LMFDB - Embedded newform 1521.4.a.bh.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 $ x^9 - 4x^8 - 46x^7 + 145x^6 + 680x^5 - 1501x^4 - 3203x^3 + 4784x^2 + 3584x - 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 169)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.16135$ 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)

$f(q)$	$=$	$q+3.16135 q^{2} +1.99415 q^{4} +13.6039 q^{5} +14.3315 q^{7} -18.9866 q^{8} +O(q^{10})$	$q+3.16135 q^{2} +1.99415 q^{4} +13.6039 q^{5} +14.3315 q^{7} -18.9866 q^{8} +43.0068 q^{10} +67.7223 q^{11} +45.3068 q^{14} -75.9766 q^{16} -0.337795 q^{17} +40.5229 q^{19} +27.1282 q^{20} +214.094 q^{22} +155.632 q^{23} +60.0670 q^{25} +28.5791 q^{28} +33.7925 q^{29} -157.397 q^{31} -88.2957 q^{32} -1.06789 q^{34} +194.964 q^{35} -58.6204 q^{37} +128.107 q^{38} -258.293 q^{40} -59.3488 q^{41} -208.311 q^{43} +135.048 q^{44} +492.006 q^{46} +221.212 q^{47} -137.609 q^{49} +189.893 q^{50} +409.639 q^{53} +921.289 q^{55} -272.106 q^{56} +106.830 q^{58} -173.587 q^{59} +560.796 q^{61} -497.586 q^{62} +328.679 q^{64} -269.074 q^{67} -0.673613 q^{68} +616.351 q^{70} -60.9754 q^{71} +282.066 q^{73} -185.320 q^{74} +80.8086 q^{76} +970.560 q^{77} +984.026 q^{79} -1033.58 q^{80} -187.622 q^{82} -1201.86 q^{83} -4.59534 q^{85} -658.543 q^{86} -1285.82 q^{88} +539.899 q^{89} +310.352 q^{92} +699.328 q^{94} +551.271 q^{95} +1587.25 q^{97} -435.030 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8}+O(q^{10}) $ 9 * q + 5 * q^2 + 37 * q^4 + 30 * q^5 - 38 * q^7 + 60 * q^8	$ 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8} - 147 q^{10} + 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} - 161 q^{19} + 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} - 344 q^{28} - 280 q^{29} - 706 q^{31} + 680 q^{32} - 216 q^{34} - 20 q^{35} - 298 q^{37} + 739 q^{38} + 13 q^{40} + 1201 q^{41} - 533 q^{43} + 355 q^{44} + 840 q^{46} + 956 q^{47} + 403 q^{49} - 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} + 2877 q^{58} + 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} + 931 q^{67} + 1536 q^{68} + 4854 q^{70} + 2046 q^{71} + 45 q^{73} + 1990 q^{74} + 3608 q^{76} + 718 q^{77} + 412 q^{79} - 787 q^{80} + 2757 q^{82} + 3709 q^{83} + 2106 q^{85} + 125 q^{86} - 636 q^{88} + 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} + 1087 q^{97} - 282 q^{98}+O(q^{100}) $ 9 * q + 5 * q^2 + 37 * q^4 + 30 * q^5 - 38 * q^7 + 60 * q^8 - 147 * q^10 + 181 * q^11 + 147 * q^14 + 269 * q^16 + 55 * q^17 - 161 * q^19 + 370 * q^20 + 340 * q^22 + 204 * q^23 + 307 * q^25 - 344 * q^28 - 280 * q^29 - 706 * q^31 + 680 * q^32 - 216 * q^34 - 20 * q^35 - 298 * q^37 + 739 * q^38 + 13 * q^40 + 1201 * q^41 - 533 * q^43 + 355 * q^44 + 840 * q^46 + 956 * q^47 + 403 * q^49 - 1156 * q^50 + 278 * q^53 - 250 * q^55 - 250 * q^56 + 2877 * q^58 + 1377 * q^59 - 136 * q^61 - 2035 * q^62 + 284 * q^64 + 931 * q^67 + 1536 * q^68 + 4854 * q^70 + 2046 * q^71 + 45 * q^73 + 1990 * q^74 + 3608 * q^76 + 718 * q^77 + 412 * q^79 - 787 * q^80 + 2757 * q^82 + 3709 * q^83 + 2106 * q^85 + 125 * q^86 - 636 * q^88 + 1663 * q^89 - 4010 * q^92 - 2531 * q^94 + 1614 * q^95 + 1087 * q^97 - 282 * 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 $

$2$	3.16135	1.11771	0.558853	−	0.829267i	$-0.311241\pi$
$2$	3.16135	1.11771	0.558853	+	0.829267i	$0.311241\pi$
$3$	0	0
$3$	0	0
$4$	1.99415	0.249268
$5$	13.6039	1.21677	0.608386	−	0.793641i	$-0.291817\pi$
$5$	13.6039	1.21677	0.608386	+	0.793641i	$0.291817\pi$
$6$	0	0
$7$	14.3315	0.773827	0.386913	−	0.922116i	$-0.373541\pi$
$7$	14.3315	0.773827	0.386913	+	0.922116i	$0.373541\pi$
$8$	−18.9866	−0.839098
$9$	0	0
$10$	43.0068	1.36000
$11$	67.7223	1.85628	0.928138	−	0.372237i	$-0.121409\pi$
$11$	67.7223	1.85628	0.928138	+	0.372237i	$0.121409\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	45.3068	0.864911
$15$	0	0
$16$	−75.9766	−1.18713
$17$	−0.337795	−0.00481925	−0.00240963	−	0.999997i	$-0.500767\pi$
$17$	−0.337795	−0.00481925	−0.00240963	+	0.999997i	$0.500767\pi$
$18$	0	0
$19$	40.5229	0.489294	0.244647	−	0.969612i	$-0.421328\pi$
$19$	40.5229	0.489294	0.244647	+	0.969612i	$0.421328\pi$
$20$	27.1282	0.303303
$21$	0	0
$22$	214.094	2.07477
$23$	155.632	1.41093	0.705466	−	0.708744i	$-0.250738\pi$
$23$	155.632	1.41093	0.705466	+	0.708744i	$0.250738\pi$
$24$	0	0
$25$	60.0670	0.480536
$26$	0	0
$27$	0	0
$28$	28.5791	0.192890
$29$	33.7925	0.216383	0.108192	−	0.994130i	$-0.465494\pi$
$29$	33.7925	0.216383	0.108192	+	0.994130i	$0.465494\pi$
$30$	0	0
$31$	−157.397	−0.911911	−0.455956	−	0.890003i	$-0.650702\pi$
$31$	−157.397	−0.911911	−0.455956	+	0.890003i	$0.650702\pi$
$32$	−88.2957	−0.487769
$33$	0	0
$34$	−1.06789	−0.00538651
$35$	194.964	0.941571
$36$	0	0
$37$	−58.6204	−0.260463	−0.130231	−	0.991484i	$-0.541572\pi$
$37$	−58.6204	−0.260463	−0.130231	+	0.991484i	$0.541572\pi$
$38$	128.107	0.546888
$39$	0	0
$40$	−258.293	−1.02099
$41$	−59.3488	−0.226066	−0.113033	−	0.993591i	$-0.536057\pi$
$41$	−59.3488	−0.226066	−0.113033	+	0.993591i	$0.536057\pi$
$42$	0	0
$43$	−208.311	−0.738769	−0.369385	−	0.929277i	$-0.620431\pi$
$43$	−208.311	−0.738769	−0.369385	+	0.929277i	$0.620431\pi$
$44$	135.048	0.462711
$45$	0	0
$46$	492.006	1.57701
$47$	221.212	0.686533	0.343266	−	0.939238i	$-0.388467\pi$
$47$	221.212	0.686533	0.343266	+	0.939238i	$0.388467\pi$
$48$	0	0
$49$	−137.609	−0.401192
$50$	189.893	0.537098
$51$	0	0
$52$	0	0
$53$	409.639	1.06167	0.530833	−	0.847477i	$-0.321879\pi$
$53$	409.639	1.06167	0.530833	+	0.847477i	$0.321879\pi$
$54$	0	0
$55$	921.289	2.25867
$56$	−272.106	−0.649316
$57$	0	0
$58$	106.830	0.241853
$59$	−173.587	−0.383036	−0.191518	−	0.981489i	$-0.561341\pi$
$59$	−173.587	−0.383036	−0.191518	+	0.981489i	$0.561341\pi$
$60$	0	0
$61$	560.796	1.17709	0.588546	−	0.808464i	$-0.299701\pi$
$61$	560.796	1.17709	0.588546	+	0.808464i	$0.299701\pi$
$62$	−497.586	−1.01925
$63$	0	0
$64$	328.679	0.641951
$65$	0	0
$66$	0	0
$67$	−269.074	−0.490635	−0.245318	−	0.969443i	$-0.578892\pi$
$67$	−269.074	−0.490635	−0.245318	+	0.969443i	$0.578892\pi$
$68$	−0.673613	−0.00120129
$69$	0	0
$70$	616.351	1.05240
$71$	−60.9754	−0.101922	−0.0509609	−	0.998701i	$-0.516228\pi$
$71$	−60.9754	−0.101922	−0.0509609	+	0.998701i	$0.516228\pi$
$72$	0	0
$73$	282.066	0.452238	0.226119	−	0.974100i	$-0.427396\pi$
$73$	282.066	0.452238	0.226119	+	0.974100i	$0.427396\pi$
$74$	−185.320	−0.291121
$75$	0	0
$76$	80.8086	0.121966
$77$	970.560	1.43644
$78$	0	0
$79$	984.026	1.40141	0.700706	−	0.713450i	$-0.252868\pi$
$79$	984.026	1.40141	0.700706	+	0.713450i	$0.252868\pi$
$80$	−1033.58	−1.44447
$81$	0	0
$82$	−187.622	−0.252676
$83$	−1201.86	−1.58942	−0.794709	−	0.606991i	$-0.792376\pi$
$83$	−1201.86	−1.58942	−0.794709	+	0.606991i	$0.792376\pi$
$84$	0	0
$85$	−4.59534	−0.00586394
$86$	−658.543	−0.825727
$87$	0	0
$88$	−1285.82	−1.55760
$89$	539.899	0.643024	0.321512	−	0.946906i	$-0.395809\pi$
$89$	539.899	0.643024	0.321512	+	0.946906i	$0.395809\pi$
$90$	0	0
$91$	0	0
$92$	310.352	0.351701
$93$	0	0
$94$	699.328	0.767342
$95$	551.271	0.595360
$96$	0	0
$97$	1587.25	1.66146	0.830728	−	0.556678i	$-0.187924\pi$
$97$	1587.25	1.66146	0.830728	+	0.556678i	$0.187924\pi$
$98$	−435.030	−0.448415
$99$	0	0
$100$	119.782	0.119782
$101$	−1160.73	−1.14353	−0.571765	−	0.820417i	$-0.693741\pi$
$101$	−1160.73	−1.14353	−0.571765	+	0.820417i	$0.693741\pi$
$102$	0	0
$103$	−82.2962	−0.0787270	−0.0393635	−	0.999225i	$-0.512533\pi$
$103$	−82.2962	−0.0787270	−0.0393635	+	0.999225i	$0.512533\pi$
$104$	0	0
$105$	0	0
$106$	1295.01	1.18663
$107$	−1322.04	−1.19446	−0.597229	−	0.802071i	$-0.703731\pi$
$107$	−1322.04	−1.19446	−0.597229	+	0.802071i	$0.703731\pi$
$108$	0	0
$109$	−2073.41	−1.82199	−0.910994	−	0.412420i	$-0.864684\pi$
$109$	−2073.41	−1.82199	−0.910994	+	0.412420i	$0.864684\pi$
$110$	2912.52	2.52453
$111$	0	0
$112$	−1088.86	−0.918636
$113$	1602.82	1.33434	0.667172	−	0.744904i	$-0.267505\pi$
$113$	1602.82	1.33434	0.667172	+	0.744904i	$0.267505\pi$
$114$	0	0
$115$	2117.20	1.71678
$116$	67.3872	0.0539375
$117$	0	0
$118$	−548.771	−0.428122
$119$	−4.84110	−0.00372927
$120$	0	0
$121$	3255.30	2.44576
$122$	1772.87	1.31564
$123$	0	0
$124$	−313.872	−0.227311
$125$	−883.344	−0.632070
$126$	0	0
$127$	614.495	0.429352	0.214676	−	0.976685i	$-0.431131\pi$
$127$	614.495	0.429352	0.214676	+	0.976685i	$0.431131\pi$
$128$	1745.43	1.20528
$129$	0	0
$130$	0	0
$131$	330.171	0.220208	0.110104	−	0.993920i	$-0.464882\pi$
$131$	330.171	0.220208	0.110104	+	0.993920i	$0.464882\pi$
$132$	0	0
$133$	580.753	0.378629
$134$	−850.636	−0.548386
$135$	0	0
$136$	6.41358	0.00404383
$137$	2302.54	1.43591	0.717953	−	0.696091i	$-0.245079\pi$
$137$	2302.54	1.43591	0.717953	+	0.696091i	$0.245079\pi$
$138$	0	0
$139$	1207.22	0.736656	0.368328	−	0.929696i	$-0.379930\pi$
$139$	1207.22	0.736656	0.368328	+	0.929696i	$0.379930\pi$
$140$	388.788	0.234704
$141$	0	0
$142$	−192.765	−0.113919
$143$	0	0
$144$	0	0
$145$	459.711	0.263289
$146$	891.711	0.505470
$147$	0	0
$148$	−116.898	−0.0649251
$149$	338.938	0.186355	0.0931774	−	0.995650i	$-0.470298\pi$
$149$	338.938	0.186355	0.0931774	+	0.995650i	$0.470298\pi$
$150$	0	0
$151$	−1694.81	−0.913386	−0.456693	−	0.889624i	$-0.650966\pi$
$151$	−1694.81	−0.913386	−0.456693	+	0.889624i	$0.650966\pi$
$152$	−769.393	−0.410566
$153$	0	0
$154$	3068.28	1.60551
$155$	−2141.21	−1.10959
$156$	0	0
$157$	9.59250	0.00487621	0.00243811	−	0.999997i	$-0.499224\pi$
$157$	9.59250	0.00487621	0.00243811	+	0.999997i	$0.499224\pi$
$158$	3110.85	1.56637
$159$	0	0
$160$	−1201.17	−0.593504
$161$	2230.43	1.09182
$162$	0	0
$163$	−122.100	−0.0586725	−0.0293363	−	0.999570i	$-0.509339\pi$
$163$	−122.100	−0.0586725	−0.0293363	+	0.999570i	$0.509339\pi$
$164$	−118.350	−0.0563512
$165$	0	0
$166$	−3799.51	−1.77650
$167$	2211.25	1.02462	0.512309	−	0.858801i	$-0.328790\pi$
$167$	2211.25	1.02462	0.512309	+	0.858801i	$0.328790\pi$
$168$	0	0
$169$	0	0
$170$	−14.5275	−0.00655416
$171$	0	0
$172$	−415.402	−0.184152
$173$	213.874	0.0939914	0.0469957	−	0.998895i	$-0.485035\pi$
$173$	213.874	0.0939914	0.0469957	+	0.998895i	$0.485035\pi$
$174$	0	0
$175$	860.848	0.371852
$176$	−5145.30	−2.20365
$177$	0	0
$178$	1706.81	0.718712
$179$	−2967.41	−1.23907	−0.619537	−	0.784967i	$-0.712680\pi$
$179$	−2967.41	−1.23907	−0.619537	+	0.784967i	$0.712680\pi$
$180$	0	0
$181$	2329.44	0.956609	0.478304	−	0.878194i	$-0.341252\pi$
$181$	2329.44	0.956609	0.478304	+	0.878194i	$0.341252\pi$
$182$	0	0
$183$	0	0
$184$	−2954.92	−1.18391
$185$	−797.467	−0.316924
$186$	0	0
$187$	−22.8762	−0.00894586
$188$	441.128	0.171131
$189$	0	0
$190$	1742.76	0.665438
$191$	4479.76	1.69709	0.848544	−	0.529125i	$-0.177480\pi$
$191$	4479.76	1.69709	0.848544	+	0.529125i	$0.177480\pi$
$192$	0	0
$193$	−1302.63	−0.485830	−0.242915	−	0.970048i	$-0.578104\pi$
$193$	−1302.63	−0.485830	−0.242915	+	0.970048i	$0.578104\pi$
$194$	5017.87	1.85702
$195$	0	0
$196$	−274.412	−0.100004
$197$	−740.954	−0.267974	−0.133987	−	0.990983i	$-0.542778\pi$
$197$	−740.954	−0.267974	−0.133987	+	0.990983i	$0.542778\pi$
$198$	0	0
$199$	−4688.74	−1.67023	−0.835115	−	0.550075i	$-0.814599\pi$
$199$	−4688.74	−1.67023	−0.835115	+	0.550075i	$0.814599\pi$
$200$	−1140.47	−0.403217
$201$	0	0
$202$	−3669.46	−1.27813
$203$	484.297	0.167443
$204$	0	0
$205$	−807.377	−0.275071
$206$	−260.167	−0.0879937
$207$	0	0
$208$	0	0
$209$	2744.30	0.908265
$210$	0	0
$211$	−3205.11	−1.04573	−0.522865	−	0.852416i	$-0.675137\pi$
$211$	−3205.11	−1.04573	−0.522865	+	0.852416i	$0.675137\pi$
$212$	816.880	0.264640
$213$	0	0
$214$	−4179.45	−1.33505
$215$	−2833.84	−0.898914
$216$	0	0
$217$	−2255.72	−0.705661
$218$	−6554.78	−2.03645
$219$	0	0
$220$	1837.18	0.563014
$221$	0	0
$222$	0	0
$223$	−1057.39	−0.317525	−0.158762	−	0.987317i	$-0.550750\pi$
$223$	−1057.39	−0.317525	−0.158762	+	0.987317i	$0.550750\pi$
$224$	−1265.41	−0.377449
$225$	0	0
$226$	5067.09	1.49141
$227$	5159.68	1.50863	0.754317	−	0.656510i	$-0.227968\pi$
$227$	5159.68	1.50863	0.754317	+	0.656510i	$0.227968\pi$
$228$	0	0
$229$	−1698.25	−0.490059	−0.245030	−	0.969516i	$-0.578798\pi$
$229$	−1698.25	−0.490059	−0.245030	+	0.969516i	$0.578798\pi$
$230$	6693.22	1.91886
$231$	0	0
$232$	−641.606	−0.181567
$233$	−3162.23	−0.889119	−0.444559	−	0.895749i	$-0.646640\pi$
$233$	−3162.23	−0.889119	−0.444559	+	0.895749i	$0.646640\pi$
$234$	0	0
$235$	3009.35	0.835354
$236$	−346.159	−0.0954788
$237$	0	0
$238$	−15.3044	−0.00416823
$239$	2350.04	0.636030	0.318015	−	0.948086i	$-0.396984\pi$
$239$	2350.04	0.636030	0.318015	+	0.948086i	$0.396984\pi$
$240$	0	0
$241$	5167.18	1.38111	0.690555	−	0.723280i	$-0.257367\pi$
$241$	5167.18	1.38111	0.690555	+	0.723280i	$0.257367\pi$
$242$	10291.2	2.73364
$243$	0	0
$244$	1118.31	0.293412
$245$	−1872.02	−0.488159
$246$	0	0
$247$	0	0
$248$	2988.43	0.765183
$249$	0	0
$250$	−2792.56	−0.706469
$251$	−2894.01	−0.727761	−0.363880	−	0.931446i	$-0.618548\pi$
$251$	−2894.01	−0.727761	−0.363880	+	0.931446i	$0.618548\pi$
$252$	0	0
$253$	10539.7	2.61908
$254$	1942.64	0.479889
$255$	0	0
$256$	2888.50	0.705201
$257$	6162.63	1.49577	0.747887	−	0.663826i	$-0.231068\pi$
$257$	6162.63	1.49577	0.747887	+	0.663826i	$0.231068\pi$
$258$	0	0
$259$	−840.116	−0.201553
$260$	0	0
$261$	0	0
$262$	1043.79	0.246127
$263$	199.427	0.0467574	0.0233787	−	0.999727i	$-0.492558\pi$
$263$	199.427	0.0467574	0.0233787	+	0.999727i	$0.492558\pi$
$264$	0	0
$265$	5572.70	1.29181
$266$	1835.96	0.423196
$267$	0	0
$268$	−536.572	−0.122300
$269$	−1108.00	−0.251138	−0.125569	−	0.992085i	$-0.540076\pi$
$269$	−1108.00	−0.251138	−0.125569	+	0.992085i	$0.540076\pi$
$270$	0	0
$271$	−406.054	−0.0910186	−0.0455093	−	0.998964i	$-0.514491\pi$
$271$	−406.054	−0.0910186	−0.0455093	+	0.998964i	$0.514491\pi$
$272$	25.6645	0.00572110
$273$	0	0
$274$	7279.13	1.60492
$275$	4067.87	0.892007
$276$	0	0
$277$	288.810	0.0626460	0.0313230	−	0.999509i	$-0.490028\pi$
$277$	288.810	0.0626460	0.0313230	+	0.999509i	$0.490028\pi$
$278$	3816.45	0.823366
$279$	0	0
$280$	−3701.71	−0.790071
$281$	−5134.93	−1.09012	−0.545061	−	0.838396i	$-0.683494\pi$
$281$	−5134.93	−1.09012	−0.545061	+	0.838396i	$0.683494\pi$
$282$	0	0
$283$	5274.40	1.10788	0.553940	−	0.832556i	$-0.313124\pi$
$283$	5274.40	1.10788	0.553940	+	0.832556i	$0.313124\pi$
$284$	−121.594	−0.0254059
$285$	0	0
$286$	0	0
$287$	−850.556	−0.174936
$288$	0	0
$289$	−4912.89	−0.999977
$290$	1453.31	0.294280
$291$	0	0
$292$	562.482	0.112729
$293$	−1445.69	−0.288253	−0.144127	−	0.989559i	$-0.546037\pi$
$293$	−1445.69	−0.288253	−0.144127	+	0.989559i	$0.546037\pi$
$294$	0	0
$295$	−2361.47	−0.466068
$296$	1113.00	0.218554
$297$	0	0
$298$	1071.50	0.208290
$299$	0	0
$300$	0	0
$301$	−2985.40	−0.571679
$302$	−5357.88	−1.02090
$303$	0	0
$304$	−3078.79	−0.580858
$305$	7629.04	1.43225
$306$	0	0
$307$	9421.42	1.75149	0.875747	−	0.482770i	$-0.160369\pi$
$307$	9421.42	1.75149	0.875747	+	0.482770i	$0.160369\pi$
$308$	1935.44	0.358058
$309$	0	0
$310$	−6769.12	−1.24019
$311$	−7885.87	−1.43784	−0.718918	−	0.695095i	$-0.755362\pi$
$311$	−7885.87	−1.43784	−0.718918	+	0.695095i	$0.755362\pi$
$312$	0	0
$313$	−550.423	−0.0993986	−0.0496993	−	0.998764i	$-0.515826\pi$
$313$	−550.423	−0.0993986	−0.0496993	+	0.998764i	$0.515826\pi$
$314$	30.3253	0.00545017
$315$	0	0
$316$	1962.29	0.349328
$317$	150.974	0.0267494	0.0133747	−	0.999911i	$-0.495743\pi$
$317$	150.974	0.0267494	0.0133747	+	0.999911i	$0.495743\pi$
$318$	0	0
$319$	2288.51	0.401667
$320$	4471.32	0.781108
$321$	0	0
$322$	7051.18	1.22033
$323$	−13.6884	−0.00235803
$324$	0	0
$325$	0	0
$326$	−386.002	−0.0655787
$327$	0	0
$328$	1126.83	0.189692
$329$	3170.29	0.531257
$330$	0	0
$331$	−7079.29	−1.17557	−0.587784	−	0.809018i	$-0.699999\pi$
$331$	−7079.29	−1.17557	−0.587784	+	0.809018i	$0.699999\pi$
$332$	−2396.69	−0.396191
$333$	0	0
$334$	6990.52	1.14522
$335$	−3660.46	−0.596992
$336$	0	0
$337$	−6633.34	−1.07223	−0.536114	−	0.844145i	$-0.680108\pi$
$337$	−6633.34	−1.07223	−0.536114	+	0.844145i	$0.680108\pi$
$338$	0	0
$339$	0	0
$340$	−9.16378	−0.00146169
$341$	−10659.2	−1.69276
$342$	0	0
$343$	−6887.83	−1.08428
$344$	3955.11	0.619900
$345$	0	0
$346$	676.130	0.105055
$347$	−6404.17	−0.990760	−0.495380	−	0.868676i	$-0.664971\pi$
$347$	−6404.17	−0.990760	−0.495380	+	0.868676i	$0.664971\pi$
$348$	0	0
$349$	−168.098	−0.0257825	−0.0128913	−	0.999917i	$-0.504104\pi$
$349$	−168.098	−0.0257825	−0.0128913	+	0.999917i	$0.504104\pi$
$350$	2721.45	0.415621
$351$	0	0
$352$	−5979.58	−0.905434
$353$	−8451.92	−1.27436	−0.637182	−	0.770713i	$-0.719900\pi$
$353$	−8451.92	−1.27436	−0.637182	+	0.770713i	$0.719900\pi$
$354$	0	0
$355$	−829.505	−0.124016
$356$	1076.64	0.160285
$357$	0	0
$358$	−9381.01	−1.38492
$359$	2631.92	0.386929	0.193464	−	0.981107i	$-0.438028\pi$
$359$	2631.92	0.386929	0.193464	+	0.981107i	$0.438028\pi$
$360$	0	0
$361$	−5216.89	−0.760591
$362$	7364.19	1.06921
$363$	0	0
$364$	0	0
$365$	3837.21	0.550271
$366$	0	0
$367$	−13269.9	−1.88742	−0.943708	−	0.330780i	$-0.892688\pi$
$367$	−13269.9	−1.88742	−0.943708	+	0.330780i	$0.892688\pi$
$368$	−11824.4	−1.67496
$369$	0	0
$370$	−2521.07	−0.354228
$371$	5870.73	0.821545
$372$	0	0
$373$	−11680.1	−1.62137	−0.810684	−	0.585483i	$-0.800905\pi$
$373$	−11680.1	−1.62137	−0.810684	+	0.585483i	$0.800905\pi$
$374$	−72.3198	−0.00999885
$375$	0	0
$376$	−4200.06	−0.576068
$377$	0	0
$378$	0	0
$379$	3378.23	0.457857	0.228928	−	0.973443i	$-0.426478\pi$
$379$	3378.23	0.457857	0.228928	+	0.973443i	$0.426478\pi$
$380$	1099.31	0.148404
$381$	0	0
$382$	14162.1	1.89685
$383$	−4380.92	−0.584476	−0.292238	−	0.956346i	$-0.594400\pi$
$383$	−4380.92	−0.584476	−0.292238	+	0.956346i	$0.594400\pi$
$384$	0	0
$385$	13203.4	1.74782
$386$	−4118.06	−0.543015
$387$	0	0
$388$	3165.22	0.414148
$389$	12484.0	1.62716	0.813578	−	0.581456i	$-0.197517\pi$
$389$	12484.0	1.62716	0.813578	+	0.581456i	$0.197517\pi$
$390$	0	0
$391$	−52.5716	−0.00679964
$392$	2612.73	0.336639
$393$	0	0
$394$	−2342.42	−0.299516
$395$	13386.6	1.70520
$396$	0	0
$397$	−9450.48	−1.19473	−0.597363	−	0.801971i	$-0.703785\pi$
$397$	−9450.48	−1.19473	−0.597363	+	0.801971i	$0.703785\pi$
$398$	−14822.8	−1.86683
$399$	0	0
$400$	−4563.68	−0.570460
$401$	2906.03	0.361895	0.180948	−	0.983493i	$-0.442084\pi$
$401$	2906.03	0.361895	0.180948	+	0.983493i	$0.442084\pi$
$402$	0	0
$403$	0	0
$404$	−2314.66	−0.285046
$405$	0	0
$406$	1531.03	0.187152
$407$	−3969.90	−0.483491
$408$	0	0
$409$	5795.93	0.700710	0.350355	−	0.936617i	$-0.386061\pi$
$409$	5795.93	0.700710	0.350355	+	0.936617i	$0.386061\pi$
$410$	−2552.40	−0.307449
$411$	0	0
$412$	−164.111	−0.0196242
$413$	−2487.76	−0.296404
$414$	0	0
$415$	−16350.1	−1.93396
$416$	0	0
$417$	0	0
$418$	8675.71	1.01517
$419$	4142.47	0.482990	0.241495	−	0.970402i	$-0.422362\pi$
$419$	4142.47	0.482990	0.241495	+	0.970402i	$0.422362\pi$
$420$	0	0
$421$	1426.92	0.165187	0.0825935	−	0.996583i	$-0.473680\pi$
$421$	1426.92	0.165187	0.0825935	+	0.996583i	$0.473680\pi$
$422$	−10132.5	−1.16882
$423$	0	0
$424$	−7777.66	−0.890841
$425$	−20.2903	−0.00231582
$426$	0	0
$427$	8037.04	0.910865
$428$	−2636.35	−0.297740
$429$	0	0
$430$	−8958.78	−1.00472
$431$	2543.08	0.284213	0.142106	−	0.989851i	$-0.454612\pi$
$431$	2543.08	0.284213	0.142106	+	0.989851i	$0.454612\pi$
$432$	0	0
$433$	−3448.44	−0.382728	−0.191364	−	0.981519i	$-0.561291\pi$
$433$	−3448.44	−0.382728	−0.191364	+	0.981519i	$0.561291\pi$
$434$	−7131.14	−0.788723
$435$	0	0
$436$	−4134.68	−0.454164
$437$	6306.65	0.690361
$438$	0	0
$439$	−12642.6	−1.37448	−0.687240	−	0.726431i	$-0.741178\pi$
$439$	−12642.6	−1.37448	−0.687240	+	0.726431i	$0.741178\pi$
$440$	−17492.2	−1.89524
$441$	0	0
$442$	0	0
$443$	−8486.59	−0.910181	−0.455091	−	0.890445i	$-0.650393\pi$
$443$	−8486.59	−0.910181	−0.455091	+	0.890445i	$0.650393\pi$
$444$	0	0
$445$	7344.74	0.782414
$446$	−3342.78	−0.354900
$447$	0	0
$448$	4710.45	0.496759
$449$	−11369.5	−1.19502	−0.597508	−	0.801863i	$-0.703842\pi$
$449$	−11369.5	−1.19502	−0.597508	+	0.801863i	$0.703842\pi$
$450$	0	0
$451$	−4019.23	−0.419642
$452$	3196.26	0.332610
$453$	0	0
$454$	16311.6	1.68621
$455$	0	0
$456$	0	0
$457$	11289.0	1.15553	0.577766	−	0.816202i	$-0.303925\pi$
$457$	11289.0	1.15553	0.577766	+	0.816202i	$0.303925\pi$
$458$	−5368.77	−0.547742
$459$	0	0
$460$	4222.01	0.427940
$461$	−3834.12	−0.387359	−0.193680	−	0.981065i	$-0.562042\pi$
$461$	−3834.12	−0.387359	−0.193680	+	0.981065i	$0.562042\pi$
$462$	0	0
$463$	1294.44	0.129930	0.0649651	−	0.997888i	$-0.479306\pi$
$463$	1294.44	0.129930	0.0649651	+	0.997888i	$0.479306\pi$
$464$	−2567.44	−0.256876
$465$	0	0
$466$	−9996.93	−0.993774
$467$	−13861.5	−1.37352	−0.686760	−	0.726884i	$-0.740968\pi$
$467$	−13861.5	−1.37352	−0.686760	+	0.726884i	$0.740968\pi$
$468$	0	0
$469$	−3856.22	−0.379667
$470$	9513.61	0.933681
$471$	0	0
$472$	3295.84	0.321405
$473$	−14107.3	−1.37136
$474$	0	0
$475$	2434.09	0.235124
$476$	−9.65386	−0.000929588 0
$477$	0	0
$478$	7429.29	0.710895
$479$	9627.66	0.918369	0.459185	−	0.888341i	$-0.348142\pi$
$479$	9627.66	0.918369	0.459185	+	0.888341i	$0.348142\pi$
$480$	0	0
$481$	0	0
$482$	16335.3	1.54368
$483$	0	0
$484$	6491.55	0.609650
$485$	21592.9	2.02162
$486$	0	0
$487$	−11952.8	−1.11219	−0.556094	−	0.831120i	$-0.687700\pi$
$487$	−11952.8	−1.11219	−0.556094	+	0.831120i	$0.687700\pi$
$488$	−10647.6	−0.987695
$489$	0	0
$490$	−5918.12	−0.545619
$491$	−5410.12	−0.497261	−0.248631	−	0.968598i	$-0.579980\pi$
$491$	−5410.12	−0.497261	−0.248631	+	0.968598i	$0.579980\pi$
$492$	0	0
$493$	−11.4149	−0.00104281
$494$	0	0
$495$	0	0
$496$	11958.4	1.08256
$497$	−873.867	−0.0788698
$498$	0	0
$499$	−14472.9	−1.29838	−0.649192	−	0.760624i	$-0.724893\pi$
$499$	−14472.9	−1.29838	−0.649192	+	0.760624i	$0.724893\pi$
$500$	−1761.52	−0.157555
$501$	0	0
$502$	−9148.97	−0.813423
$503$	601.940	0.0533582	0.0266791	−	0.999644i	$-0.491507\pi$
$503$	601.940	0.0533582	0.0266791	+	0.999644i	$0.491507\pi$
$504$	0	0
$505$	−15790.4	−1.39142
$506$	33319.8	2.92736
$507$	0	0
$508$	1225.39	0.107024
$509$	−17420.8	−1.51702	−0.758512	−	0.651659i	$-0.774073\pi$
$509$	−17420.8	−1.51702	−0.758512	+	0.651659i	$0.774073\pi$
$510$	0	0
$511$	4042.43	0.349954
$512$	−4831.90	−0.417074
$513$	0	0
$514$	19482.2	1.67184
$515$	−1119.55	−0.0957929
$516$	0	0
$517$	14981.0	1.27439
$518$	−2655.90	−0.225277
$519$	0	0
$520$	0	0
$521$	−12881.5	−1.08320	−0.541601	−	0.840636i	$-0.682182\pi$
$521$	−12881.5	−1.08320	−0.541601	+	0.840636i	$0.682182\pi$
$522$	0	0
$523$	16534.0	1.38238	0.691188	−	0.722675i	$-0.257088\pi$
$523$	16534.0	1.38238	0.691188	+	0.722675i	$0.257088\pi$
$524$	658.410	0.0548907
$525$	0	0
$526$	630.459	0.0522611
$527$	53.1678	0.00439473
$528$	0	0
$529$	12054.2	0.990730
$530$	17617.3	1.44386
$531$	0	0
$532$	1158.11	0.0943802
$533$	0	0
$534$	0	0
$535$	−17985.0	−1.45338
$536$	5108.80	0.411691
$537$	0	0
$538$	−3502.79	−0.280699
$539$	−9319.18	−0.744723
$540$	0	0
$541$	−19026.9	−1.51207	−0.756035	−	0.654531i	$-0.772866\pi$
$541$	−19026.9	−1.51207	−0.756035	+	0.654531i	$0.772866\pi$
$542$	−1283.68	−0.101732
$543$	0	0
$544$	29.8258	0.00235068
$545$	−28206.5	−2.21694
$546$	0	0
$547$	8153.61	0.637336	0.318668	−	0.947866i	$-0.396764\pi$
$547$	8153.61	0.637336	0.318668	+	0.947866i	$0.396764\pi$
$548$	4591.60	0.357926
$549$	0	0
$550$	12860.0	0.997002
$551$	1369.37	0.105875
$552$	0	0
$553$	14102.5	1.08445
$554$	913.031	0.0700198
$555$	0	0
$556$	2407.38	0.183625
$557$	−1308.06	−0.0995051	−0.0497525	−	0.998762i	$-0.515843\pi$
$557$	−1308.06	−0.0995051	−0.0497525	+	0.998762i	$0.515843\pi$
$558$	0	0
$559$	0	0
$560$	−14812.7	−1.11777
$561$	0	0
$562$	−16233.3	−1.21844
$563$	20736.1	1.55226	0.776129	−	0.630575i	$-0.217181\pi$
$563$	20736.1	1.55226	0.776129	+	0.630575i	$0.217181\pi$
$564$	0	0
$565$	21804.7	1.62359
$566$	16674.2	1.23829
$567$	0	0
$568$	1157.72	0.0855224
$569$	14982.0	1.10383	0.551913	−	0.833902i	$-0.313898\pi$
$569$	14982.0	1.10383	0.551913	+	0.833902i	$0.313898\pi$
$570$	0	0
$571$	−5668.79	−0.415467	−0.207734	−	0.978185i	$-0.566609\pi$
$571$	−5668.79	−0.415467	−0.207734	+	0.978185i	$0.566609\pi$
$572$	0	0
$573$	0	0
$574$	−2688.91	−0.195527
$575$	9348.32	0.678004
$576$	0	0
$577$	6872.94	0.495882	0.247941	−	0.968775i	$-0.420246\pi$
$577$	6872.94	0.495882	0.247941	+	0.968775i	$0.420246\pi$
$578$	−15531.4	−1.11768
$579$	0	0
$580$	916.731	0.0656297
$581$	−17224.5	−1.22993
$582$	0	0
$583$	27741.7	1.97074
$584$	−5355.49	−0.379472
$585$	0	0
$586$	−4570.34	−0.322182
$587$	−18593.4	−1.30738	−0.653691	−	0.756762i	$-0.726780\pi$
$587$	−18593.4	−1.30738	−0.653691	+	0.756762i	$0.726780\pi$
$588$	0	0
$589$	−6378.17	−0.446193
$590$	−7465.44	−0.520928
$591$	0	0
$592$	4453.77	0.309204
$593$	−15612.6	−1.08117	−0.540583	−	0.841291i	$-0.681796\pi$
$593$	−15612.6	−1.08117	−0.540583	+	0.841291i	$0.681796\pi$
$594$	0	0
$595$	−65.8580	−0.00453767
$596$	675.892	0.0464523
$597$	0	0
$598$	0	0
$599$	22979.8	1.56749	0.783747	−	0.621081i	$-0.213306\pi$
$599$	22979.8	1.56749	0.783747	+	0.621081i	$0.213306\pi$
$600$	0	0
$601$	2781.77	0.188803	0.0944015	−	0.995534i	$-0.469906\pi$
$601$	2781.77	0.188803	0.0944015	+	0.995534i	$0.469906\pi$
$602$	−9437.89	−0.638970
$603$	0	0
$604$	−3379.69	−0.227678
$605$	44284.9	2.97593
$606$	0	0
$607$	−4078.57	−0.272725	−0.136363	−	0.990659i	$-0.543541\pi$
$607$	−4078.57	−0.272725	−0.136363	+	0.990659i	$0.543541\pi$
$608$	−3578.00	−0.238663
$609$	0	0
$610$	24118.1	1.60084
$611$	0	0
$612$	0	0
$613$	−26290.1	−1.73221	−0.866107	−	0.499858i	$-0.833385\pi$
$613$	−26290.1	−1.73221	−0.866107	+	0.499858i	$0.833385\pi$
$614$	29784.4	1.95766
$615$	0	0
$616$	−18427.6	−1.20531
$617$	−4851.69	−0.316567	−0.158283	−	0.987394i	$-0.550596\pi$
$617$	−4851.69	−0.316567	−0.158283	+	0.987394i	$0.550596\pi$
$618$	0	0
$619$	4957.05	0.321875	0.160937	−	0.986965i	$-0.448548\pi$
$619$	4957.05	0.321875	0.160937	+	0.986965i	$0.448548\pi$
$620$	−4269.89	−0.276585
$621$	0	0
$622$	−24930.0	−1.60708
$623$	7737.54	0.497589
$624$	0	0
$625$	−19525.3	−1.24962
$626$	−1740.08	−0.111098
$627$	0	0
$628$	19.1289	0.00121548
$629$	19.8017	0.00125524
$630$	0	0
$631$	2725.11	0.171925	0.0859627	−	0.996298i	$-0.472603\pi$
$631$	2725.11	0.171925	0.0859627	+	0.996298i	$0.472603\pi$
$632$	−18683.3	−1.17592
$633$	0	0
$634$	477.283	0.0298980
$635$	8359.55	0.522423
$636$	0	0
$637$	0	0
$638$	7234.77	0.448946
$639$	0	0
$640$	23744.8	1.46655
$641$	−21396.3	−1.31841	−0.659207	−	0.751961i	$-0.729108\pi$
$641$	−21396.3	−1.31841	−0.659207	+	0.751961i	$0.729108\pi$
$642$	0	0
$643$	−21833.8	−1.33910	−0.669549	−	0.742768i	$-0.733513\pi$
$643$	−21833.8	−1.33910	−0.669549	+	0.742768i	$0.733513\pi$
$644$	4447.80	0.272155
$645$	0	0
$646$	−43.2740	−0.00263559
$647$	4869.06	0.295861	0.147931	−	0.988998i	$-0.452739\pi$
$647$	4869.06	0.295861	0.147931	+	0.988998i	$0.452739\pi$
$648$	0	0
$649$	−11755.7	−0.711021
$650$	0	0
$651$	0	0
$652$	−243.486	−0.0146252
$653$	16847.0	1.00961	0.504803	−	0.863235i	$-0.331565\pi$
$653$	16847.0	1.00961	0.504803	+	0.863235i	$0.331565\pi$
$654$	0	0
$655$	4491.63	0.267943
$656$	4509.12	0.268371
$657$	0	0
$658$	10022.4	0.593790
$659$	24741.0	1.46248	0.731240	−	0.682121i	$-0.238942\pi$
$659$	24741.0	1.46248	0.731240	+	0.682121i	$0.238942\pi$
$660$	0	0
$661$	16522.0	0.972210	0.486105	−	0.873900i	$-0.338417\pi$
$661$	16522.0	0.972210	0.486105	+	0.873900i	$0.338417\pi$
$662$	−22380.1	−1.31394
$663$	0	0
$664$	22819.3	1.33368
$665$	7900.53	0.460706
$666$	0	0
$667$	5259.19	0.305302
$668$	4409.55	0.255405
$669$	0	0
$670$	−11572.0	−0.667262
$671$	37978.4	2.18501
$672$	0	0
$673$	−20721.5	−1.18686	−0.593429	−	0.804886i	$-0.702226\pi$
$673$	−20721.5	−1.18686	−0.593429	+	0.804886i	$0.702226\pi$
$674$	−20970.3	−1.19844
$675$	0	0
$676$	0	0
$677$	−32407.3	−1.83975	−0.919877	−	0.392207i	$-0.871712\pi$
$677$	−32407.3	−1.83975	−0.919877	+	0.392207i	$0.871712\pi$
$678$	0	0
$679$	22747.7	1.28568
$680$	87.2500	0.00492042
$681$	0	0
$682$	−33697.6	−1.89201
$683$	10123.8	0.567168	0.283584	−	0.958947i	$-0.408477\pi$
$683$	10123.8	0.567168	0.283584	+	0.958947i	$0.408477\pi$
$684$	0	0
$685$	31323.6	1.74717
$686$	−21774.9	−1.21191
$687$	0	0
$688$	15826.7	0.877018
$689$	0	0
$690$	0	0
$691$	−29704.5	−1.63533	−0.817665	−	0.575695i	$-0.804732\pi$
$691$	−29704.5	−1.63533	−0.817665	+	0.575695i	$0.804732\pi$
$692$	426.495	0.0234291
$693$	0	0
$694$	−20245.8	−1.10738
$695$	16423.0	0.896343
$696$	0	0
$697$	20.0477	0.00108947
$698$	−531.418	−0.0288173
$699$	0	0
$700$	1716.66	0.0926908
$701$	20585.2	1.10912	0.554558	−	0.832145i	$-0.312887\pi$
$701$	20585.2	1.10912	0.554558	+	0.832145i	$0.312887\pi$
$702$	0	0
$703$	−2375.47	−0.127443
$704$	22258.9	1.19164
$705$	0	0
$706$	−26719.5	−1.42437
$707$	−16634.9	−0.884894
$708$	0	0
$709$	29660.2	1.57110	0.785552	−	0.618796i	$-0.212379\pi$
$709$	29660.2	1.57110	0.785552	+	0.618796i	$0.212379\pi$
$710$	−2622.36	−0.138613
$711$	0	0
$712$	−10250.8	−0.539560
$713$	−24495.9	−1.28664
$714$	0	0
$715$	0	0
$716$	−5917.44	−0.308862
$717$	0	0
$718$	8320.43	0.432473
$719$	−1279.68	−0.0663758	−0.0331879	−	0.999449i	$-0.510566\pi$
$719$	−1279.68	−0.0663758	−0.0331879	+	0.999449i	$0.510566\pi$
$720$	0	0
$721$	−1179.43	−0.0609211
$722$	−16492.4	−0.850118
$723$	0	0
$724$	4645.25	0.238452
$725$	2029.82	0.103980
$726$	0	0
$727$	6202.77	0.316435	0.158217	−	0.987404i	$-0.449425\pi$
$727$	6202.77	0.316435	0.158217	+	0.987404i	$0.449425\pi$
$728$	0	0
$729$	0	0
$730$	12130.8	0.615042
$731$	70.3663	0.00356032
$732$	0	0
$733$	−35501.2	−1.78891	−0.894453	−	0.447162i	$-0.852435\pi$
$733$	−35501.2	−1.78891	−0.894453	+	0.447162i	$0.852435\pi$
$734$	−41950.7	−2.10958
$735$	0	0
$736$	−13741.6	−0.688209
$737$	−18222.3	−0.910754
$738$	0	0
$739$	−3893.52	−0.193810	−0.0969050	−	0.995294i	$-0.530894\pi$
$739$	−3893.52	−0.193810	−0.0969050	+	0.995294i	$0.530894\pi$
$740$	−1590.27	−0.0789991
$741$	0	0
$742$	18559.5	0.918247
$743$	2214.99	0.109368	0.0546839	−	0.998504i	$-0.482585\pi$
$743$	2214.99	0.109368	0.0546839	+	0.998504i	$0.482585\pi$
$744$	0	0
$745$	4610.89	0.226751
$746$	−36924.8	−1.81221
$747$	0	0
$748$	−45.6186	−0.00222992
$749$	−18946.9	−0.924303
$750$	0	0
$751$	2604.34	0.126543	0.0632714	−	0.997996i	$-0.479847\pi$
$751$	2604.34	0.126543	0.0632714	+	0.997996i	$0.479847\pi$
$752$	−16806.9	−0.815006
$753$	0	0
$754$	0	0
$755$	−23056.0	−1.11138
$756$	0	0
$757$	−9685.72	−0.465037	−0.232519	−	0.972592i	$-0.574697\pi$
$757$	−9685.72	−0.465037	−0.232519	+	0.972592i	$0.574697\pi$
$758$	10679.8	0.511750
$759$	0	0
$760$	−10466.8	−0.499565
$761$	10105.6	0.481378	0.240689	−	0.970602i	$-0.422627\pi$
$761$	10105.6	0.481378	0.240689	+	0.970602i	$0.422627\pi$
$762$	0	0
$763$	−29715.0	−1.40990
$764$	8933.29	0.423030
$765$	0	0
$766$	−13849.6	−0.653273
$767$	0	0
$768$	0	0
$769$	−4573.20	−0.214452	−0.107226	−	0.994235i	$-0.534197\pi$
$769$	−4573.20	−0.214452	−0.107226	+	0.994235i	$0.534197\pi$
$770$	41740.7	1.95355
$771$	0	0
$772$	−2597.63	−0.121102
$773$	−7446.13	−0.346467	−0.173233	−	0.984881i	$-0.555421\pi$
$773$	−7446.13	−0.346467	−0.173233	+	0.984881i	$0.555421\pi$
$774$	0	0
$775$	−9454.34	−0.438206
$776$	−30136.6	−1.39412
$777$	0	0
$778$	39466.3	1.81868
$779$	−2404.99	−0.110613
$780$	0	0
$781$	−4129.39	−0.189195
$782$	−166.197	−0.00760000
$783$	0	0
$784$	10455.0	0.476269
$785$	130.496	0.00593324
$786$	0	0
$787$	3267.11	0.147980	0.0739898	−	0.997259i	$-0.476427\pi$
$787$	3267.11	0.147980	0.0739898	+	0.997259i	$0.476427\pi$
$788$	−1477.57	−0.0667973
$789$	0	0
$790$	42319.8	1.90591
$791$	22970.8	1.03255
$792$	0	0
$793$	0	0
$794$	−29876.3	−1.33535
$795$	0	0
$796$	−9350.03	−0.416336
$797$	3165.75	0.140698	0.0703492	−	0.997522i	$-0.477589\pi$
$797$	3165.75	0.140698	0.0703492	+	0.997522i	$0.477589\pi$
$798$	0	0
$799$	−74.7242	−0.00330857
$800$	−5303.66	−0.234391
$801$	0	0
$802$	9186.97	0.404493
$803$	19102.2	0.839478
$804$	0	0
$805$	30342.6	1.32849
$806$	0	0
$807$	0	0
$808$	22038.3	0.959534
$809$	−13589.7	−0.590592	−0.295296	−	0.955406i	$-0.595418\pi$
$809$	−13589.7	−0.590592	−0.295296	+	0.955406i	$0.595418\pi$
$810$	0	0
$811$	24977.7	1.08149	0.540744	−	0.841188i	$-0.318143\pi$
$811$	24977.7	1.08149	0.540744	+	0.841188i	$0.318143\pi$
$812$	965.758	0.0417383
$813$	0	0
$814$	−12550.3	−0.540401
$815$	−1661.04	−0.0713911
$816$	0	0
$817$	−8441.35	−0.361476
$818$	18323.0	0.783188
$819$	0	0
$820$	−1610.03	−0.0685666
$821$	8317.38	0.353567	0.176784	−	0.984250i	$-0.443431\pi$
$821$	8317.38	0.353567	0.176784	+	0.984250i	$0.443431\pi$
$822$	0	0
$823$	14462.9	0.612571	0.306286	−	0.951940i	$-0.400914\pi$
$823$	14462.9	0.612571	0.306286	+	0.951940i	$0.400914\pi$
$824$	1562.53	0.0660597
$825$	0	0
$826$	−7864.69	−0.331293
$827$	−17881.0	−0.751854	−0.375927	−	0.926649i	$-0.622676\pi$
$827$	−17881.0	−0.751854	−0.375927	+	0.926649i	$0.622676\pi$
$828$	0	0
$829$	351.169	0.0147124	0.00735620	−	0.999973i	$-0.497658\pi$
$829$	351.169	0.0147124	0.00735620	+	0.999973i	$0.497658\pi$
$830$	−51688.3	−2.16160
$831$	0	0
$832$	0	0
$833$	46.4836	0.00193345
$834$	0	0
$835$	30081.6	1.24673
$836$	5472.54	0.226402
$837$	0	0
$838$	13095.8	0.539841
$839$	33425.9	1.37543	0.687717	−	0.725978i	$-0.258613\pi$
$839$	33425.9	1.37543	0.687717	+	0.725978i	$0.258613\pi$
$840$	0	0
$841$	−23247.1	−0.953178
$842$	4510.99	0.184631
$843$	0	0
$844$	−6391.46	−0.260667
$845$	0	0
$846$	0	0
$847$	46653.3	1.89259
$848$	−31123.0	−1.26034
$849$	0	0
$850$	−64.1449	−0.00258841
$851$	−9123.18	−0.367495
$852$	0	0
$853$	−35097.5	−1.40881	−0.704406	−	0.709797i	$-0.748787\pi$
$853$	−35097.5	−1.40881	−0.704406	+	0.709797i	$0.748787\pi$
$854$	25407.9	1.01808
$855$	0	0
$856$	25101.2	1.00227
$857$	−15015.8	−0.598519	−0.299259	−	0.954172i	$-0.596740\pi$
$857$	−15015.8	−0.598519	−0.299259	+	0.954172i	$0.596740\pi$
$858$	0	0
$859$	19647.1	0.780383	0.390192	−	0.920734i	$-0.372409\pi$
$859$	19647.1	0.780383	0.390192	+	0.920734i	$0.372409\pi$
$860$	−5651.10	−0.224071
$861$	0	0
$862$	8039.57	0.317667
$863$	9035.14	0.356385	0.178192	−	0.983996i	$-0.442975\pi$
$863$	9035.14	0.356385	0.178192	+	0.983996i	$0.442975\pi$
$864$	0	0
$865$	2909.52	0.114366
$866$	−10901.7	−0.427778
$867$	0	0
$868$	−4498.24	−0.175899
$869$	66640.5	2.60141
$870$	0	0
$871$	0	0
$872$	39367.0	1.52883
$873$	0	0
$874$	19937.5	0.771621
$875$	−12659.6	−0.489113
$876$	0	0
$877$	−8254.16	−0.317814	−0.158907	−	0.987294i	$-0.550797\pi$
$877$	−8254.16	−0.317814	−0.158907	+	0.987294i	$0.550797\pi$
$878$	−39967.6	−1.53627
$879$	0	0
$880$	−69996.4	−2.68134
$881$	42107.4	1.61026	0.805128	−	0.593101i	$-0.202097\pi$
$881$	42107.4	1.61026	0.805128	+	0.593101i	$0.202097\pi$
$882$	0	0
$883$	−17584.7	−0.670183	−0.335091	−	0.942186i	$-0.608767\pi$
$883$	−17584.7	−0.670183	−0.335091	+	0.942186i	$0.608767\pi$
$884$	0	0
$885$	0	0
$886$	−26829.1	−1.01732
$887$	23845.8	0.902666	0.451333	−	0.892356i	$-0.350949\pi$
$887$	23845.8	0.902666	0.451333	+	0.892356i	$0.350949\pi$
$888$	0	0
$889$	8806.63	0.332244
$890$	23219.3	0.874509
$891$	0	0
$892$	−2108.59	−0.0791488
$893$	8964.14	0.335917
$894$	0	0
$895$	−40368.4	−1.50767
$896$	25014.6	0.932679
$897$	0	0
$898$	−35943.1	−1.33568
$899$	−5318.83	−0.197322
$900$	0	0
$901$	−138.374	−0.00511644
$902$	−12706.2	−0.469036
$903$	0	0
$904$	−30432.2	−1.11965
$905$	31689.6	1.16398
$906$	0	0
$907$	30565.7	1.11898	0.559492	−	0.828836i	$-0.310996\pi$
$907$	30565.7	1.11898	0.559492	+	0.828836i	$0.310996\pi$
$908$	10289.2	0.376055
$909$	0	0
$910$	0	0
$911$	−18556.9	−0.674882	−0.337441	−	0.941347i	$-0.609561\pi$
$911$	−18556.9	−0.674882	−0.337441	+	0.941347i	$0.609561\pi$
$912$	0	0
$913$	−81392.9	−2.95040
$914$	35688.6	1.29155
$915$	0	0
$916$	−3386.56	−0.122156
$917$	4731.84	0.170402
$918$	0	0
$919$	−9938.82	−0.356748	−0.178374	−	0.983963i	$-0.557084\pi$
$919$	−9938.82	−0.356748	−0.178374	+	0.983963i	$0.557084\pi$
$920$	−40198.5	−1.44055
$921$	0	0
$922$	−12121.0	−0.432954
$923$	0	0
$924$	0	0
$925$	−3521.15	−0.125162
$926$	4092.18	0.145224
$927$	0	0
$928$	−2983.73	−0.105545
$929$	5953.61	0.210260	0.105130	−	0.994458i	$-0.466474\pi$
$929$	5953.61	0.210260	0.105130	+	0.994458i	$0.466474\pi$
$930$	0	0
$931$	−5576.31	−0.196301
$932$	−6305.95	−0.221629
$933$	0	0
$934$	−43821.1	−1.53519
$935$	−311.207	−0.0108851
$936$	0	0
$937$	−24568.0	−0.856564	−0.428282	−	0.903645i	$-0.640881\pi$
$937$	−24568.0	−0.856564	−0.428282	+	0.903645i	$0.640881\pi$
$938$	−12190.9	−0.424356
$939$	0	0
$940$	6001.08	0.208227
$941$	11024.8	0.381931	0.190965	−	0.981597i	$-0.438838\pi$
$941$	11024.8	0.381931	0.190965	+	0.981597i	$0.438838\pi$
$942$	0	0
$943$	−9236.55	−0.318964
$944$	13188.6	0.454715
$945$	0	0
$946$	−44598.0	−1.53278
$947$	−6171.46	−0.211769	−0.105885	−	0.994378i	$-0.533767\pi$
$947$	−6171.46	−0.211769	−0.105885	+	0.994378i	$0.533767\pi$
$948$	0	0
$949$	0	0
$950$	7695.01	0.262799
$951$	0	0
$952$	91.9161	0.00312922
$953$	18838.4	0.640333	0.320166	−	0.947361i	$-0.396261\pi$
$953$	18838.4	0.640333	0.320166	+	0.947361i	$0.396261\pi$
$954$	0	0
$955$	60942.3	2.06497
$956$	4686.32	0.158542
$957$	0	0
$958$	30436.4	1.02647
$959$	32998.8	1.11114
$960$	0	0
$961$	−5017.33	−0.168418
$962$	0	0
$963$	0	0
$964$	10304.1	0.344267
$965$	−17720.8	−0.591144
$966$	0	0
$967$	24934.5	0.829203	0.414602	−	0.910003i	$-0.363921\pi$
$967$	24934.5	0.829203	0.414602	+	0.910003i	$0.363921\pi$
$968$	−61807.2	−2.05223
$969$	0	0
$970$	68262.8	2.25957
$971$	−32161.2	−1.06293	−0.531463	−	0.847082i	$-0.678357\pi$
$971$	−32161.2	−1.06293	−0.531463	+	0.847082i	$0.678357\pi$
$972$	0	0
$973$	17301.3	0.570044
$974$	−37787.2	−1.24310
$975$	0	0
$976$	−42607.4	−1.39737
$977$	18864.0	0.617720	0.308860	−	0.951107i	$-0.400052\pi$
$977$	18864.0	0.617720	0.308860	+	0.951107i	$0.400052\pi$
$978$	0	0
$979$	36563.1	1.19363
$980$	−3733.08	−0.121683
$981$	0	0
$982$	−17103.3	−0.555792
$983$	7883.83	0.255804	0.127902	−	0.991787i	$-0.459176\pi$
$983$	7883.83	0.255804	0.127902	+	0.991787i	$0.459176\pi$
$984$	0	0
$985$	−10079.9	−0.326063
$986$	−36.0867	−0.00116555
$987$	0	0
$988$	0	0
$989$	−32419.7	−1.04235
$990$	0	0
$991$	14172.3	0.454286	0.227143	−	0.973861i	$-0.427062\pi$
$991$	14172.3	0.454286	0.227143	+	0.973861i	$0.427062\pi$
$992$	13897.4	0.444802
$993$	0	0
$994$	−2762.60	−0.0881533
$995$	−63785.3	−2.03229
$996$	0	0
$997$	52462.7	1.66651	0.833256	−	0.552888i	$-0.186474\pi$
$997$	52462.7	1.66651	0.833256	+	0.552888i	$0.186474\pi$
$998$	−45753.8	−1.45121
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bh.1.7		9
3.2	odd	2		169.4.a.k.1.3	✓	9
13.12	even	2		1521.4.a.bg.1.3		9
39.2	even	12		169.4.e.h.147.5		36
39.5	even	4		169.4.b.g.168.14		18
39.8	even	4		169.4.b.g.168.5		18
39.11	even	12		169.4.e.h.147.14		36
39.17	odd	6		169.4.c.k.146.3		18
39.20	even	12		169.4.e.h.23.5		36
39.23	odd	6		169.4.c.k.22.3		18
39.29	odd	6		169.4.c.l.22.7		18
39.32	even	12		169.4.e.h.23.14		36
39.35	odd	6		169.4.c.l.146.7		18
39.38	odd	2		169.4.a.l.1.7	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.3	✓	9	3.2	odd	2
169.4.a.l.1.7	yes	9	39.38	odd	2
169.4.b.g.168.5		18	39.8	even	4
169.4.b.g.168.14		18	39.5	even	4
169.4.c.k.22.3		18	39.23	odd	6
169.4.c.k.146.3		18	39.17	odd	6
169.4.c.l.22.7		18	39.29	odd	6
169.4.c.l.146.7		18	39.35	odd	6
169.4.e.h.23.5		36	39.20	even	12
169.4.e.h.23.14		36	39.32	even	12
169.4.e.h.147.5		36	39.2	even	12
169.4.e.h.147.14		36	39.11	even	12
1521.4.a.bg.1.3		9	13.12	even	2
1521.4.a.bh.1.7		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)

\(f(q)\)	\(=\)	\(q+3.16135 q^{2} +1.99415 q^{4} +13.6039 q^{5} +14.3315 q^{7} -18.9866 q^{8} +O(q^{10})\)	\(q+3.16135 q^{2} +1.99415 q^{4} +13.6039 q^{5} +14.3315 q^{7} -18.9866 q^{8} +43.0068 q^{10} +67.7223 q^{11} +45.3068 q^{14} -75.9766 q^{16} -0.337795 q^{17} +40.5229 q^{19} +27.1282 q^{20} +214.094 q^{22} +155.632 q^{23} +60.0670 q^{25} +28.5791 q^{28} +33.7925 q^{29} -157.397 q^{31} -88.2957 q^{32} -1.06789 q^{34} +194.964 q^{35} -58.6204 q^{37} +128.107 q^{38} -258.293 q^{40} -59.3488 q^{41} -208.311 q^{43} +135.048 q^{44} +492.006 q^{46} +221.212 q^{47} -137.609 q^{49} +189.893 q^{50} +409.639 q^{53} +921.289 q^{55} -272.106 q^{56} +106.830 q^{58} -173.587 q^{59} +560.796 q^{61} -497.586 q^{62} +328.679 q^{64} -269.074 q^{67} -0.673613 q^{68} +616.351 q^{70} -60.9754 q^{71} +282.066 q^{73} -185.320 q^{74} +80.8086 q^{76} +970.560 q^{77} +984.026 q^{79} -1033.58 q^{80} -187.622 q^{82} -1201.86 q^{83} -4.59534 q^{85} -658.543 q^{86} -1285.82 q^{88} +539.899 q^{89} +310.352 q^{92} +699.328 q^{94} +551.271 q^{95} +1587.25 q^{97} -435.030 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8}+O(q^{10}) \) 9 * q + 5 * q^2 + 37 * q^4 + 30 * q^5 - 38 * q^7 + 60 * q^8	\( 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8} - 147 q^{10} + 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} - 161 q^{19} + 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} - 344 q^{28} - 280 q^{29} - 706 q^{31} + 680 q^{32} - 216 q^{34} - 20 q^{35} - 298 q^{37} + 739 q^{38} + 13 q^{40} + 1201 q^{41} - 533 q^{43} + 355 q^{44} + 840 q^{46} + 956 q^{47} + 403 q^{49} - 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} + 2877 q^{58} + 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} + 931 q^{67} + 1536 q^{68} + 4854 q^{70} + 2046 q^{71} + 45 q^{73} + 1990 q^{74} + 3608 q^{76} + 718 q^{77} + 412 q^{79} - 787 q^{80} + 2757 q^{82} + 3709 q^{83} + 2106 q^{85} + 125 q^{86} - 636 q^{88} + 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} + 1087 q^{97} - 282 q^{98}+O(q^{100}) \) 9 * q + 5 * q^2 + 37 * q^4 + 30 * q^5 - 38 * q^7 + 60 * q^8 - 147 * q^10 + 181 * q^11 + 147 * q^14 + 269 * q^16 + 55 * q^17 - 161 * q^19 + 370 * q^20 + 340 * q^22 + 204 * q^23 + 307 * q^25 - 344 * q^28 - 280 * q^29 - 706 * q^31 + 680 * q^32 - 216 * q^34 - 20 * q^35 - 298 * q^37 + 739 * q^38 + 13 * q^40 + 1201 * q^41 - 533 * q^43 + 355 * q^44 + 840 * q^46 + 956 * q^47 + 403 * q^49 - 1156 * q^50 + 278 * q^53 - 250 * q^55 - 250 * q^56 + 2877 * q^58 + 1377 * q^59 - 136 * q^61 - 2035 * q^62 + 284 * q^64 + 931 * q^67 + 1536 * q^68 + 4854 * q^70 + 2046 * q^71 + 45 * q^73 + 1990 * q^74 + 3608 * q^76 + 718 * q^77 + 412 * q^79 - 787 * q^80 + 2757 * q^82 + 3709 * q^83 + 2106 * q^85 + 125 * q^86 - 636 * q^88 + 1663 * q^89 - 4010 * q^92 - 2531 * q^94 + 1614 * q^95 + 1087 * q^97 - 282 * q^98

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