LMFDB - Embedded newform 1521.4.a.bg.1.7

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:	$1$
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			1.7
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	$-0.401208\pi$
$2$	1.72763	0.610811	0.305405	+	0.952222i	$0.401208\pi$
$3$	0	0
$3$	0	0
$4$	−5.01528	−0.626910
$5$	−20.8281	−1.86292	−0.931460	−	0.363844i	$-0.881464\pi$
$5$	−20.8281	−1.86292	−0.931460	+	0.363844i	$0.881464\pi$
$6$	0	0
$7$	7.56566	0.408507	0.204254	−	0.978918i	$-0.434523\pi$
$7$	7.56566	0.408507	0.204254	+	0.978918i	$0.434523\pi$
$8$	−22.4856	−0.993734
$9$	0	0
$10$	−35.9833	−1.13789
$11$	4.40295	0.120685	0.0603427	−	0.998178i	$-0.480781\pi$
$11$	4.40295	0.120685	0.0603427	+	0.998178i	$0.480781\pi$
$12$	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	$-0.390338\pi$
$19$	55.9424	0.675477	0.337738	+	0.941240i	$0.390338\pi$
$20$	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	$-0.578367\pi$
$31$	−84.1320	−0.487437	−0.243719	+	0.969846i	$0.578367\pi$
$32$	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	$-0.375413\pi$
$37$	171.716	0.762969	0.381485	+	0.924375i	$0.375413\pi$
$38$	96.6479	0.412588
$39$	0	0
$40$	468.332	1.85125
$41$	−93.5714	−0.356424	−0.178212	−	0.983992i	$-0.557031\pi$
$41$	−93.5714	−0.356424	−0.178212	+	0.983992i	$0.557031\pi$
$42$	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	$-0.638987\pi$
$47$	−272.528	−0.845794	−0.422897	+	0.906178i	$0.638987\pi$
$48$	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	$-0.626400\pi$
$59$	−350.534	−0.773486	−0.386743	+	0.922187i	$0.626400\pi$
$60$	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	$-0.843890\pi$
$67$	−967.552	−1.76426	−0.882129	+	0.471008i	$0.843890\pi$
$68$	−366.159	−0.652990
$69$	0	0
$70$	−272.237	−0.464837
$71$	402.749	0.673205	0.336603	−	0.941647i	$-0.390722\pi$
$71$	402.749	0.673205	0.336603	+	0.941647i	$0.390722\pi$
$72$	0	0
$73$	−351.621	−0.563754	−0.281877	−	0.959450i	$-0.590957\pi$
$73$	−351.621	−0.563754	−0.281877	+	0.959450i	$0.590957\pi$
$74$	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	$-0.540587\pi$
$83$	−192.314	−0.254328	−0.127164	+	0.991882i	$0.540587\pi$
$84$	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	$-0.339007\pi$
$89$	813.571	0.968970	0.484485	+	0.874799i	$0.339007\pi$
$90$	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	$-0.364630\pi$
$97$	788.293	0.825144	0.412572	+	0.910925i	$0.364630\pi$
$98$	−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	$-0.299336\pi$
$109$	1341.63	1.17894	0.589472	+	0.807789i	$0.299336\pi$
$110$	−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	$-0.583331\pi$
$137$	−830.034	−0.517625	−0.258812	+	0.965928i	$0.583331\pi$
$138$	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	$-0.458794\pi$
$149$	469.575	0.258182	0.129091	+	0.991633i	$0.458794\pi$
$150$	0	0
$151$	−1936.82	−1.04382	−0.521908	−	0.853002i	$-0.674780\pi$
$151$	−1936.82	−1.04382	−0.521908	+	0.853002i	$0.674780\pi$
$152$	−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	$-0.685815\pi$
$163$	−2293.98	−1.10232	−0.551160	+	0.834399i	$0.685815\pi$
$164$	469.287	0.223446
$165$	0	0
$166$	−332.249	−0.155346
$167$	−1126.64	−0.522050	−0.261025	−	0.965332i	$-0.584061\pi$
$167$	−1126.64	−0.522050	−0.261025	+	0.965332i	$0.584061\pi$
$168$	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	$-0.509814\pi$
$193$	−165.316	−0.0616564	−0.0308282	+	0.999525i	$0.509814\pi$
$194$	1361.88	0.504007
$195$	0	0
$196$	1433.17	0.522293
$197$	−821.276	−0.297023	−0.148511	−	0.988911i	$-0.547448\pi$
$197$	−821.276	−0.297023	−0.148511	+	0.988911i	$0.547448\pi$
$198$	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	$-0.871860\pi$
$223$	−6127.78	−1.84012	−0.920059	+	0.391780i	$0.871860\pi$
$224$	1377.62	0.410920
$225$	0	0
$226$	−2111.41	−0.621454
$227$	−6342.92	−1.85460	−0.927300	−	0.374318i	$-0.877877\pi$
$227$	−6342.92	−1.85460	−0.927300	+	0.374318i	$0.877877\pi$
$228$	0	0
$229$	−1334.67	−0.385141	−0.192571	−	0.981283i	$-0.561682\pi$
$229$	−1334.67	−0.385141	−0.192571	+	0.981283i	$0.561682\pi$
$230$	−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	$-0.669352\pi$
$239$	−3748.70	−1.01458	−0.507288	+	0.861777i	$0.669352\pi$
$240$	0	0
$241$	−3829.47	−1.02356	−0.511780	−	0.859116i	$-0.671014\pi$
$241$	−3829.47	−1.02356	−0.511780	+	0.859116i	$0.671014\pi$
$242$	−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	$-0.148195\pi$
$271$	7972.80	1.78713	0.893566	+	0.448932i	$0.148195\pi$
$272$	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	$-0.255997\pi$
$281$	6534.86	1.38732	0.693661	+	0.720302i	$0.255997\pi$
$282$	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	$-0.576707\pi$
$293$	−2393.89	−0.477313	−0.238656	+	0.971104i	$0.576707\pi$
$294$	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	$-0.475662\pi$
$307$	821.783	0.152774	0.0763870	+	0.997078i	$0.475662\pi$
$308$	−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	$-0.762777\pi$
$317$	−8295.72	−1.46982	−0.734912	+	0.678163i	$0.762777\pi$
$318$	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	$-0.522617\pi$
$331$	−855.068	−0.141990	−0.0709951	+	0.997477i	$0.522617\pi$
$332$	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	$-0.720868\pi$
$349$	−8339.20	−1.27905	−0.639523	+	0.768772i	$0.720868\pi$
$350$	4036.34	0.616433
$351$	0	0
$352$	801.726	0.121398
$353$	4321.95	0.651655	0.325827	−	0.945429i	$-0.394357\pi$
$353$	4321.95	0.651655	0.325827	+	0.945429i	$0.394357\pi$
$354$	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	$-0.608202\pi$
$359$	−4535.87	−0.666836	−0.333418	+	0.942779i	$0.608202\pi$
$360$	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	$-0.322453\pi$
$379$	7810.76	1.05861	0.529303	+	0.848433i	$0.322453\pi$
$380$	5843.67	0.788878
$381$	0	0
$382$	6927.36	0.927839
$383$	−5576.15	−0.743937	−0.371968	−	0.928245i	$-0.621317\pi$
$383$	−5576.15	−0.743937	−0.371968	+	0.928245i	$0.621317\pi$
$384$	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	$-0.632983\pi$
$397$	−6418.82	−0.811464	−0.405732	+	0.913992i	$0.632983\pi$
$398$	−66.6353	−0.00839227
$399$	0	0
$400$	393.832	0.0492290
$401$	−1553.37	−0.193446	−0.0967230	−	0.995311i	$-0.530836\pi$
$401$	−1553.37	−0.193446	−0.0967230	+	0.995311i	$0.530836\pi$
$402$	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	$-0.677666\pi$
$409$	−8761.54	−1.05924	−0.529622	+	0.848234i	$0.677666\pi$
$410$	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	$-0.417420\pi$
$421$	4431.95	0.513064	0.256532	+	0.966536i	$0.417420\pi$
$422$	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	$-0.690370\pi$
$431$	−10076.0	−1.12609	−0.563045	+	0.826426i	$0.690370\pi$
$432$	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	$-0.610547\pi$
$449$	−6476.36	−0.680709	−0.340355	+	0.940297i	$0.610547\pi$
$450$	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	$-0.484870\pi$
$457$	928.362	0.0950261	0.0475131	+	0.998871i	$0.484870\pi$
$458$	−2305.82	−0.235248
$459$	0	0
$460$	3512.36	0.356010
$461$	−10539.2	−1.06477	−0.532386	−	0.846502i	$-0.678704\pi$
$461$	−10539.2	−1.06477	−0.532386	+	0.846502i	$0.678704\pi$
$462$	0	0
$463$	4928.72	0.494724	0.247362	−	0.968923i	$-0.420436\pi$
$463$	4928.72	0.494724	0.247362	+	0.968923i	$0.420436\pi$
$464$	−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	$-0.471672\pi$
$479$	1863.51	0.177757	0.0888787	+	0.996042i	$0.471672\pi$
$480$	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	$-0.799746\pi$
$487$	−17379.2	−1.61710	−0.808548	+	0.588431i	$0.799746\pi$
$488$	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	$-0.300277\pi$
$499$	13088.2	1.17416	0.587082	+	0.809528i	$0.300277\pi$
$500$	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	$-0.497877\pi$
$509$	153.195	0.0133403	0.00667017	+	0.999978i	$0.497877\pi$
$510$	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	$-0.283491\pi$
$541$	15828.2	1.25787	0.628936	+	0.777457i	$0.283491\pi$
$542$	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	$-0.754171\pi$
$557$	−18832.8	−1.43262	−0.716311	+	0.697781i	$0.754171\pi$
$558$	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	$-0.679640\pi$
$577$	−14826.7	−1.06975	−0.534873	+	0.844933i	$0.679640\pi$
$578$	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	$-0.704042\pi$
$587$	−17009.7	−1.19602	−0.598011	+	0.801488i	$0.704042\pi$
$588$	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	$-0.397116\pi$
$593$	9173.23	0.635244	0.317622	+	0.948217i	$0.397116\pi$
$594$	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	$-0.686152\pi$
$613$	−16756.9	−1.10409	−0.552044	+	0.833815i	$0.686152\pi$
$614$	1419.74	0.0933160
$615$	0	0
$616$	−749.024	−0.0489920
$617$	11985.4	0.782035	0.391017	−	0.920383i	$-0.372123\pi$
$617$	11985.4	0.782035	0.391017	+	0.920383i	$0.372123\pi$
$618$	0	0
$619$	22471.2	1.45912	0.729560	−	0.683917i	$-0.239725\pi$
$619$	22471.2	1.45912	0.729560	+	0.683917i	$0.239725\pi$
$620$	−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	$-0.490042\pi$
$631$	991.566	0.0625572	0.0312786	+	0.999511i	$0.490042\pi$
$632$	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	$-0.257248\pi$
$643$	22527.5	1.38165	0.690823	+	0.723024i	$0.257248\pi$
$644$	−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	$-0.169752\pi$
$661$	29268.8	1.72228	0.861138	+	0.508372i	$0.169752\pi$
$662$	−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	$-0.454841\pi$
$683$	5047.76	0.282793	0.141396	+	0.989953i	$0.454841\pi$
$684$	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	$-0.649549\pi$
$691$	−16446.9	−0.905456	−0.452728	+	0.891649i	$0.649549\pi$
$692$	−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	$-0.671076\pi$
$709$	−19329.7	−1.02390	−0.511949	+	0.859016i	$0.671076\pi$
$710$	−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	$-0.573706\pi$
$733$	−9108.56	−0.458980	−0.229490	+	0.973311i	$0.573706\pi$
$734$	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	$-0.657442\pi$
$739$	−19072.7	−0.949394	−0.474697	+	0.880149i	$0.657442\pi$
$740$	17937.2	0.891059
$741$	0	0
$742$	6276.57	0.310539
$743$	−12276.5	−0.606168	−0.303084	−	0.952964i	$-0.598016\pi$
$743$	−12276.5	−0.606168	−0.303084	+	0.952964i	$0.598016\pi$
$744$	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	$-0.716613\pi$
$761$	−26417.3	−1.25838	−0.629189	+	0.777252i	$0.716613\pi$
$762$	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	$-0.235234\pi$
$769$	31524.2	1.47827	0.739136	+	0.673556i	$0.235234\pi$
$770$	−1198.65	−0.0560990
$771$	0	0
$772$	829.105	0.0386530
$773$	−21666.6	−1.00814	−0.504071	−	0.863662i	$-0.668165\pi$
$773$	−21666.6	−1.00814	−0.504071	+	0.863662i	$0.668165\pi$
$774$	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	$-0.297168\pi$
$787$	26271.2	1.18992	0.594959	+	0.803756i	$0.297168\pi$
$788$	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	$-0.206321\pi$
$811$	36823.2	1.59437	0.797186	+	0.603734i	$0.206321\pi$
$812$	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	$-0.442006\pi$
$821$	8524.53	0.362373	0.181187	+	0.983449i	$0.442006\pi$
$822$	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	$-0.157646\pi$
$827$	41849.9	1.75969	0.879845	+	0.475261i	$0.157646\pi$
$828$	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	$-0.636497\pi$
$839$	−20209.4	−0.831593	−0.415796	+	0.909458i	$0.636497\pi$
$840$	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	$-0.551060\pi$
$853$	−7958.24	−0.319443	−0.159721	+	0.987162i	$0.551060\pi$
$854$	−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	$-0.565716\pi$
$863$	−10393.8	−0.409977	−0.204989	+	0.978764i	$0.565716\pi$
$864$	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	$-0.469278\pi$
$877$	5005.59	0.192733	0.0963664	+	0.995346i	$0.469278\pi$
$878$	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	$-0.535263\pi$
$929$	−6260.88	−0.221112	−0.110556	+	0.993870i	$0.535263\pi$
$930$	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	$-0.556530\pi$
$941$	−10199.1	−0.353326	−0.176663	+	0.984271i	$0.556530\pi$
$942$	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	$-0.500552\pi$
$947$	−101.128	−0.00347015	−0.00173507	+	0.999998i	$0.500552\pi$
$948$	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	$-0.506463\pi$
$967$	−1221.07	−0.0406069	−0.0203035	+	0.999794i	$0.506463\pi$
$968$	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	$-0.329011\pi$
$977$	31253.6	1.02343	0.511714	+	0.859156i	$0.329011\pi$
$978$	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	$-0.736426\pi$
$983$	−41688.2	−1.35264	−0.676320	+	0.736608i	$0.736426\pi$
$984$	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.bg.1.7		9
3.2	odd	2		169.4.a.l.1.3	yes	9
13.12	even	2		1521.4.a.bh.1.3		9
39.2	even	12		169.4.e.h.147.7		36
39.5	even	4		169.4.b.g.168.12		18
39.8	even	4		169.4.b.g.168.7		18
39.11	even	12		169.4.e.h.147.12		36
39.17	odd	6		169.4.c.l.146.3		18
39.20	even	12		169.4.e.h.23.7		36
39.23	odd	6		169.4.c.l.22.3		18
39.29	odd	6		169.4.c.k.22.7		18
39.32	even	12		169.4.e.h.23.12		36
39.35	odd	6		169.4.c.k.146.7		18
39.38	odd	2		169.4.a.k.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.7	✓	9	39.38	odd	2
169.4.a.l.1.3	yes	9	3.2	odd	2
169.4.b.g.168.7		18	39.8	even	4
169.4.b.g.168.12		18	39.5	even	4
169.4.c.k.22.7		18	39.29	odd	6
169.4.c.k.146.7		18	39.35	odd	6
169.4.c.l.22.3		18	39.23	odd	6
169.4.c.l.146.3		18	39.17	odd	6
169.4.e.h.23.7		36	39.20	even	12
169.4.e.h.23.12		36	39.32	even	12
169.4.e.h.147.7		36	39.2	even	12
169.4.e.h.147.12		36	39.11	even	12
1521.4.a.bg.1.7		9	1.1	even	1	trivial
1521.4.a.bh.1.3		9	13.12	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
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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