$609$	0	0
$610$	−5494.04	−0.364667
$611$	−1284.86	−0.0850732
$612$	4555.03	0.300860
$613$	6906.67	0.455070	0.227535	−	0.973770i	$-0.426933\pi$
$613$	6906.67	0.455070	0.227535	+	0.973770i	$0.426933\pi$
$614$	31537.1	2.07286
$615$	−789.339	−0.0517549
$616$	0	0
$617$	12104.3	0.789789	0.394894	−	0.918727i	$-0.370781\pi$
$617$	12104.3	0.789789	0.394894	+	0.918727i	$0.370781\pi$
$618$	−1571.44	−0.102286
$619$	−14945.2	−0.970437	−0.485218	−	0.874393i	$-0.661260\pi$
$619$	−14945.2	−0.970437	−0.485218	+	0.874393i	$0.661260\pi$
$620$	4277.52	0.277080
$621$	1814.33	0.117241
$622$	25499.5	1.64379
$623$	0	0
$624$	312.577	0.0200530
$625$	14361.2	0.919116
$626$	−43098.2	−2.75168
$627$	−980.895	−0.0624772
$628$	−31807.5	−2.02111
$629$	−5236.63	−0.331953
$630$	0	0
$631$	−17711.9	−1.11743	−0.558717	−	0.829358i	$-0.688706\pi$
$631$	−17711.9	−1.11743	−0.558717	+	0.829358i	$0.688706\pi$
$632$	11696.0	0.736141
$633$	−3501.67	−0.219872
$634$	1009.28	0.0632236
$635$	1805.65	0.112842
$636$	−3016.04	−0.188041
$637$	0	0
$638$	2104.09	0.130567
$639$	−7124.02	−0.441036
$640$	4456.01	0.275218
$641$	−17994.4	−1.10879	−0.554396	−	0.832253i	$-0.687051\pi$
$641$	−17994.4	−1.10879	−0.554396	+	0.832253i	$0.687051\pi$
$642$	22233.6	1.36681
$643$	−27947.8	−1.71408	−0.857039	−	0.515251i	$-0.827699\pi$
$643$	−27947.8	−1.71408	−0.857039	+	0.515251i	$0.827699\pi$
$644$	0	0
$645$	−1417.80	−0.0865516
$646$	2646.21	0.161167
$647$	14336.2	0.871122	0.435561	−	0.900159i	$-0.356550\pi$
$647$	14336.2	0.871122	0.435561	+	0.900159i	$0.356550\pi$
$648$	3945.45	0.239185
$649$	7840.97	0.474245
$650$	13794.4	0.832402
$651$	0	0
$652$	47890.7	2.87660
$653$	4315.79	0.258637	0.129318	−	0.991603i	$-0.458721\pi$
$653$	4315.79	0.258637	0.129318	+	0.991603i	$0.458721\pi$
$654$	−3179.26	−0.190090
$655$	−3062.91	−0.182714
$656$	−705.611	−0.0419962
$657$	22119.1	1.31347
$658$	0	0
$659$	4002.23	0.236578	0.118289	−	0.992979i	$-0.462259\pi$
$659$	4002.23	0.236578	0.118289	+	0.992979i	$0.462259\pi$
$660$	−742.601	−0.0437965
$661$	12223.4	0.719268	0.359634	−	0.933094i	$-0.382902\pi$
$661$	12223.4	0.719268	0.359634	+	0.933094i	$0.382902\pi$
$662$	15550.4	0.912968
$663$	−1222.12	−0.0715887
$664$	−13133.7	−0.767603
$665$	0	0
$666$	−26030.6	−1.51451
$667$	−586.846	−0.0340671
$668$	50489.9	2.92442
$669$	8131.61	0.469935
$670$	−2214.90	−0.127715
$671$	−7117.86	−0.409511
$672$	0	0
$673$	−6121.53	−0.350621	−0.175310	−	0.984513i	$-0.556093\pi$
$673$	−6121.53	−0.350621	−0.175310	+	0.984513i	$0.556093\pi$
$674$	35707.9	2.04068
$675$	−15619.3	−0.890649
$676$	−20980.4	−1.19370
$677$	−9626.46	−0.546492	−0.273246	−	0.961944i	$-0.588097\pi$
$677$	−9626.46	−0.546492	−0.273246	+	0.961944i	$0.588097\pi$
$678$	6314.97	0.357706
$679$	0	0
$680$	788.863	0.0444875
$681$	11923.6	0.670947
$682$	8901.38	0.499782
$683$	11554.4	0.647315	0.323658	−	0.946174i	$-0.395087\pi$
$683$	11554.4	0.647315	0.323658	+	0.946174i	$0.395087\pi$
$684$	8189.34	0.457789
$685$	2984.19	0.166453
$686$	0	0
$687$	1937.63	0.107606
$688$	−1267.41	−0.0702318
$689$	−2030.12	−0.112251
$690$	332.677	0.0183548
$691$	−2991.45	−0.164689	−0.0823444	−	0.996604i	$-0.526241\pi$
$691$	−2991.45	−0.164689	−0.0823444	+	0.996604i	$0.526241\pi$
$692$	56885.2	3.12493
$693$	0	0
$694$	−3826.08	−0.209274
$695$	1283.06	0.0700279
$696$	2757.25	0.150163
$697$	2758.82	0.149925
$698$	−8475.60	−0.459608
$699$	−5236.67	−0.283361
$700$	0	0
$701$	30772.8	1.65802	0.829010	−	0.559234i	$-0.188905\pi$
$701$	30772.8	1.65802	0.829010	+	0.559234i	$0.188905\pi$
$702$	−14571.5	−0.783429
$703$	−9414.77	−0.505099
$704$	9028.84	0.483363
$705$	266.760	0.0142507
$706$	−15695.4	−0.836690
$707$	0	0
$708$	26093.7	1.38512
$709$	−21698.3	−1.14936	−0.574681	−	0.818377i	$-0.694874\pi$
$709$	−21698.3	−1.14936	−0.574681	+	0.818377i	$0.694874\pi$
$710$	−3133.21	−0.165616
$711$	−9437.91	−0.497819
$712$	−2512.93	−0.132269
$713$	−2482.66	−0.130402
$714$	0	0
$715$	−499.849	−0.0261445
$716$	32873.7	1.71585
$717$	17892.0	0.931925
$718$	10127.5	0.526402
$719$	12364.2	0.641318	0.320659	−	0.947195i	$-0.396096\pi$
$719$	12364.2	0.641318	0.320659	+	0.947195i	$0.396096\pi$
$720$	162.811	0.00842723
$721$	0	0
$722$	−26821.0	−1.38251
$723$	12862.9	0.661656
$724$	56948.7	2.92332
$725$	5052.09	0.258800
$726$	−1545.33	−0.0789979
$727$	16017.2	0.817119	0.408559	−	0.912732i	$-0.366031\pi$
$727$	16017.2	0.817119	0.408559	+	0.912732i	$0.366031\pi$
$728$	0	0
$729$	6436.85	0.327026
$730$	9728.19	0.493228
$731$	4955.34	0.250725
$732$	−23687.3	−1.19605
$733$	−10564.4	−0.532340	−0.266170	−	0.963926i	$-0.585758\pi$
$733$	−10564.4	−0.532340	−0.266170	+	0.963926i	$0.585758\pi$
$734$	3936.42	0.197951
$735$	0	0
$736$	−2405.97	−0.120496
$737$	−2869.54	−0.143420
$738$	13713.7	0.684023
$739$	1760.32	0.0876242	0.0438121	−	0.999040i	$-0.486050\pi$
$739$	1760.32	0.0876242	0.0438121	+	0.999040i	$0.486050\pi$
$740$	−7127.59	−0.354075
$741$	−2197.21	−0.108929
$742$	0	0
$743$	11383.4	0.562067	0.281034	−	0.959698i	$-0.409323\pi$
$743$	11383.4	0.562067	0.281034	+	0.959698i	$0.409323\pi$
$744$	11664.6	0.574790
$745$	3839.29	0.188806
$746$	14702.3	0.721566
$747$	10598.1	0.519095
$748$	2595.46	0.126871
$749$	0	0
$750$	−5808.05	−0.282774
$751$	−15058.3	−0.731670	−0.365835	−	0.930680i	$-0.619217\pi$
$751$	−15058.3	−0.731670	−0.365835	+	0.930680i	$0.619217\pi$
$752$	238.464	0.0115637
$753$	6086.45	0.294558
$754$	4713.18	0.227644
$755$	−4941.08	−0.238178
$756$	0	0
$757$	38073.4	1.82801	0.914004	−	0.405706i	$-0.132974\pi$
$757$	38073.4	1.82801	0.914004	+	0.405706i	$0.132974\pi$
$758$	−25848.8	−1.23861
$759$	431.003	0.0206119
$760$	1418.27	0.0676923
$761$	−15232.1	−0.725577	−0.362788	−	0.931872i	$-0.618175\pi$
$761$	−15232.1	−0.725577	−0.362788	+	0.931872i	$0.618175\pi$
$762$	12504.4	0.594473
$763$	0	0
$764$	37483.1	1.77499
$765$	−636.563	−0.0300849
$766$	7996.40	0.377182
$767$	17563.8	0.826850
$768$	12643.5	0.594053
$769$	−12013.1	−0.563332	−0.281666	−	0.959513i	$-0.590887\pi$
$769$	−12013.1	−0.563332	−0.281666	+	0.959513i	$0.590887\pi$
$770$	0	0
$771$	−8230.33	−0.384446
$772$	−22891.5	−1.06720
$773$	14258.5	0.663443	0.331721	−	0.943377i	$-0.392371\pi$
$773$	14258.5	0.663443	0.331721	+	0.943377i	$0.392371\pi$
$774$	24632.4	1.14392
$775$	21372.9	0.990630
$776$	32582.0	1.50725
$777$	0	0
$778$	40104.0	1.84807
$779$	4959.99	0.228126
$780$	−1663.43	−0.0763596
$781$	−4059.27	−0.185982
$782$	−1162.74	−0.0531706
$783$	−5336.70	−0.243574
$784$	0	0
$785$	4445.08	0.202104
$786$	−21211.2	−0.962568
$787$	14377.9	0.651227	0.325613	−	0.945503i	$-0.394429\pi$
$787$	14377.9	0.651227	0.325613	+	0.945503i	$0.394429\pi$
$788$	−40887.9	−1.84844
$789$	−2540.45	−0.114629
$790$	−4150.89	−0.186939
$791$	0	0
$792$	5080.34	0.227932
$793$	−15944.1	−0.713986
$794$	−54011.8	−2.41412
$795$	421.490	0.0188034
$796$	−59349.0	−2.64268
$797$	−38515.9	−1.71180	−0.855899	−	0.517144i	$-0.826995\pi$
$797$	−38515.9	−1.71180	−0.855899	+	0.517144i	$0.826995\pi$
$798$	0	0
$799$	−932.352	−0.0412819
$800$	20712.7	0.915381
$801$	2027.77	0.0894479
$802$	−66337.4	−2.92076
$803$	12603.5	0.553882
$804$	−9549.46	−0.418885
$805$	0	0
$806$	19939.2	0.871374
$807$	−456.217	−0.0199004
$808$	38525.1	1.67736
$809$	11438.6	0.497106	0.248553	−	0.968618i	$-0.420045\pi$
$809$	11438.6	0.497106	0.248553	+	0.968618i	$0.420045\pi$
$810$	−1400.23	−0.0607398
$811$	−15872.9	−0.687265	−0.343632	−	0.939104i	$-0.611657\pi$
$811$	−15872.9	−0.687265	−0.343632	+	0.939104i	$0.611657\pi$
$812$	0	0
$813$	4168.21	0.179810
$814$	−14832.3	−0.638662
$815$	−6692.70	−0.287650
$816$	226.821	0.00973077
$817$	8909.05	0.381503
$818$	9924.00	0.424186
$819$	0	0
$820$	3755.03	0.159916
$821$	−11988.8	−0.509638	−0.254819	−	0.966989i	$-0.582016\pi$
$821$	−11988.8	−0.509638	−0.254819	+	0.966989i	$0.582016\pi$
$822$	20666.1	0.876901
$823$	−222.224	−0.00941219	−0.00470610	−	0.999989i	$-0.501498\pi$
$823$	−222.224	−0.00941219	−0.00470610	+	0.999989i	$0.501498\pi$
$824$	2943.69	0.124452
$825$	−3710.46	−0.156584
$826$	0	0
$827$	7524.36	0.316382	0.158191	−	0.987409i	$-0.449434\pi$
$827$	7524.36	0.316382	0.158191	+	0.987409i	$0.449434\pi$
$828$	−3598.38	−0.151029
$829$	11807.1	0.494663	0.247332	−	0.968931i	$-0.420446\pi$
$829$	11807.1	0.494663	0.247332	+	0.968931i	$0.420446\pi$
$830$	4661.15	0.194929
$831$	20805.2	0.868502
$832$	20224.7	0.842746
$833$	0	0
$834$	8885.46	0.368919
$835$	−7055.93	−0.292432
$836$	4666.30	0.193047
$837$	−22577.0	−0.932349
$838$	−38873.1	−1.60244
$839$	29112.9	1.19796	0.598980	−	0.800764i	$-0.295573\pi$
$839$	29112.9	1.19796	0.598980	+	0.800764i	$0.295573\pi$
$840$	0	0
$841$	−22662.8	−0.929224
$842$	−14135.4	−0.578549
$843$	8169.73	0.333785
$844$	16658.1	0.679377
$845$	2932.00	0.119366
$846$	−4634.60	−0.188346
$847$	0	0
$848$	376.781	0.0152579
$849$	14467.2	0.584819
$850$	10009.9	0.403924
$851$	4136.83	0.166638
$852$	−13508.7	−0.543194
$853$	24892.1	0.999169	0.499584	−	0.866265i	$-0.333486\pi$
$853$	24892.1	0.999169	0.499584	+	0.866265i	$0.333486\pi$
$854$	0	0
$855$	−1144.46	−0.0457773
$856$	−41649.0	−1.66301
$857$	27299.2	1.08812	0.544062	−	0.839045i	$-0.316886\pi$
$857$	27299.2	1.08812	0.544062	+	0.839045i	$0.316886\pi$
$858$	−3461.55	−0.137733
$859$	6975.13	0.277053	0.138526	−	0.990359i	$-0.455763\pi$
$859$	6975.13	0.277053	0.138526	+	0.990359i	$0.455763\pi$
$860$	6744.73	0.267434
$861$	0	0
$862$	−32247.9	−1.27421
$863$	−4481.56	−0.176772	−0.0883858	−	0.996086i	$-0.528171\pi$
$863$	−4481.56	−0.176772	−0.0883858	+	0.996086i	$0.528171\pi$
$864$	−21879.6	−0.861526
$865$	−7949.67	−0.312482
$866$	−34465.2	−1.35240
$867$	12741.8	0.499116
$868$	0	0
$869$	−5377.73	−0.209928
$870$	−978.543	−0.0381330
$871$	−6427.80	−0.250055
$872$	5955.54	0.231284
$873$	−26291.6	−1.01928
$874$	−2090.45	−0.0809044
$875$	0	0
$876$	41942.7	1.61771
$877$	−7207.96	−0.277532	−0.138766	−	0.990325i	$-0.544314\pi$
$877$	−7207.96	−0.277532	−0.138766	+	0.990325i	$0.544314\pi$
$878$	60063.9	2.30872
$879$	20549.7	0.788536
$880$	92.7699	0.00355372
$881$	38413.7	1.46900	0.734502	−	0.678607i	$-0.237416\pi$
$881$	38413.7	1.46900	0.734502	+	0.678607i	$0.237416\pi$
$882$	0	0
$883$	−8705.04	−0.331764	−0.165882	−	0.986146i	$-0.553047\pi$
$883$	−8705.04	−0.331764	−0.165882	+	0.986146i	$0.553047\pi$
$884$	5813.86	0.221200
$885$	−3646.58	−0.138507
$886$	−54240.6	−2.05671
$887$	44100.0	1.66937	0.834687	−	0.550725i	$-0.185649\pi$
$887$	44100.0	1.66937	0.834687	+	0.550725i	$0.185649\pi$
$888$	−19436.6	−0.734514
$889$	0	0
$890$	891.834	0.0335892
$891$	−1814.09	−0.0682091
$892$	−38683.6	−1.45204
$893$	−1676.25	−0.0628146
$894$	26587.8	0.994663
$895$	−4594.08	−0.171579
$896$	0	0
$897$	965.451	0.0359370
$898$	34882.2	1.29625
$899$	7302.55	0.270916
$900$	30978.0	1.14733
$901$	−1473.15	−0.0544702
$902$	7814.09	0.288449
$903$	0	0
$904$	−11829.5	−0.435224
$905$	−7958.54	−0.292322
$906$	−34217.9	−1.25476
$907$	19921.2	0.729296	0.364648	−	0.931145i	$-0.381189\pi$
$907$	19921.2	0.729296	0.364648	+	0.931145i	$0.381189\pi$
$908$	−56722.9	−2.07314
$909$	−31087.3	−1.13432
$910$	0	0
$911$	−14446.7	−0.525402	−0.262701	−	0.964877i	$-0.584613\pi$
$911$	−14446.7	−0.525402	−0.262701	+	0.964877i	$0.584613\pi$
$912$	407.793	0.0148063
$913$	6038.81	0.218900
$914$	54294.4	1.96488
$915$	3310.29	0.119601
$916$	−9217.66	−0.332489
$917$	0	0
$918$	−10573.8	−0.380160
$919$	44381.1	1.59303	0.796516	−	0.604617i	$-0.206674\pi$
$919$	44381.1	1.59303	0.796516	+	0.604617i	$0.206674\pi$
$920$	−623.186	−0.0223324
$921$	−19001.8	−0.679839
$922$	−19467.0	−0.695347
$923$	−9092.80	−0.324261
$924$	0	0
$925$	−35613.5	−1.26591
$926$	66521.6	2.36073
$927$	−2375.37	−0.0841612
$928$	7076.97	0.250337
$929$	3455.30	0.122029	0.0610145	−	0.998137i	$-0.480566\pi$
$929$	3455.30	0.122029	0.0610145	+	0.998137i	$0.480566\pi$
$930$	−4139.74	−0.145965
$931$	0	0
$932$	24911.8	0.875550
$933$	−15364.1	−0.539117
$934$	−76183.8	−2.66896
$935$	−362.714	−0.0126866
$936$	11380.0	0.397400
$937$	14962.8	0.521678	0.260839	−	0.965382i	$-0.416001\pi$
$937$	14962.8	0.521678	0.260839	+	0.965382i	$0.416001\pi$
$938$	0	0
$939$	25967.7	0.902474
$940$	−1269.03	−0.0440331
$941$	6116.95	0.211909	0.105955	−	0.994371i	$-0.466210\pi$
$941$	6116.95	0.211909	0.105955	+	0.994371i	$0.466210\pi$
$942$	30783.0	1.06472
$943$	−2179.41	−0.0752611
$944$	−3259.77	−0.112390
$945$	0	0
$946$	14035.6	0.482384
$947$	53343.7	1.83045	0.915225	−	0.402943i	$-0.132013\pi$
$947$	53343.7	1.83045	0.915225	+	0.402943i	$0.132013\pi$
$948$	−17896.4	−0.613130
$949$	28231.9	0.965696
$950$	17996.4	0.614611
$951$	−608.117	−0.0207356
$952$	0	0
$953$	1979.62	0.0672887	0.0336443	−	0.999434i	$-0.489289\pi$
$953$	1979.62	0.0672887	0.0336443	+	0.999434i	$0.489289\pi$
$954$	−7322.82	−0.248517
$955$	−5238.24	−0.177493
$956$	−85115.6	−2.87954
$957$	−1267.76	−0.0428223
$958$	13762.8	0.464149
$959$	0	0
$960$	−4199.02	−0.141170
$961$	1102.58	0.0370104
$962$	−33224.4	−1.11351
$963$	33608.1	1.12462
$964$	−61191.2	−2.04444
$965$	3199.07	0.106717
$966$	0	0
$967$	−38892.0	−1.29336	−0.646681	−	0.762761i	$-0.723843\pi$
$967$	−38892.0	−1.29336	−0.646681	+	0.762761i	$0.723843\pi$
$968$	2894.78	0.0961175
$969$	−1594.40	−0.0528582
$970$	−11563.3	−0.382758
$971$	47826.1	1.58065	0.790325	−	0.612688i	$-0.209912\pi$
$971$	47826.1	1.58065	0.790325	+	0.612688i	$0.209912\pi$
$972$	−51803.8	−1.70947
$973$	0	0
$974$	−25550.1	−0.840533
$975$	−8311.45	−0.273005
$976$	2959.15	0.0970493
$977$	20840.5	0.682442	0.341221	−	0.939983i	$-0.389160\pi$
$977$	20840.5	0.682442	0.341221	+	0.939983i	$0.389160\pi$
$978$	−46348.1	−1.51539
$979$	1155.43	0.0377197
$980$	0	0
$981$	−4805.74	−0.156407
$982$	−35686.2	−1.15966
$983$	54430.5	1.76609	0.883044	−	0.469290i	$-0.155490\pi$
$983$	54430.5	1.76609	0.883044	+	0.469290i	$0.155490\pi$
$984$	10239.8	0.331740
$985$	5714.05	0.184837
$986$	3420.10	0.110465
$987$	0	0
$988$	10452.5	0.336579
$989$	−3914.62	−0.125862
$990$	−1803.00	−0.0578821
$991$	−35161.9	−1.12710	−0.563550	−	0.826082i	$-0.690565\pi$
$991$	−35161.9	−1.12710	−0.563550	+	0.826082i	$0.690565\pi$
$992$	29939.2	0.958238
$993$	−9369.50	−0.299428
$994$	0	0
$995$	8293.99	0.264258
$996$	20096.4	0.639335
$997$	−26961.6	−0.856451	−0.428225	−	0.903672i	$-0.640861\pi$
$997$	−26961.6	−0.856451	−0.428225	+	0.903672i	$0.640861\pi$
$998$	−31499.4	−0.999096
$999$	37619.8	1.19143

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.4.a.g.1.4		4
7.6	odd	2		77.4.a.d.1.4	✓	4
21.20	even	2		693.4.a.l.1.1		4
28.27	even	2		1232.4.a.s.1.3		4
35.34	odd	2		1925.4.a.p.1.1		4
77.76	even	2		847.4.a.d.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.4	✓	4	7.6	odd	2
539.4.a.g.1.4		4	1.1	even	1	trivial
693.4.a.l.1.1		4	21.20	even	2
847.4.a.d.1.1		4	77.76	even	2
1232.4.a.s.1.3		4	28.27	even	2
1925.4.a.p.1.1		4	35.34	odd	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:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q+4.60395 q^{2} -2.77399 q^{3} +13.1964 q^{4} -1.84418 q^{5} -12.7713 q^{6} +23.9238 q^{8} -19.3050 q^{9} -8.49053 q^{10} -11.0000 q^{11} -36.6066 q^{12} -24.6401 q^{13} +5.11574 q^{15} +4.57310 q^{16} -17.8800 q^{17} -88.8792 q^{18} -32.1459 q^{19} -24.3365 q^{20} -50.6435 q^{22} +14.1248 q^{23} -66.3643 q^{24} -121.599 q^{25} -113.442 q^{26} +128.450 q^{27} -41.5471 q^{29} +23.5526 q^{30} -175.766 q^{31} -170.336 q^{32} +30.5139 q^{33} -82.3187 q^{34} -254.756 q^{36} +292.877 q^{37} -147.998 q^{38} +68.3513 q^{39} -44.1199 q^{40} -154.296 q^{41} -277.144 q^{43} -145.160 q^{44} +35.6019 q^{45} +65.0300 q^{46} +52.1450 q^{47} -12.6857 q^{48} -559.836 q^{50} +49.5989 q^{51} -325.160 q^{52} +82.3907 q^{53} +591.375 q^{54} +20.2860 q^{55} +89.1723 q^{57} -191.281 q^{58} -712.816 q^{59} +67.5092 q^{60} +647.078 q^{61} -809.217 q^{62} -820.804 q^{64} +45.4408 q^{65} +140.484 q^{66} +260.867 q^{67} -235.951 q^{68} -39.1821 q^{69} +369.025 q^{71} -461.849 q^{72} -1145.77 q^{73} +1348.39 q^{74} +337.314 q^{75} -424.209 q^{76} +314.686 q^{78} +488.885 q^{79} -8.43362 q^{80} +164.917 q^{81} -710.372 q^{82} -548.982 q^{83} +32.9740 q^{85} -1275.96 q^{86} +115.251 q^{87} -263.162 q^{88} -105.039 q^{89} +163.910 q^{90} +186.396 q^{92} +487.572 q^{93} +240.073 q^{94} +59.2829 q^{95} +472.510 q^{96} +1361.91 q^{97} +212.355 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 14 q^{3} + 26 q^{4} - 10 q^{5} - 14 q^{6} - 18 q^{8} + 76 q^{9} + 2 q^{10} - 44 q^{11} - 70 q^{12} - 58 q^{13} + 284 q^{15} + 2 q^{16} - 4 q^{17} - 62 q^{18} - 258 q^{19} - 182 q^{20} + 22 q^{22}+ \cdots - 836 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - 14 * q^3 + 26 * q^4 - 10 * q^5 - 14 * q^6 - 18 * q^8 + 76 * q^9 + 2 * q^10 - 44 * q^11 - 70 * q^12 - 58 * q^13 + 284 * q^15 + 2 * q^16 - 4 * q^17 - 62 * q^18 - 258 * q^19 - 182 * q^20 + 22 * q^22 + 8 * q^23 + 498 * q^24 + 80 * q^25 + 482 * q^26 - 428 * q^27 - 396 * q^29 - 628 * q^30 + 56 * q^31 + 134 * q^32 + 154 * q^33 - 472 * q^34 - 418 * q^36 + 84 * q^37 + 942 * q^38 - 412 * q^39 + 1026 * q^40 - 52 * q^41 + 408 * q^43 - 286 * q^44 - 826 * q^45 + 368 * q^46 - 8 * q^47 - 982 * q^48 - 1642 * q^50 - 388 * q^51 - 2030 * q^52 + 624 * q^53 - 92 * q^54 + 110 * q^55 + 48 * q^57 + 864 * q^58 + 238 * q^59 + 1420 * q^60 + 162 * q^61 - 688 * q^62 - 902 * q^64 - 32 * q^65 + 154 * q^66 + 1340 * q^67 + 1384 * q^68 - 2416 * q^69 + 1788 * q^71 - 2622 * q^72 - 1456 * q^73 + 996 * q^74 + 806 * q^75 - 3042 * q^76 - 2632 * q^78 - 1324 * q^79 - 2342 * q^80 + 1444 * q^81 - 1984 * q^82 - 450 * q^83 - 1736 * q^85 - 4380 * q^86 - 588 * q^87 + 198 * q^88 + 3072 * q^89 + 218 * q^90 + 544 * q^92 - 1264 * q^93 + 1696 * q^94 + 24 * q^95 - 862 * q^96 + 652 * q^97 - 836 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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