LMFDB - Embedded newform 1521.4.a.bf.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:	$0$
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} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 $ x^9 - 56x^7 - 27x^6 + 945x^5 + 763x^4 - 4139x^3 - 2478x^2 + 63*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$5.06791$ 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)

$f(q)$	$=$	$q+2.82093 q^{2} -0.0423641 q^{4} +3.41089 q^{5} +13.3442 q^{7} -22.6869 q^{8} +O(q^{10})$	$q+2.82093 q^{2} -0.0423641 q^{4} +3.41089 q^{5} +13.3442 q^{7} -22.6869 q^{8} +9.62187 q^{10} -35.4529 q^{11} +37.6430 q^{14} -63.6593 q^{16} -69.6526 q^{17} +12.4014 q^{19} -0.144499 q^{20} -100.010 q^{22} +126.251 q^{23} -113.366 q^{25} -0.565314 q^{28} +179.060 q^{29} +255.935 q^{31} +1.91716 q^{32} -196.485 q^{34} +45.5155 q^{35} +207.235 q^{37} +34.9833 q^{38} -77.3825 q^{40} +117.701 q^{41} +553.224 q^{43} +1.50193 q^{44} +356.146 q^{46} -62.9185 q^{47} -164.933 q^{49} -319.797 q^{50} +147.031 q^{53} -120.926 q^{55} -302.738 q^{56} +505.115 q^{58} -274.087 q^{59} +603.039 q^{61} +721.974 q^{62} +514.683 q^{64} -741.019 q^{67} +2.95077 q^{68} +128.396 q^{70} -572.574 q^{71} +26.7155 q^{73} +584.595 q^{74} -0.525372 q^{76} -473.090 q^{77} -207.798 q^{79} -217.135 q^{80} +332.026 q^{82} +1031.37 q^{83} -237.577 q^{85} +1560.60 q^{86} +804.318 q^{88} -1229.66 q^{89} -5.34852 q^{92} -177.488 q^{94} +42.2996 q^{95} +1795.12 q^{97} -465.264 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) $ 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8	$ 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29} + 717 q^{31} - 707 q^{32} + 632 q^{34} + 418 q^{35} + 1108 q^{37} + 660 q^{38} - 1506 q^{40} - 334 q^{41} + 242 q^{43} + 307 q^{44} + 979 q^{46} + 184 q^{47} - 38 q^{49} + 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} + 1161 q^{58} - 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} + 2308 q^{67} - 2785 q^{68} - 1420 q^{70} - 96 q^{71} + 2505 q^{73} + 1191 q^{74} + 2409 q^{76} + 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 1517 q^{82} - 1539 q^{83} + 4296 q^{85} + 3763 q^{86} - 3716 q^{88} + 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} + 1445 q^{97} - 1457 q^{98}+O(q^{100}) $ 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8 - 54 * q^10 - 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 + 352 * q^19 - 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 + 940 * q^28 + 97 * q^29 + 717 * q^31 - 707 * q^32 + 632 * q^34 + 418 * q^35 + 1108 * q^37 + 660 * q^38 - 1506 * q^40 - 334 * q^41 + 242 * q^43 + 307 * q^44 + 979 * q^46 + 184 * q^47 - 38 * q^49 + 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 + 1161 * q^58 - 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 + 2308 * q^67 - 2785 * q^68 - 1420 * q^70 - 96 * q^71 + 2505 * q^73 + 1191 * q^74 + 2409 * q^76 + 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 1517 * q^82 - 1539 * q^83 + 4296 * q^85 + 3763 * q^86 - 3716 * q^88 + 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 + 1445 * q^97 - 1457 * 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 $

$2$	2.82093	0.997349	0.498674	−	0.866789i	$-0.333820\pi$
$2$	2.82093	0.997349	0.498674	+	0.866789i	$0.333820\pi$
$3$	0	0
$3$	0	0
$4$	−0.0423641	−0.00529552
$5$	3.41089	0.305079	0.152539	−	0.988297i	$-0.451255\pi$
$5$	3.41089	0.305079	0.152539	+	0.988297i	$0.451255\pi$
$6$	0	0
$7$	13.3442	0.720518	0.360259	−	0.932852i	$-0.382688\pi$
$7$	13.3442	0.720518	0.360259	+	0.932852i	$0.382688\pi$
$8$	−22.6869	−1.00263
$9$	0	0
$10$	9.62187	0.304270
$11$	−35.4529	−0.971769	−0.485885	−	0.874023i	$-0.661502\pi$
$11$	−35.4529	−0.971769	−0.485885	+	0.874023i	$0.661502\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	37.6430	0.718607
$15$	0	0
$16$	−63.6593	−0.994676
$17$	−69.6526	−0.993720	−0.496860	−	0.867831i	$-0.665514\pi$
$17$	−69.6526	−0.993720	−0.496860	+	0.867831i	$0.665514\pi$
$18$	0	0
$19$	12.4014	0.149740	0.0748701	−	0.997193i	$-0.476146\pi$
$19$	12.4014	0.149740	0.0748701	+	0.997193i	$0.476146\pi$
$20$	−0.144499	−0.00161555
$21$	0	0
$22$	−100.010	−0.969193
$23$	126.251	1.14457	0.572287	−	0.820053i	$-0.306056\pi$
$23$	126.251	1.14457	0.572287	+	0.820053i	$0.306056\pi$
$24$	0	0
$25$	−113.366	−0.906927
$26$	0	0
$27$	0	0
$28$	−0.565314	−0.00381551
$29$	179.060	1.14657	0.573286	−	0.819356i	$-0.305669\pi$
$29$	179.060	1.14657	0.573286	+	0.819356i	$0.305669\pi$
$30$	0	0
$31$	255.935	1.48281	0.741407	−	0.671055i	$-0.234159\pi$
$31$	255.935	1.48281	0.741407	+	0.671055i	$0.234159\pi$
$32$	1.91716	0.0105909
$33$	0	0
$34$	−196.485	−0.991085
$35$	45.5155	0.219815
$36$	0	0
$37$	207.235	0.920790	0.460395	−	0.887714i	$-0.347708\pi$
$37$	207.235	0.920790	0.460395	+	0.887714i	$0.347708\pi$
$38$	34.9833	0.149343
$39$	0	0
$40$	−77.3825	−0.305881
$41$	117.701	0.448337	0.224169	−	0.974550i	$-0.428033\pi$
$41$	117.701	0.448337	0.224169	+	0.974550i	$0.428033\pi$
$42$	0	0
$43$	553.224	1.96200	0.980998	−	0.194016i	$-0.0621515\pi$
$43$	553.224	1.96200	0.980998	+	0.194016i	$0.0621515\pi$
$44$	1.50193	0.00514602
$45$	0	0
$46$	356.146	1.14154
$47$	−62.9185	−0.195268	−0.0976341	−	0.995222i	$-0.531127\pi$
$47$	−62.9185	−0.195268	−0.0976341	+	0.995222i	$0.531127\pi$
$48$	0	0
$49$	−164.933	−0.480854
$50$	−319.797	−0.904522
$51$	0	0
$52$	0	0
$53$	147.031	0.381061	0.190531	−	0.981681i	$-0.438979\pi$
$53$	147.031	0.381061	0.190531	+	0.981681i	$0.438979\pi$
$54$	0	0
$55$	−120.926	−0.296466
$56$	−302.738	−0.722413
$57$	0	0
$58$	505.115	1.14353
$59$	−274.087	−0.604798	−0.302399	−	0.953181i	$-0.597787\pi$
$59$	−274.087	−0.604798	−0.302399	+	0.953181i	$0.597787\pi$
$60$	0	0
$61$	603.039	1.26576	0.632879	−	0.774251i	$-0.281873\pi$
$61$	603.039	1.26576	0.632879	+	0.774251i	$0.281873\pi$
$62$	721.974	1.47888
$63$	0	0
$64$	514.683	1.00524
$65$	0	0
$66$	0	0
$67$	−741.019	−1.35119	−0.675596	−	0.737272i	$-0.736113\pi$
$67$	−741.019	−1.35119	−0.675596	+	0.737272i	$0.736113\pi$
$68$	2.95077	0.00526226
$69$	0	0
$70$	128.396	0.219232
$71$	−572.574	−0.957071	−0.478536	−	0.878068i	$-0.658832\pi$
$71$	−572.574	−0.957071	−0.478536	+	0.878068i	$0.658832\pi$
$72$	0	0
$73$	26.7155	0.0428330	0.0214165	−	0.999771i	$-0.493182\pi$
$73$	26.7155	0.0428330	0.0214165	+	0.999771i	$0.493182\pi$
$74$	584.595	0.918349
$75$	0	0
$76$	−0.525372	−0.000792952 0
$77$	−473.090	−0.700177
$78$	0	0
$79$	−207.798	−0.295938	−0.147969	−	0.988992i	$-0.547274\pi$
$79$	−207.798	−0.295938	−0.147969	+	0.988992i	$0.547274\pi$
$80$	−217.135	−0.303455
$81$	0	0
$82$	332.026	0.447149
$83$	1031.37	1.36395	0.681976	−	0.731374i	$-0.261121\pi$
$83$	1031.37	1.36395	0.681976	+	0.731374i	$0.261121\pi$
$84$	0	0
$85$	−237.577	−0.303163
$86$	1560.60	1.95679
$87$	0	0
$88$	804.318	0.974325
$89$	−1229.66	−1.46453	−0.732266	−	0.681019i	$-0.761537\pi$
$89$	−1229.66	−1.46453	−0.732266	+	0.681019i	$0.761537\pi$
$90$	0	0
$91$	0	0
$92$	−5.34852	−0.00606111
$93$	0	0
$94$	−177.488	−0.194750
$95$	42.2996	0.0456826
$96$	0	0
$97$	1795.12	1.87903	0.939517	−	0.342502i	$-0.111274\pi$
$97$	1795.12	1.87903	0.939517	+	0.342502i	$0.111274\pi$
$98$	−465.264	−0.479579
$99$	0	0
$100$	4.80265	0.00480265
$101$	769.697	0.758295	0.379147	−	0.925336i	$-0.376217\pi$
$101$	769.697	0.758295	0.379147	+	0.925336i	$0.376217\pi$
$102$	0	0
$103$	1543.66	1.47671	0.738357	−	0.674410i	$-0.235602\pi$
$103$	1543.66	1.47671	0.738357	+	0.674410i	$0.235602\pi$
$104$	0	0
$105$	0	0
$106$	414.764	0.380051
$107$	−2027.64	−1.83196	−0.915978	−	0.401228i	$-0.868583\pi$
$107$	−2027.64	−1.83196	−0.915978	+	0.401228i	$0.868583\pi$
$108$	0	0
$109$	1644.97	1.44550	0.722751	−	0.691108i	$-0.242877\pi$
$109$	1644.97	1.44550	0.722751	+	0.691108i	$0.242877\pi$
$110$	−341.123	−0.295680
$111$	0	0
$112$	−849.481	−0.716682
$113$	654.514	0.544880	0.272440	−	0.962173i	$-0.412169\pi$
$113$	654.514	0.544880	0.272440	+	0.962173i	$0.412169\pi$
$114$	0	0
$115$	430.629	0.349186
$116$	−7.58571	−0.00607169
$117$	0	0
$118$	−773.179	−0.603194
$119$	−929.456	−0.715993
$120$	0	0
$121$	−74.0896	−0.0556646
$122$	1701.13	1.26240
$123$	0	0
$124$	−10.8425	−0.00785227
$125$	−813.039	−0.581763
$126$	0	0
$127$	1325.54	0.926161	0.463080	−	0.886316i	$-0.346744\pi$
$127$	1325.54	0.926161	0.463080	+	0.886316i	$0.346744\pi$
$128$	1436.55	0.991983
$129$	0	0
$130$	0	0
$131$	−930.114	−0.620339	−0.310170	−	0.950681i	$-0.600386\pi$
$131$	−930.114	−0.620339	−0.310170	+	0.950681i	$0.600386\pi$
$132$	0	0
$133$	165.486	0.107891
$134$	−2090.36	−1.34761
$135$	0	0
$136$	1580.20	0.996333
$137$	2751.03	1.71559	0.857797	−	0.513989i	$-0.171833\pi$
$137$	2751.03	1.71559	0.857797	+	0.513989i	$0.171833\pi$
$138$	0	0
$139$	2436.65	1.48686	0.743432	−	0.668812i	$-0.233197\pi$
$139$	2436.65	1.48686	0.743432	+	0.668812i	$0.233197\pi$
$140$	−1.92822	−0.00116403
$141$	0	0
$142$	−1615.19	−0.954534
$143$	0	0
$144$	0	0
$145$	610.753	0.349795
$146$	75.3624	0.0427194
$147$	0	0
$148$	−8.77933	−0.00487606
$149$	1517.58	0.834397	0.417198	−	0.908815i	$-0.363012\pi$
$149$	1517.58	0.834397	0.417198	+	0.908815i	$0.363012\pi$
$150$	0	0
$151$	583.642	0.314544	0.157272	−	0.987555i	$-0.449730\pi$
$151$	583.642	0.314544	0.157272	+	0.987555i	$0.449730\pi$
$152$	−281.349	−0.150134
$153$	0	0
$154$	−1334.55	−0.698321
$155$	872.965	0.452376
$156$	0	0
$157$	26.0932	0.0132641	0.00663206	−	0.999978i	$-0.497889\pi$
$157$	26.0932	0.0132641	0.00663206	+	0.999978i	$0.497889\pi$
$158$	−586.184	−0.295154
$159$	0	0
$160$	6.53922	0.00323107
$161$	1684.72	0.824686
$162$	0	0
$163$	17.0905	0.00821246	0.00410623	−	0.999992i	$-0.498693\pi$
$163$	17.0905	0.00821246	0.00410623	+	0.999992i	$0.498693\pi$
$164$	−4.98631	−0.00237418
$165$	0	0
$166$	2909.43	1.36034
$167$	−3919.48	−1.81616	−0.908079	−	0.418799i	$-0.862451\pi$
$167$	−3919.48	−1.81616	−0.908079	+	0.418799i	$0.862451\pi$
$168$	0	0
$169$	0	0
$170$	−670.188	−0.302359
$171$	0	0
$172$	−23.4368	−0.0103898
$173$	1976.35	0.868551	0.434276	−	0.900780i	$-0.357004\pi$
$173$	1976.35	0.868551	0.434276	+	0.900780i	$0.357004\pi$
$174$	0	0
$175$	−1512.77	−0.653457
$176$	2256.91	0.966596
$177$	0	0
$178$	−3468.77	−1.46065
$179$	2784.34	1.16263	0.581316	−	0.813678i	$-0.302538\pi$
$179$	2784.34	1.16263	0.581316	+	0.813678i	$0.302538\pi$
$180$	0	0
$181$	−1886.53	−0.774723	−0.387361	−	0.921928i	$-0.626613\pi$
$181$	−1886.53	−0.774723	−0.387361	+	0.921928i	$0.626613\pi$
$182$	0	0
$183$	0	0
$184$	−2864.25	−1.14758
$185$	706.855	0.280914
$186$	0	0
$187$	2469.39	0.965666
$188$	2.66549	0.00103405
$189$	0	0
$190$	119.324	0.0455615
$191$	−2447.94	−0.927364	−0.463682	−	0.886002i	$-0.653472\pi$
$191$	−2447.94	−0.927364	−0.463682	+	0.886002i	$0.653472\pi$
$192$	0	0
$193$	−1925.66	−0.718196	−0.359098	−	0.933300i	$-0.616916\pi$
$193$	−1925.66	−0.718196	−0.359098	+	0.933300i	$0.616916\pi$
$194$	5063.89	1.87405
$195$	0	0
$196$	6.98724	0.00254637
$197$	1819.95	0.658203	0.329102	−	0.944294i	$-0.393254\pi$
$197$	1819.95	0.658203	0.329102	+	0.944294i	$0.393254\pi$
$198$	0	0
$199$	−1273.97	−0.453816	−0.226908	−	0.973916i	$-0.572862\pi$
$199$	−1273.97	−0.453816	−0.226908	+	0.973916i	$0.572862\pi$
$200$	2571.92	0.909312
$201$	0	0
$202$	2171.26	0.756284
$203$	2389.40	0.826125
$204$	0	0
$205$	401.465	0.136778
$206$	4354.56	1.47280
$207$	0	0
$208$	0	0
$209$	−439.664	−0.145513
$210$	0	0
$211$	−1319.44	−0.430492	−0.215246	−	0.976560i	$-0.569055\pi$
$211$	−1319.44	−0.430492	−0.215246	+	0.976560i	$0.569055\pi$
$212$	−6.22884	−0.00201792
$213$	0	0
$214$	−5719.83	−1.82710
$215$	1886.98	0.598564
$216$	0	0
$217$	3415.24	1.06839
$218$	4640.35	1.44167
$219$	0	0
$220$	5.12292	0.00156994
$221$	0	0
$222$	0	0
$223$	1203.45	0.361384	0.180692	−	0.983540i	$-0.442166\pi$
$223$	1203.45	0.361384	0.180692	+	0.983540i	$0.442166\pi$
$224$	25.5829	0.00763095
$225$	0	0
$226$	1846.34	0.543436
$227$	4361.23	1.27518	0.637589	−	0.770377i	$-0.279932\pi$
$227$	4361.23	1.27518	0.637589	+	0.770377i	$0.279932\pi$
$228$	0	0
$229$	5384.29	1.55373	0.776864	−	0.629669i	$-0.216809\pi$
$229$	5384.29	1.55373	0.776864	+	0.629669i	$0.216809\pi$
$230$	1214.77	0.348260
$231$	0	0
$232$	−4062.32	−1.14959
$233$	4913.60	1.38155	0.690774	−	0.723070i	$-0.257270\pi$
$233$	4913.60	1.38155	0.690774	+	0.723070i	$0.257270\pi$
$234$	0	0
$235$	−214.608	−0.0595722
$236$	11.6114	0.00320272
$237$	0	0
$238$	−2621.93	−0.714094
$239$	−963.718	−0.260827	−0.130414	−	0.991460i	$-0.541631\pi$
$239$	−963.718	−0.260827	−0.130414	+	0.991460i	$0.541631\pi$
$240$	0	0
$241$	1544.48	0.412817	0.206409	−	0.978466i	$-0.433822\pi$
$241$	1544.48	0.412817	0.206409	+	0.978466i	$0.433822\pi$
$242$	−209.001	−0.0555170
$243$	0	0
$244$	−25.5472	−0.00670284
$245$	−562.568	−0.146698
$246$	0	0
$247$	0	0
$248$	−5806.38	−1.48671
$249$	0	0
$250$	−2293.52	−0.580221
$251$	−3768.17	−0.947588	−0.473794	−	0.880636i	$-0.657116\pi$
$251$	−3768.17	−0.947588	−0.473794	+	0.880636i	$0.657116\pi$
$252$	0	0
$253$	−4475.98	−1.11226
$254$	3739.25	0.923705
$255$	0	0
$256$	−65.0695	−0.0158861
$257$	−1282.68	−0.311327	−0.155664	−	0.987810i	$-0.549752\pi$
$257$	−1282.68	−0.311327	−0.155664	+	0.987810i	$0.549752\pi$
$258$	0	0
$259$	2765.38	0.663445
$260$	0	0
$261$	0	0
$262$	−2623.78	−0.618694
$263$	−5029.74	−1.17927	−0.589633	−	0.807671i	$-0.700728\pi$
$263$	−5029.74	−1.17927	−0.589633	+	0.807671i	$0.700728\pi$
$264$	0	0
$265$	501.506	0.116254
$266$	466.824	0.107604
$267$	0	0
$268$	31.3926	0.00715526
$269$	−5628.46	−1.27574	−0.637868	−	0.770146i	$-0.720184\pi$
$269$	−5628.46	−1.27574	−0.637868	+	0.770146i	$0.720184\pi$
$270$	0	0
$271$	3368.39	0.755037	0.377518	−	0.926002i	$-0.376778\pi$
$271$	3368.39	0.755037	0.377518	+	0.926002i	$0.376778\pi$
$272$	4434.03	0.988429
$273$	0	0
$274$	7760.45	1.71104
$275$	4019.15	0.881324
$276$	0	0
$277$	−6507.75	−1.41160	−0.705799	−	0.708412i	$-0.749412\pi$
$277$	−6507.75	−1.41160	−0.705799	+	0.708412i	$0.749412\pi$
$278$	6873.62	1.48292
$279$	0	0
$280$	−1032.61	−0.220393
$281$	3625.51	0.769680	0.384840	−	0.922983i	$-0.374257\pi$
$281$	3625.51	0.769680	0.384840	+	0.922983i	$0.374257\pi$
$282$	0	0
$283$	−4635.41	−0.973662	−0.486831	−	0.873496i	$-0.661847\pi$
$283$	−4635.41	−0.973662	−0.486831	+	0.873496i	$0.661847\pi$
$284$	24.2566	0.00506819
$285$	0	0
$286$	0	0
$287$	1570.62	0.323035
$288$	0	0
$289$	−61.5178	−0.0125214
$290$	1722.89	0.348867
$291$	0	0
$292$	−1.13178	−0.000226823 0
$293$	2907.07	0.579634	0.289817	−	0.957082i	$-0.406406\pi$
$293$	2907.07	0.579634	0.289817	+	0.957082i	$0.406406\pi$
$294$	0	0
$295$	−934.879	−0.184511
$296$	−4701.53	−0.923212
$297$	0	0
$298$	4280.99	0.832185
$299$	0	0
$300$	0	0
$301$	7382.32	1.41365
$302$	1646.41	0.313710
$303$	0	0
$304$	−789.461	−0.148943
$305$	2056.90	0.386156
$306$	0	0
$307$	933.950	0.173627	0.0868133	−	0.996225i	$-0.472332\pi$
$307$	933.950	0.173627	0.0868133	+	0.996225i	$0.472332\pi$
$308$	20.0421	0.00370780
$309$	0	0
$310$	2462.57	0.451176
$311$	3633.55	0.662508	0.331254	−	0.943542i	$-0.392528\pi$
$311$	3633.55	0.662508	0.331254	+	0.943542i	$0.392528\pi$
$312$	0	0
$313$	−7507.26	−1.35570	−0.677852	−	0.735198i	$-0.737089\pi$
$313$	−7507.26	−1.35570	−0.677852	+	0.735198i	$0.737089\pi$
$314$	73.6071	0.0132289
$315$	0	0
$316$	8.80319	0.00156715
$317$	−1214.60	−0.215202	−0.107601	−	0.994194i	$-0.534317\pi$
$317$	−1214.60	−0.215202	−0.107601	+	0.994194i	$0.534317\pi$
$318$	0	0
$319$	−6348.19	−1.11420
$320$	1755.52	0.306677
$321$	0	0
$322$	4752.47	0.822500
$323$	−863.786	−0.148800
$324$	0	0
$325$	0	0
$326$	48.2111	0.00819069
$327$	0	0
$328$	−2670.28	−0.449517
$329$	−839.595	−0.140694
$330$	0	0
$331$	2443.20	0.405712	0.202856	−	0.979209i	$-0.434978\pi$
$331$	2443.20	0.405712	0.202856	+	0.979209i	$0.434978\pi$
$332$	−43.6933	−0.00722283
$333$	0	0
$334$	−11056.6	−1.81134
$335$	−2527.53	−0.412220
$336$	0	0
$337$	1890.96	0.305660	0.152830	−	0.988253i	$-0.451161\pi$
$337$	1890.96	0.305660	0.152830	+	0.988253i	$0.451161\pi$
$338$	0	0
$339$	0	0
$340$	10.0647	0.00160540
$341$	−9073.64	−1.44095
$342$	0	0
$343$	−6777.95	−1.06698
$344$	−12551.0	−1.96716
$345$	0	0
$346$	5575.15	0.866248
$347$	2791.57	0.431872	0.215936	−	0.976408i	$-0.430720\pi$
$347$	2791.57	0.431872	0.215936	+	0.976408i	$0.430720\pi$
$348$	0	0
$349$	−6917.33	−1.06096	−0.530482	−	0.847696i	$-0.677989\pi$
$349$	−6917.33	−1.06096	−0.530482	+	0.847696i	$0.677989\pi$
$350$	−4267.43	−0.651724
$351$	0	0
$352$	−67.9690	−0.0102919
$353$	−7638.37	−1.15170	−0.575849	−	0.817556i	$-0.695328\pi$
$353$	−7638.37	−1.15170	−0.575849	+	0.817556i	$0.695328\pi$
$354$	0	0
$355$	−1952.99	−0.291982
$356$	52.0933	0.00775545
$357$	0	0
$358$	7854.42	1.15955
$359$	5490.97	0.807249	0.403625	−	0.914925i	$-0.367750\pi$
$359$	5490.97	0.807249	0.403625	+	0.914925i	$0.367750\pi$
$360$	0	0
$361$	−6705.21	−0.977578
$362$	−5321.77	−0.772669
$363$	0	0
$364$	0	0
$365$	91.1234	0.0130674
$366$	0	0
$367$	6737.43	0.958287	0.479143	−	0.877737i	$-0.340947\pi$
$367$	6737.43	0.958287	0.479143	+	0.877737i	$0.340947\pi$
$368$	−8037.07	−1.13848
$369$	0	0
$370$	1993.99	0.280169
$371$	1962.01	0.274561
$372$	0	0
$373$	−3930.42	−0.545602	−0.272801	−	0.962070i	$-0.587950\pi$
$373$	−3930.42	−0.545602	−0.272801	+	0.962070i	$0.587950\pi$
$374$	6965.97	0.963106
$375$	0	0
$376$	1427.43	0.195782
$377$	0	0
$378$	0	0
$379$	11897.7	1.61252	0.806258	−	0.591563i	$-0.201489\pi$
$379$	11897.7	1.61252	0.806258	+	0.591563i	$0.201489\pi$
$380$	−1.79199	−0.000241913 0
$381$	0	0
$382$	−6905.46	−0.924906
$383$	−4748.31	−0.633492	−0.316746	−	0.948510i	$-0.602590\pi$
$383$	−4748.31	−0.633492	−0.316746	+	0.948510i	$0.602590\pi$
$384$	0	0
$385$	−1613.66	−0.213609
$386$	−5432.14	−0.716292
$387$	0	0
$388$	−76.0485	−0.00995046
$389$	−11299.5	−1.47277	−0.736386	−	0.676561i	$-0.763469\pi$
$389$	−11299.5	−1.47277	−0.736386	+	0.676561i	$0.763469\pi$
$390$	0	0
$391$	−8793.73	−1.13739
$392$	3741.82	0.482119
$393$	0	0
$394$	5133.95	0.656458
$395$	−708.776	−0.0902845
$396$	0	0
$397$	13853.0	1.75129	0.875646	−	0.482953i	$-0.160436\pi$
$397$	13853.0	1.75129	0.875646	+	0.482953i	$0.160436\pi$
$398$	−3593.78	−0.452613
$399$	0	0
$400$	7216.79	0.902099
$401$	−5434.64	−0.676790	−0.338395	−	0.941004i	$-0.609884\pi$
$401$	−5434.64	−0.676790	−0.338395	+	0.941004i	$0.609884\pi$
$402$	0	0
$403$	0	0
$404$	−32.6076	−0.00401556
$405$	0	0
$406$	6740.34	0.823935
$407$	−7347.09	−0.894795
$408$	0	0
$409$	−3472.43	−0.419806	−0.209903	−	0.977722i	$-0.567315\pi$
$409$	−3472.43	−0.419806	−0.209903	+	0.977722i	$0.567315\pi$
$410$	1132.50	0.136416
$411$	0	0
$412$	−65.3959	−0.00781996
$413$	−3657.46	−0.435767
$414$	0	0
$415$	3517.90	0.416113
$416$	0	0
$417$	0	0
$418$	−1240.26	−0.145127
$419$	4261.65	0.496886	0.248443	−	0.968647i	$-0.420081\pi$
$419$	4261.65	0.496886	0.248443	+	0.968647i	$0.420081\pi$
$420$	0	0
$421$	6235.06	0.721801	0.360900	−	0.932604i	$-0.382469\pi$
$421$	6235.06	0.721801	0.360900	+	0.932604i	$0.382469\pi$
$422$	−3722.04	−0.429351
$423$	0	0
$424$	−3335.68	−0.382064
$425$	7896.22	0.901231
$426$	0	0
$427$	8047.06	0.912001
$428$	85.8992	0.00970115
$429$	0	0
$430$	5323.05	0.596977
$431$	−2520.58	−0.281698	−0.140849	−	0.990031i	$-0.544983\pi$
$431$	−2520.58	−0.281698	−0.140849	+	0.990031i	$0.544983\pi$
$432$	0	0
$433$	−1283.89	−0.142494	−0.0712468	−	0.997459i	$-0.522698\pi$
$433$	−1283.89	−0.142494	−0.0712468	+	0.997459i	$0.522698\pi$
$434$	9634.15	1.06556
$435$	0	0
$436$	−69.6878	−0.00765468
$437$	1565.69	0.171389
$438$	0	0
$439$	−16942.0	−1.84190	−0.920951	−	0.389677i	$-0.872586\pi$
$439$	−16942.0	−1.84190	−0.920951	+	0.389677i	$0.872586\pi$
$440$	2743.44	0.297246
$441$	0	0
$442$	0	0
$443$	2163.00	0.231980	0.115990	−	0.993250i	$-0.462996\pi$
$443$	2163.00	0.231980	0.115990	+	0.993250i	$0.462996\pi$
$444$	0	0
$445$	−4194.22	−0.446798
$446$	3394.83	0.360426
$447$	0	0
$448$	6868.01	0.724293
$449$	1673.35	0.175880	0.0879400	−	0.996126i	$-0.471972\pi$
$449$	1673.35	0.175880	0.0879400	+	0.996126i	$0.471972\pi$
$450$	0	0
$451$	−4172.85	−0.435680
$452$	−27.7279	−0.00288542
$453$	0	0
$454$	12302.7	1.27180
$455$	0	0
$456$	0	0
$457$	−4652.38	−0.476212	−0.238106	−	0.971239i	$-0.576527\pi$
$457$	−4652.38	−0.476212	−0.238106	+	0.971239i	$0.576527\pi$
$458$	15188.7	1.54961
$459$	0	0
$460$	−18.2432	−0.00184912
$461$	−3460.07	−0.349570	−0.174785	−	0.984607i	$-0.555923\pi$
$461$	−3460.07	−0.349570	−0.174785	+	0.984607i	$0.555923\pi$
$462$	0	0
$463$	−2105.64	−0.211355	−0.105677	−	0.994400i	$-0.533701\pi$
$463$	−2105.64	−0.211355	−0.105677	+	0.994400i	$0.533701\pi$
$464$	−11398.8	−1.14047
$465$	0	0
$466$	13860.9	1.37789
$467$	1415.88	0.140298	0.0701489	−	0.997537i	$-0.477653\pi$
$467$	1415.88	0.140298	0.0701489	+	0.997537i	$0.477653\pi$
$468$	0	0
$469$	−9888.28	−0.973557
$470$	−605.393	−0.0594143
$471$	0	0
$472$	6218.19	0.606388
$473$	−19613.4	−1.90661
$474$	0	0
$475$	−1405.89	−0.135803
$476$	39.3756	0.00379155
$477$	0	0
$478$	−2718.58	−0.260136
$479$	985.594	0.0940145	0.0470073	−	0.998895i	$-0.485032\pi$
$479$	985.594	0.0940145	0.0470073	+	0.998895i	$0.485032\pi$
$480$	0	0
$481$	0	0
$482$	4356.88	0.411723
$483$	0	0
$484$	3.13874	0.000294773 0
$485$	6122.93	0.573254
$486$	0	0
$487$	14724.7	1.37011	0.685053	−	0.728494i	$-0.259779\pi$
$487$	14724.7	1.37011	0.685053	+	0.728494i	$0.259779\pi$
$488$	−13681.1	−1.26909
$489$	0	0
$490$	−1586.96	−0.146310
$491$	16301.2	1.49830	0.749149	−	0.662401i	$-0.230463\pi$
$491$	16301.2	1.49830	0.749149	+	0.662401i	$0.230463\pi$
$492$	0	0
$493$	−12472.0	−1.13937
$494$	0	0
$495$	0	0
$496$	−16292.6	−1.47492
$497$	−7640.53	−0.689587
$498$	0	0
$499$	3230.42	0.289806	0.144903	−	0.989446i	$-0.453713\pi$
$499$	3230.42	0.289806	0.144903	+	0.989446i	$0.453713\pi$
$500$	34.4437	0.00308074
$501$	0	0
$502$	−10629.7	−0.945076
$503$	3577.57	0.317129	0.158565	−	0.987349i	$-0.449313\pi$
$503$	3577.57	0.317129	0.158565	+	0.987349i	$0.449313\pi$
$504$	0	0
$505$	2625.35	0.231340
$506$	−12626.4	−1.10931
$507$	0	0
$508$	−56.1552	−0.00490450
$509$	13864.7	1.20735	0.603674	−	0.797231i	$-0.293703\pi$
$509$	13864.7	1.20735	0.603674	+	0.797231i	$0.293703\pi$
$510$	0	0
$511$	356.496	0.0308619
$512$	−11675.9	−1.00783
$513$	0	0
$514$	−3618.34	−0.310502
$515$	5265.25	0.450514
$516$	0	0
$517$	2230.64	0.189756
$518$	7800.94	0.661686
$519$	0	0
$520$	0	0
$521$	9554.66	0.803449	0.401725	−	0.915760i	$-0.368411\pi$
$521$	9554.66	0.803449	0.401725	+	0.915760i	$0.368411\pi$
$522$	0	0
$523$	−19809.1	−1.65620	−0.828098	−	0.560583i	$-0.810577\pi$
$523$	−19809.1	−1.65620	−0.828098	+	0.560583i	$0.810577\pi$
$524$	39.4035	0.00328502
$525$	0	0
$526$	−14188.5	−1.17614
$527$	−17826.5	−1.47350
$528$	0	0
$529$	3772.38	0.310050
$530$	1414.71	0.115946
$531$	0	0
$532$	−7.01066	−0.000571336 0
$533$	0	0
$534$	0	0
$535$	−6916.05	−0.558891
$536$	16811.4	1.35475
$537$	0	0
$538$	−15877.5	−1.27235
$539$	5847.36	0.467279
$540$	0	0
$541$	−1011.14	−0.0803551	−0.0401775	−	0.999193i	$-0.512792\pi$
$541$	−1011.14	−0.0803551	−0.0401775	+	0.999193i	$0.512792\pi$
$542$	9501.98	0.753035
$543$	0	0
$544$	−133.535	−0.0105244
$545$	5610.81	0.440992
$546$	0	0
$547$	15745.2	1.23074	0.615372	−	0.788237i	$-0.289006\pi$
$547$	15745.2	1.23074	0.615372	+	0.788237i	$0.289006\pi$
$548$	−116.545	−0.00908495
$549$	0	0
$550$	11337.7	0.878987
$551$	2220.58	0.171688
$552$	0	0
$553$	−2772.89	−0.213229
$554$	−18357.9	−1.40785
$555$	0	0
$556$	−103.227	−0.00787371
$557$	−8510.94	−0.647433	−0.323716	−	0.946154i	$-0.604932\pi$
$557$	−8510.94	−0.647433	−0.323716	+	0.946154i	$0.604932\pi$
$558$	0	0
$559$	0	0
$560$	−2897.48	−0.218645
$561$	0	0
$562$	10227.3	0.767639
$563$	−16764.1	−1.25493	−0.627463	−	0.778646i	$-0.715907\pi$
$563$	−16764.1	−1.25493	−0.627463	+	0.778646i	$0.715907\pi$
$564$	0	0
$565$	2232.47	0.166232
$566$	−13076.1	−0.971080
$567$	0	0
$568$	12990.0	0.959589
$569$	−20755.4	−1.52919	−0.764597	−	0.644508i	$-0.777062\pi$
$569$	−20755.4	−1.52919	−0.764597	+	0.644508i	$0.777062\pi$
$570$	0	0
$571$	−23408.6	−1.71562	−0.857810	−	0.513968i	$-0.828175\pi$
$571$	−23408.6	−1.71562	−0.857810	+	0.513968i	$0.828175\pi$
$572$	0	0
$573$	0	0
$574$	4430.62	0.322179
$575$	−14312.6	−1.03805
$576$	0	0
$577$	10387.2	0.749436	0.374718	−	0.927139i	$-0.377739\pi$
$577$	10387.2	0.749436	0.374718	+	0.927139i	$0.377739\pi$
$578$	−173.537	−0.0124882
$579$	0	0
$580$	−25.8740	−0.00185234
$581$	13762.8	0.982752
$582$	0	0
$583$	−5212.68	−0.370304
$584$	−606.092	−0.0429457
$585$	0	0
$586$	8200.63	0.578097
$587$	2898.42	0.203800	0.101900	−	0.994795i	$-0.467508\pi$
$587$	2898.42	0.203800	0.101900	+	0.994795i	$0.467508\pi$
$588$	0	0
$589$	3173.94	0.222037
$590$	−2637.23	−0.184022
$591$	0	0
$592$	−13192.4	−0.915888
$593$	−8805.33	−0.609766	−0.304883	−	0.952390i	$-0.598617\pi$
$593$	−8805.33	−0.609766	−0.304883	+	0.952390i	$0.598617\pi$
$594$	0	0
$595$	−3170.27	−0.218434
$596$	−64.2910	−0.00441856
$597$	0	0
$598$	0	0
$599$	20236.8	1.38039	0.690194	−	0.723624i	$-0.257525\pi$
$599$	20236.8	1.38039	0.690194	+	0.723624i	$0.257525\pi$
$600$	0	0
$601$	−9884.92	−0.670906	−0.335453	−	0.942057i	$-0.608889\pi$
$601$	−9884.92	−0.670906	−0.335453	+	0.942057i	$0.608889\pi$
$602$	20825.0	1.40991
$603$	0	0
$604$	−24.7255	−0.00166567
$605$	−252.711	−0.0169821
$606$	0	0
$607$	22919.3	1.53256	0.766281	−	0.642505i	$-0.222105\pi$
$607$	22919.3	1.53256	0.766281	+	0.642505i	$0.222105\pi$
$608$	23.7754	0.00158589
$609$	0	0
$610$	5802.36	0.385132
$611$	0	0
$612$	0	0
$613$	916.052	0.0603572	0.0301786	−	0.999545i	$-0.490392\pi$
$613$	916.052	0.0603572	0.0301786	+	0.999545i	$0.490392\pi$
$614$	2634.61	0.173166
$615$	0	0
$616$	10733.0	0.702019
$617$	30179.4	1.96917	0.984584	−	0.174915i	$-0.0559650\pi$
$617$	30179.4	1.96917	0.984584	+	0.174915i	$0.0559650\pi$
$618$	0	0
$619$	5571.12	0.361748	0.180874	−	0.983506i	$-0.442107\pi$
$619$	5571.12	0.361748	0.180874	+	0.983506i	$0.442107\pi$
$620$	−36.9824	−0.00239556
$621$	0	0
$622$	10250.0	0.660751
$623$	−16408.8	−1.05522
$624$	0	0
$625$	11397.5	0.729443
$626$	−21177.5	−1.35211
$627$	0	0
$628$	−1.10542	−7.02403e−5 0
$629$	−14434.5	−0.915007
$630$	0	0
$631$	−11360.1	−0.716703	−0.358352	−	0.933587i	$-0.616661\pi$
$631$	−11360.1	−0.716703	−0.358352	+	0.933587i	$0.616661\pi$
$632$	4714.30	0.296717
$633$	0	0
$634$	−3426.31	−0.214631
$635$	4521.26	0.282552
$636$	0	0
$637$	0	0
$638$	−17907.8	−1.11125
$639$	0	0
$640$	4899.89	0.302633
$641$	24709.8	1.52259	0.761293	−	0.648408i	$-0.224565\pi$
$641$	24709.8	1.52259	0.761293	+	0.648408i	$0.224565\pi$
$642$	0	0
$643$	15226.6	0.933869	0.466934	−	0.884292i	$-0.345358\pi$
$643$	15226.6	0.933869	0.466934	+	0.884292i	$0.345358\pi$
$644$	−71.3717	−0.00436714
$645$	0	0
$646$	−2436.68	−0.148405
$647$	−16706.6	−1.01515	−0.507576	−	0.861607i	$-0.669458\pi$
$647$	−16706.6	−1.01515	−0.507576	+	0.861607i	$0.669458\pi$
$648$	0	0
$649$	9717.18	0.587724
$650$	0	0
$651$	0	0
$652$	−0.724024	−4.34892e−5 0
$653$	28205.8	1.69032	0.845158	−	0.534516i	$-0.179506\pi$
$653$	28205.8	1.69032	0.845158	+	0.534516i	$0.179506\pi$
$654$	0	0
$655$	−3172.51	−0.189252
$656$	−7492.77	−0.445951
$657$	0	0
$658$	−2368.44	−0.140321
$659$	11424.2	0.675299	0.337649	−	0.941272i	$-0.390368\pi$
$659$	11424.2	0.675299	0.337649	+	0.941272i	$0.390368\pi$
$660$	0	0
$661$	−26518.2	−1.56042	−0.780211	−	0.625516i	$-0.784888\pi$
$661$	−26518.2	−1.56042	−0.780211	+	0.625516i	$0.784888\pi$
$662$	6892.10	0.404636
$663$	0	0
$664$	−23398.7	−1.36754
$665$	564.453	0.0329151
$666$	0	0
$667$	22606.5	1.31234
$668$	166.045	0.00961749
$669$	0	0
$670$	−7129.98	−0.411127
$671$	−21379.5	−1.23002
$672$	0	0
$673$	22816.8	1.30687	0.653434	−	0.756984i	$-0.273328\pi$
$673$	22816.8	1.30687	0.653434	+	0.756984i	$0.273328\pi$
$674$	5334.27	0.304849
$675$	0	0
$676$	0	0
$677$	−17737.6	−1.00696	−0.503478	−	0.864008i	$-0.667947\pi$
$677$	−17737.6	−1.00696	−0.503478	+	0.864008i	$0.667947\pi$
$678$	0	0
$679$	23954.3	1.35388
$680$	5389.89	0.303960
$681$	0	0
$682$	−25596.1	−1.43713
$683$	18878.4	1.05763	0.528816	−	0.848737i	$-0.322636\pi$
$683$	18878.4	1.05763	0.528816	+	0.848737i	$0.322636\pi$
$684$	0	0
$685$	9383.45	0.523391
$686$	−19120.1	−1.06415
$687$	0	0
$688$	−35217.8	−1.95155
$689$	0	0
$690$	0	0
$691$	11692.8	0.643726	0.321863	−	0.946786i	$-0.395691\pi$
$691$	11692.8	0.643726	0.321863	+	0.946786i	$0.395691\pi$
$692$	−83.7265	−0.00459943
$693$	0	0
$694$	7874.82	0.430726
$695$	8311.14	0.453611
$696$	0	0
$697$	−8198.19	−0.445522
$698$	−19513.3	−1.05815
$699$	0	0
$700$	64.0873	0.00346039
$701$	−20658.1	−1.11304	−0.556522	−	0.830833i	$-0.687865\pi$
$701$	−20658.1	−1.11304	−0.556522	+	0.830833i	$0.687865\pi$
$702$	0	0
$703$	2569.99	0.137879
$704$	−18247.0	−0.976861
$705$	0	0
$706$	−21547.3	−1.14864
$707$	10271.0	0.546365
$708$	0	0
$709$	−20695.8	−1.09626	−0.548128	−	0.836394i	$-0.684659\pi$
$709$	−20695.8	−1.09626	−0.548128	+	0.836394i	$0.684659\pi$
$710$	−5509.23	−0.291208
$711$	0	0
$712$	27897.1	1.46838
$713$	32312.1	1.69719
$714$	0	0
$715$	0	0
$716$	−117.956	−0.00615674
$717$	0	0
$718$	15489.6	0.805109
$719$	26299.2	1.36411	0.682053	−	0.731302i	$-0.261087\pi$
$719$	26299.2	1.36411	0.682053	+	0.731302i	$0.261087\pi$
$720$	0	0
$721$	20598.9	1.06400
$722$	−18914.9	−0.974986
$723$	0	0
$724$	79.9213	0.00410256
$725$	−20299.3	−1.03986
$726$	0	0
$727$	−16915.6	−0.862952	−0.431476	−	0.902125i	$-0.642007\pi$
$727$	−16915.6	−0.862952	−0.431476	+	0.902125i	$0.642007\pi$
$728$	0	0
$729$	0	0
$730$	257.053	0.0130328
$731$	−38533.5	−1.94967
$732$	0	0
$733$	−13203.1	−0.665302	−0.332651	−	0.943050i	$-0.607943\pi$
$733$	−13203.1	−0.665302	−0.332651	+	0.943050i	$0.607943\pi$
$734$	19005.8	0.955746
$735$	0	0
$736$	242.044	0.0121221
$737$	26271.3	1.31305
$738$	0	0
$739$	−16813.9	−0.836954	−0.418477	−	0.908227i	$-0.637436\pi$
$739$	−16813.9	−0.836954	−0.418477	+	0.908227i	$0.637436\pi$
$740$	−29.9453	−0.00148758
$741$	0	0
$742$	5534.68	0.273834
$743$	32682.1	1.61371	0.806857	−	0.590747i	$-0.201167\pi$
$743$	32682.1	1.61371	0.806857	+	0.590747i	$0.201167\pi$
$744$	0	0
$745$	5176.30	0.254557
$746$	−11087.4	−0.544156
$747$	0	0
$748$	−104.613	−0.00511370
$749$	−27057.2	−1.31996
$750$	0	0
$751$	2157.76	0.104844	0.0524219	−	0.998625i	$-0.483306\pi$
$751$	2157.76	0.104844	0.0524219	+	0.998625i	$0.483306\pi$
$752$	4005.35	0.194229
$753$	0	0
$754$	0	0
$755$	1990.74	0.0959606
$756$	0	0
$757$	13769.6	0.661117	0.330558	−	0.943786i	$-0.392763\pi$
$757$	13769.6	0.661117	0.330558	+	0.943786i	$0.392763\pi$
$758$	33562.6	1.60824
$759$	0	0
$760$	−959.648	−0.0458028
$761$	−16905.2	−0.805272	−0.402636	−	0.915360i	$-0.631906\pi$
$761$	−16905.2	−0.805272	−0.402636	+	0.915360i	$0.631906\pi$
$762$	0	0
$763$	21950.8	1.04151
$764$	103.705	0.00491087
$765$	0	0
$766$	−13394.7	−0.631813
$767$	0	0
$768$	0	0
$769$	−1153.35	−0.0540845	−0.0270422	−	0.999634i	$-0.508609\pi$
$769$	−1153.35	−0.0540845	−0.0270422	+	0.999634i	$0.508609\pi$
$770$	−4552.01	−0.213043
$771$	0	0
$772$	81.5787	0.00380322
$773$	−20713.9	−0.963811	−0.481905	−	0.876223i	$-0.660055\pi$
$773$	−20713.9	−0.963811	−0.481905	+	0.876223i	$0.660055\pi$
$774$	0	0
$775$	−29014.3	−1.34480
$776$	−40725.7	−1.88398
$777$	0	0
$778$	−31875.1	−1.46887
$779$	1459.65	0.0671341
$780$	0	0
$781$	20299.4	0.930053
$782$	−24806.5	−1.13437
$783$	0	0
$784$	10499.5	0.478294
$785$	89.0010	0.00404660
$786$	0	0
$787$	−10870.2	−0.492353	−0.246177	−	0.969225i	$-0.579174\pi$
$787$	−10870.2	−0.492353	−0.246177	+	0.969225i	$0.579174\pi$
$788$	−77.1006	−0.00348553
$789$	0	0
$790$	−1999.41	−0.0900451
$791$	8733.95	0.392596
$792$	0	0
$793$	0	0
$794$	39078.4	1.74665
$795$	0	0
$796$	53.9706	0.00240319
$797$	−12026.6	−0.534511	−0.267255	−	0.963626i	$-0.586117\pi$
$797$	−12026.6	−0.534511	−0.267255	+	0.963626i	$0.586117\pi$
$798$	0	0
$799$	4382.43	0.194042
$800$	−217.341	−0.00960519
$801$	0	0
$802$	−15330.7	−0.674996
$803$	−947.142	−0.0416238
$804$	0	0
$805$	5746.39	0.251594
$806$	0	0
$807$	0	0
$808$	−17462.1	−0.760289
$809$	32384.0	1.40737	0.703685	−	0.710512i	$-0.251537\pi$
$809$	32384.0	1.40737	0.703685	+	0.710512i	$0.251537\pi$
$810$	0	0
$811$	−42506.8	−1.84046	−0.920231	−	0.391375i	$-0.872000\pi$
$811$	−42506.8	−1.84046	−0.920231	+	0.391375i	$0.872000\pi$
$812$	−101.225	−0.00437476
$813$	0	0
$814$	−20725.6	−0.892423
$815$	58.2937	0.00250545
$816$	0	0
$817$	6860.72	0.293790
$818$	−9795.47	−0.418693
$819$	0	0
$820$	−17.0077	−0.000724312 0
$821$	−20052.6	−0.852427	−0.426213	−	0.904623i	$-0.640153\pi$
$821$	−20052.6	−0.852427	−0.426213	+	0.904623i	$0.640153\pi$
$822$	0	0
$823$	−31625.5	−1.33948	−0.669742	−	0.742594i	$-0.733595\pi$
$823$	−31625.5	−1.33948	−0.669742	+	0.742594i	$0.733595\pi$
$824$	−35020.9	−1.48060
$825$	0	0
$826$	−10317.4	−0.434612
$827$	5440.97	0.228780	0.114390	−	0.993436i	$-0.463509\pi$
$827$	5440.97	0.228780	0.114390	+	0.993436i	$0.463509\pi$
$828$	0	0
$829$	−24845.8	−1.04093	−0.520465	−	0.853883i	$-0.674241\pi$
$829$	−24845.8	−1.04093	−0.520465	+	0.853883i	$0.674241\pi$
$830$	9923.75	0.415010
$831$	0	0
$832$	0	0
$833$	11488.0	0.477834
$834$	0	0
$835$	−13368.9	−0.554071
$836$	18.6260	0.000770566 0
$837$	0	0
$838$	12021.8	0.495568
$839$	45887.8	1.88823	0.944113	−	0.329621i	$-0.106921\pi$
$839$	45887.8	1.88823	0.944113	+	0.329621i	$0.106921\pi$
$840$	0	0
$841$	7673.40	0.314625
$842$	17588.7	0.719887
$843$	0	0
$844$	55.8969	0.00227968
$845$	0	0
$846$	0	0
$847$	−988.665	−0.0401073
$848$	−9359.88	−0.379033
$849$	0	0
$850$	22274.7	0.898842
$851$	26163.7	1.05391
$852$	0	0
$853$	33655.3	1.35092	0.675460	−	0.737397i	$-0.263945\pi$
$853$	33655.3	1.35092	0.675460	+	0.737397i	$0.263945\pi$
$854$	22700.2	0.909583
$855$	0	0
$856$	46000.9	1.83677
$857$	−41365.6	−1.64880	−0.824399	−	0.566009i	$-0.808487\pi$
$857$	−41365.6	−1.64880	−0.824399	+	0.566009i	$0.808487\pi$
$858$	0	0
$859$	14866.0	0.590480	0.295240	−	0.955423i	$-0.404600\pi$
$859$	14866.0	0.590480	0.295240	+	0.955423i	$0.404600\pi$
$860$	−79.9404	−0.00316970
$861$	0	0
$862$	−7110.37	−0.280952
$863$	15469.1	0.610167	0.305083	−	0.952326i	$-0.401316\pi$
$863$	15469.1	0.610167	0.305083	+	0.952326i	$0.401316\pi$
$864$	0	0
$865$	6741.12	0.264977
$866$	−3621.76	−0.142116
$867$	0	0
$868$	−144.684	−0.00565770
$869$	7367.05	0.287584
$870$	0	0
$871$	0	0
$872$	−37319.4	−1.44930
$873$	0	0
$874$	4416.69	0.170934
$875$	−10849.3	−0.419171
$876$	0	0
$877$	−38100.7	−1.46701	−0.733505	−	0.679684i	$-0.762117\pi$
$877$	−38100.7	−1.46701	−0.733505	+	0.679684i	$0.762117\pi$
$878$	−47792.0	−1.83702
$879$	0	0
$880$	7698.06	0.294888
$881$	−3879.84	−0.148371	−0.0741857	−	0.997244i	$-0.523636\pi$
$881$	−3879.84	−0.148371	−0.0741857	+	0.997244i	$0.523636\pi$
$882$	0	0
$883$	13046.5	0.497225	0.248613	−	0.968603i	$-0.420025\pi$
$883$	13046.5	0.497225	0.248613	+	0.968603i	$0.420025\pi$
$884$	0	0
$885$	0	0
$886$	6101.66	0.231365
$887$	32833.2	1.24288	0.621438	−	0.783463i	$-0.286549\pi$
$887$	32833.2	1.24288	0.621438	+	0.783463i	$0.286549\pi$
$888$	0	0
$889$	17688.2	0.667315
$890$	−11831.6	−0.445613
$891$	0	0
$892$	−50.9829	−0.00191372
$893$	−780.274	−0.0292395
$894$	0	0
$895$	9497.06	0.354695
$896$	19169.5	0.714741
$897$	0	0
$898$	4720.39	0.175414
$899$	45827.6	1.70015
$900$	0	0
$901$	−10241.1	−0.378668
$902$	−11771.3	−0.434525
$903$	0	0
$904$	−14848.9	−0.546314
$905$	−6434.74	−0.236352
$906$	0	0
$907$	21346.3	0.781468	0.390734	−	0.920504i	$-0.372221\pi$
$907$	21346.3	0.781468	0.390734	+	0.920504i	$0.372221\pi$
$908$	−184.760	−0.00675272
$909$	0	0
$910$	0	0
$911$	−7792.71	−0.283407	−0.141704	−	0.989909i	$-0.545258\pi$
$911$	−7792.71	−0.283407	−0.141704	+	0.989909i	$0.545258\pi$
$912$	0	0
$913$	−36565.2	−1.32545
$914$	−13124.0	−0.474950
$915$	0	0
$916$	−228.101	−0.00822779
$917$	−12411.6	−0.446965
$918$	0	0
$919$	−31352.0	−1.12536	−0.562681	−	0.826674i	$-0.690230\pi$
$919$	−31352.0	−1.12536	−0.562681	+	0.826674i	$0.690230\pi$
$920$	−9769.64	−0.350104
$921$	0	0
$922$	−9760.61	−0.348643
$923$	0	0
$924$	0	0
$925$	−23493.4	−0.835089
$926$	−5939.85	−0.210794
$927$	0	0
$928$	343.286	0.0121432
$929$	−12675.3	−0.447646	−0.223823	−	0.974630i	$-0.571854\pi$
$929$	−12675.3	−0.447646	−0.223823	+	0.974630i	$0.571854\pi$
$930$	0	0
$931$	−2045.39	−0.0720032
$932$	−208.161	−0.00731601
$933$	0	0
$934$	3994.09	0.139926
$935$	8422.80	0.294604
$936$	0	0
$937$	−12308.5	−0.429138	−0.214569	−	0.976709i	$-0.568835\pi$
$937$	−12308.5	−0.429138	−0.214569	+	0.976709i	$0.568835\pi$
$938$	−27894.1	−0.970976
$939$	0	0
$940$	9.09167	0.000315466 0
$941$	−20465.2	−0.708978	−0.354489	−	0.935060i	$-0.615345\pi$
$941$	−20465.2	−0.708978	−0.354489	+	0.935060i	$0.615345\pi$
$942$	0	0
$943$	14859.9	0.513155
$944$	17448.2	0.601578
$945$	0	0
$946$	−55328.0	−1.90155
$947$	−11170.8	−0.383318	−0.191659	−	0.981462i	$-0.561387\pi$
$947$	−11170.8	−0.383318	−0.191659	+	0.981462i	$0.561387\pi$
$948$	0	0
$949$	0	0
$950$	−3965.91	−0.135443
$951$	0	0
$952$	21086.5	0.717876
$953$	−14109.4	−0.479587	−0.239794	−	0.970824i	$-0.577080\pi$
$953$	−14109.4	−0.479587	−0.239794	+	0.970824i	$0.577080\pi$
$954$	0	0
$955$	−8349.64	−0.282919
$956$	40.8271	0.00138122
$957$	0	0
$958$	2780.29	0.0937653
$959$	36710.2	1.23612
$960$	0	0
$961$	35711.6	1.19874
$962$	0	0
$963$	0	0
$964$	−65.4307	−0.00218608
$965$	−6568.19	−0.219106
$966$	0	0
$967$	−40785.8	−1.35634	−0.678171	−	0.734904i	$-0.737227\pi$
$967$	−40785.8	−1.35634	−0.678171	+	0.734904i	$0.737227\pi$
$968$	1680.87	0.0558110
$969$	0	0
$970$	17272.4	0.571734
$971$	−4230.29	−0.139811	−0.0699055	−	0.997554i	$-0.522270\pi$
$971$	−4230.29	−0.139811	−0.0699055	+	0.997554i	$0.522270\pi$
$972$	0	0
$973$	32515.1	1.07131
$974$	41537.4	1.36647
$975$	0	0
$976$	−38389.0	−1.25902
$977$	59925.0	1.96230	0.981151	−	0.193242i	$-0.0619004\pi$
$977$	59925.0	1.96230	0.981151	+	0.193242i	$0.0619004\pi$
$978$	0	0
$979$	43594.9	1.42319
$980$	23.8327	0.000776844 0
$981$	0	0
$982$	45984.6	1.49433
$983$	23333.3	0.757087	0.378543	−	0.925584i	$-0.376425\pi$
$983$	23333.3	0.757087	0.378543	+	0.925584i	$0.376425\pi$
$984$	0	0
$985$	6207.64	0.200804
$986$	−35182.5	−1.13635
$987$	0	0
$988$	0	0
$989$	69845.2	2.24565
$990$	0	0
$991$	−32523.5	−1.04253	−0.521263	−	0.853396i	$-0.674539\pi$
$991$	−32523.5	−1.04253	−0.521263	+	0.853396i	$0.674539\pi$
$992$	490.668	0.0157044
$993$	0	0
$994$	−21553.4	−0.687759
$995$	−4345.37	−0.138450
$996$	0	0
$997$	−7114.35	−0.225992	−0.112996	−	0.993595i	$-0.536045\pi$
$997$	−7114.35	−0.225992	−0.112996	+	0.993595i	$0.536045\pi$
$998$	9112.77	0.289038
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bf.1.7		9
3.2	odd	2		507.4.a.p.1.3	yes	9
13.12	even	2		1521.4.a.bi.1.3		9
39.5	even	4		507.4.b.k.337.13		18
39.8	even	4		507.4.b.k.337.6		18
39.38	odd	2		507.4.a.o.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.7	✓	9	39.38	odd	2
507.4.a.p.1.3	yes	9	3.2	odd	2
507.4.b.k.337.6		18	39.8	even	4
507.4.b.k.337.13		18	39.5	even	4
1521.4.a.bf.1.7		9	1.1	even	1	trivial
1521.4.a.bi.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)

\(f(q)\)	\(=\)	\(q+2.82093 q^{2} -0.0423641 q^{4} +3.41089 q^{5} +13.3442 q^{7} -22.6869 q^{8} +O(q^{10})\)	\(q+2.82093 q^{2} -0.0423641 q^{4} +3.41089 q^{5} +13.3442 q^{7} -22.6869 q^{8} +9.62187 q^{10} -35.4529 q^{11} +37.6430 q^{14} -63.6593 q^{16} -69.6526 q^{17} +12.4014 q^{19} -0.144499 q^{20} -100.010 q^{22} +126.251 q^{23} -113.366 q^{25} -0.565314 q^{28} +179.060 q^{29} +255.935 q^{31} +1.91716 q^{32} -196.485 q^{34} +45.5155 q^{35} +207.235 q^{37} +34.9833 q^{38} -77.3825 q^{40} +117.701 q^{41} +553.224 q^{43} +1.50193 q^{44} +356.146 q^{46} -62.9185 q^{47} -164.933 q^{49} -319.797 q^{50} +147.031 q^{53} -120.926 q^{55} -302.738 q^{56} +505.115 q^{58} -274.087 q^{59} +603.039 q^{61} +721.974 q^{62} +514.683 q^{64} -741.019 q^{67} +2.95077 q^{68} +128.396 q^{70} -572.574 q^{71} +26.7155 q^{73} +584.595 q^{74} -0.525372 q^{76} -473.090 q^{77} -207.798 q^{79} -217.135 q^{80} +332.026 q^{82} +1031.37 q^{83} -237.577 q^{85} +1560.60 q^{86} +804.318 q^{88} -1229.66 q^{89} -5.34852 q^{92} -177.488 q^{94} +42.2996 q^{95} +1795.12 q^{97} -465.264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) \) 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8	\( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29} + 717 q^{31} - 707 q^{32} + 632 q^{34} + 418 q^{35} + 1108 q^{37} + 660 q^{38} - 1506 q^{40} - 334 q^{41} + 242 q^{43} + 307 q^{44} + 979 q^{46} + 184 q^{47} - 38 q^{49} + 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} + 1161 q^{58} - 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} + 2308 q^{67} - 2785 q^{68} - 1420 q^{70} - 96 q^{71} + 2505 q^{73} + 1191 q^{74} + 2409 q^{76} + 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 1517 q^{82} - 1539 q^{83} + 4296 q^{85} + 3763 q^{86} - 3716 q^{88} + 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} + 1445 q^{97} - 1457 q^{98}+O(q^{100}) \) 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8 - 54 * q^10 - 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 + 352 * q^19 - 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 + 940 * q^28 + 97 * q^29 + 717 * q^31 - 707 * q^32 + 632 * q^34 + 418 * q^35 + 1108 * q^37 + 660 * q^38 - 1506 * q^40 - 334 * q^41 + 242 * q^43 + 307 * q^44 + 979 * q^46 + 184 * q^47 - 38 * q^49 + 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 + 1161 * q^58 - 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 + 2308 * q^67 - 2785 * q^68 - 1420 * q^70 - 96 * q^71 + 2505 * q^73 + 1191 * q^74 + 2409 * q^76 + 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 1517 * q^82 - 1539 * q^83 + 4296 * q^85 + 3763 * q^86 - 3716 * q^88 + 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 + 1445 * q^97 - 1457 * q^98

Introduction

L-functions

Modular forms

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