LMFDB - Embedded newform 1472.4.a.bf.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,4,Mod(1,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1472.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.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:	$86.8508115285$
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:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$5.22031$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.42170 q^{3} -14.1026 q^{5} +14.0109 q^{7} +14.2382 q^{9} +O(q^{10})$	$q+6.42170 q^{3} -14.1026 q^{5} +14.0109 q^{7} +14.2382 q^{9} -55.5140 q^{11} +18.3149 q^{13} -90.5626 q^{15} +10.0273 q^{17} +161.106 q^{19} +89.9740 q^{21} +23.0000 q^{23} +73.8833 q^{25} -81.9525 q^{27} -183.185 q^{29} +144.762 q^{31} -356.494 q^{33} -197.591 q^{35} -181.411 q^{37} +117.613 q^{39} +77.7996 q^{41} +315.335 q^{43} -200.795 q^{45} +524.190 q^{47} -146.693 q^{49} +64.3923 q^{51} -73.7334 q^{53} +782.892 q^{55} +1034.57 q^{57} -132.892 q^{59} -236.683 q^{61} +199.490 q^{63} -258.288 q^{65} +493.624 q^{67} +147.699 q^{69} +806.060 q^{71} +1011.91 q^{73} +474.456 q^{75} -777.804 q^{77} +599.386 q^{79} -910.705 q^{81} -642.245 q^{83} -141.411 q^{85} -1176.36 q^{87} +883.399 q^{89} +256.609 q^{91} +929.617 q^{93} -2272.02 q^{95} -71.2938 q^{97} -790.419 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 7 q^{3} - 14 q^{5} - 16 q^{7} - 33 q^{9}+O(q^{10}) $ 4 * q + 7 * q^3 - 14 * q^5 - 16 * q^7 - 33 * q^9	$ 4 q + 7 q^{3} - 14 q^{5} - 16 q^{7} - 33 q^{9} + 8 q^{11} - 111 q^{13} - 10 q^{15} + 98 q^{17} + 96 q^{19} - 180 q^{21} + 92 q^{23} + 184 q^{25} - 155 q^{27} - 21 q^{29} + 193 q^{31} - 418 q^{33} - 752 q^{35} - 170 q^{37} + 291 q^{39} - 125 q^{41} + 2 q^{43} - 168 q^{45} + 677 q^{47} + 1220 q^{49} - 340 q^{51} + 230 q^{53} + 972 q^{55} + 1322 q^{57} - 1140 q^{59} - 754 q^{61} + 1092 q^{63} + 1318 q^{65} + 488 q^{67} + 161 q^{69} + 401 q^{71} + 1509 q^{73} + 1401 q^{75} - 736 q^{77} + 838 q^{79} - 932 q^{81} + 142 q^{83} - 112 q^{85} - 2223 q^{87} + 2342 q^{89} + 292 q^{91} + 509 q^{93} + 956 q^{95} + 1062 q^{97} - 1498 q^{99}+O(q^{100}) $ 4 * q + 7 * q^3 - 14 * q^5 - 16 * q^7 - 33 * q^9 + 8 * q^11 - 111 * q^13 - 10 * q^15 + 98 * q^17 + 96 * q^19 - 180 * q^21 + 92 * q^23 + 184 * q^25 - 155 * q^27 - 21 * q^29 + 193 * q^31 - 418 * q^33 - 752 * q^35 - 170 * q^37 + 291 * q^39 - 125 * q^41 + 2 * q^43 - 168 * q^45 + 677 * q^47 + 1220 * q^49 - 340 * q^51 + 230 * q^53 + 972 * q^55 + 1322 * q^57 - 1140 * q^59 - 754 * q^61 + 1092 * q^63 + 1318 * q^65 + 488 * q^67 + 161 * q^69 + 401 * q^71 + 1509 * q^73 + 1401 * q^75 - 736 * q^77 + 838 * q^79 - 932 * q^81 + 142 * q^83 - 112 * q^85 - 2223 * q^87 + 2342 * q^89 + 292 * q^91 + 509 * q^93 + 956 * q^95 + 1062 * q^97 - 1498 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	6.42170	1.23586	0.617928	−	0.786235i	$-0.287972\pi$
$3$	6.42170	1.23586	0.617928	+	0.786235i	$0.287972\pi$
$4$	0	0
$5$	−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	$-0.376523\pi$
$7$	14.0109	0.756520	0.378260	+	0.925699i	$0.376523\pi$
$8$	0	0
$9$	14.2382	0.527340
$10$	0	0
$11$	−55.5140	−1.52165	−0.760823	−	0.648959i	$-0.775204\pi$
$11$	−55.5140	−1.52165	−0.760823	+	0.648959i	$0.775204\pi$
$12$	0	0
$13$	18.3149	0.390742	0.195371	−	0.980729i	$-0.437409\pi$
$13$	18.3149	0.390742	0.195371	+	0.980729i	$0.437409\pi$
$14$	0	0
$15$	−90.5626	−1.55888
$16$	0	0
$17$	10.0273	0.143058	0.0715288	−	0.997439i	$-0.477212\pi$
$17$	10.0273	0.143058	0.0715288	+	0.997439i	$0.477212\pi$
$18$	0	0
$19$	161.106	1.94528	0.972639	−	0.232321i	$-0.0746318\pi$
$19$	161.106	1.94528	0.972639	+	0.232321i	$0.0746318\pi$
$20$	0	0
$21$	89.9740	0.934950
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	73.8833	0.591066
$26$	0	0
$27$	−81.9525	−0.584139
$28$	0	0
$29$	−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	$-0.362256\pi$
$31$	144.762	0.838710	0.419355	+	0.907822i	$0.362256\pi$
$32$	0	0
$33$	−356.494	−1.88054
$34$	0	0
$35$	−197.591	−0.954255
$36$	0	0
$37$	−181.411	−0.806047	−0.403023	−	0.915190i	$-0.632041\pi$
$37$	−181.411	−0.806047	−0.403023	+	0.915190i	$0.632041\pi$
$38$	0	0
$39$	117.613	0.482900
$40$	0	0
$41$	77.7996	0.296348	0.148174	−	0.988961i	$-0.452660\pi$
$41$	77.7996	0.296348	0.148174	+	0.988961i	$0.452660\pi$
$42$	0	0
$43$	315.335	1.11833	0.559164	−	0.829057i	$-0.311122\pi$
$43$	315.335	1.11833	0.559164	+	0.829057i	$0.311122\pi$
$44$	0	0
$45$	−200.795	−0.665174
$46$	0	0
$47$	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
$50$	0	0
$51$	64.3923	0.176799
$52$	0	0
$53$	−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
$56$	0	0
$57$	1034.57	2.40408
$58$	0	0
$59$	−132.892	−0.293239	−0.146619	−	0.989193i	$-0.546839\pi$
$59$	−132.892	−0.293239	−0.146619	+	0.989193i	$0.546839\pi$
$60$	0	0
$61$	−236.683	−0.496789	−0.248394	−	0.968659i	$-0.579903\pi$
$61$	−236.683	−0.496789	−0.248394	+	0.968659i	$0.579903\pi$
$62$	0	0
$63$	199.490	0.398943
$64$	0	0
$65$	−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	$0.351409\pi$
$68$	0	0
$69$	147.699	0.257694
$70$	0	0
$71$	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	$0.198813\pi$
$74$	0	0
$75$	474.456	0.730473
$76$	0	0
$77$	−777.804	−1.15116
$78$	0	0
$79$	599.386	0.853623	0.426811	−	0.904341i	$-0.359637\pi$
$79$	599.386	0.853623	0.426811	+	0.904341i	$0.359637\pi$
$80$	0	0
$81$	−910.705	−1.24925
$82$	0	0
$83$	−642.245	−0.849344	−0.424672	−	0.905347i	$-0.639610\pi$
$83$	−642.245	−0.849344	−0.424672	+	0.905347i	$0.639610\pi$
$84$	0	0
$85$	−141.411	−0.180449
$86$	0	0
$87$	−1176.36	−1.44964
$88$	0	0
$89$	883.399	1.05214	0.526068	−	0.850442i	$-0.323666\pi$
$89$	883.399	1.05214	0.526068	+	0.850442i	$0.323666\pi$
$90$	0	0
$91$	256.609	0.295604
$92$	0	0
$93$	929.617	1.03652
$94$	0	0
$95$	−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	$0.511880\pi$
$98$	0	0
$99$	−790.419	−0.802425
$100$	0	0
$101$	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	$-0.414218\pi$
$103$	556.624	0.532484	0.266242	+	0.963906i	$0.414218\pi$
$104$	0	0
$105$	−1268.87	−1.17932
$106$	0	0
$107$	647.477	0.584990	0.292495	−	0.956267i	$-0.405514\pi$
$107$	647.477	0.584990	0.292495	+	0.956267i	$0.405514\pi$
$108$	0	0
$109$	1349.13	1.18554	0.592768	−	0.805373i	$-0.298035\pi$
$109$	1349.13	1.18554	0.592768	+	0.805373i	$0.298035\pi$
$110$	0	0
$111$	−1164.96	−0.996158
$112$	0	0
$113$	−284.549	−0.236886	−0.118443	−	0.992961i	$-0.537790\pi$
$113$	−284.549	−0.236886	−0.118443	+	0.992961i	$0.537790\pi$
$114$	0	0
$115$	−324.360	−0.263015
$116$	0	0
$117$	260.771	0.206054
$118$	0	0
$119$	140.492	0.108226
$120$	0	0
$121$	1750.81	1.31541
$122$	0	0
$123$	499.605	0.366243
$124$	0	0
$125$	720.878	0.515819
$126$	0	0
$127$	753.553	0.526512	0.263256	−	0.964726i	$-0.415204\pi$
$127$	753.553	0.526512	0.263256	+	0.964726i	$0.415204\pi$
$128$	0	0
$129$	2024.98	1.38209
$130$	0	0
$131$	−769.226	−0.513035	−0.256518	−	0.966540i	$-0.582575\pi$
$131$	−769.226	−0.513035	−0.256518	+	0.966540i	$0.582575\pi$
$132$	0	0
$133$	2257.25	1.47164
$134$	0	0
$135$	1155.74	0.736819
$136$	0	0
$137$	−211.638	−0.131982	−0.0659908	−	0.997820i	$-0.521021\pi$
$137$	−211.638	−0.131982	−0.0659908	+	0.997820i	$0.521021\pi$
$138$	0	0
$139$	1998.54	1.21953	0.609763	−	0.792584i	$-0.291265\pi$
$139$	1998.54	1.21953	0.609763	+	0.792584i	$0.291265\pi$
$140$	0	0
$141$	3366.19	2.01052
$142$	0	0
$143$	−1016.73	−0.594570
$144$	0	0
$145$	2583.38	1.47957
$146$	0	0
$147$	−942.021	−0.528548
$148$	0	0
$149$	−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	$-0.543161\pi$
$151$	−501.652	−0.270357	−0.135178	+	0.990821i	$0.543161\pi$
$152$	0	0
$153$	142.771	0.0754400
$154$	0	0
$155$	−2041.52	−1.05793
$156$	0	0
$157$	−1686.36	−0.857237	−0.428619	−	0.903485i	$-0.641000\pi$
$157$	−1686.36	−0.857237	−0.428619	+	0.903485i	$0.641000\pi$
$158$	0	0
$159$	−473.493	−0.236166
$160$	0	0
$161$	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	$0.222793\pi$
$164$	0	0
$165$	5027.49	2.37206
$166$	0	0
$167$	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
$170$	0	0
$171$	2293.86	1.02582
$172$	0	0
$173$	−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
$176$	0	0
$177$	−853.393	−0.362401
$178$	0	0
$179$	810.182	0.338301	0.169150	−	0.985590i	$-0.445898\pi$
$179$	810.182	0.338301	0.169150	+	0.985590i	$0.445898\pi$
$180$	0	0
$181$	−2430.33	−0.998039	−0.499019	−	0.866591i	$-0.666306\pi$
$181$	−2430.33	−0.998039	−0.499019	+	0.866591i	$0.666306\pi$
$182$	0	0
$183$	−1519.90	−0.613960
$184$	0	0
$185$	2558.36	1.01673
$186$	0	0
$187$	−556.656	−0.217683
$188$	0	0
$189$	−1148.23	−0.441913
$190$	0	0
$191$	−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	$0.230072\pi$
$194$	0	0
$195$	−1658.65	−0.609118
$196$	0	0
$197$	−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	$-0.541157\pi$
$199$	−723.923	−0.257877	−0.128939	+	0.991653i	$0.541157\pi$
$200$	0	0
$201$	3169.90	1.11238
$202$	0	0
$203$	−2566.59	−0.887385
$204$	0	0
$205$	−1097.18	−0.373805
$206$	0	0
$207$	327.478	0.109958
$208$	0	0
$209$	−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	$0.530366\pi$
$212$	0	0
$213$	5176.27	1.66513
$214$	0	0
$215$	−4447.04	−1.41063
$216$	0	0
$217$	2028.25	0.634501
$218$	0	0
$219$	6498.21	2.00506
$220$	0	0
$221$	183.649	0.0558985
$222$	0	0
$223$	−3157.57	−0.948191	−0.474096	−	0.880473i	$-0.657225\pi$
$223$	−3157.57	−0.948191	−0.474096	+	0.880473i	$0.657225\pi$
$224$	0	0
$225$	1051.96	0.311693
$226$	0	0
$227$	−2219.80	−0.649044	−0.324522	−	0.945878i	$-0.605203\pi$
$227$	−2219.80	−0.649044	−0.324522	+	0.945878i	$0.605203\pi$
$228$	0	0
$229$	4398.24	1.26919	0.634593	−	0.772846i	$-0.281168\pi$
$229$	4398.24	1.26919	0.634593	+	0.772846i	$0.281168\pi$
$230$	0	0
$231$	−4994.82	−1.42266
$232$	0	0
$233$	2112.84	0.594065	0.297032	−	0.954867i	$-0.404003\pi$
$233$	2112.84	0.594065	0.297032	+	0.954867i	$0.404003\pi$
$234$	0	0
$235$	−7392.43	−2.05204
$236$	0	0
$237$	3849.08	1.05496
$238$	0	0
$239$	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	$0.609198\pi$
$242$	0	0
$243$	−3635.55	−0.959757
$244$	0	0
$245$	2068.76	0.539462
$246$	0	0
$247$	2950.64	0.760101
$248$	0	0
$249$	−4124.30	−1.04967
$250$	0	0
$251$	386.310	0.0971461	0.0485731	−	0.998820i	$-0.484533\pi$
$251$	386.310	0.0971461	0.0485731	+	0.998820i	$0.484533\pi$
$252$	0	0
$253$	−1276.82	−0.317285
$254$	0	0
$255$	−908.099	−0.223009
$256$	0	0
$257$	3816.61	0.926357	0.463178	−	0.886265i	$-0.346709\pi$
$257$	3816.61	0.926357	0.463178	+	0.886265i	$0.346709\pi$
$258$	0	0
$259$	−2541.73	−0.609790
$260$	0	0
$261$	−2608.22	−0.618561
$262$	0	0
$263$	−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
$266$	0	0
$267$	5672.92	1.30029
$268$	0	0
$269$	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	$-0.467914\pi$
$271$	897.873	0.201262	0.100631	+	0.994924i	$0.467914\pi$
$272$	0	0
$273$	1647.87	0.365324
$274$	0	0
$275$	−4101.56	−0.899394
$276$	0	0
$277$	−2288.67	−0.496437	−0.248219	−	0.968704i	$-0.579845\pi$
$277$	−2288.67	−0.496437	−0.248219	+	0.968704i	$0.579845\pi$
$278$	0	0
$279$	2061.15	0.442285
$280$	0	0
$281$	−2587.28	−0.549268	−0.274634	−	0.961549i	$-0.588557\pi$
$281$	−2587.28	−0.549268	−0.274634	+	0.961549i	$0.588557\pi$
$282$	0	0
$283$	1488.48	0.312653	0.156326	−	0.987705i	$-0.450035\pi$
$283$	1488.48	0.312653	0.156326	+	0.987705i	$0.450035\pi$
$284$	0	0
$285$	−14590.2	−3.03245
$286$	0	0
$287$	1090.05	0.224193
$288$	0	0
$289$	−4812.45	−0.979535
$290$	0	0
$291$	−457.827	−0.0922278
$292$	0	0
$293$	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
$296$	0	0
$297$	4549.51	0.888853
$298$	0	0
$299$	421.243	0.0814753
$300$	0	0
$301$	4418.14	0.846037
$302$	0	0
$303$	6053.66	1.14777
$304$	0	0
$305$	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	$0.624000\pi$
$308$	0	0
$309$	3574.47	0.658073
$310$	0	0
$311$	−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	$0.347299\pi$
$314$	0	0
$315$	−2813.33	−0.503217
$316$	0	0
$317$	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
$320$	0	0
$321$	4157.90	0.722964
$322$	0	0
$323$	1615.46	0.278287
$324$	0	0
$325$	1353.17	0.230954
$326$	0	0
$327$	8663.72	1.46515
$328$	0	0
$329$	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	$0.682621\pi$
$332$	0	0
$333$	−2582.96	−0.425061
$334$	0	0
$335$	−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	$0.521582\pi$
$338$	0	0
$339$	−1827.29	−0.292757
$340$	0	0
$341$	−8036.32	−1.27622
$342$	0	0
$343$	−6861.07	−1.08007
$344$	0	0
$345$	−2082.94	−0.325048
$346$	0	0
$347$	7101.86	1.09870	0.549348	−	0.835593i	$-0.314876\pi$
$347$	7101.86	1.09870	0.549348	+	0.835593i	$0.314876\pi$
$348$	0	0
$349$	2675.10	0.410300	0.205150	−	0.978731i	$-0.434232\pi$
$349$	2675.10	0.410300	0.205150	+	0.978731i	$0.434232\pi$
$350$	0	0
$351$	−1500.95	−0.228248
$352$	0	0
$353$	−8032.84	−1.21118	−0.605588	−	0.795778i	$-0.707062\pi$
$353$	−8032.84	−1.21118	−0.605588	+	0.795778i	$0.707062\pi$
$354$	0	0
$355$	−11367.5	−1.69951
$356$	0	0
$357$	902.197	0.133752
$358$	0	0
$359$	−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
$362$	0	0
$363$	11243.1	1.62565
$364$	0	0
$365$	−14270.6	−2.04646
$366$	0	0
$367$	−89.8597	−0.0127810	−0.00639052	−	0.999980i	$-0.502034\pi$
$367$	−89.8597	−0.0127810	−0.00639052	+	0.999980i	$0.502034\pi$
$368$	0	0
$369$	1107.72	0.156276
$370$	0	0
$371$	−1033.07	−0.144567
$372$	0	0
$373$	−5196.16	−0.721306	−0.360653	−	0.932700i	$-0.617446\pi$
$373$	−5196.16	−0.721306	−0.360653	+	0.932700i	$0.617446\pi$
$374$	0	0
$375$	4629.26	0.637478
$376$	0	0
$377$	−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	$0.392246\pi$
$380$	0	0
$381$	4839.09	0.650693
$382$	0	0
$383$	−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
$386$	0	0
$387$	4489.79	0.589739
$388$	0	0
$389$	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
$392$	0	0
$393$	−4939.74	−0.634038
$394$	0	0
$395$	−8452.90	−1.07674
$396$	0	0
$397$	11309.9	1.42979	0.714895	−	0.699232i	$-0.246474\pi$
$397$	11309.9	1.42979	0.714895	+	0.699232i	$0.246474\pi$
$398$	0	0
$399$	14495.4	1.81874
$400$	0	0
$401$	−14306.5	−1.78163	−0.890813	−	0.454371i	$-0.849864\pi$
$401$	−14306.5	−1.78163	−0.890813	+	0.454371i	$0.849864\pi$
$402$	0	0
$403$	2651.30	0.327719
$404$	0	0
$405$	12843.3	1.57578
$406$	0	0
$407$	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	$0.415038\pi$
$410$	0	0
$411$	−1359.08	−0.163110
$412$	0	0
$413$	−1861.94	−0.221841
$414$	0	0
$415$	9057.32	1.07134
$416$	0	0
$417$	12834.0	1.50716
$418$	0	0
$419$	12304.9	1.43468	0.717341	−	0.696722i	$-0.245359\pi$
$419$	12304.9	1.43468	0.717341	+	0.696722i	$0.245359\pi$
$420$	0	0
$421$	8353.05	0.966990	0.483495	−	0.875347i	$-0.339367\pi$
$421$	8353.05	0.966990	0.483495	+	0.875347i	$0.339367\pi$
$422$	0	0
$423$	7463.51	0.857892
$424$	0	0
$425$	740.850	0.0845565
$426$	0	0
$427$	−3316.15	−0.375831
$428$	0	0
$429$	−6529.16	−0.734803
$430$	0	0
$431$	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	$0.396073\pi$
$434$	0	0
$435$	16589.7	1.82854
$436$	0	0
$437$	3705.44	0.405619
$438$	0	0
$439$	−795.488	−0.0864842	−0.0432421	−	0.999065i	$-0.513769\pi$
$439$	−795.488	−0.0864842	−0.0432421	+	0.999065i	$0.513769\pi$
$440$	0	0
$441$	−2088.65	−0.225532
$442$	0	0
$443$	7357.81	0.789120	0.394560	−	0.918870i	$-0.370897\pi$
$443$	7357.81	0.789120	0.394560	+	0.918870i	$0.370897\pi$
$444$	0	0
$445$	−12458.2	−1.32714
$446$	0	0
$447$	−3210.65	−0.339728
$448$	0	0
$449$	−6672.34	−0.701308	−0.350654	−	0.936505i	$-0.614041\pi$
$449$	−6672.34	−0.701308	−0.350654	+	0.936505i	$0.614041\pi$
$450$	0	0
$451$	−4318.97	−0.450936
$452$	0	0
$453$	−3221.46	−0.334122
$454$	0	0
$455$	−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	$0.604925\pi$
$458$	0	0
$459$	−821.763	−0.0835656
$460$	0	0
$461$	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	$-0.511414\pi$
$463$	−714.350	−0.0717034	−0.0358517	+	0.999357i	$0.511414\pi$
$464$	0	0
$465$	−13110.0	−1.30745
$466$	0	0
$467$	3937.64	0.390176	0.195088	−	0.980786i	$-0.437501\pi$
$467$	3937.64	0.390176	0.195088	+	0.980786i	$0.437501\pi$
$468$	0	0
$469$	6916.13	0.680933
$470$	0	0
$471$	−10829.3	−1.05942
$472$	0	0
$473$	−17505.5	−1.70170
$474$	0	0
$475$	11903.1	1.14979
$476$	0	0
$477$	−1049.83	−0.100772
$478$	0	0
$479$	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
$482$	0	0
$483$	2069.40	0.194950
$484$	0	0
$485$	1005.43	0.0941322
$486$	0	0
$487$	−11821.8	−1.09999	−0.549995	−	0.835168i	$-0.685370\pi$
$487$	−11821.8	−1.09999	−0.549995	+	0.835168i	$0.685370\pi$
$488$	0	0
$489$	20443.7	1.89059
$490$	0	0
$491$	20198.1	1.85647	0.928235	−	0.371993i	$-0.121326\pi$
$491$	20198.1	1.85647	0.928235	+	0.371993i	$0.121326\pi$
$492$	0	0
$493$	−1836.85	−0.167804
$494$	0	0
$495$	11147.0	1.01216
$496$	0	0
$497$	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	$0.509049\pi$
$500$	0	0
$501$	24218.6	2.15969
$502$	0	0
$503$	−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
$506$	0	0
$507$	−11954.4	−1.04717
$508$	0	0
$509$	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
$512$	0	0
$513$	−13203.1	−1.13631
$514$	0	0
$515$	−7849.85	−0.671662
$516$	0	0
$517$	−29099.9	−2.47546
$518$	0	0
$519$	−834.439	−0.0705738
$520$	0	0
$521$	−19842.9	−1.66858	−0.834291	−	0.551324i	$-0.814123\pi$
$521$	−19842.9	−1.66858	−0.834291	+	0.551324i	$0.814123\pi$
$522$	0	0
$523$	−10894.9	−0.910897	−0.455448	−	0.890262i	$-0.650521\pi$
$523$	−10894.9	−0.910897	−0.455448	+	0.890262i	$0.650521\pi$
$524$	0	0
$525$	6647.58	0.552617
$526$	0	0
$527$	1451.57	0.119984
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−1892.14	−0.154637
$532$	0	0
$533$	1424.89	0.115795
$534$	0	0
$535$	−9131.11	−0.737892
$536$	0	0
$537$	5202.74	0.418091
$538$	0	0
$539$	8143.54	0.650774
$540$	0	0
$541$	−4405.04	−0.350069	−0.175035	−	0.984562i	$-0.556004\pi$
$541$	−4405.04	−0.350069	−0.175035	+	0.984562i	$0.556004\pi$
$542$	0	0
$543$	−15606.8	−1.23343
$544$	0	0
$545$	−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	$0.371766\pi$
$548$	0	0
$549$	−3369.93	−0.261977
$550$	0	0
$551$	−29512.2	−2.28178
$552$	0	0
$553$	8397.97	0.645783
$554$	0	0
$555$	16429.0	1.25653
$556$	0	0
$557$	−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
$560$	0	0
$561$	−3574.68	−0.269025
$562$	0	0
$563$	−20989.8	−1.57125	−0.785625	−	0.618702i	$-0.787659\pi$
$563$	−20989.8	−1.57125	−0.785625	+	0.618702i	$0.787659\pi$
$564$	0	0
$565$	4012.88	0.298802
$566$	0	0
$567$	−12759.8	−0.945084
$568$	0	0
$569$	−11942.2	−0.879868	−0.439934	−	0.898030i	$-0.644998\pi$
$569$	−11942.2	−0.879868	−0.439934	+	0.898030i	$0.644998\pi$
$570$	0	0
$571$	−13981.9	−1.02473	−0.512367	−	0.858767i	$-0.671231\pi$
$571$	−13981.9	−1.02473	−0.512367	+	0.858767i	$0.671231\pi$
$572$	0	0
$573$	−15815.8	−1.15308
$574$	0	0
$575$	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	$0.558015\pi$
$578$	0	0
$579$	25825.9	1.85369
$580$	0	0
$581$	−8998.45	−0.642545
$582$	0	0
$583$	4093.24	0.290780
$584$	0	0
$585$	−3677.55	−0.259911
$586$	0	0
$587$	−22464.4	−1.57957	−0.789784	−	0.613385i	$-0.789807\pi$
$587$	−22464.4	−1.57957	−0.789784	+	0.613385i	$0.789807\pi$
$588$	0	0
$589$	23322.0	1.63152
$590$	0	0
$591$	−11633.5	−0.809710
$592$	0	0
$593$	14073.1	0.974561	0.487281	−	0.873245i	$-0.337989\pi$
$593$	14073.1	0.974561	0.487281	+	0.873245i	$0.337989\pi$
$594$	0	0
$595$	−1981.30	−0.136513
$596$	0	0
$597$	−4648.82	−0.318699
$598$	0	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	$0.745238\pi$
$602$	0	0
$603$	7028.31	0.474651
$604$	0	0
$605$	−24690.9	−1.65922
$606$	0	0
$607$	−23022.8	−1.53949	−0.769743	−	0.638353i	$-0.779616\pi$
$607$	−23022.8	−1.53949	−0.769743	+	0.638353i	$0.779616\pi$
$608$	0	0
$609$	−16481.8	−1.09668
$610$	0	0
$611$	9600.48	0.635669
$612$	0	0
$613$	6159.74	0.405856	0.202928	−	0.979194i	$-0.434954\pi$
$613$	6159.74	0.405856	0.202928	+	0.979194i	$0.434954\pi$
$614$	0	0
$615$	−7045.73	−0.461970
$616$	0	0
$617$	26889.1	1.75448	0.877241	−	0.480050i	$-0.159381\pi$
$617$	26889.1	1.75448	0.877241	+	0.480050i	$0.159381\pi$
$618$	0	0
$619$	−6478.47	−0.420665	−0.210333	−	0.977630i	$-0.567455\pi$
$619$	−6478.47	−0.420665	−0.210333	+	0.977630i	$0.567455\pi$
$620$	0	0
$621$	−1884.91	−0.121801
$622$	0	0
$623$	12377.3	0.795962
$624$	0	0
$625$	−19401.7	−1.24171
$626$	0	0
$627$	−57433.4	−3.65817
$628$	0	0
$629$	−1819.06	−0.115311
$630$	0	0
$631$	−2956.54	−0.186526	−0.0932630	−	0.995642i	$-0.529730\pi$
$631$	−2956.54	−0.186526	−0.0932630	+	0.995642i	$0.529730\pi$
$632$	0	0
$633$	−3749.52	−0.235434
$634$	0	0
$635$	−10627.1	−0.664129
$636$	0	0
$637$	−2686.68	−0.167111
$638$	0	0
$639$	11476.8	0.710511
$640$	0	0
$641$	20604.4	1.26961	0.634807	−	0.772670i	$-0.281079\pi$
$641$	20604.4	1.26961	0.634807	+	0.772670i	$0.281079\pi$
$642$	0	0
$643$	−23773.4	−1.45806	−0.729030	−	0.684482i	$-0.760028\pi$
$643$	−23773.4	−1.45806	−0.729030	+	0.684482i	$0.760028\pi$
$644$	0	0
$645$	−28557.5	−1.74334
$646$	0	0
$647$	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
$650$	0	0
$651$	13024.8	0.784152
$652$	0	0
$653$	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
$656$	0	0
$657$	14407.8	0.855560
$658$	0	0
$659$	24408.5	1.44282	0.721411	−	0.692507i	$-0.243494\pi$
$659$	24408.5	1.44282	0.721411	+	0.692507i	$0.243494\pi$
$660$	0	0
$661$	−20341.7	−1.19697	−0.598487	−	0.801133i	$-0.704231\pi$
$661$	−20341.7	−1.19697	−0.598487	+	0.801133i	$0.704231\pi$
$662$	0	0
$663$	1179.34	0.0690826
$664$	0	0
$665$	−31833.1	−1.85629
$666$	0	0
$667$	−4213.24	−0.244584
$668$	0	0
$669$	−20277.0	−1.17183
$670$	0	0
$671$	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	$0.385588\pi$
$674$	0	0
$675$	−6054.92	−0.345265
$676$	0	0
$677$	−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
$680$	0	0
$681$	−14254.9	−0.802125
$682$	0	0
$683$	−22877.0	−1.28165	−0.640823	−	0.767688i	$-0.721407\pi$
$683$	−22877.0	−1.28165	−0.640823	+	0.767688i	$0.721407\pi$
$684$	0	0
$685$	2984.65	0.166478
$686$	0	0
$687$	28244.2	1.56853
$688$	0	0
$689$	−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	$0.264413\pi$
$692$	0	0
$693$	−11074.5	−0.607051
$694$	0	0
$695$	−28184.6	−1.53828
$696$	0	0
$697$	780.120	0.0423948
$698$	0	0
$699$	13568.0	0.734178
$700$	0	0
$701$	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
$704$	0	0
$705$	−47472.0	−2.53603
$706$	0	0
$707$	13208.0	0.702597
$708$	0	0
$709$	15069.4	0.798225	0.399112	−	0.916902i	$-0.369318\pi$
$709$	15069.4	0.798225	0.399112	+	0.916902i	$0.369318\pi$
$710$	0	0
$711$	8534.17	0.450150
$712$	0	0
$713$	3329.52	0.174883
$714$	0	0
$715$	14338.6	0.749976
$716$	0	0
$717$	9945.19	0.518006
$718$	0	0
$719$	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
$722$	0	0
$723$	−16162.9	−0.831403
$724$	0	0
$725$	−13534.3	−0.693311
$726$	0	0
$727$	−32641.3	−1.66520	−0.832598	−	0.553877i	$-0.813148\pi$
$727$	−32641.3	−1.66520	−0.832598	+	0.553877i	$0.813148\pi$
$728$	0	0
$729$	1242.61	0.0631312
$730$	0	0
$731$	3161.96	0.159985
$732$	0	0
$733$	628.010	0.0316454	0.0158227	−	0.999875i	$-0.494963\pi$
$733$	628.010	0.0316454	0.0158227	+	0.999875i	$0.494963\pi$
$734$	0	0
$735$	13284.9	0.666697
$736$	0	0
$737$	−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	$0.666868\pi$
$740$	0	0
$741$	18948.1	0.939376
$742$	0	0
$743$	−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
$746$	0	0
$747$	−9144.40	−0.447893
$748$	0	0
$749$	9071.77	0.442557
$750$	0	0
$751$	−5494.75	−0.266986	−0.133493	−	0.991050i	$-0.542619\pi$
$751$	−5494.75	−0.266986	−0.133493	+	0.991050i	$0.542619\pi$
$752$	0	0
$753$	2480.77	0.120059
$754$	0	0
$755$	7074.60	0.341021
$756$	0	0
$757$	3411.29	0.163785	0.0818926	−	0.996641i	$-0.473904\pi$
$757$	3411.29	0.163785	0.0818926	+	0.996641i	$0.473904\pi$
$758$	0	0
$759$	−8199.37	−0.392119
$760$	0	0
$761$	−15927.5	−0.758700	−0.379350	−	0.925253i	$-0.623852\pi$
$761$	−15927.5	−0.758700	−0.379350	+	0.925253i	$0.623852\pi$
$762$	0	0
$763$	18902.6	0.896882
$764$	0	0
$765$	−2013.44	−0.0951581
$766$	0	0
$767$	−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	$0.339365\pi$
$770$	0	0
$771$	24509.1	1.14484
$772$	0	0
$773$	−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
$776$	0	0
$777$	−16322.2	−0.753613
$778$	0	0
$779$	12534.0	0.576479
$780$	0	0
$781$	−44747.6	−2.05019
$782$	0	0
$783$	15012.4	0.685186
$784$	0	0
$785$	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	$0.745220\pi$
$788$	0	0
$789$	−28962.6	−1.30684
$790$	0	0
$791$	−3986.80	−0.179209
$792$	0	0
$793$	−4334.82	−0.194116
$794$	0	0
$795$	6677.49	0.297894
$796$	0	0
$797$	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
$800$	0	0
$801$	12578.0	0.554834
$802$	0	0
$803$	−56175.5	−2.46873
$804$	0	0
$805$	−4544.59	−0.198976
$806$	0	0
$807$	3536.62	0.154269
$808$	0	0
$809$	−2591.22	−0.112611	−0.0563055	−	0.998414i	$-0.517932\pi$
$809$	−2591.22	−0.112611	−0.0563055	+	0.998414i	$0.517932\pi$
$810$	0	0
$811$	37339.5	1.61673	0.808364	−	0.588683i	$-0.200353\pi$
$811$	37339.5	1.61673	0.808364	+	0.588683i	$0.200353\pi$
$812$	0	0
$813$	5765.87	0.248730
$814$	0	0
$815$	−44896.2	−1.92963
$816$	0	0
$817$	50802.4	2.17546
$818$	0	0
$819$	3653.65	0.155884
$820$	0	0
$821$	−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	$-0.407230\pi$
$823$	13568.2	0.574675	0.287338	+	0.957829i	$0.407230\pi$
$824$	0	0
$825$	−26339.0	−1.11152
$826$	0	0
$827$	−39430.4	−1.65796	−0.828978	−	0.559281i	$-0.811077\pi$
$827$	−39430.4	−1.65796	−0.828978	+	0.559281i	$0.811077\pi$
$828$	0	0
$829$	−14439.0	−0.604931	−0.302465	−	0.953160i	$-0.597810\pi$
$829$	−14439.0	−0.604931	−0.302465	+	0.953160i	$0.597810\pi$
$830$	0	0
$831$	−14697.2	−0.613525
$832$	0	0
$833$	−1470.94	−0.0611825
$834$	0	0
$835$	−53186.1	−2.20429
$836$	0	0
$837$	−11863.6	−0.489924
$838$	0	0
$839$	−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
$842$	0	0
$843$	−16614.7	−0.678816
$844$	0	0
$845$	26252.9	1.06879
$846$	0	0
$847$	24530.4	0.995131
$848$	0	0
$849$	9558.55	0.386394
$850$	0	0
$851$	−4172.45	−0.168072
$852$	0	0
$853$	−27379.7	−1.09902	−0.549510	−	0.835487i	$-0.685185\pi$
$853$	−27379.7	−1.09902	−0.549510	+	0.835487i	$0.685185\pi$
$854$	0	0
$855$	−32349.4	−1.29395
$856$	0	0
$857$	20497.8	0.817025	0.408513	−	0.912753i	$-0.366048\pi$
$857$	20497.8	0.817025	0.408513	+	0.912753i	$0.366048\pi$
$858$	0	0
$859$	−26376.7	−1.04768	−0.523842	−	0.851816i	$-0.675502\pi$
$859$	−26376.7	−1.04768	−0.523842	+	0.851816i	$0.675502\pi$
$860$	0	0
$861$	6999.94	0.277070
$862$	0	0
$863$	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
$866$	0	0
$867$	−30904.1	−1.21056
$868$	0	0
$869$	−33274.3	−1.29891
$870$	0	0
$871$	9040.67	0.351701
$872$	0	0
$873$	−1015.09	−0.0393536
$874$	0	0
$875$	10100.2	0.390227
$876$	0	0
$877$	−6671.86	−0.256890	−0.128445	−	0.991717i	$-0.540999\pi$
$877$	−6671.86	−0.256890	−0.128445	+	0.991717i	$0.540999\pi$
$878$	0	0
$879$	18117.2	0.695196
$880$	0	0
$881$	−25432.9	−0.972595	−0.486298	−	0.873793i	$-0.661653\pi$
$881$	−25432.9	−0.972595	−0.486298	+	0.873793i	$0.661653\pi$
$882$	0	0
$883$	16292.4	0.620933	0.310467	−	0.950584i	$-0.399515\pi$
$883$	16292.4	0.620933	0.310467	+	0.950584i	$0.399515\pi$
$884$	0	0
$885$	12035.1	0.457123
$886$	0	0
$887$	−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
$890$	0	0
$891$	50556.9	1.90092
$892$	0	0
$893$	84450.2	3.16463
$894$	0	0
$895$	−11425.7	−0.426724
$896$	0	0
$897$	2705.09	0.100692
$898$	0	0
$899$	−26518.1	−0.983793
$900$	0	0
$901$	−739.347	−0.0273376
$902$	0	0
$903$	28371.9	1.04558
$904$	0	0
$905$	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	$0.404855\pi$
$908$	0	0
$909$	13422.2	0.489753
$910$	0	0
$911$	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
$914$	0	0
$915$	21434.6	0.774433
$916$	0	0
$917$	−10777.6	−0.388121
$918$	0	0
$919$	10077.9	0.361739	0.180869	−	0.983507i	$-0.442109\pi$
$919$	10077.9	0.361739	0.180869	+	0.983507i	$0.442109\pi$
$920$	0	0
$921$	−26237.2	−0.938704
$922$	0	0
$923$	14762.9	0.526465
$924$	0	0
$925$	−13403.2	−0.476427
$926$	0	0
$927$	7925.32	0.280800
$928$	0	0
$929$	−7768.50	−0.274355	−0.137178	−	0.990546i	$-0.543803\pi$
$929$	−7768.50	−0.274355	−0.137178	+	0.990546i	$0.543803\pi$
$930$	0	0
$931$	−23633.2	−0.831952
$932$	0	0
$933$	−25974.8	−0.911442
$934$	0	0
$935$	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	$0.508943\pi$
$938$	0	0
$939$	32824.8	1.14078
$940$	0	0
$941$	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
$944$	0	0
$945$	16193.1	0.557418
$946$	0	0
$947$	57242.2	1.96422	0.982112	−	0.188297i	$-0.0602969\pi$
$947$	57242.2	1.96422	0.982112	+	0.188297i	$0.0602969\pi$
$948$	0	0
$949$	18533.1	0.633942
$950$	0	0
$951$	45067.1	1.53670
$952$	0	0
$953$	−14723.4	−0.500458	−0.250229	−	0.968187i	$-0.580506\pi$
$953$	−14723.4	−0.500458	−0.250229	+	0.968187i	$0.580506\pi$
$954$	0	0
$955$	34732.9	1.17689
$956$	0	0
$957$	65304.2	2.20584
$958$	0	0
$959$	−2965.25	−0.0998467
$960$	0	0
$961$	−8834.99	−0.296566
$962$	0	0
$963$	9218.90	0.308489
$964$	0	0
$965$	−56715.9	−1.89197
$966$	0	0
$967$	46140.9	1.53443	0.767214	−	0.641391i	$-0.221642\pi$
$967$	46140.9	1.53443	0.767214	+	0.641391i	$0.221642\pi$
$968$	0	0
$969$	10374.0	0.343922
$970$	0	0
$971$	15349.8	0.507310	0.253655	−	0.967295i	$-0.418367\pi$
$971$	15349.8	0.507310	0.253655	+	0.967295i	$0.418367\pi$
$972$	0	0
$973$	28001.5	0.922596
$974$	0	0
$975$	8689.62	0.285426
$976$	0	0
$977$	−1517.18	−0.0496815	−0.0248407	−	0.999691i	$-0.507908\pi$
$977$	−1517.18	−0.0496815	−0.0248407	+	0.999691i	$0.507908\pi$
$978$	0	0
$979$	−49041.0	−1.60098
$980$	0	0
$981$	19209.2	0.625181
$982$	0	0
$983$	−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
$986$	0	0
$987$	47163.4	1.52100
$988$	0	0
$989$	7252.70	0.233187
$990$	0	0
$991$	−12346.4	−0.395760	−0.197880	−	0.980226i	$-0.563406\pi$
$991$	−12346.4	−0.395760	−0.197880	+	0.980226i	$0.563406\pi$
$992$	0	0
$993$	−41978.8	−1.34155
$994$	0	0
$995$	10209.2	0.325280
$996$	0	0
$997$	43566.0	1.38390	0.691950	−	0.721945i	$-0.256752\pi$
$997$	43566.0	1.38390	0.691950	+	0.721945i	$0.256752\pi$
$998$	0	0
$999$	14867.1	0.470844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bf.1.4		4
4.3	odd	2		1472.4.a.y.1.1		4
8.3	odd	2		23.4.a.b.1.2	✓	4
8.5	even	2		368.4.a.l.1.1		4
24.11	even	2		207.4.a.e.1.3		4
40.3	even	4		575.4.b.g.24.5		8
40.19	odd	2		575.4.a.i.1.3		4
40.27	even	4		575.4.b.g.24.4		8
56.27	even	2		1127.4.a.c.1.2		4
184.91	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	8.3	odd	2
207.4.a.e.1.3		4	24.11	even	2
368.4.a.l.1.1		4	8.5	even	2
529.4.a.g.1.2		4	184.91	even	2
575.4.a.i.1.3		4	40.19	odd	2
575.4.b.g.24.4		8	40.27	even	4
575.4.b.g.24.5		8	40.3	even	4
1127.4.a.c.1.2		4	56.27	even	2
1472.4.a.y.1.1		4	4.3	odd	2
1472.4.a.bf.1.4		4	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q+6.42170 q^{3} -14.1026 q^{5} +14.0109 q^{7} +14.2382 q^{9} +O(q^{10})\)	\(q+6.42170 q^{3} -14.1026 q^{5} +14.0109 q^{7} +14.2382 q^{9} -55.5140 q^{11} +18.3149 q^{13} -90.5626 q^{15} +10.0273 q^{17} +161.106 q^{19} +89.9740 q^{21} +23.0000 q^{23} +73.8833 q^{25} -81.9525 q^{27} -183.185 q^{29} +144.762 q^{31} -356.494 q^{33} -197.591 q^{35} -181.411 q^{37} +117.613 q^{39} +77.7996 q^{41} +315.335 q^{43} -200.795 q^{45} +524.190 q^{47} -146.693 q^{49} +64.3923 q^{51} -73.7334 q^{53} +782.892 q^{55} +1034.57 q^{57} -132.892 q^{59} -236.683 q^{61} +199.490 q^{63} -258.288 q^{65} +493.624 q^{67} +147.699 q^{69} +806.060 q^{71} +1011.91 q^{73} +474.456 q^{75} -777.804 q^{77} +599.386 q^{79} -910.705 q^{81} -642.245 q^{83} -141.411 q^{85} -1176.36 q^{87} +883.399 q^{89} +256.609 q^{91} +929.617 q^{93} -2272.02 q^{95} -71.2938 q^{97} -790.419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 7 q^{3} - 14 q^{5} - 16 q^{7} - 33 q^{9}+O(q^{10}) \) 4 * q + 7 * q^3 - 14 * q^5 - 16 * q^7 - 33 * q^9	\( 4 q + 7 q^{3} - 14 q^{5} - 16 q^{7} - 33 q^{9} + 8 q^{11} - 111 q^{13} - 10 q^{15} + 98 q^{17} + 96 q^{19} - 180 q^{21} + 92 q^{23} + 184 q^{25} - 155 q^{27} - 21 q^{29} + 193 q^{31} - 418 q^{33} - 752 q^{35} - 170 q^{37} + 291 q^{39} - 125 q^{41} + 2 q^{43} - 168 q^{45} + 677 q^{47} + 1220 q^{49} - 340 q^{51} + 230 q^{53} + 972 q^{55} + 1322 q^{57} - 1140 q^{59} - 754 q^{61} + 1092 q^{63} + 1318 q^{65} + 488 q^{67} + 161 q^{69} + 401 q^{71} + 1509 q^{73} + 1401 q^{75} - 736 q^{77} + 838 q^{79} - 932 q^{81} + 142 q^{83} - 112 q^{85} - 2223 q^{87} + 2342 q^{89} + 292 q^{91} + 509 q^{93} + 956 q^{95} + 1062 q^{97} - 1498 q^{99}+O(q^{100}) \) 4 * q + 7 * q^3 - 14 * q^5 - 16 * q^7 - 33 * q^9 + 8 * q^11 - 111 * q^13 - 10 * q^15 + 98 * q^17 + 96 * q^19 - 180 * q^21 + 92 * q^23 + 184 * q^25 - 155 * q^27 - 21 * q^29 + 193 * q^31 - 418 * q^33 - 752 * q^35 - 170 * q^37 + 291 * q^39 - 125 * q^41 + 2 * q^43 - 168 * q^45 + 677 * q^47 + 1220 * q^49 - 340 * q^51 + 230 * q^53 + 972 * q^55 + 1322 * q^57 - 1140 * q^59 - 754 * q^61 + 1092 * q^63 + 1318 * q^65 + 488 * q^67 + 161 * q^69 + 401 * q^71 + 1509 * q^73 + 1401 * q^75 - 736 * q^77 + 838 * q^79 - 932 * q^81 + 142 * q^83 - 112 * q^85 - 2223 * q^87 + 2342 * q^89 + 292 * q^91 + 509 * q^93 + 956 * q^95 + 1062 * q^97 - 1498 * q^99

Introduction

L-functions

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