LMFDB - Embedded newform 1521.4.a.bi.1.9

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} - 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			1.9
Root			$-3.27560$ 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)

$f(q)$	$=$	$q+5.52257 q^{2} +22.4988 q^{4} -6.08065 q^{5} -20.2718 q^{7} +80.0709 q^{8} +O(q^{10})$	$q+5.52257 q^{2} +22.4988 q^{4} -6.08065 q^{5} -20.2718 q^{7} +80.0709 q^{8} -33.5808 q^{10} -48.8284 q^{11} -111.953 q^{14} +262.207 q^{16} +37.7513 q^{17} -120.837 q^{19} -136.807 q^{20} -269.658 q^{22} -74.8543 q^{23} -88.0257 q^{25} -456.092 q^{28} +112.710 q^{29} -113.134 q^{31} +807.490 q^{32} +208.485 q^{34} +123.266 q^{35} +85.7704 q^{37} -667.331 q^{38} -486.883 q^{40} -133.993 q^{41} -319.135 q^{43} -1098.58 q^{44} -413.388 q^{46} -401.982 q^{47} +67.9471 q^{49} -486.129 q^{50} +384.493 q^{53} +296.908 q^{55} -1623.18 q^{56} +622.450 q^{58} -121.629 q^{59} +220.043 q^{61} -624.790 q^{62} +2361.77 q^{64} -975.363 q^{67} +849.361 q^{68} +680.745 q^{70} +106.725 q^{71} -43.2566 q^{73} +473.673 q^{74} -2718.69 q^{76} +989.841 q^{77} +539.339 q^{79} -1594.39 q^{80} -739.986 q^{82} +811.183 q^{83} -229.553 q^{85} -1762.45 q^{86} -3909.73 q^{88} -1130.64 q^{89} -1684.13 q^{92} -2219.98 q^{94} +734.767 q^{95} +229.088 q^{97} +375.243 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.52257	1.95253	0.976263	−	0.216591i	$-0.0694937\pi$
$2$	5.52257	1.95253	0.976263	+	0.216591i	$0.0694937\pi$
$3$	0	0
$3$	0	0
$4$	22.4988	2.81235
$5$	−6.08065	−0.543870	−0.271935	−	0.962316i	$-0.587664\pi$
$5$	−6.08065	−0.543870	−0.271935	+	0.962316i	$0.587664\pi$
$6$	0	0
$7$	−20.2718	−1.09458	−0.547288	−	0.836944i	$-0.684340\pi$
$7$	−20.2718	−1.09458	−0.547288	+	0.836944i	$0.684340\pi$
$8$	80.0709	3.53867
$9$	0	0
$10$	−33.5808	−1.06192
$11$	−48.8284	−1.33839	−0.669196	−	0.743086i	$-0.733361\pi$
$11$	−48.8284	−1.33839	−0.669196	+	0.743086i	$0.733361\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−111.953	−2.13719
$15$	0	0
$16$	262.207	4.09698
$17$	37.7513	0.538591	0.269295	−	0.963058i	$-0.413209\pi$
$17$	37.7513	0.538591	0.269295	+	0.963058i	$0.413209\pi$
$18$	0	0
$19$	−120.837	−1.45905	−0.729524	−	0.683955i	$-0.760258\pi$
$19$	−120.837	−1.45905	−0.729524	+	0.683955i	$0.760258\pi$
$20$	−136.807	−1.52955
$21$	0	0
$22$	−269.658	−2.61324
$23$	−74.8543	−0.678617	−0.339309	−	0.940675i	$-0.610193\pi$
$23$	−74.8543	−0.678617	−0.339309	+	0.940675i	$0.610193\pi$
$24$	0	0
$25$	−88.0257	−0.704206
$26$	0	0
$27$	0	0
$28$	−456.092	−3.07834
$29$	112.710	0.721715	0.360858	−	0.932621i	$-0.382484\pi$
$29$	112.710	0.721715	0.360858	+	0.932621i	$0.382484\pi$
$30$	0	0
$31$	−113.134	−0.655465	−0.327733	−	0.944770i	$-0.606285\pi$
$31$	−113.134	−0.655465	−0.327733	+	0.944770i	$0.606285\pi$
$32$	807.490	4.46079
$33$	0	0
$34$	208.485	1.05161
$35$	123.266	0.595307
$36$	0	0
$37$	85.7704	0.381096	0.190548	−	0.981678i	$-0.438973\pi$
$37$	85.7704	0.381096	0.190548	+	0.981678i	$0.438973\pi$
$38$	−667.331	−2.84883
$39$	0	0
$40$	−486.883	−1.92457
$41$	−133.993	−0.510395	−0.255197	−	0.966889i	$-0.582140\pi$
$41$	−133.993	−0.510395	−0.255197	+	0.966889i	$0.582140\pi$
$42$	0	0
$43$	−319.135	−1.13181	−0.565903	−	0.824472i	$-0.691472\pi$
$43$	−319.135	−1.13181	−0.565903	+	0.824472i	$0.691472\pi$
$44$	−1098.58	−3.76403
$45$	0	0
$46$	−413.388	−1.32502
$47$	−401.982	−1.24755	−0.623777	−	0.781602i	$-0.714403\pi$
$47$	−401.982	−1.24755	−0.623777	+	0.781602i	$0.714403\pi$
$48$	0	0
$49$	67.9471	0.198096
$50$	−486.129	−1.37498
$51$	0	0
$52$	0	0
$53$	384.493	0.996494	0.498247	−	0.867035i	$-0.333977\pi$
$53$	384.493	0.996494	0.498247	+	0.867035i	$0.333977\pi$
$54$	0	0
$55$	296.908	0.727911
$56$	−1623.18	−3.87334
$57$	0	0
$58$	622.450	1.40917
$59$	−121.629	−0.268385	−0.134192	−	0.990955i	$-0.542844\pi$
$59$	−121.629	−0.268385	−0.134192	+	0.990955i	$0.542844\pi$
$60$	0	0
$61$	220.043	0.461864	0.230932	−	0.972970i	$-0.425823\pi$
$61$	220.043	0.461864	0.230932	+	0.972970i	$0.425823\pi$
$62$	−624.790	−1.27981
$63$	0	0
$64$	2361.77	4.61283
$65$	0	0
$66$	0	0
$67$	−975.363	−1.77850	−0.889250	−	0.457421i	$-0.848773\pi$
$67$	−975.363	−1.77850	−0.889250	+	0.457421i	$0.848773\pi$
$68$	849.361	1.51471
$69$	0	0
$70$	680.745	1.16235
$71$	106.725	0.178394	0.0891969	−	0.996014i	$-0.471570\pi$
$71$	106.725	0.178394	0.0891969	+	0.996014i	$0.471570\pi$
$72$	0	0
$73$	−43.2566	−0.0693535	−0.0346767	−	0.999399i	$-0.511040\pi$
$73$	−43.2566	−0.0693535	−0.0346767	+	0.999399i	$0.511040\pi$
$74$	473.673	0.744100
$75$	0	0
$76$	−2718.69	−4.10336
$77$	989.841	1.46497
$78$	0	0
$79$	539.339	0.768106	0.384053	−	0.923311i	$-0.374528\pi$
$79$	539.339	0.768106	0.384053	+	0.923311i	$0.374528\pi$
$80$	−1594.39	−2.22822
$81$	0	0
$82$	−739.986	−0.996558
$83$	811.183	1.07276	0.536379	−	0.843977i	$-0.319792\pi$
$83$	811.183	1.07276	0.536379	+	0.843977i	$0.319792\pi$
$84$	0	0
$85$	−229.553	−0.292923
$86$	−1762.45	−2.20988
$87$	0	0
$88$	−3909.73	−4.73612
$89$	−1130.64	−1.34661	−0.673304	−	0.739366i	$-0.735125\pi$
$89$	−1130.64	−1.34661	−0.673304	+	0.739366i	$0.735125\pi$
$90$	0	0
$91$	0	0
$92$	−1684.13	−1.90851
$93$	0	0
$94$	−2219.98	−2.43588
$95$	734.767	0.793532
$96$	0	0
$97$	229.088	0.239797	0.119899	−	0.992786i	$-0.461743\pi$
$97$	229.088	0.239797	0.119899	+	0.992786i	$0.461743\pi$
$98$	375.243	0.386788
$99$	0	0
$100$	−1980.48	−1.98048
$101$	845.077	0.832557	0.416279	−	0.909237i	$-0.363334\pi$
$101$	845.077	0.832557	0.416279	+	0.909237i	$0.363334\pi$
$102$	0	0
$103$	−1095.09	−1.04760	−0.523800	−	0.851841i	$-0.675486\pi$
$103$	−1095.09	−1.04760	−0.523800	+	0.851841i	$0.675486\pi$
$104$	0	0
$105$	0	0
$106$	2123.39	1.94568
$107$	−1183.33	−1.06913	−0.534563	−	0.845129i	$-0.679524\pi$
$107$	−1183.33	−1.06913	−0.534563	+	0.845129i	$0.679524\pi$
$108$	0	0
$109$	239.968	0.210870	0.105435	−	0.994426i	$-0.466377\pi$
$109$	239.968	0.210870	0.105435	+	0.994426i	$0.466377\pi$
$110$	1639.70	1.42126
$111$	0	0
$112$	−5315.41	−4.48446
$113$	1031.22	0.858490	0.429245	−	0.903188i	$-0.358780\pi$
$113$	1031.22	0.858490	0.429245	+	0.903188i	$0.358780\pi$
$114$	0	0
$115$	455.162	0.369079
$116$	2535.85	2.02972
$117$	0	0
$118$	−671.703	−0.524028
$119$	−765.288	−0.589528
$120$	0	0
$121$	1053.21	0.791294
$122$	1215.21	0.901800
$123$	0	0
$124$	−2545.38	−1.84340
$125$	1295.33	0.926866
$126$	0	0
$127$	−18.0201	−0.0125908	−0.00629539	−	0.999980i	$-0.502004\pi$
$127$	−18.0201	−0.0125908	−0.00629539	+	0.999980i	$0.502004\pi$
$128$	6583.12	4.54587
$129$	0	0
$130$	0	0
$131$	1282.52	0.855378	0.427689	−	0.903926i	$-0.359328\pi$
$131$	1282.52	0.855378	0.427689	+	0.903926i	$0.359328\pi$
$132$	0	0
$133$	2449.59	1.59704
$134$	−5386.51	−3.47257
$135$	0	0
$136$	3022.78	1.90589
$137$	1547.36	0.964965	0.482482	−	0.875906i	$-0.339735\pi$
$137$	1547.36	0.964965	0.482482	+	0.875906i	$0.339735\pi$
$138$	0	0
$139$	−3096.87	−1.88973	−0.944867	−	0.327455i	$-0.893809\pi$
$139$	−3096.87	−1.88973	−0.944867	+	0.327455i	$0.893809\pi$
$140$	2773.34	1.67421
$141$	0	0
$142$	589.398	0.348318
$143$	0	0
$144$	0	0
$145$	−685.351	−0.392519
$146$	−238.888	−0.135414
$147$	0	0
$148$	1929.73	1.07178
$149$	420.879	0.231408	0.115704	−	0.993284i	$-0.463088\pi$
$149$	420.879	0.231408	0.115704	+	0.993284i	$0.463088\pi$
$150$	0	0
$151$	2801.17	1.50964	0.754821	−	0.655931i	$-0.227724\pi$
$151$	2801.17	1.50964	0.754821	+	0.655931i	$0.227724\pi$
$152$	−9675.52	−5.16308
$153$	0	0
$154$	5466.47	2.86039
$155$	687.927	0.356488
$156$	0	0
$157$	−2344.20	−1.19164	−0.595820	−	0.803118i	$-0.703173\pi$
$157$	−2344.20	−1.19164	−0.595820	+	0.803118i	$0.703173\pi$
$158$	2978.54	1.49975
$159$	0	0
$160$	−4910.06	−2.42609
$161$	1517.43	0.742798
$162$	0	0
$163$	−3658.41	−1.75797	−0.878984	−	0.476851i	$-0.841778\pi$
$163$	−3658.41	−1.75797	−0.878984	+	0.476851i	$0.841778\pi$
$164$	−3014.68	−1.43541
$165$	0	0
$166$	4479.82	2.09459
$167$	1987.85	0.921105	0.460552	−	0.887632i	$-0.347651\pi$
$167$	1987.85	0.921105	0.460552	+	0.887632i	$0.347651\pi$
$168$	0	0
$169$	0	0
$170$	−1267.72	−0.571940
$171$	0	0
$172$	−7180.17	−3.18304
$173$	3577.11	1.57204	0.786019	−	0.618202i	$-0.212139\pi$
$173$	3577.11	1.57204	0.786019	+	0.618202i	$0.212139\pi$
$174$	0	0
$175$	1784.44	0.770807
$176$	−12803.1	−5.48337
$177$	0	0
$178$	−6244.07	−2.62928
$179$	−770.144	−0.321582	−0.160791	−	0.986988i	$-0.551405\pi$
$179$	−770.144	−0.321582	−0.160791	+	0.986988i	$0.551405\pi$
$180$	0	0
$181$	−1895.69	−0.778482	−0.389241	−	0.921136i	$-0.627263\pi$
$181$	−1895.69	−0.778482	−0.389241	+	0.921136i	$0.627263\pi$
$182$	0	0
$183$	0	0
$184$	−5993.65	−2.40140
$185$	−521.539	−0.207267
$186$	0	0
$187$	−1843.34	−0.720846
$188$	−9044.12	−3.50857
$189$	0	0
$190$	4057.81	1.54939
$191$	−2203.83	−0.834887	−0.417443	−	0.908703i	$-0.637074\pi$
$191$	−2203.83	−0.834887	−0.417443	+	0.908703i	$0.637074\pi$
$192$	0	0
$193$	−1016.38	−0.379069	−0.189534	−	0.981874i	$-0.560698\pi$
$193$	−1016.38	−0.379069	−0.189534	+	0.981874i	$0.560698\pi$
$194$	1265.15	0.468210
$195$	0	0
$196$	1528.73	0.557117
$197$	4318.58	1.56186	0.780930	−	0.624619i	$-0.214746\pi$
$197$	4318.58	1.56186	0.780930	+	0.624619i	$0.214746\pi$
$198$	0	0
$199$	1363.40	0.485672	0.242836	−	0.970067i	$-0.421922\pi$
$199$	1363.40	0.485672	0.242836	+	0.970067i	$0.421922\pi$
$200$	−7048.30	−2.49195
$201$	0	0
$202$	4667.00	1.62559
$203$	−2284.84	−0.789972
$204$	0	0
$205$	814.764	0.277588
$206$	−6047.74	−2.04546
$207$	0	0
$208$	0	0
$209$	5900.28	1.95278
$210$	0	0
$211$	5288.34	1.72542	0.862711	−	0.505697i	$-0.168765\pi$
$211$	5288.34	1.72542	0.862711	+	0.505697i	$0.168765\pi$
$212$	8650.64	2.80249
$213$	0	0
$214$	−6535.00	−2.08749
$215$	1940.55	0.615555
$216$	0	0
$217$	2293.43	0.717457
$218$	1325.24	0.411728
$219$	0	0
$220$	6680.09	2.04714
$221$	0	0
$222$	0	0
$223$	1258.01	0.377769	0.188884	−	0.981999i	$-0.439513\pi$
$223$	1258.01	0.377769	0.188884	+	0.981999i	$0.439513\pi$
$224$	−16369.3	−4.88268
$225$	0	0
$226$	5695.01	1.67622
$227$	1135.21	0.331922	0.165961	−	0.986132i	$-0.446927\pi$
$227$	1135.21	0.331922	0.165961	+	0.986132i	$0.446927\pi$
$228$	0	0
$229$	218.002	0.0629083	0.0314541	−	0.999505i	$-0.489986\pi$
$229$	218.002	0.0629083	0.0314541	+	0.999505i	$0.489986\pi$
$230$	2513.67	0.720636
$231$	0	0
$232$	9024.80	2.55391
$233$	2.56367	0.000720823 0	0.000360412	−	1.00000i	$-0.499885\pi$
$233$	2.56367	0.000720823 0	0.000360412		1.00000i	$0.499885\pi$
$234$	0	0
$235$	2444.31	0.678507
$236$	−2736.50	−0.754793
$237$	0	0
$238$	−4226.36	−1.15107
$239$	3971.13	1.07477	0.537387	−	0.843336i	$-0.319411\pi$
$239$	3971.13	1.07477	0.537387	+	0.843336i	$0.319411\pi$
$240$	0	0
$241$	658.032	0.175882	0.0879411	−	0.996126i	$-0.471971\pi$
$241$	658.032	0.175882	0.0879411	+	0.996126i	$0.471971\pi$
$242$	5816.44	1.54502
$243$	0	0
$244$	4950.72	1.29892
$245$	−413.162	−0.107739
$246$	0	0
$247$	0	0
$248$	−9058.72	−2.31947
$249$	0	0
$250$	7153.58	1.80973
$251$	−7091.01	−1.78319	−0.891595	−	0.452833i	$-0.850413\pi$
$251$	−7091.01	−1.78319	−0.891595	+	0.452833i	$0.850413\pi$
$252$	0	0
$253$	3655.01	0.908256
$254$	−99.5176	−0.0245838
$255$	0	0
$256$	17461.6	4.26309
$257$	−792.530	−0.192361	−0.0961803	−	0.995364i	$-0.530663\pi$
$257$	−792.530	−0.192361	−0.0961803	+	0.995364i	$0.530663\pi$
$258$	0	0
$259$	−1738.72	−0.417139
$260$	0	0
$261$	0	0
$262$	7082.83	1.67015
$263$	337.467	0.0791221	0.0395610	−	0.999217i	$-0.487404\pi$
$263$	337.467	0.0791221	0.0395610	+	0.999217i	$0.487404\pi$
$264$	0	0
$265$	−2337.97	−0.541963
$266$	13528.0	3.11826
$267$	0	0
$268$	−21944.5	−5.00177
$269$	3523.93	0.798728	0.399364	−	0.916792i	$-0.369231\pi$
$269$	3523.93	0.798728	0.399364	+	0.916792i	$0.369231\pi$
$270$	0	0
$271$	−8313.62	−1.86353	−0.931765	−	0.363063i	$-0.881731\pi$
$271$	−8313.62	−1.86353	−0.931765	+	0.363063i	$0.881731\pi$
$272$	9898.65	2.20660
$273$	0	0
$274$	8545.43	1.88412
$275$	4298.16	0.942504
$276$	0	0
$277$	−2500.33	−0.542347	−0.271174	−	0.962530i	$-0.587412\pi$
$277$	−2500.33	−0.542347	−0.271174	+	0.962530i	$0.587412\pi$
$278$	−17102.7	−3.68975
$279$	0	0
$280$	9870.01	2.10659
$281$	−3053.13	−0.648165	−0.324083	−	0.946029i	$-0.605056\pi$
$281$	−3053.13	−0.648165	−0.324083	+	0.946029i	$0.605056\pi$
$282$	0	0
$283$	1158.80	0.243404	0.121702	−	0.992567i	$-0.461165\pi$
$283$	1158.80	0.243404	0.121702	+	0.992567i	$0.461165\pi$
$284$	2401.19	0.501706
$285$	0	0
$286$	0	0
$287$	2716.28	0.558666
$288$	0	0
$289$	−3487.84	−0.709920
$290$	−3784.90	−0.766403
$291$	0	0
$292$	−973.223	−0.195046
$293$	−8357.45	−1.66637	−0.833187	−	0.552991i	$-0.813486\pi$
$293$	−8357.45	−1.66637	−0.833187	+	0.552991i	$0.813486\pi$
$294$	0	0
$295$	739.581	0.145966
$296$	6867.71	1.34857
$297$	0	0
$298$	2324.33	0.451829
$299$	0	0
$300$	0	0
$301$	6469.46	1.23885
$302$	15469.7	2.94761
$303$	0	0
$304$	−31684.3	−5.97769
$305$	−1338.01	−0.251194
$306$	0	0
$307$	8429.30	1.56705	0.783527	−	0.621357i	$-0.213418\pi$
$307$	8429.30	1.56705	0.783527	+	0.621357i	$0.213418\pi$
$308$	22270.3	4.12002
$309$	0	0
$310$	3799.13	0.696051
$311$	−3602.87	−0.656914	−0.328457	−	0.944519i	$-0.606529\pi$
$311$	−3602.87	−0.656914	−0.328457	+	0.944519i	$0.606529\pi$
$312$	0	0
$313$	6626.98	1.19674	0.598369	−	0.801220i	$-0.295816\pi$
$313$	6626.98	1.19674	0.598369	+	0.801220i	$0.295816\pi$
$314$	−12946.0	−2.32671
$315$	0	0
$316$	12134.5	2.16019
$317$	3983.87	0.705857	0.352928	−	0.935650i	$-0.385186\pi$
$317$	3983.87	0.705857	0.352928	+	0.935650i	$0.385186\pi$
$318$	0	0
$319$	−5503.46	−0.965938
$320$	−14361.1	−2.50878
$321$	0	0
$322$	8380.14	1.45033
$323$	−4561.76	−0.785829
$324$	0	0
$325$	0	0
$326$	−20203.8	−3.43248
$327$	0	0
$328$	−10728.9	−1.80612
$329$	8148.91	1.36554
$330$	0	0
$331$	−4172.82	−0.692927	−0.346464	−	0.938063i	$-0.612618\pi$
$331$	−4172.82	−0.692927	−0.346464	+	0.938063i	$0.612618\pi$
$332$	18250.7	3.01697
$333$	0	0
$334$	10978.1	1.79848
$335$	5930.84	0.967272
$336$	0	0
$337$	4157.51	0.672030	0.336015	−	0.941857i	$-0.390921\pi$
$337$	4157.51	0.672030	0.336015	+	0.941857i	$0.390921\pi$
$338$	0	0
$339$	0	0
$340$	−5164.66	−0.823804
$341$	5524.14	0.877270
$342$	0	0
$343$	5575.83	0.877744
$344$	−25553.4	−4.00509
$345$	0	0
$346$	19754.9	3.06944
$347$	−4667.98	−0.722162	−0.361081	−	0.932535i	$-0.617592\pi$
$347$	−4667.98	−0.722162	−0.361081	+	0.932535i	$0.617592\pi$
$348$	0	0
$349$	−11905.6	−1.82606	−0.913028	−	0.407896i	$-0.866263\pi$
$349$	−11905.6	−1.82606	−0.913028	+	0.407896i	$0.866263\pi$
$350$	9854.72	1.50502
$351$	0	0
$352$	−39428.4	−5.97029
$353$	−1662.12	−0.250610	−0.125305	−	0.992118i	$-0.539991\pi$
$353$	−1662.12	−0.250610	−0.125305	+	0.992118i	$0.539991\pi$
$354$	0	0
$355$	−648.959	−0.0970229
$356$	−25438.2	−3.78714
$357$	0	0
$358$	−4253.18	−0.627898
$359$	12754.4	1.87507	0.937535	−	0.347890i	$-0.113102\pi$
$359$	12754.4	1.87507	0.937535	+	0.347890i	$0.113102\pi$
$360$	0	0
$361$	7742.58	1.12882
$362$	−10469.1	−1.52001
$363$	0	0
$364$	0	0
$365$	263.028	0.0377192
$366$	0	0
$367$	4463.81	0.634901	0.317451	−	0.948275i	$-0.397173\pi$
$367$	4463.81	0.634901	0.317451	+	0.948275i	$0.397173\pi$
$368$	−19627.3	−2.78028
$369$	0	0
$370$	−2880.24	−0.404693
$371$	−7794.38	−1.09074
$372$	0	0
$373$	9454.46	1.31242	0.656211	−	0.754578i	$-0.272158\pi$
$373$	9454.46	1.31242	0.656211	+	0.754578i	$0.272158\pi$
$374$	−10180.0	−1.40747
$375$	0	0
$376$	−32187.0	−4.41468
$377$	0	0
$378$	0	0
$379$	−14085.9	−1.90908	−0.954542	−	0.298077i	$-0.903655\pi$
$379$	−14085.9	−1.90908	−0.954542	+	0.298077i	$0.903655\pi$
$380$	16531.4	2.23169
$381$	0	0
$382$	−12170.8	−1.63014
$383$	−8663.84	−1.15588	−0.577939	−	0.816080i	$-0.696143\pi$
$383$	−8663.84	−1.15588	−0.577939	+	0.816080i	$0.696143\pi$
$384$	0	0
$385$	−6018.87	−0.796754
$386$	−5613.01	−0.740141
$387$	0	0
$388$	5154.20	0.674395
$389$	−5085.58	−0.662852	−0.331426	−	0.943481i	$-0.607530\pi$
$389$	−5085.58	−0.662852	−0.331426	+	0.943481i	$0.607530\pi$
$390$	0	0
$391$	−2825.85	−0.365497
$392$	5440.58	0.700997
$393$	0	0
$394$	23849.7	3.04957
$395$	−3279.53	−0.417750
$396$	0	0
$397$	−12983.7	−1.64140	−0.820700	−	0.571360i	$-0.806416\pi$
$397$	−12983.7	−1.64140	−0.820700	+	0.571360i	$0.806416\pi$
$398$	7529.48	0.948288
$399$	0	0
$400$	−23080.9	−2.88512
$401$	6493.66	0.808673	0.404337	−	0.914610i	$-0.367502\pi$
$401$	6493.66	0.808673	0.404337	+	0.914610i	$0.367502\pi$
$402$	0	0
$403$	0	0
$404$	19013.2	2.34145
$405$	0	0
$406$	−12618.2	−1.54244
$407$	−4188.03	−0.510056
$408$	0	0
$409$	12546.2	1.51680	0.758398	−	0.651792i	$-0.225983\pi$
$409$	12546.2	1.51680	0.758398	+	0.651792i	$0.225983\pi$
$410$	4499.59	0.541998
$411$	0	0
$412$	−24638.3	−2.94622
$413$	2465.63	0.293767
$414$	0	0
$415$	−4932.52	−0.583440
$416$	0	0
$417$	0	0
$418$	32584.7	3.81285
$419$	−7945.58	−0.926413	−0.463206	−	0.886250i	$-0.653301\pi$
$419$	−7945.58	−0.926413	−0.463206	+	0.886250i	$0.653301\pi$
$420$	0	0
$421$	284.758	0.0329650	0.0164825	−	0.999864i	$-0.494753\pi$
$421$	284.758	0.0329650	0.0164825	+	0.999864i	$0.494753\pi$
$422$	29205.2	3.36893
$423$	0	0
$424$	30786.7	3.52626
$425$	−3323.09	−0.379279
$426$	0	0
$427$	−4460.68	−0.505545
$428$	−26623.4	−3.00676
$429$	0	0
$430$	10716.8	1.20189
$431$	−12512.1	−1.39835	−0.699173	−	0.714952i	$-0.746448\pi$
$431$	−12512.1	−1.39835	−0.699173	+	0.714952i	$0.746448\pi$
$432$	0	0
$433$	−13217.2	−1.46692	−0.733460	−	0.679732i	$-0.762096\pi$
$433$	−13217.2	−1.46692	−0.733460	+	0.679732i	$0.762096\pi$
$434$	12665.6	1.40085
$435$	0	0
$436$	5399.01	0.593040
$437$	9045.16	0.990135
$438$	0	0
$439$	4635.40	0.503953	0.251977	−	0.967733i	$-0.418919\pi$
$439$	4635.40	0.503953	0.251977	+	0.967733i	$0.418919\pi$
$440$	23773.7	2.57583
$441$	0	0
$442$	0	0
$443$	2945.92	0.315947	0.157974	−	0.987443i	$-0.449504\pi$
$443$	2945.92	0.315947	0.157974	+	0.987443i	$0.449504\pi$
$444$	0	0
$445$	6875.05	0.732379
$446$	6947.45	0.737603
$447$	0	0
$448$	−47877.3	−5.04909
$449$	−8668.29	−0.911095	−0.455548	−	0.890211i	$-0.650557\pi$
$449$	−8668.29	−0.911095	−0.455548	+	0.890211i	$0.650557\pi$
$450$	0	0
$451$	6542.66	0.683108
$452$	23201.3	2.41438
$453$	0	0
$454$	6269.27	0.648087
$455$	0	0
$456$	0	0
$457$	5195.36	0.531791	0.265896	−	0.964002i	$-0.414332\pi$
$457$	5195.36	0.531791	0.265896	+	0.964002i	$0.414332\pi$
$458$	1203.93	0.122830
$459$	0	0
$460$	10240.6	1.03798
$461$	−2931.19	−0.296136	−0.148068	−	0.988977i	$-0.547306\pi$
$461$	−2931.19	−0.296136	−0.148068	+	0.988977i	$0.547306\pi$
$462$	0	0
$463$	396.589	0.0398079	0.0199039	−	0.999802i	$-0.493664\pi$
$463$	396.589	0.0398079	0.0199039	+	0.999802i	$0.493664\pi$
$464$	29553.4	2.95685
$465$	0	0
$466$	14.1581	0.00140743
$467$	−16627.5	−1.64760	−0.823801	−	0.566880i	$-0.808150\pi$
$467$	−16627.5	−1.64760	−0.823801	+	0.566880i	$0.808150\pi$
$468$	0	0
$469$	19772.4	1.94670
$470$	13498.9	1.32480
$471$	0	0
$472$	−9738.91	−0.949724
$473$	15582.9	1.51480
$474$	0	0
$475$	10636.8	1.02747
$476$	−17218.1	−1.65796
$477$	0	0
$478$	21930.9	2.09852
$479$	8903.28	0.849272	0.424636	−	0.905364i	$-0.360402\pi$
$479$	8903.28	0.849272	0.424636	+	0.905364i	$0.360402\pi$
$480$	0	0
$481$	0	0
$482$	3634.03	0.343414
$483$	0	0
$484$	23696.1	2.22540
$485$	−1393.00	−0.130418
$486$	0	0
$487$	469.526	0.0436884	0.0218442	−	0.999761i	$-0.493046\pi$
$487$	469.526	0.0436884	0.0218442	+	0.999761i	$0.493046\pi$
$488$	17619.1	1.63438
$489$	0	0
$490$	−2281.72	−0.210362
$491$	18823.8	1.73015	0.865077	−	0.501639i	$-0.167269\pi$
$491$	18823.8	1.73015	0.865077	+	0.501639i	$0.167269\pi$
$492$	0	0
$493$	4254.96	0.388709
$494$	0	0
$495$	0	0
$496$	−29664.4	−2.68543
$497$	−2163.52	−0.195265
$498$	0	0
$499$	5651.95	0.507046	0.253523	−	0.967329i	$-0.418411\pi$
$499$	5651.95	0.507046	0.253523	+	0.967329i	$0.418411\pi$
$500$	29143.5	2.60667
$501$	0	0
$502$	−39160.7	−3.48172
$503$	−8022.35	−0.711131	−0.355565	−	0.934651i	$-0.615712\pi$
$503$	−8022.35	−0.711131	−0.355565	+	0.934651i	$0.615712\pi$
$504$	0	0
$505$	−5138.62	−0.452803
$506$	20185.1	1.77339
$507$	0	0
$508$	−405.432	−0.0354097
$509$	3940.56	0.343148	0.171574	−	0.985171i	$-0.445115\pi$
$509$	3940.56	0.343148	0.171574	+	0.985171i	$0.445115\pi$
$510$	0	0
$511$	876.890	0.0759126
$512$	43768.2	3.77793
$513$	0	0
$514$	−4376.81	−0.375589
$515$	6658.88	0.569758
$516$	0	0
$517$	19628.1	1.66972
$518$	−9602.22	−0.814474
$519$	0	0
$520$	0	0
$521$	−8644.84	−0.726943	−0.363472	−	0.931605i	$-0.618409\pi$
$521$	−8644.84	−0.726943	−0.363472	+	0.931605i	$0.618409\pi$
$522$	0	0
$523$	1956.76	0.163600	0.0818001	−	0.996649i	$-0.473933\pi$
$523$	1956.76	0.163600	0.0818001	+	0.996649i	$0.473933\pi$
$524$	28855.3	2.40563
$525$	0	0
$526$	1863.69	0.154488
$527$	−4270.95	−0.353028
$528$	0	0
$529$	−6563.84	−0.539479
$530$	−12911.6	−1.05820
$531$	0	0
$532$	55112.8	4.49144
$533$	0	0
$534$	0	0
$535$	7195.39	0.581465
$536$	−78098.2	−6.29352
$537$	0	0
$538$	19461.2	1.55954
$539$	−3317.75	−0.265131
$540$	0	0
$541$	−7005.37	−0.556718	−0.278359	−	0.960477i	$-0.589790\pi$
$541$	−7005.37	−0.556718	−0.278359	+	0.960477i	$0.589790\pi$
$542$	−45912.6	−3.63859
$543$	0	0
$544$	30483.8	2.40254
$545$	−1459.16	−0.114686
$546$	0	0
$547$	23216.7	1.81476	0.907382	−	0.420307i	$-0.138077\pi$
$547$	23216.7	1.81476	0.907382	+	0.420307i	$0.138077\pi$
$548$	34813.9	2.71382
$549$	0	0
$550$	23736.9	1.84026
$551$	−13619.6	−1.05302
$552$	0	0
$553$	−10933.4	−0.840751
$554$	−13808.2	−1.05895
$555$	0	0
$556$	−69675.9	−5.31460
$557$	−14912.9	−1.13443	−0.567217	−	0.823568i	$-0.691980\pi$
$557$	−14912.9	−1.13443	−0.567217	+	0.823568i	$0.691980\pi$
$558$	0	0
$559$	0	0
$560$	32321.1	2.43896
$561$	0	0
$562$	−16861.1	−1.26556
$563$	−10854.3	−0.812531	−0.406266	−	0.913755i	$-0.633169\pi$
$563$	−10854.3	−0.812531	−0.406266	+	0.913755i	$0.633169\pi$
$564$	0	0
$565$	−6270.50	−0.466906
$566$	6399.55	0.475253
$567$	0	0
$568$	8545.58	0.631276
$569$	−17892.5	−1.31827	−0.659134	−	0.752026i	$-0.729077\pi$
$569$	−17892.5	−1.31827	−0.659134	+	0.752026i	$0.729077\pi$
$570$	0	0
$571$	17319.8	1.26937	0.634686	−	0.772770i	$-0.281130\pi$
$571$	17319.8	1.26937	0.634686	+	0.772770i	$0.281130\pi$
$572$	0	0
$573$	0	0
$574$	15000.9	1.09081
$575$	6589.10	0.477886
$576$	0	0
$577$	9738.48	0.702632	0.351316	−	0.936257i	$-0.385734\pi$
$577$	9738.48	0.702632	0.351316	+	0.936257i	$0.385734\pi$
$578$	−19261.8	−1.38614
$579$	0	0
$580$	−15419.6	−1.10390
$581$	−16444.2	−1.17421
$582$	0	0
$583$	−18774.2	−1.33370
$584$	−3463.59	−0.245419
$585$	0	0
$586$	−46154.7	−3.25364
$587$	−5960.31	−0.419095	−0.209547	−	0.977799i	$-0.567199\pi$
$587$	−5960.31	−0.419095	−0.209547	+	0.977799i	$0.567199\pi$
$588$	0	0
$589$	13670.7	0.956355
$590$	4084.39	0.285003
$591$	0	0
$592$	22489.6	1.56134
$593$	−17870.2	−1.23751	−0.618753	−	0.785586i	$-0.712362\pi$
$593$	−17870.2	−1.23751	−0.618753	+	0.785586i	$0.712362\pi$
$594$	0	0
$595$	4653.45	0.320627
$596$	9469.28	0.650800
$597$	0	0
$598$	0	0
$599$	17943.7	1.22397	0.611987	−	0.790868i	$-0.290371\pi$
$599$	17943.7	1.22397	0.611987	+	0.790868i	$0.290371\pi$
$600$	0	0
$601$	6548.20	0.444437	0.222219	−	0.974997i	$-0.428670\pi$
$601$	6548.20	0.444437	0.222219	+	0.974997i	$0.428670\pi$
$602$	35728.1	2.41888
$603$	0	0
$604$	63023.0	4.24565
$605$	−6404.21	−0.430361
$606$	0	0
$607$	−14886.0	−0.995391	−0.497696	−	0.867352i	$-0.665820\pi$
$607$	−14886.0	−0.995391	−0.497696	+	0.867352i	$0.665820\pi$
$608$	−97574.6	−6.50851
$609$	0	0
$610$	−7389.24	−0.490462
$611$	0	0
$612$	0	0
$613$	8774.65	0.578148	0.289074	−	0.957307i	$-0.406653\pi$
$613$	8774.65	0.578148	0.289074	+	0.957307i	$0.406653\pi$
$614$	46551.5	3.05971
$615$	0	0
$616$	79257.4	5.18405
$617$	−3810.41	−0.248625	−0.124312	−	0.992243i	$-0.539673\pi$
$617$	−3810.41	−0.248625	−0.124312	+	0.992243i	$0.539673\pi$
$618$	0	0
$619$	16392.6	1.06441	0.532207	−	0.846614i	$-0.321363\pi$
$619$	16392.6	1.06441	0.532207	+	0.846614i	$0.321363\pi$
$620$	15477.5	1.00257
$621$	0	0
$622$	−19897.1	−1.28264
$623$	22920.2	1.47396
$624$	0	0
$625$	3126.74	0.200112
$626$	36598.0	2.33666
$627$	0	0
$628$	−52741.8	−3.35131
$629$	3237.95	0.205255
$630$	0	0
$631$	11077.3	0.698858	0.349429	−	0.936963i	$-0.386376\pi$
$631$	11077.3	0.698858	0.349429	+	0.936963i	$0.386376\pi$
$632$	43185.4	2.71807
$633$	0	0
$634$	22001.2	1.37820
$635$	109.574	0.00684775
$636$	0	0
$637$	0	0
$638$	−30393.2	−1.88602
$639$	0	0
$640$	−40029.6	−2.47236
$641$	2143.45	0.132077	0.0660383	−	0.997817i	$-0.478964\pi$
$641$	2143.45	0.132077	0.0660383	+	0.997817i	$0.478964\pi$
$642$	0	0
$643$	−9774.93	−0.599511	−0.299755	−	0.954016i	$-0.596905\pi$
$643$	−9774.93	−0.599511	−0.299755	+	0.954016i	$0.596905\pi$
$644$	34140.5	2.08901
$645$	0	0
$646$	−25192.6	−1.53435
$647$	22447.6	1.36400	0.681999	−	0.731353i	$-0.261111\pi$
$647$	22447.6	1.36400	0.681999	+	0.731353i	$0.261111\pi$
$648$	0	0
$649$	5938.93	0.359204
$650$	0	0
$651$	0	0
$652$	−82310.0	−4.94403
$653$	−10582.0	−0.634157	−0.317079	−	0.948399i	$-0.602702\pi$
$653$	−10582.0	−0.634157	−0.317079	+	0.948399i	$0.602702\pi$
$654$	0	0
$655$	−7798.57	−0.465214
$656$	−35133.9	−2.09108
$657$	0	0
$658$	45003.0	2.66626
$659$	15114.6	0.893445	0.446722	−	0.894673i	$-0.352591\pi$
$659$	15114.6	0.893445	0.446722	+	0.894673i	$0.352591\pi$
$660$	0	0
$661$	−2229.61	−0.131198	−0.0655990	−	0.997846i	$-0.520896\pi$
$661$	−2229.61	−0.131198	−0.0655990	+	0.997846i	$0.520896\pi$
$662$	−23044.7	−1.35296
$663$	0	0
$664$	64952.1	3.79613
$665$	−14895.1	−0.868581
$666$	0	0
$667$	−8436.83	−0.489768
$668$	44724.3	2.59047
$669$	0	0
$670$	32753.5	1.88862
$671$	−10744.4	−0.618155
$672$	0	0
$673$	−3588.22	−0.205521	−0.102761	−	0.994706i	$-0.532768\pi$
$673$	−3588.22	−0.205521	−0.102761	+	0.994706i	$0.532768\pi$
$674$	22960.2	1.31216
$675$	0	0
$676$	0	0
$677$	26458.4	1.50204	0.751019	−	0.660280i	$-0.229562\pi$
$677$	26458.4	1.50204	0.751019	+	0.660280i	$0.229562\pi$
$678$	0	0
$679$	−4644.03	−0.262476
$680$	−18380.5	−1.03656
$681$	0	0
$682$	30507.5	1.71289
$683$	−23598.4	−1.32206	−0.661031	−	0.750358i	$-0.729881\pi$
$683$	−23598.4	−1.32206	−0.661031	+	0.750358i	$0.729881\pi$
$684$	0	0
$685$	−9408.97	−0.524815
$686$	30792.9	1.71382
$687$	0	0
$688$	−83679.4	−4.63699
$689$	0	0
$690$	0	0
$691$	16433.6	0.904722	0.452361	−	0.891835i	$-0.350582\pi$
$691$	16433.6	0.904722	0.452361	+	0.891835i	$0.350582\pi$
$692$	80480.8	4.42113
$693$	0	0
$694$	−25779.2	−1.41004
$695$	18831.0	1.02777
$696$	0	0
$697$	−5058.41	−0.274894
$698$	−65749.7	−3.56542
$699$	0	0
$700$	40147.9	2.16778
$701$	−13899.4	−0.748892	−0.374446	−	0.927249i	$-0.622167\pi$
$701$	−13899.4	−0.748892	−0.374446	+	0.927249i	$0.622167\pi$
$702$	0	0
$703$	−10364.2	−0.556038
$704$	−115321.	−6.17377
$705$	0	0
$706$	−9179.16	−0.489323
$707$	−17131.3	−0.911297
$708$	0	0
$709$	−32829.6	−1.73899	−0.869494	−	0.493944i	$-0.835555\pi$
$709$	−32829.6	−1.73899	−0.869494	+	0.493944i	$0.835555\pi$
$710$	−3583.92	−0.189440
$711$	0	0
$712$	−90531.7	−4.76519
$713$	8468.55	0.444810
$714$	0	0
$715$	0	0
$716$	−17327.3	−0.904403
$717$	0	0
$718$	70437.0	3.66112
$719$	4683.94	0.242951	0.121475	−	0.992594i	$-0.461237\pi$
$719$	4683.94	0.242951	0.121475	+	0.992594i	$0.461237\pi$
$720$	0	0
$721$	22199.5	1.14668
$722$	42759.0	2.20405
$723$	0	0
$724$	−42650.7	−2.18937
$725$	−9921.39	−0.508236
$726$	0	0
$727$	−35368.3	−1.80432	−0.902159	−	0.431405i	$-0.858018\pi$
$727$	−35368.3	−1.80432	−0.902159	+	0.431405i	$0.858018\pi$
$728$	0	0
$729$	0	0
$730$	1452.59	0.0736478
$731$	−12047.8	−0.609580
$732$	0	0
$733$	−33988.4	−1.71268	−0.856338	−	0.516416i	$-0.827266\pi$
$733$	−33988.4	−1.71268	−0.856338	+	0.516416i	$0.827266\pi$
$734$	24651.7	1.23966
$735$	0	0
$736$	−60444.0	−3.02717
$737$	47625.4	2.38033
$738$	0	0
$739$	7777.31	0.387135	0.193568	−	0.981087i	$-0.437994\pi$
$739$	7777.31	0.387135	0.193568	+	0.981087i	$0.437994\pi$
$740$	−11734.0	−0.582907
$741$	0	0
$742$	−43045.0	−2.12969
$743$	−7022.08	−0.346723	−0.173361	−	0.984858i	$-0.555463\pi$
$743$	−7022.08	−0.346723	−0.173361	+	0.984858i	$0.555463\pi$
$744$	0	0
$745$	−2559.22	−0.125856
$746$	52212.9	2.56254
$747$	0	0
$748$	−41472.9	−2.02727
$749$	23988.2	1.17024
$750$	0	0
$751$	8214.69	0.399146	0.199573	−	0.979883i	$-0.436045\pi$
$751$	8214.69	0.399146	0.199573	+	0.979883i	$0.436045\pi$
$752$	−105402.	−5.11121
$753$	0	0
$754$	0	0
$755$	−17032.9	−0.821048
$756$	0	0
$757$	−20846.9	−1.00092	−0.500459	−	0.865760i	$-0.666835\pi$
$757$	−20846.9	−1.00092	−0.500459	+	0.865760i	$0.666835\pi$
$758$	−77790.3	−3.72753
$759$	0	0
$760$	58833.5	2.80804
$761$	−3563.06	−0.169725	−0.0848627	−	0.996393i	$-0.527045\pi$
$761$	−3563.06	−0.169725	−0.0848627	+	0.996393i	$0.527045\pi$
$762$	0	0
$763$	−4864.60	−0.230813
$764$	−49583.6	−2.34800
$765$	0	0
$766$	−47846.7	−2.25688
$767$	0	0
$768$	0	0
$769$	−32557.9	−1.52675	−0.763373	−	0.645958i	$-0.776458\pi$
$769$	−32557.9	−1.52675	−0.763373	+	0.645958i	$0.776458\pi$
$770$	−33239.7	−1.55568
$771$	0	0
$772$	−22867.3	−1.06608
$773$	20126.4	0.936477	0.468238	−	0.883602i	$-0.344889\pi$
$773$	20126.4	0.936477	0.468238	+	0.883602i	$0.344889\pi$
$774$	0	0
$775$	9958.68	0.461583
$776$	18343.2	0.848562
$777$	0	0
$778$	−28085.5	−1.29423
$779$	16191.3	0.744690
$780$	0	0
$781$	−5211.22	−0.238761
$782$	−15606.0	−0.713642
$783$	0	0
$784$	17816.2	0.811597
$785$	14254.3	0.648097
$786$	0	0
$787$	−18828.4	−0.852806	−0.426403	−	0.904533i	$-0.640219\pi$
$787$	−18828.4	−0.852806	−0.426403	+	0.904533i	$0.640219\pi$
$788$	97163.1	4.39250
$789$	0	0
$790$	−18111.5	−0.815667
$791$	−20904.8	−0.939682
$792$	0	0
$793$	0	0
$794$	−71703.7	−3.20487
$795$	0	0
$796$	30674.9	1.36588
$797$	−7288.52	−0.323930	−0.161965	−	0.986796i	$-0.551783\pi$
$797$	−7288.52	−0.323930	−0.161965	+	0.986796i	$0.551783\pi$
$798$	0	0
$799$	−15175.3	−0.671921
$800$	−71079.9	−3.14132
$801$	0	0
$802$	35861.7	1.57896
$803$	2112.15	0.0928221
$804$	0	0
$805$	−9226.97	−0.403985
$806$	0	0
$807$	0	0
$808$	67666.1	2.94614
$809$	33799.0	1.46886	0.734430	−	0.678684i	$-0.237449\pi$
$809$	33799.0	1.46886	0.734430	+	0.678684i	$0.237449\pi$
$810$	0	0
$811$	29055.3	1.25804	0.629019	−	0.777390i	$-0.283457\pi$
$811$	29055.3	1.25804	0.629019	+	0.777390i	$0.283457\pi$
$812$	−51406.2	−2.22168
$813$	0	0
$814$	−23128.7	−0.995898
$815$	22245.5	0.956106
$816$	0	0
$817$	38563.3	1.65136
$818$	69287.3	2.96158
$819$	0	0
$820$	18331.2	0.780676
$821$	−8921.98	−0.379268	−0.189634	−	0.981855i	$-0.560730\pi$
$821$	−8921.98	−0.379268	−0.189634	+	0.981855i	$0.560730\pi$
$822$	0	0
$823$	−9148.48	−0.387480	−0.193740	−	0.981053i	$-0.562062\pi$
$823$	−9148.48	−0.387480	−0.193740	+	0.981053i	$0.562062\pi$
$824$	−87685.1	−3.70711
$825$	0	0
$826$	13616.6	0.573588
$827$	−20206.0	−0.849615	−0.424807	−	0.905284i	$-0.639658\pi$
$827$	−20206.0	−0.849615	−0.424807	+	0.905284i	$0.639658\pi$
$828$	0	0
$829$	−10832.4	−0.453830	−0.226915	−	0.973915i	$-0.572864\pi$
$829$	−10832.4	−0.453830	−0.226915	+	0.973915i	$0.572864\pi$
$830$	−27240.2	−1.13918
$831$	0	0
$832$	0	0
$833$	2565.09	0.106693
$834$	0	0
$835$	−12087.4	−0.500961
$836$	132749.	5.49190
$837$	0	0
$838$	−43880.1	−1.80884
$839$	7047.44	0.289994	0.144997	−	0.989432i	$-0.453683\pi$
$839$	7047.44	0.289994	0.144997	+	0.989432i	$0.453683\pi$
$840$	0	0
$841$	−11685.4	−0.479127
$842$	1572.60	0.0643650
$843$	0	0
$844$	118981.	4.85250
$845$	0	0
$846$	0	0
$847$	−21350.5	−0.866132
$848$	100817.	4.08262
$849$	0	0
$850$	−18352.0	−0.740551
$851$	−6420.28	−0.258618
$852$	0	0
$853$	24085.5	0.966790	0.483395	−	0.875402i	$-0.339404\pi$
$853$	24085.5	0.966790	0.483395	+	0.875402i	$0.339404\pi$
$854$	−24634.5	−0.987089
$855$	0	0
$856$	−94749.9	−3.78328
$857$	4161.31	0.165866	0.0829332	−	0.996555i	$-0.473571\pi$
$857$	4161.31	0.165866	0.0829332	+	0.996555i	$0.473571\pi$
$858$	0	0
$859$	27387.5	1.08784	0.543918	−	0.839139i	$-0.316940\pi$
$859$	27387.5	1.08784	0.543918	+	0.839139i	$0.316940\pi$
$860$	43660.1	1.73116
$861$	0	0
$862$	−69099.1	−2.73031
$863$	−19420.4	−0.766022	−0.383011	−	0.923744i	$-0.625113\pi$
$863$	−19420.4	−0.766022	−0.383011	+	0.923744i	$0.625113\pi$
$864$	0	0
$865$	−21751.1	−0.854984
$866$	−72992.8	−2.86420
$867$	0	0
$868$	51599.5	2.01774
$869$	−26335.1	−1.02803
$870$	0	0
$871$	0	0
$872$	19214.5	0.746198
$873$	0	0
$874$	49952.6	1.93326
$875$	−26258.8	−1.01453
$876$	0	0
$877$	−9908.85	−0.381526	−0.190763	−	0.981636i	$-0.561096\pi$
$877$	−9908.85	−0.381526	−0.190763	+	0.981636i	$0.561096\pi$
$878$	25599.3	0.983981
$879$	0	0
$880$	77851.4	2.98224
$881$	−20323.3	−0.777197	−0.388598	−	0.921407i	$-0.627041\pi$
$881$	−20323.3	−0.777197	−0.388598	+	0.921407i	$0.627041\pi$
$882$	0	0
$883$	12443.3	0.474238	0.237119	−	0.971481i	$-0.423797\pi$
$883$	12443.3	0.474238	0.237119	+	0.971481i	$0.423797\pi$
$884$	0	0
$885$	0	0
$886$	16269.0	0.616895
$887$	−42313.5	−1.60174	−0.800872	−	0.598835i	$-0.795631\pi$
$887$	−42313.5	−1.60174	−0.800872	+	0.598835i	$0.795631\pi$
$888$	0	0
$889$	365.301	0.0137816
$890$	37968.0	1.42999
$891$	0	0
$892$	28303.7	1.06242
$893$	48574.3	1.82024
$894$	0	0
$895$	4682.97	0.174899
$896$	−133452.	−4.97580
$897$	0	0
$898$	−47871.3	−1.77894
$899$	−12751.3	−0.473059
$900$	0	0
$901$	14515.1	0.536702
$902$	36132.3	1.33379
$903$	0	0
$904$	82571.0	3.03791
$905$	11527.0	0.423393
$906$	0	0
$907$	−11002.3	−0.402786	−0.201393	−	0.979511i	$-0.564547\pi$
$907$	−11002.3	−0.402786	−0.201393	+	0.979511i	$0.564547\pi$
$908$	25540.8	0.933483
$909$	0	0
$910$	0	0
$911$	40803.4	1.48395	0.741974	−	0.670428i	$-0.233890\pi$
$911$	40803.4	1.48395	0.741974	+	0.670428i	$0.233890\pi$
$912$	0	0
$913$	−39608.8	−1.43577
$914$	28691.8	1.03834
$915$	0	0
$916$	4904.80	0.176920
$917$	−25999.1	−0.936276
$918$	0	0
$919$	2642.69	0.0948579	0.0474290	−	0.998875i	$-0.484897\pi$
$919$	2642.69	0.0948579	0.0474290	+	0.998875i	$0.484897\pi$
$920$	36445.3	1.30605
$921$	0	0
$922$	−16187.7	−0.578214
$923$	0	0
$924$	0	0
$925$	−7550.00	−0.268370
$926$	2190.19	0.0777258
$927$	0	0
$928$	91012.3	3.21942
$929$	−12687.7	−0.448083	−0.224042	−	0.974580i	$-0.571925\pi$
$929$	−12687.7	−0.448083	−0.224042	+	0.974580i	$0.571925\pi$
$930$	0	0
$931$	−8210.52	−0.289032
$932$	57.6796	0.00202721
$933$	0	0
$934$	−91826.7	−3.21698
$935$	11208.7	0.392046
$936$	0	0
$937$	−23066.2	−0.804205	−0.402103	−	0.915595i	$-0.631721\pi$
$937$	−23066.2	−0.804205	−0.402103	+	0.915595i	$0.631721\pi$
$938$	109195.	3.80099
$939$	0	0
$940$	54994.1	1.90820
$941$	−12660.7	−0.438605	−0.219303	−	0.975657i	$-0.570378\pi$
$941$	−12660.7	−0.438605	−0.219303	+	0.975657i	$0.570378\pi$
$942$	0	0
$943$	10029.9	0.346362
$944$	−31891.8	−1.09957
$945$	0	0
$946$	86057.5	2.95769
$947$	−21059.1	−0.722629	−0.361315	−	0.932444i	$-0.617672\pi$
$947$	−21059.1	−0.722629	−0.361315	+	0.932444i	$0.617672\pi$
$948$	0	0
$949$	0	0
$950$	58742.3	2.00616
$951$	0	0
$952$	−61277.3	−2.08614
$953$	−35646.2	−1.21164	−0.605820	−	0.795602i	$-0.707155\pi$
$953$	−35646.2	−1.21164	−0.605820	+	0.795602i	$0.707155\pi$
$954$	0	0
$955$	13400.7	0.454070
$956$	89345.8	3.02265
$957$	0	0
$958$	49169.0	1.65822
$959$	−31367.9	−1.05623
$960$	0	0
$961$	−16991.7	−0.570365
$962$	0	0
$963$	0	0
$964$	14805.0	0.494643
$965$	6180.22	0.206164
$966$	0	0
$967$	53452.1	1.77756	0.888782	−	0.458329i	$-0.151552\pi$
$967$	53452.1	1.77756	0.888782	+	0.458329i	$0.151552\pi$
$968$	84331.7	2.80013
$969$	0	0
$970$	−7692.95	−0.254645
$971$	−6502.60	−0.214911	−0.107455	−	0.994210i	$-0.534270\pi$
$971$	−6502.60	−0.214911	−0.107455	+	0.994210i	$0.534270\pi$
$972$	0	0
$973$	62779.2	2.06846
$974$	2592.99	0.0853027
$975$	0	0
$976$	57696.9	1.89225
$977$	4351.84	0.142505	0.0712527	−	0.997458i	$-0.477300\pi$
$977$	4351.84	0.142505	0.0712527	+	0.997458i	$0.477300\pi$
$978$	0	0
$979$	55207.5	1.80229
$980$	−9295.67	−0.302999
$981$	0	0
$982$	103956.	3.37817
$983$	−48698.2	−1.58009	−0.790047	−	0.613046i	$-0.789944\pi$
$983$	−48698.2	−1.58009	−0.790047	+	0.613046i	$0.789944\pi$
$984$	0	0
$985$	−26259.8	−0.849448
$986$	23498.3	0.758964
$987$	0	0
$988$	0	0
$989$	23888.6	0.768063
$990$	0	0
$991$	46630.0	1.49470	0.747351	−	0.664429i	$-0.231325\pi$
$991$	46630.0	1.49470	0.747351	+	0.664429i	$0.231325\pi$
$992$	−91354.3	−2.92389
$993$	0	0
$994$	−11948.2	−0.381261
$995$	−8290.35	−0.264143
$996$	0	0
$997$	37175.5	1.18090	0.590452	−	0.807073i	$-0.298950\pi$
$997$	37175.5	1.18090	0.590452	+	0.807073i	$0.298950\pi$
$998$	31213.3	0.990021
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bi.1.9		9
3.2	odd	2		507.4.a.o.1.1	✓	9
13.12	even	2		1521.4.a.bf.1.1		9
39.5	even	4		507.4.b.k.337.18		18
39.8	even	4		507.4.b.k.337.1		18
39.38	odd	2		507.4.a.p.1.9	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.1	✓	9	3.2	odd	2
507.4.a.p.1.9	yes	9	39.38	odd	2
507.4.b.k.337.1		18	39.8	even	4
507.4.b.k.337.18		18	39.5	even	4
1521.4.a.bf.1.1		9	13.12	even	2
1521.4.a.bi.1.9		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+5.52257 q^{2} +22.4988 q^{4} -6.08065 q^{5} -20.2718 q^{7} +80.0709 q^{8} +O(q^{10})\)	\(q+5.52257 q^{2} +22.4988 q^{4} -6.08065 q^{5} -20.2718 q^{7} +80.0709 q^{8} -33.5808 q^{10} -48.8284 q^{11} -111.953 q^{14} +262.207 q^{16} +37.7513 q^{17} -120.837 q^{19} -136.807 q^{20} -269.658 q^{22} -74.8543 q^{23} -88.0257 q^{25} -456.092 q^{28} +112.710 q^{29} -113.134 q^{31} +807.490 q^{32} +208.485 q^{34} +123.266 q^{35} +85.7704 q^{37} -667.331 q^{38} -486.883 q^{40} -133.993 q^{41} -319.135 q^{43} -1098.58 q^{44} -413.388 q^{46} -401.982 q^{47} +67.9471 q^{49} -486.129 q^{50} +384.493 q^{53} +296.908 q^{55} -1623.18 q^{56} +622.450 q^{58} -121.629 q^{59} +220.043 q^{61} -624.790 q^{62} +2361.77 q^{64} -975.363 q^{67} +849.361 q^{68} +680.745 q^{70} +106.725 q^{71} -43.2566 q^{73} +473.673 q^{74} -2718.69 q^{76} +989.841 q^{77} +539.339 q^{79} -1594.39 q^{80} -739.986 q^{82} +811.183 q^{83} -229.553 q^{85} -1762.45 q^{86} -3909.73 q^{88} -1130.64 q^{89} -1684.13 q^{92} -2219.98 q^{94} +734.767 q^{95} +229.088 q^{97} +375.243 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8}+O(q^{10}) \) 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

Varieties

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Database

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