LMFDB - Embedded newform 1521.4.a.bf.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:	$0$
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			1.9
Root			$5.39246$ 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)

$f(q)$	$=$	$q+4.83750 q^{2} +15.4014 q^{4} -21.1983 q^{5} +16.2806 q^{7} +35.8043 q^{8} +O(q^{10})$	$q+4.83750 q^{2} +15.4014 q^{4} -21.1983 q^{5} +16.2806 q^{7} +35.8043 q^{8} -102.547 q^{10} -30.7532 q^{11} +78.7575 q^{14} +49.9922 q^{16} -46.2371 q^{17} +144.865 q^{19} -326.483 q^{20} -148.769 q^{22} -8.38045 q^{23} +324.366 q^{25} +250.744 q^{28} +242.958 q^{29} +87.9353 q^{31} -44.5973 q^{32} -223.672 q^{34} -345.121 q^{35} -49.6950 q^{37} +700.783 q^{38} -758.990 q^{40} +107.947 q^{41} -35.4166 q^{43} -473.643 q^{44} -40.5404 q^{46} +374.815 q^{47} -77.9418 q^{49} +1569.12 q^{50} +348.583 q^{53} +651.915 q^{55} +582.916 q^{56} +1175.31 q^{58} +679.430 q^{59} -230.403 q^{61} +425.387 q^{62} -615.677 q^{64} +295.642 q^{67} -712.117 q^{68} -1669.52 q^{70} +329.215 q^{71} -48.9973 q^{73} -240.399 q^{74} +2231.12 q^{76} -500.681 q^{77} -107.942 q^{79} -1059.75 q^{80} +522.192 q^{82} +515.654 q^{83} +980.147 q^{85} -171.328 q^{86} -1101.10 q^{88} +984.453 q^{89} -129.071 q^{92} +1813.17 q^{94} -3070.88 q^{95} -487.072 q^{97} -377.043 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.83750	1.71031	0.855157	−	0.518368i	$-0.173460\pi$
$2$	4.83750	1.71031	0.855157	+	0.518368i	$0.173460\pi$
$3$	0	0
$3$	0	0
$4$	15.4014	1.92518
$5$	−21.1983	−1.89603	−0.948015	−	0.318225i	$-0.896913\pi$
$5$	−21.1983	−1.89603	−0.948015	+	0.318225i	$0.896913\pi$
$6$	0	0
$7$	16.2806	0.879070	0.439535	−	0.898225i	$-0.355143\pi$
$7$	16.2806	0.879070	0.439535	+	0.898225i	$0.355143\pi$
$8$	35.8043	1.58234
$9$	0	0
$10$	−102.547	−3.24281
$11$	−30.7532	−0.842950	−0.421475	−	0.906840i	$-0.638487\pi$
$11$	−30.7532	−0.842950	−0.421475	+	0.906840i	$0.638487\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	78.7575	1.50349
$15$	0	0
$16$	49.9922	0.781128
$17$	−46.2371	−0.659656	−0.329828	−	0.944041i	$-0.606991\pi$
$17$	−46.2371	−0.659656	−0.329828	+	0.944041i	$0.606991\pi$
$18$	0	0
$19$	144.865	1.74917	0.874585	−	0.484873i	$-0.161134\pi$
$19$	144.865	1.74917	0.874585	+	0.484873i	$0.161134\pi$
$20$	−326.483	−3.65019
$21$	0	0
$22$	−148.769	−1.44171
$23$	−8.38045	−0.0759758	−0.0379879	−	0.999278i	$-0.512095\pi$
$23$	−8.38045	−0.0759758	−0.0379879	+	0.999278i	$0.512095\pi$
$24$	0	0
$25$	324.366	2.59493
$26$	0	0
$27$	0	0
$28$	250.744	1.69237
$29$	242.958	1.55573	0.777865	−	0.628431i	$-0.216303\pi$
$29$	242.958	1.55573	0.777865	+	0.628431i	$0.216303\pi$
$30$	0	0
$31$	87.9353	0.509472	0.254736	−	0.967011i	$-0.418011\pi$
$31$	87.9353	0.509472	0.254736	+	0.967011i	$0.418011\pi$
$32$	−44.5973	−0.246368
$33$	0	0
$34$	−223.672	−1.12822
$35$	−345.121	−1.66674
$36$	0	0
$37$	−49.6950	−0.220806	−0.110403	−	0.993887i	$-0.535214\pi$
$37$	−49.6950	−0.220806	−0.110403	+	0.993887i	$0.535214\pi$
$38$	700.783	2.99163
$39$	0	0
$40$	−758.990	−3.00017
$41$	107.947	0.411181	0.205591	−	0.978638i	$-0.434088\pi$
$41$	107.947	0.411181	0.205591	+	0.978638i	$0.434088\pi$
$42$	0	0
$43$	−35.4166	−0.125604	−0.0628021	−	0.998026i	$-0.520004\pi$
$43$	−35.4166	−0.125604	−0.0628021	+	0.998026i	$0.520004\pi$
$44$	−473.643	−1.62283
$45$	0	0
$46$	−40.5404	−0.129943
$47$	374.815	1.16324	0.581621	−	0.813460i	$-0.302419\pi$
$47$	374.815	1.16324	0.581621	+	0.813460i	$0.302419\pi$
$48$	0	0
$49$	−77.9418	−0.227236
$50$	1569.12	4.43815
$51$	0	0
$52$	0	0
$53$	348.583	0.903426	0.451713	−	0.892163i	$-0.350813\pi$
$53$	348.583	0.903426	0.451713	+	0.892163i	$0.350813\pi$
$54$	0	0
$55$	651.915	1.59826
$56$	582.916	1.39099
$57$	0	0
$58$	1175.31	2.66079
$59$	679.430	1.49923	0.749613	−	0.661877i	$-0.230240\pi$
$59$	679.430	1.49923	0.749613	+	0.661877i	$0.230240\pi$
$60$	0	0
$61$	−230.403	−0.483608	−0.241804	−	0.970325i	$-0.577739\pi$
$61$	−230.403	−0.483608	−0.241804	+	0.970325i	$0.577739\pi$
$62$	425.387	0.871358
$63$	0	0
$64$	−615.677	−1.20249
$65$	0	0
$66$	0	0
$67$	295.642	0.539082	0.269541	−	0.962989i	$-0.413128\pi$
$67$	295.642	0.539082	0.269541	+	0.962989i	$0.413128\pi$
$68$	−712.117	−1.26995
$69$	0	0
$70$	−1669.52	−2.85066
$71$	329.215	0.550290	0.275145	−	0.961403i	$-0.411274\pi$
$71$	329.215	0.550290	0.275145	+	0.961403i	$0.411274\pi$
$72$	0	0
$73$	−48.9973	−0.0785575	−0.0392787	−	0.999228i	$-0.512506\pi$
$73$	−48.9973	−0.0785575	−0.0392787	+	0.999228i	$0.512506\pi$
$74$	−240.399	−0.377647
$75$	0	0
$76$	2231.12	3.36746
$77$	−500.681	−0.741012
$78$	0	0
$79$	−107.942	−0.153727	−0.0768636	−	0.997042i	$-0.524491\pi$
$79$	−107.942	−0.153727	−0.0768636	+	0.997042i	$0.524491\pi$
$80$	−1059.75	−1.48104
$81$	0	0
$82$	522.192	0.703250
$83$	515.654	0.681932	0.340966	−	0.940076i	$-0.389246\pi$
$83$	515.654	0.681932	0.340966	+	0.940076i	$0.389246\pi$
$84$	0	0
$85$	980.147	1.25073
$86$	−171.328	−0.214823
$87$	0	0
$88$	−1101.10	−1.33384
$89$	984.453	1.17249	0.586246	−	0.810133i	$-0.300605\pi$
$89$	984.453	1.17249	0.586246	+	0.810133i	$0.300605\pi$
$90$	0	0
$91$	0	0
$92$	−129.071	−0.146267
$93$	0	0
$94$	1813.17	1.98951
$95$	−3070.88	−3.31648
$96$	0	0
$97$	−487.072	−0.509842	−0.254921	−	0.966962i	$-0.582049\pi$
$97$	−487.072	−0.509842	−0.254921	+	0.966962i	$0.582049\pi$
$98$	−377.043	−0.388644
$99$	0	0
$100$	4995.70	4.99570
$101$	−766.375	−0.755021	−0.377511	−	0.926005i	$-0.623220\pi$
$101$	−766.375	−0.755021	−0.377511	+	0.926005i	$0.623220\pi$
$102$	0	0
$103$	1229.87	1.17653	0.588266	−	0.808668i	$-0.299811\pi$
$103$	1229.87	1.17653	0.588266	+	0.808668i	$0.299811\pi$
$104$	0	0
$105$	0	0
$106$	1686.27	1.54514
$107$	−76.8261	−0.0694117	−0.0347059	−	0.999398i	$-0.511049\pi$
$107$	−76.8261	−0.0694117	−0.0347059	+	0.999398i	$0.511049\pi$
$108$	0	0
$109$	626.001	0.550092	0.275046	−	0.961431i	$-0.411307\pi$
$109$	626.001	0.550092	0.275046	+	0.961431i	$0.411307\pi$
$110$	3153.64	2.73353
$111$	0	0
$112$	813.904	0.686667
$113$	1343.21	1.11822	0.559108	−	0.829095i	$-0.311144\pi$
$113$	1343.21	1.11822	0.559108	+	0.829095i	$0.311144\pi$
$114$	0	0
$115$	177.651	0.144052
$116$	3741.90	2.99506
$117$	0	0
$118$	3286.74	2.56415
$119$	−752.769	−0.579884
$120$	0	0
$121$	−385.238	−0.289435
$122$	−1114.58	−0.827122
$123$	0	0
$124$	1354.33	0.980824
$125$	−4226.22	−3.02404
$126$	0	0
$127$	−2146.69	−1.49990	−0.749952	−	0.661492i	$-0.769923\pi$
$127$	−2146.69	−1.49990	−0.749952	+	0.661492i	$0.769923\pi$
$128$	−2621.56	−1.81028
$129$	0	0
$130$	0	0
$131$	798.626	0.532644	0.266322	−	0.963884i	$-0.414192\pi$
$131$	798.626	0.532644	0.266322	+	0.963884i	$0.414192\pi$
$132$	0	0
$133$	2358.48	1.53764
$134$	1430.17	0.921999
$135$	0	0
$136$	−1655.49	−1.04380
$137$	601.153	0.374891	0.187445	−	0.982275i	$-0.439979\pi$
$137$	601.153	0.374891	0.187445	+	0.982275i	$0.439979\pi$
$138$	0	0
$139$	−2134.18	−1.30229	−0.651146	−	0.758953i	$-0.725711\pi$
$139$	−2134.18	−1.30229	−0.651146	+	0.758953i	$0.725711\pi$
$140$	−5315.35	−3.20878
$141$	0	0
$142$	1592.58	0.941169
$143$	0	0
$144$	0	0
$145$	−5150.29	−2.94971
$146$	−237.024	−0.134358
$147$	0	0
$148$	−765.373	−0.425090
$149$	−3439.93	−1.89134	−0.945670	−	0.325127i	$-0.894593\pi$
$149$	−3439.93	−1.89134	−0.945670	+	0.325127i	$0.894593\pi$
$150$	0	0
$151$	2224.04	1.19861	0.599305	−	0.800521i	$-0.295444\pi$
$151$	2224.04	1.19861	0.599305	+	0.800521i	$0.295444\pi$
$152$	5186.78	2.76779
$153$	0	0
$154$	−2422.05	−1.26736
$155$	−1864.07	−0.965975
$156$	0	0
$157$	2465.32	1.25321	0.626604	−	0.779338i	$-0.284444\pi$
$157$	2465.32	1.25321	0.626604	+	0.779338i	$0.284444\pi$
$158$	−522.171	−0.262922
$159$	0	0
$160$	945.386	0.467121
$161$	−136.439	−0.0667881
$162$	0	0
$163$	−243.565	−0.117040	−0.0585199	−	0.998286i	$-0.518638\pi$
$163$	−243.565	−0.117040	−0.0585199	+	0.998286i	$0.518638\pi$
$164$	1662.53	0.791597
$165$	0	0
$166$	2494.47	1.16632
$167$	−409.099	−0.189563	−0.0947814	−	0.995498i	$-0.530215\pi$
$167$	−409.099	−0.189563	−0.0947814	+	0.995498i	$0.530215\pi$
$168$	0	0
$169$	0	0
$170$	4741.46	2.13914
$171$	0	0
$172$	−545.466	−0.241810
$173$	2618.97	1.15096	0.575481	−	0.817815i	$-0.304815\pi$
$173$	2618.97	1.15096	0.575481	+	0.817815i	$0.304815\pi$
$174$	0	0
$175$	5280.88	2.28113
$176$	−1537.42	−0.658452
$177$	0	0
$178$	4762.29	2.00533
$179$	−163.311	−0.0681925	−0.0340963	−	0.999419i	$-0.510855\pi$
$179$	−163.311	−0.0681925	−0.0340963	+	0.999419i	$0.510855\pi$
$180$	0	0
$181$	−3313.07	−1.36054	−0.680272	−	0.732960i	$-0.738138\pi$
$181$	−3313.07	−1.36054	−0.680272	+	0.732960i	$0.738138\pi$
$182$	0	0
$183$	0	0
$184$	−300.056	−0.120220
$185$	1053.45	0.418654
$186$	0	0
$187$	1421.94	0.556057
$188$	5772.68	2.23945
$189$	0	0
$190$	−14855.4	−5.67222
$191$	4281.90	1.62213	0.811066	−	0.584954i	$-0.198888\pi$
$191$	4281.90	1.62213	0.811066	+	0.584954i	$0.198888\pi$
$192$	0	0
$193$	1877.33	0.700171	0.350086	−	0.936718i	$-0.386152\pi$
$193$	1877.33	0.700171	0.350086	+	0.936718i	$0.386152\pi$
$194$	−2356.21	−0.871990
$195$	0	0
$196$	−1200.41	−0.437468
$197$	1991.26	0.720158	0.360079	−	0.932922i	$-0.382750\pi$
$197$	1991.26	0.720158	0.360079	+	0.932922i	$0.382750\pi$
$198$	0	0
$199$	1345.05	0.479137	0.239568	−	0.970879i	$-0.422994\pi$
$199$	1345.05	0.479137	0.239568	+	0.970879i	$0.422994\pi$
$200$	11613.7	4.10607
$201$	0	0
$202$	−3707.34	−1.29132
$203$	3955.51	1.36760
$204$	0	0
$205$	−2288.28	−0.779612
$206$	5949.50	2.01224
$207$	0	0
$208$	0	0
$209$	−4455.05	−1.47446
$210$	0	0
$211$	−288.763	−0.0942147	−0.0471073	−	0.998890i	$-0.515000\pi$
$211$	−288.763	−0.0942147	−0.0471073	+	0.998890i	$0.515000\pi$
$212$	5368.67	1.73925
$213$	0	0
$214$	−371.646	−0.118716
$215$	750.771	0.238150
$216$	0	0
$217$	1431.64	0.447862
$218$	3028.28	0.940830
$219$	0	0
$220$	10040.4	3.07693
$221$	0	0
$222$	0	0
$223$	−1798.41	−0.540046	−0.270023	−	0.962854i	$-0.587031\pi$
$223$	−1798.41	−0.540046	−0.270023	+	0.962854i	$0.587031\pi$
$224$	−726.072	−0.216575
$225$	0	0
$226$	6497.77	1.91250
$227$	−2486.10	−0.726909	−0.363454	−	0.931612i	$-0.618403\pi$
$227$	−2486.10	−0.726909	−0.363454	+	0.931612i	$0.618403\pi$
$228$	0	0
$229$	6074.71	1.75296	0.876480	−	0.481438i	$-0.159885\pi$
$229$	6074.71	1.75296	0.876480	+	0.481438i	$0.159885\pi$
$230$	859.386	0.246375
$231$	0	0
$232$	8698.95	2.46170
$233$	−6367.16	−1.79024	−0.895121	−	0.445822i	$-0.852911\pi$
$233$	−6367.16	−1.79024	−0.895121	+	0.445822i	$0.852911\pi$
$234$	0	0
$235$	−7945.42	−2.20554
$236$	10464.2	2.88627
$237$	0	0
$238$	−3641.52	−0.991784
$239$	1886.43	0.510556	0.255278	−	0.966868i	$-0.417833\pi$
$239$	1886.43	0.510556	0.255278	+	0.966868i	$0.417833\pi$
$240$	0	0
$241$	5847.91	1.56306	0.781529	−	0.623869i	$-0.214440\pi$
$241$	5847.91	1.56306	0.781529	+	0.623869i	$0.214440\pi$
$242$	−1863.59	−0.495025
$243$	0	0
$244$	−3548.53	−0.931031
$245$	1652.23	0.430845
$246$	0	0
$247$	0	0
$248$	3148.46	0.806160
$249$	0	0
$250$	−20444.3	−5.17205
$251$	2388.65	0.600678	0.300339	−	0.953832i	$-0.402900\pi$
$251$	2388.65	0.600678	0.300339	+	0.953832i	$0.402900\pi$
$252$	0	0
$253$	257.726	0.0640438
$254$	−10384.6	−2.56531
$255$	0	0
$256$	−7756.38	−1.89365
$257$	−4447.21	−1.07941	−0.539707	−	0.841853i	$-0.681465\pi$
$257$	−4447.21	−1.07941	−0.539707	+	0.841853i	$0.681465\pi$
$258$	0	0
$259$	−809.064	−0.194104
$260$	0	0
$261$	0	0
$262$	3863.36	0.910988
$263$	−7590.87	−1.77975	−0.889873	−	0.456208i	$-0.849207\pi$
$263$	−7590.87	−1.77975	−0.889873	+	0.456208i	$0.849207\pi$
$264$	0	0
$265$	−7389.36	−1.71292
$266$	11409.2	2.62985
$267$	0	0
$268$	4553.31	1.03783
$269$	557.911	0.126455	0.0632275	−	0.997999i	$-0.479861\pi$
$269$	557.911	0.126455	0.0632275	+	0.997999i	$0.479861\pi$
$270$	0	0
$271$	−2707.41	−0.606877	−0.303439	−	0.952851i	$-0.598135\pi$
$271$	−2707.41	−0.606877	−0.303439	+	0.952851i	$0.598135\pi$
$272$	−2311.50	−0.515276
$273$	0	0
$274$	2908.08	0.641181
$275$	−9975.31	−2.18740
$276$	0	0
$277$	−1231.35	−0.267092	−0.133546	−	0.991043i	$-0.542636\pi$
$277$	−1231.35	−0.267092	−0.133546	+	0.991043i	$0.542636\pi$
$278$	−10324.1	−2.22733
$279$	0	0
$280$	−12356.8	−2.63736
$281$	4601.44	0.976864	0.488432	−	0.872602i	$-0.337569\pi$
$281$	4601.44	0.976864	0.488432	+	0.872602i	$0.337569\pi$
$282$	0	0
$283$	−54.5677	−0.0114619	−0.00573094	−	0.999984i	$-0.501824\pi$
$283$	−54.5677	−0.0114619	−0.00573094	+	0.999984i	$0.501824\pi$
$284$	5070.37	1.05941
$285$	0	0
$286$	0	0
$287$	1757.44	0.361457
$288$	0	0
$289$	−2775.13	−0.564854
$290$	−24914.5	−5.04494
$291$	0	0
$292$	−754.627	−0.151237
$293$	−4745.24	−0.946144	−0.473072	−	0.881024i	$-0.656855\pi$
$293$	−4745.24	−0.946144	−0.473072	+	0.881024i	$0.656855\pi$
$294$	0	0
$295$	−14402.7	−2.84258
$296$	−1779.30	−0.349390
$297$	0	0
$298$	−16640.6	−3.23479
$299$	0	0
$300$	0	0
$301$	−576.604	−0.110415
$302$	10758.8	2.05000
$303$	0	0
$304$	7242.10	1.36633
$305$	4884.15	0.916936
$306$	0	0
$307$	1028.82	0.191264	0.0956320	−	0.995417i	$-0.469513\pi$
$307$	1028.82	0.191264	0.0956320	+	0.995417i	$0.469513\pi$
$308$	−7711.20	−1.42658
$309$	0	0
$310$	−9017.46	−1.65212
$311$	2437.84	0.444492	0.222246	−	0.974991i	$-0.428661\pi$
$311$	2437.84	0.444492	0.222246	+	0.974991i	$0.428661\pi$
$312$	0	0
$313$	7934.20	1.43280	0.716402	−	0.697688i	$-0.245788\pi$
$313$	7934.20	1.43280	0.716402	+	0.697688i	$0.245788\pi$
$314$	11926.0	2.14338
$315$	0	0
$316$	−1662.46	−0.295952
$317$	−6942.72	−1.23010	−0.615050	−	0.788488i	$-0.710864\pi$
$317$	−6942.72	−1.23010	−0.615050	+	0.788488i	$0.710864\pi$
$318$	0	0
$319$	−7471.75	−1.31140
$320$	13051.3	2.27997
$321$	0	0
$322$	−660.023	−0.114229
$323$	−6698.12	−1.15385
$324$	0	0
$325$	0	0
$326$	−1178.25	−0.200175
$327$	0	0
$328$	3864.96	0.650630
$329$	6102.21	1.02257
$330$	0	0
$331$	−5439.27	−0.903230	−0.451615	−	0.892213i	$-0.649152\pi$
$331$	−5439.27	−0.903230	−0.451615	+	0.892213i	$0.649152\pi$
$332$	7941.79	1.31284
$333$	0	0
$334$	−1979.01	−0.324212
$335$	−6267.10	−1.02211
$336$	0	0
$337$	−1663.87	−0.268952	−0.134476	−	0.990917i	$-0.542935\pi$
$337$	−1663.87	−0.268952	−0.134476	+	0.990917i	$0.542935\pi$
$338$	0	0
$339$	0	0
$340$	15095.6	2.40787
$341$	−2704.29	−0.429460
$342$	0	0
$343$	−6853.19	−1.07883
$344$	−1268.07	−0.198749
$345$	0	0
$346$	12669.3	1.96851
$347$	−9268.10	−1.43383	−0.716913	−	0.697163i	$-0.754446\pi$
$347$	−9268.10	−1.43383	−0.716913	+	0.697163i	$0.754446\pi$
$348$	0	0
$349$	5239.45	0.803614	0.401807	−	0.915724i	$-0.368382\pi$
$349$	5239.45	0.803614	0.401807	+	0.915724i	$0.368382\pi$
$350$	25546.3	3.90144
$351$	0	0
$352$	1371.51	0.207676
$353$	−1218.01	−0.183649	−0.0918244	−	0.995775i	$-0.529270\pi$
$353$	−1218.01	−0.183649	−0.0918244	+	0.995775i	$0.529270\pi$
$354$	0	0
$355$	−6978.78	−1.04337
$356$	15162.0	2.25726
$357$	0	0
$358$	−790.018	−0.116631
$359$	−5316.06	−0.781534	−0.390767	−	0.920490i	$-0.627790\pi$
$359$	−5316.06	−0.781534	−0.390767	+	0.920490i	$0.627790\pi$
$360$	0	0
$361$	14126.7	2.05959
$362$	−16027.0	−2.32696
$363$	0	0
$364$	0	0
$365$	1038.66	0.148947
$366$	0	0
$367$	−289.101	−0.0411197	−0.0205599	−	0.999789i	$-0.506545\pi$
$367$	−289.101	−0.0411197	−0.0205599	+	0.999789i	$0.506545\pi$
$368$	−418.957	−0.0593469
$369$	0	0
$370$	5096.05	0.716030
$371$	5675.15	0.794175
$372$	0	0
$373$	6163.98	0.855654	0.427827	−	0.903861i	$-0.359279\pi$
$373$	6163.98	0.855654	0.427827	+	0.903861i	$0.359279\pi$
$374$	6878.64	0.951032
$375$	0	0
$376$	13420.0	1.84065
$377$	0	0
$378$	0	0
$379$	−2440.03	−0.330702	−0.165351	−	0.986235i	$-0.552876\pi$
$379$	−2440.03	−0.330702	−0.165351	+	0.986235i	$0.552876\pi$
$380$	−47295.9	−6.38481
$381$	0	0
$382$	20713.7	2.77436
$383$	6577.28	0.877502	0.438751	−	0.898609i	$-0.355421\pi$
$383$	6577.28	0.877502	0.438751	+	0.898609i	$0.355421\pi$
$384$	0	0
$385$	10613.6	1.40498
$386$	9081.58	1.19751
$387$	0	0
$388$	−7501.59	−0.981535
$389$	−4964.94	−0.647127	−0.323563	−	0.946206i	$-0.604881\pi$
$389$	−4964.94	−0.647127	−0.323563	+	0.946206i	$0.604881\pi$
$390$	0	0
$391$	387.488	0.0501179
$392$	−2790.65	−0.359565
$393$	0	0
$394$	9632.70	1.23170
$395$	2288.19	0.291471
$396$	0	0
$397$	−12073.9	−1.52637	−0.763187	−	0.646178i	$-0.776366\pi$
$397$	−12073.9	−1.52637	−0.763187	+	0.646178i	$0.776366\pi$
$398$	6506.69	0.819475
$399$	0	0
$400$	16215.8	2.02697
$401$	4916.90	0.612315	0.306158	−	0.951981i	$-0.400957\pi$
$401$	4916.90	0.612315	0.306158	+	0.951981i	$0.400957\pi$
$402$	0	0
$403$	0	0
$404$	−11803.3	−1.45355
$405$	0	0
$406$	19134.8	2.33902
$407$	1528.28	0.186128
$408$	0	0
$409$	15350.9	1.85588	0.927939	−	0.372733i	$-0.121579\pi$
$409$	15350.9	1.85588	0.927939	+	0.372733i	$0.121579\pi$
$410$	−11069.6	−1.33338
$411$	0	0
$412$	18941.7	2.26503
$413$	11061.5	1.31792
$414$	0	0
$415$	−10931.0	−1.29296
$416$	0	0
$417$	0	0
$418$	−21551.3	−2.52179
$419$	5488.58	0.639939	0.319970	−	0.947428i	$-0.396327\pi$
$419$	5488.58	0.639939	0.319970	+	0.947428i	$0.396327\pi$
$420$	0	0
$421$	−927.681	−0.107393	−0.0536964	−	0.998557i	$-0.517100\pi$
$421$	−927.681	−0.107393	−0.0536964	+	0.998557i	$0.517100\pi$
$422$	−1396.89	−0.161137
$423$	0	0
$424$	12480.8	1.42953
$425$	−14997.8	−1.71176
$426$	0	0
$427$	−3751.10	−0.425126
$428$	−1183.23	−0.133630
$429$	0	0
$430$	3631.85	0.407311
$431$	−11002.3	−1.22961	−0.614806	−	0.788678i	$-0.710766\pi$
$431$	−11002.3	−1.22961	−0.614806	+	0.788678i	$0.710766\pi$
$432$	0	0
$433$	−9596.76	−1.06511	−0.532553	−	0.846397i	$-0.678767\pi$
$433$	−9596.76	−1.06511	−0.532553	+	0.846397i	$0.678767\pi$
$434$	6925.56	0.765985
$435$	0	0
$436$	9641.30	1.05902
$437$	−1214.03	−0.132895
$438$	0	0
$439$	−10346.3	−1.12483	−0.562416	−	0.826854i	$-0.690128\pi$
$439$	−10346.3	−1.12483	−0.562416	+	0.826854i	$0.690128\pi$
$440$	23341.4	2.52899
$441$	0	0
$442$	0	0
$443$	−5650.38	−0.606000	−0.303000	−	0.952991i	$-0.597988\pi$
$443$	−5650.38	−0.606000	−0.303000	+	0.952991i	$0.597988\pi$
$444$	0	0
$445$	−20868.7	−2.22308
$446$	−8699.80	−0.923649
$447$	0	0
$448$	−10023.6	−1.05708
$449$	11987.8	1.25999	0.629997	−	0.776597i	$-0.283056\pi$
$449$	11987.8	1.25999	0.629997	+	0.776597i	$0.283056\pi$
$450$	0	0
$451$	−3319.71	−0.346605
$452$	20687.3	2.15276
$453$	0	0
$454$	−12026.5	−1.24324
$455$	0	0
$456$	0	0
$457$	16437.6	1.68253	0.841265	−	0.540623i	$-0.181811\pi$
$457$	16437.6	1.68253	0.841265	+	0.540623i	$0.181811\pi$
$458$	29386.4	2.99811
$459$	0	0
$460$	2736.08	0.277326
$461$	8847.70	0.893880	0.446940	−	0.894564i	$-0.352514\pi$
$461$	8847.70	0.893880	0.446940	+	0.894564i	$0.352514\pi$
$462$	0	0
$463$	−10269.6	−1.03081	−0.515407	−	0.856945i	$-0.672359\pi$
$463$	−10269.6	−1.03081	−0.515407	+	0.856945i	$0.672359\pi$
$464$	12146.0	1.21523
$465$	0	0
$466$	−30801.1	−3.06188
$467$	−14730.4	−1.45962	−0.729811	−	0.683649i	$-0.760392\pi$
$467$	−14730.4	−1.45962	−0.729811	+	0.683649i	$0.760392\pi$
$468$	0	0
$469$	4813.24	0.473891
$470$	−38436.0	−3.77217
$471$	0	0
$472$	24326.6	2.37229
$473$	1089.18	0.105878
$474$	0	0
$475$	46989.2	4.53897
$476$	−11593.7	−1.11638
$477$	0	0
$478$	9125.59	0.873211
$479$	635.077	0.0605792	0.0302896	−	0.999541i	$-0.490357\pi$
$479$	635.077	0.0605792	0.0302896	+	0.999541i	$0.490357\pi$
$480$	0	0
$481$	0	0
$482$	28289.3	2.67332
$483$	0	0
$484$	−5933.22	−0.557214
$485$	10325.1	0.966675
$486$	0	0
$487$	16611.1	1.54563	0.772815	−	0.634632i	$-0.218848\pi$
$487$	16611.1	1.54563	0.772815	+	0.634632i	$0.218848\pi$
$488$	−8249.43	−0.765234
$489$	0	0
$490$	7992.66	0.736881
$491$	2584.14	0.237516	0.118758	−	0.992923i	$-0.462109\pi$
$491$	2584.14	0.237516	0.118758	+	0.992923i	$0.462109\pi$
$492$	0	0
$493$	−11233.7	−1.02625
$494$	0	0
$495$	0	0
$496$	4396.08	0.397963
$497$	5359.82	0.483744
$498$	0	0
$499$	−2183.88	−0.195919	−0.0979597	−	0.995190i	$-0.531232\pi$
$499$	−2183.88	−0.195919	−0.0979597	+	0.995190i	$0.531232\pi$
$500$	−65089.8	−5.82180
$501$	0	0
$502$	11555.1	1.02735
$503$	17214.9	1.52600	0.762998	−	0.646401i	$-0.223727\pi$
$503$	17214.9	1.52600	0.762998	+	0.646401i	$0.223727\pi$
$504$	0	0
$505$	16245.8	1.43154
$506$	1246.75	0.109535
$507$	0	0
$508$	−33062.0	−2.88758
$509$	20260.6	1.76431	0.882157	−	0.470955i	$-0.156091\pi$
$509$	20260.6	1.76431	0.882157	+	0.470955i	$0.156091\pi$
$510$	0	0
$511$	−797.705	−0.0690575
$512$	−16549.0	−1.42846
$513$	0	0
$514$	−21513.4	−1.84614
$515$	−26071.1	−2.23074
$516$	0	0
$517$	−11526.8	−0.980554
$518$	−3913.85	−0.331978
$519$	0	0
$520$	0	0
$521$	20731.5	1.74331	0.871653	−	0.490124i	$-0.163048\pi$
$521$	20731.5	1.74331	0.871653	+	0.490124i	$0.163048\pi$
$522$	0	0
$523$	−944.353	−0.0789554	−0.0394777	−	0.999220i	$-0.512569\pi$
$523$	−944.353	−0.0789554	−0.0394777	+	0.999220i	$0.512569\pi$
$524$	12300.0	1.02543
$525$	0	0
$526$	−36720.9	−3.04393
$527$	−4065.87	−0.336076
$528$	0	0
$529$	−12096.8	−0.994228
$530$	−35746.0	−2.92964
$531$	0	0
$532$	36324.0	2.96023
$533$	0	0
$534$	0	0
$535$	1628.58	0.131607
$536$	10585.3	0.853012
$537$	0	0
$538$	2698.89	0.216278
$539$	2396.96	0.191548
$540$	0	0
$541$	−4883.06	−0.388058	−0.194029	−	0.980996i	$-0.562156\pi$
$541$	−4883.06	−0.388058	−0.194029	+	0.980996i	$0.562156\pi$
$542$	−13097.1	−1.03795
$543$	0	0
$544$	2062.05	0.162518
$545$	−13270.1	−1.04299
$546$	0	0
$547$	−16269.1	−1.27169	−0.635847	−	0.771815i	$-0.719349\pi$
$547$	−16269.1	−1.27169	−0.635847	+	0.771815i	$0.719349\pi$
$548$	9258.61	0.721730
$549$	0	0
$550$	−48255.6	−3.74114
$551$	35196.0	2.72124
$552$	0	0
$553$	−1757.37	−0.135137
$554$	−5956.63	−0.456811
$555$	0	0
$556$	−32869.3	−2.50714
$557$	−15661.0	−1.19134	−0.595670	−	0.803229i	$-0.703113\pi$
$557$	−15661.0	−1.19134	−0.595670	+	0.803229i	$0.703113\pi$
$558$	0	0
$559$	0	0
$560$	−17253.3	−1.30194
$561$	0	0
$562$	22259.5	1.67074
$563$	9915.18	0.742230	0.371115	−	0.928587i	$-0.378976\pi$
$563$	9915.18	0.742230	0.371115	+	0.928587i	$0.378976\pi$
$564$	0	0
$565$	−28473.7	−2.12017
$566$	−263.971	−0.0196034
$567$	0	0
$568$	11787.3	0.870748
$569$	11299.1	0.832482	0.416241	−	0.909254i	$-0.363347\pi$
$569$	11299.1	0.832482	0.416241	+	0.909254i	$0.363347\pi$
$570$	0	0
$571$	−17619.6	−1.29134	−0.645672	−	0.763615i	$-0.723423\pi$
$571$	−17619.6	−1.29134	−0.645672	+	0.763615i	$0.723423\pi$
$572$	0	0
$573$	0	0
$574$	8501.61	0.618206
$575$	−2718.33	−0.197152
$576$	0	0
$577$	−22153.5	−1.59837	−0.799186	−	0.601083i	$-0.794736\pi$
$577$	−22153.5	−1.59837	−0.799186	+	0.601083i	$0.794736\pi$
$578$	−13424.7	−0.966078
$579$	0	0
$580$	−79321.7	−5.67872
$581$	8395.15	0.599466
$582$	0	0
$583$	−10720.1	−0.761543
$584$	−1754.31	−0.124305
$585$	0	0
$586$	−22955.1	−1.61820
$587$	9891.28	0.695497	0.347748	−	0.937588i	$-0.386946\pi$
$587$	9891.28	0.695497	0.347748	+	0.937588i	$0.386946\pi$
$588$	0	0
$589$	12738.7	0.891153
$590$	−69673.3	−4.86170
$591$	0	0
$592$	−2484.36	−0.172477
$593$	9746.23	0.674924	0.337462	−	0.941339i	$-0.390432\pi$
$593$	9746.23	0.674924	0.337462	+	0.941339i	$0.390432\pi$
$594$	0	0
$595$	15957.4	1.09948
$596$	−52979.7	−3.64116
$597$	0	0
$598$	0	0
$599$	8460.53	0.577109	0.288554	−	0.957464i	$-0.406825\pi$
$599$	8460.53	0.577109	0.288554	+	0.957464i	$0.406825\pi$
$600$	0	0
$601$	6792.99	0.461052	0.230526	−	0.973066i	$-0.425955\pi$
$601$	6792.99	0.461052	0.230526	+	0.973066i	$0.425955\pi$
$602$	−2789.32	−0.188844
$603$	0	0
$604$	34253.4	2.30754
$605$	8166.38	0.548778
$606$	0	0
$607$	19073.4	1.27539	0.637697	−	0.770287i	$-0.279887\pi$
$607$	19073.4	1.27539	0.637697	+	0.770287i	$0.279887\pi$
$608$	−6460.58	−0.430939
$609$	0	0
$610$	23627.1	1.56825
$611$	0	0
$612$	0	0
$613$	−14465.2	−0.953091	−0.476545	−	0.879150i	$-0.658111\pi$
$613$	−14465.2	−0.953091	−0.476545	+	0.879150i	$0.658111\pi$
$614$	4976.93	0.327122
$615$	0	0
$616$	−17926.6	−1.17254
$617$	−12897.3	−0.841536	−0.420768	−	0.907168i	$-0.638239\pi$
$617$	−12897.3	−0.841536	−0.420768	+	0.907168i	$0.638239\pi$
$618$	0	0
$619$	−27735.8	−1.80096	−0.900481	−	0.434895i	$-0.856786\pi$
$619$	−27735.8	−1.80096	−0.900481	+	0.434895i	$0.856786\pi$
$620$	−28709.4	−1.85967
$621$	0	0
$622$	11793.0	0.760222
$623$	16027.5	1.03070
$624$	0	0
$625$	49042.7	3.13873
$626$	38381.7	2.45055
$627$	0	0
$628$	37969.3	2.41265
$629$	2297.75	0.145656
$630$	0	0
$631$	28560.1	1.80184	0.900918	−	0.433990i	$-0.142895\pi$
$631$	28560.1	1.80184	0.900918	+	0.433990i	$0.142895\pi$
$632$	−3864.80	−0.243249
$633$	0	0
$634$	−33585.4	−2.10386
$635$	45506.1	2.84386
$636$	0	0
$637$	0	0
$638$	−36144.6	−2.24291
$639$	0	0
$640$	55572.5	3.43234
$641$	−1862.53	−0.114767	−0.0573833	−	0.998352i	$-0.518276\pi$
$641$	−1862.53	−0.114767	−0.0573833	+	0.998352i	$0.518276\pi$
$642$	0	0
$643$	29495.2	1.80899	0.904494	−	0.426487i	$-0.140249\pi$
$643$	29495.2	1.80899	0.904494	+	0.426487i	$0.140249\pi$
$644$	−2101.35	−0.128579
$645$	0	0
$646$	−32402.2	−1.97345
$647$	−27780.7	−1.68805	−0.844027	−	0.536301i	$-0.819821\pi$
$647$	−27780.7	−1.68805	−0.844027	+	0.536301i	$0.819821\pi$
$648$	0	0
$649$	−20894.7	−1.26377
$650$	0	0
$651$	0	0
$652$	−3751.25	−0.225322
$653$	−276.678	−0.0165808	−0.00829039	−	0.999966i	$-0.502639\pi$
$653$	−276.678	−0.0165808	−0.00829039	+	0.999966i	$0.502639\pi$
$654$	0	0
$655$	−16929.5	−1.00991
$656$	5396.49	0.321185
$657$	0	0
$658$	29519.5	1.74892
$659$	−18114.9	−1.07080	−0.535398	−	0.844600i	$-0.679838\pi$
$659$	−18114.9	−1.07080	−0.535398	+	0.844600i	$0.679838\pi$
$660$	0	0
$661$	27094.4	1.59433	0.797163	−	0.603764i	$-0.206333\pi$
$661$	27094.4	1.59433	0.797163	+	0.603764i	$0.206333\pi$
$662$	−26312.5	−1.54481
$663$	0	0
$664$	18462.6	1.07905
$665$	−49995.8	−2.91542
$666$	0	0
$667$	−2036.10	−0.118198
$668$	−6300.70	−0.364942
$669$	0	0
$670$	−30317.1	−1.74814
$671$	7085.64	0.407657
$672$	0	0
$673$	2410.31	0.138054	0.0690272	−	0.997615i	$-0.478010\pi$
$673$	2410.31	0.138054	0.0690272	+	0.997615i	$0.478010\pi$
$674$	−8048.99	−0.459993
$675$	0	0
$676$	0	0
$677$	−12508.3	−0.710091	−0.355046	−	0.934849i	$-0.615535\pi$
$677$	−12508.3	−0.710091	−0.355046	+	0.934849i	$0.615535\pi$
$678$	0	0
$679$	−7929.83	−0.448187
$680$	35093.5	1.97908
$681$	0	0
$682$	−13082.0	−0.734511
$683$	16793.8	0.940844	0.470422	−	0.882442i	$-0.344102\pi$
$683$	16793.8	0.940844	0.470422	+	0.882442i	$0.344102\pi$
$684$	0	0
$685$	−12743.4	−0.710804
$686$	−33152.3	−1.84513
$687$	0	0
$688$	−1770.56	−0.0981131
$689$	0	0
$690$	0	0
$691$	−26026.9	−1.43286	−0.716432	−	0.697657i	$-0.754226\pi$
$691$	−26026.9	−1.43286	−0.716432	+	0.697657i	$0.754226\pi$
$692$	40335.8	2.21580
$693$	0	0
$694$	−44834.4	−2.45229
$695$	45240.8	2.46918
$696$	0	0
$697$	−4991.14	−0.271238
$698$	25345.8	1.37443
$699$	0	0
$700$	81333.0	4.39157
$701$	−5079.08	−0.273658	−0.136829	−	0.990595i	$-0.543691\pi$
$701$	−5079.08	−0.273658	−0.136829	+	0.990595i	$0.543691\pi$
$702$	0	0
$703$	−7199.04	−0.386226
$704$	18934.1	1.01364
$705$	0	0
$706$	−5892.11	−0.314097
$707$	−12477.1	−0.663717
$708$	0	0
$709$	−2530.72	−0.134052	−0.0670262	−	0.997751i	$-0.521351\pi$
$709$	−2530.72	−0.134052	−0.0670262	+	0.997751i	$0.521351\pi$
$710$	−33759.9	−1.78449
$711$	0	0
$712$	35247.7	1.85529
$713$	−736.937	−0.0387076
$714$	0	0
$715$	0	0
$716$	−2515.22	−0.131283
$717$	0	0
$718$	−25716.4	−1.33667
$719$	−9503.67	−0.492944	−0.246472	−	0.969150i	$-0.579271\pi$
$719$	−9503.67	−0.492944	−0.246472	+	0.969150i	$0.579271\pi$
$720$	0	0
$721$	20023.0	1.03425
$722$	68338.2	3.52255
$723$	0	0
$724$	−51026.0	−2.61929
$725$	78807.4	4.03701
$726$	0	0
$727$	33343.1	1.70100	0.850500	−	0.525975i	$-0.176300\pi$
$727$	33343.1	1.70100	0.850500	+	0.525975i	$0.176300\pi$
$728$	0	0
$729$	0	0
$730$	5024.50	0.254747
$731$	1637.56	0.0828556
$732$	0	0
$733$	32288.1	1.62699	0.813497	−	0.581569i	$-0.197561\pi$
$733$	32288.1	1.62699	0.813497	+	0.581569i	$0.197561\pi$
$734$	−1398.53	−0.0703277
$735$	0	0
$736$	373.746	0.0187180
$737$	−9091.96	−0.454419
$738$	0	0
$739$	−14168.8	−0.705288	−0.352644	−	0.935758i	$-0.614717\pi$
$739$	−14168.8	−0.705288	−0.352644	+	0.935758i	$0.614717\pi$
$740$	16224.6	0.805983
$741$	0	0
$742$	27453.5	1.35829
$743$	−13248.6	−0.654165	−0.327083	−	0.944996i	$-0.606066\pi$
$743$	−13248.6	−0.654165	−0.327083	+	0.944996i	$0.606066\pi$
$744$	0	0
$745$	72920.5	3.58604
$746$	29818.3	1.46344
$747$	0	0
$748$	21899.9	1.07051
$749$	−1250.78	−0.0610178
$750$	0	0
$751$	5113.98	0.248484	0.124242	−	0.992252i	$-0.460350\pi$
$751$	5113.98	0.248484	0.124242	+	0.992252i	$0.460350\pi$
$752$	18737.8	0.908641
$753$	0	0
$754$	0	0
$755$	−47145.9	−2.27260
$756$	0	0
$757$	27380.4	1.31461	0.657303	−	0.753626i	$-0.271697\pi$
$757$	27380.4	1.31461	0.657303	+	0.753626i	$0.271697\pi$
$758$	−11803.6	−0.565604
$759$	0	0
$760$	−109951.	−5.24781
$761$	10613.8	0.505586	0.252793	−	0.967520i	$-0.418651\pi$
$761$	10613.8	0.505586	0.252793	+	0.967520i	$0.418651\pi$
$762$	0	0
$763$	10191.7	0.483569
$764$	65947.3	3.12289
$765$	0	0
$766$	31817.6	1.50080
$767$	0	0
$768$	0	0
$769$	9172.22	0.430115	0.215058	−	0.976601i	$-0.431006\pi$
$769$	9172.22	0.430115	0.215058	+	0.976601i	$0.431006\pi$
$770$	51343.2	2.40296
$771$	0	0
$772$	28913.5	1.34795
$773$	−14952.1	−0.695718	−0.347859	−	0.937547i	$-0.613091\pi$
$773$	−14952.1	−0.695718	−0.347859	+	0.937547i	$0.613091\pi$
$774$	0	0
$775$	28523.2	1.32205
$776$	−17439.3	−0.806745
$777$	0	0
$778$	−24017.9	−1.10679
$779$	15637.7	0.719226
$780$	0	0
$781$	−10124.4	−0.463867
$782$	1874.47	0.0857174
$783$	0	0
$784$	−3896.48	−0.177500
$785$	−52260.4	−2.37612
$786$	0	0
$787$	−2671.52	−0.121003	−0.0605015	−	0.998168i	$-0.519270\pi$
$787$	−2671.52	−0.121003	−0.0605015	+	0.998168i	$0.519270\pi$
$788$	30668.1	1.38643
$789$	0	0
$790$	11069.1	0.498508
$791$	21868.3	0.982991
$792$	0	0
$793$	0	0
$794$	−58407.4	−2.61058
$795$	0	0
$796$	20715.7	0.922423
$797$	12949.7	0.575536	0.287768	−	0.957700i	$-0.407087\pi$
$797$	12949.7	0.575536	0.287768	+	0.957700i	$0.407087\pi$
$798$	0	0
$799$	−17330.4	−0.767339
$800$	−14465.9	−0.639307
$801$	0	0
$802$	23785.5	1.04725
$803$	1506.82	0.0662200
$804$	0	0
$805$	2892.27	0.126632
$806$	0	0
$807$	0	0
$808$	−27439.6	−1.19470
$809$	11640.4	0.505876	0.252938	−	0.967482i	$-0.418603\pi$
$809$	11640.4	0.505876	0.252938	+	0.967482i	$0.418603\pi$
$810$	0	0
$811$	18135.3	0.785225	0.392613	−	0.919704i	$-0.371571\pi$
$811$	18135.3	0.785225	0.392613	+	0.919704i	$0.371571\pi$
$812$	60920.4	2.63286
$813$	0	0
$814$	7393.06	0.318337
$815$	5163.16	0.221911
$816$	0	0
$817$	−5130.61	−0.219703
$818$	74260.0	3.17413
$819$	0	0
$820$	−35242.8	−1.50089
$821$	10789.1	0.458640	0.229320	−	0.973351i	$-0.426350\pi$
$821$	10789.1	0.458640	0.229320	+	0.973351i	$0.426350\pi$
$822$	0	0
$823$	−12284.7	−0.520312	−0.260156	−	0.965567i	$-0.583774\pi$
$823$	−12284.7	−0.520312	−0.260156	+	0.965567i	$0.583774\pi$
$824$	44034.7	1.86168
$825$	0	0
$826$	53510.2	2.25407
$827$	36077.2	1.51696	0.758480	−	0.651696i	$-0.225942\pi$
$827$	36077.2	1.51696	0.758480	+	0.651696i	$0.225942\pi$
$828$	0	0
$829$	−8861.83	−0.371271	−0.185636	−	0.982619i	$-0.559434\pi$
$829$	−8861.83	−0.371271	−0.185636	+	0.982619i	$0.559434\pi$
$830$	−52878.5	−2.21137
$831$	0	0
$832$	0	0
$833$	3603.80	0.149897
$834$	0	0
$835$	8672.18	0.359417
$836$	−68614.1	−2.83860
$837$	0	0
$838$	26551.0	1.09450
$839$	5833.37	0.240036	0.120018	−	0.992772i	$-0.461705\pi$
$839$	5833.37	0.240036	0.120018	+	0.992772i	$0.461705\pi$
$840$	0	0
$841$	34639.6	1.42030
$842$	−4487.66	−0.183676
$843$	0	0
$844$	−4447.37	−0.181380
$845$	0	0
$846$	0	0
$847$	−6271.92	−0.254434
$848$	17426.4	0.705692
$849$	0	0
$850$	−72551.7	−2.92765
$851$	416.466	0.0167759
$852$	0	0
$853$	28649.5	1.14999	0.574995	−	0.818157i	$-0.305004\pi$
$853$	28649.5	1.14999	0.574995	+	0.818157i	$0.305004\pi$
$854$	−18146.0	−0.727098
$855$	0	0
$856$	−2750.71	−0.109833
$857$	−32336.1	−1.28889	−0.644445	−	0.764651i	$-0.722911\pi$
$857$	−32336.1	−1.28889	−0.644445	+	0.764651i	$0.722911\pi$
$858$	0	0
$859$	−13878.5	−0.551254	−0.275627	−	0.961265i	$-0.588885\pi$
$859$	−13878.5	−0.551254	−0.275627	+	0.961265i	$0.588885\pi$
$860$	11562.9	0.458480
$861$	0	0
$862$	−53223.7	−2.10302
$863$	−32934.7	−1.29908	−0.649542	−	0.760326i	$-0.725039\pi$
$863$	−32934.7	−1.29908	−0.649542	+	0.760326i	$0.725039\pi$
$864$	0	0
$865$	−55517.5	−2.18226
$866$	−46424.3	−1.82167
$867$	0	0
$868$	22049.3	0.862213
$869$	3319.57	0.129584
$870$	0	0
$871$	0	0
$872$	22413.5	0.870434
$873$	0	0
$874$	−5872.87	−0.227292
$875$	−68805.4	−2.65834
$876$	0	0
$877$	18375.7	0.707531	0.353765	−	0.935334i	$-0.384901\pi$
$877$	18375.7	0.707531	0.353765	+	0.935334i	$0.384901\pi$
$878$	−50050.2	−1.92382
$879$	0	0
$880$	32590.7	1.24845
$881$	−46883.4	−1.79290	−0.896448	−	0.443150i	$-0.853861\pi$
$881$	−46883.4	−1.79290	−0.896448	+	0.443150i	$0.853861\pi$
$882$	0	0
$883$	1050.05	0.0400194	0.0200097	−	0.999800i	$-0.493630\pi$
$883$	1050.05	0.0400194	0.0200097	+	0.999800i	$0.493630\pi$
$884$	0	0
$885$	0	0
$886$	−27333.7	−1.03645
$887$	−27916.8	−1.05677	−0.528385	−	0.849005i	$-0.677202\pi$
$887$	−27916.8	−1.05677	−0.528385	+	0.849005i	$0.677202\pi$
$888$	0	0
$889$	−34949.4	−1.31852
$890$	−100952.	−3.80217
$891$	0	0
$892$	−27698.0	−1.03968
$893$	54297.4	2.03471
$894$	0	0
$895$	3461.92	0.129295
$896$	−42680.6	−1.59136
$897$	0	0
$898$	57990.8	2.15499
$899$	21364.6	0.792601
$900$	0	0
$901$	−16117.5	−0.595950
$902$	−16059.1	−0.592804
$903$	0	0
$904$	48092.7	1.76940
$905$	70231.3	2.57963
$906$	0	0
$907$	23220.1	0.850066	0.425033	−	0.905178i	$-0.360263\pi$
$907$	23220.1	0.850066	0.425033	+	0.905178i	$0.360263\pi$
$908$	−38289.5	−1.39943
$909$	0	0
$910$	0	0
$911$	−16344.9	−0.594435	−0.297217	−	0.954810i	$-0.596059\pi$
$911$	−16344.9	−0.594435	−0.297217	+	0.954810i	$0.596059\pi$
$912$	0	0
$913$	−15858.0	−0.574834
$914$	79516.7	2.87766
$915$	0	0
$916$	93559.0	3.37476
$917$	13002.1	0.468231
$918$	0	0
$919$	−32931.7	−1.18206	−0.591032	−	0.806648i	$-0.701279\pi$
$919$	−32931.7	−1.18206	−0.591032	+	0.806648i	$0.701279\pi$
$920$	6360.67	0.227940
$921$	0	0
$922$	42800.8	1.52882
$923$	0	0
$924$	0	0
$925$	−16119.4	−0.572975
$926$	−49679.0	−1.76302
$927$	0	0
$928$	−10835.3	−0.383282
$929$	−56305.5	−1.98851	−0.994253	−	0.107056i	$-0.965858\pi$
$929$	−56305.5	−1.98851	−0.994253	+	0.107056i	$0.965858\pi$
$930$	0	0
$931$	−11291.0	−0.397473
$932$	−98063.3	−3.44653
$933$	0	0
$934$	−71258.5	−2.49641
$935$	−30142.7	−1.05430
$936$	0	0
$937$	7095.61	0.247389	0.123695	−	0.992320i	$-0.460526\pi$
$937$	7095.61	0.247389	0.123695	+	0.992320i	$0.460526\pi$
$938$	23284.0	0.810502
$939$	0	0
$940$	−122371.	−4.24606
$941$	−5185.17	−0.179630	−0.0898150	−	0.995958i	$-0.528628\pi$
$941$	−5185.17	−0.179630	−0.0898150	+	0.995958i	$0.528628\pi$
$942$	0	0
$943$	−904.641	−0.0312398
$944$	33966.2	1.17109
$945$	0	0
$946$	5268.89	0.181085
$947$	38451.7	1.31944	0.659722	−	0.751510i	$-0.270674\pi$
$947$	38451.7	1.31944	0.659722	+	0.751510i	$0.270674\pi$
$948$	0	0
$949$	0	0
$950$	227310.	7.76307
$951$	0	0
$952$	−26952.4	−0.917575
$953$	20930.5	0.711445	0.355722	−	0.934592i	$-0.384235\pi$
$953$	20930.5	0.711445	0.355722	+	0.934592i	$0.384235\pi$
$954$	0	0
$955$	−90768.8	−3.07561
$956$	29053.7	0.982910
$957$	0	0
$958$	3072.19	0.103609
$959$	9787.14	0.329555
$960$	0	0
$961$	−22058.4	−0.740438
$962$	0	0
$963$	0	0
$964$	90066.1	3.00916
$965$	−39796.1	−1.32755
$966$	0	0
$967$	50788.1	1.68897	0.844486	−	0.535578i	$-0.179906\pi$
$967$	50788.1	1.68897	0.844486	+	0.535578i	$0.179906\pi$
$968$	−13793.2	−0.457986
$969$	0	0
$970$	49947.6	1.65332
$971$	−15277.5	−0.504922	−0.252461	−	0.967607i	$-0.581240\pi$
$971$	−15277.5	−0.504922	−0.252461	+	0.967607i	$0.581240\pi$
$972$	0	0
$973$	−34745.7	−1.14481
$974$	80356.3	2.64351
$975$	0	0
$976$	−11518.4	−0.377760
$977$	1293.19	0.0423469	0.0211734	−	0.999776i	$-0.493260\pi$
$977$	1293.19	0.0423469	0.0211734	+	0.999776i	$0.493260\pi$
$978$	0	0
$979$	−30275.1	−0.988353
$980$	25446.7	0.829453
$981$	0	0
$982$	12500.8	0.406227
$983$	8474.63	0.274973	0.137487	−	0.990504i	$-0.456098\pi$
$983$	8474.63	0.274973	0.137487	+	0.990504i	$0.456098\pi$
$984$	0	0
$985$	−42211.1	−1.36544
$986$	−54343.0	−1.75521
$987$	0	0
$988$	0	0
$989$	296.807	0.00954289
$990$	0	0
$991$	−7080.71	−0.226969	−0.113484	−	0.993540i	$-0.536201\pi$
$991$	−7080.71	−0.226969	−0.113484	+	0.993540i	$0.536201\pi$
$992$	−3921.68	−0.125518
$993$	0	0
$994$	25928.1	0.827354
$995$	−28512.8	−0.908458
$996$	0	0
$997$	19423.1	0.616985	0.308493	−	0.951227i	$-0.400175\pi$
$997$	19423.1	0.616985	0.308493	+	0.951227i	$0.400175\pi$
$998$	−10564.5	−0.335084
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bf.1.9		9
3.2	odd	2		507.4.a.p.1.1	yes	9
13.12	even	2		1521.4.a.bi.1.1		9
39.5	even	4		507.4.b.k.337.17		18
39.8	even	4		507.4.b.k.337.2		18
39.38	odd	2		507.4.a.o.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.9	✓	9	39.38	odd	2
507.4.a.p.1.1	yes	9	3.2	odd	2
507.4.b.k.337.2		18	39.8	even	4
507.4.b.k.337.17		18	39.5	even	4
1521.4.a.bf.1.9		9	1.1	even	1	trivial
1521.4.a.bi.1.1		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)

\(f(q)\)	\(=\)	\(q+4.83750 q^{2} +15.4014 q^{4} -21.1983 q^{5} +16.2806 q^{7} +35.8043 q^{8} +O(q^{10})\)	\(q+4.83750 q^{2} +15.4014 q^{4} -21.1983 q^{5} +16.2806 q^{7} +35.8043 q^{8} -102.547 q^{10} -30.7532 q^{11} +78.7575 q^{14} +49.9922 q^{16} -46.2371 q^{17} +144.865 q^{19} -326.483 q^{20} -148.769 q^{22} -8.38045 q^{23} +324.366 q^{25} +250.744 q^{28} +242.958 q^{29} +87.9353 q^{31} -44.5973 q^{32} -223.672 q^{34} -345.121 q^{35} -49.6950 q^{37} +700.783 q^{38} -758.990 q^{40} +107.947 q^{41} -35.4166 q^{43} -473.643 q^{44} -40.5404 q^{46} +374.815 q^{47} -77.9418 q^{49} +1569.12 q^{50} +348.583 q^{53} +651.915 q^{55} +582.916 q^{56} +1175.31 q^{58} +679.430 q^{59} -230.403 q^{61} +425.387 q^{62} -615.677 q^{64} +295.642 q^{67} -712.117 q^{68} -1669.52 q^{70} +329.215 q^{71} -48.9973 q^{73} -240.399 q^{74} +2231.12 q^{76} -500.681 q^{77} -107.942 q^{79} -1059.75 q^{80} +522.192 q^{82} +515.654 q^{83} +980.147 q^{85} -171.328 q^{86} -1101.10 q^{88} +984.453 q^{89} -129.071 q^{92} +1813.17 q^{94} -3070.88 q^{95} -487.072 q^{97} -377.043 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) \) 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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