LMFDB - Embedded newform 1521.4.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 $ x^9 - 4x^8 - 46x^7 + 145x^6 + 680x^5 - 1501x^4 - 3203x^3 + 4784x^2 + 3584x - 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 169)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.72763$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.72763 q^{2} -5.01528 q^{4} +20.8281 q^{5} -7.56566 q^{7} +22.4856 q^{8} +O(q^{10})$	$q-1.72763 q^{2} -5.01528 q^{4} +20.8281 q^{5} -7.56566 q^{7} +22.4856 q^{8} -35.9833 q^{10} -4.40295 q^{11} +13.0707 q^{14} +1.27533 q^{16} +73.0087 q^{17} -55.9424 q^{19} -104.459 q^{20} +7.60668 q^{22} +33.6244 q^{23} +308.809 q^{25} +37.9439 q^{28} -121.429 q^{29} +84.1320 q^{31} -182.088 q^{32} -126.132 q^{34} -157.578 q^{35} -171.716 q^{37} +96.6479 q^{38} +468.332 q^{40} +93.5714 q^{41} +441.776 q^{43} +22.0820 q^{44} -58.0907 q^{46} +272.528 q^{47} -285.761 q^{49} -533.508 q^{50} +480.202 q^{53} -91.7049 q^{55} -170.119 q^{56} +209.785 q^{58} +350.534 q^{59} -484.467 q^{61} -145.349 q^{62} +304.379 q^{64} +967.552 q^{67} -366.159 q^{68} +272.237 q^{70} -402.749 q^{71} +351.621 q^{73} +296.662 q^{74} +280.567 q^{76} +33.3112 q^{77} -820.078 q^{79} +26.5626 q^{80} -161.657 q^{82} +192.314 q^{83} +1520.63 q^{85} -763.226 q^{86} -99.0031 q^{88} -813.571 q^{89} -168.636 q^{92} -470.829 q^{94} -1165.17 q^{95} -788.293 q^{97} +493.690 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8}+O(q^{10}) $ 9 * q + 5 * q^2 + 37 * q^4 + 30 * q^5 - 38 * q^7 + 60 * q^8	$ 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8} - 147 q^{10} + 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} - 161 q^{19} + 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} - 344 q^{28} - 280 q^{29} - 706 q^{31} + 680 q^{32} - 216 q^{34} - 20 q^{35} - 298 q^{37} + 739 q^{38} + 13 q^{40} + 1201 q^{41} - 533 q^{43} + 355 q^{44} + 840 q^{46} + 956 q^{47} + 403 q^{49} - 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} + 2877 q^{58} + 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} + 931 q^{67} + 1536 q^{68} + 4854 q^{70} + 2046 q^{71} + 45 q^{73} + 1990 q^{74} + 3608 q^{76} + 718 q^{77} + 412 q^{79} - 787 q^{80} + 2757 q^{82} + 3709 q^{83} + 2106 q^{85} + 125 q^{86} - 636 q^{88} + 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} + 1087 q^{97} - 282 q^{98}+O(q^{100}) $ 9 * q + 5 * q^2 + 37 * q^4 + 30 * q^5 - 38 * q^7 + 60 * q^8 - 147 * q^10 + 181 * q^11 + 147 * q^14 + 269 * q^16 + 55 * q^17 - 161 * q^19 + 370 * q^20 + 340 * q^22 + 204 * q^23 + 307 * q^25 - 344 * q^28 - 280 * q^29 - 706 * q^31 + 680 * q^32 - 216 * q^34 - 20 * q^35 - 298 * q^37 + 739 * q^38 + 13 * q^40 + 1201 * q^41 - 533 * q^43 + 355 * q^44 + 840 * q^46 + 956 * q^47 + 403 * q^49 - 1156 * q^50 + 278 * q^53 - 250 * q^55 - 250 * q^56 + 2877 * q^58 + 1377 * q^59 - 136 * q^61 - 2035 * q^62 + 284 * q^64 + 931 * q^67 + 1536 * q^68 + 4854 * q^70 + 2046 * q^71 + 45 * q^73 + 1990 * q^74 + 3608 * q^76 + 718 * q^77 + 412 * q^79 - 787 * q^80 + 2757 * q^82 + 3709 * q^83 + 2106 * q^85 + 125 * q^86 - 636 * q^88 + 1663 * q^89 - 4010 * q^92 - 2531 * q^94 + 1614 * q^95 + 1087 * q^97 - 282 * 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 $

$2$	−1.72763	−0.610811	−0.305405	−	0.952222i	$-0.598792\pi$
$2$	−1.72763	−0.610811	−0.305405	+	0.952222i	$0.598792\pi$
$3$	0	0
$3$	0	0
$4$	−5.01528	−0.626910
$5$	20.8281	1.86292	0.931460	−	0.363844i	$-0.118536\pi$
$5$	20.8281	1.86292	0.931460	+	0.363844i	$0.118536\pi$
$6$	0	0
$7$	−7.56566	−0.408507	−0.204254	−	0.978918i	$-0.565477\pi$
$7$	−7.56566	−0.408507	−0.204254	+	0.978918i	$0.565477\pi$
$8$	22.4856	0.993734
$9$	0	0
$10$	−35.9833	−1.13789
$11$	−4.40295	−0.120685	−0.0603427	−	0.998178i	$-0.519219\pi$
$11$	−4.40295	−0.120685	−0.0603427	+	0.998178i	$0.519219\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	13.0707	0.249521
$15$	0	0
$16$	1.27533	0.0199270
$17$	73.0087	1.04160	0.520800	−	0.853679i	$-0.325634\pi$
$17$	73.0087	1.04160	0.520800	+	0.853679i	$0.325634\pi$
$18$	0	0
$19$	−55.9424	−0.675477	−0.337738	−	0.941240i	$-0.609662\pi$
$19$	−55.9424	−0.675477	−0.337738	+	0.941240i	$0.609662\pi$
$20$	−104.459	−1.16788
$21$	0	0
$22$	7.60668	0.0737159
$23$	33.6244	0.304834	0.152417	−	0.988316i	$-0.451294\pi$
$23$	33.6244	0.304834	0.152417	+	0.988316i	$0.451294\pi$
$24$	0	0
$25$	308.809	2.47047
$26$	0	0
$27$	0	0
$28$	37.9439	0.256098
$29$	−121.429	−0.777546	−0.388773	−	0.921334i	$-0.627101\pi$
$29$	−121.429	−0.777546	−0.388773	+	0.921334i	$0.627101\pi$
$30$	0	0
$31$	84.1320	0.487437	0.243719	−	0.969846i	$-0.421633\pi$
$31$	84.1320	0.487437	0.243719	+	0.969846i	$0.421633\pi$
$32$	−182.088	−1.00591
$33$	0	0
$34$	−126.132	−0.636220
$35$	−157.578	−0.761016
$36$	0	0
$37$	−171.716	−0.762969	−0.381485	−	0.924375i	$-0.624587\pi$
$37$	−171.716	−0.762969	−0.381485	+	0.924375i	$0.624587\pi$
$38$	96.6479	0.412588
$39$	0	0
$40$	468.332	1.85125
$41$	93.5714	0.356424	0.178212	−	0.983992i	$-0.442969\pi$
$41$	93.5714	0.356424	0.178212	+	0.983992i	$0.442969\pi$
$42$	0	0
$43$	441.776	1.56675	0.783374	−	0.621551i	$-0.213497\pi$
$43$	441.776	1.56675	0.783374	+	0.621551i	$0.213497\pi$
$44$	22.0820	0.0756589
$45$	0	0
$46$	−58.0907	−0.186196
$47$	272.528	0.845794	0.422897	−	0.906178i	$-0.361013\pi$
$47$	272.528	0.845794	0.422897	+	0.906178i	$0.361013\pi$
$48$	0	0
$49$	−285.761	−0.833122
$50$	−533.508	−1.50899
$51$	0	0
$52$	0	0
$53$	480.202	1.24454	0.622272	−	0.782801i	$-0.286210\pi$
$53$	480.202	1.24454	0.622272	+	0.782801i	$0.286210\pi$
$54$	0	0
$55$	−91.7049	−0.224827
$56$	−170.119	−0.405948
$57$	0	0
$58$	209.785	0.474934
$59$	350.534	0.773486	0.386743	−	0.922187i	$-0.373600\pi$
$59$	350.534	0.773486	0.386743	+	0.922187i	$0.373600\pi$
$60$	0	0
$61$	−484.467	−1.01688	−0.508439	−	0.861098i	$-0.669777\pi$
$61$	−484.467	−1.01688	−0.508439	+	0.861098i	$0.669777\pi$
$62$	−145.349	−0.297732
$63$	0	0
$64$	304.379	0.594491
$65$	0	0
$66$	0	0
$67$	967.552	1.76426	0.882129	−	0.471008i	$-0.156110\pi$
$67$	967.552	1.76426	0.882129	+	0.471008i	$0.156110\pi$
$68$	−366.159	−0.652990
$69$	0	0
$70$	272.237	0.464837
$71$	−402.749	−0.673205	−0.336603	−	0.941647i	$-0.609278\pi$
$71$	−402.749	−0.673205	−0.336603	+	0.941647i	$0.609278\pi$
$72$	0	0
$73$	351.621	0.563754	0.281877	−	0.959450i	$-0.409043\pi$
$73$	351.621	0.563754	0.281877	+	0.959450i	$0.409043\pi$
$74$	296.662	0.466030
$75$	0	0
$76$	280.567	0.423463
$77$	33.3112	0.0493009
$78$	0	0
$79$	−820.078	−1.16792	−0.583962	−	0.811781i	$-0.698498\pi$
$79$	−820.078	−1.16792	−0.583962	+	0.811781i	$0.698498\pi$
$80$	26.5626	0.0371224
$81$	0	0
$82$	−161.657	−0.217708
$83$	192.314	0.254328	0.127164	−	0.991882i	$-0.459413\pi$
$83$	192.314	0.254328	0.127164	+	0.991882i	$0.459413\pi$
$84$	0	0
$85$	1520.63	1.94042
$86$	−763.226	−0.956986
$87$	0	0
$88$	−99.0031	−0.119929
$89$	−813.571	−0.968970	−0.484485	−	0.874799i	$-0.660993\pi$
$89$	−813.571	−0.968970	−0.484485	+	0.874799i	$0.660993\pi$
$90$	0	0
$91$	0	0
$92$	−168.636	−0.191103
$93$	0	0
$94$	−470.829	−0.516620
$95$	−1165.17	−1.25836
$96$	0	0
$97$	−788.293	−0.825144	−0.412572	−	0.910925i	$-0.635370\pi$
$97$	−788.293	−0.825144	−0.412572	+	0.910925i	$0.635370\pi$
$98$	493.690	0.508880
$99$	0	0
$100$	−1548.76	−1.54876
$101$	1593.06	1.56946	0.784730	−	0.619838i	$-0.212802\pi$
$101$	1593.06	1.56946	0.784730	+	0.619838i	$0.212802\pi$
$102$	0	0
$103$	−134.659	−0.128819	−0.0644094	−	0.997924i	$-0.520516\pi$
$103$	−134.659	−0.128819	−0.0644094	+	0.997924i	$0.520516\pi$
$104$	0	0
$105$	0	0
$106$	−829.613	−0.760180
$107$	779.219	0.704018	0.352009	−	0.935997i	$-0.385499\pi$
$107$	779.219	0.704018	0.352009	+	0.935997i	$0.385499\pi$
$108$	0	0
$109$	−1341.63	−1.17894	−0.589472	−	0.807789i	$-0.700664\pi$
$109$	−1341.63	−1.17894	−0.589472	+	0.807789i	$0.700664\pi$
$110$	158.433	0.137327
$111$	0	0
$112$	−9.64870	−0.00814032
$113$	−1222.14	−1.01742	−0.508712	−	0.860937i	$-0.669878\pi$
$113$	−1222.14	−1.01742	−0.508712	+	0.860937i	$0.669878\pi$
$114$	0	0
$115$	700.332	0.567881
$116$	609.002	0.487452
$117$	0	0
$118$	−605.595	−0.472454
$119$	−552.359	−0.425501
$120$	0	0
$121$	−1311.61	−0.985435
$122$	836.981	0.621120
$123$	0	0
$124$	−421.946	−0.305579
$125$	3828.38	2.73937
$126$	0	0
$127$	448.886	0.313640	0.156820	−	0.987627i	$-0.449876\pi$
$127$	448.886	0.313640	0.156820	+	0.987627i	$0.449876\pi$
$128$	930.851	0.642784
$129$	0	0
$130$	0	0
$131$	−1787.67	−1.19229	−0.596144	−	0.802877i	$-0.703301\pi$
$131$	−1787.67	−1.19229	−0.596144	+	0.802877i	$0.703301\pi$
$132$	0	0
$133$	423.241	0.275937
$134$	−1671.58	−1.07763
$135$	0	0
$136$	1641.65	1.03507
$137$	830.034	0.517625	0.258812	−	0.965928i	$-0.416669\pi$
$137$	830.034	0.517625	0.258812	+	0.965928i	$0.416669\pi$
$138$	0	0
$139$	989.347	0.603707	0.301854	−	0.953354i	$-0.402395\pi$
$139$	989.347	0.603707	0.301854	+	0.953354i	$0.402395\pi$
$140$	790.299	0.477089
$141$	0	0
$142$	695.803	0.411201
$143$	0	0
$144$	0	0
$145$	−2529.14	−1.44851
$146$	−607.471	−0.344347
$147$	0	0
$148$	861.202	0.478313
$149$	−469.575	−0.258182	−0.129091	−	0.991633i	$-0.541206\pi$
$149$	−469.575	−0.258182	−0.129091	+	0.991633i	$0.541206\pi$
$150$	0	0
$151$	1936.82	1.04382	0.521908	−	0.853002i	$-0.325220\pi$
$151$	1936.82	1.04382	0.521908	+	0.853002i	$0.325220\pi$
$152$	−1257.90	−0.671244
$153$	0	0
$154$	−57.5496	−0.0301135
$155$	1752.31	0.908056
$156$	0	0
$157$	−2891.58	−1.46989	−0.734946	−	0.678125i	$-0.762793\pi$
$157$	−2891.58	−1.46989	−0.734946	+	0.678125i	$0.762793\pi$
$158$	1416.79	0.713380
$159$	0	0
$160$	−3792.55	−1.87392
$161$	−254.391	−0.124527
$162$	0	0
$163$	2293.98	1.10232	0.551160	−	0.834399i	$-0.314185\pi$
$163$	2293.98	1.10232	0.551160	+	0.834399i	$0.314185\pi$
$164$	−469.287	−0.223446
$165$	0	0
$166$	−332.249	−0.155346
$167$	1126.64	0.522050	0.261025	−	0.965332i	$-0.415939\pi$
$167$	1126.64	0.522050	0.261025	+	0.965332i	$0.415939\pi$
$168$	0	0
$169$	0	0
$170$	−2627.09	−1.18523
$171$	0	0
$172$	−2215.63	−0.982210
$173$	511.900	0.224966	0.112483	−	0.993654i	$-0.464120\pi$
$173$	511.900	0.224966	0.112483	+	0.993654i	$0.464120\pi$
$174$	0	0
$175$	−2336.34	−1.00920
$176$	−5.61520	−0.00240490
$177$	0	0
$178$	1405.55	0.591857
$179$	−363.391	−0.151738	−0.0758690	−	0.997118i	$-0.524173\pi$
$179$	−363.391	−0.151738	−0.0758690	+	0.997118i	$0.524173\pi$
$180$	0	0
$181$	780.933	0.320698	0.160349	−	0.987060i	$-0.448738\pi$
$181$	780.933	0.320698	0.160349	+	0.987060i	$0.448738\pi$
$182$	0	0
$183$	0	0
$184$	756.066	0.302924
$185$	−3576.50	−1.42135
$186$	0	0
$187$	−321.453	−0.125706
$188$	−1366.81	−0.530237
$189$	0	0
$190$	2012.99	0.768619
$191$	4009.74	1.51903	0.759515	−	0.650490i	$-0.225437\pi$
$191$	4009.74	1.51903	0.759515	+	0.650490i	$0.225437\pi$
$192$	0	0
$193$	165.316	0.0616564	0.0308282	−	0.999525i	$-0.490186\pi$
$193$	165.316	0.0616564	0.0308282	+	0.999525i	$0.490186\pi$
$194$	1361.88	0.504007
$195$	0	0
$196$	1433.17	0.522293
$197$	821.276	0.297023	0.148511	−	0.988911i	$-0.452552\pi$
$197$	821.276	0.297023	0.148511	+	0.988911i	$0.452552\pi$
$198$	0	0
$199$	−38.5703	−0.0137396	−0.00686978	−	0.999976i	$-0.502187\pi$
$199$	−38.5703	−0.0137396	−0.00686978	+	0.999976i	$0.502187\pi$
$200$	6943.76	2.45499
$201$	0	0
$202$	−2752.23	−0.958643
$203$	918.693	0.317633
$204$	0	0
$205$	1948.91	0.663990
$206$	232.641	0.0786839
$207$	0	0
$208$	0	0
$209$	246.311	0.0815202
$210$	0	0
$211$	5044.01	1.64571	0.822853	−	0.568255i	$-0.192381\pi$
$211$	5044.01	1.64571	0.822853	+	0.568255i	$0.192381\pi$
$212$	−2408.35	−0.780217
$213$	0	0
$214$	−1346.20	−0.430022
$215$	9201.33	2.91872
$216$	0	0
$217$	−636.514	−0.199122
$218$	2317.85	0.720112
$219$	0	0
$220$	459.926	0.140946
$221$	0	0
$222$	0	0
$223$	6127.78	1.84012	0.920059	−	0.391780i	$-0.128140\pi$
$223$	6127.78	1.84012	0.920059	+	0.391780i	$0.128140\pi$
$224$	1377.62	0.410920
$225$	0	0
$226$	2111.41	0.621454
$227$	6342.92	1.85460	0.927300	−	0.374318i	$-0.122123\pi$
$227$	6342.92	1.85460	0.927300	+	0.374318i	$0.122123\pi$
$228$	0	0
$229$	1334.67	0.385141	0.192571	−	0.981283i	$-0.438318\pi$
$229$	1334.67	0.385141	0.192571	+	0.981283i	$0.438318\pi$
$230$	−1209.92	−0.346868
$231$	0	0
$232$	−2730.41	−0.772674
$233$	5392.39	1.51617	0.758084	−	0.652157i	$-0.226136\pi$
$233$	5392.39	1.51617	0.758084	+	0.652157i	$0.226136\pi$
$234$	0	0
$235$	5676.24	1.57565
$236$	−1758.03	−0.484907
$237$	0	0
$238$	954.274	0.259901
$239$	3748.70	1.01458	0.507288	−	0.861777i	$-0.330648\pi$
$239$	3748.70	1.01458	0.507288	+	0.861777i	$0.330648\pi$
$240$	0	0
$241$	3829.47	1.02356	0.511780	−	0.859116i	$-0.328986\pi$
$241$	3829.47	1.02356	0.511780	+	0.859116i	$0.328986\pi$
$242$	2265.99	0.601914
$243$	0	0
$244$	2429.74	0.637492
$245$	−5951.85	−1.55204
$246$	0	0
$247$	0	0
$248$	1891.76	0.484383
$249$	0	0
$250$	−6614.04	−1.67323
$251$	476.577	0.119846	0.0599229	−	0.998203i	$-0.480915\pi$
$251$	476.577	0.119846	0.0599229	+	0.998203i	$0.480915\pi$
$252$	0	0
$253$	−148.047	−0.0367890
$254$	−775.511	−0.191574
$255$	0	0
$256$	−4043.20	−0.987110
$257$	−254.676	−0.0618142	−0.0309071	−	0.999522i	$-0.509840\pi$
$257$	−254.676	−0.0618142	−0.0309071	+	0.999522i	$0.509840\pi$
$258$	0	0
$259$	1299.14	0.311679
$260$	0	0
$261$	0	0
$262$	3088.44	0.728262
$263$	−2244.97	−0.526352	−0.263176	−	0.964748i	$-0.584770\pi$
$263$	−2244.97	−0.526352	−0.263176	+	0.964748i	$0.584770\pi$
$264$	0	0
$265$	10001.7	2.31848
$266$	−731.205	−0.168545
$267$	0	0
$268$	−4852.55	−1.10603
$269$	−4585.32	−1.03930	−0.519650	−	0.854379i	$-0.673938\pi$
$269$	−4585.32	−1.03930	−0.519650	+	0.854379i	$0.673938\pi$
$270$	0	0
$271$	−7972.80	−1.78713	−0.893566	−	0.448932i	$-0.851805\pi$
$271$	−7972.80	−1.78713	−0.893566	+	0.448932i	$0.851805\pi$
$272$	93.1099	0.0207560
$273$	0	0
$274$	−1433.99	−0.316171
$275$	−1359.67	−0.298149
$276$	0	0
$277$	5308.79	1.15153	0.575766	−	0.817615i	$-0.304704\pi$
$277$	5308.79	1.15153	0.575766	+	0.817615i	$0.304704\pi$
$278$	−1709.23	−0.368751
$279$	0	0
$280$	−3543.25	−0.756248
$281$	−6534.86	−1.38732	−0.693661	−	0.720302i	$-0.744003\pi$
$281$	−6534.86	−1.38732	−0.693661	+	0.720302i	$0.744003\pi$
$282$	0	0
$283$	4192.60	0.880652	0.440326	−	0.897838i	$-0.354863\pi$
$283$	4192.60	0.880652	0.440326	+	0.897838i	$0.354863\pi$
$284$	2019.90	0.422039
$285$	0	0
$286$	0	0
$287$	−707.930	−0.145602
$288$	0	0
$289$	417.265	0.0849308
$290$	4369.42	0.884763
$291$	0	0
$292$	−1763.48	−0.353424
$293$	2393.89	0.477313	0.238656	−	0.971104i	$-0.423293\pi$
$293$	2393.89	0.477313	0.238656	+	0.971104i	$0.423293\pi$
$294$	0	0
$295$	7300.96	1.44094
$296$	−3861.13	−0.758189
$297$	0	0
$298$	811.253	0.157700
$299$	0	0
$300$	0	0
$301$	−3342.32	−0.640028
$302$	−3346.12	−0.637574
$303$	0	0
$304$	−71.3448	−0.0134602
$305$	−10090.5	−1.89436
$306$	0	0
$307$	−821.783	−0.152774	−0.0763870	−	0.997078i	$-0.524338\pi$
$307$	−821.783	−0.152774	−0.0763870	+	0.997078i	$0.524338\pi$
$308$	−167.065	−0.0309072
$309$	0	0
$310$	−3027.34	−0.554650
$311$	5490.98	1.00117	0.500587	−	0.865686i	$-0.333118\pi$
$311$	5490.98	1.00117	0.500587	+	0.865686i	$0.333118\pi$
$312$	0	0
$313$	−315.481	−0.0569714	−0.0284857	−	0.999594i	$-0.509069\pi$
$313$	−315.481	−0.0569714	−0.0284857	+	0.999594i	$0.509069\pi$
$314$	4995.59	0.897826
$315$	0	0
$316$	4112.92	0.732184
$317$	8295.72	1.46982	0.734912	−	0.678163i	$-0.237223\pi$
$317$	8295.72	1.46982	0.734912	+	0.678163i	$0.237223\pi$
$318$	0	0
$319$	534.647	0.0938385
$320$	6339.64	1.10749
$321$	0	0
$322$	439.494	0.0760623
$323$	−4084.28	−0.703577
$324$	0	0
$325$	0	0
$326$	−3963.15	−0.673309
$327$	0	0
$328$	2104.01	0.354191
$329$	−2061.86	−0.345513
$330$	0	0
$331$	855.068	0.141990	0.0709951	−	0.997477i	$-0.477383\pi$
$331$	855.068	0.141990	0.0709951	+	0.997477i	$0.477383\pi$
$332$	−964.511	−0.159441
$333$	0	0
$334$	−1946.43	−0.318874
$335$	20152.2	3.28667
$336$	0	0
$337$	−3400.09	−0.549598	−0.274799	−	0.961502i	$-0.588611\pi$
$337$	−3400.09	−0.549598	−0.274799	+	0.961502i	$0.588611\pi$
$338$	0	0
$339$	0	0
$340$	−7626.39	−1.21647
$341$	−370.429	−0.0588265
$342$	0	0
$343$	4756.99	0.748844
$344$	9933.60	1.55693
$345$	0	0
$346$	−884.375	−0.137411
$347$	−3544.33	−0.548327	−0.274164	−	0.961683i	$-0.588401\pi$
$347$	−3544.33	−0.548327	−0.274164	+	0.961683i	$0.588401\pi$
$348$	0	0
$349$	8339.20	1.27905	0.639523	−	0.768772i	$-0.279132\pi$
$349$	8339.20	1.27905	0.639523	+	0.768772i	$0.279132\pi$
$350$	4036.34	0.616433
$351$	0	0
$352$	801.726	0.121398
$353$	−4321.95	−0.651655	−0.325827	−	0.945429i	$-0.605643\pi$
$353$	−4321.95	−0.651655	−0.325827	+	0.945429i	$0.605643\pi$
$354$	0	0
$355$	−8388.49	−1.25413
$356$	4080.29	0.607458
$357$	0	0
$358$	627.806	0.0926832
$359$	4535.87	0.666836	0.333418	−	0.942779i	$-0.391798\pi$
$359$	4535.87	0.666836	0.333418	+	0.942779i	$0.391798\pi$
$360$	0	0
$361$	−3729.45	−0.543731
$362$	−1349.17	−0.195886
$363$	0	0
$364$	0	0
$365$	7323.58	1.05023
$366$	0	0
$367$	225.362	0.0320540	0.0160270	−	0.999872i	$-0.494898\pi$
$367$	225.362	0.0320540	0.0160270	+	0.999872i	$0.494898\pi$
$368$	42.8821	0.00607442
$369$	0	0
$370$	6178.89	0.868176
$371$	−3633.04	−0.508405
$372$	0	0
$373$	2929.26	0.406625	0.203313	−	0.979114i	$-0.434829\pi$
$373$	2929.26	0.406625	0.203313	+	0.979114i	$0.434829\pi$
$374$	555.354	0.0767825
$375$	0	0
$376$	6127.97	0.840495
$377$	0	0
$378$	0	0
$379$	−7810.76	−1.05861	−0.529303	−	0.848433i	$-0.677547\pi$
$379$	−7810.76	−1.05861	−0.529303	+	0.848433i	$0.677547\pi$
$380$	5843.67	0.788878
$381$	0	0
$382$	−6927.36	−0.927839
$383$	5576.15	0.743937	0.371968	−	0.928245i	$-0.378683\pi$
$383$	5576.15	0.743937	0.371968	+	0.928245i	$0.378683\pi$
$384$	0	0
$385$	693.809	0.0918435
$386$	−285.605	−0.0376604
$387$	0	0
$388$	3953.51	0.517292
$389$	−12425.7	−1.61956	−0.809778	−	0.586737i	$-0.800412\pi$
$389$	−12425.7	−1.61956	−0.809778	+	0.586737i	$0.800412\pi$
$390$	0	0
$391$	2454.87	0.317515
$392$	−6425.51	−0.827902
$393$	0	0
$394$	−1418.86	−0.181425
$395$	−17080.6	−2.17575
$396$	0	0
$397$	6418.82	0.811464	0.405732	−	0.913992i	$-0.367017\pi$
$397$	6418.82	0.811464	0.405732	+	0.913992i	$0.367017\pi$
$398$	66.6353	0.00839227
$399$	0	0
$400$	393.832	0.0492290
$401$	1553.37	0.193446	0.0967230	−	0.995311i	$-0.469164\pi$
$401$	1553.37	0.193446	0.0967230	+	0.995311i	$0.469164\pi$
$402$	0	0
$403$	0	0
$404$	−7989.65	−0.983911
$405$	0	0
$406$	−1587.16	−0.194014
$407$	756.055	0.0920792
$408$	0	0
$409$	8761.54	1.05924	0.529622	−	0.848234i	$-0.322334\pi$
$409$	8761.54	1.05924	0.529622	+	0.848234i	$0.322334\pi$
$410$	−3367.01	−0.405572
$411$	0	0
$412$	675.353	0.0807578
$413$	−2652.03	−0.315975
$414$	0	0
$415$	4005.54	0.473793
$416$	0	0
$417$	0	0
$418$	−425.536	−0.0497934
$419$	−8261.71	−0.963272	−0.481636	−	0.876371i	$-0.659957\pi$
$419$	−8261.71	−0.963272	−0.481636	+	0.876371i	$0.659957\pi$
$420$	0	0
$421$	−4431.95	−0.513064	−0.256532	−	0.966536i	$-0.582580\pi$
$421$	−4431.95	−0.513064	−0.256532	+	0.966536i	$0.582580\pi$
$422$	−8714.20	−1.00521
$423$	0	0
$424$	10797.6	1.23674
$425$	22545.7	2.57324
$426$	0	0
$427$	3665.31	0.415402
$428$	−3908.00	−0.441356
$429$	0	0
$430$	−15896.5	−1.78279
$431$	10076.0	1.12609	0.563045	−	0.826426i	$-0.309630\pi$
$431$	10076.0	1.12609	0.563045	+	0.826426i	$0.309630\pi$
$432$	0	0
$433$	99.6760	0.0110626	0.00553132	−	0.999985i	$-0.498239\pi$
$433$	99.6760	0.0110626	0.00553132	+	0.999985i	$0.498239\pi$
$434$	1099.66	0.121626
$435$	0	0
$436$	6728.66	0.739092
$437$	−1881.03	−0.205908
$438$	0	0
$439$	11428.8	1.24252	0.621258	−	0.783606i	$-0.286622\pi$
$439$	11428.8	1.24252	0.621258	+	0.783606i	$0.286622\pi$
$440$	−2062.04	−0.223418
$441$	0	0
$442$	0	0
$443$	4786.94	0.513396	0.256698	−	0.966492i	$-0.417365\pi$
$443$	4786.94	0.513396	0.256698	+	0.966492i	$0.417365\pi$
$444$	0	0
$445$	−16945.1	−1.80511
$446$	−10586.6	−1.12396
$447$	0	0
$448$	−2302.83	−0.242854
$449$	6476.36	0.680709	0.340355	−	0.940297i	$-0.389453\pi$
$449$	6476.36	0.680709	0.340355	+	0.940297i	$0.389453\pi$
$450$	0	0
$451$	−411.990	−0.0430152
$452$	6129.36	0.637834
$453$	0	0
$454$	−10958.2	−1.13281
$455$	0	0
$456$	0	0
$457$	−928.362	−0.0950261	−0.0475131	−	0.998871i	$-0.515130\pi$
$457$	−928.362	−0.0950261	−0.0475131	+	0.998871i	$0.515130\pi$
$458$	−2305.82	−0.235248
$459$	0	0
$460$	−3512.36	−0.356010
$461$	10539.2	1.06477	0.532386	−	0.846502i	$-0.321296\pi$
$461$	10539.2	1.06477	0.532386	+	0.846502i	$0.321296\pi$
$462$	0	0
$463$	−4928.72	−0.494724	−0.247362	−	0.968923i	$-0.579564\pi$
$463$	−4928.72	−0.494724	−0.247362	+	0.968923i	$0.579564\pi$
$464$	−154.862	−0.0154942
$465$	0	0
$466$	−9316.07	−0.926092
$467$	−326.459	−0.0323484	−0.0161742	−	0.999869i	$-0.505149\pi$
$467$	−326.459	−0.0323484	−0.0161742	+	0.999869i	$0.505149\pi$
$468$	0	0
$469$	−7320.17	−0.720712
$470$	−9806.46	−0.962422
$471$	0	0
$472$	7881.99	0.768640
$473$	−1945.12	−0.189083
$474$	0	0
$475$	−17275.5	−1.66874
$476$	2770.24	0.266751
$477$	0	0
$478$	−6476.38	−0.619713
$479$	−1863.51	−0.177757	−0.0888787	−	0.996042i	$-0.528328\pi$
$479$	−1863.51	−0.177757	−0.0888787	+	0.996042i	$0.528328\pi$
$480$	0	0
$481$	0	0
$482$	−6615.92	−0.625201
$483$	0	0
$484$	6578.12	0.617779
$485$	−16418.6	−1.53718
$486$	0	0
$487$	17379.2	1.61710	0.808548	−	0.588431i	$-0.200254\pi$
$487$	17379.2	1.61710	0.808548	+	0.588431i	$0.200254\pi$
$488$	−10893.5	−1.01051
$489$	0	0
$490$	10282.6	0.948002
$491$	−19258.2	−1.77008	−0.885041	−	0.465513i	$-0.845870\pi$
$491$	−19258.2	−1.77008	−0.885041	+	0.465513i	$0.845870\pi$
$492$	0	0
$493$	−8865.39	−0.809892
$494$	0	0
$495$	0	0
$496$	107.296	0.00971315
$497$	3047.07	0.275009
$498$	0	0
$499$	−13088.2	−1.17416	−0.587082	−	0.809528i	$-0.699723\pi$
$499$	−13088.2	−1.17416	−0.587082	+	0.809528i	$0.699723\pi$
$500$	−19200.4	−1.71734
$501$	0	0
$502$	−823.351	−0.0732031
$503$	−18837.0	−1.66978	−0.834890	−	0.550416i	$-0.814469\pi$
$503$	−18837.0	−1.66978	−0.834890	+	0.550416i	$0.814469\pi$
$504$	0	0
$505$	33180.4	2.92378
$506$	255.770	0.0224711
$507$	0	0
$508$	−2251.29	−0.196624
$509$	−153.195	−0.0133403	−0.00667017	−	0.999978i	$-0.502123\pi$
$509$	−153.195	−0.0133403	−0.00667017	+	0.999978i	$0.502123\pi$
$510$	0	0
$511$	−2660.24	−0.230298
$512$	−461.635	−0.0398468
$513$	0	0
$514$	439.987	0.0377568
$515$	−2804.69	−0.239979
$516$	0	0
$517$	−1199.93	−0.102075
$518$	−2244.44	−0.190377
$519$	0	0
$520$	0	0
$521$	10847.8	0.912192	0.456096	−	0.889931i	$-0.349247\pi$
$521$	10847.8	0.912192	0.456096	+	0.889931i	$0.349247\pi$
$522$	0	0
$523$	19849.9	1.65961	0.829804	−	0.558055i	$-0.188452\pi$
$523$	19849.9	1.65961	0.829804	+	0.558055i	$0.188452\pi$
$524$	8965.69	0.747458
$525$	0	0
$526$	3878.48	0.321501
$527$	6142.36	0.507714
$528$	0	0
$529$	−11036.4	−0.907076
$530$	−17279.2	−1.41615
$531$	0	0
$532$	−2122.67	−0.172988
$533$	0	0
$534$	0	0
$535$	16229.6	1.31153
$536$	21756.0	1.75320
$537$	0	0
$538$	7921.75	0.634815
$539$	1258.19	0.100546
$540$	0	0
$541$	−15828.2	−1.25787	−0.628936	−	0.777457i	$-0.716509\pi$
$541$	−15828.2	−1.25787	−0.628936	+	0.777457i	$0.716509\pi$
$542$	13774.1	1.09160
$543$	0	0
$544$	−13294.0	−1.04775
$545$	−27943.6	−2.19628
$546$	0	0
$547$	6963.82	0.544335	0.272168	−	0.962250i	$-0.412259\pi$
$547$	6963.82	0.544335	0.272168	+	0.962250i	$0.412259\pi$
$548$	−4162.85	−0.324504
$549$	0	0
$550$	2349.01	0.182113
$551$	6793.04	0.525215
$552$	0	0
$553$	6204.43	0.477106
$554$	−9171.64	−0.703368
$555$	0	0
$556$	−4961.85	−0.378470
$557$	18832.8	1.43262	0.716311	−	0.697781i	$-0.245829\pi$
$557$	18832.8	1.43262	0.716311	+	0.697781i	$0.245829\pi$
$558$	0	0
$559$	0	0
$560$	−200.964	−0.0151648
$561$	0	0
$562$	11289.8	0.847391
$563$	19661.5	1.47182	0.735911	−	0.677079i	$-0.236754\pi$
$563$	19661.5	1.47182	0.735911	+	0.677079i	$0.236754\pi$
$564$	0	0
$565$	−25454.8	−1.89538
$566$	−7243.28	−0.537912
$567$	0	0
$568$	−9056.08	−0.668987
$569$	−2153.41	−0.158656	−0.0793282	−	0.996849i	$-0.525278\pi$
$569$	−2153.41	−0.158656	−0.0793282	+	0.996849i	$0.525278\pi$
$570$	0	0
$571$	4437.31	0.325212	0.162606	−	0.986691i	$-0.448010\pi$
$571$	4437.31	0.325212	0.162606	+	0.986691i	$0.448010\pi$
$572$	0	0
$573$	0	0
$574$	1223.04	0.0889352
$575$	10383.5	0.753082
$576$	0	0
$577$	14826.7	1.06975	0.534873	−	0.844933i	$-0.320360\pi$
$577$	14826.7	1.06975	0.534873	+	0.844933i	$0.320360\pi$
$578$	−720.881	−0.0518767
$579$	0	0
$580$	12684.3	0.908084
$581$	−1454.99	−0.103895
$582$	0	0
$583$	−2114.30	−0.150198
$584$	7906.41	0.560222
$585$	0	0
$586$	−4135.77	−0.291548
$587$	17009.7	1.19602	0.598011	−	0.801488i	$-0.295958\pi$
$587$	17009.7	1.19602	0.598011	+	0.801488i	$0.295958\pi$
$588$	0	0
$589$	−4706.54	−0.329252
$590$	−12613.4	−0.880143
$591$	0	0
$592$	−218.994	−0.0152037
$593$	−9173.23	−0.635244	−0.317622	−	0.948217i	$-0.602884\pi$
$593$	−9173.23	−0.635244	−0.317622	+	0.948217i	$0.602884\pi$
$594$	0	0
$595$	−11504.6	−0.792675
$596$	2355.05	0.161857
$597$	0	0
$598$	0	0
$599$	−2983.22	−0.203491	−0.101745	−	0.994810i	$-0.532443\pi$
$599$	−2983.22	−0.203491	−0.101745	+	0.994810i	$0.532443\pi$
$600$	0	0
$601$	−18196.0	−1.23499	−0.617496	−	0.786574i	$-0.711853\pi$
$601$	−18196.0	−1.23499	−0.617496	+	0.786574i	$0.711853\pi$
$602$	5774.31	0.390936
$603$	0	0
$604$	−9713.71	−0.654380
$605$	−27318.4	−1.83579
$606$	0	0
$607$	−16672.6	−1.11486	−0.557428	−	0.830225i	$-0.688212\pi$
$607$	−16672.6	−1.11486	−0.557428	+	0.830225i	$0.688212\pi$
$608$	10186.5	0.679466
$609$	0	0
$610$	17432.7	1.15710
$611$	0	0
$612$	0	0
$613$	16756.9	1.10409	0.552044	−	0.833815i	$-0.313848\pi$
$613$	16756.9	1.10409	0.552044	+	0.833815i	$0.313848\pi$
$614$	1419.74	0.0933160
$615$	0	0
$616$	749.024	0.0489920
$617$	−11985.4	−0.782035	−0.391017	−	0.920383i	$-0.627877\pi$
$617$	−11985.4	−0.782035	−0.391017	+	0.920383i	$0.627877\pi$
$618$	0	0
$619$	−22471.2	−1.45912	−0.729560	−	0.683917i	$-0.760275\pi$
$619$	−22471.2	−1.45912	−0.729560	+	0.683917i	$0.760275\pi$
$620$	−8788.31	−0.569270
$621$	0	0
$622$	−9486.40	−0.611527
$623$	6155.20	0.395832
$624$	0	0
$625$	41136.7	2.63275
$626$	545.035	0.0347987
$627$	0	0
$628$	14502.1	0.921491
$629$	−12536.7	−0.794709
$630$	0	0
$631$	−991.566	−0.0625572	−0.0312786	−	0.999511i	$-0.509958\pi$
$631$	−991.566	−0.0625572	−0.0312786	+	0.999511i	$0.509958\pi$
$632$	−18440.0	−1.16061
$633$	0	0
$634$	−14332.0	−0.897784
$635$	9349.44	0.584285
$636$	0	0
$637$	0	0
$638$	−923.674	−0.0573175
$639$	0	0
$640$	19387.8	1.19746
$641$	8483.76	0.522759	0.261379	−	0.965236i	$-0.415823\pi$
$641$	8483.76	0.522759	0.261379	+	0.965236i	$0.415823\pi$
$642$	0	0
$643$	−22527.5	−1.38165	−0.690823	−	0.723024i	$-0.742752\pi$
$643$	−22527.5	−1.38165	−0.690823	+	0.723024i	$0.742752\pi$
$644$	1275.84	0.0780671
$645$	0	0
$646$	7056.14	0.429752
$647$	−5486.17	−0.333359	−0.166680	−	0.986011i	$-0.553305\pi$
$647$	−5486.17	−0.333359	−0.166680	+	0.986011i	$0.553305\pi$
$648$	0	0
$649$	−1543.39	−0.0933485
$650$	0	0
$651$	0	0
$652$	−11505.0	−0.691056
$653$	−20436.1	−1.22470	−0.612348	−	0.790588i	$-0.709775\pi$
$653$	−20436.1	−1.22470	−0.612348	+	0.790588i	$0.709775\pi$
$654$	0	0
$655$	−37233.8	−2.22114
$656$	119.334	0.00710246
$657$	0	0
$658$	3562.13	0.211043
$659$	8613.12	0.509134	0.254567	−	0.967055i	$-0.418067\pi$
$659$	8613.12	0.509134	0.254567	+	0.967055i	$0.418067\pi$
$660$	0	0
$661$	−29268.8	−1.72228	−0.861138	−	0.508372i	$-0.830248\pi$
$661$	−29268.8	−1.72228	−0.861138	+	0.508372i	$0.830248\pi$
$662$	−1477.24	−0.0867291
$663$	0	0
$664$	4324.31	0.252735
$665$	8815.30	0.514049
$666$	0	0
$667$	−4082.99	−0.237022
$668$	−5650.44	−0.327279
$669$	0	0
$670$	−34815.7	−2.00753
$671$	2133.08	0.122722
$672$	0	0
$673$	−22540.5	−1.29104	−0.645522	−	0.763741i	$-0.723360\pi$
$673$	−22540.5	−1.29104	−0.645522	+	0.763741i	$0.723360\pi$
$674$	5874.10	0.335700
$675$	0	0
$676$	0	0
$677$	18727.0	1.06313	0.531563	−	0.847019i	$-0.321605\pi$
$677$	18727.0	1.06313	0.531563	+	0.847019i	$0.321605\pi$
$678$	0	0
$679$	5963.96	0.337078
$680$	34192.3	1.92826
$681$	0	0
$682$	639.965	0.0359319
$683$	−5047.76	−0.282793	−0.141396	−	0.989953i	$-0.545159\pi$
$683$	−5047.76	−0.282793	−0.141396	+	0.989953i	$0.545159\pi$
$684$	0	0
$685$	17288.0	0.964293
$686$	−8218.34	−0.457402
$687$	0	0
$688$	563.408	0.0312206
$689$	0	0
$690$	0	0
$691$	16446.9	0.905456	0.452728	−	0.891649i	$-0.350451\pi$
$691$	16446.9	0.905456	0.452728	+	0.891649i	$0.350451\pi$
$692$	−2567.32	−0.141033
$693$	0	0
$694$	6123.30	0.334924
$695$	20606.2	1.12466
$696$	0	0
$697$	6831.52	0.371252
$698$	−14407.1	−0.781255
$699$	0	0
$700$	11717.4	0.632681
$701$	−14841.3	−0.799639	−0.399820	−	0.916594i	$-0.630927\pi$
$701$	−14841.3	−0.799639	−0.399820	+	0.916594i	$0.630927\pi$
$702$	0	0
$703$	9606.18	0.515368
$704$	−1340.17	−0.0717464
$705$	0	0
$706$	7466.74	0.398038
$707$	−12052.6	−0.641136
$708$	0	0
$709$	19329.7	1.02390	0.511949	−	0.859016i	$-0.328924\pi$
$709$	19329.7	1.02390	0.511949	+	0.859016i	$0.328924\pi$
$710$	14492.2	0.766034
$711$	0	0
$712$	−18293.7	−0.962899
$713$	2828.89	0.148587
$714$	0	0
$715$	0	0
$716$	1822.51	0.0951261
$717$	0	0
$718$	−7836.32	−0.407310
$719$	−21340.1	−1.10689	−0.553443	−	0.832887i	$-0.686686\pi$
$719$	−21340.1	−1.10689	−0.553443	+	0.832887i	$0.686686\pi$
$720$	0	0
$721$	1018.78	0.0526234
$722$	6443.12	0.332117
$723$	0	0
$724$	−3916.60	−0.201049
$725$	−37498.4	−1.92090
$726$	0	0
$727$	−15092.3	−0.769934	−0.384967	−	0.922930i	$-0.625787\pi$
$727$	−15092.3	−0.769934	−0.384967	+	0.922930i	$0.625787\pi$
$728$	0	0
$729$	0	0
$730$	−12652.5	−0.641491
$731$	32253.4	1.63192
$732$	0	0
$733$	9108.56	0.458980	0.229490	−	0.973311i	$-0.426294\pi$
$733$	9108.56	0.458980	0.229490	+	0.973311i	$0.426294\pi$
$734$	−389.343	−0.0195789
$735$	0	0
$736$	−6122.62	−0.306634
$737$	−4260.08	−0.212920
$738$	0	0
$739$	19072.7	0.949394	0.474697	−	0.880149i	$-0.342558\pi$
$739$	19072.7	0.949394	0.474697	+	0.880149i	$0.342558\pi$
$740$	17937.2	0.891059
$741$	0	0
$742$	6276.57	0.310539
$743$	12276.5	0.606168	0.303084	−	0.952964i	$-0.401984\pi$
$743$	12276.5	0.606168	0.303084	+	0.952964i	$0.401984\pi$
$744$	0	0
$745$	−9780.33	−0.480971
$746$	−5060.68	−0.248371
$747$	0	0
$748$	1612.18	0.0788063
$749$	−5895.30	−0.287596
$750$	0	0
$751$	28665.5	1.39283	0.696417	−	0.717637i	$-0.254776\pi$
$751$	28665.5	1.39283	0.696417	+	0.717637i	$0.254776\pi$
$752$	347.563	0.0168541
$753$	0	0
$754$	0	0
$755$	40340.3	1.94455
$756$	0	0
$757$	−28617.8	−1.37402	−0.687008	−	0.726650i	$-0.741076\pi$
$757$	−28617.8	−1.37402	−0.687008	+	0.726650i	$0.741076\pi$
$758$	13494.1	0.646608
$759$	0	0
$760$	−26199.6	−1.25047
$761$	26417.3	1.25838	0.629189	−	0.777252i	$-0.283387\pi$
$761$	26417.3	1.25838	0.629189	+	0.777252i	$0.283387\pi$
$762$	0	0
$763$	10150.3	0.481607
$764$	−20110.0	−0.952295
$765$	0	0
$766$	−9633.54	−0.454404
$767$	0	0
$768$	0	0
$769$	−31524.2	−1.47827	−0.739136	−	0.673556i	$-0.764766\pi$
$769$	−31524.2	−1.47827	−0.739136	+	0.673556i	$0.764766\pi$
$770$	−1198.65	−0.0560990
$771$	0	0
$772$	−829.105	−0.0386530
$773$	21666.6	1.00814	0.504071	−	0.863662i	$-0.331835\pi$
$773$	21666.6	1.00814	0.504071	+	0.863662i	$0.331835\pi$
$774$	0	0
$775$	25980.7	1.20420
$776$	−17725.3	−0.819974
$777$	0	0
$778$	21467.0	0.989242
$779$	−5234.61	−0.240756
$780$	0	0
$781$	1773.28	0.0812460
$782$	−4241.12	−0.193941
$783$	0	0
$784$	−364.438	−0.0166016
$785$	−60226.0	−2.73829
$786$	0	0
$787$	−26271.2	−1.18992	−0.594959	−	0.803756i	$-0.702832\pi$
$787$	−26271.2	−1.18992	−0.594959	+	0.803756i	$0.702832\pi$
$788$	−4118.93	−0.186207
$789$	0	0
$790$	29509.1	1.32897
$791$	9246.28	0.415626
$792$	0	0
$793$	0	0
$794$	−11089.4	−0.495651
$795$	0	0
$796$	193.441	0.00861348
$797$	336.983	0.0149769	0.00748843	−	0.999972i	$-0.497616\pi$
$797$	336.983	0.0149769	0.00748843	+	0.999972i	$0.497616\pi$
$798$	0	0
$799$	19896.9	0.880980
$800$	−56230.5	−2.48506
$801$	0	0
$802$	−2683.66	−0.118159
$803$	−1548.17	−0.0680369
$804$	0	0
$805$	−5298.47	−0.231983
$806$	0	0
$807$	0	0
$808$	35821.0	1.55963
$809$	−6478.70	−0.281556	−0.140778	−	0.990041i	$-0.544960\pi$
$809$	−6478.70	−0.281556	−0.140778	+	0.990041i	$0.544960\pi$
$810$	0	0
$811$	−36823.2	−1.59437	−0.797186	−	0.603734i	$-0.793679\pi$
$811$	−36823.2	−1.59437	−0.797186	+	0.603734i	$0.793679\pi$
$812$	−4607.50	−0.199128
$813$	0	0
$814$	−1306.19	−0.0562430
$815$	47779.2	2.05354
$816$	0	0
$817$	−24714.0	−1.05830
$818$	−15136.7	−0.646997
$819$	0	0
$820$	−9774.34	−0.416262
$821$	−8524.53	−0.362373	−0.181187	−	0.983449i	$-0.557994\pi$
$821$	−8524.53	−0.362373	−0.181187	+	0.983449i	$0.557994\pi$
$822$	0	0
$823$	−13313.1	−0.563869	−0.281935	−	0.959434i	$-0.590976\pi$
$823$	−13313.1	−0.563869	−0.281935	+	0.959434i	$0.590976\pi$
$824$	−3027.89	−0.128012
$825$	0	0
$826$	4581.73	0.193001
$827$	−41849.9	−1.75969	−0.879845	−	0.475261i	$-0.842354\pi$
$827$	−41849.9	−1.75969	−0.879845	+	0.475261i	$0.842354\pi$
$828$	0	0
$829$	−13017.2	−0.545362	−0.272681	−	0.962104i	$-0.587910\pi$
$829$	−13017.2	−0.545362	−0.272681	+	0.962104i	$0.587910\pi$
$830$	−6920.10	−0.289398
$831$	0	0
$832$	0	0
$833$	−20863.0	−0.867780
$834$	0	0
$835$	23465.8	0.972537
$836$	−1235.32	−0.0511058
$837$	0	0
$838$	14273.2	0.588377
$839$	20209.4	0.831593	0.415796	−	0.909458i	$-0.363503\pi$
$839$	20209.4	0.831593	0.415796	+	0.909458i	$0.363503\pi$
$840$	0	0
$841$	−9643.94	−0.395422
$842$	7656.78	0.313385
$843$	0	0
$844$	−25297.1	−1.03171
$845$	0	0
$846$	0	0
$847$	9923.23	0.402557
$848$	612.414	0.0248000
$849$	0	0
$850$	−38950.7	−1.57176
$851$	−5773.84	−0.232579
$852$	0	0
$853$	7958.24	0.319443	0.159721	−	0.987162i	$-0.448940\pi$
$853$	7958.24	0.319443	0.159721	+	0.987162i	$0.448940\pi$
$854$	−6332.31	−0.253732
$855$	0	0
$856$	17521.2	0.699607
$857$	2144.65	0.0854840	0.0427420	−	0.999086i	$-0.486391\pi$
$857$	2144.65	0.0854840	0.0427420	+	0.999086i	$0.486391\pi$
$858$	0	0
$859$	−41723.5	−1.65726	−0.828632	−	0.559794i	$-0.810880\pi$
$859$	−41723.5	−1.65726	−0.828632	+	0.559794i	$0.810880\pi$
$860$	−46147.3	−1.82978
$861$	0	0
$862$	−17407.7	−0.687828
$863$	10393.8	0.409977	0.204989	−	0.978764i	$-0.434284\pi$
$863$	10393.8	0.409977	0.204989	+	0.978764i	$0.434284\pi$
$864$	0	0
$865$	10661.9	0.419093
$866$	−172.204	−0.00675718
$867$	0	0
$868$	3192.30	0.124831
$869$	3610.76	0.140951
$870$	0	0
$871$	0	0
$872$	−30167.4	−1.17156
$873$	0	0
$874$	3249.73	0.125771
$875$	−28964.2	−1.11905
$876$	0	0
$877$	−5005.59	−0.192733	−0.0963664	−	0.995346i	$-0.530722\pi$
$877$	−5005.59	−0.192733	−0.0963664	+	0.995346i	$0.530722\pi$
$878$	−19744.7	−0.758942
$879$	0	0
$880$	−116.954	−0.00448013
$881$	14597.1	0.558218	0.279109	−	0.960259i	$-0.409961\pi$
$881$	14597.1	0.558218	0.279109	+	0.960259i	$0.409961\pi$
$882$	0	0
$883$	−5629.11	−0.214535	−0.107268	−	0.994230i	$-0.534210\pi$
$883$	−5629.11	−0.214535	−0.107268	+	0.994230i	$0.534210\pi$
$884$	0	0
$885$	0	0
$886$	−8270.07	−0.313588
$887$	−17014.9	−0.644085	−0.322043	−	0.946725i	$-0.604369\pi$
$887$	−17014.9	−0.644085	−0.322043	+	0.946725i	$0.604369\pi$
$888$	0	0
$889$	−3396.12	−0.128124
$890$	29275.0	1.10258
$891$	0	0
$892$	−30732.5	−1.15359
$893$	−15245.9	−0.571315
$894$	0	0
$895$	−7568.73	−0.282676
$896$	−7042.51	−0.262582
$897$	0	0
$898$	−11188.8	−0.415784
$899$	−10216.1	−0.379005
$900$	0	0
$901$	35058.9	1.29632
$902$	711.768	0.0262741
$903$	0	0
$904$	−27480.5	−1.01105
$905$	16265.3	0.597434
$906$	0	0
$907$	−19185.3	−0.702358	−0.351179	−	0.936308i	$-0.614219\pi$
$907$	−19185.3	−0.702358	−0.351179	+	0.936308i	$0.614219\pi$
$908$	−31811.5	−1.16267
$909$	0	0
$910$	0	0
$911$	−30427.5	−1.10659	−0.553297	−	0.832984i	$-0.686631\pi$
$911$	−30427.5	−1.10659	−0.553297	+	0.832984i	$0.686631\pi$
$912$	0	0
$913$	−846.750	−0.0306937
$914$	1603.87	0.0580430
$915$	0	0
$916$	−6693.74	−0.241449
$917$	13524.9	0.487059
$918$	0	0
$919$	−39750.8	−1.42683	−0.713415	−	0.700742i	$-0.752853\pi$
$919$	−39750.8	−1.42683	−0.713415	+	0.700742i	$0.752853\pi$
$920$	15747.4	0.564322
$921$	0	0
$922$	−18207.9	−0.650374
$923$	0	0
$924$	0	0
$925$	−53027.2	−1.88489
$926$	8515.03	0.302183
$927$	0	0
$928$	22110.9	0.782138
$929$	6260.88	0.221112	0.110556	−	0.993870i	$-0.464737\pi$
$929$	6260.88	0.221112	0.110556	+	0.993870i	$0.464737\pi$
$930$	0	0
$931$	15986.1	0.562755
$932$	−27044.4	−0.950502
$933$	0	0
$934$	564.002	0.0197588
$935$	−6695.26	−0.234180
$936$	0	0
$937$	−24497.3	−0.854101	−0.427050	−	0.904228i	$-0.640447\pi$
$937$	−24497.3	−0.854101	−0.427050	+	0.904228i	$0.640447\pi$
$938$	12646.6	0.440219
$939$	0	0
$940$	−28467.9	−0.987789
$941$	10199.1	0.353326	0.176663	−	0.984271i	$-0.443470\pi$
$941$	10199.1	0.353326	0.176663	+	0.984271i	$0.443470\pi$
$942$	0	0
$943$	3146.28	0.108650
$944$	447.046	0.0154133
$945$	0	0
$946$	3360.45	0.115494
$947$	101.128	0.00347015	0.00173507	−	0.999998i	$-0.499448\pi$
$947$	101.128	0.00347015	0.00173507	+	0.999998i	$0.499448\pi$
$948$	0	0
$949$	0	0
$950$	29845.7	1.01929
$951$	0	0
$952$	−12420.1	−0.422835
$953$	42197.1	1.43431	0.717155	−	0.696914i	$-0.245444\pi$
$953$	42197.1	1.43431	0.717155	+	0.696914i	$0.245444\pi$
$954$	0	0
$955$	83515.2	2.82983
$956$	−18800.8	−0.636048
$957$	0	0
$958$	3219.46	0.108576
$959$	−6279.75	−0.211453
$960$	0	0
$961$	−22712.8	−0.762405
$962$	0	0
$963$	0	0
$964$	−19205.9	−0.641681
$965$	3443.21	0.114861
$966$	0	0
$967$	1221.07	0.0406069	0.0203035	−	0.999794i	$-0.493537\pi$
$967$	1221.07	0.0406069	0.0203035	+	0.999794i	$0.493537\pi$
$968$	−29492.5	−0.979260
$969$	0	0
$970$	28365.4	0.938924
$971$	42915.7	1.41836	0.709181	−	0.705026i	$-0.249065\pi$
$971$	42915.7	1.41836	0.709181	+	0.705026i	$0.249065\pi$
$972$	0	0
$973$	−7485.06	−0.246619
$974$	−30024.8	−0.987739
$975$	0	0
$976$	−617.853	−0.0202633
$977$	−31253.6	−1.02343	−0.511714	−	0.859156i	$-0.670989\pi$
$977$	−31253.6	−1.02343	−0.511714	+	0.859156i	$0.670989\pi$
$978$	0	0
$979$	3582.11	0.116941
$980$	29850.2	0.972989
$981$	0	0
$982$	33271.1	1.08119
$983$	41688.2	1.35264	0.676320	−	0.736608i	$-0.263574\pi$
$983$	41688.2	1.35264	0.676320	+	0.736608i	$0.263574\pi$
$984$	0	0
$985$	17105.6	0.553329
$986$	15316.1	0.494691
$987$	0	0
$988$	0	0
$989$	14854.4	0.477597
$990$	0	0
$991$	20371.9	0.653013	0.326506	−	0.945195i	$-0.394129\pi$
$991$	20371.9	0.653013	0.326506	+	0.945195i	$0.394129\pi$
$992$	−15319.5	−0.490316
$993$	0	0
$994$	−5264.21	−0.167979
$995$	−803.344	−0.0255957
$996$	0	0
$997$	18696.5	0.593907	0.296954	−	0.954892i	$-0.404029\pi$
$997$	18696.5	0.593907	0.296954	+	0.954892i	$0.404029\pi$
$998$	22611.6	0.717191
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bh.1.3		9
3.2	odd	2		169.4.a.k.1.7	✓	9
13.12	even	2		1521.4.a.bg.1.7		9
39.2	even	12		169.4.e.h.147.12		36
39.5	even	4		169.4.b.g.168.7		18
39.8	even	4		169.4.b.g.168.12		18
39.11	even	12		169.4.e.h.147.7		36
39.17	odd	6		169.4.c.k.146.7		18
39.20	even	12		169.4.e.h.23.12		36
39.23	odd	6		169.4.c.k.22.7		18
39.29	odd	6		169.4.c.l.22.3		18
39.32	even	12		169.4.e.h.23.7		36
39.35	odd	6		169.4.c.l.146.3		18
39.38	odd	2		169.4.a.l.1.3	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.7	✓	9	3.2	odd	2
169.4.a.l.1.3	yes	9	39.38	odd	2
169.4.b.g.168.7		18	39.5	even	4
169.4.b.g.168.12		18	39.8	even	4
169.4.c.k.22.7		18	39.23	odd	6
169.4.c.k.146.7		18	39.17	odd	6
169.4.c.l.22.3		18	39.29	odd	6
169.4.c.l.146.3		18	39.35	odd	6
169.4.e.h.23.7		36	39.32	even	12
169.4.e.h.23.12		36	39.20	even	12
169.4.e.h.147.7		36	39.11	even	12
169.4.e.h.147.12		36	39.2	even	12
1521.4.a.bg.1.7		9	13.12	even	2
1521.4.a.bh.1.3		9	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-1.72763 q^{2} -5.01528 q^{4} +20.8281 q^{5} -7.56566 q^{7} +22.4856 q^{8} +O(q^{10})\)	\(q-1.72763 q^{2} -5.01528 q^{4} +20.8281 q^{5} -7.56566 q^{7} +22.4856 q^{8} -35.9833 q^{10} -4.40295 q^{11} +13.0707 q^{14} +1.27533 q^{16} +73.0087 q^{17} -55.9424 q^{19} -104.459 q^{20} +7.60668 q^{22} +33.6244 q^{23} +308.809 q^{25} +37.9439 q^{28} -121.429 q^{29} +84.1320 q^{31} -182.088 q^{32} -126.132 q^{34} -157.578 q^{35} -171.716 q^{37} +96.6479 q^{38} +468.332 q^{40} +93.5714 q^{41} +441.776 q^{43} +22.0820 q^{44} -58.0907 q^{46} +272.528 q^{47} -285.761 q^{49} -533.508 q^{50} +480.202 q^{53} -91.7049 q^{55} -170.119 q^{56} +209.785 q^{58} +350.534 q^{59} -484.467 q^{61} -145.349 q^{62} +304.379 q^{64} +967.552 q^{67} -366.159 q^{68} +272.237 q^{70} -402.749 q^{71} +351.621 q^{73} +296.662 q^{74} +280.567 q^{76} +33.3112 q^{77} -820.078 q^{79} +26.5626 q^{80} -161.657 q^{82} +192.314 q^{83} +1520.63 q^{85} -763.226 q^{86} -99.0031 q^{88} -813.571 q^{89} -168.636 q^{92} -470.829 q^{94} -1165.17 q^{95} -788.293 q^{97} +493.690 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8}+O(q^{10}) \) 9 * q + 5 * q^2 + 37 * q^4 + 30 * q^5 - 38 * q^7 + 60 * q^8	\( 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8} - 147 q^{10} + 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} - 161 q^{19} + 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} - 344 q^{28} - 280 q^{29} - 706 q^{31} + 680 q^{32} - 216 q^{34} - 20 q^{35} - 298 q^{37} + 739 q^{38} + 13 q^{40} + 1201 q^{41} - 533 q^{43} + 355 q^{44} + 840 q^{46} + 956 q^{47} + 403 q^{49} - 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} + 2877 q^{58} + 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} + 931 q^{67} + 1536 q^{68} + 4854 q^{70} + 2046 q^{71} + 45 q^{73} + 1990 q^{74} + 3608 q^{76} + 718 q^{77} + 412 q^{79} - 787 q^{80} + 2757 q^{82} + 3709 q^{83} + 2106 q^{85} + 125 q^{86} - 636 q^{88} + 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} + 1087 q^{97} - 282 q^{98}+O(q^{100}) \) 9 * q + 5 * q^2 + 37 * q^4 + 30 * q^5 - 38 * q^7 + 60 * q^8 - 147 * q^10 + 181 * q^11 + 147 * q^14 + 269 * q^16 + 55 * q^17 - 161 * q^19 + 370 * q^20 + 340 * q^22 + 204 * q^23 + 307 * q^25 - 344 * q^28 - 280 * q^29 - 706 * q^31 + 680 * q^32 - 216 * q^34 - 20 * q^35 - 298 * q^37 + 739 * q^38 + 13 * q^40 + 1201 * q^41 - 533 * q^43 + 355 * q^44 + 840 * q^46 + 956 * q^47 + 403 * q^49 - 1156 * q^50 + 278 * q^53 - 250 * q^55 - 250 * q^56 + 2877 * q^58 + 1377 * q^59 - 136 * q^61 - 2035 * q^62 + 284 * q^64 + 931 * q^67 + 1536 * q^68 + 4854 * q^70 + 2046 * q^71 + 45 * q^73 + 1990 * q^74 + 3608 * q^76 + 718 * q^77 + 412 * q^79 - 787 * q^80 + 2757 * q^82 + 3709 * q^83 + 2106 * q^85 + 125 * q^86 - 636 * q^88 + 1663 * q^89 - 4010 * q^92 - 2531 * q^94 + 1614 * q^95 + 1087 * q^97 - 282 * q^98

Introduction

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