LMFDB - Embedded newform 207.4.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,4,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	207.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:	$12.2133953712$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.334189.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 16x^{2} - 5x + 4 $ x^4 - 2x^3 - 16x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.22031$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0323756 q^{2} -7.99895 q^{4} -14.1026 q^{5} -14.0109 q^{7} -0.517976 q^{8} +O(q^{10})$	\(q+0.0323756 q^{2} -7.99895 q^{4} -14.1026 q^{5} -14.0109 q^{7} -0.517976 q^{8} -0.456580 q^{10} +55.5140 q^{11} -18.3149 q^{13} -0.453613 q^{14} +63.9748 q^{16} -10.0273 q^{17} +161.106 q^{19} +112.806 q^{20} +1.79730 q^{22} +23.0000 q^{23} +73.8833 q^{25} -0.592956 q^{26} +112.073 q^{28} -183.185 q^{29} -144.762 q^{31} +6.21503 q^{32} -0.324640 q^{34} +197.591 q^{35} +181.411 q^{37} +5.21591 q^{38} +7.30481 q^{40} -77.7996 q^{41} +315.335 q^{43} -444.054 q^{44} +0.744639 q^{46} +524.190 q^{47} -146.693 q^{49} +2.39202 q^{50} +146.500 q^{52} -73.7334 q^{53} -782.892 q^{55} +7.25733 q^{56} -5.93071 q^{58} +132.892 q^{59} +236.683 q^{61} -4.68676 q^{62} -511.598 q^{64} +258.288 q^{65} +493.624 q^{67} +80.2079 q^{68} +6.39712 q^{70} +806.060 q^{71} +1011.91 q^{73} +5.87328 q^{74} -1288.68 q^{76} -777.804 q^{77} -599.386 q^{79} -902.212 q^{80} -2.51881 q^{82} +642.245 q^{83} +141.411 q^{85} +10.2092 q^{86} -28.7549 q^{88} -883.399 q^{89} +256.609 q^{91} -183.976 q^{92} +16.9710 q^{94} -2272.02 q^{95} -71.2938 q^{97} -4.74929 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 20 q^{4} - 14 q^{5} + 16 q^{7} + 63 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 20 * q^4 - 14 * q^5 + 16 * q^7 + 63 * q^8	$ 4 q - 2 q^{2} + 20 q^{4} - 14 q^{5} + 16 q^{7} + 63 q^{8} - 70 q^{10} - 8 q^{11} + 111 q^{13} + 144 q^{14} + 64 q^{16} - 98 q^{17} + 96 q^{19} - 140 q^{20} + 220 q^{22} + 92 q^{23} + 184 q^{25} + 229 q^{26} + 282 q^{28} - 21 q^{29} - 193 q^{31} + 432 q^{32} + 666 q^{34} + 752 q^{35} + 170 q^{37} - 748 q^{38} - 26 q^{40} + 125 q^{41} + 2 q^{43} - 830 q^{44} - 46 q^{46} + 677 q^{47} + 1220 q^{49} - 414 q^{50} + 2247 q^{52} + 230 q^{53} - 972 q^{55} + 2174 q^{56} - 1835 q^{58} + 1140 q^{59} + 754 q^{61} - 443 q^{62} - 805 q^{64} - 1318 q^{65} + 488 q^{67} - 284 q^{68} - 3820 q^{70} + 401 q^{71} + 1509 q^{73} - 1366 q^{74} - 3832 q^{76} - 736 q^{77} - 838 q^{79} - 2846 q^{80} - 949 q^{82} - 142 q^{83} + 112 q^{85} - 918 q^{86} - 404 q^{88} - 2342 q^{89} + 292 q^{91} + 460 q^{92} + 1567 q^{94} + 956 q^{95} + 1062 q^{97} - 2478 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 20 * q^4 - 14 * q^5 + 16 * q^7 + 63 * q^8 - 70 * q^10 - 8 * q^11 + 111 * q^13 + 144 * q^14 + 64 * q^16 - 98 * q^17 + 96 * q^19 - 140 * q^20 + 220 * q^22 + 92 * q^23 + 184 * q^25 + 229 * q^26 + 282 * q^28 - 21 * q^29 - 193 * q^31 + 432 * q^32 + 666 * q^34 + 752 * q^35 + 170 * q^37 - 748 * q^38 - 26 * q^40 + 125 * q^41 + 2 * q^43 - 830 * q^44 - 46 * q^46 + 677 * q^47 + 1220 * q^49 - 414 * q^50 + 2247 * q^52 + 230 * q^53 - 972 * q^55 + 2174 * q^56 - 1835 * q^58 + 1140 * q^59 + 754 * q^61 - 443 * q^62 - 805 * q^64 - 1318 * q^65 + 488 * q^67 - 284 * q^68 - 3820 * q^70 + 401 * q^71 + 1509 * q^73 - 1366 * q^74 - 3832 * q^76 - 736 * q^77 - 838 * q^79 - 2846 * q^80 - 949 * q^82 - 142 * q^83 + 112 * q^85 - 918 * q^86 - 404 * q^88 - 2342 * q^89 + 292 * q^91 + 460 * q^92 + 1567 * q^94 + 956 * q^95 + 1062 * q^97 - 2478 * 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 $

$2$	0.0323756	0.0114465	0.00572325	−	0.999984i	$-0.498178\pi$
$2$	0.0323756	0.0114465	0.00572325	+	0.999984i	$0.498178\pi$
$3$	0	0
$3$	0	0
$4$	−7.99895	−0.999869
$5$	−14.1026	−1.26137	−0.630687	−	0.776037i	$-0.717227\pi$
$5$	−14.1026	−1.26137	−0.630687	+	0.776037i	$0.717227\pi$
$6$	0	0
$7$	−14.0109	−0.756520	−0.378260	−	0.925699i	$-0.623477\pi$
$7$	−14.0109	−0.756520	−0.378260	+	0.925699i	$0.623477\pi$
$8$	−0.517976	−0.0228915
$9$	0	0
$10$	−0.456580	−0.0144383
$11$	55.5140	1.52165	0.760823	−	0.648959i	$-0.224796\pi$
$11$	55.5140	1.52165	0.760823	+	0.648959i	$0.224796\pi$
$12$	0	0
$13$	−18.3149	−0.390742	−0.195371	−	0.980729i	$-0.562591\pi$
$13$	−18.3149	−0.390742	−0.195371	+	0.980729i	$0.562591\pi$
$14$	−0.453613	−0.00865951
$15$	0	0
$16$	63.9748	0.999607
$17$	−10.0273	−0.143058	−0.0715288	−	0.997439i	$-0.522788\pi$
$17$	−10.0273	−0.143058	−0.0715288	+	0.997439i	$0.522788\pi$
$18$	0	0
$19$	161.106	1.94528	0.972639	−	0.232321i	$-0.0746318\pi$
$19$	161.106	1.94528	0.972639	+	0.232321i	$0.0746318\pi$
$20$	112.806	1.26121
$21$	0	0
$22$	1.79730	0.0174175
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	73.8833	0.591066
$26$	−0.592956	−0.00447263
$27$	0	0
$28$	112.073	0.756421
$29$	−183.185	−1.17298	−0.586492	−	0.809955i	$-0.699491\pi$
$29$	−183.185	−1.17298	−0.586492	+	0.809955i	$0.699491\pi$
$30$	0	0
$31$	−144.762	−0.838710	−0.419355	−	0.907822i	$-0.637744\pi$
$31$	−144.762	−0.838710	−0.419355	+	0.907822i	$0.637744\pi$
$32$	6.21503	0.0343335
$33$	0	0
$34$	−0.324640	−0.00163751
$35$	197.591	0.954255
$36$	0	0
$37$	181.411	0.806047	0.403023	−	0.915190i	$-0.367959\pi$
$37$	181.411	0.806047	0.403023	+	0.915190i	$0.367959\pi$
$38$	5.21591	0.0222666
$39$	0	0
$40$	7.30481	0.0288748
$41$	−77.7996	−0.296348	−0.148174	−	0.988961i	$-0.547340\pi$
$41$	−77.7996	−0.296348	−0.148174	+	0.988961i	$0.547340\pi$
$42$	0	0
$43$	315.335	1.11833	0.559164	−	0.829057i	$-0.311122\pi$
$43$	315.335	1.11833	0.559164	+	0.829057i	$0.311122\pi$
$44$	−444.054	−1.52145
$45$	0	0
$46$	0.744639	0.00238676
$47$	524.190	1.62683	0.813414	−	0.581685i	$-0.197607\pi$
$47$	524.190	1.62683	0.813414	+	0.581685i	$0.197607\pi$
$48$	0	0
$49$	−146.693	−0.427678
$50$	2.39202	0.00676565
$51$	0	0
$52$	146.500	0.390690
$53$	−73.7334	−0.191095	−0.0955477	−	0.995425i	$-0.530460\pi$
$53$	−73.7334	−0.191095	−0.0955477	+	0.995425i	$0.530460\pi$
$54$	0	0
$55$	−782.892	−1.91937
$56$	7.25733	0.0173179
$57$	0	0
$58$	−5.93071	−0.0134266
$59$	132.892	0.293239	0.146619	−	0.989193i	$-0.453161\pi$
$59$	132.892	0.293239	0.146619	+	0.989193i	$0.453161\pi$
$60$	0	0
$61$	236.683	0.496789	0.248394	−	0.968659i	$-0.420097\pi$
$61$	236.683	0.496789	0.248394	+	0.968659i	$0.420097\pi$
$62$	−4.68676	−0.00960030
$63$	0	0
$64$	−511.598	−0.999214
$65$	258.288	0.492872
$66$	0	0
$67$	493.624	0.900086	0.450043	−	0.893007i	$-0.351409\pi$
$67$	493.624	0.900086	0.450043	+	0.893007i	$0.351409\pi$
$68$	80.2079	0.143039
$69$	0	0
$70$	6.39712	0.0109229
$71$	806.060	1.34735	0.673674	−	0.739029i	$-0.264715\pi$
$71$	806.060	1.34735	0.673674	+	0.739029i	$0.264715\pi$
$72$	0	0
$73$	1011.91	1.62241	0.811203	−	0.584764i	$-0.198813\pi$
$73$	1011.91	1.62241	0.811203	+	0.584764i	$0.198813\pi$
$74$	5.87328	0.00922642
$75$	0	0
$76$	−1288.68	−1.94502
$77$	−777.804	−1.15116
$78$	0	0
$79$	−599.386	−0.853623	−0.426811	−	0.904341i	$-0.640363\pi$
$79$	−599.386	−0.853623	−0.426811	+	0.904341i	$0.640363\pi$
$80$	−902.212	−1.26088
$81$	0	0
$82$	−2.51881	−0.00339215
$83$	642.245	0.849344	0.424672	−	0.905347i	$-0.360390\pi$
$83$	642.245	0.849344	0.424672	+	0.905347i	$0.360390\pi$
$84$	0	0
$85$	141.411	0.180449
$86$	10.2092	0.0128009
$87$	0	0
$88$	−28.7549	−0.0348328
$89$	−883.399	−1.05214	−0.526068	−	0.850442i	$-0.676334\pi$
$89$	−883.399	−1.05214	−0.526068	+	0.850442i	$0.676334\pi$
$90$	0	0
$91$	256.609	0.295604
$92$	−183.976	−0.208487
$93$	0	0
$94$	16.9710	0.0186215
$95$	−2272.02	−2.45373
$96$	0	0
$97$	−71.2938	−0.0746266	−0.0373133	−	0.999304i	$-0.511880\pi$
$97$	−71.2938	−0.0746266	−0.0373133	+	0.999304i	$0.511880\pi$
$98$	−4.74929	−0.00489542
$99$	0	0
$100$	−590.989	−0.590989
$101$	942.689	0.928723	0.464361	−	0.885646i	$-0.346284\pi$
$101$	942.689	0.928723	0.464361	+	0.885646i	$0.346284\pi$
$102$	0	0
$103$	−556.624	−0.532484	−0.266242	−	0.963906i	$-0.585782\pi$
$103$	−556.624	−0.532484	−0.266242	+	0.963906i	$0.585782\pi$
$104$	9.48668	0.00894467
$105$	0	0
$106$	−2.38716	−0.00218738
$107$	−647.477	−0.584990	−0.292495	−	0.956267i	$-0.594486\pi$
$107$	−647.477	−0.584990	−0.292495	+	0.956267i	$0.594486\pi$
$108$	0	0
$109$	−1349.13	−1.18554	−0.592768	−	0.805373i	$-0.701965\pi$
$109$	−1349.13	−1.18554	−0.592768	+	0.805373i	$0.701965\pi$
$110$	−25.3466	−0.0219700
$111$	0	0
$112$	−896.348	−0.756222
$113$	284.549	0.236886	0.118443	−	0.992961i	$-0.462210\pi$
$113$	284.549	0.236886	0.118443	+	0.992961i	$0.462210\pi$
$114$	0	0
$115$	−324.360	−0.263015
$116$	1465.28	1.17283
$117$	0	0
$118$	4.30247	0.00335656
$119$	140.492	0.108226
$120$	0	0
$121$	1750.81	1.31541
$122$	7.66275	0.00568650
$123$	0	0
$124$	1157.94	0.838600
$125$	720.878	0.515819
$126$	0	0
$127$	−753.553	−0.526512	−0.263256	−	0.964726i	$-0.584796\pi$
$127$	−753.553	−0.526512	−0.263256	+	0.964726i	$0.584796\pi$
$128$	−66.2835	−0.0457710
$129$	0	0
$130$	8.36223	0.00564166
$131$	769.226	0.513035	0.256518	−	0.966540i	$-0.417425\pi$
$131$	769.226	0.513035	0.256518	+	0.966540i	$0.417425\pi$
$132$	0	0
$133$	−2257.25	−1.47164
$134$	15.9814	0.0103028
$135$	0	0
$136$	5.19390	0.00327481
$137$	211.638	0.131982	0.0659908	−	0.997820i	$-0.478979\pi$
$137$	211.638	0.131982	0.0659908	+	0.997820i	$0.478979\pi$
$138$	0	0
$139$	1998.54	1.21953	0.609763	−	0.792584i	$-0.291265\pi$
$139$	1998.54	1.21953	0.609763	+	0.792584i	$0.291265\pi$
$140$	−1580.52	−0.954130
$141$	0	0
$142$	26.0967	0.0154224
$143$	−1016.73	−0.594570
$144$	0	0
$145$	2583.38	1.47957
$146$	32.7614	0.0185709
$147$	0	0
$148$	−1451.10	−0.805941
$149$	−499.968	−0.274893	−0.137446	−	0.990509i	$-0.543889\pi$
$149$	−499.968	−0.274893	−0.137446	+	0.990509i	$0.543889\pi$
$150$	0	0
$151$	501.652	0.270357	0.135178	−	0.990821i	$-0.456839\pi$
$151$	501.652	0.270357	0.135178	+	0.990821i	$0.456839\pi$
$152$	−83.4491	−0.0445304
$153$	0	0
$154$	−25.1819	−0.0131767
$155$	2041.52	1.05793
$156$	0	0
$157$	1686.36	0.857237	0.428619	−	0.903485i	$-0.359000\pi$
$157$	1686.36	0.857237	0.428619	+	0.903485i	$0.359000\pi$
$158$	−19.4055	−0.00977100
$159$	0	0
$160$	−87.6481	−0.0433074
$161$	−322.252	−0.157745
$162$	0	0
$163$	3183.54	1.52978	0.764890	−	0.644161i	$-0.222793\pi$
$163$	3183.54	1.52978	0.764890	+	0.644161i	$0.222793\pi$
$164$	622.315	0.296309
$165$	0	0
$166$	20.7931	0.00972202
$167$	3771.37	1.74753	0.873764	−	0.486350i	$-0.161672\pi$
$167$	3771.37	1.74753	0.873764	+	0.486350i	$0.161672\pi$
$168$	0	0
$169$	−1861.56	−0.847321
$170$	4.57827	0.00206551
$171$	0	0
$172$	−2522.35	−1.11818
$173$	−129.941	−0.0571052	−0.0285526	−	0.999592i	$-0.509090\pi$
$173$	−129.941	−0.0571052	−0.0285526	+	0.999592i	$0.509090\pi$
$174$	0	0
$175$	−1035.17	−0.447153
$176$	3551.50	1.52105
$177$	0	0
$178$	−28.6006	−0.0120433
$179$	−810.182	−0.338301	−0.169150	−	0.985590i	$-0.554102\pi$
$179$	−810.182	−0.338301	−0.169150	+	0.985590i	$0.554102\pi$
$180$	0	0
$181$	2430.33	0.998039	0.499019	−	0.866591i	$-0.333694\pi$
$181$	2430.33	0.998039	0.499019	+	0.866591i	$0.333694\pi$
$182$	8.30788	0.00338363
$183$	0	0
$184$	−11.9134	−0.00477321
$185$	−2558.36	−1.01673
$186$	0	0
$187$	−556.656	−0.217683
$188$	−4192.97	−1.62661
$189$	0	0
$190$	−73.5579	−0.0280866
$191$	−2462.87	−0.933022	−0.466511	−	0.884515i	$-0.654489\pi$
$191$	−2462.87	−0.933022	−0.466511	+	0.884515i	$0.654489\pi$
$192$	0	0
$193$	4021.66	1.49992	0.749962	−	0.661481i	$-0.230072\pi$
$193$	4021.66	1.49992	0.749962	+	0.661481i	$0.230072\pi$
$194$	−2.30818	−0.000854214 0
$195$	0	0
$196$	1173.39	0.427622
$197$	−1811.60	−0.655182	−0.327591	−	0.944820i	$-0.606237\pi$
$197$	−1811.60	−0.655182	−0.327591	+	0.944820i	$0.606237\pi$
$198$	0	0
$199$	723.923	0.257877	0.128939	−	0.991653i	$-0.458843\pi$
$199$	723.923	0.257877	0.128939	+	0.991653i	$0.458843\pi$
$200$	−38.2698	−0.0135304
$201$	0	0
$202$	30.5201	0.0106306
$203$	2566.59	0.887385
$204$	0	0
$205$	1097.18	0.373805
$206$	−18.0211	−0.00609508
$207$	0	0
$208$	−1171.69	−0.390588
$209$	8943.65	2.96003
$210$	0	0
$211$	−583.883	−0.190503	−0.0952515	−	0.995453i	$-0.530366\pi$
$211$	−583.883	−0.190503	−0.0952515	+	0.995453i	$0.530366\pi$
$212$	589.790	0.191070
$213$	0	0
$214$	−20.9625	−0.00669610
$215$	−4447.04	−1.41063
$216$	0	0
$217$	2028.25	0.634501
$218$	−43.6790	−0.0135703
$219$	0	0
$220$	6262.31	1.91911
$221$	183.649	0.0558985
$222$	0	0
$223$	3157.57	0.948191	0.474096	−	0.880473i	$-0.342775\pi$
$223$	3157.57	0.948191	0.474096	+	0.880473i	$0.342775\pi$
$224$	−87.0785	−0.0259740
$225$	0	0
$226$	9.21245	0.00271152
$227$	2219.80	0.649044	0.324522	−	0.945878i	$-0.394797\pi$
$227$	2219.80	0.649044	0.324522	+	0.945878i	$0.394797\pi$
$228$	0	0
$229$	−4398.24	−1.26919	−0.634593	−	0.772846i	$-0.718832\pi$
$229$	−4398.24	−1.26919	−0.634593	+	0.772846i	$0.718832\pi$
$230$	−10.5013	−0.00301060
$231$	0	0
$232$	94.8852	0.0268514
$233$	−2112.84	−0.594065	−0.297032	−	0.954867i	$-0.595997\pi$
$233$	−2112.84	−0.594065	−0.297032	+	0.954867i	$0.595997\pi$
$234$	0	0
$235$	−7392.43	−2.05204
$236$	−1063.00	−0.293200
$237$	0	0
$238$	4.54852	0.00123881
$239$	1548.69	0.419147	0.209574	−	0.977793i	$-0.432792\pi$
$239$	1548.69	0.419147	0.209574	+	0.977793i	$0.432792\pi$
$240$	0	0
$241$	−2516.92	−0.672734	−0.336367	−	0.941731i	$-0.609198\pi$
$241$	−2516.92	−0.672734	−0.336367	+	0.941731i	$0.609198\pi$
$242$	56.6834	0.0150568
$243$	0	0
$244$	−1893.21	−0.496724
$245$	2068.76	0.539462
$246$	0	0
$247$	−2950.64	−0.760101
$248$	74.9832	0.0191993
$249$	0	0
$250$	23.3389	0.00590432
$251$	−386.310	−0.0971461	−0.0485731	−	0.998820i	$-0.515467\pi$
$251$	−386.310	−0.0971461	−0.0485731	+	0.998820i	$0.515467\pi$
$252$	0	0
$253$	1276.82	0.317285
$254$	−24.3967	−0.00602672
$255$	0	0
$256$	4090.63	0.998690
$257$	−3816.61	−0.926357	−0.463178	−	0.886265i	$-0.653291\pi$
$257$	−3816.61	−0.926357	−0.463178	+	0.886265i	$0.653291\pi$
$258$	0	0
$259$	−2541.73	−0.609790
$260$	−2066.03	−0.492807
$261$	0	0
$262$	24.9042	0.00587246
$263$	−4510.11	−1.05744	−0.528718	−	0.848798i	$-0.677327\pi$
$263$	−4510.11	−1.05744	−0.528718	+	0.848798i	$0.677327\pi$
$264$	0	0
$265$	1039.83	0.241043
$266$	−73.0798	−0.0168452
$267$	0	0
$268$	−3948.47	−0.899968
$269$	550.730	0.124827	0.0624137	−	0.998050i	$-0.480120\pi$
$269$	550.730	0.124827	0.0624137	+	0.998050i	$0.480120\pi$
$270$	0	0
$271$	−897.873	−0.201262	−0.100631	−	0.994924i	$-0.532086\pi$
$271$	−897.873	−0.201262	−0.100631	+	0.994924i	$0.532086\pi$
$272$	−641.495	−0.143001
$273$	0	0
$274$	6.85192	0.00151073
$275$	4101.56	0.899394
$276$	0	0
$277$	2288.67	0.496437	0.248219	−	0.968704i	$-0.420155\pi$
$277$	2288.67	0.496437	0.248219	+	0.968704i	$0.420155\pi$
$278$	64.7040	0.0139593
$279$	0	0
$280$	−102.347	−0.0218443
$281$	2587.28	0.549268	0.274634	−	0.961549i	$-0.411443\pi$
$281$	2587.28	0.549268	0.274634	+	0.961549i	$0.411443\pi$
$282$	0	0
$283$	1488.48	0.312653	0.156326	−	0.987705i	$-0.450035\pi$
$283$	1488.48	0.312653	0.156326	+	0.987705i	$0.450035\pi$
$284$	−6447.63	−1.34717
$285$	0	0
$286$	−32.9174	−0.00680576
$287$	1090.05	0.224193
$288$	0	0
$289$	−4812.45	−0.979535
$290$	83.6384	0.0169359
$291$	0	0
$292$	−8094.26	−1.62219
$293$	2821.24	0.562522	0.281261	−	0.959631i	$-0.409247\pi$
$293$	2821.24	0.562522	0.281261	+	0.959631i	$0.409247\pi$
$294$	0	0
$295$	−1874.12	−0.369884
$296$	−93.9664	−0.0184516
$297$	0	0
$298$	−16.1868	−0.00314656
$299$	−421.243	−0.0814753
$300$	0	0
$301$	−4418.14	−0.846037
$302$	16.2413	0.00309464
$303$	0	0
$304$	10306.7	1.94451
$305$	−3337.84	−0.626637
$306$	0	0
$307$	−4085.72	−0.759558	−0.379779	−	0.925077i	$-0.624000\pi$
$307$	−4085.72	−0.759558	−0.379779	+	0.925077i	$0.624000\pi$
$308$	6221.61	1.15100
$309$	0	0
$310$	66.0954	0.0121096
$311$	−4044.84	−0.737498	−0.368749	−	0.929529i	$-0.620214\pi$
$311$	−4044.84	−0.737498	−0.368749	+	0.929529i	$0.620214\pi$
$312$	0	0
$313$	5111.54	0.923072	0.461536	−	0.887122i	$-0.347299\pi$
$313$	5111.54	0.923072	0.461536	+	0.887122i	$0.347299\pi$
$314$	54.5970	0.00981238
$315$	0	0
$316$	4794.46	0.853511
$317$	7017.95	1.24343	0.621715	−	0.783244i	$-0.286436\pi$
$317$	7017.95	1.24343	0.621715	+	0.783244i	$0.286436\pi$
$318$	0	0
$319$	−10169.3	−1.78487
$320$	7214.85	1.26038
$321$	0	0
$322$	−10.4331	−0.00180563
$323$	−1615.46	−0.278287
$324$	0	0
$325$	−1353.17	−0.230954
$326$	103.069	0.0175106
$327$	0	0
$328$	40.2983	0.00678385
$329$	−7344.39	−1.23073
$330$	0	0
$331$	−6537.02	−1.08552	−0.542760	−	0.839888i	$-0.682621\pi$
$331$	−6537.02	−1.08552	−0.542760	+	0.839888i	$0.682621\pi$
$332$	−5137.28	−0.849232
$333$	0	0
$334$	122.100	0.0200031
$335$	−6961.38	−1.13535
$336$	0	0
$337$	−838.254	−0.135497	−0.0677486	−	0.997702i	$-0.521582\pi$
$337$	−838.254	−0.135497	−0.0677486	+	0.997702i	$0.521582\pi$
$338$	−60.2693	−0.00969887
$339$	0	0
$340$	−1131.14	−0.180426
$341$	−8036.32	−1.27622
$342$	0	0
$343$	6861.07	1.08007
$344$	−163.336	−0.0256002
$345$	0	0
$346$	−4.20691	−0.000653655 0
$347$	−7101.86	−1.09870	−0.549348	−	0.835593i	$-0.685124\pi$
$347$	−7101.86	−1.09870	−0.549348	+	0.835593i	$0.685124\pi$
$348$	0	0
$349$	−2675.10	−0.410300	−0.205150	−	0.978731i	$-0.565768\pi$
$349$	−2675.10	−0.410300	−0.205150	+	0.978731i	$0.565768\pi$
$350$	−33.5144	−0.00511835
$351$	0	0
$352$	345.021	0.0522435
$353$	8032.84	1.21118	0.605588	−	0.795778i	$-0.292938\pi$
$353$	8032.84	1.21118	0.605588	+	0.795778i	$0.292938\pi$
$354$	0	0
$355$	−11367.5	−1.69951
$356$	7066.27	1.05200
$357$	0	0
$358$	−26.2301	−0.00387236
$359$	−4828.72	−0.709889	−0.354944	−	0.934887i	$-0.615500\pi$
$359$	−4828.72	−0.709889	−0.354944	+	0.934887i	$0.615500\pi$
$360$	0	0
$361$	19096.2	2.78411
$362$	78.6834	0.0114241
$363$	0	0
$364$	−2052.60	−0.295565
$365$	−14270.6	−2.04646
$366$	0	0
$367$	89.8597	0.0127810	0.00639052	−	0.999980i	$-0.497966\pi$
$367$	89.8597	0.0127810	0.00639052	+	0.999980i	$0.497966\pi$
$368$	1471.42	0.208432
$369$	0	0
$370$	−82.8285	−0.0116380
$371$	1033.07	0.144567
$372$	0	0
$373$	5196.16	0.721306	0.360653	−	0.932700i	$-0.382554\pi$
$373$	5196.16	0.721306	0.360653	+	0.932700i	$0.382554\pi$
$374$	−18.0221	−0.00249171
$375$	0	0
$376$	−271.518	−0.0372406
$377$	3355.01	0.458333
$378$	0	0
$379$	4900.54	0.664179	0.332090	−	0.943248i	$-0.392246\pi$
$379$	4900.54	0.664179	0.332090	+	0.943248i	$0.392246\pi$
$380$	18173.7	2.45340
$381$	0	0
$382$	−79.7370	−0.0106798
$383$	−9973.73	−1.33064	−0.665318	−	0.746560i	$-0.731704\pi$
$383$	−9973.73	−1.33064	−0.665318	+	0.746560i	$0.731704\pi$
$384$	0	0
$385$	10969.1	1.45204
$386$	130.204	0.0171689
$387$	0	0
$388$	570.275	0.0746169
$389$	6277.96	0.818266	0.409133	−	0.912475i	$-0.365831\pi$
$389$	6277.96	0.818266	0.409133	+	0.912475i	$0.365831\pi$
$390$	0	0
$391$	−230.628	−0.0298296
$392$	75.9837	0.00979019
$393$	0	0
$394$	−58.6515	−0.00749955
$395$	8452.90	1.07674
$396$	0	0
$397$	−11309.9	−1.42979	−0.714895	−	0.699232i	$-0.753526\pi$
$397$	−11309.9	−1.42979	−0.714895	+	0.699232i	$0.753526\pi$
$398$	23.4375	0.00295179
$399$	0	0
$400$	4726.67	0.590834
$401$	14306.5	1.78163	0.890813	−	0.454371i	$-0.150136\pi$
$401$	14306.5	1.78163	0.890813	+	0.454371i	$0.150136\pi$
$402$	0	0
$403$	2651.30	0.327719
$404$	−7540.52	−0.928601
$405$	0	0
$406$	83.0949	0.0101575
$407$	10070.8	1.22652
$408$	0	0
$409$	4363.34	0.527514	0.263757	−	0.964589i	$-0.415038\pi$
$409$	4363.34	0.527514	0.263757	+	0.964589i	$0.415038\pi$
$410$	35.5218	0.00427877
$411$	0	0
$412$	4452.41	0.532414
$413$	−1861.94	−0.221841
$414$	0	0
$415$	−9057.32	−1.07134
$416$	−113.828	−0.0134155
$417$	0	0
$418$	289.556	0.0338820
$419$	−12304.9	−1.43468	−0.717341	−	0.696722i	$-0.754641\pi$
$419$	−12304.9	−1.43468	−0.717341	+	0.696722i	$0.754641\pi$
$420$	0	0
$421$	−8353.05	−0.966990	−0.483495	−	0.875347i	$-0.660633\pi$
$421$	−8353.05	−0.966990	−0.483495	+	0.875347i	$0.660633\pi$
$422$	−18.9036	−0.00218059
$423$	0	0
$424$	38.1921	0.00437446
$425$	−740.850	−0.0845565
$426$	0	0
$427$	−3316.15	−0.375831
$428$	5179.14	0.584914
$429$	0	0
$430$	−143.976	−0.0161468
$431$	15959.9	1.78367	0.891835	−	0.452361i	$-0.149418\pi$
$431$	15959.9	1.78367	0.891835	+	0.452361i	$0.149418\pi$
$432$	0	0
$433$	5779.60	0.641454	0.320727	−	0.947172i	$-0.396073\pi$
$433$	5779.60	0.641454	0.320727	+	0.947172i	$0.396073\pi$
$434$	65.6659	0.00726282
$435$	0	0
$436$	10791.6	1.18538
$437$	3705.44	0.405619
$438$	0	0
$439$	795.488	0.0864842	0.0432421	−	0.999065i	$-0.486231\pi$
$439$	795.488	0.0864842	0.0432421	+	0.999065i	$0.486231\pi$
$440$	405.519	0.0439372
$441$	0	0
$442$	5.94575	0.000639843 0
$443$	−7357.81	−0.789120	−0.394560	−	0.918870i	$-0.629103\pi$
$443$	−7357.81	−0.789120	−0.394560	+	0.918870i	$0.629103\pi$
$444$	0	0
$445$	12458.2	1.32714
$446$	102.228	0.0108535
$447$	0	0
$448$	7167.96	0.755925
$449$	6672.34	0.701308	0.350654	−	0.936505i	$-0.385959\pi$
$449$	6672.34	0.701308	0.350654	+	0.936505i	$0.385959\pi$
$450$	0	0
$451$	−4318.97	−0.450936
$452$	−2276.09	−0.236855
$453$	0	0
$454$	71.8673	0.00742929
$455$	−3618.86	−0.372867
$456$	0	0
$457$	−6324.72	−0.647392	−0.323696	−	0.946161i	$-0.604925\pi$
$457$	−6324.72	−0.647392	−0.323696	+	0.946161i	$0.604925\pi$
$458$	−142.396	−0.0145278
$459$	0	0
$460$	2594.54	0.262980
$461$	641.556	0.0648162	0.0324081	−	0.999475i	$-0.489682\pi$
$461$	641.556	0.0648162	0.0324081	+	0.999475i	$0.489682\pi$
$462$	0	0
$463$	714.350	0.0717034	0.0358517	−	0.999357i	$-0.488586\pi$
$463$	714.350	0.0717034	0.0358517	+	0.999357i	$0.488586\pi$
$464$	−11719.2	−1.17252
$465$	0	0
$466$	−68.4047	−0.00679997
$467$	−3937.64	−0.390176	−0.195088	−	0.980786i	$-0.562499\pi$
$467$	−3937.64	−0.390176	−0.195088	+	0.980786i	$0.562499\pi$
$468$	0	0
$469$	−6916.13	−0.680933
$470$	−239.335	−0.0234887
$471$	0	0
$472$	−68.8349	−0.00671268
$473$	17505.5	1.70170
$474$	0	0
$475$	11903.1	1.14979
$476$	−1123.79	−0.108212
$477$	0	0
$478$	50.1397	0.00479777
$479$	2252.09	0.214824	0.107412	−	0.994215i	$-0.465744\pi$
$479$	2252.09	0.214824	0.107412	+	0.994215i	$0.465744\pi$
$480$	0	0
$481$	−3322.52	−0.314956
$482$	−81.4868	−0.00770046
$483$	0	0
$484$	−14004.6	−1.31523
$485$	1005.43	0.0941322
$486$	0	0
$487$	11821.8	1.09999	0.549995	−	0.835168i	$-0.314630\pi$
$487$	11821.8	1.09999	0.549995	+	0.835168i	$0.314630\pi$
$488$	−122.596	−0.0113723
$489$	0	0
$490$	66.9773	0.00617496
$491$	−20198.1	−1.85647	−0.928235	−	0.371993i	$-0.878674\pi$
$491$	−20198.1	−1.85647	−0.928235	+	0.371993i	$0.878674\pi$
$492$	0	0
$493$	1836.85	0.167804
$494$	−95.5289	−0.00870051
$495$	0	0
$496$	−9261.12	−0.838380
$497$	−11293.7	−1.01930
$498$	0	0
$499$	−633.684	−0.0568488	−0.0284244	−	0.999596i	$-0.509049\pi$
$499$	−633.684	−0.0568488	−0.0284244	+	0.999596i	$0.509049\pi$
$500$	−5766.27	−0.515751
$501$	0	0
$502$	−12.5070	−0.00111198
$503$	−10068.6	−0.892518	−0.446259	−	0.894904i	$-0.647244\pi$
$503$	−10068.6	−0.892518	−0.446259	+	0.894904i	$0.647244\pi$
$504$	0	0
$505$	−13294.4	−1.17147
$506$	41.3379	0.00363181
$507$	0	0
$508$	6027.63	0.526443
$509$	4287.40	0.373351	0.186675	−	0.982422i	$-0.440229\pi$
$509$	4287.40	0.373351	0.186675	+	0.982422i	$0.440229\pi$
$510$	0	0
$511$	−14177.9	−1.22738
$512$	662.705	0.0572026
$513$	0	0
$514$	−123.565	−0.0106035
$515$	7849.85	0.671662
$516$	0	0
$517$	29099.9	2.47546
$518$	−82.2902	−0.00697997
$519$	0	0
$520$	−133.787	−0.0112826
$521$	19842.9	1.66858	0.834291	−	0.551324i	$-0.185877\pi$
$521$	19842.9	1.66858	0.834291	+	0.551324i	$0.185877\pi$
$522$	0	0
$523$	−10894.9	−0.910897	−0.455448	−	0.890262i	$-0.650521\pi$
$523$	−10894.9	−0.910897	−0.455448	+	0.890262i	$0.650521\pi$
$524$	−6153.00	−0.512968
$525$	0	0
$526$	−146.018	−0.0121039
$527$	1451.57	0.119984
$528$	0	0
$529$	529.000	0.0434783
$530$	33.6652	0.00275910
$531$	0	0
$532$	18055.6	1.47145
$533$	1424.89	0.115795
$534$	0	0
$535$	9131.11	0.737892
$536$	−255.685	−0.0206043
$537$	0	0
$538$	17.8302	0.00142884
$539$	−8143.54	−0.650774
$540$	0	0
$541$	4405.04	0.350069	0.175035	−	0.984562i	$-0.443996\pi$
$541$	4405.04	0.350069	0.175035	+	0.984562i	$0.443996\pi$
$542$	−29.0692	−0.00230374
$543$	0	0
$544$	−62.3200	−0.00491167
$545$	19026.3	1.49541
$546$	0	0
$547$	10031.2	0.784099	0.392050	−	0.919944i	$-0.371766\pi$
$547$	10031.2	0.784099	0.392050	+	0.919944i	$0.371766\pi$
$548$	−1692.88	−0.131964
$549$	0	0
$550$	132.790	0.0102949
$551$	−29512.2	−2.28178
$552$	0	0
$553$	8397.97	0.645783
$554$	74.0972	0.00568247
$555$	0	0
$556$	−15986.2	−1.21937
$557$	−23278.3	−1.77079	−0.885396	−	0.464837i	$-0.846113\pi$
$557$	−23278.3	−1.77079	−0.885396	+	0.464837i	$0.846113\pi$
$558$	0	0
$559$	−5775.32	−0.436977
$560$	12640.8	0.953880
$561$	0	0
$562$	83.7648	0.00628720
$563$	20989.8	1.57125	0.785625	−	0.618702i	$-0.212341\pi$
$563$	20989.8	1.57125	0.785625	+	0.618702i	$0.212341\pi$
$564$	0	0
$565$	−4012.88	−0.298802
$566$	48.1904	0.00357878
$567$	0	0
$568$	−417.520	−0.0308428
$569$	11942.2	0.879868	0.439934	−	0.898030i	$-0.355002\pi$
$569$	11942.2	0.879868	0.439934	+	0.898030i	$0.355002\pi$
$570$	0	0
$571$	−13981.9	−1.02473	−0.512367	−	0.858767i	$-0.671231\pi$
$571$	−13981.9	−1.02473	−0.512367	+	0.858767i	$0.671231\pi$
$572$	8132.81	0.594492
$573$	0	0
$574$	35.2909	0.00256623
$575$	1699.32	0.123246
$576$	0	0
$577$	−5024.27	−0.362501	−0.181251	−	0.983437i	$-0.558015\pi$
$577$	−5024.27	−0.362501	−0.181251	+	0.983437i	$0.558015\pi$
$578$	−155.806	−0.0112123
$579$	0	0
$580$	−20664.3	−1.47938
$581$	−8998.45	−0.642545
$582$	0	0
$583$	−4093.24	−0.290780
$584$	−524.148	−0.0371393
$585$	0	0
$586$	91.3395	0.00643891
$587$	22464.4	1.57957	0.789784	−	0.613385i	$-0.210193\pi$
$587$	22464.4	1.57957	0.789784	+	0.613385i	$0.210193\pi$
$588$	0	0
$589$	−23322.0	−1.63152
$590$	−60.6760	−0.00423388
$591$	0	0
$592$	11605.7	0.805730
$593$	−14073.1	−0.974561	−0.487281	−	0.873245i	$-0.662011\pi$
$593$	−14073.1	−0.974561	−0.487281	+	0.873245i	$0.662011\pi$
$594$	0	0
$595$	−1981.30	−0.136513
$596$	3999.22	0.274857
$597$	0	0
$598$	−13.6380	−0.000932607 0
$599$	8952.63	0.610675	0.305338	−	0.952244i	$-0.401231\pi$
$599$	8952.63	0.610675	0.305338	+	0.952244i	$0.401231\pi$
$600$	0	0
$601$	−20522.5	−1.39290	−0.696449	−	0.717607i	$-0.745238\pi$
$601$	−20522.5	−1.39290	−0.696449	+	0.717607i	$0.745238\pi$
$602$	−143.040	−0.00968417
$603$	0	0
$604$	−4012.69	−0.270321
$605$	−24690.9	−1.65922
$606$	0	0
$607$	23022.8	1.53949	0.769743	−	0.638353i	$-0.220384\pi$
$607$	23022.8	1.53949	0.769743	+	0.638353i	$0.220384\pi$
$608$	1001.28	0.0667883
$609$	0	0
$610$	−108.065	−0.00717281
$611$	−9600.48	−0.635669
$612$	0	0
$613$	−6159.74	−0.405856	−0.202928	−	0.979194i	$-0.565046\pi$
$613$	−6159.74	−0.405856	−0.202928	+	0.979194i	$0.565046\pi$
$614$	−132.278	−0.00869429
$615$	0	0
$616$	402.884	0.0263517
$617$	−26889.1	−1.75448	−0.877241	−	0.480050i	$-0.840619\pi$
$617$	−26889.1	−1.75448	−0.877241	+	0.480050i	$0.840619\pi$
$618$	0	0
$619$	−6478.47	−0.420665	−0.210333	−	0.977630i	$-0.567455\pi$
$619$	−6478.47	−0.420665	−0.210333	+	0.977630i	$0.567455\pi$
$620$	−16330.0	−1.05779
$621$	0	0
$622$	−130.954	−0.00844178
$623$	12377.3	0.795962
$624$	0	0
$625$	−19401.7	−1.24171
$626$	165.489	0.0105659
$627$	0	0
$628$	−13489.1	−0.857125
$629$	−1819.06	−0.115311
$630$	0	0
$631$	2956.54	0.186526	0.0932630	−	0.995642i	$-0.470270\pi$
$631$	2956.54	0.186526	0.0932630	+	0.995642i	$0.470270\pi$
$632$	310.468	0.0195407
$633$	0	0
$634$	227.210	0.0142329
$635$	10627.1	0.664129
$636$	0	0
$637$	2686.68	0.167111
$638$	−329.238	−0.0204305
$639$	0	0
$640$	934.770	0.0577344
$641$	−20604.4	−1.26961	−0.634807	−	0.772670i	$-0.718921\pi$
$641$	−20604.4	−1.26961	−0.634807	+	0.772670i	$0.718921\pi$
$642$	0	0
$643$	−23773.4	−1.45806	−0.729030	−	0.684482i	$-0.760028\pi$
$643$	−23773.4	−1.45806	−0.729030	+	0.684482i	$0.760028\pi$
$644$	2577.68	0.157725
$645$	0	0
$646$	−52.3015	−0.00318541
$647$	24907.8	1.51349	0.756744	−	0.653712i	$-0.226789\pi$
$647$	24907.8	1.51349	0.756744	+	0.653712i	$0.226789\pi$
$648$	0	0
$649$	7377.38	0.446206
$650$	−43.8096	−0.00264362
$651$	0	0
$652$	−25465.0	−1.52958
$653$	18523.9	1.11010	0.555050	−	0.831817i	$-0.312699\pi$
$653$	18523.9	1.11010	0.555050	+	0.831817i	$0.312699\pi$
$654$	0	0
$655$	−10848.1	−0.647130
$656$	−4977.22	−0.296231
$657$	0	0
$658$	−237.779	−0.0140875
$659$	−24408.5	−1.44282	−0.721411	−	0.692507i	$-0.756506\pi$
$659$	−24408.5	−1.44282	−0.721411	+	0.692507i	$0.756506\pi$
$660$	0	0
$661$	20341.7	1.19697	0.598487	−	0.801133i	$-0.295769\pi$
$661$	20341.7	1.19697	0.598487	+	0.801133i	$0.295769\pi$
$662$	−211.640	−0.0124254
$663$	0	0
$664$	−332.667	−0.0194428
$665$	31833.1	1.85629
$666$	0	0
$667$	−4213.24	−0.244584
$668$	−30167.0	−1.74730
$669$	0	0
$670$	−225.379	−0.0129957
$671$	13139.2	0.755937
$672$	0	0
$673$	12282.3	0.703490	0.351745	−	0.936096i	$-0.385588\pi$
$673$	12282.3	0.703490	0.351745	+	0.936096i	$0.385588\pi$
$674$	−27.1390	−0.00155097
$675$	0	0
$676$	14890.6	0.847210
$677$	−21868.0	−1.24144	−0.620721	−	0.784031i	$-0.713160\pi$
$677$	−21868.0	−1.24144	−0.620721	+	0.784031i	$0.713160\pi$
$678$	0	0
$679$	998.893	0.0564565
$680$	−73.2475	−0.00413076
$681$	0	0
$682$	−260.181	−0.0146083
$683$	22877.0	1.28165	0.640823	−	0.767688i	$-0.278593\pi$
$683$	22877.0	1.28165	0.640823	+	0.767688i	$0.278593\pi$
$684$	0	0
$685$	−2984.65	−0.166478
$686$	222.131	0.0123630
$687$	0	0
$688$	20173.5	1.11789
$689$	1350.42	0.0746689
$690$	0	0
$691$	24499.1	1.34875	0.674376	−	0.738388i	$-0.264413\pi$
$691$	24499.1	1.34875	0.674376	+	0.738388i	$0.264413\pi$
$692$	1039.39	0.0570977
$693$	0	0
$694$	−229.927	−0.0125762
$695$	−28184.6	−1.53828
$696$	0	0
$697$	780.120	0.0423948
$698$	−86.6080	−0.00469650
$699$	0	0
$700$	8280.31	0.447095
$701$	25020.5	1.34809	0.674044	−	0.738691i	$-0.264556\pi$
$701$	25020.5	1.34809	0.674044	+	0.738691i	$0.264556\pi$
$702$	0	0
$703$	29226.4	1.56799
$704$	−28400.8	−1.52045
$705$	0	0
$706$	260.068	0.0138637
$707$	−13208.0	−0.702597
$708$	0	0
$709$	−15069.4	−0.798225	−0.399112	−	0.916902i	$-0.630682\pi$
$709$	−15069.4	−0.798225	−0.399112	+	0.916902i	$0.630682\pi$
$710$	−368.031	−0.0194535
$711$	0	0
$712$	457.579	0.0240850
$713$	−3329.52	−0.174883
$714$	0	0
$715$	14338.6	0.749976
$716$	6480.61	0.338256
$717$	0	0
$718$	−156.333	−0.00812575
$719$	23894.6	1.23938	0.619692	−	0.784845i	$-0.287257\pi$
$719$	23894.6	1.23938	0.619692	+	0.784845i	$0.287257\pi$
$720$	0	0
$721$	7798.83	0.402835
$722$	618.251	0.0318683
$723$	0	0
$724$	−19440.1	−0.997908
$725$	−13534.3	−0.693311
$726$	0	0
$727$	32641.3	1.66520	0.832598	−	0.553877i	$-0.186852\pi$
$727$	32641.3	1.66520	0.832598	+	0.553877i	$0.186852\pi$
$728$	−132.917	−0.00676682
$729$	0	0
$730$	−462.020	−0.0234249
$731$	−3161.96	−0.159985
$732$	0	0
$733$	−628.010	−0.0316454	−0.0158227	−	0.999875i	$-0.505037\pi$
$733$	−628.010	−0.0316454	−0.0158227	+	0.999875i	$0.505037\pi$
$734$	2.90926	0.000146298 0
$735$	0	0
$736$	142.946	0.00715904
$737$	27403.0	1.36961
$738$	0	0
$739$	−20111.4	−1.00109	−0.500547	−	0.865709i	$-0.666868\pi$
$739$	−20111.4	−1.00109	−0.500547	+	0.865709i	$0.666868\pi$
$740$	20464.2	1.01659
$741$	0	0
$742$	33.4464	0.00165479
$743$	−19508.4	−0.963251	−0.481625	−	0.876377i	$-0.659953\pi$
$743$	−19508.4	−0.963251	−0.481625	+	0.876377i	$0.659953\pi$
$744$	0	0
$745$	7050.85	0.346743
$746$	168.229	0.00825643
$747$	0	0
$748$	4452.66	0.217654
$749$	9071.77	0.442557
$750$	0	0
$751$	5494.75	0.266986	0.133493	−	0.991050i	$-0.457381\pi$
$751$	5494.75	0.266986	0.133493	+	0.991050i	$0.457381\pi$
$752$	33534.9	1.62619
$753$	0	0
$754$	108.620	0.00524632
$755$	−7074.60	−0.341021
$756$	0	0
$757$	−3411.29	−0.163785	−0.0818926	−	0.996641i	$-0.526096\pi$
$757$	−3411.29	−0.163785	−0.0818926	+	0.996641i	$0.526096\pi$
$758$	158.658	0.00760253
$759$	0	0
$760$	1176.85	0.0561695
$761$	15927.5	0.758700	0.379350	−	0.925253i	$-0.376148\pi$
$761$	15927.5	0.758700	0.379350	+	0.925253i	$0.376148\pi$
$762$	0	0
$763$	18902.6	0.896882
$764$	19700.4	0.932899
$765$	0	0
$766$	−322.906	−0.0152311
$767$	−2433.91	−0.114581
$768$	0	0
$769$	20621.3	0.967000	0.483500	−	0.875344i	$-0.339365\pi$
$769$	20621.3	0.967000	0.483500	+	0.875344i	$0.339365\pi$
$770$	355.130	0.0166208
$771$	0	0
$772$	−32169.1	−1.49973
$773$	−18163.8	−0.845156	−0.422578	−	0.906326i	$-0.638875\pi$
$773$	−18163.8	−0.845156	−0.422578	+	0.906326i	$0.638875\pi$
$774$	0	0
$775$	−10695.5	−0.495733
$776$	36.9284	0.00170832
$777$	0	0
$778$	203.253	0.00936628
$779$	−12534.0	−0.576479
$780$	0	0
$781$	44747.6	2.05019
$782$	−7.46672	−0.000341444 0
$783$	0	0
$784$	−9384.69	−0.427510
$785$	−23782.1	−1.08130
$786$	0	0
$787$	−30750.8	−1.39282	−0.696408	−	0.717646i	$-0.745220\pi$
$787$	−30750.8	−1.39282	−0.696408	+	0.717646i	$0.745220\pi$
$788$	14490.9	0.655096
$789$	0	0
$790$	273.668	0.0123249
$791$	−3986.80	−0.179209
$792$	0	0
$793$	−4334.82	−0.194116
$794$	−366.164	−0.0163661
$795$	0	0
$796$	−5790.63	−0.257843
$797$	32871.8	1.46095	0.730475	−	0.682939i	$-0.239299\pi$
$797$	32871.8	1.46095	0.730475	+	0.682939i	$0.239299\pi$
$798$	0	0
$799$	−5256.21	−0.232730
$800$	459.187	0.0202934
$801$	0	0
$802$	463.181	0.0203934
$803$	56175.5	2.46873
$804$	0	0
$805$	4544.59	0.198976
$806$	85.8375	0.00375124
$807$	0	0
$808$	−488.290	−0.0212599
$809$	2591.22	0.112611	0.0563055	−	0.998414i	$-0.482068\pi$
$809$	2591.22	0.112611	0.0563055	+	0.998414i	$0.482068\pi$
$810$	0	0
$811$	37339.5	1.61673	0.808364	−	0.588683i	$-0.200353\pi$
$811$	37339.5	1.61673	0.808364	+	0.588683i	$0.200353\pi$
$812$	−20530.0	−0.887269
$813$	0	0
$814$	326.050	0.0140393
$815$	−44896.2	−1.92963
$816$	0	0
$817$	50802.4	2.17546
$818$	141.266	0.00603819
$819$	0	0
$820$	−8776.26	−0.373756
$821$	−20494.9	−0.871226	−0.435613	−	0.900134i	$-0.643468\pi$
$821$	−20494.9	−0.871226	−0.435613	+	0.900134i	$0.643468\pi$
$822$	0	0
$823$	−13568.2	−0.574675	−0.287338	−	0.957829i	$-0.592770\pi$
$823$	−13568.2	−0.574675	−0.287338	+	0.957829i	$0.592770\pi$
$824$	288.318	0.0121894
$825$	0	0
$826$	−60.2816	−0.00253930
$827$	39430.4	1.65796	0.828978	−	0.559281i	$-0.188923\pi$
$827$	39430.4	1.65796	0.828978	+	0.559281i	$0.188923\pi$
$828$	0	0
$829$	14439.0	0.604931	0.302465	−	0.953160i	$-0.402190\pi$
$829$	14439.0	0.604931	0.302465	+	0.953160i	$0.402190\pi$
$830$	−293.236	−0.0122631
$831$	0	0
$832$	9369.86	0.390434
$833$	1470.94	0.0611825
$834$	0	0
$835$	−53186.1	−2.20429
$836$	−71539.8	−2.95964
$837$	0	0
$838$	−398.378	−0.0164221
$839$	−33140.6	−1.36369	−0.681847	−	0.731495i	$-0.738823\pi$
$839$	−33140.6	−1.36369	−0.681847	+	0.731495i	$0.738823\pi$
$840$	0	0
$841$	9167.56	0.375889
$842$	−270.435	−0.0110687
$843$	0	0
$844$	4670.45	0.190478
$845$	26252.9	1.06879
$846$	0	0
$847$	−24530.4	−0.995131
$848$	−4717.08	−0.191020
$849$	0	0
$850$	−23.9855	−0.000967877 0
$851$	4172.45	0.168072
$852$	0	0
$853$	27379.7	1.09902	0.549510	−	0.835487i	$-0.314815\pi$
$853$	27379.7	1.09902	0.549510	+	0.835487i	$0.314815\pi$
$854$	−107.362	−0.00430195
$855$	0	0
$856$	335.378	0.0133913
$857$	−20497.8	−0.817025	−0.408513	−	0.912753i	$-0.633952\pi$
$857$	−20497.8	−0.817025	−0.408513	+	0.912753i	$0.633952\pi$
$858$	0	0
$859$	−26376.7	−1.04768	−0.523842	−	0.851816i	$-0.675502\pi$
$859$	−26376.7	−1.04768	−0.523842	+	0.851816i	$0.675502\pi$
$860$	35571.6	1.41045
$861$	0	0
$862$	516.712	0.0204168
$863$	32543.2	1.28364	0.641821	−	0.766854i	$-0.278179\pi$
$863$	32543.2	1.28364	0.641821	+	0.766854i	$0.278179\pi$
$864$	0	0
$865$	1832.50	0.0720311
$866$	187.118	0.00734241
$867$	0	0
$868$	−16223.9	−0.634418
$869$	−33274.3	−1.29891
$870$	0	0
$871$	−9040.67	−0.351701
$872$	698.818	0.0271387
$873$	0	0
$874$	119.966	0.00464292
$875$	−10100.2	−0.390227
$876$	0	0
$877$	6671.86	0.256890	0.128445	−	0.991717i	$-0.459001\pi$
$877$	6671.86	0.256890	0.128445	+	0.991717i	$0.459001\pi$
$878$	25.7544	0.000989942 0
$879$	0	0
$880$	−50085.4	−1.91861
$881$	25432.9	0.972595	0.486298	−	0.873793i	$-0.338347\pi$
$881$	25432.9	0.972595	0.486298	+	0.873793i	$0.338347\pi$
$882$	0	0
$883$	16292.4	0.620933	0.310467	−	0.950584i	$-0.399515\pi$
$883$	16292.4	0.620933	0.310467	+	0.950584i	$0.399515\pi$
$884$	−1469.00	−0.0558912
$885$	0	0
$886$	−238.214	−0.00903267
$887$	−8139.80	−0.308126	−0.154063	−	0.988061i	$-0.549236\pi$
$887$	−8139.80	−0.308126	−0.154063	+	0.988061i	$0.549236\pi$
$888$	0	0
$889$	10558.0	0.398317
$890$	403.343	0.0151911
$891$	0	0
$892$	−25257.3	−0.948067
$893$	84450.2	3.16463
$894$	0	0
$895$	11425.7	0.426724
$896$	928.695	0.0346267
$897$	0	0
$898$	216.021	0.00802753
$899$	26518.1	0.983793
$900$	0	0
$901$	739.347	0.0273376
$902$	−139.829	−0.00516164
$903$	0	0
$904$	−147.390	−0.00542268
$905$	−34274.0	−1.25890
$906$	0	0
$907$	16087.6	0.588954	0.294477	−	0.955659i	$-0.404855\pi$
$907$	16087.6	0.588954	0.294477	+	0.955659i	$0.404855\pi$
$908$	−17756.0	−0.648959
$909$	0	0
$910$	−117.163	−0.00426803
$911$	36379.0	1.32304	0.661521	−	0.749927i	$-0.269911\pi$
$911$	36379.0	1.32304	0.661521	+	0.749927i	$0.269911\pi$
$912$	0	0
$913$	35653.6	1.29240
$914$	−204.767	−0.00741037
$915$	0	0
$916$	35181.3	1.26902
$917$	−10777.6	−0.388121
$918$	0	0
$919$	−10077.9	−0.361739	−0.180869	−	0.983507i	$-0.557891\pi$
$919$	−10077.9	−0.361739	−0.180869	+	0.983507i	$0.557891\pi$
$920$	168.011	0.00602081
$921$	0	0
$922$	20.7708	0.000741919 0
$923$	−14762.9	−0.526465
$924$	0	0
$925$	13403.2	0.476427
$926$	23.1275	0.000820753 0
$927$	0	0
$928$	−1138.50	−0.0402726
$929$	7768.50	0.274355	0.137178	−	0.990546i	$-0.456197\pi$
$929$	7768.50	0.274355	0.137178	+	0.990546i	$0.456197\pi$
$930$	0	0
$931$	−23633.2	−0.831952
$932$	16900.5	0.593987
$933$	0	0
$934$	−127.484	−0.00446615
$935$	7850.30	0.274580
$936$	0	0
$937$	−1611.36	−0.0561804	−0.0280902	−	0.999605i	$-0.508943\pi$
$937$	−1611.36	−0.0561804	−0.0280902	+	0.999605i	$0.508943\pi$
$938$	−223.914	−0.00779430
$939$	0	0
$940$	59131.7	2.05177
$941$	1974.16	0.0683909	0.0341954	−	0.999415i	$-0.489113\pi$
$941$	1974.16	0.0683909	0.0341954	+	0.999415i	$0.489113\pi$
$942$	0	0
$943$	−1789.39	−0.0617927
$944$	8501.76	0.293124
$945$	0	0
$946$	566.751	0.0194785
$947$	−57242.2	−1.96422	−0.982112	−	0.188297i	$-0.939703\pi$
$947$	−57242.2	−1.96422	−0.982112	+	0.188297i	$0.939703\pi$
$948$	0	0
$949$	−18533.1	−0.633942
$950$	385.369	0.0131611
$951$	0	0
$952$	−72.7715	−0.00247746
$953$	14723.4	0.500458	0.250229	−	0.968187i	$-0.419494\pi$
$953$	14723.4	0.500458	0.250229	+	0.968187i	$0.419494\pi$
$954$	0	0
$955$	34732.9	1.17689
$956$	−12387.9	−0.419092
$957$	0	0
$958$	72.9128	0.00245898
$959$	−2965.25	−0.0998467
$960$	0	0
$961$	−8834.99	−0.296566
$962$	−107.569	−0.00360515
$963$	0	0
$964$	20132.7	0.672646
$965$	−56715.9	−1.89197
$966$	0	0
$967$	−46140.9	−1.53443	−0.767214	−	0.641391i	$-0.778358\pi$
$967$	−46140.9	−1.53443	−0.767214	+	0.641391i	$0.778358\pi$
$968$	−906.875	−0.0301117
$969$	0	0
$970$	32.5513	0.00107748
$971$	−15349.8	−0.507310	−0.253655	−	0.967295i	$-0.581633\pi$
$971$	−15349.8	−0.507310	−0.253655	+	0.967295i	$0.581633\pi$
$972$	0	0
$973$	−28001.5	−0.922596
$974$	382.737	0.0125910
$975$	0	0
$976$	15141.7	0.496594
$977$	1517.18	0.0496815	0.0248407	−	0.999691i	$-0.492092\pi$
$977$	1517.18	0.0496815	0.0248407	+	0.999691i	$0.492092\pi$
$978$	0	0
$979$	−49041.0	−1.60098
$980$	−16547.9	−0.539391
$981$	0	0
$982$	−653.926	−0.0212501
$983$	−22648.4	−0.734866	−0.367433	−	0.930050i	$-0.619763\pi$
$983$	−22648.4	−0.734866	−0.367433	+	0.930050i	$0.619763\pi$
$984$	0	0
$985$	25548.2	0.826430
$986$	59.4691	0.00192077
$987$	0	0
$988$	23602.1	0.760002
$989$	7252.70	0.233187
$990$	0	0
$991$	12346.4	0.395760	0.197880	−	0.980226i	$-0.436594\pi$
$991$	12346.4	0.395760	0.197880	+	0.980226i	$0.436594\pi$
$992$	−899.700	−0.0287959
$993$	0	0
$994$	−365.639	−0.0116674
$995$	−10209.2	−0.325280
$996$	0	0
$997$	−43566.0	−1.38390	−0.691950	−	0.721945i	$-0.743248\pi$
$997$	−43566.0	−1.38390	−0.691950	+	0.721945i	$0.743248\pi$
$998$	−20.5159	−0.000650721 0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.e.1.3		4
3.2	odd	2		23.4.a.b.1.2	✓	4
12.11	even	2		368.4.a.l.1.1		4
15.2	even	4		575.4.b.g.24.4		8
15.8	even	4		575.4.b.g.24.5		8
15.14	odd	2		575.4.a.i.1.3		4
21.20	even	2		1127.4.a.c.1.2		4
24.5	odd	2		1472.4.a.y.1.1		4
24.11	even	2		1472.4.a.bf.1.4		4
69.68	even	2		529.4.a.g.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.a.b.1.2	✓	4	3.2	odd	2
207.4.a.e.1.3		4	1.1	even	1	trivial
368.4.a.l.1.1		4	12.11	even	2
529.4.a.g.1.2		4	69.68	even	2
575.4.a.i.1.3		4	15.14	odd	2
575.4.b.g.24.4		8	15.2	even	4
575.4.b.g.24.5		8	15.8	even	4
1127.4.a.c.1.2		4	21.20	even	2
1472.4.a.y.1.1		4	24.5	odd	2
1472.4.a.bf.1.4		4	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0323756 q^{2} -7.99895 q^{4} -14.1026 q^{5} -14.0109 q^{7} -0.517976 q^{8} +O(q^{10})\)	\(q+0.0323756 q^{2} -7.99895 q^{4} -14.1026 q^{5} -14.0109 q^{7} -0.517976 q^{8} -0.456580 q^{10} +55.5140 q^{11} -18.3149 q^{13} -0.453613 q^{14} +63.9748 q^{16} -10.0273 q^{17} +161.106 q^{19} +112.806 q^{20} +1.79730 q^{22} +23.0000 q^{23} +73.8833 q^{25} -0.592956 q^{26} +112.073 q^{28} -183.185 q^{29} -144.762 q^{31} +6.21503 q^{32} -0.324640 q^{34} +197.591 q^{35} +181.411 q^{37} +5.21591 q^{38} +7.30481 q^{40} -77.7996 q^{41} +315.335 q^{43} -444.054 q^{44} +0.744639 q^{46} +524.190 q^{47} -146.693 q^{49} +2.39202 q^{50} +146.500 q^{52} -73.7334 q^{53} -782.892 q^{55} +7.25733 q^{56} -5.93071 q^{58} +132.892 q^{59} +236.683 q^{61} -4.68676 q^{62} -511.598 q^{64} +258.288 q^{65} +493.624 q^{67} +80.2079 q^{68} +6.39712 q^{70} +806.060 q^{71} +1011.91 q^{73} +5.87328 q^{74} -1288.68 q^{76} -777.804 q^{77} -599.386 q^{79} -902.212 q^{80} -2.51881 q^{82} +642.245 q^{83} +141.411 q^{85} +10.2092 q^{86} -28.7549 q^{88} -883.399 q^{89} +256.609 q^{91} -183.976 q^{92} +16.9710 q^{94} -2272.02 q^{95} -71.2938 q^{97} -4.74929 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 20 q^{4} - 14 q^{5} + 16 q^{7} + 63 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 20 * q^4 - 14 * q^5 + 16 * q^7 + 63 * q^8	\( 4 q - 2 q^{2} + 20 q^{4} - 14 q^{5} + 16 q^{7} + 63 q^{8} - 70 q^{10} - 8 q^{11} + 111 q^{13} + 144 q^{14} + 64 q^{16} - 98 q^{17} + 96 q^{19} - 140 q^{20} + 220 q^{22} + 92 q^{23} + 184 q^{25} + 229 q^{26} + 282 q^{28} - 21 q^{29} - 193 q^{31} + 432 q^{32} + 666 q^{34} + 752 q^{35} + 170 q^{37} - 748 q^{38} - 26 q^{40} + 125 q^{41} + 2 q^{43} - 830 q^{44} - 46 q^{46} + 677 q^{47} + 1220 q^{49} - 414 q^{50} + 2247 q^{52} + 230 q^{53} - 972 q^{55} + 2174 q^{56} - 1835 q^{58} + 1140 q^{59} + 754 q^{61} - 443 q^{62} - 805 q^{64} - 1318 q^{65} + 488 q^{67} - 284 q^{68} - 3820 q^{70} + 401 q^{71} + 1509 q^{73} - 1366 q^{74} - 3832 q^{76} - 736 q^{77} - 838 q^{79} - 2846 q^{80} - 949 q^{82} - 142 q^{83} + 112 q^{85} - 918 q^{86} - 404 q^{88} - 2342 q^{89} + 292 q^{91} + 460 q^{92} + 1567 q^{94} + 956 q^{95} + 1062 q^{97} - 2478 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 20 * q^4 - 14 * q^5 + 16 * q^7 + 63 * q^8 - 70 * q^10 - 8 * q^11 + 111 * q^13 + 144 * q^14 + 64 * q^16 - 98 * q^17 + 96 * q^19 - 140 * q^20 + 220 * q^22 + 92 * q^23 + 184 * q^25 + 229 * q^26 + 282 * q^28 - 21 * q^29 - 193 * q^31 + 432 * q^32 + 666 * q^34 + 752 * q^35 + 170 * q^37 - 748 * q^38 - 26 * q^40 + 125 * q^41 + 2 * q^43 - 830 * q^44 - 46 * q^46 + 677 * q^47 + 1220 * q^49 - 414 * q^50 + 2247 * q^52 + 230 * q^53 - 972 * q^55 + 2174 * q^56 - 1835 * q^58 + 1140 * q^59 + 754 * q^61 - 443 * q^62 - 805 * q^64 - 1318 * q^65 + 488 * q^67 - 284 * q^68 - 3820 * q^70 + 401 * q^71 + 1509 * q^73 - 1366 * q^74 - 3832 * q^76 - 736 * q^77 - 838 * q^79 - 2846 * q^80 - 949 * q^82 - 142 * q^83 + 112 * q^85 - 918 * q^86 - 404 * q^88 - 2342 * q^89 + 292 * q^91 + 460 * q^92 + 1567 * q^94 + 956 * q^95 + 1062 * q^97 - 2478 * q^98

Introduction

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