LMFDB - Embedded newform 1521.4.a.u.1.2

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:	$3$
Coefficient field:	3.3.3144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.526440$ 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)

$f(q)$	$=$	$q+1.52644 q^{2} -5.66998 q^{4} +19.3400 q^{5} -4.84136 q^{7} -20.8664 q^{8} +O(q^{10})$	$q+1.52644 q^{2} -5.66998 q^{4} +19.3400 q^{5} -4.84136 q^{7} -20.8664 q^{8} +29.5213 q^{10} -61.0728 q^{11} -7.39005 q^{14} +13.5085 q^{16} +41.7885 q^{17} +107.561 q^{19} -109.657 q^{20} -93.2239 q^{22} -28.5138 q^{23} +249.034 q^{25} +27.4504 q^{28} +89.8886 q^{29} -183.108 q^{31} +187.551 q^{32} +63.7876 q^{34} -93.6318 q^{35} -418.029 q^{37} +164.185 q^{38} -403.555 q^{40} -142.674 q^{41} -71.0935 q^{43} +346.281 q^{44} -43.5246 q^{46} +323.711 q^{47} -319.561 q^{49} +380.136 q^{50} +25.1047 q^{53} -1181.15 q^{55} +101.022 q^{56} +137.210 q^{58} -684.508 q^{59} +308.125 q^{61} -279.503 q^{62} +178.217 q^{64} -672.808 q^{67} -236.940 q^{68} -142.923 q^{70} -326.837 q^{71} -24.3058 q^{73} -638.095 q^{74} -609.869 q^{76} +295.675 q^{77} +166.810 q^{79} +261.255 q^{80} -217.783 q^{82} -201.093 q^{83} +808.188 q^{85} -108.520 q^{86} +1274.37 q^{88} +108.834 q^{89} +161.673 q^{92} +494.126 q^{94} +2080.23 q^{95} -1157.95 q^{97} -487.791 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 10 * q^4 + 4 * q^5 - 30 * q^7 - 6 * q^8	$ 3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8} - 4 q^{10} - 16 q^{11} + 176 q^{14} - 110 q^{16} + 146 q^{17} - 94 q^{19} - 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 80 q^{28} + 2 q^{29} - 302 q^{31} + 154 q^{32} - 164 q^{34} - 80 q^{35} - 374 q^{37} - 312 q^{38} - 516 q^{40} + 480 q^{41} - 260 q^{43} + 712 q^{44} + 1104 q^{46} - 24 q^{47} + 447 q^{49} + 814 q^{50} + 678 q^{53} - 1552 q^{55} - 96 q^{56} + 628 q^{58} - 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 750 q^{64} - 74 q^{67} + 460 q^{68} - 1216 q^{70} - 948 q^{71} + 222 q^{73} - 1724 q^{74} - 2392 q^{76} - 112 q^{77} - 24 q^{79} + 1100 q^{80} + 564 q^{82} - 796 q^{83} + 248 q^{85} + 1800 q^{86} + 1608 q^{88} + 1436 q^{89} + 1296 q^{92} - 1920 q^{94} + 4032 q^{95} - 3242 q^{97} - 5070 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 10 * q^4 + 4 * q^5 - 30 * q^7 - 6 * q^8 - 4 * q^10 - 16 * q^11 + 176 * q^14 - 110 * q^16 + 146 * q^17 - 94 * q^19 - 244 * q^20 - 56 * q^22 + 48 * q^23 + 145 * q^25 - 80 * q^28 + 2 * q^29 - 302 * q^31 + 154 * q^32 - 164 * q^34 - 80 * q^35 - 374 * q^37 - 312 * q^38 - 516 * q^40 + 480 * q^41 - 260 * q^43 + 712 * q^44 + 1104 * q^46 - 24 * q^47 + 447 * q^49 + 814 * q^50 + 678 * q^53 - 1552 * q^55 - 96 * q^56 + 628 * q^58 - 1788 * q^59 + 230 * q^61 - 1952 * q^62 - 750 * q^64 - 74 * q^67 + 460 * q^68 - 1216 * q^70 - 948 * q^71 + 222 * q^73 - 1724 * q^74 - 2392 * q^76 - 112 * q^77 - 24 * q^79 + 1100 * q^80 + 564 * q^82 - 796 * q^83 + 248 * q^85 + 1800 * q^86 + 1608 * q^88 + 1436 * q^89 + 1296 * q^92 - 1920 * q^94 + 4032 * q^95 - 3242 * q^97 - 5070 * 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 $

$2$	1.52644	0.539678	0.269839	−	0.962905i	$-0.413030\pi$
$2$	1.52644	0.539678	0.269839	+	0.962905i	$0.413030\pi$
$3$	0	0
$3$	0	0
$4$	−5.66998	−0.708748
$5$	19.3400	1.72982	0.864909	−	0.501928i	$-0.167376\pi$
$5$	19.3400	1.72982	0.864909	+	0.501928i	$0.167376\pi$
$6$	0	0
$7$	−4.84136	−0.261409	−0.130704	−	0.991421i	$-0.541724\pi$
$7$	−4.84136	−0.261409	−0.130704	+	0.991421i	$0.541724\pi$
$8$	−20.8664	−0.922173
$9$	0	0
$10$	29.5213	0.933545
$11$	−61.0728	−1.67401	−0.837006	−	0.547194i	$-0.815696\pi$
$11$	−61.0728	−1.67401	−0.837006	+	0.547194i	$0.815696\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−7.39005	−0.141077
$15$	0	0
$16$	13.5085	0.211071
$17$	41.7885	0.596188	0.298094	−	0.954537i	$-0.403649\pi$
$17$	41.7885	0.596188	0.298094	+	0.954537i	$0.403649\pi$
$18$	0	0
$19$	107.561	1.29875	0.649374	−	0.760469i	$-0.275031\pi$
$19$	107.561	1.29875	0.649374	+	0.760469i	$0.275031\pi$
$20$	−109.657	−1.22601
$21$	0	0
$22$	−93.2239	−0.903427
$23$	−28.5138	−0.258502	−0.129251	−	0.991612i	$-0.541257\pi$
$23$	−28.5138	−0.258502	−0.129251	+	0.991612i	$0.541257\pi$
$24$	0	0
$25$	249.034	1.99227
$26$	0	0
$27$	0	0
$28$	27.4504	0.185273
$29$	89.8886	0.575583	0.287791	−	0.957693i	$-0.407079\pi$
$29$	89.8886	0.575583	0.287791	+	0.957693i	$0.407079\pi$
$30$	0	0
$31$	−183.108	−1.06087	−0.530437	−	0.847724i	$-0.677972\pi$
$31$	−183.108	−1.06087	−0.530437	+	0.847724i	$0.677972\pi$
$32$	187.551	1.03608
$33$	0	0
$34$	63.7876	0.321750
$35$	−93.6318	−0.452190
$36$	0	0
$37$	−418.029	−1.85739	−0.928696	−	0.370843i	$-0.879069\pi$
$37$	−418.029	−1.85739	−0.928696	+	0.370843i	$0.879069\pi$
$38$	164.185	0.700905
$39$	0	0
$40$	−403.555	−1.59519
$41$	−142.674	−0.543460	−0.271730	−	0.962373i	$-0.587596\pi$
$41$	−142.674	−0.543460	−0.271730	+	0.962373i	$0.587596\pi$
$42$	0	0
$43$	−71.0935	−0.252132	−0.126066	−	0.992022i	$-0.540235\pi$
$43$	−71.0935	−0.252132	−0.126066	+	0.992022i	$0.540235\pi$
$44$	346.281	1.18645
$45$	0	0
$46$	−43.5246	−0.139508
$47$	323.711	1.00464	0.502321	−	0.864681i	$-0.332480\pi$
$47$	323.711	1.00464	0.502321	+	0.864681i	$0.332480\pi$
$48$	0	0
$49$	−319.561	−0.931665
$50$	380.136	1.07519
$51$	0	0
$52$	0	0
$53$	25.1047	0.0650641	0.0325321	−	0.999471i	$-0.489643\pi$
$53$	25.1047	0.0650641	0.0325321	+	0.999471i	$0.489643\pi$
$54$	0	0
$55$	−1181.15	−2.89574
$56$	101.022	0.241064
$57$	0	0
$58$	137.210	0.310629
$59$	−684.508	−1.51043	−0.755215	−	0.655477i	$-0.772467\pi$
$59$	−684.508	−1.51043	−0.755215	+	0.655477i	$0.772467\pi$
$60$	0	0
$61$	308.125	0.646744	0.323372	−	0.946272i	$-0.395184\pi$
$61$	308.125	0.646744	0.323372	+	0.946272i	$0.395184\pi$
$62$	−279.503	−0.572531
$63$	0	0
$64$	178.217	0.348081
$65$	0	0
$66$	0	0
$67$	−672.808	−1.22681	−0.613407	−	0.789767i	$-0.710202\pi$
$67$	−672.808	−1.22681	−0.613407	+	0.789767i	$0.710202\pi$
$68$	−236.940	−0.422547
$69$	0	0
$70$	−142.923	−0.244037
$71$	−326.837	−0.546315	−0.273158	−	0.961969i	$-0.588068\pi$
$71$	−326.837	−0.546315	−0.273158	+	0.961969i	$0.588068\pi$
$72$	0	0
$73$	−24.3058	−0.0389695	−0.0194847	−	0.999810i	$-0.506203\pi$
$73$	−24.3058	−0.0389695	−0.0194847	+	0.999810i	$0.506203\pi$
$74$	−638.095	−1.00239
$75$	0	0
$76$	−609.869	−0.920484
$77$	295.675	0.437602
$78$	0	0
$79$	166.810	0.237565	0.118783	−	0.992920i	$-0.462101\pi$
$79$	166.810	0.237565	0.118783	+	0.992920i	$0.462101\pi$
$80$	261.255	0.365115
$81$	0	0
$82$	−217.783	−0.293294
$83$	−201.093	−0.265938	−0.132969	−	0.991120i	$-0.542451\pi$
$83$	−201.093	−0.265938	−0.132969	+	0.991120i	$0.542451\pi$
$84$	0	0
$85$	808.188	1.03130
$86$	−108.520	−0.136070
$87$	0	0
$88$	1274.37	1.54373
$89$	108.834	0.129622	0.0648109	−	0.997898i	$-0.479356\pi$
$89$	108.834	0.129622	0.0648109	+	0.997898i	$0.479356\pi$
$90$	0	0
$91$	0	0
$92$	161.673	0.183212
$93$	0	0
$94$	494.126	0.542183
$95$	2080.23	2.24660
$96$	0	0
$97$	−1157.95	−1.21208	−0.606041	−	0.795434i	$-0.707243\pi$
$97$	−1157.95	−1.21208	−0.606041	+	0.795434i	$0.707243\pi$
$98$	−487.791	−0.502799
$99$	0	0
$100$	−1412.02	−1.41202
$101$	−1702.75	−1.67752	−0.838761	−	0.544500i	$-0.816719\pi$
$101$	−1702.75	−1.67752	−0.838761	+	0.544500i	$0.816719\pi$
$102$	0	0
$103$	−1455.14	−1.39203	−0.696015	−	0.718027i	$-0.745045\pi$
$103$	−1455.14	−1.39203	−0.696015	+	0.718027i	$0.745045\pi$
$104$	0	0
$105$	0	0
$106$	38.3209	0.0351137
$107$	822.762	0.743359	0.371679	−	0.928361i	$-0.378782\pi$
$107$	822.762	0.743359	0.371679	+	0.928361i	$0.378782\pi$
$108$	0	0
$109$	−457.264	−0.401816	−0.200908	−	0.979610i	$-0.564389\pi$
$109$	−457.264	−0.401816	−0.200908	+	0.979610i	$0.564389\pi$
$110$	−1802.95	−1.56277
$111$	0	0
$112$	−65.3998	−0.0551759
$113$	381.693	0.317758	0.158879	−	0.987298i	$-0.449212\pi$
$113$	381.693	0.317758	0.158879	+	0.987298i	$0.449212\pi$
$114$	0	0
$115$	−551.456	−0.447161
$116$	−509.667	−0.407943
$117$	0	0
$118$	−1044.86	−0.815146
$119$	−202.313	−0.155849
$120$	0	0
$121$	2398.88	1.80232
$122$	470.334	0.349033
$123$	0	0
$124$	1038.22	0.751892
$125$	2398.82	1.71645
$126$	0	0
$127$	−1129.09	−0.788905	−0.394452	−	0.918916i	$-0.629066\pi$
$127$	−1129.09	−0.788905	−0.394452	+	0.918916i	$0.629066\pi$
$128$	−1228.37	−0.848232
$129$	0	0
$130$	0	0
$131$	852.761	0.568749	0.284374	−	0.958713i	$-0.408214\pi$
$131$	852.761	0.568749	0.284374	+	0.958713i	$0.408214\pi$
$132$	0	0
$133$	−520.742	−0.339504
$134$	−1027.00	−0.662085
$135$	0	0
$136$	−871.975	−0.549789
$137$	−488.903	−0.304889	−0.152445	−	0.988312i	$-0.548715\pi$
$137$	−488.903	−0.304889	−0.152445	+	0.988312i	$0.548715\pi$
$138$	0	0
$139$	407.123	0.248430	0.124215	−	0.992255i	$-0.460359\pi$
$139$	407.123	0.248430	0.124215	+	0.992255i	$0.460359\pi$
$140$	530.890	0.320489
$141$	0	0
$142$	−498.897	−0.294834
$143$	0	0
$144$	0	0
$145$	1738.44	0.995654
$146$	−37.1013	−0.0210310
$147$	0	0
$148$	2370.21	1.31642
$149$	1717.63	0.944388	0.472194	−	0.881495i	$-0.343462\pi$
$149$	1717.63	0.944388	0.472194	+	0.881495i	$0.343462\pi$
$150$	0	0
$151$	−1341.79	−0.723133	−0.361567	−	0.932346i	$-0.617758\pi$
$151$	−1341.79	−0.723133	−0.361567	+	0.932346i	$0.617758\pi$
$152$	−2244.41	−1.19767
$153$	0	0
$154$	451.331	0.236164
$155$	−3541.30	−1.83512
$156$	0	0
$157$	−760.546	−0.386612	−0.193306	−	0.981138i	$-0.561921\pi$
$157$	−760.546	−0.386612	−0.193306	+	0.981138i	$0.561921\pi$
$158$	254.626	0.128209
$159$	0	0
$160$	3627.23	1.79224
$161$	138.046	0.0675746
$162$	0	0
$163$	−2712.09	−1.30323	−0.651616	−	0.758549i	$-0.725909\pi$
$163$	−2712.09	−1.30323	−0.651616	+	0.758549i	$0.725909\pi$
$164$	808.957	0.385176
$165$	0	0
$166$	−306.957	−0.143521
$167$	1551.69	0.719004	0.359502	−	0.933144i	$-0.382947\pi$
$167$	1551.69	0.719004	0.359502	+	0.933144i	$0.382947\pi$
$168$	0	0
$169$	0	0
$170$	1233.65	0.556568
$171$	0	0
$172$	403.099	0.178698
$173$	3970.26	1.74482	0.872409	−	0.488777i	$-0.162557\pi$
$173$	3970.26	1.74482	0.872409	+	0.488777i	$0.162557\pi$
$174$	0	0
$175$	−1205.66	−0.520798
$176$	−825.004	−0.353335
$177$	0	0
$178$	166.128	0.0699540
$179$	2690.95	1.12364	0.561818	−	0.827261i	$-0.310102\pi$
$179$	2690.95	1.12364	0.561818	+	0.827261i	$0.310102\pi$
$180$	0	0
$181$	−4371.10	−1.79503	−0.897517	−	0.440980i	$-0.854631\pi$
$181$	−4371.10	−1.79503	−0.897517	+	0.440980i	$0.854631\pi$
$182$	0	0
$183$	0	0
$184$	594.980	0.238383
$185$	−8084.66	−3.21295
$186$	0	0
$187$	−2552.14	−0.998026
$188$	−1835.44	−0.712038
$189$	0	0
$190$	3175.34	1.21244
$191$	−1408.47	−0.533578	−0.266789	−	0.963755i	$-0.585963\pi$
$191$	−1408.47	−0.533578	−0.266789	+	0.963755i	$0.585963\pi$
$192$	0	0
$193$	4131.69	1.54096	0.770481	−	0.637463i	$-0.220016\pi$
$193$	4131.69	1.54096	0.770481	+	0.637463i	$0.220016\pi$
$194$	−1767.54	−0.654134
$195$	0	0
$196$	1811.91	0.660316
$197$	−3401.23	−1.23009	−0.615045	−	0.788492i	$-0.710862\pi$
$197$	−3401.23	−1.23009	−0.615045	+	0.788492i	$0.710862\pi$
$198$	0	0
$199$	−3520.74	−1.25416	−0.627081	−	0.778954i	$-0.715751\pi$
$199$	−3520.74	−1.25416	−0.627081	+	0.778954i	$0.715751\pi$
$200$	−5196.45	−1.83722
$201$	0	0
$202$	−2599.14	−0.905321
$203$	−435.183	−0.150463
$204$	0	0
$205$	−2759.30	−0.940088
$206$	−2221.18	−0.751248
$207$	0	0
$208$	0	0
$209$	−6569.05	−2.17412
$210$	0	0
$211$	−2245.22	−0.732545	−0.366272	−	0.930508i	$-0.619366\pi$
$211$	−2245.22	−0.732545	−0.366272	+	0.930508i	$0.619366\pi$
$212$	−142.343	−0.0461141
$213$	0	0
$214$	1255.90	0.401174
$215$	−1374.95	−0.436142
$216$	0	0
$217$	886.490	0.277322
$218$	−697.986	−0.216851
$219$	0	0
$220$	6697.07	2.05235
$221$	0	0
$222$	0	0
$223$	−3431.26	−1.03038	−0.515188	−	0.857077i	$-0.672278\pi$
$223$	−3431.26	−1.03038	−0.515188	+	0.857077i	$0.672278\pi$
$224$	−908.003	−0.270842
$225$	0	0
$226$	582.631	0.171487
$227$	−4757.91	−1.39116	−0.695581	−	0.718448i	$-0.744853\pi$
$227$	−4757.91	−1.39116	−0.695581	+	0.718448i	$0.744853\pi$
$228$	0	0
$229$	4368.93	1.26073	0.630364	−	0.776300i	$-0.282906\pi$
$229$	4368.93	1.26073	0.630364	+	0.776300i	$0.282906\pi$
$230$	−841.764	−0.241323
$231$	0	0
$232$	−1875.65	−0.530787
$233$	3642.00	1.02401	0.512007	−	0.858981i	$-0.328902\pi$
$233$	3642.00	1.02401	0.512007	+	0.858981i	$0.328902\pi$
$234$	0	0
$235$	6260.57	1.73785
$236$	3881.15	1.07051
$237$	0	0
$238$	−308.819	−0.0841082
$239$	2236.17	0.605213	0.302606	−	0.953116i	$-0.402143\pi$
$239$	2236.17	0.605213	0.302606	+	0.953116i	$0.402143\pi$
$240$	0	0
$241$	−6538.78	−1.74772	−0.873858	−	0.486181i	$-0.838390\pi$
$241$	−6538.78	−1.74772	−0.873858	+	0.486181i	$0.838390\pi$
$242$	3661.75	0.972670
$243$	0	0
$244$	−1747.06	−0.458378
$245$	−6180.30	−1.61161
$246$	0	0
$247$	0	0
$248$	3820.80	0.978310
$249$	0	0
$250$	3661.65	0.926332
$251$	−2507.12	−0.630470	−0.315235	−	0.949014i	$-0.602083\pi$
$251$	−2507.12	−0.630470	−0.315235	+	0.949014i	$0.602083\pi$
$252$	0	0
$253$	1741.42	0.432735
$254$	−1723.49	−0.425755
$255$	0	0
$256$	−3300.77	−0.805853
$257$	808.131	0.196147	0.0980735	−	0.995179i	$-0.468732\pi$
$257$	808.131	0.196147	0.0980735	+	0.995179i	$0.468732\pi$
$258$	0	0
$259$	2023.83	0.485539
$260$	0	0
$261$	0	0
$262$	1301.69	0.306941
$263$	−2940.70	−0.689472	−0.344736	−	0.938700i	$-0.612032\pi$
$263$	−2940.70	−0.689472	−0.344736	+	0.938700i	$0.612032\pi$
$264$	0	0
$265$	485.525	0.112549
$266$	−794.881	−0.183223
$267$	0	0
$268$	3814.81	0.869502
$269$	−7111.50	−1.61188	−0.805940	−	0.591997i	$-0.798340\pi$
$269$	−7111.50	−1.61188	−0.805940	+	0.591997i	$0.798340\pi$
$270$	0	0
$271$	−2034.96	−0.456145	−0.228072	−	0.973644i	$-0.573242\pi$
$271$	−2034.96	−0.456145	−0.228072	+	0.973644i	$0.573242\pi$
$272$	564.502	0.125838
$273$	0	0
$274$	−746.281	−0.164542
$275$	−15209.2	−3.33509
$276$	0	0
$277$	2723.20	0.590689	0.295345	−	0.955391i	$-0.404566\pi$
$277$	2723.20	0.590689	0.295345	+	0.955391i	$0.404566\pi$
$278$	621.449	0.134072
$279$	0	0
$280$	1953.76	0.416998
$281$	3265.56	0.693263	0.346632	−	0.938001i	$-0.387325\pi$
$281$	3265.56	0.693263	0.346632	+	0.938001i	$0.387325\pi$
$282$	0	0
$283$	1144.02	0.240299	0.120150	−	0.992756i	$-0.461663\pi$
$283$	1144.02	0.240299	0.120150	+	0.992756i	$0.461663\pi$
$284$	1853.16	0.387200
$285$	0	0
$286$	0	0
$287$	690.735	0.142065
$288$	0	0
$289$	−3166.72	−0.644560
$290$	2653.63	0.537333
$291$	0	0
$292$	137.813	0.0276195
$293$	−1677.35	−0.334444	−0.167222	−	0.985919i	$-0.553480\pi$
$293$	−1677.35	−0.334444	−0.167222	+	0.985919i	$0.553480\pi$
$294$	0	0
$295$	−13238.4	−2.61277
$296$	8722.75	1.71284
$297$	0	0
$298$	2621.86	0.509666
$299$	0	0
$300$	0	0
$301$	344.190	0.0659095
$302$	−2048.16	−0.390259
$303$	0	0
$304$	1452.99	0.274128
$305$	5959.13	1.11875
$306$	0	0
$307$	−7207.70	−1.33995	−0.669975	−	0.742383i	$-0.733695\pi$
$307$	−7207.70	−1.33995	−0.669975	+	0.742383i	$0.733695\pi$
$308$	−1676.47	−0.310149
$309$	0	0
$310$	−5405.57	−0.990374
$311$	−412.963	−0.0752958	−0.0376479	−	0.999291i	$-0.511987\pi$
$311$	−412.963	−0.0752958	−0.0376479	+	0.999291i	$0.511987\pi$
$312$	0	0
$313$	2936.39	0.530270	0.265135	−	0.964211i	$-0.414583\pi$
$313$	2936.39	0.530270	0.265135	+	0.964211i	$0.414583\pi$
$314$	−1160.93	−0.208646
$315$	0	0
$316$	−945.812	−0.168374
$317$	377.956	0.0669657	0.0334828	−	0.999439i	$-0.489340\pi$
$317$	377.956	0.0669657	0.0334828	+	0.999439i	$0.489340\pi$
$318$	0	0
$319$	−5489.75	−0.963533
$320$	3446.71	0.602116
$321$	0	0
$322$	210.718	0.0364685
$323$	4494.81	0.774298
$324$	0	0
$325$	0	0
$326$	−4139.83	−0.703326
$327$	0	0
$328$	2977.09	0.501165
$329$	−1567.20	−0.262622
$330$	0	0
$331$	4428.17	0.735330	0.367665	−	0.929958i	$-0.380157\pi$
$331$	4428.17	0.735330	0.367665	+	0.929958i	$0.380157\pi$
$332$	1140.20	0.188483
$333$	0	0
$334$	2368.57	0.388031
$335$	−13012.1	−2.12217
$336$	0	0
$337$	−1768.76	−0.285907	−0.142953	−	0.989729i	$-0.545660\pi$
$337$	−1768.76	−0.285907	−0.142953	+	0.989729i	$0.545660\pi$
$338$	0	0
$339$	0	0
$340$	−4582.41	−0.730930
$341$	11182.9	1.77592
$342$	0	0
$343$	3207.70	0.504955
$344$	1483.47	0.232509
$345$	0	0
$346$	6060.36	0.941639
$347$	−2412.97	−0.373300	−0.186650	−	0.982426i	$-0.559763\pi$
$347$	−2412.97	−0.373300	−0.186650	+	0.982426i	$0.559763\pi$
$348$	0	0
$349$	9967.45	1.52878	0.764392	−	0.644752i	$-0.223039\pi$
$349$	9967.45	1.52878	0.764392	+	0.644752i	$0.223039\pi$
$350$	−1840.37	−0.281063
$351$	0	0
$352$	−11454.3	−1.73442
$353$	4516.30	0.680959	0.340479	−	0.940252i	$-0.389411\pi$
$353$	4516.30	0.680959	0.340479	+	0.940252i	$0.389411\pi$
$354$	0	0
$355$	−6321.01	−0.945027
$356$	−617.084	−0.0918691
$357$	0	0
$358$	4107.57	0.606401
$359$	12159.8	1.78767	0.893833	−	0.448400i	$-0.148006\pi$
$359$	12159.8	1.78767	0.893833	+	0.448400i	$0.148006\pi$
$360$	0	0
$361$	4710.38	0.686745
$362$	−6672.22	−0.968740
$363$	0	0
$364$	0	0
$365$	−470.072	−0.0674102
$366$	0	0
$367$	−2674.25	−0.380367	−0.190183	−	0.981749i	$-0.560908\pi$
$367$	−2674.25	−0.380367	−0.190183	+	0.981749i	$0.560908\pi$
$368$	−385.180	−0.0545622
$369$	0	0
$370$	−12340.7	−1.73396
$371$	−121.541	−0.0170083
$372$	0	0
$373$	9601.74	1.33287	0.666433	−	0.745564i	$-0.267820\pi$
$373$	9601.74	1.33287	0.666433	+	0.745564i	$0.267820\pi$
$374$	−3895.69	−0.538613
$375$	0	0
$376$	−6754.69	−0.926454
$377$	0	0
$378$	0	0
$379$	−9019.65	−1.22245	−0.611225	−	0.791457i	$-0.709323\pi$
$379$	−9019.65	−1.22245	−0.611225	+	0.791457i	$0.709323\pi$
$380$	−11794.9	−1.59227
$381$	0	0
$382$	−2149.95	−0.287960
$383$	−4015.34	−0.535703	−0.267852	−	0.963460i	$-0.586314\pi$
$383$	−4015.34	−0.535703	−0.267852	+	0.963460i	$0.586314\pi$
$384$	0	0
$385$	5718.35	0.756972
$386$	6306.78	0.831623
$387$	0	0
$388$	6565.55	0.859060
$389$	2725.35	0.355221	0.177610	−	0.984101i	$-0.443163\pi$
$389$	2725.35	0.355221	0.177610	+	0.984101i	$0.443163\pi$
$390$	0	0
$391$	−1191.55	−0.154115
$392$	6668.09	0.859157
$393$	0	0
$394$	−5191.78	−0.663853
$395$	3226.11	0.410945
$396$	0	0
$397$	4391.59	0.555182	0.277591	−	0.960699i	$-0.410464\pi$
$397$	4391.59	0.555182	0.277591	+	0.960699i	$0.410464\pi$
$398$	−5374.19	−0.676844
$399$	0	0
$400$	3364.09	0.420511
$401$	3762.48	0.468552	0.234276	−	0.972170i	$-0.424728\pi$
$401$	3762.48	0.468552	0.234276	+	0.972170i	$0.424728\pi$
$402$	0	0
$403$	0	0
$404$	9654.55	1.18894
$405$	0	0
$406$	−664.281	−0.0812013
$407$	25530.2	3.10930
$408$	0	0
$409$	6797.81	0.821833	0.410917	−	0.911673i	$-0.365209\pi$
$409$	6797.81	0.821833	0.410917	+	0.911673i	$0.365209\pi$
$410$	−4211.91	−0.507345
$411$	0	0
$412$	8250.61	0.986599
$413$	3313.95	0.394840
$414$	0	0
$415$	−3889.14	−0.460025
$416$	0	0
$417$	0	0
$418$	−10027.3	−1.17332
$419$	12594.2	1.46841	0.734207	−	0.678925i	$-0.237554\pi$
$419$	12594.2	1.46841	0.734207	+	0.678925i	$0.237554\pi$
$420$	0	0
$421$	−6888.04	−0.797393	−0.398697	−	0.917083i	$-0.630537\pi$
$421$	−6888.04	−0.797393	−0.398697	+	0.917083i	$0.630537\pi$
$422$	−3427.19	−0.395338
$423$	0	0
$424$	−523.845	−0.0600004
$425$	10406.8	1.18777
$426$	0	0
$427$	−1491.74	−0.169065
$428$	−4665.04	−0.526854
$429$	0	0
$430$	−2098.77	−0.235376
$431$	−7384.53	−0.825291	−0.412645	−	0.910892i	$-0.635395\pi$
$431$	−7384.53	−0.825291	−0.412645	+	0.910892i	$0.635395\pi$
$432$	0	0
$433$	9068.33	1.00646	0.503229	−	0.864153i	$-0.332145\pi$
$433$	9068.33	1.00646	0.503229	+	0.864153i	$0.332145\pi$
$434$	1353.17	0.149665
$435$	0	0
$436$	2592.68	0.284786
$437$	−3066.97	−0.335728
$438$	0	0
$439$	16875.4	1.83466	0.917331	−	0.398125i	$-0.130339\pi$
$439$	16875.4	1.83466	0.917331	+	0.398125i	$0.130339\pi$
$440$	24646.3	2.67037
$441$	0	0
$442$	0	0
$443$	6766.18	0.725668	0.362834	−	0.931854i	$-0.381809\pi$
$443$	6766.18	0.725668	0.362834	+	0.931854i	$0.381809\pi$
$444$	0	0
$445$	2104.84	0.224222
$446$	−5237.61	−0.556072
$447$	0	0
$448$	−862.814	−0.0909914
$449$	140.944	0.0148141	0.00740706	−	0.999973i	$-0.497642\pi$
$449$	140.944	0.0148141	0.00740706	+	0.999973i	$0.497642\pi$
$450$	0	0
$451$	8713.47	0.909759
$452$	−2164.19	−0.225210
$453$	0	0
$454$	−7262.67	−0.750779
$455$	0	0
$456$	0	0
$457$	17733.1	1.81514	0.907571	−	0.419898i	$-0.137934\pi$
$457$	17733.1	1.81514	0.907571	+	0.419898i	$0.137934\pi$
$458$	6668.90	0.680387
$459$	0	0
$460$	3126.74	0.316924
$461$	2293.37	0.231699	0.115849	−	0.993267i	$-0.463041\pi$
$461$	2293.37	0.231699	0.115849	+	0.993267i	$0.463041\pi$
$462$	0	0
$463$	−13770.9	−1.38226	−0.691129	−	0.722731i	$-0.742887\pi$
$463$	−13770.9	−1.38226	−0.691129	+	0.722731i	$0.742887\pi$
$464$	1214.27	0.121489
$465$	0	0
$466$	5559.29	0.552638
$467$	3477.37	0.344568	0.172284	−	0.985047i	$-0.444885\pi$
$467$	3477.37	0.344568	0.172284	+	0.985047i	$0.444885\pi$
$468$	0	0
$469$	3257.31	0.320700
$470$	9556.38	0.937879
$471$	0	0
$472$	14283.2	1.39288
$473$	4341.88	0.422072
$474$	0	0
$475$	26786.4	2.58746
$476$	1147.11	0.110458
$477$	0	0
$478$	3413.38	0.326620
$479$	−3137.39	−0.299271	−0.149636	−	0.988741i	$-0.547810\pi$
$479$	−3137.39	−0.299271	−0.149636	+	0.988741i	$0.547810\pi$
$480$	0	0
$481$	0	0
$482$	−9981.05	−0.943204
$483$	0	0
$484$	−13601.6	−1.27739
$485$	−22394.7	−2.09668
$486$	0	0
$487$	−5996.52	−0.557964	−0.278982	−	0.960296i	$-0.589997\pi$
$487$	−5996.52	−0.557964	−0.278982	+	0.960296i	$0.589997\pi$
$488$	−6429.46	−0.596410
$489$	0	0
$490$	−9433.86	−0.869752
$491$	9401.49	0.864121	0.432060	−	0.901845i	$-0.357787\pi$
$491$	9401.49	0.864121	0.432060	+	0.901845i	$0.357787\pi$
$492$	0	0
$493$	3756.31	0.343156
$494$	0	0
$495$	0	0
$496$	−2473.52	−0.223920
$497$	1582.34	0.142812
$498$	0	0
$499$	5052.33	0.453253	0.226626	−	0.973982i	$-0.427230\pi$
$499$	5052.33	0.453253	0.226626	+	0.973982i	$0.427230\pi$
$500$	−13601.2	−1.21653
$501$	0	0
$502$	−3826.96	−0.340251
$503$	−8184.02	−0.725462	−0.362731	−	0.931894i	$-0.618156\pi$
$503$	−8184.02	−0.725462	−0.362731	+	0.931894i	$0.618156\pi$
$504$	0	0
$505$	−32931.1	−2.90181
$506$	2658.17	0.233537
$507$	0	0
$508$	6401.94	0.559134
$509$	6039.12	0.525892	0.262946	−	0.964811i	$-0.415306\pi$
$509$	6039.12	0.525892	0.262946	+	0.964811i	$0.415306\pi$
$510$	0	0
$511$	117.673	0.0101870
$512$	4788.54	0.413331
$513$	0	0
$514$	1233.56	0.105856
$515$	−28142.3	−2.40796
$516$	0	0
$517$	−19770.0	−1.68178
$518$	3089.25	0.262035
$519$	0	0
$520$	0	0
$521$	14602.5	1.22792	0.613960	−	0.789337i	$-0.289576\pi$
$521$	14602.5	1.22792	0.613960	+	0.789337i	$0.289576\pi$
$522$	0	0
$523$	8910.70	0.745005	0.372502	−	0.928031i	$-0.378500\pi$
$523$	8910.70	0.745005	0.372502	+	0.928031i	$0.378500\pi$
$524$	−4835.14	−0.403099
$525$	0	0
$526$	−4488.80	−0.372093
$527$	−7651.79	−0.632481
$528$	0	0
$529$	−11354.0	−0.933177
$530$	741.124	0.0607403
$531$	0	0
$532$	2952.60	0.240623
$533$	0	0
$534$	0	0
$535$	15912.2	1.28588
$536$	14039.1	1.13134
$537$	0	0
$538$	−10855.3	−0.869897
$539$	19516.5	1.55962
$540$	0	0
$541$	13313.6	1.05803	0.529017	−	0.848611i	$-0.322561\pi$
$541$	13313.6	1.05803	0.529017	+	0.848611i	$0.322561\pi$
$542$	−3106.25	−0.246171
$543$	0	0
$544$	7837.48	0.617701
$545$	−8843.47	−0.695069
$546$	0	0
$547$	−4116.94	−0.321806	−0.160903	−	0.986970i	$-0.551441\pi$
$547$	−4116.94	−0.321806	−0.160903	+	0.986970i	$0.551441\pi$
$548$	2772.07	0.216090
$549$	0	0
$550$	−23215.9	−1.79987
$551$	9668.52	0.747537
$552$	0	0
$553$	−807.590	−0.0621016
$554$	4156.79	0.318782
$555$	0	0
$556$	−2308.38	−0.176074
$557$	−6888.37	−0.524003	−0.262002	−	0.965067i	$-0.584383\pi$
$557$	−6888.37	−0.524003	−0.262002	+	0.965067i	$0.584383\pi$
$558$	0	0
$559$	0	0
$560$	−1264.83	−0.0954443
$561$	0	0
$562$	4984.68	0.374139
$563$	−10537.1	−0.788782	−0.394391	−	0.918943i	$-0.629045\pi$
$563$	−10537.1	−0.788782	−0.394391	+	0.918943i	$0.629045\pi$
$564$	0	0
$565$	7381.92	0.549664
$566$	1746.27	0.129684
$567$	0	0
$568$	6819.91	0.503798
$569$	−26930.1	−1.98413	−0.992065	−	0.125722i	$-0.959875\pi$
$569$	−26930.1	−1.98413	−0.992065	+	0.125722i	$0.959875\pi$
$570$	0	0
$571$	−3125.60	−0.229076	−0.114538	−	0.993419i	$-0.536539\pi$
$571$	−3125.60	−0.229076	−0.114538	+	0.993419i	$0.536539\pi$
$572$	0	0
$573$	0	0
$574$	1054.36	0.0766696
$575$	−7100.91	−0.515006
$576$	0	0
$577$	−4787.13	−0.345391	−0.172696	−	0.984975i	$-0.555248\pi$
$577$	−4787.13	−0.345391	−0.172696	+	0.984975i	$0.555248\pi$
$578$	−4833.81	−0.347855
$579$	0	0
$580$	−9856.94	−0.705668
$581$	973.566	0.0695186
$582$	0	0
$583$	−1533.22	−0.108918
$584$	507.174	0.0359366
$585$	0	0
$586$	−2560.38	−0.180492
$587$	18380.8	1.29243	0.646214	−	0.763156i	$-0.276351\pi$
$587$	18380.8	1.29243	0.646214	+	0.763156i	$0.276351\pi$
$588$	0	0
$589$	−19695.3	−1.37781
$590$	−20207.6	−1.41005
$591$	0	0
$592$	−5646.96	−0.392042
$593$	−13831.7	−0.957843	−0.478922	−	0.877858i	$-0.658972\pi$
$593$	−13831.7	−0.957843	−0.478922	+	0.877858i	$0.658972\pi$
$594$	0	0
$595$	−3912.73	−0.269590
$596$	−9738.94	−0.669333
$597$	0	0
$598$	0	0
$599$	−12248.6	−0.835502	−0.417751	−	0.908562i	$-0.637182\pi$
$599$	−12248.6	−0.835502	−0.417751	+	0.908562i	$0.637182\pi$
$600$	0	0
$601$	9719.56	0.659682	0.329841	−	0.944036i	$-0.393005\pi$
$601$	9719.56	0.659682	0.329841	+	0.944036i	$0.393005\pi$
$602$	525.385	0.0355699
$603$	0	0
$604$	7607.91	0.512519
$605$	46394.3	3.11768
$606$	0	0
$607$	1607.83	0.107512	0.0537560	−	0.998554i	$-0.482881\pi$
$607$	1607.83	0.107512	0.0537560	+	0.998554i	$0.482881\pi$
$608$	20173.2	1.34561
$609$	0	0
$610$	9096.25	0.603764
$611$	0	0
$612$	0	0
$613$	−14731.1	−0.970610	−0.485305	−	0.874345i	$-0.661291\pi$
$613$	−14731.1	−0.970610	−0.485305	+	0.874345i	$0.661291\pi$
$614$	−11002.1	−0.723142
$615$	0	0
$616$	−6169.68	−0.403545
$617$	27951.8	1.82382	0.911909	−	0.410392i	$-0.134608\pi$
$617$	27951.8	1.82382	0.911909	+	0.410392i	$0.134608\pi$
$618$	0	0
$619$	−16200.2	−1.05192	−0.525961	−	0.850509i	$-0.676294\pi$
$619$	−16200.2	−1.05192	−0.525961	+	0.850509i	$0.676294\pi$
$620$	20079.1	1.30064
$621$	0	0
$622$	−630.364	−0.0406355
$623$	−526.903	−0.0338843
$624$	0	0
$625$	15263.8	0.976880
$626$	4482.22	0.286175
$627$	0	0
$628$	4312.28	0.274011
$629$	−17468.8	−1.10735
$630$	0	0
$631$	−12731.8	−0.803239	−0.401619	−	0.915807i	$-0.631553\pi$
$631$	−12731.8	−0.803239	−0.401619	+	0.915807i	$0.631553\pi$
$632$	−3480.73	−0.219076
$633$	0	0
$634$	576.927	0.0361399
$635$	−21836.6	−1.36466
$636$	0	0
$637$	0	0
$638$	−8379.77	−0.519997
$639$	0	0
$640$	−23756.7	−1.46729
$641$	11556.9	0.712119	0.356059	−	0.934463i	$-0.384120\pi$
$641$	11556.9	0.712119	0.356059	+	0.934463i	$0.384120\pi$
$642$	0	0
$643$	9181.25	0.563100	0.281550	−	0.959547i	$-0.409152\pi$
$643$	9181.25	0.563100	0.281550	+	0.959547i	$0.409152\pi$
$644$	−782.716	−0.0478934
$645$	0	0
$646$	6861.06	0.417871
$647$	−5244.11	−0.318651	−0.159326	−	0.987226i	$-0.550932\pi$
$647$	−5244.11	−0.318651	−0.159326	+	0.987226i	$0.550932\pi$
$648$	0	0
$649$	41804.8	2.52848
$650$	0	0
$651$	0	0
$652$	15377.5	0.923663
$653$	16421.4	0.984106	0.492053	−	0.870565i	$-0.336247\pi$
$653$	16421.4	0.984106	0.492053	+	0.870565i	$0.336247\pi$
$654$	0	0
$655$	16492.4	0.983832
$656$	−1927.31	−0.114709
$657$	0	0
$658$	−2392.24	−0.141732
$659$	−1838.11	−0.108653	−0.0543266	−	0.998523i	$-0.517301\pi$
$659$	−1838.11	−0.108653	−0.0543266	+	0.998523i	$0.517301\pi$
$660$	0	0
$661$	5500.93	0.323694	0.161847	−	0.986816i	$-0.448255\pi$
$661$	5500.93	0.323694	0.161847	+	0.986816i	$0.448255\pi$
$662$	6759.34	0.396841
$663$	0	0
$664$	4196.10	0.245241
$665$	−10071.1	−0.587281
$666$	0	0
$667$	−2563.07	−0.148789
$668$	−8798.08	−0.509593
$669$	0	0
$670$	−19862.2	−1.14529
$671$	−18818.0	−1.08266
$672$	0	0
$673$	−25986.7	−1.48843	−0.744216	−	0.667939i	$-0.767177\pi$
$673$	−25986.7	−1.48843	−0.744216	+	0.667939i	$0.767177\pi$
$674$	−2699.91	−0.154298
$675$	0	0
$676$	0	0
$677$	11691.3	0.663714	0.331857	−	0.943330i	$-0.392325\pi$
$677$	11691.3	0.663714	0.331857	+	0.943330i	$0.392325\pi$
$678$	0	0
$679$	5606.05	0.316849
$680$	−16864.0	−0.951035
$681$	0	0
$682$	17070.0	0.958423
$683$	−11111.5	−0.622501	−0.311251	−	0.950328i	$-0.600748\pi$
$683$	−11111.5	−0.622501	−0.311251	+	0.950328i	$0.600748\pi$
$684$	0	0
$685$	−9455.37	−0.527403
$686$	4896.36	0.272513
$687$	0	0
$688$	−960.370	−0.0532177
$689$	0	0
$690$	0	0
$691$	7542.55	0.415242	0.207621	−	0.978209i	$-0.433428\pi$
$691$	7542.55	0.415242	0.207621	+	0.978209i	$0.433428\pi$
$692$	−22511.3	−1.23664
$693$	0	0
$694$	−3683.26	−0.201462
$695$	7873.75	0.429738
$696$	0	0
$697$	−5962.11	−0.324005
$698$	15214.7	0.825051
$699$	0	0
$700$	6836.10	0.369114
$701$	8231.17	0.443491	0.221745	−	0.975105i	$-0.428825\pi$
$701$	8231.17	0.443491	0.221745	+	0.975105i	$0.428825\pi$
$702$	0	0
$703$	−44963.6	−2.41228
$704$	−10884.2	−0.582691
$705$	0	0
$706$	6893.86	0.367498
$707$	8243.61	0.438519
$708$	0	0
$709$	−28044.6	−1.48553	−0.742764	−	0.669554i	$-0.766485\pi$
$709$	−28044.6	−1.48553	−0.742764	+	0.669554i	$0.766485\pi$
$710$	−9648.64	−0.510010
$711$	0	0
$712$	−2270.96	−0.119534
$713$	5221.09	0.274238
$714$	0	0
$715$	0	0
$716$	−15257.6	−0.796374
$717$	0	0
$718$	18561.3	0.964764
$719$	−29686.4	−1.53980	−0.769901	−	0.638163i	$-0.779694\pi$
$719$	−29686.4	−1.53980	−0.769901	+	0.638163i	$0.779694\pi$
$720$	0	0
$721$	7044.86	0.363889
$722$	7190.12	0.370621
$723$	0	0
$724$	24784.0	1.27223
$725$	22385.3	1.14672
$726$	0	0
$727$	−27654.5	−1.41080	−0.705398	−	0.708812i	$-0.749232\pi$
$727$	−27654.5	−1.41080	−0.705398	+	0.708812i	$0.749232\pi$
$728$	0	0
$729$	0	0
$730$	−717.537	−0.0363798
$731$	−2970.89	−0.150318
$732$	0	0
$733$	13077.5	0.658975	0.329488	−	0.944160i	$-0.393124\pi$
$733$	13077.5	0.658975	0.329488	+	0.944160i	$0.393124\pi$
$734$	−4082.08	−0.205276
$735$	0	0
$736$	−5347.79	−0.267829
$737$	41090.3	2.05370
$738$	0	0
$739$	4218.33	0.209978	0.104989	−	0.994473i	$-0.466519\pi$
$739$	4218.33	0.209978	0.104989	+	0.994473i	$0.466519\pi$
$740$	45839.9	2.27717
$741$	0	0
$742$	−185.525	−0.00917903
$743$	7725.54	0.381457	0.190728	−	0.981643i	$-0.438915\pi$
$743$	7725.54	0.381457	0.190728	+	0.981643i	$0.438915\pi$
$744$	0	0
$745$	33218.9	1.63362
$746$	14656.5	0.719319
$747$	0	0
$748$	14470.6	0.707349
$749$	−3983.29	−0.194321
$750$	0	0
$751$	7506.12	0.364717	0.182358	−	0.983232i	$-0.441627\pi$
$751$	7506.12	0.364717	0.182358	+	0.983232i	$0.441627\pi$
$752$	4372.87	0.212051
$753$	0	0
$754$	0	0
$755$	−25950.1	−1.25089
$756$	0	0
$757$	−2741.62	−0.131632	−0.0658162	−	0.997832i	$-0.520965\pi$
$757$	−2741.62	−0.131632	−0.0658162	+	0.997832i	$0.520965\pi$
$758$	−13768.0	−0.659729
$759$	0	0
$760$	−43406.9	−2.07175
$761$	29740.0	1.41666	0.708328	−	0.705883i	$-0.249450\pi$
$761$	29740.0	1.41666	0.708328	+	0.705883i	$0.249450\pi$
$762$	0	0
$763$	2213.78	0.105038
$764$	7986.01	0.378172
$765$	0	0
$766$	−6129.18	−0.289107
$767$	0	0
$768$	0	0
$769$	19896.3	0.933004	0.466502	−	0.884520i	$-0.345514\pi$
$769$	19896.3	0.933004	0.466502	+	0.884520i	$0.345514\pi$
$770$	8728.72	0.408521
$771$	0	0
$772$	−23426.6	−1.09215
$773$	−13601.3	−0.632866	−0.316433	−	0.948615i	$-0.602485\pi$
$773$	−13601.3	−0.632866	−0.316433	+	0.948615i	$0.602485\pi$
$774$	0	0
$775$	−45600.1	−2.11355
$776$	24162.2	1.11775
$777$	0	0
$778$	4160.09	0.191705
$779$	−15346.1	−0.705818
$780$	0	0
$781$	19960.8	0.914539
$782$	−1818.83	−0.0831727
$783$	0	0
$784$	−4316.81	−0.196648
$785$	−14708.9	−0.668770
$786$	0	0
$787$	1498.29	0.0678631	0.0339315	−	0.999424i	$-0.489197\pi$
$787$	1498.29	0.0678631	0.0339315	+	0.999424i	$0.489197\pi$
$788$	19284.9	0.871824
$789$	0	0
$790$	4924.46	0.221778
$791$	−1847.91	−0.0830647
$792$	0	0
$793$	0	0
$794$	6703.49	0.299620
$795$	0	0
$796$	19962.5	0.888885
$797$	3713.30	0.165034	0.0825168	−	0.996590i	$-0.473704\pi$
$797$	3713.30	0.165034	0.0825168	+	0.996590i	$0.473704\pi$
$798$	0	0
$799$	13527.4	0.598955
$800$	46706.7	2.06416
$801$	0	0
$802$	5743.20	0.252867
$803$	1484.42	0.0652354
$804$	0	0
$805$	2669.80	0.116892
$806$	0	0
$807$	0	0
$808$	35530.2	1.54697
$809$	34527.6	1.50053	0.750263	−	0.661139i	$-0.229927\pi$
$809$	34527.6	1.50053	0.750263	+	0.661139i	$0.229927\pi$
$810$	0	0
$811$	37279.2	1.61412	0.807059	−	0.590471i	$-0.201058\pi$
$811$	37279.2	1.61412	0.807059	+	0.590471i	$0.201058\pi$
$812$	2467.48	0.106640
$813$	0	0
$814$	38970.3	1.67802
$815$	−52451.6	−2.25436
$816$	0	0
$817$	−7646.90	−0.327455
$818$	10376.4	0.443525
$819$	0	0
$820$	15645.2	0.666285
$821$	13877.9	0.589943	0.294972	−	0.955506i	$-0.404690\pi$
$821$	13877.9	0.589943	0.294972	+	0.955506i	$0.404690\pi$
$822$	0	0
$823$	18945.1	0.802410	0.401205	−	0.915988i	$-0.368592\pi$
$823$	18945.1	0.802410	0.401205	+	0.915988i	$0.368592\pi$
$824$	30363.5	1.28369
$825$	0	0
$826$	5058.55	0.213086
$827$	−7804.75	−0.328171	−0.164086	−	0.986446i	$-0.552467\pi$
$827$	−7804.75	−0.328171	−0.164086	+	0.986446i	$0.552467\pi$
$828$	0	0
$829$	5784.85	0.242360	0.121180	−	0.992631i	$-0.461332\pi$
$829$	5784.85	0.242360	0.121180	+	0.992631i	$0.461332\pi$
$830$	−5936.54	−0.248265
$831$	0	0
$832$	0	0
$833$	−13354.0	−0.555448
$834$	0	0
$835$	30009.7	1.24375
$836$	37246.4	1.54090
$837$	0	0
$838$	19224.3	0.792471
$839$	−5011.42	−0.206214	−0.103107	−	0.994670i	$-0.532878\pi$
$839$	−5011.42	−0.206214	−0.103107	+	0.994670i	$0.532878\pi$
$840$	0	0
$841$	−16309.0	−0.668704
$842$	−10514.2	−0.430336
$843$	0	0
$844$	12730.3	0.519190
$845$	0	0
$846$	0	0
$847$	−11613.9	−0.471142
$848$	339.128	0.0137332
$849$	0	0
$850$	15885.3	0.641013
$851$	11919.6	0.480138
$852$	0	0
$853$	22059.0	0.885446	0.442723	−	0.896659i	$-0.354013\pi$
$853$	22059.0	0.885446	0.442723	+	0.896659i	$0.354013\pi$
$854$	−2277.06	−0.0912404
$855$	0	0
$856$	−17168.1	−0.685506
$857$	−13956.2	−0.556283	−0.278141	−	0.960540i	$-0.589718\pi$
$857$	−13956.2	−0.556283	−0.278141	+	0.960540i	$0.589718\pi$
$858$	0	0
$859$	12498.5	0.496442	0.248221	−	0.968703i	$-0.420154\pi$
$859$	12498.5	0.496442	0.248221	+	0.968703i	$0.420154\pi$
$860$	7795.92	0.309115
$861$	0	0
$862$	−11272.0	−0.445391
$863$	−38631.2	−1.52378	−0.761890	−	0.647707i	$-0.775728\pi$
$863$	−38631.2	−1.52378	−0.761890	+	0.647707i	$0.775728\pi$
$864$	0	0
$865$	76784.7	3.01822
$866$	13842.3	0.543163
$867$	0	0
$868$	−5026.38	−0.196551
$869$	−10187.6	−0.397687
$870$	0	0
$871$	0	0
$872$	9541.46	0.370544
$873$	0	0
$874$	−4681.55	−0.181185
$875$	−11613.5	−0.448696
$876$	0	0
$877$	856.756	0.0329881	0.0164941	−	0.999864i	$-0.494750\pi$
$877$	856.756	0.0329881	0.0164941	+	0.999864i	$0.494750\pi$
$878$	25759.2	0.990127
$879$	0	0
$880$	−15955.6	−0.611206
$881$	−33638.6	−1.28640	−0.643198	−	0.765700i	$-0.722393\pi$
$881$	−33638.6	−1.28640	−0.643198	+	0.765700i	$0.722393\pi$
$882$	0	0
$883$	−31109.1	−1.18562	−0.592811	−	0.805342i	$-0.701982\pi$
$883$	−31109.1	−1.18562	−0.592811	+	0.805342i	$0.701982\pi$
$884$	0	0
$885$	0	0
$886$	10328.2	0.391627
$887$	−26080.3	−0.987248	−0.493624	−	0.869675i	$-0.664328\pi$
$887$	−26080.3	−0.987248	−0.493624	+	0.869675i	$0.664328\pi$
$888$	0	0
$889$	5466.35	0.206227
$890$	3212.91	0.121008
$891$	0	0
$892$	19455.2	0.730277
$893$	34818.8	1.30478
$894$	0	0
$895$	52042.8	1.94369
$896$	5946.99	0.221736
$897$	0	0
$898$	215.142	0.00799486
$899$	−16459.3	−0.610621
$900$	0	0
$901$	1049.09	0.0387905
$902$	13300.6	0.490977
$903$	0	0
$904$	−7964.56	−0.293028
$905$	−84536.9	−3.10508
$906$	0	0
$907$	20169.0	0.738369	0.369184	−	0.929356i	$-0.379637\pi$
$907$	20169.0	0.738369	0.369184	+	0.929356i	$0.379637\pi$
$908$	26977.3	0.985983
$909$	0	0
$910$	0	0
$911$	−19982.2	−0.726716	−0.363358	−	0.931650i	$-0.618370\pi$
$911$	−19982.2	−0.726716	−0.363358	+	0.931650i	$0.618370\pi$
$912$	0	0
$913$	12281.3	0.445184
$914$	27068.5	0.979593
$915$	0	0
$916$	−24771.7	−0.893538
$917$	−4128.52	−0.148676
$918$	0	0
$919$	38513.1	1.38241	0.691203	−	0.722661i	$-0.257081\pi$
$919$	38513.1	1.38241	0.691203	+	0.722661i	$0.257081\pi$
$920$	11506.9	0.412360
$921$	0	0
$922$	3500.70	0.125043
$923$	0	0
$924$	0	0
$925$	−104103.	−3.70043
$926$	−21020.4	−0.745975
$927$	0	0
$928$	16858.7	0.596352
$929$	23218.9	0.820009	0.410005	−	0.912083i	$-0.365527\pi$
$929$	23218.9	0.820009	0.410005	+	0.912083i	$0.365527\pi$
$930$	0	0
$931$	−34372.3	−1.21000
$932$	−20650.1	−0.725768
$933$	0	0
$934$	5307.99	0.185956
$935$	−49358.3	−1.72640
$936$	0	0
$937$	−11112.9	−0.387452	−0.193726	−	0.981056i	$-0.562057\pi$
$937$	−11112.9	−0.387452	−0.193726	+	0.981056i	$0.562057\pi$
$938$	4972.08	0.173075
$939$	0	0
$940$	−35497.3	−1.23170
$941$	45570.4	1.57869	0.789347	−	0.613947i	$-0.210419\pi$
$941$	45570.4	1.57869	0.789347	+	0.613947i	$0.210419\pi$
$942$	0	0
$943$	4068.16	0.140485
$944$	−9246.71	−0.318808
$945$	0	0
$946$	6627.62	0.227783
$947$	34903.9	1.19770	0.598852	−	0.800860i	$-0.295624\pi$
$947$	34903.9	1.19770	0.598852	+	0.800860i	$0.295624\pi$
$948$	0	0
$949$	0	0
$950$	40887.8	1.39640
$951$	0	0
$952$	4221.55	0.143720
$953$	2886.52	0.0981151	0.0490575	−	0.998796i	$-0.484378\pi$
$953$	2886.52	0.0981151	0.0490575	+	0.998796i	$0.484378\pi$
$954$	0	0
$955$	−27239.8	−0.922994
$956$	−12679.0	−0.428943
$957$	0	0
$958$	−4789.03	−0.161510
$959$	2366.96	0.0797008
$960$	0	0
$961$	3737.42	0.125455
$962$	0	0
$963$	0	0
$964$	37074.8	1.23869
$965$	79906.8	2.66559
$966$	0	0
$967$	−11593.8	−0.385556	−0.192778	−	0.981242i	$-0.561750\pi$
$967$	−11593.8	−0.385556	−0.192778	+	0.981242i	$0.561750\pi$
$968$	−50056.1	−1.66205
$969$	0	0
$970$	−34184.1	−1.13153
$971$	4952.12	0.163667	0.0818337	−	0.996646i	$-0.473922\pi$
$971$	4952.12	0.163667	0.0818337	+	0.996646i	$0.473922\pi$
$972$	0	0
$973$	−1971.03	−0.0649418
$974$	−9153.33	−0.301121
$975$	0	0
$976$	4162.32	0.136509
$977$	−19650.1	−0.643462	−0.321731	−	0.946831i	$-0.604265\pi$
$977$	−19650.1	−0.643462	−0.321731	+	0.946831i	$0.604265\pi$
$978$	0	0
$979$	−6646.77	−0.216988
$980$	35042.2	1.14223
$981$	0	0
$982$	14350.8	0.466347
$983$	56818.4	1.84357	0.921783	−	0.387707i	$-0.126733\pi$
$983$	56818.4	1.84357	0.921783	+	0.387707i	$0.126733\pi$
$984$	0	0
$985$	−65779.7	−2.12783
$986$	5733.78	0.185193
$987$	0	0
$988$	0	0
$989$	2027.15	0.0651764
$990$	0	0
$991$	−19120.4	−0.612897	−0.306448	−	0.951887i	$-0.599141\pi$
$991$	−19120.4	−0.612897	−0.306448	+	0.951887i	$0.599141\pi$
$992$	−34342.1	−1.09915
$993$	0	0
$994$	2415.34	0.0770723
$995$	−68090.9	−2.16947
$996$	0	0
$997$	38887.9	1.23530	0.617650	−	0.786453i	$-0.288085\pi$
$997$	38887.9	1.23530	0.617650	+	0.786453i	$0.288085\pi$
$998$	7712.07	0.244611
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.u.1.2		3
3.2	odd	2		507.4.a.h.1.2		3
13.12	even	2		117.4.a.f.1.2		3
39.5	even	4		507.4.b.g.337.4		6
39.8	even	4		507.4.b.g.337.3		6
39.38	odd	2		39.4.a.c.1.2	✓	3
52.51	odd	2		1872.4.a.bk.1.1		3
156.155	even	2		624.4.a.t.1.3		3
195.194	odd	2		975.4.a.l.1.2		3
273.272	even	2		1911.4.a.k.1.2		3
312.77	odd	2		2496.4.a.bl.1.1		3
312.155	even	2		2496.4.a.bp.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.2	✓	3	39.38	odd	2
117.4.a.f.1.2		3	13.12	even	2
507.4.a.h.1.2		3	3.2	odd	2
507.4.b.g.337.3		6	39.8	even	4
507.4.b.g.337.4		6	39.5	even	4
624.4.a.t.1.3		3	156.155	even	2
975.4.a.l.1.2		3	195.194	odd	2
1521.4.a.u.1.2		3	1.1	even	1	trivial
1872.4.a.bk.1.1		3	52.51	odd	2
1911.4.a.k.1.2		3	273.272	even	2
2496.4.a.bl.1.1		3	312.77	odd	2
2496.4.a.bp.1.1		3	312.155	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)

\(f(q)\)	\(=\)	\(q+1.52644 q^{2} -5.66998 q^{4} +19.3400 q^{5} -4.84136 q^{7} -20.8664 q^{8} +O(q^{10})\)	\(q+1.52644 q^{2} -5.66998 q^{4} +19.3400 q^{5} -4.84136 q^{7} -20.8664 q^{8} +29.5213 q^{10} -61.0728 q^{11} -7.39005 q^{14} +13.5085 q^{16} +41.7885 q^{17} +107.561 q^{19} -109.657 q^{20} -93.2239 q^{22} -28.5138 q^{23} +249.034 q^{25} +27.4504 q^{28} +89.8886 q^{29} -183.108 q^{31} +187.551 q^{32} +63.7876 q^{34} -93.6318 q^{35} -418.029 q^{37} +164.185 q^{38} -403.555 q^{40} -142.674 q^{41} -71.0935 q^{43} +346.281 q^{44} -43.5246 q^{46} +323.711 q^{47} -319.561 q^{49} +380.136 q^{50} +25.1047 q^{53} -1181.15 q^{55} +101.022 q^{56} +137.210 q^{58} -684.508 q^{59} +308.125 q^{61} -279.503 q^{62} +178.217 q^{64} -672.808 q^{67} -236.940 q^{68} -142.923 q^{70} -326.837 q^{71} -24.3058 q^{73} -638.095 q^{74} -609.869 q^{76} +295.675 q^{77} +166.810 q^{79} +261.255 q^{80} -217.783 q^{82} -201.093 q^{83} +808.188 q^{85} -108.520 q^{86} +1274.37 q^{88} +108.834 q^{89} +161.673 q^{92} +494.126 q^{94} +2080.23 q^{95} -1157.95 q^{97} -487.791 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 10 * q^4 + 4 * q^5 - 30 * q^7 - 6 * q^8	\( 3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8} - 4 q^{10} - 16 q^{11} + 176 q^{14} - 110 q^{16} + 146 q^{17} - 94 q^{19} - 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 80 q^{28} + 2 q^{29} - 302 q^{31} + 154 q^{32} - 164 q^{34} - 80 q^{35} - 374 q^{37} - 312 q^{38} - 516 q^{40} + 480 q^{41} - 260 q^{43} + 712 q^{44} + 1104 q^{46} - 24 q^{47} + 447 q^{49} + 814 q^{50} + 678 q^{53} - 1552 q^{55} - 96 q^{56} + 628 q^{58} - 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 750 q^{64} - 74 q^{67} + 460 q^{68} - 1216 q^{70} - 948 q^{71} + 222 q^{73} - 1724 q^{74} - 2392 q^{76} - 112 q^{77} - 24 q^{79} + 1100 q^{80} + 564 q^{82} - 796 q^{83} + 248 q^{85} + 1800 q^{86} + 1608 q^{88} + 1436 q^{89} + 1296 q^{92} - 1920 q^{94} + 4032 q^{95} - 3242 q^{97} - 5070 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 10 * q^4 + 4 * q^5 - 30 * q^7 - 6 * q^8 - 4 * q^10 - 16 * q^11 + 176 * q^14 - 110 * q^16 + 146 * q^17 - 94 * q^19 - 244 * q^20 - 56 * q^22 + 48 * q^23 + 145 * q^25 - 80 * q^28 + 2 * q^29 - 302 * q^31 + 154 * q^32 - 164 * q^34 - 80 * q^35 - 374 * q^37 - 312 * q^38 - 516 * q^40 + 480 * q^41 - 260 * q^43 + 712 * q^44 + 1104 * q^46 - 24 * q^47 + 447 * q^49 + 814 * q^50 + 678 * q^53 - 1552 * q^55 - 96 * q^56 + 628 * q^58 - 1788 * q^59 + 230 * q^61 - 1952 * q^62 - 750 * q^64 - 74 * q^67 + 460 * q^68 - 1216 * q^70 - 948 * q^71 + 222 * q^73 - 1724 * q^74 - 2392 * q^76 - 112 * q^77 - 24 * q^79 + 1100 * q^80 + 564 * q^82 - 796 * q^83 + 248 * q^85 + 1800 * q^86 + 1608 * q^88 + 1436 * q^89 + 1296 * q^92 - 1920 * q^94 + 4032 * q^95 - 3242 * q^97 - 5070 * q^98

Introduction

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