LMFDB - Embedded newform 1521.4.a.bg.1.6

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			1.6
Root			$1.39012$ 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)

$f(q)$	$=$	$q+0.390115 q^{2} -7.84781 q^{4} +7.52136 q^{5} +19.5446 q^{7} -6.18247 q^{8} +O(q^{10})$	$q+0.390115 q^{2} -7.84781 q^{4} +7.52136 q^{5} +19.5446 q^{7} -6.18247 q^{8} +2.93420 q^{10} -45.8243 q^{11} +7.62463 q^{14} +60.3706 q^{16} -86.5200 q^{17} +148.737 q^{19} -59.0262 q^{20} -17.8767 q^{22} +91.5351 q^{23} -68.4292 q^{25} -153.382 q^{28} -258.901 q^{29} +31.2317 q^{31} +73.0113 q^{32} -33.7528 q^{34} +147.002 q^{35} +148.436 q^{37} +58.0245 q^{38} -46.5006 q^{40} -95.9135 q^{41} -80.9930 q^{43} +359.620 q^{44} +35.7093 q^{46} -94.3777 q^{47} +38.9898 q^{49} -26.6953 q^{50} -493.555 q^{53} -344.661 q^{55} -120.834 q^{56} -101.001 q^{58} +575.686 q^{59} -40.2865 q^{61} +12.1840 q^{62} -454.482 q^{64} +601.141 q^{67} +678.992 q^{68} +57.3476 q^{70} +518.800 q^{71} -1055.21 q^{73} +57.9072 q^{74} -1167.26 q^{76} -895.615 q^{77} -320.840 q^{79} +454.069 q^{80} -37.4173 q^{82} +32.4841 q^{83} -650.748 q^{85} -31.5966 q^{86} +283.307 q^{88} -450.795 q^{89} -718.350 q^{92} -36.8182 q^{94} +1118.70 q^{95} +231.743 q^{97} +15.2105 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.390115	0.137927	0.0689633	−	0.997619i	$-0.478031\pi$
$2$	0.390115	0.137927	0.0689633	+	0.997619i	$0.478031\pi$
$3$	0	0
$3$	0	0
$4$	−7.84781	−0.980976
$5$	7.52136	0.672731	0.336365	−	0.941732i	$-0.390802\pi$
$5$	7.52136	0.672731	0.336365	+	0.941732i	$0.390802\pi$
$6$	0	0
$7$	19.5446	1.05531	0.527654	−	0.849460i	$-0.323072\pi$
$7$	19.5446	1.05531	0.527654	+	0.849460i	$0.323072\pi$
$8$	−6.18247	−0.273229
$9$	0	0
$10$	2.93420	0.0927875
$11$	−45.8243	−1.25605	−0.628024	−	0.778194i	$-0.716136\pi$
$11$	−45.8243	−1.25605	−0.628024	+	0.778194i	$0.716136\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	7.62463	0.145555
$15$	0	0
$16$	60.3706	0.943291
$17$	−86.5200	−1.23436	−0.617182	−	0.786821i	$-0.711726\pi$
$17$	−86.5200	−1.23436	−0.617182	+	0.786821i	$0.711726\pi$
$18$	0	0
$19$	148.737	1.79593	0.897963	−	0.440072i	$-0.145047\pi$
$19$	148.737	1.79593	0.897963	+	0.440072i	$0.145047\pi$
$20$	−59.0262	−0.659933
$21$	0	0
$22$	−17.8767	−0.173242
$23$	91.5351	0.829843	0.414922	−	0.909857i	$-0.363809\pi$
$23$	91.5351	0.829843	0.414922	+	0.909857i	$0.363809\pi$
$24$	0	0
$25$	−68.4292	−0.547433
$26$	0	0
$27$	0	0
$28$	−153.382	−1.03523
$29$	−258.901	−1.65782	−0.828909	−	0.559383i	$-0.811038\pi$
$29$	−258.901	−1.65782	−0.828909	+	0.559383i	$0.811038\pi$
$30$	0	0
$31$	31.2317	0.180948	0.0904739	−	0.995899i	$-0.471162\pi$
$31$	31.2317	0.180948	0.0904739	+	0.995899i	$0.471162\pi$
$32$	73.0113	0.403334
$33$	0	0
$34$	−33.7528	−0.170252
$35$	147.002	0.709938
$36$	0	0
$37$	148.436	0.659533	0.329767	−	0.944062i	$-0.393030\pi$
$37$	148.436	0.659533	0.329767	+	0.944062i	$0.393030\pi$
$38$	58.0245	0.247706
$39$	0	0
$40$	−46.5006	−0.183810
$41$	−95.9135	−0.365346	−0.182673	−	0.983174i	$-0.558475\pi$
$41$	−95.9135	−0.365346	−0.182673	+	0.983174i	$0.558475\pi$
$42$	0	0
$43$	−80.9930	−0.287240	−0.143620	−	0.989633i	$-0.545874\pi$
$43$	−80.9930	−0.287240	−0.143620	+	0.989633i	$0.545874\pi$
$44$	359.620	1.23215
$45$	0	0
$46$	35.7093	0.114457
$47$	−94.3777	−0.292902	−0.146451	−	0.989218i	$-0.546785\pi$
$47$	−94.3777	−0.292902	−0.146451	+	0.989218i	$0.546785\pi$
$48$	0	0
$49$	38.9898	0.113673
$50$	−26.6953	−0.0755056
$51$	0	0
$52$	0	0
$53$	−493.555	−1.27915	−0.639575	−	0.768729i	$-0.720890\pi$
$53$	−493.555	−1.27915	−0.639575	+	0.768729i	$0.720890\pi$
$54$	0	0
$55$	−344.661	−0.844983
$56$	−120.834	−0.288341
$57$	0	0
$58$	−101.001	−0.228657
$59$	575.686	1.27030	0.635152	−	0.772387i	$-0.280938\pi$
$59$	575.686	1.27030	0.635152	+	0.772387i	$0.280938\pi$
$60$	0	0
$61$	−40.2865	−0.0845599	−0.0422800	−	0.999106i	$-0.513462\pi$
$61$	−40.2865	−0.0845599	−0.0422800	+	0.999106i	$0.513462\pi$
$62$	12.1840	0.0249575
$63$	0	0
$64$	−454.482	−0.887660
$65$	0	0
$66$	0	0
$67$	601.141	1.09613	0.548067	−	0.836434i	$-0.315364\pi$
$67$	601.141	1.09613	0.548067	+	0.836434i	$0.315364\pi$
$68$	678.992	1.21088
$69$	0	0
$70$	57.3476	0.0979193
$71$	518.800	0.867187	0.433593	−	0.901109i	$-0.357245\pi$
$71$	518.800	0.867187	0.433593	+	0.901109i	$0.357245\pi$
$72$	0	0
$73$	−1055.21	−1.69182	−0.845908	−	0.533328i	$-0.820941\pi$
$73$	−1055.21	−1.69182	−0.845908	+	0.533328i	$0.820941\pi$
$74$	57.9072	0.0909672
$75$	0	0
$76$	−1167.26	−1.76176
$77$	−895.615	−1.32552
$78$	0	0
$79$	−320.840	−0.456928	−0.228464	−	0.973552i	$-0.573370\pi$
$79$	−320.840	−0.456928	−0.228464	+	0.973552i	$0.573370\pi$
$80$	454.069	0.634581
$81$	0	0
$82$	−37.4173	−0.0503909
$83$	32.4841	0.0429590	0.0214795	−	0.999769i	$-0.493162\pi$
$83$	32.4841	0.0429590	0.0214795	+	0.999769i	$0.493162\pi$
$84$	0	0
$85$	−650.748	−0.830394
$86$	−31.5966	−0.0396180
$87$	0	0
$88$	283.307	0.343189
$89$	−450.795	−0.536901	−0.268450	−	0.963293i	$-0.586512\pi$
$89$	−450.795	−0.536901	−0.268450	+	0.963293i	$0.586512\pi$
$90$	0	0
$91$	0	0
$92$	−718.350	−0.814057
$93$	0	0
$94$	−36.8182	−0.0403990
$95$	1118.70	1.20817
$96$	0	0
$97$	231.743	0.242576	0.121288	−	0.992617i	$-0.461298\pi$
$97$	231.743	0.242576	0.121288	+	0.992617i	$0.461298\pi$
$98$	15.2105	0.0156785
$99$	0	0
$100$	537.019	0.537019
$101$	−570.125	−0.561679	−0.280840	−	0.959755i	$-0.590613\pi$
$101$	−570.125	−0.561679	−0.280840	+	0.959755i	$0.590613\pi$
$102$	0	0
$103$	969.551	0.927502	0.463751	−	0.885965i	$-0.346503\pi$
$103$	969.551	0.927502	0.463751	+	0.885965i	$0.346503\pi$
$104$	0	0
$105$	0	0
$106$	−192.543	−0.176429
$107$	343.156	0.310039	0.155019	−	0.987911i	$-0.450456\pi$
$107$	343.156	0.310039	0.155019	+	0.987911i	$0.450456\pi$
$108$	0	0
$109$	−83.1640	−0.0730795	−0.0365398	−	0.999332i	$-0.511634\pi$
$109$	−83.1640	−0.0730795	−0.0365398	+	0.999332i	$0.511634\pi$
$110$	−134.457	−0.116546
$111$	0	0
$112$	1179.92	0.995461
$113$	2116.18	1.76171	0.880856	−	0.473384i	$-0.156968\pi$
$113$	2116.18	1.76171	0.880856	+	0.473384i	$0.156968\pi$
$114$	0	0
$115$	688.469	0.558261
$116$	2031.81	1.62628
$117$	0	0
$118$	224.584	0.175209
$119$	−1691.00	−1.30263
$120$	0	0
$121$	768.863	0.577658
$122$	−15.7164	−0.0116631
$123$	0	0
$124$	−245.101	−0.177506
$125$	−1454.85	−1.04101
$126$	0	0
$127$	−1176.69	−0.822161	−0.411081	−	0.911599i	$-0.634848\pi$
$127$	−1176.69	−0.822161	−0.411081	+	0.911599i	$0.634848\pi$
$128$	−761.391	−0.525766
$129$	0	0
$130$	0	0
$131$	−775.336	−0.517110	−0.258555	−	0.965996i	$-0.583246\pi$
$131$	−775.336	−0.517110	−0.258555	+	0.965996i	$0.583246\pi$
$132$	0	0
$133$	2907.00	1.89525
$134$	234.514	0.151186
$135$	0	0
$136$	534.907	0.337264
$137$	−2542.62	−1.58563	−0.792814	−	0.609464i	$-0.791385\pi$
$137$	−2542.62	−1.58563	−0.792814	+	0.609464i	$0.791385\pi$
$138$	0	0
$139$	−286.317	−0.174713	−0.0873564	−	0.996177i	$-0.527842\pi$
$139$	−286.317	−0.174713	−0.0873564	+	0.996177i	$0.527842\pi$
$140$	−1153.64	−0.696432
$141$	0	0
$142$	202.392	0.119608
$143$	0	0
$144$	0	0
$145$	−1947.29	−1.11527
$146$	−411.652	−0.233346
$147$	0	0
$148$	−1164.90	−0.646987
$149$	−2354.13	−1.29435	−0.647173	−	0.762343i	$-0.724049\pi$
$149$	−2354.13	−1.29435	−0.647173	+	0.762343i	$0.724049\pi$
$150$	0	0
$151$	−165.158	−0.0890089	−0.0445045	−	0.999009i	$-0.514171\pi$
$151$	−165.158	−0.0890089	−0.0445045	+	0.999009i	$0.514171\pi$
$152$	−919.562	−0.490699
$153$	0	0
$154$	−349.393	−0.182824
$155$	234.905	0.121729
$156$	0	0
$157$	−3095.72	−1.57367	−0.786833	−	0.617166i	$-0.788281\pi$
$157$	−3095.72	−1.57367	−0.786833	+	0.617166i	$0.788281\pi$
$158$	−125.165	−0.0630226
$159$	0	0
$160$	549.144	0.271335
$161$	1789.01	0.875740
$162$	0	0
$163$	−299.310	−0.143827	−0.0719134	−	0.997411i	$-0.522911\pi$
$163$	−299.310	−0.143827	−0.0719134	+	0.997411i	$0.522911\pi$
$164$	752.711	0.358395
$165$	0	0
$166$	12.6726	0.00592519
$167$	−3005.17	−1.39250	−0.696249	−	0.717801i	$-0.745149\pi$
$167$	−3005.17	−1.39250	−0.696249	+	0.717801i	$0.745149\pi$
$168$	0	0
$169$	0	0
$170$	−253.867	−0.114533
$171$	0	0
$172$	635.618	0.281776
$173$	−455.853	−0.200334	−0.100167	−	0.994971i	$-0.531938\pi$
$173$	−455.853	−0.200334	−0.100167	+	0.994971i	$0.531938\pi$
$174$	0	0
$175$	−1337.42	−0.577710
$176$	−2766.44	−1.18482
$177$	0	0
$178$	−175.862	−0.0740529
$179$	−1364.19	−0.569635	−0.284818	−	0.958582i	$-0.591933\pi$
$179$	−1364.19	−0.569635	−0.284818	+	0.958582i	$0.591933\pi$
$180$	0	0
$181$	−2026.11	−0.832041	−0.416021	−	0.909355i	$-0.636576\pi$
$181$	−2026.11	−0.832041	−0.416021	+	0.909355i	$0.636576\pi$
$182$	0	0
$183$	0	0
$184$	−565.914	−0.226738
$185$	1116.44	0.443688
$186$	0	0
$187$	3964.71	1.55042
$188$	740.658	0.287330
$189$	0	0
$190$	436.423	0.166639
$191$	−2161.06	−0.818686	−0.409343	−	0.912380i	$-0.634242\pi$
$191$	−2161.06	−0.818686	−0.409343	+	0.912380i	$0.634242\pi$
$192$	0	0
$193$	−1207.88	−0.450491	−0.225246	−	0.974302i	$-0.572318\pi$
$193$	−1207.88	−0.450491	−0.225246	+	0.974302i	$0.572318\pi$
$194$	90.4063	0.0334577
$195$	0	0
$196$	−305.985	−0.111510
$197$	−4926.76	−1.78181	−0.890906	−	0.454187i	$-0.849930\pi$
$197$	−4926.76	−1.78181	−0.890906	+	0.454187i	$0.849930\pi$
$198$	0	0
$199$	1009.45	0.359589	0.179795	−	0.983704i	$-0.442457\pi$
$199$	1009.45	0.359589	0.179795	+	0.983704i	$0.442457\pi$
$200$	423.061	0.149575
$201$	0	0
$202$	−222.415	−0.0774705
$203$	−5060.11	−1.74951
$204$	0	0
$205$	−721.400	−0.245779
$206$	378.237	0.127927
$207$	0	0
$208$	0	0
$209$	−6815.76	−2.25577
$210$	0	0
$211$	−4911.32	−1.60241	−0.801206	−	0.598388i	$-0.795808\pi$
$211$	−4911.32	−1.60241	−0.801206	+	0.598388i	$0.795808\pi$
$212$	3873.32	1.25482
$213$	0	0
$214$	133.871	0.0427626
$215$	−609.178	−0.193235
$216$	0	0
$217$	610.410	0.190956
$218$	−32.4435	−0.0100796
$219$	0	0
$220$	2704.83	0.828908
$221$	0	0
$222$	0	0
$223$	−1364.58	−0.409771	−0.204885	−	0.978786i	$-0.565682\pi$
$223$	−1364.58	−0.409771	−0.204885	+	0.978786i	$0.565682\pi$
$224$	1426.97	0.425641
$225$	0	0
$226$	825.554	0.242987
$227$	4169.88	1.21923	0.609614	−	0.792699i	$-0.291325\pi$
$227$	4169.88	1.21923	0.609614	+	0.792699i	$0.291325\pi$
$228$	0	0
$229$	−3506.89	−1.01197	−0.505987	−	0.862541i	$-0.668872\pi$
$229$	−3506.89	−1.01197	−0.505987	+	0.862541i	$0.668872\pi$
$230$	268.582	0.0769991
$231$	0	0
$232$	1600.65	0.452965
$233$	570.253	0.160337	0.0801684	−	0.996781i	$-0.474454\pi$
$233$	570.253	0.160337	0.0801684	+	0.996781i	$0.474454\pi$
$234$	0	0
$235$	−709.848	−0.197044
$236$	−4517.87	−1.24614
$237$	0	0
$238$	−659.683	−0.179668
$239$	231.056	0.0625347	0.0312674	−	0.999511i	$-0.490046\pi$
$239$	231.056	0.0625347	0.0312674	+	0.999511i	$0.490046\pi$
$240$	0	0
$241$	−3088.50	−0.825508	−0.412754	−	0.910842i	$-0.635433\pi$
$241$	−3088.50	−0.825508	−0.412754	+	0.910842i	$0.635433\pi$
$242$	299.945	0.0796744
$243$	0	0
$244$	316.161	0.0829513
$245$	293.256	0.0764713
$246$	0	0
$247$	0	0
$248$	−193.089	−0.0494403
$249$	0	0
$250$	−567.559	−0.143582
$251$	2445.64	0.615009	0.307504	−	0.951547i	$-0.400506\pi$
$251$	2445.64	0.615009	0.307504	+	0.951547i	$0.400506\pi$
$252$	0	0
$253$	−4194.53	−1.04232
$254$	−459.045	−0.113398
$255$	0	0
$256$	3338.83	0.815143
$257$	−3273.99	−0.794654	−0.397327	−	0.917677i	$-0.630062\pi$
$257$	−3273.99	−0.794654	−0.397327	+	0.917677i	$0.630062\pi$
$258$	0	0
$259$	2901.12	0.696010
$260$	0	0
$261$	0	0
$262$	−302.471	−0.0713233
$263$	4631.27	1.08584	0.542921	−	0.839784i	$-0.317318\pi$
$263$	4631.27	1.08584	0.542921	+	0.839784i	$0.317318\pi$
$264$	0	0
$265$	−3712.20	−0.860524
$266$	1134.06	0.261406
$267$	0	0
$268$	−4717.64	−1.07528
$269$	2838.51	0.643373	0.321686	−	0.946846i	$-0.395750\pi$
$269$	2838.51	0.643373	0.321686	+	0.946846i	$0.395750\pi$
$270$	0	0
$271$	−7052.31	−1.58080	−0.790401	−	0.612589i	$-0.790128\pi$
$271$	−7052.31	−1.58080	−0.790401	+	0.612589i	$0.790128\pi$
$272$	−5223.26	−1.16436
$273$	0	0
$274$	−991.917	−0.218700
$275$	3135.72	0.687603
$276$	0	0
$277$	−1938.31	−0.420439	−0.210220	−	0.977654i	$-0.567418\pi$
$277$	−1938.31	−0.420439	−0.210220	+	0.977654i	$0.567418\pi$
$278$	−111.697	−0.0240975
$279$	0	0
$280$	−908.834	−0.193976
$281$	−3290.74	−0.698609	−0.349305	−	0.937009i	$-0.613582\pi$
$281$	−3290.74	−0.698609	−0.349305	+	0.937009i	$0.613582\pi$
$282$	0	0
$283$	−7918.08	−1.66318	−0.831592	−	0.555388i	$-0.812570\pi$
$283$	−7918.08	−1.66318	−0.831592	+	0.555388i	$0.812570\pi$
$284$	−4071.45	−0.850690
$285$	0	0
$286$	0	0
$287$	−1874.59	−0.385552
$288$	0	0
$289$	2572.71	0.523653
$290$	−759.667	−0.153825
$291$	0	0
$292$	8281.06	1.65963
$293$	5675.43	1.13161	0.565806	−	0.824539i	$-0.308565\pi$
$293$	5675.43	1.13161	0.565806	+	0.824539i	$0.308565\pi$
$294$	0	0
$295$	4329.94	0.854572
$296$	−917.702	−0.180204
$297$	0	0
$298$	−918.381	−0.178525
$299$	0	0
$300$	0	0
$301$	−1582.97	−0.303126
$302$	−64.4306	−0.0122767
$303$	0	0
$304$	8979.33	1.69408
$305$	−303.009	−0.0568861
$306$	0	0
$307$	4338.86	0.806618	0.403309	−	0.915064i	$-0.367860\pi$
$307$	4338.86	0.806618	0.403309	+	0.915064i	$0.367860\pi$
$308$	7028.62	1.30030
$309$	0	0
$310$	91.6401	0.0167897
$311$	5234.75	0.954454	0.477227	−	0.878780i	$-0.341642\pi$
$311$	5234.75	0.954454	0.477227	+	0.878780i	$0.341642\pi$
$312$	0	0
$313$	2167.86	0.391484	0.195742	−	0.980655i	$-0.437288\pi$
$313$	2167.86	0.391484	0.195742	+	0.980655i	$0.437288\pi$
$314$	−1207.69	−0.217050
$315$	0	0
$316$	2517.89	0.448236
$317$	4863.71	0.861744	0.430872	−	0.902413i	$-0.358206\pi$
$317$	4863.71	0.861744	0.430872	+	0.902413i	$0.358206\pi$
$318$	0	0
$319$	11864.0	2.08230
$320$	−3418.32	−0.597156
$321$	0	0
$322$	697.922	0.120788
$323$	−12868.7	−2.21682
$324$	0	0
$325$	0	0
$326$	−116.765	−0.0198375
$327$	0	0
$328$	592.983	0.0998231
$329$	−1844.57	−0.309102
$330$	0	0
$331$	2685.26	0.445908	0.222954	−	0.974829i	$-0.428430\pi$
$331$	2685.26	0.445908	0.222954	+	0.974829i	$0.428430\pi$
$332$	−254.929	−0.0421417
$333$	0	0
$334$	−1172.36	−0.192062
$335$	4521.39	0.737404
$336$	0	0
$337$	−6518.36	−1.05364	−0.526821	−	0.849976i	$-0.676616\pi$
$337$	−6518.36	−1.05364	−0.526821	+	0.849976i	$0.676616\pi$
$338$	0	0
$339$	0	0
$340$	5106.95	0.814597
$341$	−1431.17	−0.227279
$342$	0	0
$343$	−5941.75	−0.935347
$344$	500.737	0.0784824
$345$	0	0
$346$	−177.835	−0.0276314
$347$	−75.7963	−0.0117261	−0.00586305	−	0.999983i	$-0.501866\pi$
$347$	−75.7963	−0.0117261	−0.00586305	+	0.999983i	$0.501866\pi$
$348$	0	0
$349$	3682.09	0.564750	0.282375	−	0.959304i	$-0.408878\pi$
$349$	3682.09	0.564750	0.282375	+	0.959304i	$0.408878\pi$
$350$	−521.747	−0.0796816
$351$	0	0
$352$	−3345.69	−0.506607
$353$	10031.5	1.51254	0.756268	−	0.654261i	$-0.227020\pi$
$353$	10031.5	1.51254	0.756268	+	0.654261i	$0.227020\pi$
$354$	0	0
$355$	3902.08	0.583383
$356$	3537.75	0.526687
$357$	0	0
$358$	−532.193	−0.0785678
$359$	−6869.76	−1.00995	−0.504975	−	0.863134i	$-0.668498\pi$
$359$	−6869.76	−1.00995	−0.504975	+	0.863134i	$0.668498\pi$
$360$	0	0
$361$	15263.7	2.22535
$362$	−790.416	−0.114761
$363$	0	0
$364$	0	0
$365$	−7936.59	−1.13814
$366$	0	0
$367$	8883.29	1.26350	0.631749	−	0.775173i	$-0.282337\pi$
$367$	8883.29	1.26350	0.631749	+	0.775173i	$0.282337\pi$
$368$	5526.03	0.782784
$369$	0	0
$370$	435.541	0.0611964
$371$	−9646.31	−1.34990
$372$	0	0
$373$	−5454.50	−0.757166	−0.378583	−	0.925567i	$-0.623589\pi$
$373$	−5454.50	−0.757166	−0.378583	+	0.925567i	$0.623589\pi$
$374$	1546.70	0.213844
$375$	0	0
$376$	583.487	0.0800294
$377$	0	0
$378$	0	0
$379$	−6642.15	−0.900222	−0.450111	−	0.892973i	$-0.648616\pi$
$379$	−6642.15	−0.900222	−0.450111	+	0.892973i	$0.648616\pi$
$380$	−8779.37	−1.18519
$381$	0	0
$382$	−843.064	−0.112919
$383$	1262.33	0.168413	0.0842066	−	0.996448i	$-0.473164\pi$
$383$	1262.33	0.168413	0.0842066	+	0.996448i	$0.473164\pi$
$384$	0	0
$385$	−6736.24	−0.891716
$386$	−471.211	−0.0621347
$387$	0	0
$388$	−1818.67	−0.237962
$389$	2793.42	0.364093	0.182046	−	0.983290i	$-0.441728\pi$
$389$	2793.42	0.364093	0.182046	+	0.983290i	$0.441728\pi$
$390$	0	0
$391$	−7919.62	−1.02433
$392$	−241.053	−0.0310588
$393$	0	0
$394$	−1922.00	−0.245759
$395$	−2413.15	−0.307390
$396$	0	0
$397$	−6848.91	−0.865836	−0.432918	−	0.901433i	$-0.642516\pi$
$397$	−6848.91	−0.865836	−0.432918	+	0.901433i	$0.642516\pi$
$398$	393.803	0.0495969
$399$	0	0
$400$	−4131.11	−0.516389
$401$	11024.5	1.37291	0.686456	−	0.727171i	$-0.259165\pi$
$401$	11024.5	1.37291	0.686456	+	0.727171i	$0.259165\pi$
$402$	0	0
$403$	0	0
$404$	4474.24	0.550994
$405$	0	0
$406$	−1974.03	−0.241304
$407$	−6801.97	−0.828406
$408$	0	0
$409$	−3530.56	−0.426833	−0.213417	−	0.976961i	$-0.568459\pi$
$409$	−3530.56	−0.426833	−0.213417	+	0.976961i	$0.568459\pi$
$410$	−281.429	−0.0338995
$411$	0	0
$412$	−7608.86	−0.909858
$413$	11251.5	1.34056
$414$	0	0
$415$	244.325	0.0288998
$416$	0	0
$417$	0	0
$418$	−2658.93	−0.311131
$419$	4179.22	0.487275	0.243637	−	0.969866i	$-0.421659\pi$
$419$	4179.22	0.487275	0.243637	+	0.969866i	$0.421659\pi$
$420$	0	0
$421$	6209.31	0.718820	0.359410	−	0.933180i	$-0.382978\pi$
$421$	6209.31	0.718820	0.359410	+	0.933180i	$0.382978\pi$
$422$	−1915.98	−0.221015
$423$	0	0
$424$	3051.39	0.349501
$425$	5920.49	0.675732
$426$	0	0
$427$	−787.382	−0.0892367
$428$	−2693.03	−0.304141
$429$	0	0
$430$	−237.650	−0.0266523
$431$	−11880.2	−1.32773	−0.663864	−	0.747854i	$-0.731084\pi$
$431$	−11880.2	−1.32773	−0.663864	+	0.747854i	$0.731084\pi$
$432$	0	0
$433$	8725.71	0.968432	0.484216	−	0.874949i	$-0.339105\pi$
$433$	8725.71	0.968432	0.484216	+	0.874949i	$0.339105\pi$
$434$	238.130	0.0263378
$435$	0	0
$436$	652.655	0.0716893
$437$	13614.7	1.49034
$438$	0	0
$439$	−1200.37	−0.130503	−0.0652514	−	0.997869i	$-0.520785\pi$
$439$	−1200.37	−0.130503	−0.0652514	+	0.997869i	$0.520785\pi$
$440$	2130.86	0.230874
$441$	0	0
$442$	0	0
$443$	−2258.86	−0.242261	−0.121130	−	0.992637i	$-0.538652\pi$
$443$	−2258.86	−0.242261	−0.121130	+	0.992637i	$0.538652\pi$
$444$	0	0
$445$	−3390.59	−0.361190
$446$	−532.343	−0.0565183
$447$	0	0
$448$	−8882.65	−0.936754
$449$	14662.4	1.54112	0.770561	−	0.637367i	$-0.219976\pi$
$449$	14662.4	1.54112	0.770561	+	0.637367i	$0.219976\pi$
$450$	0	0
$451$	4395.16	0.458892
$452$	−16607.4	−1.72820
$453$	0	0
$454$	1626.73	0.168164
$455$	0	0
$456$	0	0
$457$	9334.94	0.955514	0.477757	−	0.878492i	$-0.341450\pi$
$457$	9334.94	0.955514	0.477757	+	0.878492i	$0.341450\pi$
$458$	−1368.09	−0.139578
$459$	0	0
$460$	−5402.97	−0.547641
$461$	−10736.9	−1.08474	−0.542370	−	0.840140i	$-0.682473\pi$
$461$	−10736.9	−1.08474	−0.542370	+	0.840140i	$0.682473\pi$
$462$	0	0
$463$	10650.0	1.06900	0.534501	−	0.845168i	$-0.320499\pi$
$463$	10650.0	1.06900	0.534501	+	0.845168i	$0.320499\pi$
$464$	−15630.0	−1.56380
$465$	0	0
$466$	222.464	0.0221147
$467$	2638.11	0.261407	0.130703	−	0.991422i	$-0.458276\pi$
$467$	2638.11	0.261407	0.130703	+	0.991422i	$0.458276\pi$
$468$	0	0
$469$	11749.0	1.15676
$470$	−276.923	−0.0271776
$471$	0	0
$472$	−3559.16	−0.347084
$473$	3711.45	0.360787
$474$	0	0
$475$	−10177.9	−0.983149
$476$	13270.6	1.27785
$477$	0	0
$478$	90.1386	0.00862520
$479$	3225.31	0.307658	0.153829	−	0.988097i	$-0.450840\pi$
$479$	3225.31	0.307658	0.153829	+	0.988097i	$0.450840\pi$
$480$	0	0
$481$	0	0
$482$	−1204.87	−0.113860
$483$	0	0
$484$	−6033.89	−0.566669
$485$	1743.02	0.163189
$486$	0	0
$487$	−1496.39	−0.139236	−0.0696181	−	0.997574i	$-0.522178\pi$
$487$	−1496.39	−0.139236	−0.0696181	+	0.997574i	$0.522178\pi$
$488$	249.070	0.0231043
$489$	0	0
$490$	114.404	0.0105474
$491$	518.408	0.0476485	0.0238243	−	0.999716i	$-0.492416\pi$
$491$	518.408	0.0476485	0.0238243	+	0.999716i	$0.492416\pi$
$492$	0	0
$493$	22400.1	2.04635
$494$	0	0
$495$	0	0
$496$	1885.48	0.170686
$497$	10139.7	0.915148
$498$	0	0
$499$	−2405.01	−0.215757	−0.107879	−	0.994164i	$-0.534406\pi$
$499$	−2405.01	−0.215757	−0.107879	+	0.994164i	$0.534406\pi$
$500$	11417.4	1.02120
$501$	0	0
$502$	954.080	0.0848260
$503$	−2413.76	−0.213965	−0.106983	−	0.994261i	$-0.534119\pi$
$503$	−2413.76	−0.213965	−0.106983	+	0.994261i	$0.534119\pi$
$504$	0	0
$505$	−4288.12	−0.377859
$506$	−1636.35	−0.143764
$507$	0	0
$508$	9234.45	0.806520
$509$	−15678.3	−1.36528	−0.682641	−	0.730754i	$-0.739169\pi$
$509$	−15678.3	−1.36528	−0.682641	+	0.730754i	$0.739169\pi$
$510$	0	0
$511$	−20623.6	−1.78539
$512$	7393.65	0.638196
$513$	0	0
$514$	−1277.24	−0.109604
$515$	7292.34	0.623959
$516$	0	0
$517$	4324.79	0.367899
$518$	1131.77	0.0959983
$519$	0	0
$520$	0	0
$521$	11691.5	0.983135	0.491568	−	0.870839i	$-0.336424\pi$
$521$	11691.5	0.983135	0.491568	+	0.870839i	$0.336424\pi$
$522$	0	0
$523$	−3878.95	−0.324311	−0.162156	−	0.986765i	$-0.551845\pi$
$523$	−3878.95	−0.324311	−0.162156	+	0.986765i	$0.551845\pi$
$524$	6084.69	0.507273
$525$	0	0
$526$	1806.73	0.149767
$527$	−2702.17	−0.223355
$528$	0	0
$529$	−3788.32	−0.311360
$530$	−1448.19	−0.118689
$531$	0	0
$532$	−22813.6	−1.85920
$533$	0	0
$534$	0	0
$535$	2581.00	0.208573
$536$	−3716.54	−0.299496
$537$	0	0
$538$	1107.35	0.0887382
$539$	−1786.68	−0.142779
$540$	0	0
$541$	−16353.0	−1.29958	−0.649788	−	0.760115i	$-0.725142\pi$
$541$	−16353.0	−1.29958	−0.649788	+	0.760115i	$0.725142\pi$
$542$	−2751.22	−0.218035
$543$	0	0
$544$	−6316.93	−0.497861
$545$	−625.506	−0.0491628
$546$	0	0
$547$	2748.67	0.214853	0.107426	−	0.994213i	$-0.465739\pi$
$547$	2748.67	0.214853	0.107426	+	0.994213i	$0.465739\pi$
$548$	19954.0	1.55546
$549$	0	0
$550$	1223.29	0.0948387
$551$	−38508.1	−2.97732
$552$	0	0
$553$	−6270.68	−0.482200
$554$	−756.163	−0.0579897
$555$	0	0
$556$	2246.96	0.171389
$557$	16765.6	1.27537	0.637686	−	0.770297i	$-0.279892\pi$
$557$	16765.6	1.27537	0.637686	+	0.770297i	$0.279892\pi$
$558$	0	0
$559$	0	0
$560$	8874.58	0.669677
$561$	0	0
$562$	−1283.77	−0.0963568
$563$	−18492.4	−1.38430	−0.692151	−	0.721752i	$-0.743337\pi$
$563$	−18492.4	−1.38430	−0.692151	+	0.721752i	$0.743337\pi$
$564$	0	0
$565$	15916.5	1.18516
$566$	−3088.96	−0.229397
$567$	0	0
$568$	−3207.47	−0.236941
$569$	−1563.28	−0.115178	−0.0575888	−	0.998340i	$-0.518341\pi$
$569$	−1563.28	−0.115178	−0.0575888	+	0.998340i	$0.518341\pi$
$570$	0	0
$571$	9165.98	0.671776	0.335888	−	0.941902i	$-0.390964\pi$
$571$	9165.98	0.671776	0.335888	+	0.941902i	$0.390964\pi$
$572$	0	0
$573$	0	0
$574$	−731.305	−0.0531778
$575$	−6263.67	−0.454284
$576$	0	0
$577$	−18762.5	−1.35372	−0.676858	−	0.736114i	$-0.736659\pi$
$577$	−18762.5	−1.35372	−0.676858	+	0.736114i	$0.736659\pi$
$578$	1003.65	0.0722257
$579$	0	0
$580$	15281.9	1.09405
$581$	634.888	0.0453349
$582$	0	0
$583$	22616.8	1.60667
$584$	6523.79	0.462254
$585$	0	0
$586$	2214.07	0.156079
$587$	12646.3	0.889212	0.444606	−	0.895726i	$-0.353344\pi$
$587$	12646.3	0.889212	0.444606	+	0.895726i	$0.353344\pi$
$588$	0	0
$589$	4645.31	0.324969
$590$	1689.18	0.117868
$591$	0	0
$592$	8961.17	0.622132
$593$	−9662.74	−0.669142	−0.334571	−	0.942371i	$-0.608591\pi$
$593$	−9662.74	−0.669142	−0.334571	+	0.942371i	$0.608591\pi$
$594$	0	0
$595$	−12718.6	−0.876321
$596$	18474.7	1.26972
$597$	0	0
$598$	0	0
$599$	26968.7	1.83959	0.919794	−	0.392402i	$-0.128356\pi$
$599$	26968.7	1.83959	0.919794	+	0.392402i	$0.128356\pi$
$600$	0	0
$601$	11280.1	0.765600	0.382800	−	0.923831i	$-0.374960\pi$
$601$	11280.1	0.765600	0.382800	+	0.923831i	$0.374960\pi$
$602$	−617.542	−0.0418092
$603$	0	0
$604$	1296.13	0.0873156
$605$	5782.89	0.388608
$606$	0	0
$607$	12052.6	0.805931	0.402965	−	0.915215i	$-0.367979\pi$
$607$	12052.6	0.805931	0.402965	+	0.915215i	$0.367979\pi$
$608$	10859.5	0.724358
$609$	0	0
$610$	−118.208	−0.00784610
$611$	0	0
$612$	0	0
$613$	−11259.7	−0.741883	−0.370941	−	0.928656i	$-0.620965\pi$
$613$	−11259.7	−0.741883	−0.370941	+	0.928656i	$0.620965\pi$
$614$	1692.65	0.111254
$615$	0	0
$616$	5537.12	0.362170
$617$	−24058.2	−1.56976	−0.784882	−	0.619645i	$-0.787277\pi$
$617$	−24058.2	−1.56976	−0.784882	+	0.619645i	$0.787277\pi$
$618$	0	0
$619$	2793.41	0.181384	0.0906919	−	0.995879i	$-0.471092\pi$
$619$	2793.41	0.181384	0.0906919	+	0.995879i	$0.471092\pi$
$620$	−1843.49	−0.119413
$621$	0	0
$622$	2042.15	0.131645
$623$	−8810.59	−0.566595
$624$	0	0
$625$	−2388.81	−0.152884
$626$	845.714	0.0539960
$627$	0	0
$628$	24294.6	1.54373
$629$	−12842.7	−0.814104
$630$	0	0
$631$	24850.0	1.56777	0.783883	−	0.620908i	$-0.213236\pi$
$631$	24850.0	1.56777	0.783883	+	0.620908i	$0.213236\pi$
$632$	1983.59	0.124846
$633$	0	0
$634$	1897.41	0.118857
$635$	−8850.32	−0.553093
$636$	0	0
$637$	0	0
$638$	4628.31	0.287205
$639$	0	0
$640$	−5726.69	−0.353699
$641$	7794.71	0.480300	0.240150	−	0.970736i	$-0.422803\pi$
$641$	7794.71	0.480300	0.240150	+	0.970736i	$0.422803\pi$
$642$	0	0
$643$	−27709.9	−1.69949	−0.849744	−	0.527195i	$-0.823244\pi$
$643$	−27709.9	−1.69949	−0.849744	+	0.527195i	$0.823244\pi$
$644$	−14039.8	−0.859080
$645$	0	0
$646$	−5020.28	−0.305759
$647$	−11150.9	−0.677571	−0.338786	−	0.940864i	$-0.610016\pi$
$647$	−11150.9	−0.677571	−0.338786	+	0.940864i	$0.610016\pi$
$648$	0	0
$649$	−26380.4	−1.59556
$650$	0	0
$651$	0	0
$652$	2348.93	0.141091
$653$	−29137.5	−1.74615	−0.873077	−	0.487583i	$-0.837879\pi$
$653$	−29137.5	−1.74615	−0.873077	+	0.487583i	$0.837879\pi$
$654$	0	0
$655$	−5831.58	−0.347876
$656$	−5790.36	−0.344627
$657$	0	0
$658$	−719.595	−0.0426333
$659$	5300.12	0.313298	0.156649	−	0.987654i	$-0.449931\pi$
$659$	5300.12	0.313298	0.156649	+	0.987654i	$0.449931\pi$
$660$	0	0
$661$	27412.5	1.61305	0.806523	−	0.591203i	$-0.201347\pi$
$661$	27412.5	1.61305	0.806523	+	0.591203i	$0.201347\pi$
$662$	1047.56	0.0615025
$663$	0	0
$664$	−200.832	−0.0117377
$665$	21864.6	1.27499
$666$	0	0
$667$	−23698.6	−1.37573
$668$	23584.0	1.36601
$669$	0	0
$670$	1763.87	0.101708
$671$	1846.10	0.106211
$672$	0	0
$673$	−21282.1	−1.21896	−0.609482	−	0.792800i	$-0.708623\pi$
$673$	−21282.1	−1.21896	−0.609482	+	0.792800i	$0.708623\pi$
$674$	−2542.91	−0.145325
$675$	0	0
$676$	0	0
$677$	13544.2	0.768904	0.384452	−	0.923145i	$-0.374390\pi$
$677$	13544.2	0.768904	0.384452	+	0.923145i	$0.374390\pi$
$678$	0	0
$679$	4529.31	0.255992
$680$	4023.23	0.226888
$681$	0	0
$682$	−558.322	−0.0313479
$683$	10350.1	0.579849	0.289924	−	0.957050i	$-0.406370\pi$
$683$	10350.1	0.579849	0.289924	+	0.957050i	$0.406370\pi$
$684$	0	0
$685$	−19124.0	−1.06670
$686$	−2317.97	−0.129009
$687$	0	0
$688$	−4889.60	−0.270951
$689$	0	0
$690$	0	0
$691$	25714.8	1.41569	0.707843	−	0.706370i	$-0.249669\pi$
$691$	25714.8	1.41569	0.707843	+	0.706370i	$0.249669\pi$
$692$	3577.45	0.196523
$693$	0	0
$694$	−29.5693	−0.00161734
$695$	−2153.49	−0.117535
$696$	0	0
$697$	8298.43	0.450969
$698$	1436.44	0.0778940
$699$	0	0
$700$	10495.8	0.566720
$701$	−7431.30	−0.400394	−0.200197	−	0.979756i	$-0.564158\pi$
$701$	−7431.30	−0.400394	−0.200197	+	0.979756i	$0.564158\pi$
$702$	0	0
$703$	22077.9	1.18447
$704$	20826.3	1.11494
$705$	0	0
$706$	3913.46	0.208619
$707$	−11142.8	−0.592744
$708$	0	0
$709$	19986.8	1.05870	0.529351	−	0.848403i	$-0.322435\pi$
$709$	19986.8	1.05870	0.529351	+	0.848403i	$0.322435\pi$
$710$	1522.26	0.0804641
$711$	0	0
$712$	2787.03	0.146697
$713$	2858.80	0.150158
$714$	0	0
$715$	0	0
$716$	10705.9	0.558798
$717$	0	0
$718$	−2680.00	−0.139299
$719$	−36101.9	−1.87256	−0.936281	−	0.351251i	$-0.885756\pi$
$719$	−36101.9	−1.87256	−0.936281	+	0.351251i	$0.885756\pi$
$720$	0	0
$721$	18949.5	0.978800
$722$	5954.59	0.306935
$723$	0	0
$724$	15900.5	0.816213
$725$	17716.4	0.907545
$726$	0	0
$727$	−1751.90	−0.0893735	−0.0446868	−	0.999001i	$-0.514229\pi$
$727$	−1751.90	−0.0893735	−0.0446868	+	0.999001i	$0.514229\pi$
$728$	0	0
$729$	0	0
$730$	−3096.18	−0.156979
$731$	7007.52	0.354559
$732$	0	0
$733$	20031.3	1.00938	0.504688	−	0.863302i	$-0.331607\pi$
$733$	20031.3	1.00938	0.504688	+	0.863302i	$0.331607\pi$
$734$	3465.51	0.174270
$735$	0	0
$736$	6683.10	0.334704
$737$	−27546.8	−1.37680
$738$	0	0
$739$	−18632.9	−0.927502	−0.463751	−	0.885966i	$-0.653497\pi$
$739$	−18632.9	−0.927502	−0.463751	+	0.885966i	$0.653497\pi$
$740$	−8761.62	−0.435248
$741$	0	0
$742$	−3763.17	−0.186187
$743$	4907.69	0.242323	0.121161	−	0.992633i	$-0.461338\pi$
$743$	4907.69	0.242323	0.121161	+	0.992633i	$0.461338\pi$
$744$	0	0
$745$	−17706.2	−0.870747
$746$	−2127.88	−0.104433
$747$	0	0
$748$	−31114.3	−1.52093
$749$	6706.84	0.327186
$750$	0	0
$751$	31156.9	1.51389	0.756945	−	0.653479i	$-0.226691\pi$
$751$	31156.9	1.51389	0.756945	+	0.653479i	$0.226691\pi$
$752$	−5697.64	−0.276292
$753$	0	0
$754$	0	0
$755$	−1242.21	−0.0598790
$756$	0	0
$757$	−11047.0	−0.530396	−0.265198	−	0.964194i	$-0.585437\pi$
$757$	−11047.0	−0.530396	−0.265198	+	0.964194i	$0.585437\pi$
$758$	−2591.20	−0.124165
$759$	0	0
$760$	−6916.35	−0.330109
$761$	27606.1	1.31501	0.657503	−	0.753452i	$-0.271613\pi$
$761$	27606.1	1.31501	0.657503	+	0.753452i	$0.271613\pi$
$762$	0	0
$763$	−1625.40	−0.0771213
$764$	16959.6	0.803112
$765$	0	0
$766$	492.456	0.0232287
$767$	0	0
$768$	0	0
$769$	3694.72	0.173257	0.0866287	−	0.996241i	$-0.472391\pi$
$769$	3694.72	0.173257	0.0866287	+	0.996241i	$0.472391\pi$
$770$	−2627.91	−0.122991
$771$	0	0
$772$	9479.18	0.441921
$773$	19808.0	0.921662	0.460831	−	0.887488i	$-0.347551\pi$
$773$	19808.0	0.921662	0.460831	+	0.887488i	$0.347551\pi$
$774$	0	0
$775$	−2137.16	−0.0990569
$776$	−1432.74	−0.0662789
$777$	0	0
$778$	1089.76	0.0502181
$779$	−14265.9	−0.656133
$780$	0	0
$781$	−23773.6	−1.08923
$782$	−3089.56	−0.141282
$783$	0	0
$784$	2353.84	0.107227
$785$	−23284.0	−1.05865
$786$	0	0
$787$	4116.79	0.186464	0.0932322	−	0.995644i	$-0.470280\pi$
$787$	4116.79	0.186464	0.0932322	+	0.995644i	$0.470280\pi$
$788$	38664.3	1.74792
$789$	0	0
$790$	−941.408	−0.0423972
$791$	41359.8	1.85915
$792$	0	0
$793$	0	0
$794$	−2671.86	−0.119422
$795$	0	0
$796$	−7922.00	−0.352748
$797$	−25359.3	−1.12707	−0.563533	−	0.826094i	$-0.690558\pi$
$797$	−25359.3	−1.12707	−0.563533	+	0.826094i	$0.690558\pi$
$798$	0	0
$799$	8165.55	0.361548
$800$	−4996.10	−0.220799
$801$	0	0
$802$	4300.83	0.189361
$803$	48354.1	2.12500
$804$	0	0
$805$	13455.8	0.589137
$806$	0	0
$807$	0	0
$808$	3524.78	0.153467
$809$	5558.73	0.241576	0.120788	−	0.992678i	$-0.461458\pi$
$809$	5558.73	0.241576	0.120788	+	0.992678i	$0.461458\pi$
$810$	0	0
$811$	−15021.4	−0.650399	−0.325199	−	0.945646i	$-0.605431\pi$
$811$	−15021.4	−0.650399	−0.325199	+	0.945646i	$0.605431\pi$
$812$	39710.8	1.71623
$813$	0	0
$814$	−2653.55	−0.114259
$815$	−2251.22	−0.0967567
$816$	0	0
$817$	−12046.6	−0.515862
$818$	−1377.32	−0.0588717
$819$	0	0
$820$	5661.41	0.241104
$821$	−15901.6	−0.675966	−0.337983	−	0.941152i	$-0.609745\pi$
$821$	−15901.6	−0.675966	−0.337983	+	0.941152i	$0.609745\pi$
$822$	0	0
$823$	−40279.3	−1.70601	−0.853006	−	0.521901i	$-0.825223\pi$
$823$	−40279.3	−1.70601	−0.853006	+	0.521901i	$0.825223\pi$
$824$	−5994.23	−0.253421
$825$	0	0
$826$	4389.39	0.184899
$827$	−5251.09	−0.220796	−0.110398	−	0.993887i	$-0.535213\pi$
$827$	−5251.09	−0.220796	−0.110398	+	0.993887i	$0.535213\pi$
$828$	0	0
$829$	33964.4	1.42296	0.711479	−	0.702708i	$-0.248026\pi$
$829$	33964.4	1.42296	0.711479	+	0.702708i	$0.248026\pi$
$830$	95.3148	0.00398606
$831$	0	0
$832$	0	0
$833$	−3373.40	−0.140314
$834$	0	0
$835$	−22603.0	−0.936776
$836$	53488.8	2.21286
$837$	0	0
$838$	1630.38	0.0672081
$839$	15485.0	0.637188	0.318594	−	0.947891i	$-0.396789\pi$
$839$	15485.0	0.637188	0.318594	+	0.947891i	$0.396789\pi$
$840$	0	0
$841$	42640.8	1.74836
$842$	2422.35	0.0991444
$843$	0	0
$844$	38543.1	1.57193
$845$	0	0
$846$	0	0
$847$	15027.1	0.609606
$848$	−29796.2	−1.20661
$849$	0	0
$850$	2309.67	0.0932014
$851$	13587.1	0.547309
$852$	0	0
$853$	20057.8	0.805118	0.402559	−	0.915394i	$-0.368121\pi$
$853$	20057.8	0.805118	0.402559	+	0.915394i	$0.368121\pi$
$854$	−307.170	−0.0123081
$855$	0	0
$856$	−2121.55	−0.0847117
$857$	8066.23	0.321514	0.160757	−	0.986994i	$-0.448606\pi$
$857$	8066.23	0.321514	0.160757	+	0.986994i	$0.448606\pi$
$858$	0	0
$859$	39719.0	1.57764	0.788821	−	0.614623i	$-0.210692\pi$
$859$	39719.0	1.57764	0.788821	+	0.614623i	$0.210692\pi$
$860$	4780.71	0.189559
$861$	0	0
$862$	−4634.66	−0.183129
$863$	24473.8	0.965351	0.482676	−	0.875799i	$-0.339665\pi$
$863$	24473.8	0.965351	0.482676	+	0.875799i	$0.339665\pi$
$864$	0	0
$865$	−3428.63	−0.134771
$866$	3404.03	0.133572
$867$	0	0
$868$	−4790.38	−0.187323
$869$	14702.3	0.573924
$870$	0	0
$871$	0	0
$872$	514.159	0.0199675
$873$	0	0
$874$	5311.28	0.205557
$875$	−28434.4	−1.09858
$876$	0	0
$877$	−25326.6	−0.975162	−0.487581	−	0.873078i	$-0.662121\pi$
$877$	−25326.6	−0.975162	−0.487581	+	0.873078i	$0.662121\pi$
$878$	−468.284	−0.0179998
$879$	0	0
$880$	−20807.4	−0.797064
$881$	−1327.44	−0.0507634	−0.0253817	−	0.999678i	$-0.508080\pi$
$881$	−1327.44	−0.0507634	−0.0253817	+	0.999678i	$0.508080\pi$
$882$	0	0
$883$	−2112.05	−0.0804941	−0.0402470	−	0.999190i	$-0.512814\pi$
$883$	−2112.05	−0.0804941	−0.0402470	+	0.999190i	$0.512814\pi$
$884$	0	0
$885$	0	0
$886$	−881.214	−0.0334142
$887$	40935.4	1.54958	0.774790	−	0.632219i	$-0.217856\pi$
$887$	40935.4	1.54958	0.774790	+	0.632219i	$0.217856\pi$
$888$	0	0
$889$	−22997.9	−0.867632
$890$	−1322.72	−0.0498177
$891$	0	0
$892$	10708.9	0.401975
$893$	−14037.4	−0.526030
$894$	0	0
$895$	−10260.6	−0.383211
$896$	−14881.0	−0.554845
$897$	0	0
$898$	5720.04	0.212562
$899$	−8085.93	−0.299979
$900$	0	0
$901$	42702.3	1.57894
$902$	1714.62	0.0632934
$903$	0	0
$904$	−13083.2	−0.481351
$905$	−15239.1	−0.559740
$906$	0	0
$907$	153.968	0.00563663	0.00281831	−	0.999996i	$-0.499103\pi$
$907$	153.968	0.00563663	0.00281831	+	0.999996i	$0.499103\pi$
$908$	−32724.4	−1.19603
$909$	0	0
$910$	0	0
$911$	733.607	0.0266800	0.0133400	−	0.999911i	$-0.495754\pi$
$911$	733.607	0.0266800	0.0133400	+	0.999911i	$0.495754\pi$
$912$	0	0
$913$	−1488.56	−0.0539586
$914$	3641.70	0.131791
$915$	0	0
$916$	27521.4	0.992722
$917$	−15153.6	−0.545710
$918$	0	0
$919$	−51106.4	−1.83443	−0.917216	−	0.398390i	$-0.869569\pi$
$919$	−51106.4	−1.83443	−0.917216	+	0.398390i	$0.869569\pi$
$920$	−4256.44	−0.152533
$921$	0	0
$922$	−4188.61	−0.149615
$923$	0	0
$924$	0	0
$925$	−10157.4	−0.361051
$926$	4154.73	0.147444
$927$	0	0
$928$	−18902.7	−0.668655
$929$	−30645.2	−1.08228	−0.541139	−	0.840933i	$-0.682007\pi$
$929$	−30645.2	−1.08228	−0.541139	+	0.840933i	$0.682007\pi$
$930$	0	0
$931$	5799.22	0.204148
$932$	−4475.23	−0.157287
$933$	0	0
$934$	1029.17	0.0360550
$935$	29820.0	1.04302
$936$	0	0
$937$	−24422.2	−0.851482	−0.425741	−	0.904845i	$-0.639987\pi$
$937$	−24422.2	−0.851482	−0.425741	+	0.904845i	$0.639987\pi$
$938$	4583.48	0.159548
$939$	0	0
$940$	5570.75	0.193296
$941$	−16475.9	−0.570774	−0.285387	−	0.958412i	$-0.592122\pi$
$941$	−16475.9	−0.570774	−0.285387	+	0.958412i	$0.592122\pi$
$942$	0	0
$943$	−8779.46	−0.303180
$944$	34754.5	1.19827
$945$	0	0
$946$	1447.89	0.0497622
$947$	−10122.1	−0.347332	−0.173666	−	0.984805i	$-0.555561\pi$
$947$	−10122.1	−0.347332	−0.173666	+	0.984805i	$0.555561\pi$
$948$	0	0
$949$	0	0
$950$	−3970.57	−0.135602
$951$	0	0
$952$	10454.5	0.355917
$953$	−38097.2	−1.29495	−0.647475	−	0.762086i	$-0.724175\pi$
$953$	−38097.2	−1.29495	−0.647475	+	0.762086i	$0.724175\pi$
$954$	0	0
$955$	−16254.1	−0.550756
$956$	−1813.29	−0.0613451
$957$	0	0
$958$	1258.24	0.0424342
$959$	−49694.5	−1.67332
$960$	0	0
$961$	−28815.6	−0.967258
$962$	0	0
$963$	0	0
$964$	24237.9	0.809804
$965$	−9084.87	−0.303059
$966$	0	0
$967$	44515.5	1.48037	0.740187	−	0.672401i	$-0.234737\pi$
$967$	44515.5	1.48037	0.740187	+	0.672401i	$0.234737\pi$
$968$	−4753.47	−0.157833
$969$	0	0
$970$	679.978	0.0225080
$971$	25444.3	0.840934	0.420467	−	0.907308i	$-0.361866\pi$
$971$	25444.3	0.840934	0.420467	+	0.907308i	$0.361866\pi$
$972$	0	0
$973$	−5595.94	−0.184376
$974$	−583.765	−0.0192044
$975$	0	0
$976$	−2432.12	−0.0797646
$977$	39467.9	1.29242	0.646208	−	0.763161i	$-0.276354\pi$
$977$	39467.9	1.29242	0.646208	+	0.763161i	$0.276354\pi$
$978$	0	0
$979$	20657.4	0.674374
$980$	−2301.42	−0.0750165
$981$	0	0
$982$	202.239	0.00657200
$983$	−14970.4	−0.485740	−0.242870	−	0.970059i	$-0.578089\pi$
$983$	−14970.4	−0.485740	−0.242870	+	0.970059i	$0.578089\pi$
$984$	0	0
$985$	−37055.9	−1.19868
$986$	8738.63	0.282246
$987$	0	0
$988$	0	0
$989$	−7413.71	−0.238364
$990$	0	0
$991$	58422.4	1.87270	0.936352	−	0.351063i	$-0.114180\pi$
$991$	58422.4	1.87270	0.936352	+	0.351063i	$0.114180\pi$
$992$	2280.27	0.0729825
$993$	0	0
$994$	3955.66	0.126223
$995$	7592.46	0.241907
$996$	0	0
$997$	20131.6	0.639493	0.319746	−	0.947503i	$-0.396402\pi$
$997$	20131.6	0.639493	0.319746	+	0.947503i	$0.396402\pi$
$998$	−938.230	−0.0297587
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bg.1.6		9
3.2	odd	2		169.4.a.l.1.4	yes	9
13.12	even	2		1521.4.a.bh.1.4		9
39.2	even	12		169.4.e.h.147.8		36
39.5	even	4		169.4.b.g.168.11		18
39.8	even	4		169.4.b.g.168.8		18
39.11	even	12		169.4.e.h.147.11		36
39.17	odd	6		169.4.c.l.146.4		18
39.20	even	12		169.4.e.h.23.8		36
39.23	odd	6		169.4.c.l.22.4		18
39.29	odd	6		169.4.c.k.22.6		18
39.32	even	12		169.4.e.h.23.11		36
39.35	odd	6		169.4.c.k.146.6		18
39.38	odd	2		169.4.a.k.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.6	✓	9	39.38	odd	2
169.4.a.l.1.4	yes	9	3.2	odd	2
169.4.b.g.168.8		18	39.8	even	4
169.4.b.g.168.11		18	39.5	even	4
169.4.c.k.22.6		18	39.29	odd	6
169.4.c.k.146.6		18	39.35	odd	6
169.4.c.l.22.4		18	39.23	odd	6
169.4.c.l.146.4		18	39.17	odd	6
169.4.e.h.23.8		36	39.20	even	12
169.4.e.h.23.11		36	39.32	even	12
169.4.e.h.147.8		36	39.2	even	12
169.4.e.h.147.11		36	39.11	even	12
1521.4.a.bg.1.6		9	1.1	even	1	trivial
1521.4.a.bh.1.4		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)

\(f(q)\)	\(=\)	\(q+0.390115 q^{2} -7.84781 q^{4} +7.52136 q^{5} +19.5446 q^{7} -6.18247 q^{8} +O(q^{10})\)	\(q+0.390115 q^{2} -7.84781 q^{4} +7.52136 q^{5} +19.5446 q^{7} -6.18247 q^{8} +2.93420 q^{10} -45.8243 q^{11} +7.62463 q^{14} +60.3706 q^{16} -86.5200 q^{17} +148.737 q^{19} -59.0262 q^{20} -17.8767 q^{22} +91.5351 q^{23} -68.4292 q^{25} -153.382 q^{28} -258.901 q^{29} +31.2317 q^{31} +73.0113 q^{32} -33.7528 q^{34} +147.002 q^{35} +148.436 q^{37} +58.0245 q^{38} -46.5006 q^{40} -95.9135 q^{41} -80.9930 q^{43} +359.620 q^{44} +35.7093 q^{46} -94.3777 q^{47} +38.9898 q^{49} -26.6953 q^{50} -493.555 q^{53} -344.661 q^{55} -120.834 q^{56} -101.001 q^{58} +575.686 q^{59} -40.2865 q^{61} +12.1840 q^{62} -454.482 q^{64} +601.141 q^{67} +678.992 q^{68} +57.3476 q^{70} +518.800 q^{71} -1055.21 q^{73} +57.9072 q^{74} -1167.26 q^{76} -895.615 q^{77} -320.840 q^{79} +454.069 q^{80} -37.4173 q^{82} +32.4841 q^{83} -650.748 q^{85} -31.5966 q^{86} +283.307 q^{88} -450.795 q^{89} -718.350 q^{92} -36.8182 q^{94} +1118.70 q^{95} +231.743 q^{97} +15.2105 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + 37 q^{4} - 30 q^{5} + 38 q^{7} - 60 q^{8}+O(q^{10}) \) 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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