LMFDB - Embedded newform 549.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [549,2,Mod(1,549)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(549, 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("549.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 549 = 3^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	549.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:	$4.38378707097$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 61)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.311108 q^{2} -1.90321 q^{4} +2.52543 q^{5} -3.21432 q^{7} +1.21432 q^{8} +O(q^{10})$	\(q-0.311108 q^{2} -1.90321 q^{4} +2.52543 q^{5} -3.21432 q^{7} +1.21432 q^{8} -0.785680 q^{10} -4.31111 q^{11} +1.42864 q^{13} +1.00000 q^{14} +3.42864 q^{16} -1.52543 q^{17} -6.70964 q^{19} -4.80642 q^{20} +1.34122 q^{22} -1.68889 q^{23} +1.37778 q^{25} -0.444461 q^{26} +6.11753 q^{28} -3.52543 q^{29} +1.65878 q^{31} -3.49532 q^{32} +0.474572 q^{34} -8.11753 q^{35} -8.70964 q^{37} +2.08742 q^{38} +3.06668 q^{40} +7.85728 q^{41} -2.47457 q^{43} +8.20495 q^{44} +0.525428 q^{46} -7.47949 q^{47} +3.33185 q^{49} -0.428639 q^{50} -2.71900 q^{52} +0.622216 q^{53} -10.8874 q^{55} -3.90321 q^{56} +1.09679 q^{58} -11.9699 q^{59} +1.00000 q^{61} -0.516060 q^{62} -5.76986 q^{64} +3.60793 q^{65} +5.34767 q^{67} +2.90321 q^{68} +2.52543 q^{70} -2.34122 q^{71} -6.95407 q^{73} +2.70964 q^{74} +12.7699 q^{76} +13.8573 q^{77} +13.3017 q^{79} +8.65878 q^{80} -2.44446 q^{82} +11.6128 q^{83} -3.85236 q^{85} +0.769859 q^{86} -5.23506 q^{88} +10.2351 q^{89} -4.59210 q^{91} +3.21432 q^{92} +2.32693 q^{94} -16.9447 q^{95} +12.1017 q^{97} -1.03657 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} + q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + q^4 + q^5 - 3 * q^7 - 3 * q^8	$ 3 q - q^{2} + q^{4} + q^{5} - 3 q^{7} - 3 q^{8} - 9 q^{10} - 13 q^{11} - 9 q^{13} + 3 q^{14} - 3 q^{16} + 2 q^{17} - q^{20} + 11 q^{22} - 5 q^{23} + 4 q^{25} - q^{26} + 5 q^{28} - 4 q^{29} - 2 q^{31} + 3 q^{32} + 8 q^{34} - 11 q^{35} - 6 q^{37} - 14 q^{38} + 9 q^{40} - 3 q^{41} - 14 q^{43} - 9 q^{44} - 5 q^{46} + 4 q^{47} - 10 q^{49} + 12 q^{50} - 15 q^{52} + 2 q^{53} - 13 q^{55} - 5 q^{56} + 10 q^{58} - 29 q^{59} + 3 q^{61} + 32 q^{62} - 11 q^{64} + 17 q^{65} + 9 q^{67} + 2 q^{68} + q^{70} - 14 q^{71} - q^{73} - 12 q^{74} + 32 q^{76} + 15 q^{77} + 13 q^{79} + 19 q^{80} - 7 q^{82} + 8 q^{83} - 18 q^{85} - 4 q^{86} + 11 q^{88} + 4 q^{89} - 7 q^{91} + 3 q^{92} + 20 q^{94} - 4 q^{95} + 10 q^{97} + 4 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 + q^5 - 3 * q^7 - 3 * q^8 - 9 * q^10 - 13 * q^11 - 9 * q^13 + 3 * q^14 - 3 * q^16 + 2 * q^17 - q^20 + 11 * q^22 - 5 * q^23 + 4 * q^25 - q^26 + 5 * q^28 - 4 * q^29 - 2 * q^31 + 3 * q^32 + 8 * q^34 - 11 * q^35 - 6 * q^37 - 14 * q^38 + 9 * q^40 - 3 * q^41 - 14 * q^43 - 9 * q^44 - 5 * q^46 + 4 * q^47 - 10 * q^49 + 12 * q^50 - 15 * q^52 + 2 * q^53 - 13 * q^55 - 5 * q^56 + 10 * q^58 - 29 * q^59 + 3 * q^61 + 32 * q^62 - 11 * q^64 + 17 * q^65 + 9 * q^67 + 2 * q^68 + q^70 - 14 * q^71 - q^73 - 12 * q^74 + 32 * q^76 + 15 * q^77 + 13 * q^79 + 19 * q^80 - 7 * q^82 + 8 * q^83 - 18 * q^85 - 4 * q^86 + 11 * q^88 + 4 * q^89 - 7 * q^91 + 3 * q^92 + 20 * q^94 - 4 * q^95 + 10 * q^97 + 4 * 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 $

$2$	−0.311108	−0.219986	−0.109993	−	0.993932i	$-0.535083\pi$
$2$	−0.311108	−0.219986	−0.109993	+	0.993932i	$0.535083\pi$
$3$	0	0
$3$	0	0
$4$	−1.90321	−0.951606
$5$	2.52543	1.12941	0.564703	−	0.825294i	$-0.308991\pi$
$5$	2.52543	1.12941	0.564703	+	0.825294i	$0.308991\pi$
$6$	0	0
$7$	−3.21432	−1.21490	−0.607449	−	0.794358i	$-0.707807\pi$
$7$	−3.21432	−1.21490	−0.607449	+	0.794358i	$0.707807\pi$
$8$	1.21432	0.429327
$9$	0	0
$10$	−0.785680	−0.248454
$11$	−4.31111	−1.29985	−0.649924	−	0.759999i	$-0.725199\pi$
$11$	−4.31111	−1.29985	−0.649924	+	0.759999i	$0.725199\pi$
$12$	0	0
$13$	1.42864	0.396233	0.198117	−	0.980178i	$-0.436518\pi$
$13$	1.42864	0.396233	0.198117	+	0.980178i	$0.436518\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	3.42864	0.857160
$17$	−1.52543	−0.369971	−0.184985	−	0.982741i	$-0.559224\pi$
$17$	−1.52543	−0.369971	−0.184985	+	0.982741i	$0.559224\pi$
$18$	0	0
$19$	−6.70964	−1.53930	−0.769648	−	0.638468i	$-0.779568\pi$
$19$	−6.70964	−1.53930	−0.769648	+	0.638468i	$0.779568\pi$
$20$	−4.80642	−1.07475
$21$	0	0
$22$	1.34122	0.285949
$23$	−1.68889	−0.352158	−0.176079	−	0.984376i	$-0.556341\pi$
$23$	−1.68889	−0.352158	−0.176079	+	0.984376i	$0.556341\pi$
$24$	0	0
$25$	1.37778	0.275557
$26$	−0.444461	−0.0871660
$27$	0	0
$28$	6.11753	1.15610
$29$	−3.52543	−0.654655	−0.327328	−	0.944911i	$-0.606148\pi$
$29$	−3.52543	−0.654655	−0.327328	+	0.944911i	$0.606148\pi$
$30$	0	0
$31$	1.65878	0.297926	0.148963	−	0.988843i	$-0.452407\pi$
$31$	1.65878	0.297926	0.148963	+	0.988843i	$0.452407\pi$
$32$	−3.49532	−0.617890
$33$	0	0
$34$	0.474572	0.0813885
$35$	−8.11753	−1.37211
$36$	0	0
$37$	−8.70964	−1.43186	−0.715928	−	0.698174i	$-0.753996\pi$
$37$	−8.70964	−1.43186	−0.715928	+	0.698174i	$0.753996\pi$
$38$	2.08742	0.338624
$39$	0	0
$40$	3.06668	0.484884
$41$	7.85728	1.22710	0.613550	−	0.789656i	$-0.289741\pi$
$41$	7.85728	1.22710	0.613550	+	0.789656i	$0.289741\pi$
$42$	0	0
$43$	−2.47457	−0.377369	−0.188684	−	0.982038i	$-0.560422\pi$
$43$	−2.47457	−0.377369	−0.188684	+	0.982038i	$0.560422\pi$
$44$	8.20495	1.23694
$45$	0	0
$46$	0.525428	0.0774701
$47$	−7.47949	−1.09100	−0.545498	−	0.838112i	$-0.683660\pi$
$47$	−7.47949	−1.09100	−0.545498	+	0.838112i	$0.683660\pi$
$48$	0	0
$49$	3.33185	0.475979
$50$	−0.428639	−0.0606188
$51$	0	0
$52$	−2.71900	−0.377058
$53$	0.622216	0.0854679	0.0427339	−	0.999086i	$-0.486393\pi$
$53$	0.622216	0.0854679	0.0427339	+	0.999086i	$0.486393\pi$
$54$	0	0
$55$	−10.8874	−1.46806
$56$	−3.90321	−0.521589
$57$	0	0
$58$	1.09679	0.144015
$59$	−11.9699	−1.55835	−0.779173	−	0.626808i	$-0.784361\pi$
$59$	−11.9699	−1.55835	−0.779173	+	0.626808i	$0.784361\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−0.516060	−0.0655396
$63$	0	0
$64$	−5.76986	−0.721232
$65$	3.60793	0.447508
$66$	0	0
$67$	5.34767	0.653322	0.326661	−	0.945142i	$-0.394076\pi$
$67$	5.34767	0.653322	0.326661	+	0.945142i	$0.394076\pi$
$68$	2.90321	0.352066
$69$	0	0
$70$	2.52543	0.301846
$71$	−2.34122	−0.277852	−0.138926	−	0.990303i	$-0.544365\pi$
$71$	−2.34122	−0.277852	−0.138926	+	0.990303i	$0.544365\pi$
$72$	0	0
$73$	−6.95407	−0.813912	−0.406956	−	0.913448i	$-0.633410\pi$
$73$	−6.95407	−0.813912	−0.406956	+	0.913448i	$0.633410\pi$
$74$	2.70964	0.314989
$75$	0	0
$76$	12.7699	1.46480
$77$	13.8573	1.57918
$78$	0	0
$79$	13.3017	1.49656	0.748281	−	0.663382i	$-0.230879\pi$
$79$	13.3017	1.49656	0.748281	+	0.663382i	$0.230879\pi$
$80$	8.65878	0.968081
$81$	0	0
$82$	−2.44446	−0.269946
$83$	11.6128	1.27468	0.637338	−	0.770585i	$-0.280036\pi$
$83$	11.6128	1.27468	0.637338	+	0.770585i	$0.280036\pi$
$84$	0	0
$85$	−3.85236	−0.417847
$86$	0.769859	0.0830160
$87$	0	0
$88$	−5.23506	−0.558060
$89$	10.2351	1.08491	0.542457	−	0.840083i	$-0.317494\pi$
$89$	10.2351	1.08491	0.542457	+	0.840083i	$0.317494\pi$
$90$	0	0
$91$	−4.59210	−0.481383
$92$	3.21432	0.335116
$93$	0	0
$94$	2.32693	0.240004
$95$	−16.9447	−1.73849
$96$	0	0
$97$	12.1017	1.22874	0.614371	−	0.789017i	$-0.289410\pi$
$97$	12.1017	1.22874	0.614371	+	0.789017i	$0.289410\pi$
$98$	−1.03657	−0.104709
$99$	0	0
$100$	−2.62222	−0.262222
$101$	−3.33185	−0.331532	−0.165766	−	0.986165i	$-0.553010\pi$
$101$	−3.33185	−0.331532	−0.165766	+	0.986165i	$0.553010\pi$
$102$	0	0
$103$	−2.70964	−0.266988	−0.133494	−	0.991050i	$-0.542620\pi$
$103$	−2.70964	−0.266988	−0.133494	+	0.991050i	$0.542620\pi$
$104$	1.73483	0.170114
$105$	0	0
$106$	−0.193576	−0.0188018
$107$	4.66370	0.450857	0.225429	−	0.974260i	$-0.427622\pi$
$107$	4.66370	0.450857	0.225429	+	0.974260i	$0.427622\pi$
$108$	0	0
$109$	4.24443	0.406543	0.203271	−	0.979122i	$-0.434843\pi$
$109$	4.24443	0.406543	0.203271	+	0.979122i	$0.434843\pi$
$110$	3.38715	0.322952
$111$	0	0
$112$	−11.0207	−1.04136
$113$	7.29529	0.686283	0.343141	−	0.939284i	$-0.388509\pi$
$113$	7.29529	0.686283	0.343141	+	0.939284i	$0.388509\pi$
$114$	0	0
$115$	−4.26517	−0.397730
$116$	6.70964	0.622974
$117$	0	0
$118$	3.72393	0.342815
$119$	4.90321	0.449477
$120$	0	0
$121$	7.58565	0.689605
$122$	−0.311108	−0.0281664
$123$	0	0
$124$	−3.15701	−0.283508
$125$	−9.14764	−0.818190
$126$	0	0
$127$	−3.85236	−0.341841	−0.170921	−	0.985285i	$-0.554674\pi$
$127$	−3.85236	−0.341841	−0.170921	+	0.985285i	$0.554674\pi$
$128$	8.78568	0.776552
$129$	0	0
$130$	−1.12245	−0.0984457
$131$	3.95407	0.345468	0.172734	−	0.984968i	$-0.444740\pi$
$131$	3.95407	0.345468	0.172734	+	0.984968i	$0.444740\pi$
$132$	0	0
$133$	21.5669	1.87009
$134$	−1.66370	−0.143722
$135$	0	0
$136$	−1.85236	−0.158838
$137$	−3.41435	−0.291708	−0.145854	−	0.989306i	$-0.546593\pi$
$137$	−3.41435	−0.291708	−0.145854	+	0.989306i	$0.546593\pi$
$138$	0	0
$139$	1.21432	0.102997	0.0514986	−	0.998673i	$-0.483600\pi$
$139$	1.21432	0.102997	0.0514986	+	0.998673i	$0.483600\pi$
$140$	15.4494	1.30571
$141$	0	0
$142$	0.728372	0.0611236
$143$	−6.15902	−0.515043
$144$	0	0
$145$	−8.90321	−0.739372
$146$	2.16346	0.179050
$147$	0	0
$148$	16.5763	1.36256
$149$	−7.42864	−0.608578	−0.304289	−	0.952580i	$-0.598419\pi$
$149$	−7.42864	−0.608578	−0.304289	+	0.952580i	$0.598419\pi$
$150$	0	0
$151$	11.9382	0.971521	0.485760	−	0.874092i	$-0.338543\pi$
$151$	11.9382	0.971521	0.485760	+	0.874092i	$0.338543\pi$
$152$	−8.14764	−0.660861
$153$	0	0
$154$	−4.31111	−0.347399
$155$	4.18913	0.336479
$156$	0	0
$157$	−14.2494	−1.13722	−0.568611	−	0.822606i	$-0.692519\pi$
$157$	−14.2494	−1.13722	−0.568611	+	0.822606i	$0.692519\pi$
$158$	−4.13828	−0.329224
$159$	0	0
$160$	−8.82717	−0.697849
$161$	5.42864	0.427837
$162$	0	0
$163$	23.3733	1.83074	0.915371	−	0.402612i	$-0.131898\pi$
$163$	23.3733	1.83074	0.915371	+	0.402612i	$0.131898\pi$
$164$	−14.9541	−1.16772
$165$	0	0
$166$	−3.61285	−0.280411
$167$	21.8938	1.69420	0.847098	−	0.531436i	$-0.178348\pi$
$167$	21.8938	1.69420	0.847098	+	0.531436i	$0.178348\pi$
$168$	0	0
$169$	−10.9590	−0.842999
$170$	1.19850	0.0919206
$171$	0	0
$172$	4.70964	0.359106
$173$	−1.05086	−0.0798950	−0.0399475	−	0.999202i	$-0.512719\pi$
$173$	−1.05086	−0.0798950	−0.0399475	+	0.999202i	$0.512719\pi$
$174$	0	0
$175$	−4.42864	−0.334774
$176$	−14.7812	−1.11418
$177$	0	0
$178$	−3.18421	−0.238666
$179$	−18.5620	−1.38739	−0.693694	−	0.720270i	$-0.744018\pi$
$179$	−18.5620	−1.38739	−0.693694	+	0.720270i	$0.744018\pi$
$180$	0	0
$181$	−6.48886	−0.482313	−0.241157	−	0.970486i	$-0.577527\pi$
$181$	−6.48886	−0.482313	−0.241157	+	0.970486i	$0.577527\pi$
$182$	1.42864	0.105898
$183$	0	0
$184$	−2.05086	−0.151191
$185$	−21.9956	−1.61715
$186$	0	0
$187$	6.57628	0.480905
$188$	14.2351	1.03820
$189$	0	0
$190$	5.27163	0.382444
$191$	−25.8780	−1.87247	−0.936234	−	0.351377i	$-0.885713\pi$
$191$	−25.8780	−1.87247	−0.936234	+	0.351377i	$0.885713\pi$
$192$	0	0
$193$	−6.75557	−0.486276	−0.243138	−	0.969992i	$-0.578177\pi$
$193$	−6.75557	−0.486276	−0.243138	+	0.969992i	$0.578177\pi$
$194$	−3.76494	−0.270307
$195$	0	0
$196$	−6.34122	−0.452944
$197$	7.34122	0.523040	0.261520	−	0.965198i	$-0.415776\pi$
$197$	7.34122	0.523040	0.261520	+	0.965198i	$0.415776\pi$
$198$	0	0
$199$	−12.1936	−0.864380	−0.432190	−	0.901783i	$-0.642259\pi$
$199$	−12.1936	−0.864380	−0.432190	+	0.901783i	$0.642259\pi$
$200$	1.67307	0.118304
$201$	0	0
$202$	1.03657	0.0729325
$203$	11.3319	0.795340
$204$	0	0
$205$	19.8430	1.38589
$206$	0.842989	0.0587338
$207$	0	0
$208$	4.89829	0.339635
$209$	28.9260	2.00085
$210$	0	0
$211$	22.8113	1.57040	0.785199	−	0.619244i	$-0.212561\pi$
$211$	22.8113	1.57040	0.785199	+	0.619244i	$0.212561\pi$
$212$	−1.18421	−0.0813318
$213$	0	0
$214$	−1.45091	−0.0991825
$215$	−6.24935	−0.426202
$216$	0	0
$217$	−5.33185	−0.361950
$218$	−1.32048	−0.0894339
$219$	0	0
$220$	20.7210	1.39701
$221$	−2.17929	−0.146595
$222$	0	0
$223$	−17.1082	−1.14565	−0.572824	−	0.819679i	$-0.694152\pi$
$223$	−17.1082	−1.14565	−0.572824	+	0.819679i	$0.694152\pi$
$224$	11.2351	0.750674
$225$	0	0
$226$	−2.26962	−0.150973
$227$	−12.8129	−0.850421	−0.425210	−	0.905095i	$-0.639800\pi$
$227$	−12.8129	−0.850421	−0.425210	+	0.905095i	$0.639800\pi$
$228$	0	0
$229$	8.00492	0.528980	0.264490	−	0.964388i	$-0.414796\pi$
$229$	8.00492	0.528980	0.264490	+	0.964388i	$0.414796\pi$
$230$	1.32693	0.0874951
$231$	0	0
$232$	−4.28100	−0.281061
$233$	2.23951	0.146715	0.0733576	−	0.997306i	$-0.476629\pi$
$233$	2.23951	0.146715	0.0733576	+	0.997306i	$0.476629\pi$
$234$	0	0
$235$	−18.8889	−1.23218
$236$	22.7812	1.48293
$237$	0	0
$238$	−1.52543	−0.0988788
$239$	10.3412	0.668918	0.334459	−	0.942410i	$-0.391446\pi$
$239$	10.3412	0.668918	0.334459	+	0.942410i	$0.391446\pi$
$240$	0	0
$241$	−17.7511	−1.14345	−0.571725	−	0.820445i	$-0.693726\pi$
$241$	−17.7511	−1.14345	−0.571725	+	0.820445i	$0.693726\pi$
$242$	−2.35996	−0.151704
$243$	0	0
$244$	−1.90321	−0.121841
$245$	8.41435	0.537573
$246$	0	0
$247$	−9.58565	−0.609920
$248$	2.01429	0.127908
$249$	0	0
$250$	2.84590	0.179991
$251$	−28.1891	−1.77928	−0.889641	−	0.456660i	$-0.849045\pi$
$251$	−28.1891	−1.77928	−0.889641	+	0.456660i	$0.849045\pi$
$252$	0	0
$253$	7.28100	0.457752
$254$	1.19850	0.0752005
$255$	0	0
$256$	8.80642	0.550401
$257$	−5.05086	−0.315064	−0.157532	−	0.987514i	$-0.550354\pi$
$257$	−5.05086	−0.315064	−0.157532	+	0.987514i	$0.550354\pi$
$258$	0	0
$259$	27.9956	1.73956
$260$	−6.86665	−0.425851
$261$	0	0
$262$	−1.23014	−0.0759984
$263$	15.8207	0.975547	0.487774	−	0.872970i	$-0.337809\pi$
$263$	15.8207	0.975547	0.487774	+	0.872970i	$0.337809\pi$
$264$	0	0
$265$	1.57136	0.0965279
$266$	−6.70964	−0.411394
$267$	0	0
$268$	−10.1778	−0.621705
$269$	−14.3368	−0.874129	−0.437064	−	0.899430i	$-0.643982\pi$
$269$	−14.3368	−0.874129	−0.437064	+	0.899430i	$0.643982\pi$
$270$	0	0
$271$	−20.4844	−1.24434	−0.622170	−	0.782882i	$-0.713749\pi$
$271$	−20.4844	−1.24434	−0.622170	+	0.782882i	$0.713749\pi$
$272$	−5.23014	−0.317124
$273$	0	0
$274$	1.06223	0.0641717
$275$	−5.93978	−0.358182
$276$	0	0
$277$	−14.3225	−0.860555	−0.430277	−	0.902697i	$-0.641584\pi$
$277$	−14.3225	−0.860555	−0.430277	+	0.902697i	$0.641584\pi$
$278$	−0.377784	−0.0226580
$279$	0	0
$280$	−9.85728	−0.589085
$281$	−13.6731	−0.815667	−0.407834	−	0.913056i	$-0.633716\pi$
$281$	−13.6731	−0.815667	−0.407834	+	0.913056i	$0.633716\pi$
$282$	0	0
$283$	8.96343	0.532821	0.266410	−	0.963860i	$-0.414162\pi$
$283$	8.96343	0.532821	0.266410	+	0.963860i	$0.414162\pi$
$284$	4.45584	0.264405
$285$	0	0
$286$	1.91612	0.113302
$287$	−25.2558	−1.49080
$288$	0	0
$289$	−14.6731	−0.863122
$290$	2.76986	0.162652
$291$	0	0
$292$	13.2351	0.774523
$293$	27.4608	1.60427	0.802137	−	0.597140i	$-0.203696\pi$
$293$	27.4608	1.60427	0.802137	+	0.597140i	$0.203696\pi$
$294$	0	0
$295$	−30.2291	−1.76001
$296$	−10.5763	−0.614734
$297$	0	0
$298$	2.31111	0.133879
$299$	−2.41282	−0.139537
$300$	0	0
$301$	7.95407	0.458465
$302$	−3.71408	−0.213721
$303$	0	0
$304$	−23.0049	−1.31942
$305$	2.52543	0.144606
$306$	0	0
$307$	−15.5368	−0.886732	−0.443366	−	0.896341i	$-0.646216\pi$
$307$	−15.5368	−0.886732	−0.443366	+	0.896341i	$0.646216\pi$
$308$	−26.3733	−1.50276
$309$	0	0
$310$	−1.30327	−0.0740208
$311$	−13.7906	−0.781993	−0.390997	−	0.920392i	$-0.627870\pi$
$311$	−13.7906	−0.781993	−0.390997	+	0.920392i	$0.627870\pi$
$312$	0	0
$313$	−14.5620	−0.823092	−0.411546	−	0.911389i	$-0.635011\pi$
$313$	−14.5620	−0.823092	−0.411546	+	0.911389i	$0.635011\pi$
$314$	4.43309	0.250173
$315$	0	0
$316$	−25.3160	−1.42414
$317$	−14.8158	−0.832138	−0.416069	−	0.909333i	$-0.636592\pi$
$317$	−14.8158	−0.832138	−0.416069	+	0.909333i	$0.636592\pi$
$318$	0	0
$319$	15.1985	0.850953
$320$	−14.5714	−0.814564
$321$	0	0
$322$	−1.68889	−0.0941183
$323$	10.2351	0.569494
$324$	0	0
$325$	1.96836	0.109185
$326$	−7.27163	−0.402738
$327$	0	0
$328$	9.54125	0.526827
$329$	24.0415	1.32545
$330$	0	0
$331$	−2.16346	−0.118915	−0.0594574	−	0.998231i	$-0.518937\pi$
$331$	−2.16346	−0.118915	−0.0594574	+	0.998231i	$0.518937\pi$
$332$	−22.1017	−1.21299
$333$	0	0
$334$	−6.81135	−0.372700
$335$	13.5052	0.737866
$336$	0	0
$337$	−26.8069	−1.46026	−0.730132	−	0.683306i	$-0.760542\pi$
$337$	−26.8069	−1.46026	−0.730132	+	0.683306i	$0.760542\pi$
$338$	3.40943	0.185448
$339$	0	0
$340$	7.33185	0.397625
$341$	−7.15118	−0.387258
$342$	0	0
$343$	11.7906	0.636633
$344$	−3.00492	−0.162015
$345$	0	0
$346$	0.326929	0.0175758
$347$	−8.34122	−0.447780	−0.223890	−	0.974614i	$-0.571876\pi$
$347$	−8.34122	−0.447780	−0.223890	+	0.974614i	$0.571876\pi$
$348$	0	0
$349$	2.60793	0.139599	0.0697995	−	0.997561i	$-0.477764\pi$
$349$	2.60793	0.139599	0.0697995	+	0.997561i	$0.477764\pi$
$350$	1.37778	0.0736457
$351$	0	0
$352$	15.0687	0.803164
$353$	−32.7101	−1.74098	−0.870492	−	0.492183i	$-0.836199\pi$
$353$	−32.7101	−1.74098	−0.870492	+	0.492183i	$0.836199\pi$
$354$	0	0
$355$	−5.91258	−0.313807
$356$	−19.4795	−1.03241
$357$	0	0
$358$	5.77478	0.305207
$359$	24.2306	1.27884	0.639422	−	0.768856i	$-0.279174\pi$
$359$	24.2306	1.27884	0.639422	+	0.768856i	$0.279174\pi$
$360$	0	0
$361$	26.0192	1.36943
$362$	2.01874	0.106102
$363$	0	0
$364$	8.73975	0.458087
$365$	−17.5620	−0.919237
$366$	0	0
$367$	−12.7556	−0.665835	−0.332918	−	0.942956i	$-0.608033\pi$
$367$	−12.7556	−0.665835	−0.332918	+	0.942956i	$0.608033\pi$
$368$	−5.79060	−0.301856
$369$	0	0
$370$	6.84299	0.355750
$371$	−2.00000	−0.103835
$372$	0	0
$373$	10.5477	0.546139	0.273070	−	0.961994i	$-0.411961\pi$
$373$	10.5477	0.546139	0.273070	+	0.961994i	$0.411961\pi$
$374$	−2.04593	−0.105793
$375$	0	0
$376$	−9.08250	−0.468394
$377$	−5.03657	−0.259396
$378$	0	0
$379$	26.2623	1.34900	0.674501	−	0.738274i	$-0.264359\pi$
$379$	26.2623	1.34900	0.674501	+	0.738274i	$0.264359\pi$
$380$	32.2494	1.65436
$381$	0	0
$382$	8.05086	0.411918
$383$	25.5970	1.30795	0.653973	−	0.756517i	$-0.273101\pi$
$383$	25.5970	1.30795	0.653973	+	0.756517i	$0.273101\pi$
$384$	0	0
$385$	34.9956	1.78354
$386$	2.10171	0.106974
$387$	0	0
$388$	−23.0321	−1.16928
$389$	20.2351	1.02596	0.512979	−	0.858401i	$-0.328542\pi$
$389$	20.2351	1.02596	0.512979	+	0.858401i	$0.328542\pi$
$390$	0	0
$391$	2.57628	0.130288
$392$	4.04593	0.204350
$393$	0	0
$394$	−2.28391	−0.115062
$395$	33.5926	1.69023
$396$	0	0
$397$	17.3002	0.868273	0.434136	−	0.900847i	$-0.357054\pi$
$397$	17.3002	0.868273	0.434136	+	0.900847i	$0.357054\pi$
$398$	3.79352	0.190152
$399$	0	0
$400$	4.72393	0.236196
$401$	4.81579	0.240489	0.120245	−	0.992744i	$-0.461632\pi$
$401$	4.81579	0.240489	0.120245	+	0.992744i	$0.461632\pi$
$402$	0	0
$403$	2.36980	0.118048
$404$	6.34122	0.315487
$405$	0	0
$406$	−3.52543	−0.174964
$407$	37.5482	1.86119
$408$	0	0
$409$	3.93978	0.194809	0.0974047	−	0.995245i	$-0.468946\pi$
$409$	3.93978	0.194809	0.0974047	+	0.995245i	$0.468946\pi$
$410$	−6.17331	−0.304878
$411$	0	0
$412$	5.15701	0.254068
$413$	38.4750	1.89323
$414$	0	0
$415$	29.3274	1.43963
$416$	−4.99355	−0.244829
$417$	0	0
$418$	−8.99909	−0.440160
$419$	−36.3511	−1.77587	−0.887933	−	0.459973i	$-0.847859\pi$
$419$	−36.3511	−1.77587	−0.887933	+	0.459973i	$0.847859\pi$
$420$	0	0
$421$	−1.52543	−0.0743448	−0.0371724	−	0.999309i	$-0.511835\pi$
$421$	−1.52543	−0.0743448	−0.0371724	+	0.999309i	$0.511835\pi$
$422$	−7.09679	−0.345466
$423$	0	0
$424$	0.755569	0.0366937
$425$	−2.10171	−0.101948
$426$	0	0
$427$	−3.21432	−0.155552
$428$	−8.87601	−0.429038
$429$	0	0
$430$	1.94422	0.0937587
$431$	3.53972	0.170502	0.0852511	−	0.996359i	$-0.472831\pi$
$431$	3.53972	0.170502	0.0852511	+	0.996359i	$0.472831\pi$
$432$	0	0
$433$	−5.76494	−0.277045	−0.138523	−	0.990359i	$-0.544235\pi$
$433$	−5.76494	−0.277045	−0.138523	+	0.990359i	$0.544235\pi$
$434$	1.65878	0.0796240
$435$	0	0
$436$	−8.07805	−0.386869
$437$	11.3319	0.542076
$438$	0	0
$439$	−12.6079	−0.601743	−0.300872	−	0.953665i	$-0.597278\pi$
$439$	−12.6079	−0.601743	−0.300872	+	0.953665i	$0.597278\pi$
$440$	−13.2208	−0.630276
$441$	0	0
$442$	0.677993	0.0322488
$443$	14.7971	0.703029	0.351515	−	0.936182i	$-0.385667\pi$
$443$	14.7971	0.703029	0.351515	+	0.936182i	$0.385667\pi$
$444$	0	0
$445$	25.8479	1.22531
$446$	5.32248	0.252027
$447$	0	0
$448$	18.5462	0.876224
$449$	2.91258	0.137453	0.0687266	−	0.997636i	$-0.478106\pi$
$449$	2.91258	0.137453	0.0687266	+	0.997636i	$0.478106\pi$
$450$	0	0
$451$	−33.8736	−1.59504
$452$	−13.8845	−0.653071
$453$	0	0
$454$	3.98619	0.187081
$455$	−11.5970	−0.543677
$456$	0	0
$457$	32.2810	1.51004	0.755021	−	0.655701i	$-0.227627\pi$
$457$	32.2810	1.51004	0.755021	+	0.655701i	$0.227627\pi$
$458$	−2.49039	−0.116368
$459$	0	0
$460$	8.11753	0.378482
$461$	−28.8385	−1.34314	−0.671572	−	0.740939i	$-0.734381\pi$
$461$	−28.8385	−1.34314	−0.671572	+	0.740939i	$0.734381\pi$
$462$	0	0
$463$	11.7333	0.545292	0.272646	−	0.962114i	$-0.412101\pi$
$463$	11.7333	0.545292	0.272646	+	0.962114i	$0.412101\pi$
$464$	−12.0874	−0.561144
$465$	0	0
$466$	−0.696729	−0.0322753
$467$	9.34767	0.432559	0.216279	−	0.976332i	$-0.430608\pi$
$467$	9.34767	0.432559	0.216279	+	0.976332i	$0.430608\pi$
$468$	0	0
$469$	−17.1891	−0.793720
$470$	5.87649	0.271062
$471$	0	0
$472$	−14.5353	−0.669040
$473$	10.6681	0.490522
$474$	0	0
$475$	−9.24443	−0.424164
$476$	−9.33185	−0.427725
$477$	0	0
$478$	−3.21723	−0.147153
$479$	−25.0923	−1.14650	−0.573249	−	0.819381i	$-0.694317\pi$
$479$	−25.0923	−1.14650	−0.573249	+	0.819381i	$0.694317\pi$
$480$	0	0
$481$	−12.4429	−0.567349
$482$	5.52251	0.251544
$483$	0	0
$484$	−14.4371	−0.656232
$485$	30.5620	1.38775
$486$	0	0
$487$	26.6178	1.20617	0.603083	−	0.797678i	$-0.293939\pi$
$487$	26.6178	1.20617	0.603083	+	0.797678i	$0.293939\pi$
$488$	1.21432	0.0549697
$489$	0	0
$490$	−2.61777	−0.118259
$491$	−18.1891	−0.820864	−0.410432	−	0.911891i	$-0.634622\pi$
$491$	−18.1891	−0.820864	−0.410432	+	0.911891i	$0.634622\pi$
$492$	0	0
$493$	5.37778	0.242203
$494$	2.98217	0.134174
$495$	0	0
$496$	5.68736	0.255370
$497$	7.52543	0.337562
$498$	0	0
$499$	−18.2148	−0.815406	−0.407703	−	0.913115i	$-0.633670\pi$
$499$	−18.2148	−0.815406	−0.407703	+	0.913115i	$0.633670\pi$
$500$	17.4099	0.778595
$501$	0	0
$502$	8.76986	0.391418
$503$	22.9447	1.02305	0.511527	−	0.859267i	$-0.329080\pi$
$503$	22.9447	1.02305	0.511527	+	0.859267i	$0.329080\pi$
$504$	0	0
$505$	−8.41435	−0.374434
$506$	−2.26517	−0.100699
$507$	0	0
$508$	7.33185	0.325298
$509$	25.9541	1.15039	0.575197	−	0.818015i	$-0.304925\pi$
$509$	25.9541	1.15039	0.575197	+	0.818015i	$0.304925\pi$
$510$	0	0
$511$	22.3526	0.988821
$512$	−20.3111	−0.897633
$513$	0	0
$514$	1.57136	0.0693097
$515$	−6.84299	−0.301538
$516$	0	0
$517$	32.2449	1.41813
$518$	−8.70964	−0.382679
$519$	0	0
$520$	4.38118	0.192127
$521$	39.6400	1.73666	0.868331	−	0.495985i	$-0.165193\pi$
$521$	39.6400	1.73666	0.868331	+	0.495985i	$0.165193\pi$
$522$	0	0
$523$	−32.8829	−1.43787	−0.718935	−	0.695077i	$-0.755370\pi$
$523$	−32.8829	−1.43787	−0.718935	+	0.695077i	$0.755370\pi$
$524$	−7.52543	−0.328750
$525$	0	0
$526$	−4.92195	−0.214607
$527$	−2.53035	−0.110224
$528$	0	0
$529$	−20.1476	−0.875984
$530$	−0.488863	−0.0212348
$531$	0	0
$532$	−41.0464	−1.77959
$533$	11.2252	0.486218
$534$	0	0
$535$	11.7778	0.509201
$536$	6.49378	0.280489
$537$	0	0
$538$	4.46028	0.192296
$539$	−14.3640	−0.618700
$540$	0	0
$541$	−38.1847	−1.64169	−0.820844	−	0.571153i	$-0.806496\pi$
$541$	−38.1847	−1.64169	−0.820844	+	0.571153i	$0.806496\pi$
$542$	6.37286	0.273738
$543$	0	0
$544$	5.33185	0.228601
$545$	10.7190	0.459152
$546$	0	0
$547$	−5.74912	−0.245814	−0.122907	−	0.992418i	$-0.539222\pi$
$547$	−5.74912	−0.245814	−0.122907	+	0.992418i	$0.539222\pi$
$548$	6.49823	0.277591
$549$	0	0
$550$	1.84791	0.0787952
$551$	23.6543	1.00771
$552$	0	0
$553$	−42.7560	−1.81817
$554$	4.45584	0.189310
$555$	0	0
$556$	−2.31111	−0.0980128
$557$	13.5111	0.572485	0.286243	−	0.958157i	$-0.407594\pi$
$557$	13.5111	0.572485	0.286243	+	0.958157i	$0.407594\pi$
$558$	0	0
$559$	−3.53527	−0.149526
$560$	−27.8321	−1.17612
$561$	0	0
$562$	4.25380	0.179436
$563$	14.2208	0.599334	0.299667	−	0.954044i	$-0.403124\pi$
$563$	14.2208	0.599334	0.299667	+	0.954044i	$0.403124\pi$
$564$	0	0
$565$	18.4237	0.775092
$566$	−2.78859	−0.117213
$567$	0	0
$568$	−2.84299	−0.119289
$569$	−6.33185	−0.265445	−0.132723	−	0.991153i	$-0.542372\pi$
$569$	−6.33185	−0.265445	−0.132723	+	0.991153i	$0.542372\pi$
$570$	0	0
$571$	−2.22522	−0.0931225	−0.0465613	−	0.998915i	$-0.514826\pi$
$571$	−2.22522	−0.0931225	−0.0465613	+	0.998915i	$0.514826\pi$
$572$	11.7219	0.490118
$573$	0	0
$574$	7.85728	0.327956
$575$	−2.32693	−0.0970397
$576$	0	0
$577$	14.0272	0.583960	0.291980	−	0.956424i	$-0.405686\pi$
$577$	14.0272	0.583960	0.291980	+	0.956424i	$0.405686\pi$
$578$	4.56491	0.189875
$579$	0	0
$580$	16.9447	0.703590
$581$	−37.3274	−1.54860
$582$	0	0
$583$	−2.68244	−0.111095
$584$	−8.44446	−0.349434
$585$	0	0
$586$	−8.54326	−0.352919
$587$	5.36349	0.221375	0.110688	−	0.993855i	$-0.464695\pi$
$587$	5.36349	0.221375	0.110688	+	0.993855i	$0.464695\pi$
$588$	0	0
$589$	−11.1298	−0.458596
$590$	9.40451	0.387177
$591$	0	0
$592$	−29.8622	−1.22733
$593$	2.70964	0.111271	0.0556357	−	0.998451i	$-0.482281\pi$
$593$	2.70964	0.111271	0.0556357	+	0.998451i	$0.482281\pi$
$594$	0	0
$595$	12.3827	0.507641
$596$	14.1383	0.579126
$597$	0	0
$598$	0.750647	0.0306962
$599$	18.2148	0.744236	0.372118	−	0.928185i	$-0.378632\pi$
$599$	18.2148	0.744236	0.372118	+	0.928185i	$0.378632\pi$
$600$	0	0
$601$	22.8479	0.931986	0.465993	−	0.884788i	$-0.345697\pi$
$601$	22.8479	0.931986	0.465993	+	0.884788i	$0.345697\pi$
$602$	−2.47457	−0.100856
$603$	0	0
$604$	−22.7210	−0.924505
$605$	19.1570	0.778843
$606$	0	0
$607$	−6.38715	−0.259247	−0.129623	−	0.991563i	$-0.541377\pi$
$607$	−6.38715	−0.259247	−0.129623	+	0.991563i	$0.541377\pi$
$608$	23.4523	0.951116
$609$	0	0
$610$	−0.785680	−0.0318113
$611$	−10.6855	−0.432289
$612$	0	0
$613$	34.9403	1.41122	0.705612	−	0.708599i	$-0.250672\pi$
$613$	34.9403	1.41122	0.705612	+	0.708599i	$0.250672\pi$
$614$	4.83362	0.195069
$615$	0	0
$616$	16.8272	0.677986
$617$	−19.6731	−0.792008	−0.396004	−	0.918249i	$-0.629603\pi$
$617$	−19.6731	−0.792008	−0.396004	+	0.918249i	$0.629603\pi$
$618$	0	0
$619$	9.39069	0.377444	0.188722	−	0.982031i	$-0.439566\pi$
$619$	9.39069	0.377444	0.188722	+	0.982031i	$0.439566\pi$
$620$	−7.97280	−0.320195
$621$	0	0
$622$	4.29036	0.172028
$623$	−32.8988	−1.31806
$624$	0	0
$625$	−29.9906	−1.19963
$626$	4.53035	0.181069
$627$	0	0
$628$	27.1195	1.08219
$629$	13.2859	0.529744
$630$	0	0
$631$	−14.7210	−0.586034	−0.293017	−	0.956107i	$-0.594659\pi$
$631$	−14.7210	−0.586034	−0.293017	+	0.956107i	$0.594659\pi$
$632$	16.1526	0.642515
$633$	0	0
$634$	4.60931	0.183059
$635$	−9.72885	−0.386078
$636$	0	0
$637$	4.76001	0.188599
$638$	−4.72837	−0.187198
$639$	0	0
$640$	22.1876	0.877042
$641$	−15.1111	−0.596852	−0.298426	−	0.954433i	$-0.596462\pi$
$641$	−15.1111	−0.596852	−0.298426	+	0.954433i	$0.596462\pi$
$642$	0	0
$643$	−4.14764	−0.163567	−0.0817835	−	0.996650i	$-0.526062\pi$
$643$	−4.14764	−0.163567	−0.0817835	+	0.996650i	$0.526062\pi$
$644$	−10.3319	−0.407132
$645$	0	0
$646$	−3.18421	−0.125281
$647$	11.1827	0.439636	0.219818	−	0.975541i	$-0.429454\pi$
$647$	11.1827	0.439636	0.219818	+	0.975541i	$0.429454\pi$
$648$	0	0
$649$	51.6035	2.02561
$650$	−0.612371	−0.0240192
$651$	0	0
$652$	−44.4844	−1.74214
$653$	−4.44738	−0.174039	−0.0870196	−	0.996207i	$-0.527734\pi$
$653$	−4.44738	−0.174039	−0.0870196	+	0.996207i	$0.527734\pi$
$654$	0	0
$655$	9.98571	0.390174
$656$	26.9398	1.05182
$657$	0	0
$658$	−7.47949	−0.291581
$659$	−32.1606	−1.25280	−0.626399	−	0.779503i	$-0.715472\pi$
$659$	−32.1606	−1.25280	−0.626399	+	0.779503i	$0.715472\pi$
$660$	0	0
$661$	23.7288	0.922945	0.461473	−	0.887154i	$-0.347321\pi$
$661$	23.7288	0.922945	0.461473	+	0.887154i	$0.347321\pi$
$662$	0.673071	0.0261596
$663$	0	0
$664$	14.1017	0.547252
$665$	54.4657	2.11209
$666$	0	0
$667$	5.95407	0.230542
$668$	−41.6686	−1.61221
$669$	0	0
$670$	−4.20156	−0.162320
$671$	−4.31111	−0.166428
$672$	0	0
$673$	48.0183	1.85097	0.925485	−	0.378785i	$-0.123658\pi$
$673$	48.0183	1.85097	0.925485	+	0.378785i	$0.123658\pi$
$674$	8.33984	0.321238
$675$	0	0
$676$	20.8573	0.802203
$677$	13.4608	0.517339	0.258669	−	0.965966i	$-0.416716\pi$
$677$	13.4608	0.517339	0.258669	+	0.965966i	$0.416716\pi$
$678$	0	0
$679$	−38.8988	−1.49280
$680$	−4.67799	−0.179393
$681$	0	0
$682$	2.22479	0.0851916
$683$	−37.5254	−1.43587	−0.717935	−	0.696110i	$-0.754913\pi$
$683$	−37.5254	−1.43587	−0.717935	+	0.696110i	$0.754913\pi$
$684$	0	0
$685$	−8.62269	−0.329456
$686$	−3.66815	−0.140051
$687$	0	0
$688$	−8.48442	−0.323465
$689$	0.888922	0.0338652
$690$	0	0
$691$	−13.8983	−0.528716	−0.264358	−	0.964425i	$-0.585160\pi$
$691$	−13.8983	−0.528716	−0.264358	+	0.964425i	$0.585160\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	2.59502	0.0985056
$695$	3.06668	0.116326
$696$	0	0
$697$	−11.9857	−0.453991
$698$	−0.811346	−0.0307099
$699$	0	0
$700$	8.42864	0.318573
$701$	−38.0830	−1.43837	−0.719187	−	0.694817i	$-0.755486\pi$
$701$	−38.0830	−1.43837	−0.719187	+	0.694817i	$0.755486\pi$
$702$	0	0
$703$	58.4385	2.20405
$704$	24.8745	0.937492
$705$	0	0
$706$	10.1764	0.382993
$707$	10.7096	0.402777
$708$	0	0
$709$	−42.6307	−1.60103	−0.800514	−	0.599314i	$-0.795440\pi$
$709$	−42.6307	−1.60103	−0.800514	+	0.599314i	$0.795440\pi$
$710$	1.83945	0.0690333
$711$	0	0
$712$	12.4286	0.465783
$713$	−2.80150	−0.104917
$714$	0	0
$715$	−15.5542	−0.581692
$716$	35.3274	1.32025
$717$	0	0
$718$	−7.53833	−0.281328
$719$	−50.5990	−1.88703	−0.943513	−	0.331336i	$-0.892501\pi$
$719$	−50.5990	−1.88703	−0.943513	+	0.331336i	$0.892501\pi$
$720$	0	0
$721$	8.70964	0.324364
$722$	−8.09478	−0.301257
$723$	0	0
$724$	12.3497	0.458972
$725$	−4.85728	−0.180395
$726$	0	0
$727$	−4.89877	−0.181685	−0.0908426	−	0.995865i	$-0.528956\pi$
$727$	−4.89877	−0.181685	−0.0908426	+	0.995865i	$0.528956\pi$
$728$	−5.57628	−0.206671
$729$	0	0
$730$	5.46367	0.202220
$731$	3.77478	0.139615
$732$	0	0
$733$	−49.9086	−1.84342	−0.921708	−	0.387884i	$-0.873206\pi$
$733$	−49.9086	−1.84342	−0.921708	+	0.387884i	$0.873206\pi$
$734$	3.96836	0.146475
$735$	0	0
$736$	5.90321	0.217595
$737$	−23.0544	−0.849220
$738$	0	0
$739$	−8.60639	−0.316591	−0.158296	−	0.987392i	$-0.550600\pi$
$739$	−8.60639	−0.316591	−0.158296	+	0.987392i	$0.550600\pi$
$740$	41.8622	1.53889
$741$	0	0
$742$	0.622216	0.0228423
$743$	−45.0908	−1.65422	−0.827111	−	0.562039i	$-0.810017\pi$
$743$	−45.0908	−1.65422	−0.827111	+	0.562039i	$0.810017\pi$
$744$	0	0
$745$	−18.7605	−0.687331
$746$	−3.28147	−0.120143
$747$	0	0
$748$	−12.5161	−0.457632
$749$	−14.9906	−0.547746
$750$	0	0
$751$	−8.59364	−0.313586	−0.156793	−	0.987631i	$-0.550116\pi$
$751$	−8.59364	−0.313586	−0.156793	+	0.987631i	$0.550116\pi$
$752$	−25.6445	−0.935158
$753$	0	0
$754$	1.56691	0.0570637
$755$	30.1492	1.09724
$756$	0	0
$757$	−1.36980	−0.0497862	−0.0248931	−	0.999690i	$-0.507925\pi$
$757$	−1.36980	−0.0497862	−0.0248931	+	0.999690i	$0.507925\pi$
$758$	−8.17039	−0.296762
$759$	0	0
$760$	−20.5763	−0.746380
$761$	40.2494	1.45904	0.729519	−	0.683961i	$-0.239744\pi$
$761$	40.2494	1.45904	0.729519	+	0.683961i	$0.239744\pi$
$762$	0	0
$763$	−13.6430	−0.493908
$764$	49.2514	1.78185
$765$	0	0
$766$	−7.96343	−0.287731
$767$	−17.1007	−0.617469
$768$	0	0
$769$	−0.626661	−0.0225980	−0.0112990	−	0.999936i	$-0.503597\pi$
$769$	−0.626661	−0.0225980	−0.0112990	+	0.999936i	$0.503597\pi$
$770$	−10.8874	−0.392354
$771$	0	0
$772$	12.8573	0.462744
$773$	−29.0923	−1.04638	−0.523189	−	0.852216i	$-0.675258\pi$
$773$	−29.0923	−1.04638	−0.523189	+	0.852216i	$0.675258\pi$
$774$	0	0
$775$	2.28544	0.0820955
$776$	14.6953	0.527532
$777$	0	0
$778$	−6.29529	−0.225697
$779$	−52.7195	−1.88887
$780$	0	0
$781$	10.0932	0.361165
$782$	−0.801502	−0.0286616
$783$	0	0
$784$	11.4237	0.407990
$785$	−35.9857	−1.28439
$786$	0	0
$787$	29.5165	1.05215	0.526075	−	0.850438i	$-0.323663\pi$
$787$	29.5165	1.05215	0.526075	+	0.850438i	$0.323663\pi$
$788$	−13.9719	−0.497728
$789$	0	0
$790$	−10.4509	−0.371827
$791$	−23.4494	−0.833764
$792$	0	0
$793$	1.42864	0.0507325
$794$	−5.38223	−0.191008
$795$	0	0
$796$	23.2070	0.822549
$797$	43.7324	1.54908	0.774540	−	0.632525i	$-0.217981\pi$
$797$	43.7324	1.54908	0.774540	+	0.632525i	$0.217981\pi$
$798$	0	0
$799$	11.4094	0.403637
$800$	−4.81579	−0.170264
$801$	0	0
$802$	−1.49823	−0.0529044
$803$	29.9797	1.05796
$804$	0	0
$805$	13.7096	0.483201
$806$	−0.737263	−0.0259690
$807$	0	0
$808$	−4.04593	−0.142335
$809$	−56.3546	−1.98132	−0.990661	−	0.136347i	$-0.956464\pi$
$809$	−56.3546	−1.98132	−0.990661	+	0.136347i	$0.956464\pi$
$810$	0	0
$811$	−39.0908	−1.37266	−0.686332	−	0.727288i	$-0.740780\pi$
$811$	−39.0908	−1.37266	−0.686332	+	0.727288i	$0.740780\pi$
$812$	−21.5669	−0.756850
$813$	0	0
$814$	−11.6815	−0.409437
$815$	59.0277	2.06765
$816$	0	0
$817$	16.6035	0.580882
$818$	−1.22570	−0.0428554
$819$	0	0
$820$	−37.7654	−1.31883
$821$	−27.0420	−0.943771	−0.471886	−	0.881660i	$-0.656426\pi$
$821$	−27.0420	−0.943771	−0.471886	+	0.881660i	$0.656426\pi$
$822$	0	0
$823$	−37.0464	−1.29136	−0.645678	−	0.763610i	$-0.723425\pi$
$823$	−37.0464	−1.29136	−0.645678	+	0.763610i	$0.723425\pi$
$824$	−3.29036	−0.114625
$825$	0	0
$826$	−11.9699	−0.416486
$827$	−22.2810	−0.774786	−0.387393	−	0.921915i	$-0.626624\pi$
$827$	−22.2810	−0.774786	−0.387393	+	0.921915i	$0.626624\pi$
$828$	0	0
$829$	29.2636	1.01637	0.508184	−	0.861248i	$-0.330317\pi$
$829$	29.2636	1.01637	0.508184	+	0.861248i	$0.330317\pi$
$830$	−9.12399	−0.316698
$831$	0	0
$832$	−8.24305	−0.285776
$833$	−5.08250	−0.176098
$834$	0	0
$835$	55.2913	1.91343
$836$	−55.0522	−1.90402
$837$	0	0
$838$	11.3091	0.390666
$839$	6.29529	0.217337	0.108669	−	0.994078i	$-0.465341\pi$
$839$	6.29529	0.217337	0.108669	+	0.994078i	$0.465341\pi$
$840$	0	0
$841$	−16.5714	−0.571426
$842$	0.474572	0.0163548
$843$	0	0
$844$	−43.4148	−1.49440
$845$	−27.6761	−0.952088
$846$	0	0
$847$	−24.3827	−0.837800
$848$	2.13335	0.0732596
$849$	0	0
$850$	0.653858	0.0224272
$851$	14.7096	0.504240
$852$	0	0
$853$	22.4019	0.767027	0.383514	−	0.923535i	$-0.374714\pi$
$853$	22.4019	0.767027	0.383514	+	0.923535i	$0.374714\pi$
$854$	1.00000	0.0342193
$855$	0	0
$856$	5.66323	0.193565
$857$	8.62269	0.294546	0.147273	−	0.989096i	$-0.452950\pi$
$857$	8.62269	0.294546	0.147273	+	0.989096i	$0.452950\pi$
$858$	0	0
$859$	−32.3225	−1.10283	−0.551414	−	0.834231i	$-0.685912\pi$
$859$	−32.3225	−1.10283	−0.551414	+	0.834231i	$0.685912\pi$
$860$	11.8938	0.405577
$861$	0	0
$862$	−1.10123	−0.0375082
$863$	48.2163	1.64130	0.820651	−	0.571429i	$-0.193611\pi$
$863$	48.2163	1.64130	0.820651	+	0.571429i	$0.193611\pi$
$864$	0	0
$865$	−2.65386	−0.0902339
$866$	1.79352	0.0609462
$867$	0	0
$868$	10.1476	0.344433
$869$	−57.3452	−1.94530
$870$	0	0
$871$	7.63990	0.258868
$872$	5.15410	0.174540
$873$	0	0
$874$	−3.52543	−0.119249
$875$	29.4035	0.994018
$876$	0	0
$877$	33.8479	1.14296	0.571481	−	0.820615i	$-0.306369\pi$
$877$	33.8479	1.14296	0.571481	+	0.820615i	$0.306369\pi$
$878$	3.92242	0.132375
$879$	0	0
$880$	−37.3289	−1.25836
$881$	8.79213	0.296215	0.148107	−	0.988971i	$-0.452682\pi$
$881$	8.79213	0.296215	0.148107	+	0.988971i	$0.452682\pi$
$882$	0	0
$883$	10.8044	0.363598	0.181799	−	0.983336i	$-0.441808\pi$
$883$	10.8044	0.363598	0.181799	+	0.983336i	$0.441808\pi$
$884$	4.14764	0.139500
$885$	0	0
$886$	−4.60348	−0.154657
$887$	−14.2623	−0.478880	−0.239440	−	0.970911i	$-0.576964\pi$
$887$	−14.2623	−0.478880	−0.239440	+	0.970911i	$0.576964\pi$
$888$	0	0
$889$	12.3827	0.415303
$890$	−8.04149	−0.269551
$891$	0	0
$892$	32.5605	1.09020
$893$	50.1847	1.67937
$894$	0	0
$895$	−46.8770	−1.56692
$896$	−28.2400	−0.943432
$897$	0	0
$898$	−0.906126	−0.0302378
$899$	−5.84791	−0.195039
$900$	0	0
$901$	−0.949145	−0.0316206
$902$	10.5383	0.350888
$903$	0	0
$904$	8.85881	0.294640
$905$	−16.3872	−0.544727
$906$	0	0
$907$	−33.8908	−1.12532	−0.562662	−	0.826687i	$-0.690223\pi$
$907$	−33.8908	−1.12532	−0.562662	+	0.826687i	$0.690223\pi$
$908$	24.3856	0.809265
$909$	0	0
$910$	3.60793	0.119602
$911$	−1.18865	−0.0393819	−0.0196909	−	0.999806i	$-0.506268\pi$
$911$	−1.18865	−0.0393819	−0.0196909	+	0.999806i	$0.506268\pi$
$912$	0	0
$913$	−50.0642	−1.65688
$914$	−10.0429	−0.332189
$915$	0	0
$916$	−15.2351	−0.503381
$917$	−12.7096	−0.419709
$918$	0	0
$919$	47.6829	1.57291	0.786457	−	0.617645i	$-0.211913\pi$
$919$	47.6829	1.57291	0.786457	+	0.617645i	$0.211913\pi$
$920$	−5.17929	−0.170756
$921$	0	0
$922$	8.97190	0.295474
$923$	−3.34476	−0.110094
$924$	0	0
$925$	−12.0000	−0.394558
$926$	−3.65032	−0.119957
$927$	0	0
$928$	12.3225	0.404505
$929$	3.56199	0.116865	0.0584326	−	0.998291i	$-0.481390\pi$
$929$	3.56199	0.116865	0.0584326	+	0.998291i	$0.481390\pi$
$930$	0	0
$931$	−22.3555	−0.732672
$932$	−4.26226	−0.139615
$933$	0	0
$934$	−2.90813	−0.0951571
$935$	16.6079	0.543137
$936$	0	0
$937$	19.6035	0.640418	0.320209	−	0.947347i	$-0.396247\pi$
$937$	19.6035	0.640418	0.320209	+	0.947347i	$0.396247\pi$
$938$	5.34767	0.174608
$939$	0	0
$940$	35.9496	1.17255
$941$	8.50315	0.277195	0.138597	−	0.990349i	$-0.455741\pi$
$941$	8.50315	0.277195	0.138597	+	0.990349i	$0.455741\pi$
$942$	0	0
$943$	−13.2701	−0.432134
$944$	−41.0404	−1.33575
$945$	0	0
$946$	−3.31894	−0.107908
$947$	58.7037	1.90761	0.953806	−	0.300422i	$-0.0971276\pi$
$947$	58.7037	1.90761	0.953806	+	0.300422i	$0.0971276\pi$
$948$	0	0
$949$	−9.93485	−0.322499
$950$	2.87601	0.0933102
$951$	0	0
$952$	5.95407	0.192972
$953$	36.4371	1.18031	0.590157	−	0.807289i	$-0.299066\pi$
$953$	36.4371	1.18031	0.590157	+	0.807289i	$0.299066\pi$
$954$	0	0
$955$	−65.3531	−2.11478
$956$	−19.6815	−0.636546
$957$	0	0
$958$	7.80642	0.252214
$959$	10.9748	0.354395
$960$	0	0
$961$	−28.2484	−0.911240
$962$	3.87109	0.124809
$963$	0	0
$964$	33.7841	1.08811
$965$	−17.0607	−0.549203
$966$	0	0
$967$	−12.9349	−0.415957	−0.207978	−	0.978133i	$-0.566688\pi$
$967$	−12.9349	−0.415957	−0.207978	+	0.978133i	$0.566688\pi$
$968$	9.21141	0.296066
$969$	0	0
$970$	−9.50807	−0.305286
$971$	39.7690	1.27625	0.638123	−	0.769934i	$-0.279711\pi$
$971$	39.7690	1.27625	0.638123	+	0.769934i	$0.279711\pi$
$972$	0	0
$973$	−3.90321	−0.125131
$974$	−8.28100	−0.265340
$975$	0	0
$976$	3.42864	0.109748
$977$	−6.72393	−0.215117	−0.107559	−	0.994199i	$-0.534303\pi$
$977$	−6.72393	−0.215117	−0.107559	+	0.994199i	$0.534303\pi$
$978$	0	0
$979$	−44.1245	−1.41022
$980$	−16.0143	−0.511558
$981$	0	0
$982$	5.65878	0.180579
$983$	12.2395	0.390380	0.195190	−	0.980765i	$-0.437468\pi$
$983$	12.2395	0.390380	0.195190	+	0.980765i	$0.437468\pi$
$984$	0	0
$985$	18.5397	0.590725
$986$	−1.67307	−0.0532814
$987$	0	0
$988$	18.2435	0.580404
$989$	4.17929	0.132894
$990$	0	0
$991$	−7.36794	−0.234050	−0.117025	−	0.993129i	$-0.537336\pi$
$991$	−7.36794	−0.234050	−0.117025	+	0.993129i	$0.537336\pi$
$992$	−5.79796	−0.184085
$993$	0	0
$994$	−2.34122	−0.0742590
$995$	−30.7940	−0.976235
$996$	0	0
$997$	30.7195	0.972896	0.486448	−	0.873710i	$-0.338292\pi$
$997$	30.7195	0.972896	0.486448	+	0.873710i	$0.338292\pi$
$998$	5.66677	0.179378
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	549.2.a.g.1.2		3
3.2	odd	2		61.2.a.b.1.2	✓	3
4.3	odd	2		8784.2.a.bn.1.3		3
12.11	even	2		976.2.a.f.1.1		3
15.2	even	4		1525.2.b.b.1099.4		6
15.8	even	4		1525.2.b.b.1099.3		6
15.14	odd	2		1525.2.a.d.1.2		3
21.20	even	2		2989.2.a.i.1.2		3
24.5	odd	2		3904.2.a.r.1.1		3
24.11	even	2		3904.2.a.w.1.3		3
33.32	even	2		7381.2.a.f.1.2		3
183.182	odd	2		3721.2.a.c.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.2.a.b.1.2	✓	3	3.2	odd	2
549.2.a.g.1.2		3	1.1	even	1	trivial
976.2.a.f.1.1		3	12.11	even	2
1525.2.a.d.1.2		3	15.14	odd	2
1525.2.b.b.1099.3		6	15.8	even	4
1525.2.b.b.1099.4		6	15.2	even	4
2989.2.a.i.1.2		3	21.20	even	2
3721.2.a.c.1.2		3	183.182	odd	2
3904.2.a.r.1.1		3	24.5	odd	2
3904.2.a.w.1.3		3	24.11	even	2
7381.2.a.f.1.2		3	33.32	even	2
8784.2.a.bn.1.3		3	4.3	odd	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:	\( N \)	\(=\)	\( 549 = 3^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	549.a (trivial)

\(f(q)\)	\(=\)	\(q-0.311108 q^{2} -1.90321 q^{4} +2.52543 q^{5} -3.21432 q^{7} +1.21432 q^{8} +O(q^{10})\)	\(q-0.311108 q^{2} -1.90321 q^{4} +2.52543 q^{5} -3.21432 q^{7} +1.21432 q^{8} -0.785680 q^{10} -4.31111 q^{11} +1.42864 q^{13} +1.00000 q^{14} +3.42864 q^{16} -1.52543 q^{17} -6.70964 q^{19} -4.80642 q^{20} +1.34122 q^{22} -1.68889 q^{23} +1.37778 q^{25} -0.444461 q^{26} +6.11753 q^{28} -3.52543 q^{29} +1.65878 q^{31} -3.49532 q^{32} +0.474572 q^{34} -8.11753 q^{35} -8.70964 q^{37} +2.08742 q^{38} +3.06668 q^{40} +7.85728 q^{41} -2.47457 q^{43} +8.20495 q^{44} +0.525428 q^{46} -7.47949 q^{47} +3.33185 q^{49} -0.428639 q^{50} -2.71900 q^{52} +0.622216 q^{53} -10.8874 q^{55} -3.90321 q^{56} +1.09679 q^{58} -11.9699 q^{59} +1.00000 q^{61} -0.516060 q^{62} -5.76986 q^{64} +3.60793 q^{65} +5.34767 q^{67} +2.90321 q^{68} +2.52543 q^{70} -2.34122 q^{71} -6.95407 q^{73} +2.70964 q^{74} +12.7699 q^{76} +13.8573 q^{77} +13.3017 q^{79} +8.65878 q^{80} -2.44446 q^{82} +11.6128 q^{83} -3.85236 q^{85} +0.769859 q^{86} -5.23506 q^{88} +10.2351 q^{89} -4.59210 q^{91} +3.21432 q^{92} +2.32693 q^{94} -16.9447 q^{95} +12.1017 q^{97} -1.03657 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} + q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + q^4 + q^5 - 3 * q^7 - 3 * q^8	\( 3 q - q^{2} + q^{4} + q^{5} - 3 q^{7} - 3 q^{8} - 9 q^{10} - 13 q^{11} - 9 q^{13} + 3 q^{14} - 3 q^{16} + 2 q^{17} - q^{20} + 11 q^{22} - 5 q^{23} + 4 q^{25} - q^{26} + 5 q^{28} - 4 q^{29} - 2 q^{31} + 3 q^{32} + 8 q^{34} - 11 q^{35} - 6 q^{37} - 14 q^{38} + 9 q^{40} - 3 q^{41} - 14 q^{43} - 9 q^{44} - 5 q^{46} + 4 q^{47} - 10 q^{49} + 12 q^{50} - 15 q^{52} + 2 q^{53} - 13 q^{55} - 5 q^{56} + 10 q^{58} - 29 q^{59} + 3 q^{61} + 32 q^{62} - 11 q^{64} + 17 q^{65} + 9 q^{67} + 2 q^{68} + q^{70} - 14 q^{71} - q^{73} - 12 q^{74} + 32 q^{76} + 15 q^{77} + 13 q^{79} + 19 q^{80} - 7 q^{82} + 8 q^{83} - 18 q^{85} - 4 q^{86} + 11 q^{88} + 4 q^{89} - 7 q^{91} + 3 q^{92} + 20 q^{94} - 4 q^{95} + 10 q^{97} + 4 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 + q^5 - 3 * q^7 - 3 * q^8 - 9 * q^10 - 13 * q^11 - 9 * q^13 + 3 * q^14 - 3 * q^16 + 2 * q^17 - q^20 + 11 * q^22 - 5 * q^23 + 4 * q^25 - q^26 + 5 * q^28 - 4 * q^29 - 2 * q^31 + 3 * q^32 + 8 * q^34 - 11 * q^35 - 6 * q^37 - 14 * q^38 + 9 * q^40 - 3 * q^41 - 14 * q^43 - 9 * q^44 - 5 * q^46 + 4 * q^47 - 10 * q^49 + 12 * q^50 - 15 * q^52 + 2 * q^53 - 13 * q^55 - 5 * q^56 + 10 * q^58 - 29 * q^59 + 3 * q^61 + 32 * q^62 - 11 * q^64 + 17 * q^65 + 9 * q^67 + 2 * q^68 + q^70 - 14 * q^71 - q^73 - 12 * q^74 + 32 * q^76 + 15 * q^77 + 13 * q^79 + 19 * q^80 - 7 * q^82 + 8 * q^83 - 18 * q^85 - 4 * q^86 + 11 * q^88 + 4 * q^89 - 7 * q^91 + 3 * q^92 + 20 * q^94 - 4 * q^95 + 10 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more