LMFDB - Embedded newform 1521.4.a.bj.1.3

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} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 $ x^9 - x^8 - 48x^7 + 29x^6 + 772x^5 - 150x^4 - 4745x^3 - 966x^2 + 9428*x + 5144
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			1.3
Root			$2.86460$ 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)

$f(q)$	$=$	$q-1.86460 q^{2} -4.52327 q^{4} -2.36060 q^{5} +4.86461 q^{7} +23.3509 q^{8} +O(q^{10})$	$q-1.86460 q^{2} -4.52327 q^{4} -2.36060 q^{5} +4.86461 q^{7} +23.3509 q^{8} +4.40158 q^{10} -35.1633 q^{11} -9.07054 q^{14} -7.35391 q^{16} +33.1102 q^{17} +104.145 q^{19} +10.6776 q^{20} +65.5656 q^{22} +86.3184 q^{23} -119.428 q^{25} -22.0039 q^{28} +118.190 q^{29} -262.589 q^{31} -173.095 q^{32} -61.7373 q^{34} -11.4834 q^{35} +59.6265 q^{37} -194.189 q^{38} -55.1222 q^{40} -76.9932 q^{41} -344.731 q^{43} +159.053 q^{44} -160.949 q^{46} +415.931 q^{47} -319.336 q^{49} +222.685 q^{50} -141.710 q^{53} +83.0067 q^{55} +113.593 q^{56} -220.376 q^{58} +598.746 q^{59} +791.010 q^{61} +489.624 q^{62} +381.584 q^{64} -22.1123 q^{67} -149.766 q^{68} +21.4120 q^{70} -599.641 q^{71} -776.246 q^{73} -111.180 q^{74} -471.076 q^{76} -171.056 q^{77} -1276.41 q^{79} +17.3597 q^{80} +143.562 q^{82} +493.338 q^{83} -78.1601 q^{85} +642.785 q^{86} -821.095 q^{88} +1251.31 q^{89} -390.441 q^{92} -775.545 q^{94} -245.845 q^{95} +76.4923 q^{97} +595.433 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8}+O(q^{10}) $ 9 * q + 8 * q^2 + 32 * q^4 + 41 * q^5 - q^7 + 111 * q^8	$ 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8} + 198 q^{10} + 37 q^{11} - 98 q^{14} + 32 q^{16} + 134 q^{17} + 72 q^{19} + 356 q^{20} + 274 q^{22} - 226 q^{23} + 612 q^{25} - 132 q^{28} + 547 q^{29} + 521 q^{31} + 721 q^{32} + 100 q^{34} - 138 q^{35} - 584 q^{37} + 416 q^{38} + 1342 q^{40} + 482 q^{41} + 158 q^{43} + 1453 q^{44} - 1537 q^{46} + 1500 q^{47} + 642 q^{49} + 2777 q^{50} - 1399 q^{53} - 1408 q^{55} + 616 q^{56} - 1455 q^{58} + 1541 q^{59} + 2092 q^{61} + 293 q^{62} + 2481 q^{64} - 252 q^{67} + 1579 q^{68} - 2492 q^{70} + 2352 q^{71} - 903 q^{73} - 1037 q^{74} + 485 q^{76} + 1686 q^{77} - 115 q^{79} + 5701 q^{80} - 5147 q^{82} + 1207 q^{83} - 4308 q^{85} + 5691 q^{86} - 484 q^{88} + 2336 q^{89} - 2087 q^{92} - 468 q^{94} + 222 q^{95} - 2155 q^{97} + 5593 q^{98}+O(q^{100}) $ 9 * q + 8 * q^2 + 32 * q^4 + 41 * q^5 - q^7 + 111 * q^8 + 198 * q^10 + 37 * q^11 - 98 * q^14 + 32 * q^16 + 134 * q^17 + 72 * q^19 + 356 * q^20 + 274 * q^22 - 226 * q^23 + 612 * q^25 - 132 * q^28 + 547 * q^29 + 521 * q^31 + 721 * q^32 + 100 * q^34 - 138 * q^35 - 584 * q^37 + 416 * q^38 + 1342 * q^40 + 482 * q^41 + 158 * q^43 + 1453 * q^44 - 1537 * q^46 + 1500 * q^47 + 642 * q^49 + 2777 * q^50 - 1399 * q^53 - 1408 * q^55 + 616 * q^56 - 1455 * q^58 + 1541 * q^59 + 2092 * q^61 + 293 * q^62 + 2481 * q^64 - 252 * q^67 + 1579 * q^68 - 2492 * q^70 + 2352 * q^71 - 903 * q^73 - 1037 * q^74 + 485 * q^76 + 1686 * q^77 - 115 * q^79 + 5701 * q^80 - 5147 * q^82 + 1207 * q^83 - 4308 * q^85 + 5691 * q^86 - 484 * q^88 + 2336 * q^89 - 2087 * q^92 - 468 * q^94 + 222 * q^95 - 2155 * q^97 + 5593 * 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 $

$2$	−1.86460	−0.659236	−0.329618	−	0.944114i	$-0.606920\pi$
$2$	−1.86460	−0.659236	−0.329618	+	0.944114i	$0.606920\pi$
$3$	0	0
$3$	0	0
$4$	−4.52327	−0.565408
$5$	−2.36060	−0.211139	−0.105569	−	0.994412i	$-0.533667\pi$
$5$	−2.36060	−0.211139	−0.105569	+	0.994412i	$0.533667\pi$
$6$	0	0
$7$	4.86461	0.262664	0.131332	−	0.991338i	$-0.458075\pi$
$7$	4.86461	0.262664	0.131332	+	0.991338i	$0.458075\pi$
$8$	23.3509	1.03197
$9$	0	0
$10$	4.40158	0.139190
$11$	−35.1633	−0.963832	−0.481916	−	0.876218i	$-0.660059\pi$
$11$	−35.1633	−0.963832	−0.481916	+	0.876218i	$0.660059\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−9.07054	−0.173157
$15$	0	0
$16$	−7.35391	−0.114905
$17$	33.1102	0.472377	0.236189	−	0.971707i	$-0.424102\pi$
$17$	33.1102	0.472377	0.236189	+	0.971707i	$0.424102\pi$
$18$	0	0
$19$	104.145	1.25750	0.628751	−	0.777607i	$-0.283566\pi$
$19$	104.145	1.25750	0.628751	+	0.777607i	$0.283566\pi$
$20$	10.6776	0.119380
$21$	0	0
$22$	65.5656	0.635392
$23$	86.3184	0.782549	0.391275	−	0.920274i	$-0.372034\pi$
$23$	86.3184	0.782549	0.391275	+	0.920274i	$0.372034\pi$
$24$	0	0
$25$	−119.428	−0.955420
$26$	0	0
$27$	0	0
$28$	−22.0039	−0.148512
$29$	118.190	0.756802	0.378401	−	0.925642i	$-0.376474\pi$
$29$	118.190	0.756802	0.378401	+	0.925642i	$0.376474\pi$
$30$	0	0
$31$	−262.589	−1.52137	−0.760684	−	0.649122i	$-0.775136\pi$
$31$	−262.589	−1.52137	−0.760684	+	0.649122i	$0.775136\pi$
$32$	−173.095	−0.956224
$33$	0	0
$34$	−61.7373	−0.311408
$35$	−11.4834	−0.0554586
$36$	0	0
$37$	59.6265	0.264933	0.132467	−	0.991187i	$-0.457710\pi$
$37$	59.6265	0.264933	0.132467	+	0.991187i	$0.457710\pi$
$38$	−194.189	−0.828990
$39$	0	0
$40$	−55.1222	−0.217890
$41$	−76.9932	−0.293276	−0.146638	−	0.989190i	$-0.546845\pi$
$41$	−76.9932	−0.293276	−0.146638	+	0.989190i	$0.546845\pi$
$42$	0	0
$43$	−344.731	−1.22258	−0.611290	−	0.791406i	$-0.709349\pi$
$43$	−344.731	−1.22258	−0.611290	+	0.791406i	$0.709349\pi$
$44$	159.053	0.544959
$45$	0	0
$46$	−160.949	−0.515884
$47$	415.931	1.29085	0.645423	−	0.763825i	$-0.276681\pi$
$47$	415.931	1.29085	0.645423	+	0.763825i	$0.276681\pi$
$48$	0	0
$49$	−319.336	−0.931008
$50$	222.685	0.629847
$51$	0	0
$52$	0	0
$53$	−141.710	−0.367271	−0.183636	−	0.982994i	$-0.558787\pi$
$53$	−141.710	−0.367271	−0.183636	+	0.982994i	$0.558787\pi$
$54$	0	0
$55$	83.0067	0.203502
$56$	113.593	0.271062
$57$	0	0
$58$	−220.376	−0.498911
$59$	598.746	1.32119	0.660594	−	0.750744i	$-0.270305\pi$
$59$	598.746	1.32119	0.660594	+	0.750744i	$0.270305\pi$
$60$	0	0
$61$	791.010	1.66030	0.830151	−	0.557539i	$-0.188254\pi$
$61$	791.010	1.66030	0.830151	+	0.557539i	$0.188254\pi$
$62$	489.624	1.00294
$63$	0	0
$64$	381.584	0.745281
$65$	0	0
$66$	0	0
$67$	−22.1123	−0.0403201	−0.0201601	−	0.999797i	$-0.506418\pi$
$67$	−22.1123	−0.0403201	−0.0201601	+	0.999797i	$0.506418\pi$
$68$	−149.766	−0.267086
$69$	0	0
$70$	21.4120	0.0365603
$71$	−599.641	−1.00231	−0.501157	−	0.865356i	$-0.667092\pi$
$71$	−599.641	−1.00231	−0.501157	+	0.865356i	$0.667092\pi$
$72$	0	0
$73$	−776.246	−1.24456	−0.622279	−	0.782795i	$-0.713793\pi$
$73$	−776.246	−1.24456	−0.622279	+	0.782795i	$0.713793\pi$
$74$	−111.180	−0.174653
$75$	0	0
$76$	−471.076	−0.711002
$77$	−171.056	−0.253164
$78$	0	0
$79$	−1276.41	−1.81781	−0.908904	−	0.417006i	$-0.863080\pi$
$79$	−1276.41	−1.81781	−0.908904	+	0.417006i	$0.863080\pi$
$80$	17.3597	0.0242609
$81$	0	0
$82$	143.562	0.193338
$83$	493.338	0.652420	0.326210	−	0.945297i	$-0.394228\pi$
$83$	493.338	0.652420	0.326210	+	0.945297i	$0.394228\pi$
$84$	0	0
$85$	−78.1601	−0.0997371
$86$	642.785	0.805969
$87$	0	0
$88$	−821.095	−0.994648
$89$	1251.31	1.49032	0.745161	−	0.666885i	$-0.232373\pi$
$89$	1251.31	1.49032	0.745161	+	0.666885i	$0.232373\pi$
$90$	0	0
$91$	0	0
$92$	−390.441	−0.442460
$93$	0	0
$94$	−775.545	−0.850972
$95$	−245.845	−0.265507
$96$	0	0
$97$	76.4923	0.0800682	0.0400341	−	0.999198i	$-0.487253\pi$
$97$	76.4923	0.0800682	0.0400341	+	0.999198i	$0.487253\pi$
$98$	595.433	0.613753
$99$	0	0
$100$	540.203	0.540203
$101$	−1218.67	−1.20062	−0.600310	−	0.799768i	$-0.704956\pi$
$101$	−1218.67	−1.20062	−0.600310	+	0.799768i	$0.704956\pi$
$102$	0	0
$103$	−989.436	−0.946524	−0.473262	−	0.880922i	$-0.656924\pi$
$103$	−989.436	−0.946524	−0.473262	+	0.880922i	$0.656924\pi$
$104$	0	0
$105$	0	0
$106$	264.232	0.242118
$107$	1126.11	1.01743	0.508717	−	0.860934i	$-0.330120\pi$
$107$	1126.11	1.01743	0.508717	+	0.860934i	$0.330120\pi$
$108$	0	0
$109$	44.7158	0.0392935	0.0196468	−	0.999807i	$-0.493746\pi$
$109$	44.7158	0.0392935	0.0196468	+	0.999807i	$0.493746\pi$
$110$	−154.774	−0.134156
$111$	0	0
$112$	−35.7739	−0.0301814
$113$	−1570.11	−1.30711	−0.653554	−	0.756880i	$-0.726723\pi$
$113$	−1570.11	−1.30711	−0.653554	+	0.756880i	$0.726723\pi$
$114$	0	0
$115$	−203.764	−0.165226
$116$	−534.603	−0.427902
$117$	0	0
$118$	−1116.42	−0.870974
$119$	161.068	0.124076
$120$	0	0
$121$	−94.5390	−0.0710285
$122$	−1474.92	−1.09453
$123$	0	0
$124$	1187.76	0.860194
$125$	576.996	0.412865
$126$	0	0
$127$	691.316	0.483027	0.241513	−	0.970397i	$-0.422356\pi$
$127$	691.316	0.483027	0.241513	+	0.970397i	$0.422356\pi$
$128$	673.258	0.464907
$129$	0	0
$130$	0	0
$131$	−2768.65	−1.84655	−0.923274	−	0.384143i	$-0.874497\pi$
$131$	−2768.65	−1.84655	−0.923274	+	0.384143i	$0.874497\pi$
$132$	0	0
$133$	506.625	0.330301
$134$	41.2306	0.0265804
$135$	0	0
$136$	773.153	0.487480
$137$	1345.44	0.839041	0.419520	−	0.907746i	$-0.362198\pi$
$137$	1345.44	0.839041	0.419520	+	0.907746i	$0.362198\pi$
$138$	0	0
$139$	1859.48	1.13467	0.567334	−	0.823488i	$-0.307975\pi$
$139$	1859.48	1.13467	0.567334	+	0.823488i	$0.307975\pi$
$140$	51.9425	0.0313567
$141$	0	0
$142$	1118.09	0.660761
$143$	0	0
$144$	0	0
$145$	−278.999	−0.159790
$146$	1447.39	0.820457
$147$	0	0
$148$	−269.706	−0.149796
$149$	1425.71	0.783885	0.391942	−	0.919990i	$-0.371803\pi$
$149$	1425.71	0.783885	0.391942	+	0.919990i	$0.371803\pi$
$150$	0	0
$151$	1692.79	0.912299	0.456149	−	0.889903i	$-0.349228\pi$
$151$	1692.79	0.912299	0.456149	+	0.889903i	$0.349228\pi$
$152$	2431.88	1.29771
$153$	0	0
$154$	318.951	0.166895
$155$	619.869	0.321220
$156$	0	0
$157$	−527.380	−0.268086	−0.134043	−	0.990976i	$-0.542796\pi$
$157$	−527.380	−0.268086	−0.134043	+	0.990976i	$0.542796\pi$
$158$	2379.99	1.19836
$159$	0	0
$160$	408.609	0.201896
$161$	419.905	0.205548
$162$	0	0
$163$	2970.91	1.42760	0.713802	−	0.700348i	$-0.246972\pi$
$163$	2970.91	1.42760	0.713802	+	0.700348i	$0.246972\pi$
$164$	348.261	0.165821
$165$	0	0
$166$	−919.877	−0.430098
$167$	−406.803	−0.188499	−0.0942497	−	0.995549i	$-0.530045\pi$
$167$	−406.803	−0.188499	−0.0942497	+	0.995549i	$0.530045\pi$
$168$	0	0
$169$	0	0
$170$	145.737	0.0657503
$171$	0	0
$172$	1559.31	0.691257
$173$	2892.70	1.27126	0.635629	−	0.771994i	$-0.280741\pi$
$173$	2892.70	1.27126	0.635629	+	0.771994i	$0.280741\pi$
$174$	0	0
$175$	−580.968	−0.250955
$176$	258.588	0.110749
$177$	0	0
$178$	−2333.19	−0.982473
$179$	−3772.31	−1.57517	−0.787586	−	0.616204i	$-0.788670\pi$
$179$	−3772.31	−1.57517	−0.787586	+	0.616204i	$0.788670\pi$
$180$	0	0
$181$	−1673.88	−0.687395	−0.343698	−	0.939080i	$-0.611679\pi$
$181$	−1673.88	−0.687395	−0.343698	+	0.939080i	$0.611679\pi$
$182$	0	0
$183$	0	0
$184$	2015.61	0.807570
$185$	−140.754	−0.0559377
$186$	0	0
$187$	−1164.27	−0.455292
$188$	−1881.37	−0.729856
$189$	0	0
$190$	458.403	0.175032
$191$	5024.84	1.90359	0.951793	−	0.306740i	$-0.0992383\pi$
$191$	5024.84	1.90359	0.951793	+	0.306740i	$0.0992383\pi$
$192$	0	0
$193$	2398.57	0.894575	0.447287	−	0.894390i	$-0.352390\pi$
$193$	2398.57	0.894575	0.447287	+	0.894390i	$0.352390\pi$
$194$	−142.628	−0.0527838
$195$	0	0
$196$	1444.44	0.526400
$197$	4579.71	1.65630	0.828150	−	0.560507i	$-0.189394\pi$
$197$	4579.71	1.65630	0.828150	+	0.560507i	$0.189394\pi$
$198$	0	0
$199$	177.539	0.0632431	0.0316215	−	0.999500i	$-0.489933\pi$
$199$	177.539	0.0632431	0.0316215	+	0.999500i	$0.489933\pi$
$200$	−2788.74	−0.985968
$201$	0	0
$202$	2272.34	0.791491
$203$	574.946	0.198785
$204$	0	0
$205$	181.750	0.0619220
$206$	1844.90	0.623983
$207$	0	0
$208$	0	0
$209$	−3662.09	−1.21202
$210$	0	0
$211$	4363.80	1.42378	0.711888	−	0.702293i	$-0.247841\pi$
$211$	4363.80	1.42378	0.711888	+	0.702293i	$0.247841\pi$
$212$	640.992	0.207658
$213$	0	0
$214$	−2099.75	−0.670729
$215$	813.773	0.258134
$216$	0	0
$217$	−1277.39	−0.399609
$218$	−83.3770	−0.0259037
$219$	0	0
$220$	−375.462	−0.115062
$221$	0	0
$222$	0	0
$223$	−1691.78	−0.508026	−0.254013	−	0.967201i	$-0.581751\pi$
$223$	−1691.78	−0.508026	−0.254013	+	0.967201i	$0.581751\pi$
$224$	−842.039	−0.251166
$225$	0	0
$226$	2927.62	0.861692
$227$	3553.37	1.03897	0.519484	−	0.854480i	$-0.326124\pi$
$227$	3553.37	1.03897	0.519484	+	0.854480i	$0.326124\pi$
$228$	0	0
$229$	3780.28	1.09086	0.545432	−	0.838155i	$-0.316365\pi$
$229$	3780.28	1.09086	0.545432	+	0.838155i	$0.316365\pi$
$230$	379.937	0.108923
$231$	0	0
$232$	2759.83	0.780999
$233$	6588.54	1.85249	0.926243	−	0.376926i	$-0.123019\pi$
$233$	6588.54	1.85249	0.926243	+	0.376926i	$0.123019\pi$
$234$	0	0
$235$	−981.849	−0.272548
$236$	−2708.29	−0.747010
$237$	0	0
$238$	−300.328	−0.0817956
$239$	−3252.33	−0.880233	−0.440117	−	0.897941i	$-0.645063\pi$
$239$	−3252.33	−0.880233	−0.440117	+	0.897941i	$0.645063\pi$
$240$	0	0
$241$	−3173.99	−0.848360	−0.424180	−	0.905578i	$-0.639438\pi$
$241$	−3173.99	−0.848360	−0.424180	+	0.905578i	$0.639438\pi$
$242$	176.277	0.0468245
$243$	0	0
$244$	−3577.95	−0.938749
$245$	753.825	0.196572
$246$	0	0
$247$	0	0
$248$	−6131.69	−1.57001
$249$	0	0
$250$	−1075.87	−0.272175
$251$	1493.79	0.375647	0.187823	−	0.982203i	$-0.439857\pi$
$251$	1493.79	0.375647	0.187823	+	0.982203i	$0.439857\pi$
$252$	0	0
$253$	−3035.24	−0.754246
$254$	−1289.03	−0.318428
$255$	0	0
$256$	−4308.03	−1.05177
$257$	−2979.55	−0.723188	−0.361594	−	0.932336i	$-0.617767\pi$
$257$	−2979.55	−0.723188	−0.361594	+	0.932336i	$0.617767\pi$
$258$	0	0
$259$	290.059	0.0695884
$260$	0	0
$261$	0	0
$262$	5162.42	1.21731
$263$	−1090.01	−0.255562	−0.127781	−	0.991802i	$-0.540786\pi$
$263$	−1090.01	−0.255562	−0.127781	+	0.991802i	$0.540786\pi$
$264$	0	0
$265$	334.521	0.0775451
$266$	−944.653	−0.217746
$267$	0	0
$268$	100.020	0.0227973
$269$	4586.58	1.03959	0.519793	−	0.854292i	$-0.326009\pi$
$269$	4586.58	1.03959	0.519793	+	0.854292i	$0.326009\pi$
$270$	0	0
$271$	5480.65	1.22851	0.614254	−	0.789108i	$-0.289457\pi$
$271$	5480.65	1.22851	0.614254	+	0.789108i	$0.289457\pi$
$272$	−243.490	−0.0542784
$273$	0	0
$274$	−2508.71	−0.553126
$275$	4199.47	0.920864
$276$	0	0
$277$	3856.51	0.836518	0.418259	−	0.908328i	$-0.362640\pi$
$277$	3856.51	0.836518	0.418259	+	0.908328i	$0.362640\pi$
$278$	−3467.18	−0.748013
$279$	0	0
$280$	−268.148	−0.0572317
$281$	5331.62	1.13188	0.565939	−	0.824447i	$-0.308514\pi$
$281$	5331.62	1.13188	0.565939	+	0.824447i	$0.308514\pi$
$282$	0	0
$283$	5430.68	1.14071	0.570354	−	0.821399i	$-0.306806\pi$
$283$	5430.68	1.14071	0.570354	+	0.821399i	$0.306806\pi$
$284$	2712.34	0.566717
$285$	0	0
$286$	0	0
$287$	−374.542	−0.0770331
$288$	0	0
$289$	−3816.71	−0.776860
$290$	520.221	0.105339
$291$	0	0
$292$	3511.17	0.703684
$293$	−950.151	−0.189449	−0.0947243	−	0.995504i	$-0.530197\pi$
$293$	−950.151	−0.189449	−0.0947243	+	0.995504i	$0.530197\pi$
$294$	0	0
$295$	−1413.40	−0.278954
$296$	1392.33	0.273404
$297$	0	0
$298$	−2658.38	−0.516765
$299$	0	0
$300$	0	0
$301$	−1676.98	−0.321128
$302$	−3156.37	−0.601420
$303$	0	0
$304$	−765.874	−0.144493
$305$	−1867.26	−0.350554
$306$	0	0
$307$	9212.70	1.71269	0.856346	−	0.516403i	$-0.172729\pi$
$307$	9212.70	1.71269	0.856346	+	0.516403i	$0.172729\pi$
$308$	773.731	0.143141
$309$	0	0
$310$	−1155.81	−0.211759
$311$	−2395.47	−0.436767	−0.218383	−	0.975863i	$-0.570078\pi$
$311$	−2395.47	−0.436767	−0.218383	+	0.975863i	$0.570078\pi$
$312$	0	0
$313$	−3760.86	−0.679157	−0.339578	−	0.940578i	$-0.610284\pi$
$313$	−3760.86	−0.679157	−0.339578	+	0.940578i	$0.610284\pi$
$314$	983.353	0.176732
$315$	0	0
$316$	5773.52	1.02780
$317$	1509.19	0.267396	0.133698	−	0.991022i	$-0.457315\pi$
$317$	1509.19	0.267396	0.133698	+	0.991022i	$0.457315\pi$
$318$	0	0
$319$	−4155.94	−0.729430
$320$	−900.769	−0.157358
$321$	0	0
$322$	−782.955	−0.135504
$323$	3448.27	0.594015
$324$	0	0
$325$	0	0
$326$	−5539.55	−0.941127
$327$	0	0
$328$	−1797.86	−0.302653
$329$	2023.34	0.339059
$330$	0	0
$331$	−1508.07	−0.250426	−0.125213	−	0.992130i	$-0.539961\pi$
$331$	−1508.07	−0.250426	−0.125213	+	0.992130i	$0.539961\pi$
$332$	−2231.50	−0.368884
$333$	0	0
$334$	758.525	0.124265
$335$	52.1984	0.00851314
$336$	0	0
$337$	8030.49	1.29807	0.649033	−	0.760760i	$-0.275173\pi$
$337$	8030.49	1.29807	0.649033	+	0.760760i	$0.275173\pi$
$338$	0	0
$339$	0	0
$340$	353.539	0.0563922
$341$	9233.51	1.46634
$342$	0	0
$343$	−3222.00	−0.507206
$344$	−8049.77	−1.26167
$345$	0	0
$346$	−5393.72	−0.838059
$347$	−6414.73	−0.992394	−0.496197	−	0.868210i	$-0.665271\pi$
$347$	−6414.73	−0.992394	−0.496197	+	0.868210i	$0.665271\pi$
$348$	0	0
$349$	−3381.63	−0.518667	−0.259333	−	0.965788i	$-0.583503\pi$
$349$	−3381.63	−0.518667	−0.259333	+	0.965788i	$0.583503\pi$
$350$	1083.27	0.165438
$351$	0	0
$352$	6086.60	0.921639
$353$	6521.67	0.983325	0.491662	−	0.870786i	$-0.336389\pi$
$353$	6521.67	0.983325	0.491662	+	0.870786i	$0.336389\pi$
$354$	0	0
$355$	1415.52	0.211627
$356$	−5660.01	−0.842640
$357$	0	0
$358$	7033.85	1.03841
$359$	2259.11	0.332121	0.166060	−	0.986116i	$-0.446895\pi$
$359$	2259.11	0.332121	0.166060	+	0.986116i	$0.446895\pi$
$360$	0	0
$361$	3987.22	0.581312
$362$	3121.12	0.453155
$363$	0	0
$364$	0	0
$365$	1832.41	0.262775
$366$	0	0
$367$	3404.86	0.484284	0.242142	−	0.970241i	$-0.422150\pi$
$367$	3404.86	0.484284	0.242142	+	0.970241i	$0.422150\pi$
$368$	−634.778	−0.0899187
$369$	0	0
$370$	262.451	0.0368761
$371$	−689.363	−0.0964689
$372$	0	0
$373$	11445.0	1.58874	0.794372	−	0.607431i	$-0.207800\pi$
$373$	11445.0	1.58874	0.794372	+	0.607431i	$0.207800\pi$
$374$	2170.89	0.300145
$375$	0	0
$376$	9712.36	1.33212
$377$	0	0
$378$	0	0
$379$	3587.91	0.486276	0.243138	−	0.969992i	$-0.421823\pi$
$379$	3587.91	0.486276	0.243138	+	0.969992i	$0.421823\pi$
$380$	1112.02	0.150120
$381$	0	0
$382$	−9369.32	−1.25491
$383$	11338.6	1.51273	0.756367	−	0.654148i	$-0.226972\pi$
$383$	11338.6	1.51273	0.756367	+	0.654148i	$0.226972\pi$
$384$	0	0
$385$	403.795	0.0534527
$386$	−4472.37	−0.589735
$387$	0	0
$388$	−345.995	−0.0452713
$389$	−4424.87	−0.576735	−0.288367	−	0.957520i	$-0.593112\pi$
$389$	−4424.87	−0.576735	−0.288367	+	0.957520i	$0.593112\pi$
$390$	0	0
$391$	2858.02	0.369658
$392$	−7456.77	−0.960775
$393$	0	0
$394$	−8539.33	−1.09189
$395$	3013.09	0.383810
$396$	0	0
$397$	−11640.3	−1.47156	−0.735780	−	0.677221i	$-0.763184\pi$
$397$	−11640.3	−1.47156	−0.735780	+	0.677221i	$0.763184\pi$
$398$	−331.038	−0.0416921
$399$	0	0
$400$	878.260	0.109782
$401$	10195.9	1.26972	0.634861	−	0.772626i	$-0.281057\pi$
$401$	10195.9	1.26972	0.634861	+	0.772626i	$0.281057\pi$
$402$	0	0
$403$	0	0
$404$	5512.39	0.678840
$405$	0	0
$406$	−1072.04	−0.131046
$407$	−2096.67	−0.255351
$408$	0	0
$409$	2195.22	0.265395	0.132697	−	0.991157i	$-0.457636\pi$
$409$	2195.22	0.265395	0.132697	+	0.991157i	$0.457636\pi$
$410$	−338.892	−0.0408212
$411$	0	0
$412$	4475.48	0.535173
$413$	2912.66	0.347028
$414$	0	0
$415$	−1164.57	−0.137751
$416$	0	0
$417$	0	0
$418$	6828.34	0.799007
$419$	7413.03	0.864320	0.432160	−	0.901797i	$-0.357752\pi$
$419$	7413.03	0.864320	0.432160	+	0.901797i	$0.357752\pi$
$420$	0	0
$421$	−2671.47	−0.309263	−0.154631	−	0.987972i	$-0.549419\pi$
$421$	−2671.47	−0.309263	−0.154631	+	0.987972i	$0.549419\pi$
$422$	−8136.75	−0.938603
$423$	0	0
$424$	−3309.05	−0.379014
$425$	−3954.27	−0.451319
$426$	0	0
$427$	3847.95	0.436101
$428$	−5093.71	−0.575266
$429$	0	0
$430$	−1517.36	−0.170171
$431$	−1243.06	−0.138924	−0.0694619	−	0.997585i	$-0.522128\pi$
$431$	−1243.06	−0.138924	−0.0694619	+	0.997585i	$0.522128\pi$
$432$	0	0
$433$	−965.341	−0.107139	−0.0535697	−	0.998564i	$-0.517060\pi$
$433$	−965.341	−0.107139	−0.0535697	+	0.998564i	$0.517060\pi$
$434$	2381.83	0.263436
$435$	0	0
$436$	−202.261	−0.0222169
$437$	8989.65	0.984057
$438$	0	0
$439$	−5443.94	−0.591857	−0.295928	−	0.955210i	$-0.595629\pi$
$439$	−5443.94	−0.591857	−0.295928	+	0.955210i	$0.595629\pi$
$440$	1938.28	0.210009
$441$	0	0
$442$	0	0
$443$	9917.30	1.06362	0.531812	−	0.846862i	$-0.321511\pi$
$443$	9917.30	1.06362	0.531812	+	0.846862i	$0.321511\pi$
$444$	0	0
$445$	−2953.85	−0.314665
$446$	3154.49	0.334909
$447$	0	0
$448$	1856.26	0.195759
$449$	6519.89	0.685284	0.342642	−	0.939466i	$-0.388678\pi$
$449$	6519.89	0.685284	0.342642	+	0.939466i	$0.388678\pi$
$450$	0	0
$451$	2707.34	0.282669
$452$	7102.01	0.739050
$453$	0	0
$454$	−6625.62	−0.684924
$455$	0	0
$456$	0	0
$457$	−8414.81	−0.861331	−0.430665	−	0.902512i	$-0.641721\pi$
$457$	−8414.81	−0.861331	−0.430665	+	0.902512i	$0.641721\pi$
$458$	−7048.71	−0.719137
$459$	0	0
$460$	921.677	0.0934204
$461$	−3286.81	−0.332065	−0.166033	−	0.986120i	$-0.553096\pi$
$461$	−3286.81	−0.332065	−0.166033	+	0.986120i	$0.553096\pi$
$462$	0	0
$463$	−10342.8	−1.03817	−0.519085	−	0.854723i	$-0.673727\pi$
$463$	−10342.8	−1.03817	−0.519085	+	0.854723i	$0.673727\pi$
$464$	−869.156	−0.0869603
$465$	0	0
$466$	−12285.0	−1.22122
$467$	6152.69	0.609662	0.304831	−	0.952406i	$-0.401400\pi$
$467$	6152.69	0.609662	0.304831	+	0.952406i	$0.401400\pi$
$468$	0	0
$469$	−107.568	−0.0105906
$470$	1830.75	0.179673
$471$	0	0
$472$	13981.2	1.36343
$473$	12121.9	1.17836
$474$	0	0
$475$	−12437.8	−1.20144
$476$	−728.555	−0.0701539
$477$	0	0
$478$	6064.30	0.580281
$479$	−18873.6	−1.80032	−0.900161	−	0.435557i	$-0.856552\pi$
$479$	−18873.6	−1.80032	−0.900161	+	0.435557i	$0.856552\pi$
$480$	0	0
$481$	0	0
$482$	5918.22	0.559269
$483$	0	0
$484$	427.625	0.0401601
$485$	−180.568	−0.0169055
$486$	0	0
$487$	8546.70	0.795252	0.397626	−	0.917548i	$-0.369834\pi$
$487$	8546.70	0.795252	0.397626	+	0.917548i	$0.369834\pi$
$488$	18470.8	1.71339
$489$	0	0
$490$	−1405.58	−0.129587
$491$	−861.293	−0.0791642	−0.0395821	−	0.999216i	$-0.512603\pi$
$491$	−861.293	−0.0791642	−0.0395821	+	0.999216i	$0.512603\pi$
$492$	0	0
$493$	3913.29	0.357496
$494$	0	0
$495$	0	0
$496$	1931.06	0.174813
$497$	−2917.02	−0.263272
$498$	0	0
$499$	3549.02	0.318389	0.159194	−	0.987247i	$-0.449110\pi$
$499$	3549.02	0.318389	0.159194	+	0.987247i	$0.449110\pi$
$500$	−2609.91	−0.233437
$501$	0	0
$502$	−2785.33	−0.247640
$503$	−4578.66	−0.405870	−0.202935	−	0.979192i	$-0.565048\pi$
$503$	−4578.66	−0.405870	−0.202935	+	0.979192i	$0.565048\pi$
$504$	0	0
$505$	2876.81	0.253497
$506$	5659.52	0.497226
$507$	0	0
$508$	−3127.01	−0.273107
$509$	−11402.5	−0.992945	−0.496473	−	0.868052i	$-0.665372\pi$
$509$	−11402.5	−0.992945	−0.496473	+	0.868052i	$0.665372\pi$
$510$	0	0
$511$	−3776.13	−0.326901
$512$	2646.69	0.228453
$513$	0	0
$514$	5555.67	0.476751
$515$	2335.67	0.199848
$516$	0	0
$517$	−14625.5	−1.24416
$518$	−540.844	−0.0458752
$519$	0	0
$520$	0	0
$521$	−4027.26	−0.338651	−0.169326	−	0.985560i	$-0.554159\pi$
$521$	−4027.26	−0.338651	−0.169326	+	0.985560i	$0.554159\pi$
$522$	0	0
$523$	2560.78	0.214102	0.107051	−	0.994254i	$-0.465859\pi$
$523$	2560.78	0.214102	0.107051	+	0.994254i	$0.465859\pi$
$524$	12523.3	1.04405
$525$	0	0
$526$	2032.43	0.168476
$527$	−8694.39	−0.718660
$528$	0	0
$529$	−4716.13	−0.387617
$530$	−623.748	−0.0511205
$531$	0	0
$532$	−2291.60	−0.186755
$533$	0	0
$534$	0	0
$535$	−2658.31	−0.214820
$536$	−516.342	−0.0416093
$537$	0	0
$538$	−8552.13	−0.685332
$539$	11228.9	0.897335
$540$	0	0
$541$	−15461.8	−1.22875	−0.614376	−	0.789013i	$-0.710592\pi$
$541$	−15461.8	−1.22875	−0.614376	+	0.789013i	$0.710592\pi$
$542$	−10219.2	−0.809876
$543$	0	0
$544$	−5731.22	−0.451698
$545$	−105.556	−0.00829639
$546$	0	0
$547$	−5589.73	−0.436928	−0.218464	−	0.975845i	$-0.570105\pi$
$547$	−5589.73	−0.436928	−0.218464	+	0.975845i	$0.570105\pi$
$548$	−6085.78	−0.474401
$549$	0	0
$550$	−7830.34	−0.607067
$551$	12308.9	0.951680
$552$	0	0
$553$	−6209.21	−0.477473
$554$	−7190.85	−0.551462
$555$	0	0
$556$	−8410.91	−0.641550
$557$	−15283.9	−1.16265	−0.581327	−	0.813670i	$-0.697466\pi$
$557$	−15283.9	−1.16265	−0.581327	+	0.813670i	$0.697466\pi$
$558$	0	0
$559$	0	0
$560$	84.4479	0.00637246
$561$	0	0
$562$	−9941.34	−0.746175
$563$	−20834.6	−1.55964	−0.779818	−	0.626007i	$-0.784688\pi$
$563$	−20834.6	−1.55964	−0.779818	+	0.626007i	$0.784688\pi$
$564$	0	0
$565$	3706.40	0.275981
$566$	−10126.0	−0.751995
$567$	0	0
$568$	−14002.2	−1.03436
$569$	2621.05	0.193111	0.0965556	−	0.995328i	$-0.469217\pi$
$569$	2621.05	0.193111	0.0965556	+	0.995328i	$0.469217\pi$
$570$	0	0
$571$	−5483.96	−0.401921	−0.200960	−	0.979599i	$-0.564406\pi$
$571$	−5483.96	−0.401921	−0.200960	+	0.979599i	$0.564406\pi$
$572$	0	0
$573$	0	0
$574$	698.370	0.0507830
$575$	−10308.8	−0.747664
$576$	0	0
$577$	26965.7	1.94558	0.972789	−	0.231692i	$-0.0744262\pi$
$577$	26965.7	1.94558	0.972789	+	0.231692i	$0.0744262\pi$
$578$	7116.64	0.512134
$579$	0	0
$580$	1261.99	0.0903468
$581$	2399.89	0.171367
$582$	0	0
$583$	4983.00	0.353987
$584$	−18126.0	−1.28435
$585$	0	0
$586$	1771.65	0.124891
$587$	−521.211	−0.0366485	−0.0183243	−	0.999832i	$-0.505833\pi$
$587$	−521.211	−0.0366485	−0.0183243	+	0.999832i	$0.505833\pi$
$588$	0	0
$589$	−27347.4	−1.91312
$590$	2635.43	0.183896
$591$	0	0
$592$	−438.488	−0.0304421
$593$	−4324.86	−0.299496	−0.149748	−	0.988724i	$-0.547846\pi$
$593$	−4324.86	−0.299496	−0.149748	+	0.988724i	$0.547846\pi$
$594$	0	0
$595$	−380.218	−0.0261974
$596$	−6448.87	−0.443215
$597$	0	0
$598$	0	0
$599$	22325.2	1.52284	0.761422	−	0.648257i	$-0.224502\pi$
$599$	22325.2	1.52284	0.761422	+	0.648257i	$0.224502\pi$
$600$	0	0
$601$	1909.74	0.129617	0.0648086	−	0.997898i	$-0.479356\pi$
$601$	1909.74	0.129617	0.0648086	+	0.997898i	$0.479356\pi$
$602$	3126.90	0.211699
$603$	0	0
$604$	−7656.93	−0.515821
$605$	223.169	0.0149969
$606$	0	0
$607$	6709.42	0.448644	0.224322	−	0.974515i	$-0.427983\pi$
$607$	6709.42	0.448644	0.224322	+	0.974515i	$0.427983\pi$
$608$	−18027.0	−1.20245
$609$	0	0
$610$	3481.69	0.231098
$611$	0	0
$612$	0	0
$613$	−18099.7	−1.19256	−0.596281	−	0.802776i	$-0.703356\pi$
$613$	−18099.7	−1.19256	−0.596281	+	0.802776i	$0.703356\pi$
$614$	−17178.0	−1.12907
$615$	0	0
$616$	−3994.30	−0.261258
$617$	−1455.96	−0.0949994	−0.0474997	−	0.998871i	$-0.515125\pi$
$617$	−1455.96	−0.0949994	−0.0474997	+	0.998871i	$0.515125\pi$
$618$	0	0
$619$	19567.5	1.27057	0.635287	−	0.772276i	$-0.280882\pi$
$619$	19567.5	1.27057	0.635287	+	0.772276i	$0.280882\pi$
$620$	−2803.83	−0.181620
$621$	0	0
$622$	4466.59	0.287932
$623$	6087.13	0.391454
$624$	0	0
$625$	13566.4	0.868249
$626$	7012.49	0.447724
$627$	0	0
$628$	2385.48	0.151578
$629$	1974.25	0.125148
$630$	0	0
$631$	31211.1	1.96909	0.984544	−	0.175136i	$-0.0560367\pi$
$631$	31211.1	1.96909	0.984544	+	0.175136i	$0.0560367\pi$
$632$	−29805.2	−1.87593
$633$	0	0
$634$	−2814.03	−0.176277
$635$	−1631.92	−0.101986
$636$	0	0
$637$	0	0
$638$	7749.17	0.480866
$639$	0	0
$640$	−1589.29	−0.0981600
$641$	−19040.1	−1.17323	−0.586614	−	0.809866i	$-0.699540\pi$
$641$	−19040.1	−1.17323	−0.586614	+	0.809866i	$0.699540\pi$
$642$	0	0
$643$	−2224.04	−0.136404	−0.0682020	−	0.997672i	$-0.521726\pi$
$643$	−2224.04	−0.136404	−0.0682020	+	0.997672i	$0.521726\pi$
$644$	−1899.34	−0.116218
$645$	0	0
$646$	−6429.65	−0.391596
$647$	22766.2	1.38335	0.691677	−	0.722207i	$-0.256872\pi$
$647$	22766.2	1.38335	0.691677	+	0.722207i	$0.256872\pi$
$648$	0	0
$649$	−21053.9	−1.27340
$650$	0	0
$651$	0	0
$652$	−13438.2	−0.807179
$653$	26796.6	1.60587	0.802935	−	0.596066i	$-0.203270\pi$
$653$	26796.6	1.60587	0.802935	+	0.596066i	$0.203270\pi$
$654$	0	0
$655$	6535.67	0.389878
$656$	566.201	0.0336989
$657$	0	0
$658$	−3772.72	−0.223520
$659$	17395.3	1.02826	0.514132	−	0.857711i	$-0.328114\pi$
$659$	17395.3	1.02826	0.514132	+	0.857711i	$0.328114\pi$
$660$	0	0
$661$	−28033.4	−1.64958	−0.824790	−	0.565439i	$-0.808707\pi$
$661$	−28033.4	−1.64958	−0.824790	+	0.565439i	$0.808707\pi$
$662$	2811.94	0.165089
$663$	0	0
$664$	11519.9	0.673280
$665$	−1195.94	−0.0697392
$666$	0	0
$667$	10201.9	0.592235
$668$	1840.08	0.106579
$669$	0	0
$670$	−97.3291	−0.00561216
$671$	−27814.5	−1.60025
$672$	0	0
$673$	20885.3	1.19624	0.598120	−	0.801407i	$-0.295915\pi$
$673$	20885.3	1.19624	0.598120	+	0.801407i	$0.295915\pi$
$674$	−14973.6	−0.855732
$675$	0	0
$676$	0	0
$677$	−8938.84	−0.507456	−0.253728	−	0.967276i	$-0.581657\pi$
$677$	−8938.84	−0.507456	−0.253728	+	0.967276i	$0.581657\pi$
$678$	0	0
$679$	372.105	0.0210310
$680$	−1825.11	−0.102926
$681$	0	0
$682$	−17216.8	−0.966665
$683$	−18061.8	−1.01188	−0.505940	−	0.862569i	$-0.668854\pi$
$683$	−18061.8	−1.01188	−0.505940	+	0.862569i	$0.668854\pi$
$684$	0	0
$685$	−3176.05	−0.177154
$686$	6007.74	0.334368
$687$	0	0
$688$	2535.12	0.140480
$689$	0	0
$690$	0	0
$691$	1933.75	0.106459	0.0532295	−	0.998582i	$-0.483049\pi$
$691$	1933.75	0.106459	0.0532295	+	0.998582i	$0.483049\pi$
$692$	−13084.4	−0.718780
$693$	0	0
$694$	11960.9	0.654222
$695$	−4389.49	−0.239572
$696$	0	0
$697$	−2549.26	−0.138537
$698$	6305.39	0.341923
$699$	0	0
$700$	2627.87	0.141892
$701$	−8894.15	−0.479212	−0.239606	−	0.970870i	$-0.577018\pi$
$701$	−8894.15	−0.479212	−0.239606	+	0.970870i	$0.577018\pi$
$702$	0	0
$703$	6209.81	0.333154
$704$	−13417.8	−0.718326
$705$	0	0
$706$	−12160.3	−0.648243
$707$	−5928.37	−0.315360
$708$	0	0
$709$	20617.1	1.09209	0.546044	−	0.837757i	$-0.316133\pi$
$709$	20617.1	1.09209	0.546044	+	0.837757i	$0.316133\pi$
$710$	−2639.37	−0.139512
$711$	0	0
$712$	29219.2	1.53797
$713$	−22666.3	−1.19055
$714$	0	0
$715$	0	0
$716$	17063.2	0.890616
$717$	0	0
$718$	−4212.34	−0.218946
$719$	−12900.0	−0.669107	−0.334553	−	0.942377i	$-0.608585\pi$
$719$	−12900.0	−0.669107	−0.334553	+	0.942377i	$0.608585\pi$
$720$	0	0
$721$	−4813.21	−0.248618
$722$	−7434.56	−0.383221
$723$	0	0
$724$	7571.41	0.388659
$725$	−14115.1	−0.723064
$726$	0	0
$727$	−3681.32	−0.187803	−0.0939013	−	0.995582i	$-0.529934\pi$
$727$	−3681.32	−0.187803	−0.0939013	+	0.995582i	$0.529934\pi$
$728$	0	0
$729$	0	0
$730$	−3416.71	−0.173230
$731$	−11414.1	−0.577519
$732$	0	0
$733$	14330.0	0.722088	0.361044	−	0.932549i	$-0.382420\pi$
$733$	14330.0	0.722088	0.361044	+	0.932549i	$0.382420\pi$
$734$	−6348.70	−0.319257
$735$	0	0
$736$	−14941.3	−0.748292
$737$	777.542	0.0388618
$738$	0	0
$739$	−3980.74	−0.198151	−0.0990757	−	0.995080i	$-0.531589\pi$
$739$	−3980.74	−0.198151	−0.0990757	+	0.995080i	$0.531589\pi$
$740$	636.670	0.0316276
$741$	0	0
$742$	1285.39	0.0635957
$743$	−34167.8	−1.68707	−0.843536	−	0.537073i	$-0.819530\pi$
$743$	−34167.8	−1.68707	−0.843536	+	0.537073i	$0.819530\pi$
$744$	0	0
$745$	−3365.54	−0.165508
$746$	−21340.4	−1.04736
$747$	0	0
$748$	5266.29	0.257426
$749$	5478.10	0.267243
$750$	0	0
$751$	−1667.87	−0.0810407	−0.0405204	−	0.999179i	$-0.512902\pi$
$751$	−1667.87	−0.0810407	−0.0405204	+	0.999179i	$0.512902\pi$
$752$	−3058.72	−0.148325
$753$	0	0
$754$	0	0
$755$	−3996.00	−0.192622
$756$	0	0
$757$	26775.5	1.28556	0.642782	−	0.766049i	$-0.277780\pi$
$757$	26775.5	1.28556	0.642782	+	0.766049i	$0.277780\pi$
$758$	−6690.02	−0.320571
$759$	0	0
$760$	−5740.71	−0.273997
$761$	−25360.3	−1.20803	−0.604014	−	0.796974i	$-0.706433\pi$
$761$	−25360.3	−1.20803	−0.604014	+	0.796974i	$0.706433\pi$
$762$	0	0
$763$	217.525	0.0103210
$764$	−22728.7	−1.07630
$765$	0	0
$766$	−21142.0	−0.997248
$767$	0	0
$768$	0	0
$769$	21088.5	0.988906	0.494453	−	0.869204i	$-0.335368\pi$
$769$	21088.5	0.988906	0.494453	+	0.869204i	$0.335368\pi$
$770$	−752.916	−0.0352379
$771$	0	0
$772$	−10849.4	−0.505800
$773$	36492.1	1.69797	0.848985	−	0.528416i	$-0.177214\pi$
$773$	36492.1	1.69797	0.848985	+	0.528416i	$0.177214\pi$
$774$	0	0
$775$	31360.4	1.45355
$776$	1786.16	0.0826283
$777$	0	0
$778$	8250.61	0.380204
$779$	−8018.47	−0.368795
$780$	0	0
$781$	21085.4	0.966063
$782$	−5329.07	−0.243692
$783$	0	0
$784$	2348.37	0.106977
$785$	1244.94	0.0566034
$786$	0	0
$787$	−886.171	−0.0401380	−0.0200690	−	0.999799i	$-0.506389\pi$
$787$	−886.171	−0.0401380	−0.0200690	+	0.999799i	$0.506389\pi$
$788$	−20715.3	−0.936485
$789$	0	0
$790$	−5618.20	−0.253021
$791$	−7637.95	−0.343330
$792$	0	0
$793$	0	0
$794$	21704.5	0.970105
$795$	0	0
$796$	−803.054	−0.0357582
$797$	−40373.6	−1.79436	−0.897181	−	0.441664i	$-0.854388\pi$
$797$	−40373.6	−1.79436	−0.897181	+	0.441664i	$0.854388\pi$
$798$	0	0
$799$	13771.6	0.609767
$800$	20672.3	0.913596
$801$	0	0
$802$	−19011.3	−0.837046
$803$	27295.4	1.19954
$804$	0	0
$805$	−991.229	−0.0433990
$806$	0	0
$807$	0	0
$808$	−28457.1	−1.23901
$809$	−12136.2	−0.527425	−0.263712	−	0.964601i	$-0.584947\pi$
$809$	−12136.2	−0.527425	−0.263712	+	0.964601i	$0.584947\pi$
$810$	0	0
$811$	17625.8	0.763163	0.381582	−	0.924335i	$-0.375380\pi$
$811$	17625.8	0.763163	0.381582	+	0.924335i	$0.375380\pi$
$812$	−2600.63	−0.112395
$813$	0	0
$814$	3909.44	0.168337
$815$	−7013.13	−0.301423
$816$	0	0
$817$	−35902.1	−1.53740
$818$	−4093.20	−0.174958
$819$	0	0
$820$	−822.106	−0.0350112
$821$	4685.38	0.199173	0.0995865	−	0.995029i	$-0.468248\pi$
$821$	4685.38	0.199173	0.0995865	+	0.995029i	$0.468248\pi$
$822$	0	0
$823$	13493.1	0.571493	0.285747	−	0.958305i	$-0.407758\pi$
$823$	13493.1	0.571493	0.285747	+	0.958305i	$0.407758\pi$
$824$	−23104.2	−0.976788
$825$	0	0
$826$	−5430.95	−0.228773
$827$	−9203.58	−0.386989	−0.193494	−	0.981101i	$-0.561982\pi$
$827$	−9203.58	−0.386989	−0.193494	+	0.981101i	$0.561982\pi$
$828$	0	0
$829$	20033.6	0.839320	0.419660	−	0.907681i	$-0.362149\pi$
$829$	20033.6	0.839320	0.419660	+	0.907681i	$0.362149\pi$
$830$	2171.47	0.0908105
$831$	0	0
$832$	0	0
$833$	−10573.3	−0.439787
$834$	0	0
$835$	960.301	0.0397995
$836$	16564.6	0.685287
$837$	0	0
$838$	−13822.3	−0.569791
$839$	−22834.8	−0.939625	−0.469813	−	0.882766i	$-0.655679\pi$
$839$	−22834.8	−0.939625	−0.469813	+	0.882766i	$0.655679\pi$
$840$	0	0
$841$	−10420.2	−0.427250
$842$	4981.22	0.203877
$843$	0	0
$844$	−19738.6	−0.805014
$845$	0	0
$846$	0	0
$847$	−459.895	−0.0186566
$848$	1042.12	0.0422012
$849$	0	0
$850$	7373.14	0.297525
$851$	5146.86	0.207323
$852$	0	0
$853$	−23922.2	−0.960233	−0.480117	−	0.877205i	$-0.659406\pi$
$853$	−23922.2	−0.960233	−0.480117	+	0.877205i	$0.659406\pi$
$854$	−7174.89	−0.287494
$855$	0	0
$856$	26295.7	1.04996
$857$	13915.6	0.554665	0.277332	−	0.960774i	$-0.410550\pi$
$857$	13915.6	0.554665	0.277332	+	0.960774i	$0.410550\pi$
$858$	0	0
$859$	−41260.9	−1.63889	−0.819443	−	0.573161i	$-0.805717\pi$
$859$	−41260.9	−1.63889	−0.819443	+	0.573161i	$0.805717\pi$
$860$	−3680.91	−0.145951
$861$	0	0
$862$	2317.81	0.0915835
$863$	47277.7	1.86483	0.932417	−	0.361384i	$-0.117696\pi$
$863$	47277.7	1.86483	0.932417	+	0.361384i	$0.117696\pi$
$864$	0	0
$865$	−6828.51	−0.268412
$866$	1799.97	0.0706300
$867$	0	0
$868$	5777.99	0.225942
$869$	44882.7	1.75206
$870$	0	0
$871$	0	0
$872$	1044.15	0.0405499
$873$	0	0
$874$	−16762.1	−0.648726
$875$	2806.86	0.108445
$876$	0	0
$877$	31051.0	1.19558	0.597788	−	0.801655i	$-0.296047\pi$
$877$	31051.0	1.19558	0.597788	+	0.801655i	$0.296047\pi$
$878$	10150.8	0.390173
$879$	0	0
$880$	−610.424	−0.0233834
$881$	24283.5	0.928638	0.464319	−	0.885668i	$-0.346299\pi$
$881$	24283.5	0.928638	0.464319	+	0.885668i	$0.346299\pi$
$882$	0	0
$883$	15754.0	0.600412	0.300206	−	0.953874i	$-0.402945\pi$
$883$	15754.0	0.600412	0.300206	+	0.953874i	$0.402945\pi$
$884$	0	0
$885$	0	0
$886$	−18491.8	−0.701179
$887$	−16246.0	−0.614982	−0.307491	−	0.951551i	$-0.599489\pi$
$887$	−16246.0	−0.614982	−0.307491	+	0.951551i	$0.599489\pi$
$888$	0	0
$889$	3362.98	0.126874
$890$	5507.74	0.207438
$891$	0	0
$892$	7652.36	0.287242
$893$	43317.2	1.62324
$894$	0	0
$895$	8904.93	0.332580
$896$	3275.13	0.122114
$897$	0	0
$898$	−12157.0	−0.451764
$899$	−31035.3	−1.15137
$900$	0	0
$901$	−4692.05	−0.173490
$902$	−5048.11	−0.186345
$903$	0	0
$904$	−36663.4	−1.34890
$905$	3951.37	0.145136
$906$	0	0
$907$	−29623.9	−1.08451	−0.542253	−	0.840216i	$-0.682428\pi$
$907$	−29623.9	−1.08451	−0.542253	+	0.840216i	$0.682428\pi$
$908$	−16072.9	−0.587441
$909$	0	0
$910$	0	0
$911$	1244.48	0.0452594	0.0226297	−	0.999744i	$-0.492796\pi$
$911$	1244.48	0.0452594	0.0226297	+	0.999744i	$0.492796\pi$
$912$	0	0
$913$	−17347.4	−0.628823
$914$	15690.3	0.567820
$915$	0	0
$916$	−17099.2	−0.616784
$917$	−13468.4	−0.485022
$918$	0	0
$919$	−27386.4	−0.983020	−0.491510	−	0.870872i	$-0.663555\pi$
$919$	−27386.4	−0.983020	−0.491510	+	0.870872i	$0.663555\pi$
$920$	−4758.06	−0.170509
$921$	0	0
$922$	6128.58	0.218909
$923$	0	0
$924$	0	0
$925$	−7121.04	−0.253123
$926$	19285.3	0.684399
$927$	0	0
$928$	−20458.0	−0.723672
$929$	−40144.3	−1.41775	−0.708876	−	0.705333i	$-0.750797\pi$
$929$	−40144.3	−1.41775	−0.708876	+	0.705333i	$0.750797\pi$
$930$	0	0
$931$	−33257.3	−1.17074
$932$	−29801.7	−1.04741
$933$	0	0
$934$	−11472.3	−0.401911
$935$	2748.37	0.0961298
$936$	0	0
$937$	38928.8	1.35725	0.678627	−	0.734483i	$-0.262575\pi$
$937$	38928.8	1.35725	0.678627	+	0.734483i	$0.262575\pi$
$938$	200.571	0.00698173
$939$	0	0
$940$	4441.16	0.154101
$941$	20870.7	0.723025	0.361513	−	0.932367i	$-0.382260\pi$
$941$	20870.7	0.723025	0.361513	+	0.932367i	$0.382260\pi$
$942$	0	0
$943$	−6645.93	−0.229503
$944$	−4403.12	−0.151811
$945$	0	0
$946$	−22602.5	−0.776818
$947$	−35530.9	−1.21922	−0.609609	−	0.792702i	$-0.708673\pi$
$947$	−35530.9	−1.21922	−0.609609	+	0.792702i	$0.708673\pi$
$948$	0	0
$949$	0	0
$950$	23191.5	0.792034
$951$	0	0
$952$	3761.09	0.128044
$953$	24920.3	0.847060	0.423530	−	0.905882i	$-0.360791\pi$
$953$	24920.3	0.847060	0.423530	+	0.905882i	$0.360791\pi$
$954$	0	0
$955$	−11861.7	−0.401921
$956$	14711.2	0.497691
$957$	0	0
$958$	35191.6	1.18684
$959$	6545.03	0.220386
$960$	0	0
$961$	39162.1	1.31456
$962$	0	0
$963$	0	0
$964$	14356.8	0.479670
$965$	−5662.07	−0.188879
$966$	0	0
$967$	−34035.7	−1.13187	−0.565934	−	0.824451i	$-0.691484\pi$
$967$	−34035.7	−1.13187	−0.565934	+	0.824451i	$0.691484\pi$
$968$	−2207.57	−0.0732995
$969$	0	0
$970$	336.687	0.0111447
$971$	−27039.2	−0.893646	−0.446823	−	0.894622i	$-0.647445\pi$
$971$	−27039.2	−0.893646	−0.446823	+	0.894622i	$0.647445\pi$
$972$	0	0
$973$	9045.62	0.298036
$974$	−15936.2	−0.524259
$975$	0	0
$976$	−5817.02	−0.190777
$977$	57783.1	1.89216	0.946082	−	0.323926i	$-0.105003\pi$
$977$	57783.1	1.89216	0.946082	+	0.323926i	$0.105003\pi$
$978$	0	0
$979$	−44000.3	−1.43642
$980$	−3409.75	−0.111143
$981$	0	0
$982$	1605.97	0.0521879
$983$	52953.3	1.71816	0.859078	−	0.511845i	$-0.171038\pi$
$983$	52953.3	1.71816	0.859078	+	0.511845i	$0.171038\pi$
$984$	0	0
$985$	−10810.9	−0.349709
$986$	−7296.71	−0.235674
$987$	0	0
$988$	0	0
$989$	−29756.6	−0.956730
$990$	0	0
$991$	9632.95	0.308780	0.154390	−	0.988010i	$-0.450659\pi$
$991$	9632.95	0.308780	0.154390	+	0.988010i	$0.450659\pi$
$992$	45452.9	1.45477
$993$	0	0
$994$	5439.07	0.173558
$995$	−419.098	−0.0133531
$996$	0	0
$997$	11816.6	0.375362	0.187681	−	0.982230i	$-0.439903\pi$
$997$	11816.6	0.375362	0.187681	+	0.982230i	$0.439903\pi$
$998$	−6617.50	−0.209893
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bj.1.3		9
3.2	odd	2		507.4.a.n.1.7	✓	9
13.12	even	2		1521.4.a.be.1.7		9
39.5	even	4		507.4.b.j.337.7		18
39.8	even	4		507.4.b.j.337.12		18
39.38	odd	2		507.4.a.q.1.3	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.7	✓	9	3.2	odd	2
507.4.a.q.1.3	yes	9	39.38	odd	2
507.4.b.j.337.7		18	39.5	even	4
507.4.b.j.337.12		18	39.8	even	4
1521.4.a.be.1.7		9	13.12	even	2
1521.4.a.bj.1.3		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)

\(f(q)\)	\(=\)	\(q-1.86460 q^{2} -4.52327 q^{4} -2.36060 q^{5} +4.86461 q^{7} +23.3509 q^{8} +O(q^{10})\)	\(q-1.86460 q^{2} -4.52327 q^{4} -2.36060 q^{5} +4.86461 q^{7} +23.3509 q^{8} +4.40158 q^{10} -35.1633 q^{11} -9.07054 q^{14} -7.35391 q^{16} +33.1102 q^{17} +104.145 q^{19} +10.6776 q^{20} +65.5656 q^{22} +86.3184 q^{23} -119.428 q^{25} -22.0039 q^{28} +118.190 q^{29} -262.589 q^{31} -173.095 q^{32} -61.7373 q^{34} -11.4834 q^{35} +59.6265 q^{37} -194.189 q^{38} -55.1222 q^{40} -76.9932 q^{41} -344.731 q^{43} +159.053 q^{44} -160.949 q^{46} +415.931 q^{47} -319.336 q^{49} +222.685 q^{50} -141.710 q^{53} +83.0067 q^{55} +113.593 q^{56} -220.376 q^{58} +598.746 q^{59} +791.010 q^{61} +489.624 q^{62} +381.584 q^{64} -22.1123 q^{67} -149.766 q^{68} +21.4120 q^{70} -599.641 q^{71} -776.246 q^{73} -111.180 q^{74} -471.076 q^{76} -171.056 q^{77} -1276.41 q^{79} +17.3597 q^{80} +143.562 q^{82} +493.338 q^{83} -78.1601 q^{85} +642.785 q^{86} -821.095 q^{88} +1251.31 q^{89} -390.441 q^{92} -775.545 q^{94} -245.845 q^{95} +76.4923 q^{97} +595.433 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8}+O(q^{10}) \) 9 * q + 8 * q^2 + 32 * q^4 + 41 * q^5 - q^7 + 111 * q^8	\( 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8} + 198 q^{10} + 37 q^{11} - 98 q^{14} + 32 q^{16} + 134 q^{17} + 72 q^{19} + 356 q^{20} + 274 q^{22} - 226 q^{23} + 612 q^{25} - 132 q^{28} + 547 q^{29} + 521 q^{31} + 721 q^{32} + 100 q^{34} - 138 q^{35} - 584 q^{37} + 416 q^{38} + 1342 q^{40} + 482 q^{41} + 158 q^{43} + 1453 q^{44} - 1537 q^{46} + 1500 q^{47} + 642 q^{49} + 2777 q^{50} - 1399 q^{53} - 1408 q^{55} + 616 q^{56} - 1455 q^{58} + 1541 q^{59} + 2092 q^{61} + 293 q^{62} + 2481 q^{64} - 252 q^{67} + 1579 q^{68} - 2492 q^{70} + 2352 q^{71} - 903 q^{73} - 1037 q^{74} + 485 q^{76} + 1686 q^{77} - 115 q^{79} + 5701 q^{80} - 5147 q^{82} + 1207 q^{83} - 4308 q^{85} + 5691 q^{86} - 484 q^{88} + 2336 q^{89} - 2087 q^{92} - 468 q^{94} + 222 q^{95} - 2155 q^{97} + 5593 q^{98}+O(q^{100}) \) 9 * q + 8 * q^2 + 32 * q^4 + 41 * q^5 - q^7 + 111 * q^8 + 198 * q^10 + 37 * q^11 - 98 * q^14 + 32 * q^16 + 134 * q^17 + 72 * q^19 + 356 * q^20 + 274 * q^22 - 226 * q^23 + 612 * q^25 - 132 * q^28 + 547 * q^29 + 521 * q^31 + 721 * q^32 + 100 * q^34 - 138 * q^35 - 584 * q^37 + 416 * q^38 + 1342 * q^40 + 482 * q^41 + 158 * q^43 + 1453 * q^44 - 1537 * q^46 + 1500 * q^47 + 642 * q^49 + 2777 * q^50 - 1399 * q^53 - 1408 * q^55 + 616 * q^56 - 1455 * q^58 + 1541 * q^59 + 2092 * q^61 + 293 * q^62 + 2481 * q^64 - 252 * q^67 + 1579 * q^68 - 2492 * q^70 + 2352 * q^71 - 903 * q^73 - 1037 * q^74 + 485 * q^76 + 1686 * q^77 - 115 * q^79 + 5701 * q^80 - 5147 * q^82 + 1207 * q^83 - 4308 * q^85 + 5691 * q^86 - 484 * q^88 + 2336 * q^89 - 2087 * q^92 - 468 * q^94 + 222 * q^95 - 2155 * q^97 + 5593 * q^98

Introduction

L-functions

Modular forms

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Database

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