LMFDB - Embedded newform 2601.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.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:	$20.7690895657$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.3418281.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 $ x^6 - 9x^4 - 4x^3 + 18x^2 + 12x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 867)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.60714$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.74819 q^{2} +5.55252 q^{4} -0.973936 q^{5} +1.60333 q^{7} -9.76298 q^{8} +O(q^{10})$	$q-2.74819 q^{2} +5.55252 q^{4} -0.973936 q^{5} +1.60333 q^{7} -9.76298 q^{8} +2.67656 q^{10} +1.27043 q^{11} -3.74255 q^{13} -4.40623 q^{14} +15.7254 q^{16} -1.85988 q^{19} -5.40780 q^{20} -3.49137 q^{22} +6.88702 q^{23} -4.05145 q^{25} +10.2852 q^{26} +8.90250 q^{28} -2.80434 q^{29} +0.571419 q^{31} -23.6905 q^{32} -1.56154 q^{35} +7.49455 q^{37} +5.11131 q^{38} +9.50852 q^{40} -9.63099 q^{41} +5.17395 q^{43} +7.05407 q^{44} -18.9268 q^{46} +3.14540 q^{47} -4.42935 q^{49} +11.1341 q^{50} -20.7806 q^{52} -1.85361 q^{53} -1.23731 q^{55} -15.6532 q^{56} +7.70683 q^{58} +5.56300 q^{59} -14.3539 q^{61} -1.57037 q^{62} +33.6549 q^{64} +3.64501 q^{65} +1.49173 q^{67} +4.29139 q^{70} -5.69078 q^{71} -8.70537 q^{73} -20.5964 q^{74} -10.3270 q^{76} +2.03691 q^{77} -14.3455 q^{79} -15.3156 q^{80} +26.4677 q^{82} +14.2609 q^{83} -14.2190 q^{86} -12.4032 q^{88} +1.75856 q^{89} -6.00053 q^{91} +38.2403 q^{92} -8.64413 q^{94} +1.81141 q^{95} +6.22393 q^{97} +12.1727 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} + 9 q^{4} - 3 q^{5} - 3 q^{7} - 12 q^{8}+O(q^{10}) $ 6 * q - 3 * q^2 + 9 * q^4 - 3 * q^5 - 3 * q^7 - 12 * q^8	$ 6 q - 3 q^{2} + 9 q^{4} - 3 q^{5} - 3 q^{7} - 12 q^{8} - 12 q^{10} - 9 q^{11} + 9 q^{13} - 6 q^{14} + 15 q^{16} + 9 q^{19} + 6 q^{20} + 18 q^{22} - 9 q^{23} + 15 q^{25} + 12 q^{26} + 15 q^{28} - 6 q^{29} - 24 q^{31} - 42 q^{32} + 3 q^{37} + 6 q^{38} + 3 q^{40} - 18 q^{41} - 3 q^{44} - 15 q^{46} - 24 q^{47} + 21 q^{49} - 12 q^{50} - 18 q^{52} - 24 q^{53} - 24 q^{55} - 54 q^{56} - 3 q^{58} + 9 q^{59} - 21 q^{61} - 30 q^{62} + 24 q^{64} + 9 q^{65} - 6 q^{67} - 3 q^{70} - 27 q^{71} - 18 q^{73} - 36 q^{74} - 3 q^{76} - 33 q^{77} - 24 q^{79} - 3 q^{80} + 15 q^{82} - 6 q^{83} - 6 q^{86} + 24 q^{88} - 39 q^{91} - 15 q^{94} - 42 q^{95} + 33 q^{97} + 57 q^{98}+O(q^{100}) $ 6 * q - 3 * q^2 + 9 * q^4 - 3 * q^5 - 3 * q^7 - 12 * q^8 - 12 * q^10 - 9 * q^11 + 9 * q^13 - 6 * q^14 + 15 * q^16 + 9 * q^19 + 6 * q^20 + 18 * q^22 - 9 * q^23 + 15 * q^25 + 12 * q^26 + 15 * q^28 - 6 * q^29 - 24 * q^31 - 42 * q^32 + 3 * q^37 + 6 * q^38 + 3 * q^40 - 18 * q^41 - 3 * q^44 - 15 * q^46 - 24 * q^47 + 21 * q^49 - 12 * q^50 - 18 * q^52 - 24 * q^53 - 24 * q^55 - 54 * q^56 - 3 * q^58 + 9 * q^59 - 21 * q^61 - 30 * q^62 + 24 * q^64 + 9 * q^65 - 6 * q^67 - 3 * q^70 - 27 * q^71 - 18 * q^73 - 36 * q^74 - 3 * q^76 - 33 * q^77 - 24 * q^79 - 3 * q^80 + 15 * q^82 - 6 * q^83 - 6 * q^86 + 24 * q^88 - 39 * q^91 - 15 * q^94 - 42 * q^95 + 33 * q^97 + 57 * 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 $

$2$	−2.74819	−1.94326	−0.971630	−	0.236506i	$-0.923998\pi$
$2$	−2.74819	−1.94326	−0.971630	+	0.236506i	$0.923998\pi$
$3$	0	0
$3$	0	0
$4$	5.55252	2.77626
$5$	−0.973936	−0.435557	−0.217779	−	0.975998i	$-0.569881\pi$
$5$	−0.973936	−0.435557	−0.217779	+	0.975998i	$0.569881\pi$
$6$	0	0
$7$	1.60333	0.606000	0.303000	−	0.952991i	$-0.402012\pi$
$7$	1.60333	0.606000	0.303000	+	0.952991i	$0.402012\pi$
$8$	−9.76298	−3.45174
$9$	0	0
$10$	2.67656	0.846401
$11$	1.27043	0.383048	0.191524	−	0.981488i	$-0.438657\pi$
$11$	1.27043	0.383048	0.191524	+	0.981488i	$0.438657\pi$
$12$	0	0
$13$	−3.74255	−1.03800	−0.518999	−	0.854775i	$-0.673695\pi$
$13$	−3.74255	−1.03800	−0.518999	+	0.854775i	$0.673695\pi$
$14$	−4.40623	−1.17762
$15$	0	0
$16$	15.7254	3.93136
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−1.85988	−0.426687	−0.213343	−	0.976977i	$-0.568435\pi$
$19$	−1.85988	−0.426687	−0.213343	+	0.976977i	$0.568435\pi$
$20$	−5.40780	−1.20922
$21$	0	0
$22$	−3.49137	−0.744362
$23$	6.88702	1.43604	0.718021	−	0.696021i	$-0.245048\pi$
$23$	6.88702	1.43604	0.718021	+	0.696021i	$0.245048\pi$
$24$	0	0
$25$	−4.05145	−0.810290
$26$	10.2852	2.01710
$27$	0	0
$28$	8.90250	1.68241
$29$	−2.80434	−0.520752	−0.260376	−	0.965507i	$-0.583847\pi$
$29$	−2.80434	−0.520752	−0.260376	+	0.965507i	$0.583847\pi$
$30$	0	0
$31$	0.571419	0.102630	0.0513150	−	0.998683i	$-0.483659\pi$
$31$	0.571419	0.102630	0.0513150	+	0.998683i	$0.483659\pi$
$32$	−23.6905	−4.18792
$33$	0	0
$34$	0	0
$35$	−1.56154	−0.263948
$36$	0	0
$37$	7.49455	1.23210	0.616048	−	0.787708i	$-0.288733\pi$
$37$	7.49455	1.23210	0.616048	+	0.787708i	$0.288733\pi$
$38$	5.11131	0.829163
$39$	0	0
$40$	9.50852	1.50343
$41$	−9.63099	−1.50411	−0.752054	−	0.659102i	$-0.770937\pi$
$41$	−9.63099	−1.50411	−0.752054	+	0.659102i	$0.770937\pi$
$42$	0	0
$43$	5.17395	0.789020	0.394510	−	0.918892i	$-0.370914\pi$
$43$	5.17395	0.789020	0.394510	+	0.918892i	$0.370914\pi$
$44$	7.05407	1.06344
$45$	0	0
$46$	−18.9268	−2.79060
$47$	3.14540	0.458803	0.229402	−	0.973332i	$-0.426323\pi$
$47$	3.14540	0.458803	0.229402	+	0.973332i	$0.426323\pi$
$48$	0	0
$49$	−4.42935	−0.632764
$50$	11.1341	1.57460
$51$	0	0
$52$	−20.7806	−2.88175
$53$	−1.85361	−0.254613	−0.127306	−	0.991863i	$-0.540633\pi$
$53$	−1.85361	−0.254613	−0.127306	+	0.991863i	$0.540633\pi$
$54$	0	0
$55$	−1.23731	−0.166839
$56$	−15.6532	−2.09175
$57$	0	0
$58$	7.70683	1.01196
$59$	5.56300	0.724241	0.362120	−	0.932131i	$-0.382053\pi$
$59$	5.56300	0.724241	0.362120	+	0.932131i	$0.382053\pi$
$60$	0	0
$61$	−14.3539	−1.83783	−0.918915	−	0.394457i	$-0.870933\pi$
$61$	−14.3539	−1.83783	−0.918915	+	0.394457i	$0.870933\pi$
$62$	−1.57037	−0.199437
$63$	0	0
$64$	33.6549	4.20686
$65$	3.64501	0.452107
$66$	0	0
$67$	1.49173	0.182244	0.0911221	−	0.995840i	$-0.470955\pi$
$67$	1.49173	0.182244	0.0911221	+	0.995840i	$0.470955\pi$
$68$	0	0
$69$	0	0
$70$	4.29139	0.512919
$71$	−5.69078	−0.675371	−0.337686	−	0.941259i	$-0.609644\pi$
$71$	−5.69078	−0.675371	−0.337686	+	0.941259i	$0.609644\pi$
$72$	0	0
$73$	−8.70537	−1.01889	−0.509443	−	0.860504i	$-0.670149\pi$
$73$	−8.70537	−1.01889	−0.509443	+	0.860504i	$0.670149\pi$
$74$	−20.5964	−2.39428
$75$	0	0
$76$	−10.3270	−1.18459
$77$	2.03691	0.232127
$78$	0	0
$79$	−14.3455	−1.61399	−0.806997	−	0.590556i	$-0.798908\pi$
$79$	−14.3455	−1.61399	−0.806997	+	0.590556i	$0.798908\pi$
$80$	−15.3156	−1.71233
$81$	0	0
$82$	26.4677	2.92287
$83$	14.2609	1.56534	0.782670	−	0.622437i	$-0.213858\pi$
$83$	14.2609	1.56534	0.782670	+	0.622437i	$0.213858\pi$
$84$	0	0
$85$	0	0
$86$	−14.2190	−1.53327
$87$	0	0
$88$	−12.4032	−1.32218
$89$	1.75856	0.186407	0.0932037	−	0.995647i	$-0.470289\pi$
$89$	1.75856	0.186407	0.0932037	+	0.995647i	$0.470289\pi$
$90$	0	0
$91$	−6.00053	−0.629026
$92$	38.2403	3.98683
$93$	0	0
$94$	−8.64413	−0.891574
$95$	1.81141	0.185846
$96$	0	0
$97$	6.22393	0.631944	0.315972	−	0.948768i	$-0.397669\pi$
$97$	6.22393	0.631944	0.315972	+	0.948768i	$0.397669\pi$
$98$	12.1727	1.22963
$99$	0	0
$100$	−22.4958	−2.24958
$101$	−9.20928	−0.916357	−0.458179	−	0.888860i	$-0.651498\pi$
$101$	−9.20928	−0.916357	−0.458179	+	0.888860i	$0.651498\pi$
$102$	0	0
$103$	3.55736	0.350517	0.175258	−	0.984522i	$-0.443924\pi$
$103$	3.55736	0.350517	0.175258	+	0.984522i	$0.443924\pi$
$104$	36.5385	3.58289
$105$	0	0
$106$	5.09406	0.494778
$107$	−16.4671	−1.59193	−0.795966	−	0.605342i	$-0.793036\pi$
$107$	−16.4671	−1.59193	−0.795966	+	0.605342i	$0.793036\pi$
$108$	0	0
$109$	−2.12543	−0.203579	−0.101790	−	0.994806i	$-0.532457\pi$
$109$	−2.12543	−0.203579	−0.101790	+	0.994806i	$0.532457\pi$
$110$	3.40037	0.324212
$111$	0	0
$112$	25.2130	2.38241
$113$	1.92721	0.181297	0.0906483	−	0.995883i	$-0.471106\pi$
$113$	1.92721	0.181297	0.0906483	+	0.995883i	$0.471106\pi$
$114$	0	0
$115$	−6.70751	−0.625479
$116$	−15.5711	−1.44574
$117$	0	0
$118$	−15.2882	−1.40739
$119$	0	0
$120$	0	0
$121$	−9.38601	−0.853274
$122$	39.4472	3.57138
$123$	0	0
$124$	3.17282	0.284927
$125$	8.81553	0.788485
$126$	0	0
$127$	14.1318	1.25400	0.626999	−	0.779020i	$-0.284283\pi$
$127$	14.1318	1.25400	0.626999	+	0.779020i	$0.284283\pi$
$128$	−45.1089	−3.98710
$129$	0	0
$130$	−10.0171	−0.878562
$131$	14.8086	1.29383	0.646917	−	0.762560i	$-0.276058\pi$
$131$	14.8086	1.29383	0.646917	+	0.762560i	$0.276058\pi$
$132$	0	0
$133$	−2.98200	−0.258572
$134$	−4.09956	−0.354148
$135$	0	0
$136$	0	0
$137$	−9.28141	−0.792965	−0.396482	−	0.918042i	$-0.629769\pi$
$137$	−9.28141	−0.792965	−0.396482	+	0.918042i	$0.629769\pi$
$138$	0	0
$139$	11.4702	0.972891	0.486445	−	0.873711i	$-0.338293\pi$
$139$	11.4702	0.972891	0.486445	+	0.873711i	$0.338293\pi$
$140$	−8.67046	−0.732788
$141$	0	0
$142$	15.6393	1.31242
$143$	−4.75464	−0.397603
$144$	0	0
$145$	2.73124	0.226817
$146$	23.9240	1.97996
$147$	0	0
$148$	41.6136	3.42062
$149$	−14.7349	−1.20713	−0.603564	−	0.797315i	$-0.706253\pi$
$149$	−14.7349	−1.20713	−0.603564	+	0.797315i	$0.706253\pi$
$150$	0	0
$151$	3.92360	0.319298	0.159649	−	0.987174i	$-0.448964\pi$
$151$	3.92360	0.319298	0.159649	+	0.987174i	$0.448964\pi$
$152$	18.1580	1.47281
$153$	0	0
$154$	−5.59780	−0.451084
$155$	−0.556526	−0.0447012
$156$	0	0
$157$	21.5890	1.72299	0.861494	−	0.507768i	$-0.169529\pi$
$157$	21.5890	1.72299	0.861494	+	0.507768i	$0.169529\pi$
$158$	39.4240	3.13641
$159$	0	0
$160$	23.0730	1.82408
$161$	11.0421	0.870242
$162$	0	0
$163$	−12.1051	−0.948145	−0.474073	−	0.880486i	$-0.657217\pi$
$163$	−12.1051	−0.948145	−0.474073	+	0.880486i	$0.657217\pi$
$164$	−53.4763	−4.17580
$165$	0	0
$166$	−39.1916	−3.04186
$167$	8.51775	0.659123	0.329562	−	0.944134i	$-0.393099\pi$
$167$	8.51775	0.659123	0.329562	+	0.944134i	$0.393099\pi$
$168$	0	0
$169$	1.00670	0.0774385
$170$	0	0
$171$	0	0
$172$	28.7285	2.19053
$173$	−3.81941	−0.290385	−0.145192	−	0.989403i	$-0.546380\pi$
$173$	−3.81941	−0.290385	−0.145192	+	0.989403i	$0.546380\pi$
$174$	0	0
$175$	−6.49579	−0.491036
$176$	19.9780	1.50590
$177$	0	0
$178$	−4.83286	−0.362238
$179$	−14.8155	−1.10736	−0.553681	−	0.832729i	$-0.686777\pi$
$179$	−14.8155	−1.10736	−0.553681	+	0.832729i	$0.686777\pi$
$180$	0	0
$181$	−8.45308	−0.628313	−0.314156	−	0.949371i	$-0.601722\pi$
$181$	−8.45308	−0.628313	−0.314156	+	0.949371i	$0.601722\pi$
$182$	16.4906	1.22236
$183$	0	0
$184$	−67.2379	−4.95684
$185$	−7.29921	−0.536649
$186$	0	0
$187$	0	0
$188$	17.4649	1.27376
$189$	0	0
$190$	−4.97808	−0.361148
$191$	16.9002	1.22285	0.611426	−	0.791302i	$-0.290596\pi$
$191$	16.9002	1.22285	0.611426	+	0.791302i	$0.290596\pi$
$192$	0	0
$193$	5.78951	0.416738	0.208369	−	0.978050i	$-0.433185\pi$
$193$	5.78951	0.416738	0.208369	+	0.978050i	$0.433185\pi$
$194$	−17.1045	−1.22803
$195$	0	0
$196$	−24.5940	−1.75672
$197$	−13.7643	−0.980664	−0.490332	−	0.871536i	$-0.663124\pi$
$197$	−13.7643	−0.980664	−0.490332	+	0.871536i	$0.663124\pi$
$198$	0	0
$199$	−13.9310	−0.987544	−0.493772	−	0.869591i	$-0.664382\pi$
$199$	−13.9310	−0.987544	−0.493772	+	0.869591i	$0.664382\pi$
$200$	39.5542	2.79691
$201$	0	0
$202$	25.3088	1.78072
$203$	−4.49626	−0.315576
$204$	0	0
$205$	9.37996	0.655125
$206$	−9.77628	−0.681146
$207$	0	0
$208$	−58.8533	−4.08074
$209$	−2.36285	−0.163442
$210$	0	0
$211$	−4.47974	−0.308398	−0.154199	−	0.988040i	$-0.549280\pi$
$211$	−4.47974	−0.308398	−0.154199	+	0.988040i	$0.549280\pi$
$212$	−10.2922	−0.706871
$213$	0	0
$214$	45.2545	3.09354
$215$	−5.03909	−0.343663
$216$	0	0
$217$	0.916171	0.0621938
$218$	5.84108	0.395608
$219$	0	0
$220$	−6.87021	−0.463190
$221$	0	0
$222$	0	0
$223$	−18.0311	−1.20745	−0.603725	−	0.797193i	$-0.706317\pi$
$223$	−18.0311	−1.20745	−0.603725	+	0.797193i	$0.706317\pi$
$224$	−37.9835	−2.53788
$225$	0	0
$226$	−5.29633	−0.352306
$227$	−4.98499	−0.330865	−0.165433	−	0.986221i	$-0.552902\pi$
$227$	−4.98499	−0.330865	−0.165433	+	0.986221i	$0.552902\pi$
$228$	0	0
$229$	4.90015	0.323811	0.161905	−	0.986806i	$-0.448236\pi$
$229$	4.90015	0.323811	0.161905	+	0.986806i	$0.448236\pi$
$230$	18.4335	1.21547
$231$	0	0
$232$	27.3787	1.79750
$233$	−20.3716	−1.33459	−0.667294	−	0.744794i	$-0.732548\pi$
$233$	−20.3716	−1.33459	−0.667294	+	0.744794i	$0.732548\pi$
$234$	0	0
$235$	−3.06341	−0.199835
$236$	30.8887	2.01068
$237$	0	0
$238$	0	0
$239$	10.5715	0.683817	0.341908	−	0.939733i	$-0.388927\pi$
$239$	10.5715	0.683817	0.341908	+	0.939733i	$0.388927\pi$
$240$	0	0
$241$	20.4243	1.31565	0.657824	−	0.753172i	$-0.271477\pi$
$241$	20.4243	1.31565	0.657824	+	0.753172i	$0.271477\pi$
$242$	25.7945	1.65813
$243$	0	0
$244$	−79.7004	−5.10229
$245$	4.31390	0.275605
$246$	0	0
$247$	6.96071	0.442900
$248$	−5.57876	−0.354252
$249$	0	0
$250$	−24.2267	−1.53223
$251$	−7.57844	−0.478347	−0.239174	−	0.970977i	$-0.576876\pi$
$251$	−7.57844	−0.478347	−0.239174	+	0.970977i	$0.576876\pi$
$252$	0	0
$253$	8.74946	0.550074
$254$	−38.8369	−2.43685
$255$	0	0
$256$	56.6579	3.54112
$257$	−13.3407	−0.832169	−0.416085	−	0.909326i	$-0.636598\pi$
$257$	−13.3407	−0.832169	−0.416085	+	0.909326i	$0.636598\pi$
$258$	0	0
$259$	12.0162	0.746651
$260$	20.2390	1.25517
$261$	0	0
$262$	−40.6968	−2.51426
$263$	0.998798	0.0615885	0.0307943	−	0.999526i	$-0.490196\pi$
$263$	0.998798	0.0615885	0.0307943	+	0.999526i	$0.490196\pi$
$264$	0	0
$265$	1.80529	0.110898
$266$	8.19509	0.502473
$267$	0	0
$268$	8.28288	0.505957
$269$	−19.0477	−1.16136	−0.580679	−	0.814132i	$-0.697213\pi$
$269$	−19.0477	−1.16136	−0.580679	+	0.814132i	$0.697213\pi$
$270$	0	0
$271$	−11.8501	−0.719841	−0.359920	−	0.932983i	$-0.617196\pi$
$271$	−11.8501	−0.719841	−0.359920	+	0.932983i	$0.617196\pi$
$272$	0	0
$273$	0	0
$274$	25.5070	1.54094
$275$	−5.14707	−0.310380
$276$	0	0
$277$	−15.5321	−0.933231	−0.466615	−	0.884460i	$-0.654527\pi$
$277$	−15.5321	−0.933231	−0.466615	+	0.884460i	$0.654527\pi$
$278$	−31.5223	−1.89058
$279$	0	0
$280$	15.2452	0.911078
$281$	−10.4376	−0.622652	−0.311326	−	0.950303i	$-0.600773\pi$
$281$	−10.4376	−0.622652	−0.311326	+	0.950303i	$0.600773\pi$
$282$	0	0
$283$	−0.729929	−0.0433898	−0.0216949	−	0.999765i	$-0.506906\pi$
$283$	−0.729929	−0.0433898	−0.0216949	+	0.999765i	$0.506906\pi$
$284$	−31.5982	−1.87501
$285$	0	0
$286$	13.0666	0.772646
$287$	−15.4416	−0.911489
$288$	0	0
$289$	0	0
$290$	−7.50596	−0.440765
$291$	0	0
$292$	−48.3367	−2.82869
$293$	−28.1210	−1.64285	−0.821423	−	0.570319i	$-0.806820\pi$
$293$	−28.1210	−1.64285	−0.821423	+	0.570319i	$0.806820\pi$
$294$	0	0
$295$	−5.41800	−0.315448
$296$	−73.1692	−4.25287
$297$	0	0
$298$	40.4941	2.34576
$299$	−25.7750	−1.49061
$300$	0	0
$301$	8.29553	0.478146
$302$	−10.7828	−0.620479
$303$	0	0
$304$	−29.2475	−1.67746
$305$	13.9798	0.800480
$306$	0	0
$307$	−28.8163	−1.64463	−0.822317	−	0.569029i	$-0.807319\pi$
$307$	−28.8163	−1.64463	−0.822317	+	0.569029i	$0.807319\pi$
$308$	11.3100	0.644446
$309$	0	0
$310$	1.52944	0.0868661
$311$	−29.9417	−1.69784	−0.848918	−	0.528524i	$-0.822746\pi$
$311$	−29.9417	−1.69784	−0.848918	+	0.528524i	$0.822746\pi$
$312$	0	0
$313$	−27.8939	−1.57665	−0.788327	−	0.615256i	$-0.789053\pi$
$313$	−27.8939	−1.57665	−0.788327	+	0.615256i	$0.789053\pi$
$314$	−59.3305	−3.34821
$315$	0	0
$316$	−79.6536	−4.48087
$317$	−3.78669	−0.212682	−0.106341	−	0.994330i	$-0.533913\pi$
$317$	−3.78669	−0.212682	−0.106341	+	0.994330i	$0.533913\pi$
$318$	0	0
$319$	−3.56271	−0.199473
$320$	−32.7777	−1.83233
$321$	0	0
$322$	−30.3458	−1.69111
$323$	0	0
$324$	0	0
$325$	15.1628	0.841079
$326$	33.2671	1.84249
$327$	0	0
$328$	94.0272	5.19178
$329$	5.04310	0.278035
$330$	0	0
$331$	30.7956	1.69268	0.846341	−	0.532642i	$-0.178801\pi$
$331$	30.7956	1.69268	0.846341	+	0.532642i	$0.178801\pi$
$332$	79.1840	4.34579
$333$	0	0
$334$	−23.4084	−1.28085
$335$	−1.45285	−0.0793778
$336$	0	0
$337$	−12.5820	−0.685385	−0.342693	−	0.939448i	$-0.611339\pi$
$337$	−12.5820	−0.685385	−0.342693	+	0.939448i	$0.611339\pi$
$338$	−2.76660	−0.150483
$339$	0	0
$340$	0	0
$341$	0.725947	0.0393122
$342$	0	0
$343$	−18.3250	−0.989455
$344$	−50.5132	−2.72349
$345$	0	0
$346$	10.4965	0.564293
$347$	−7.64090	−0.410185	−0.205093	−	0.978743i	$-0.565750\pi$
$347$	−7.64090	−0.410185	−0.205093	+	0.978743i	$0.565750\pi$
$348$	0	0
$349$	19.5231	1.04505	0.522525	−	0.852624i	$-0.324990\pi$
$349$	19.5231	1.04505	0.522525	+	0.852624i	$0.324990\pi$
$350$	17.8516	0.954210
$351$	0	0
$352$	−30.0970	−1.60418
$353$	−23.7008	−1.26147	−0.630734	−	0.775999i	$-0.717246\pi$
$353$	−23.7008	−1.26147	−0.630734	+	0.775999i	$0.717246\pi$
$354$	0	0
$355$	5.54245	0.294163
$356$	9.76446	0.517516
$357$	0	0
$358$	40.7157	2.15189
$359$	10.5122	0.554815	0.277407	−	0.960752i	$-0.410525\pi$
$359$	10.5122	0.554815	0.277407	+	0.960752i	$0.410525\pi$
$360$	0	0
$361$	−15.5408	−0.817938
$362$	23.2306	1.22098
$363$	0	0
$364$	−33.3181	−1.74634
$365$	8.47847	0.443783
$366$	0	0
$367$	12.1454	0.633983	0.316992	−	0.948428i	$-0.397327\pi$
$367$	12.1454	0.633983	0.316992	+	0.948428i	$0.397327\pi$
$368$	108.301	5.64560
$369$	0	0
$370$	20.0596	1.04285
$371$	−2.97194	−0.154295
$372$	0	0
$373$	2.78161	0.144026	0.0720131	−	0.997404i	$-0.477058\pi$
$373$	2.78161	0.144026	0.0720131	+	0.997404i	$0.477058\pi$
$374$	0	0
$375$	0	0
$376$	−30.7085	−1.58367
$377$	10.4954	0.540539
$378$	0	0
$379$	−7.26396	−0.373125	−0.186562	−	0.982443i	$-0.559735\pi$
$379$	−7.26396	−0.373125	−0.186562	+	0.982443i	$0.559735\pi$
$380$	10.0579	0.515958
$381$	0	0
$382$	−46.4447	−2.37632
$383$	−4.78317	−0.244409	−0.122204	−	0.992505i	$-0.538996\pi$
$383$	−4.78317	−0.244409	−0.122204	+	0.992505i	$0.538996\pi$
$384$	0	0
$385$	−1.98382	−0.101105
$386$	−15.9106	−0.809830
$387$	0	0
$388$	34.5585	1.75444
$389$	−22.4276	−1.13712	−0.568562	−	0.822640i	$-0.692500\pi$
$389$	−22.4276	−1.13712	−0.568562	+	0.822640i	$0.692500\pi$
$390$	0	0
$391$	0	0
$392$	43.2437	2.18413
$393$	0	0
$394$	37.8268	1.90568
$395$	13.9716	0.702986
$396$	0	0
$397$	16.7854	0.842435	0.421218	−	0.906960i	$-0.361603\pi$
$397$	16.7854	0.842435	0.421218	+	0.906960i	$0.361603\pi$
$398$	38.2850	1.91905
$399$	0	0
$400$	−63.7108	−3.18554
$401$	−25.3654	−1.26669	−0.633343	−	0.773871i	$-0.718318\pi$
$401$	−25.3654	−1.26669	−0.633343	+	0.773871i	$0.718318\pi$
$402$	0	0
$403$	−2.13857	−0.106530
$404$	−51.1347	−2.54405
$405$	0	0
$406$	12.3566	0.613246
$407$	9.52128	0.471952
$408$	0	0
$409$	−6.63544	−0.328101	−0.164051	−	0.986452i	$-0.552456\pi$
$409$	−6.63544	−0.328101	−0.164051	+	0.986452i	$0.552456\pi$
$410$	−25.7779	−1.27308
$411$	0	0
$412$	19.7523	0.973126
$413$	8.91930	0.438890
$414$	0	0
$415$	−13.8892	−0.681795
$416$	88.6628	4.34705
$417$	0	0
$418$	6.49354	0.317610
$419$	−0.473185	−0.0231166	−0.0115583	−	0.999933i	$-0.503679\pi$
$419$	−0.473185	−0.0231166	−0.0115583	+	0.999933i	$0.503679\pi$
$420$	0	0
$421$	−3.73379	−0.181974	−0.0909868	−	0.995852i	$-0.529002\pi$
$421$	−3.73379	−0.181974	−0.0909868	+	0.995852i	$0.529002\pi$
$422$	12.3111	0.599297
$423$	0	0
$424$	18.0967	0.878855
$425$	0	0
$426$	0	0
$427$	−23.0140	−1.11372
$428$	−91.4337	−4.41962
$429$	0	0
$430$	13.8484	0.667827
$431$	−8.81746	−0.424722	−0.212361	−	0.977191i	$-0.568115\pi$
$431$	−8.81746	−0.424722	−0.212361	+	0.977191i	$0.568115\pi$
$432$	0	0
$433$	−19.6452	−0.944090	−0.472045	−	0.881574i	$-0.656484\pi$
$433$	−19.6452	−0.944090	−0.472045	+	0.881574i	$0.656484\pi$
$434$	−2.51781	−0.120859
$435$	0	0
$436$	−11.8015	−0.565189
$437$	−12.8091	−0.612740
$438$	0	0
$439$	−32.6773	−1.55960	−0.779801	−	0.626027i	$-0.784680\pi$
$439$	−32.6773	−1.55960	−0.779801	+	0.626027i	$0.784680\pi$
$440$	12.0799	0.575886
$441$	0	0
$442$	0	0
$443$	25.8884	1.23000	0.614998	−	0.788529i	$-0.289157\pi$
$443$	25.8884	1.23000	0.614998	+	0.788529i	$0.289157\pi$
$444$	0	0
$445$	−1.71273	−0.0811911
$446$	49.5527	2.34639
$447$	0	0
$448$	53.9597	2.54936
$449$	3.73605	0.176315	0.0881575	−	0.996107i	$-0.471902\pi$
$449$	3.73605	0.176315	0.0881575	+	0.996107i	$0.471902\pi$
$450$	0	0
$451$	−12.2355	−0.576146
$452$	10.7009	0.503326
$453$	0	0
$454$	13.6997	0.642957
$455$	5.84413	0.273977
$456$	0	0
$457$	19.1540	0.895989	0.447994	−	0.894036i	$-0.352138\pi$
$457$	19.1540	0.895989	0.447994	+	0.894036i	$0.352138\pi$
$458$	−13.4665	−0.629249
$459$	0	0
$460$	−37.2436	−1.73649
$461$	7.45279	0.347111	0.173556	−	0.984824i	$-0.444474\pi$
$461$	7.45279	0.347111	0.173556	+	0.984824i	$0.444474\pi$
$462$	0	0
$463$	11.8349	0.550017	0.275008	−	0.961442i	$-0.411319\pi$
$463$	11.8349	0.550017	0.275008	+	0.961442i	$0.411319\pi$
$464$	−44.0994	−2.04726
$465$	0	0
$466$	55.9850	2.59345
$467$	−14.3709	−0.665007	−0.332504	−	0.943102i	$-0.607893\pi$
$467$	−14.3709	−0.665007	−0.332504	+	0.943102i	$0.607893\pi$
$468$	0	0
$469$	2.39173	0.110440
$470$	8.41883	0.388332
$471$	0	0
$472$	−54.3115	−2.49989
$473$	6.57313	0.302233
$474$	0	0
$475$	7.53523	0.345740
$476$	0	0
$477$	0	0
$478$	−29.0526	−1.32883
$479$	17.2155	0.786598	0.393299	−	0.919411i	$-0.371334\pi$
$479$	17.2155	0.786598	0.393299	+	0.919411i	$0.371334\pi$
$480$	0	0
$481$	−28.0487	−1.27891
$482$	−56.1299	−2.55664
$483$	0	0
$484$	−52.1160	−2.36891
$485$	−6.06171	−0.275248
$486$	0	0
$487$	−0.844548	−0.0382701	−0.0191351	−	0.999817i	$-0.506091\pi$
$487$	−0.844548	−0.0382701	−0.0191351	+	0.999817i	$0.506091\pi$
$488$	140.137	6.34370
$489$	0	0
$490$	−11.8554	−0.535572
$491$	32.5155	1.46740	0.733701	−	0.679473i	$-0.237791\pi$
$491$	32.5155	1.46740	0.733701	+	0.679473i	$0.237791\pi$
$492$	0	0
$493$	0	0
$494$	−19.1293	−0.860669
$495$	0	0
$496$	8.98583	0.403475
$497$	−9.12417	−0.409275
$498$	0	0
$499$	29.1216	1.30366	0.651830	−	0.758365i	$-0.274002\pi$
$499$	29.1216	1.30366	0.651830	+	0.758365i	$0.274002\pi$
$500$	48.9484	2.18904
$501$	0	0
$502$	20.8270	0.929553
$503$	15.0131	0.669402	0.334701	−	0.942324i	$-0.391365\pi$
$503$	15.0131	0.669402	0.334701	+	0.942324i	$0.391365\pi$
$504$	0	0
$505$	8.96924	0.399126
$506$	−24.0451	−1.06894
$507$	0	0
$508$	78.4674	3.48143
$509$	12.5115	0.554563	0.277282	−	0.960789i	$-0.410567\pi$
$509$	12.5115	0.554563	0.277282	+	0.960789i	$0.410567\pi$
$510$	0	0
$511$	−13.9575	−0.617445
$512$	−65.4886	−2.89421
$513$	0	0
$514$	36.6627	1.61712
$515$	−3.46464	−0.152670
$516$	0	0
$517$	3.99600	0.175744
$518$	−33.0227	−1.45094
$519$	0	0
$520$	−35.5861	−1.56056
$521$	−34.4115	−1.50759	−0.753797	−	0.657107i	$-0.771780\pi$
$521$	−34.4115	−1.50759	−0.753797	+	0.657107i	$0.771780\pi$
$522$	0	0
$523$	7.31112	0.319693	0.159846	−	0.987142i	$-0.448900\pi$
$523$	7.31112	0.319693	0.159846	+	0.987142i	$0.448900\pi$
$524$	82.2252	3.59202
$525$	0	0
$526$	−2.74488	−0.119683
$527$	0	0
$528$	0	0
$529$	24.4310	1.06222
$530$	−4.96128	−0.215504
$531$	0	0
$532$	−16.5576	−0.717864
$533$	36.0445	1.56126
$534$	0	0
$535$	16.0379	0.693377
$536$	−14.5638	−0.629059
$537$	0	0
$538$	52.3466	2.25682
$539$	−5.62716	−0.242379
$540$	0	0
$541$	13.6378	0.586333	0.293167	−	0.956061i	$-0.405291\pi$
$541$	13.6378	0.586333	0.293167	+	0.956061i	$0.405291\pi$
$542$	32.5662	1.39884
$543$	0	0
$544$	0	0
$545$	2.07003	0.0886705
$546$	0	0
$547$	−24.0905	−1.03004	−0.515018	−	0.857179i	$-0.672215\pi$
$547$	−24.0905	−1.03004	−0.515018	+	0.857179i	$0.672215\pi$
$548$	−51.5352	−2.20148
$549$	0	0
$550$	14.1451	0.603149
$551$	5.21574	0.222198
$552$	0	0
$553$	−23.0005	−0.978080
$554$	42.6850	1.81351
$555$	0	0
$556$	63.6886	2.70100
$557$	16.7565	0.709995	0.354997	−	0.934867i	$-0.384482\pi$
$557$	16.7565	0.709995	0.354997	+	0.934867i	$0.384482\pi$
$558$	0	0
$559$	−19.3638	−0.819001
$560$	−24.5558	−1.03767
$561$	0	0
$562$	28.6843	1.20998
$563$	−29.0537	−1.22447	−0.612234	−	0.790677i	$-0.709729\pi$
$563$	−29.0537	−1.22447	−0.612234	+	0.790677i	$0.709729\pi$
$564$	0	0
$565$	−1.87698	−0.0789650
$566$	2.00598	0.0843176
$567$	0	0
$568$	55.5590	2.33120
$569$	44.4780	1.86461	0.932306	−	0.361669i	$-0.117793\pi$
$569$	44.4780	1.86461	0.932306	+	0.361669i	$0.117793\pi$
$570$	0	0
$571$	9.72434	0.406951	0.203475	−	0.979080i	$-0.434776\pi$
$571$	9.72434	0.406951	0.203475	+	0.979080i	$0.434776\pi$
$572$	−26.4002	−1.10385
$573$	0	0
$574$	42.4364	1.77126
$575$	−27.9024	−1.16361
$576$	0	0
$577$	−8.06446	−0.335728	−0.167864	−	0.985810i	$-0.553687\pi$
$577$	−8.06446	−0.335728	−0.167864	+	0.985810i	$0.553687\pi$
$578$	0	0
$579$	0	0
$580$	15.1653	0.629704
$581$	22.8649	0.948596
$582$	0	0
$583$	−2.35487	−0.0975289
$584$	84.9904	3.51693
$585$	0	0
$586$	77.2817	3.19248
$587$	−5.41771	−0.223613	−0.111806	−	0.993730i	$-0.535664\pi$
$587$	−5.41771	−0.223613	−0.111806	+	0.993730i	$0.535664\pi$
$588$	0	0
$589$	−1.06277	−0.0437908
$590$	14.8897	0.612998
$591$	0	0
$592$	117.855	4.84382
$593$	−44.1000	−1.81097	−0.905485	−	0.424378i	$-0.860493\pi$
$593$	−44.1000	−1.81097	−0.905485	+	0.424378i	$0.860493\pi$
$594$	0	0
$595$	0	0
$596$	−81.8157	−3.35130
$597$	0	0
$598$	70.8346	2.89664
$599$	8.34870	0.341119	0.170559	−	0.985347i	$-0.445443\pi$
$599$	8.34870	0.341119	0.170559	+	0.985347i	$0.445443\pi$
$600$	0	0
$601$	−6.71518	−0.273918	−0.136959	−	0.990577i	$-0.543733\pi$
$601$	−6.71518	−0.273918	−0.136959	+	0.990577i	$0.543733\pi$
$602$	−22.7976	−0.929162
$603$	0	0
$604$	21.7859	0.886454
$605$	9.14137	0.371650
$606$	0	0
$607$	−23.2211	−0.942515	−0.471257	−	0.881996i	$-0.656200\pi$
$607$	−23.2211	−0.942515	−0.471257	+	0.881996i	$0.656200\pi$
$608$	44.0615	1.78693
$609$	0	0
$610$	−38.4190	−1.55554
$611$	−11.7718	−0.476237
$612$	0	0
$613$	28.2852	1.14243	0.571213	−	0.820802i	$-0.306473\pi$
$613$	28.2852	1.14243	0.571213	+	0.820802i	$0.306473\pi$
$614$	79.1926	3.19595
$615$	0	0
$616$	−19.8863	−0.801242
$617$	14.1037	0.567794	0.283897	−	0.958855i	$-0.408373\pi$
$617$	14.1037	0.567794	0.283897	+	0.958855i	$0.408373\pi$
$618$	0	0
$619$	19.8067	0.796098	0.398049	−	0.917364i	$-0.369687\pi$
$619$	19.8067	0.796098	0.398049	+	0.917364i	$0.369687\pi$
$620$	−3.09012	−0.124102
$621$	0	0
$622$	82.2853	3.29934
$623$	2.81955	0.112963
$624$	0	0
$625$	11.6715	0.466860
$626$	76.6575	3.06385
$627$	0	0
$628$	119.873	4.78346
$629$	0	0
$630$	0	0
$631$	−14.7180	−0.585912	−0.292956	−	0.956126i	$-0.594639\pi$
$631$	−14.7180	−0.585912	−0.292956	+	0.956126i	$0.594639\pi$
$632$	140.055	5.57108
$633$	0	0
$634$	10.4065	0.413296
$635$	−13.7635	−0.546188
$636$	0	0
$637$	16.5771	0.656807
$638$	9.79097	0.387628
$639$	0	0
$640$	43.9332	1.73661
$641$	28.7891	1.13710	0.568550	−	0.822649i	$-0.307505\pi$
$641$	28.7891	1.13710	0.568550	+	0.822649i	$0.307505\pi$
$642$	0	0
$643$	42.8930	1.69154	0.845768	−	0.533551i	$-0.179143\pi$
$643$	42.8930	1.69154	0.845768	+	0.533551i	$0.179143\pi$
$644$	61.3117	2.41602
$645$	0	0
$646$	0	0
$647$	6.60497	0.259668	0.129834	−	0.991536i	$-0.458556\pi$
$647$	6.60497	0.259668	0.129834	+	0.991536i	$0.458556\pi$
$648$	0	0
$649$	7.06739	0.277419
$650$	−41.6701	−1.63443
$651$	0	0
$652$	−67.2139	−2.63230
$653$	31.2035	1.22109	0.610544	−	0.791982i	$-0.290951\pi$
$653$	31.2035	1.22109	0.610544	+	0.791982i	$0.290951\pi$
$654$	0	0
$655$	−14.4226	−0.563539
$656$	−151.452	−5.91319
$657$	0	0
$658$	−13.8594	−0.540294
$659$	44.9910	1.75260	0.876301	−	0.481764i	$-0.160004\pi$
$659$	44.9910	1.75260	0.876301	+	0.481764i	$0.160004\pi$
$660$	0	0
$661$	−34.9734	−1.36031	−0.680154	−	0.733069i	$-0.738087\pi$
$661$	−34.9734	−1.36031	−0.680154	+	0.733069i	$0.738087\pi$
$662$	−84.6321	−3.28932
$663$	0	0
$664$	−139.229	−5.40314
$665$	2.90428	0.112623
$666$	0	0
$667$	−19.3135	−0.747822
$668$	47.2950	1.82990
$669$	0	0
$670$	3.99271	0.154252
$671$	−18.2356	−0.703977
$672$	0	0
$673$	−30.4358	−1.17321	−0.586607	−	0.809872i	$-0.699537\pi$
$673$	−30.4358	−1.17321	−0.586607	+	0.809872i	$0.699537\pi$
$674$	34.5777	1.33188
$675$	0	0
$676$	5.58972	0.214989
$677$	−19.7489	−0.759013	−0.379507	−	0.925189i	$-0.623906\pi$
$677$	−19.7489	−0.759013	−0.379507	+	0.925189i	$0.623906\pi$
$678$	0	0
$679$	9.97899	0.382958
$680$	0	0
$681$	0	0
$682$	−1.99504	−0.0763939
$683$	−30.3630	−1.16181	−0.580903	−	0.813973i	$-0.697300\pi$
$683$	−30.3630	−1.16181	−0.580903	+	0.813973i	$0.697300\pi$
$684$	0	0
$685$	9.03950	0.345382
$686$	50.3604	1.92277
$687$	0	0
$688$	81.3627	3.10192
$689$	6.93722	0.264287
$690$	0	0
$691$	−23.0502	−0.876872	−0.438436	−	0.898762i	$-0.644467\pi$
$691$	−23.0502	−0.876872	−0.438436	+	0.898762i	$0.644467\pi$
$692$	−21.2074	−0.806183
$693$	0	0
$694$	20.9986	0.797096
$695$	−11.1712	−0.423750
$696$	0	0
$697$	0	0
$698$	−53.6532	−2.03080
$699$	0	0
$700$	−36.0680	−1.36324
$701$	−7.14153	−0.269732	−0.134866	−	0.990864i	$-0.543060\pi$
$701$	−7.14153	−0.269732	−0.134866	+	0.990864i	$0.543060\pi$
$702$	0	0
$703$	−13.9390	−0.525719
$704$	42.7561	1.61143
$705$	0	0
$706$	65.1343	2.45136
$707$	−14.7655	−0.555313
$708$	0	0
$709$	4.09970	0.153967	0.0769837	−	0.997032i	$-0.475471\pi$
$709$	4.09970	0.153967	0.0769837	+	0.997032i	$0.475471\pi$
$710$	−15.2317	−0.571635
$711$	0	0
$712$	−17.1688	−0.643429
$713$	3.93538	0.147381
$714$	0	0
$715$	4.63071	0.173179
$716$	−82.2633	−3.07433
$717$	0	0
$718$	−28.8896	−1.07815
$719$	−34.6999	−1.29409	−0.647044	−	0.762453i	$-0.723995\pi$
$719$	−34.6999	−1.29409	−0.647044	+	0.762453i	$0.723995\pi$
$720$	0	0
$721$	5.70360	0.212413
$722$	42.7091	1.58947
$723$	0	0
$724$	−46.9359	−1.74436
$725$	11.3616	0.421960
$726$	0	0
$727$	−26.5639	−0.985201	−0.492601	−	0.870255i	$-0.663954\pi$
$727$	−26.5639	−0.985201	−0.492601	+	0.870255i	$0.663954\pi$
$728$	58.5831	2.17123
$729$	0	0
$730$	−23.3004	−0.862387
$731$	0	0
$732$	0	0
$733$	37.7168	1.39310	0.696551	−	0.717508i	$-0.254717\pi$
$733$	37.7168	1.39310	0.696551	+	0.717508i	$0.254717\pi$
$734$	−33.3777	−1.23199
$735$	0	0
$736$	−163.157	−6.01404
$737$	1.89514	0.0698083
$738$	0	0
$739$	−23.9658	−0.881597	−0.440798	−	0.897606i	$-0.645305\pi$
$739$	−23.9658	−0.881597	−0.440798	+	0.897606i	$0.645305\pi$
$740$	−40.5290	−1.48988
$741$	0	0
$742$	8.16743	0.299836
$743$	−3.68672	−0.135253	−0.0676263	−	0.997711i	$-0.521543\pi$
$743$	−3.68672	−0.135253	−0.0676263	+	0.997711i	$0.521543\pi$
$744$	0	0
$745$	14.3508	0.525773
$746$	−7.64438	−0.279880
$747$	0	0
$748$	0	0
$749$	−26.4021	−0.964710
$750$	0	0
$751$	−36.5811	−1.33486	−0.667432	−	0.744670i	$-0.732607\pi$
$751$	−36.5811	−1.33486	−0.667432	+	0.744670i	$0.732607\pi$
$752$	49.4628	1.80372
$753$	0	0
$754$	−28.8432	−1.05041
$755$	−3.82133	−0.139073
$756$	0	0
$757$	−27.8039	−1.01055	−0.505274	−	0.862959i	$-0.668609\pi$
$757$	−27.8039	−1.01055	−0.505274	+	0.862959i	$0.668609\pi$
$758$	19.9627	0.725078
$759$	0	0
$760$	−17.6847	−0.641493
$761$	9.87946	0.358130	0.179065	−	0.983837i	$-0.442693\pi$
$761$	9.87946	0.358130	0.179065	+	0.983837i	$0.442693\pi$
$762$	0	0
$763$	−3.40776	−0.123369
$764$	93.8385	3.39496
$765$	0	0
$766$	13.1450	0.474950
$767$	−20.8198	−0.751760
$768$	0	0
$769$	−22.1066	−0.797184	−0.398592	−	0.917128i	$-0.630501\pi$
$769$	−22.1066	−0.797184	−0.398592	+	0.917128i	$0.630501\pi$
$770$	5.45190	0.196473
$771$	0	0
$772$	32.1464	1.15697
$773$	2.17272	0.0781472	0.0390736	−	0.999236i	$-0.487559\pi$
$773$	2.17272	0.0781472	0.0390736	+	0.999236i	$0.487559\pi$
$774$	0	0
$775$	−2.31508	−0.0831600
$776$	−60.7641	−2.18131
$777$	0	0
$778$	61.6352	2.20973
$779$	17.9125	0.641783
$780$	0	0
$781$	−7.22972	−0.258700
$782$	0	0
$783$	0	0
$784$	−69.6535	−2.48762
$785$	−21.0263	−0.750460
$786$	0	0
$787$	35.5971	1.26890	0.634450	−	0.772964i	$-0.281227\pi$
$787$	35.5971	1.26890	0.634450	+	0.772964i	$0.281227\pi$
$788$	−76.4264	−2.72258
$789$	0	0
$790$	−38.3965	−1.36609
$791$	3.08994	0.109866
$792$	0	0
$793$	53.7202	1.90766
$794$	−46.1294	−1.63707
$795$	0	0
$796$	−77.3523	−2.74168
$797$	39.7769	1.40897	0.704486	−	0.709718i	$-0.251178\pi$
$797$	39.7769	1.40897	0.704486	+	0.709718i	$0.251178\pi$
$798$	0	0
$799$	0	0
$800$	95.9807	3.39343
$801$	0	0
$802$	69.7087	2.46150
$803$	−11.0595	−0.390283
$804$	0	0
$805$	−10.7543	−0.379040
$806$	5.87718	0.207015
$807$	0	0
$808$	89.9100	3.16302
$809$	47.8030	1.68066	0.840331	−	0.542073i	$-0.182360\pi$
$809$	47.8030	1.68066	0.840331	+	0.542073i	$0.182360\pi$
$810$	0	0
$811$	−23.1361	−0.812419	−0.406209	−	0.913780i	$-0.633150\pi$
$811$	−23.1361	−0.812419	−0.406209	+	0.913780i	$0.633150\pi$
$812$	−24.9656	−0.876121
$813$	0	0
$814$	−26.1662	−0.917126
$815$	11.7896	0.412972
$816$	0	0
$817$	−9.62295	−0.336664
$818$	18.2354	0.637586
$819$	0	0
$820$	52.0824	1.81880
$821$	−5.93172	−0.207019	−0.103509	−	0.994628i	$-0.533007\pi$
$821$	−5.93172	−0.207019	−0.103509	+	0.994628i	$0.533007\pi$
$822$	0	0
$823$	38.3577	1.33706	0.668532	−	0.743683i	$-0.266923\pi$
$823$	38.3577	1.33706	0.668532	+	0.743683i	$0.266923\pi$
$824$	−34.7304	−1.20989
$825$	0	0
$826$	−24.5119	−0.852877
$827$	−13.8448	−0.481430	−0.240715	−	0.970596i	$-0.577382\pi$
$827$	−13.8448	−0.481430	−0.240715	+	0.970596i	$0.577382\pi$
$828$	0	0
$829$	−8.51046	−0.295581	−0.147790	−	0.989019i	$-0.547216\pi$
$829$	−8.51046	−0.295581	−0.147790	+	0.989019i	$0.547216\pi$
$830$	38.1701	1.32490
$831$	0	0
$832$	−125.955	−4.36671
$833$	0	0
$834$	0	0
$835$	−8.29574	−0.287086
$836$	−13.1198	−0.453756
$837$	0	0
$838$	1.30040	0.0449216
$839$	−17.7846	−0.613992	−0.306996	−	0.951711i	$-0.599324\pi$
$839$	−17.7846	−0.613992	−0.306996	+	0.951711i	$0.599324\pi$
$840$	0	0
$841$	−21.1357	−0.728817
$842$	10.2611	0.353622
$843$	0	0
$844$	−24.8738	−0.856193
$845$	−0.980461	−0.0337289
$846$	0	0
$847$	−15.0488	−0.517084
$848$	−29.1488	−1.00097
$849$	0	0
$850$	0	0
$851$	51.6151	1.76934
$852$	0	0
$853$	−7.51126	−0.257181	−0.128590	−	0.991698i	$-0.541045\pi$
$853$	−7.51126	−0.257181	−0.128590	+	0.991698i	$0.541045\pi$
$854$	63.2467	2.16426
$855$	0	0
$856$	160.768	5.49493
$857$	−40.6449	−1.38840	−0.694202	−	0.719780i	$-0.744242\pi$
$857$	−40.6449	−1.38840	−0.694202	+	0.719780i	$0.744242\pi$
$858$	0	0
$859$	−9.64918	−0.329226	−0.164613	−	0.986358i	$-0.552637\pi$
$859$	−9.64918	−0.329226	−0.164613	+	0.986358i	$0.552637\pi$
$860$	−27.9797	−0.954099
$861$	0	0
$862$	24.2320	0.825346
$863$	55.6122	1.89306	0.946530	−	0.322615i	$-0.104562\pi$
$863$	55.6122	1.89306	0.946530	+	0.322615i	$0.104562\pi$
$864$	0	0
$865$	3.71986	0.126479
$866$	53.9888	1.83461
$867$	0	0
$868$	5.08706	0.172666
$869$	−18.2249	−0.618237
$870$	0	0
$871$	−5.58289	−0.189169
$872$	20.7506	0.702702
$873$	0	0
$874$	35.2017	1.19071
$875$	14.1342	0.477822
$876$	0	0
$877$	−6.12694	−0.206892	−0.103446	−	0.994635i	$-0.532987\pi$
$877$	−6.12694	−0.206892	−0.103446	+	0.994635i	$0.532987\pi$
$878$	89.8032	3.03071
$879$	0	0
$880$	−19.4573	−0.655906
$881$	−50.3089	−1.69495	−0.847476	−	0.530834i	$-0.821879\pi$
$881$	−50.3089	−1.69495	−0.847476	+	0.530834i	$0.821879\pi$
$882$	0	0
$883$	17.5033	0.589033	0.294516	−	0.955646i	$-0.404841\pi$
$883$	17.5033	0.589033	0.294516	+	0.955646i	$0.404841\pi$
$884$	0	0
$885$	0	0
$886$	−71.1462	−2.39020
$887$	−51.4337	−1.72697	−0.863487	−	0.504371i	$-0.831724\pi$
$887$	−51.4337	−1.72697	−0.863487	+	0.504371i	$0.831724\pi$
$888$	0	0
$889$	22.6579	0.759923
$890$	4.70689	0.157775
$891$	0	0
$892$	−100.118	−3.35219
$893$	−5.85008	−0.195765
$894$	0	0
$895$	14.4293	0.482320
$896$	−72.3243	−2.41619
$897$	0	0
$898$	−10.2673	−0.342626
$899$	−1.60245	−0.0534448
$900$	0	0
$901$	0	0
$902$	33.6253	1.11960
$903$	0	0
$904$	−18.8153	−0.625788
$905$	8.23276	0.273666
$906$	0	0
$907$	9.48750	0.315027	0.157514	−	0.987517i	$-0.449652\pi$
$907$	9.48750	0.315027	0.157514	+	0.987517i	$0.449652\pi$
$908$	−27.6793	−0.918568
$909$	0	0
$910$	−16.0607	−0.532409
$911$	−34.7482	−1.15126	−0.575630	−	0.817710i	$-0.695243\pi$
$911$	−34.7482	−1.15126	−0.575630	+	0.817710i	$0.695243\pi$
$912$	0	0
$913$	18.1175	0.599600
$914$	−52.6389	−1.74114
$915$	0	0
$916$	27.2082	0.898983
$917$	23.7430	0.784064
$918$	0	0
$919$	31.3770	1.03503	0.517515	−	0.855674i	$-0.326857\pi$
$919$	31.3770	1.03503	0.517515	+	0.855674i	$0.326857\pi$
$920$	65.4853	2.15899
$921$	0	0
$922$	−20.4817	−0.674528
$923$	21.2980	0.701034
$924$	0	0
$925$	−30.3638	−0.998355
$926$	−32.5246	−1.06883
$927$	0	0
$928$	66.4360	2.18087
$929$	35.7794	1.17388	0.586942	−	0.809629i	$-0.300332\pi$
$929$	35.7794	1.17388	0.586942	+	0.809629i	$0.300332\pi$
$930$	0	0
$931$	8.23807	0.269992
$932$	−113.114	−3.70517
$933$	0	0
$934$	39.4940	1.29228
$935$	0	0
$936$	0	0
$937$	27.0422	0.883430	0.441715	−	0.897156i	$-0.354370\pi$
$937$	27.0422	0.883430	0.441715	+	0.897156i	$0.354370\pi$
$938$	−6.57293	−0.214614
$939$	0	0
$940$	−17.0097	−0.554794
$941$	46.5730	1.51824	0.759118	−	0.650953i	$-0.225630\pi$
$941$	46.5730	1.51824	0.759118	+	0.650953i	$0.225630\pi$
$942$	0	0
$943$	−66.3288	−2.15996
$944$	87.4806	2.84725
$945$	0	0
$946$	−18.0642	−0.587317
$947$	30.4381	0.989106	0.494553	−	0.869148i	$-0.335332\pi$
$947$	30.4381	0.989106	0.494553	+	0.869148i	$0.335332\pi$
$948$	0	0
$949$	32.5803	1.05760
$950$	−20.7082	−0.671863
$951$	0	0
$952$	0	0
$953$	−44.8788	−1.45377	−0.726883	−	0.686761i	$-0.759032\pi$
$953$	−44.8788	−1.45377	−0.726883	+	0.686761i	$0.759032\pi$
$954$	0	0
$955$	−16.4597	−0.532622
$956$	58.6987	1.89845
$957$	0	0
$958$	−47.3115	−1.52857
$959$	−14.8811	−0.480537
$960$	0	0
$961$	−30.6735	−0.989467
$962$	77.0832	2.48526
$963$	0	0
$964$	113.407	3.65258
$965$	−5.63861	−0.181513
$966$	0	0
$967$	−4.42726	−0.142371	−0.0711855	−	0.997463i	$-0.522678\pi$
$967$	−4.42726	−0.142371	−0.0711855	+	0.997463i	$0.522678\pi$
$968$	91.6355	2.94528
$969$	0	0
$970$	16.6587	0.534878
$971$	39.6727	1.27316	0.636579	−	0.771212i	$-0.280349\pi$
$971$	39.6727	1.27316	0.636579	+	0.771212i	$0.280349\pi$
$972$	0	0
$973$	18.3905	0.589572
$974$	2.32097	0.0743688
$975$	0	0
$976$	−225.722	−7.22517
$977$	36.9476	1.18206	0.591029	−	0.806650i	$-0.298722\pi$
$977$	36.9476	1.18206	0.591029	+	0.806650i	$0.298722\pi$
$978$	0	0
$979$	2.23413	0.0714030
$980$	23.9530	0.765151
$981$	0	0
$982$	−89.3585	−2.85154
$983$	−20.2089	−0.644564	−0.322282	−	0.946644i	$-0.604450\pi$
$983$	−20.2089	−0.644564	−0.322282	+	0.946644i	$0.604450\pi$
$984$	0	0
$985$	13.4055	0.427135
$986$	0	0
$987$	0	0
$988$	38.6495	1.22960
$989$	35.6331	1.13307
$990$	0	0
$991$	28.0215	0.890134	0.445067	−	0.895497i	$-0.353180\pi$
$991$	28.0215	0.890134	0.445067	+	0.895497i	$0.353180\pi$
$992$	−13.5372	−0.429806
$993$	0	0
$994$	25.0749	0.795328
$995$	13.5679	0.430132
$996$	0	0
$997$	−4.70005	−0.148852	−0.0744260	−	0.997227i	$-0.523712\pi$
$997$	−4.70005	−0.148852	−0.0744260	+	0.997227i	$0.523712\pi$
$998$	−80.0314	−2.53335
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bh.1.1		6
3.2	odd	2		867.2.a.o.1.6	✓	6
17.16	even	2		2601.2.a.bi.1.1		6
51.2	odd	8		867.2.e.k.616.11		24
51.5	even	16		867.2.h.m.688.2		48
51.8	odd	8		867.2.e.k.829.2		24
51.11	even	16		867.2.h.m.733.12		48
51.14	even	16		867.2.h.m.757.12		48
51.20	even	16		867.2.h.m.757.11		48
51.23	even	16		867.2.h.m.733.11		48
51.26	odd	8		867.2.e.k.829.1		24
51.29	even	16		867.2.h.m.688.1		48
51.32	odd	8		867.2.e.k.616.12		24
51.38	odd	4		867.2.d.g.577.2		12
51.41	even	16		867.2.h.m.712.2		48
51.44	even	16		867.2.h.m.712.1		48
51.47	odd	4		867.2.d.g.577.1		12
51.50	odd	2		867.2.a.p.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
867.2.a.o.1.6	✓	6	3.2	odd	2
867.2.a.p.1.6	yes	6	51.50	odd	2
867.2.d.g.577.1		12	51.47	odd	4
867.2.d.g.577.2		12	51.38	odd	4
867.2.e.k.616.11		24	51.2	odd	8
867.2.e.k.616.12		24	51.32	odd	8
867.2.e.k.829.1		24	51.26	odd	8
867.2.e.k.829.2		24	51.8	odd	8
867.2.h.m.688.1		48	51.29	even	16
867.2.h.m.688.2		48	51.5	even	16
867.2.h.m.712.1		48	51.44	even	16
867.2.h.m.712.2		48	51.41	even	16
867.2.h.m.733.11		48	51.23	even	16
867.2.h.m.733.12		48	51.11	even	16
867.2.h.m.757.11		48	51.20	even	16
867.2.h.m.757.12		48	51.14	even	16
2601.2.a.bh.1.1		6	1.1	even	1	trivial
2601.2.a.bi.1.1		6	17.16	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:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q-2.74819 q^{2} +5.55252 q^{4} -0.973936 q^{5} +1.60333 q^{7} -9.76298 q^{8} +O(q^{10})\)	\(q-2.74819 q^{2} +5.55252 q^{4} -0.973936 q^{5} +1.60333 q^{7} -9.76298 q^{8} +2.67656 q^{10} +1.27043 q^{11} -3.74255 q^{13} -4.40623 q^{14} +15.7254 q^{16} -1.85988 q^{19} -5.40780 q^{20} -3.49137 q^{22} +6.88702 q^{23} -4.05145 q^{25} +10.2852 q^{26} +8.90250 q^{28} -2.80434 q^{29} +0.571419 q^{31} -23.6905 q^{32} -1.56154 q^{35} +7.49455 q^{37} +5.11131 q^{38} +9.50852 q^{40} -9.63099 q^{41} +5.17395 q^{43} +7.05407 q^{44} -18.9268 q^{46} +3.14540 q^{47} -4.42935 q^{49} +11.1341 q^{50} -20.7806 q^{52} -1.85361 q^{53} -1.23731 q^{55} -15.6532 q^{56} +7.70683 q^{58} +5.56300 q^{59} -14.3539 q^{61} -1.57037 q^{62} +33.6549 q^{64} +3.64501 q^{65} +1.49173 q^{67} +4.29139 q^{70} -5.69078 q^{71} -8.70537 q^{73} -20.5964 q^{74} -10.3270 q^{76} +2.03691 q^{77} -14.3455 q^{79} -15.3156 q^{80} +26.4677 q^{82} +14.2609 q^{83} -14.2190 q^{86} -12.4032 q^{88} +1.75856 q^{89} -6.00053 q^{91} +38.2403 q^{92} -8.64413 q^{94} +1.81141 q^{95} +6.22393 q^{97} +12.1727 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 9 q^{4} - 3 q^{5} - 3 q^{7} - 12 q^{8}+O(q^{10}) \) 6 * q - 3 * q^2 + 9 * q^4 - 3 * q^5 - 3 * q^7 - 12 * q^8	\( 6 q - 3 q^{2} + 9 q^{4} - 3 q^{5} - 3 q^{7} - 12 q^{8} - 12 q^{10} - 9 q^{11} + 9 q^{13} - 6 q^{14} + 15 q^{16} + 9 q^{19} + 6 q^{20} + 18 q^{22} - 9 q^{23} + 15 q^{25} + 12 q^{26} + 15 q^{28} - 6 q^{29} - 24 q^{31} - 42 q^{32} + 3 q^{37} + 6 q^{38} + 3 q^{40} - 18 q^{41} - 3 q^{44} - 15 q^{46} - 24 q^{47} + 21 q^{49} - 12 q^{50} - 18 q^{52} - 24 q^{53} - 24 q^{55} - 54 q^{56} - 3 q^{58} + 9 q^{59} - 21 q^{61} - 30 q^{62} + 24 q^{64} + 9 q^{65} - 6 q^{67} - 3 q^{70} - 27 q^{71} - 18 q^{73} - 36 q^{74} - 3 q^{76} - 33 q^{77} - 24 q^{79} - 3 q^{80} + 15 q^{82} - 6 q^{83} - 6 q^{86} + 24 q^{88} - 39 q^{91} - 15 q^{94} - 42 q^{95} + 33 q^{97} + 57 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 + 9 * q^4 - 3 * q^5 - 3 * q^7 - 12 * q^8 - 12 * q^10 - 9 * q^11 + 9 * q^13 - 6 * q^14 + 15 * q^16 + 9 * q^19 + 6 * q^20 + 18 * q^22 - 9 * q^23 + 15 * q^25 + 12 * q^26 + 15 * q^28 - 6 * q^29 - 24 * q^31 - 42 * q^32 + 3 * q^37 + 6 * q^38 + 3 * q^40 - 18 * q^41 - 3 * q^44 - 15 * q^46 - 24 * q^47 + 21 * q^49 - 12 * q^50 - 18 * q^52 - 24 * q^53 - 24 * q^55 - 54 * q^56 - 3 * q^58 + 9 * q^59 - 21 * q^61 - 30 * q^62 + 24 * q^64 + 9 * q^65 - 6 * q^67 - 3 * q^70 - 27 * q^71 - 18 * q^73 - 36 * q^74 - 3 * q^76 - 33 * q^77 - 24 * q^79 - 3 * q^80 + 15 * q^82 - 6 * q^83 - 6 * q^86 + 24 * q^88 - 39 * q^91 - 15 * q^94 - 42 * q^95 + 33 * q^97 + 57 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more