LMFDB - Embedded newform 1449.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,2,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1449.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1449.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:	$11.5703232529$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.32973$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32973 q^{2} -0.231826 q^{4} -3.17434 q^{5} -1.00000 q^{7} +2.96772 q^{8} +O(q^{10})$	\(q-1.32973 q^{2} -0.231826 q^{4} -3.17434 q^{5} -1.00000 q^{7} +2.96772 q^{8} +4.22101 q^{10} -5.06562 q^{11} +4.07644 q^{13} +1.32973 q^{14} -3.48261 q^{16} -4.22101 q^{17} -5.06562 q^{19} +0.735894 q^{20} +6.73589 q^{22} -1.00000 q^{23} +5.07644 q^{25} -5.42055 q^{26} +0.231826 q^{28} -6.68466 q^{29} -2.22101 q^{31} -1.30452 q^{32} +5.61279 q^{34} +3.17434 q^{35} -1.91023 q^{37} +6.73589 q^{38} -9.42055 q^{40} +1.37639 q^{41} -3.39535 q^{43} +1.17434 q^{44} +1.32973 q^{46} +5.81484 q^{47} +1.00000 q^{49} -6.75028 q^{50} -0.945023 q^{52} +6.57594 q^{53} +16.0800 q^{55} -2.96772 q^{56} +8.88877 q^{58} +5.67027 q^{59} -14.4862 q^{61} +2.95333 q^{62} +8.69987 q^{64} -12.9400 q^{65} +13.8842 q^{67} +0.978538 q^{68} -4.22101 q^{70} +2.95333 q^{71} -2.02520 q^{73} +2.54009 q^{74} +1.17434 q^{76} +5.06562 q^{77} +7.14831 q^{79} +11.0550 q^{80} -1.83023 q^{82} +12.7226 q^{83} +13.3989 q^{85} +4.51489 q^{86} -15.0333 q^{88} +17.0629 q^{89} -4.07644 q^{91} +0.231826 q^{92} -7.73215 q^{94} +16.0800 q^{95} +2.08977 q^{97} -1.32973 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 9 * q^8	$ 4 q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 9 q^{8} + 2 q^{10} - q^{11} + 7 q^{13} + 2 q^{14} + 8 q^{16} - 2 q^{17} - q^{19} - 13 q^{20} + 11 q^{22} - 4 q^{23} + 11 q^{25} + 19 q^{26} - 4 q^{28} - 2 q^{29} + 6 q^{31} - 20 q^{32} + 23 q^{34} + 5 q^{35} + 16 q^{37} + 11 q^{38} + 3 q^{40} - 5 q^{41} + 9 q^{43} - 3 q^{44} + 2 q^{46} + 21 q^{47} + 4 q^{49} + 17 q^{50} - 24 q^{52} - 10 q^{53} + 17 q^{55} + 9 q^{56} + q^{58} + 26 q^{59} + 2 q^{61} + 19 q^{62} + 27 q^{64} - 26 q^{65} + 5 q^{67} - 7 q^{68} - 2 q^{70} + 19 q^{71} + 10 q^{73} - 9 q^{74} - 3 q^{76} + q^{77} - 6 q^{79} + 24 q^{80} - 31 q^{82} + 2 q^{83} - 7 q^{85} + 17 q^{86} - 20 q^{88} + 17 q^{89} - 7 q^{91} - 4 q^{92} - 44 q^{94} + 17 q^{95} + 32 q^{97} - 2 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 9 * q^8 + 2 * q^10 - q^11 + 7 * q^13 + 2 * q^14 + 8 * q^16 - 2 * q^17 - q^19 - 13 * q^20 + 11 * q^22 - 4 * q^23 + 11 * q^25 + 19 * q^26 - 4 * q^28 - 2 * q^29 + 6 * q^31 - 20 * q^32 + 23 * q^34 + 5 * q^35 + 16 * q^37 + 11 * q^38 + 3 * q^40 - 5 * q^41 + 9 * q^43 - 3 * q^44 + 2 * q^46 + 21 * q^47 + 4 * q^49 + 17 * q^50 - 24 * q^52 - 10 * q^53 + 17 * q^55 + 9 * q^56 + q^58 + 26 * q^59 + 2 * q^61 + 19 * q^62 + 27 * q^64 - 26 * q^65 + 5 * q^67 - 7 * q^68 - 2 * q^70 + 19 * q^71 + 10 * q^73 - 9 * q^74 - 3 * q^76 + q^77 - 6 * q^79 + 24 * q^80 - 31 * q^82 + 2 * q^83 - 7 * q^85 + 17 * q^86 - 20 * q^88 + 17 * q^89 - 7 * q^91 - 4 * q^92 - 44 * q^94 + 17 * q^95 + 32 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.32973	−0.940259	−0.470130	−	0.882597i	$-0.655793\pi$
$2$	−1.32973	−0.940259	−0.470130	+	0.882597i	$0.655793\pi$
$3$	0	0
$3$	0	0
$4$	−0.231826	−0.115913
$5$	−3.17434	−1.41961	−0.709804	−	0.704399i	$-0.751217\pi$
$5$	−3.17434	−1.41961	−0.709804	+	0.704399i	$0.751217\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.96772	1.04925
$9$	0	0
$10$	4.22101	1.33480
$11$	−5.06562	−1.52734	−0.763671	−	0.645606i	$-0.776605\pi$
$11$	−5.06562	−1.52734	−0.763671	+	0.645606i	$0.776605\pi$
$12$	0	0
$13$	4.07644	1.13060	0.565300	−	0.824885i	$-0.308760\pi$
$13$	4.07644	1.13060	0.565300	+	0.824885i	$0.308760\pi$
$14$	1.32973	0.355385
$15$	0	0
$16$	−3.48261	−0.870651
$17$	−4.22101	−1.02374	−0.511872	−	0.859062i	$-0.671048\pi$
$17$	−4.22101	−1.02374	−0.511872	+	0.859062i	$0.671048\pi$
$18$	0	0
$19$	−5.06562	−1.16213	−0.581067	−	0.813856i	$-0.697364\pi$
$19$	−5.06562	−1.16213	−0.581067	+	0.813856i	$0.697364\pi$
$20$	0.735894	0.164551
$21$	0	0
$22$	6.73589	1.43610
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	5.07644	1.01529
$26$	−5.42055	−1.06306
$27$	0	0
$28$	0.231826	0.0438109
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	−2.22101	−0.398905	−0.199452	−	0.979908i	$-0.563916\pi$
$31$	−2.22101	−0.398905	−0.199452	+	0.979908i	$0.563916\pi$
$32$	−1.30452	−0.230609
$33$	0	0
$34$	5.61279	0.962585
$35$	3.17434	0.536562
$36$	0	0
$37$	−1.91023	−0.314041	−0.157020	−	0.987595i	$-0.550189\pi$
$37$	−1.91023	−0.314041	−0.157020	+	0.987595i	$0.550189\pi$
$38$	6.73589	1.09271
$39$	0	0
$40$	−9.42055	−1.48952
$41$	1.37639	0.214957	0.107478	−	0.994207i	$-0.465722\pi$
$41$	1.37639	0.214957	0.107478	+	0.994207i	$0.465722\pi$
$42$	0	0
$43$	−3.39535	−0.517786	−0.258893	−	0.965906i	$-0.583358\pi$
$43$	−3.39535	−0.517786	−0.258893	+	0.965906i	$0.583358\pi$
$44$	1.17434	0.177039
$45$	0	0
$46$	1.32973	0.196058
$47$	5.81484	0.848182	0.424091	−	0.905620i	$-0.360594\pi$
$47$	5.81484	0.848182	0.424091	+	0.905620i	$0.360594\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−6.75028	−0.954634
$51$	0	0
$52$	−0.945023	−0.131051
$53$	6.57594	0.903275	0.451637	−	0.892202i	$-0.350840\pi$
$53$	6.57594	0.903275	0.451637	+	0.892202i	$0.350840\pi$
$54$	0	0
$55$	16.0800	2.16823
$56$	−2.96772	−0.396578
$57$	0	0
$58$	8.88877	1.16715
$59$	5.67027	0.738207	0.369103	−	0.929388i	$-0.379665\pi$
$59$	5.67027	0.738207	0.369103	+	0.929388i	$0.379665\pi$
$60$	0	0
$61$	−14.4862	−1.85476	−0.927382	−	0.374115i	$-0.877946\pi$
$61$	−14.4862	−1.85476	−0.927382	+	0.374115i	$0.877946\pi$
$62$	2.95333	0.375074
$63$	0	0
$64$	8.69987	1.08748
$65$	−12.9400	−1.60501
$66$	0	0
$67$	13.8842	1.69623	0.848113	−	0.529816i	$-0.177739\pi$
$67$	13.8842	1.69623	0.848113	+	0.529816i	$0.177739\pi$
$68$	0.978538	0.118665
$69$	0	0
$70$	−4.22101	−0.504507
$71$	2.95333	0.350496	0.175248	−	0.984524i	$-0.443927\pi$
$71$	2.95333	0.350496	0.175248	+	0.984524i	$0.443927\pi$
$72$	0	0
$73$	−2.02520	−0.237032	−0.118516	−	0.992952i	$-0.537814\pi$
$73$	−2.02520	−0.237032	−0.118516	+	0.992952i	$0.537814\pi$
$74$	2.54009	0.295280
$75$	0	0
$76$	1.17434	0.134706
$77$	5.06562	0.577281
$78$	0	0
$79$	7.14831	0.804248	0.402124	−	0.915585i	$-0.368272\pi$
$79$	7.14831	0.804248	0.402124	+	0.915585i	$0.368272\pi$
$80$	11.0550	1.23598
$81$	0	0
$82$	−1.83023	−0.202115
$83$	12.7226	1.39648	0.698242	−	0.715862i	$-0.253966\pi$
$83$	12.7226	1.39648	0.698242	+	0.715862i	$0.253966\pi$
$84$	0	0
$85$	13.3989	1.45332
$86$	4.51489	0.486853
$87$	0	0
$88$	−15.0333	−1.60256
$89$	17.0629	1.80867	0.904334	−	0.426826i	$-0.140368\pi$
$89$	17.0629	1.80867	0.904334	+	0.426826i	$0.140368\pi$
$90$	0	0
$91$	−4.07644	−0.427327
$92$	0.231826	0.0241695
$93$	0	0
$94$	−7.73215	−0.797511
$95$	16.0800	1.64977
$96$	0	0
$97$	2.08977	0.212184	0.106092	−	0.994356i	$-0.466166\pi$
$97$	2.08977	0.212184	0.106092	+	0.994356i	$0.466166\pi$
$98$	−1.32973	−0.134323
$99$	0	0
$100$	−1.17685	−0.117685
$101$	−13.4312	−1.33645	−0.668227	−	0.743957i	$-0.732946\pi$
$101$	−13.4312	−1.33645	−0.668227	+	0.743957i	$0.732946\pi$
$102$	0	0
$103$	2.72964	0.268960	0.134480	−	0.990916i	$-0.457064\pi$
$103$	2.72964	0.268960	0.134480	+	0.990916i	$0.457064\pi$
$104$	12.0977	1.18628
$105$	0	0
$106$	−8.74420	−0.849312
$107$	2.36558	0.228689	0.114344	−	0.993441i	$-0.463523\pi$
$107$	2.36558	0.228689	0.114344	+	0.993441i	$0.463523\pi$
$108$	0	0
$109$	−18.1179	−1.73538	−0.867691	−	0.497104i	$-0.834397\pi$
$109$	−18.1179	−1.73538	−0.867691	+	0.497104i	$0.834397\pi$
$110$	−21.3820	−2.03870
$111$	0	0
$112$	3.48261	0.329075
$113$	6.08001	0.571959	0.285979	−	0.958236i	$-0.407681\pi$
$113$	6.08001	0.571959	0.285979	+	0.958236i	$0.407681\pi$
$114$	0	0
$115$	3.17434	0.296009
$116$	1.54968	0.143884
$117$	0	0
$118$	−7.53992	−0.694106
$119$	4.22101	0.386939
$120$	0	0
$121$	14.6605	1.33277
$122$	19.2627	1.74396
$123$	0	0
$124$	0.514886	0.0462382
$125$	−0.242644	−0.0217027
$126$	0	0
$127$	5.42763	0.481624	0.240812	−	0.970572i	$-0.422586\pi$
$127$	5.42763	0.481624	0.240812	+	0.970572i	$0.422586\pi$
$128$	−8.95941	−0.791907
$129$	0	0
$130$	17.2067	1.50913
$131$	0.585698	0.0511727	0.0255863	−	0.999673i	$-0.491855\pi$
$131$	0.585698	0.0511727	0.0255863	+	0.999673i	$0.491855\pi$
$132$	0	0
$133$	5.06562	0.439245
$134$	−18.4622	−1.59489
$135$	0	0
$136$	−12.5268	−1.07416
$137$	−19.6605	−1.67971	−0.839856	−	0.542810i	$-0.817360\pi$
$137$	−19.6605	−1.67971	−0.839856	+	0.542810i	$0.817360\pi$
$138$	0	0
$139$	−9.98667	−0.847059	−0.423529	−	0.905882i	$-0.639209\pi$
$139$	−9.98667	−0.847059	−0.423529	+	0.905882i	$0.639209\pi$
$140$	−0.735894	−0.0621944
$141$	0	0
$142$	−3.92713	−0.329557
$143$	−20.6497	−1.72681
$144$	0	0
$145$	21.2194	1.76217
$146$	2.69297	0.222872
$147$	0	0
$148$	0.442841	0.0364013
$149$	−2.01064	−0.164718	−0.0823592	−	0.996603i	$-0.526245\pi$
$149$	−2.01064	−0.164718	−0.0823592	+	0.996603i	$0.526245\pi$
$150$	0	0
$151$	−7.43951	−0.605418	−0.302709	−	0.953083i	$-0.597891\pi$
$151$	−7.43951	−0.605418	−0.302709	+	0.953083i	$0.597891\pi$
$152$	−15.0333	−1.21936
$153$	0	0
$154$	−6.73589	−0.542794
$155$	7.05023	0.566288
$156$	0	0
$157$	17.5041	1.39698	0.698488	−	0.715621i	$-0.253856\pi$
$157$	17.5041	1.39698	0.698488	+	0.715621i	$0.253856\pi$
$158$	−9.50530	−0.756201
$159$	0	0
$160$	4.14100	0.327375
$161$	1.00000	0.0788110
$162$	0	0
$163$	−3.95439	−0.309732	−0.154866	−	0.987935i	$-0.549495\pi$
$163$	−3.95439	−0.309732	−0.154866	+	0.987935i	$0.549495\pi$
$164$	−0.319083	−0.0249162
$165$	0	0
$166$	−16.9175	−1.31306
$167$	3.08726	0.238899	0.119450	−	0.992840i	$-0.461887\pi$
$167$	3.08726	0.238899	0.119450	+	0.992840i	$0.461887\pi$
$168$	0	0
$169$	3.61736	0.278258
$170$	−17.8169	−1.36649
$171$	0	0
$172$	0.787129	0.0600180
$173$	23.9265	1.81910	0.909549	−	0.415596i	$-0.136427\pi$
$173$	23.9265	1.81910	0.909549	+	0.415596i	$0.136427\pi$
$174$	0	0
$175$	−5.07644	−0.383743
$176$	17.6416	1.32978
$177$	0	0
$178$	−22.6891	−1.70062
$179$	17.5014	1.30811	0.654057	−	0.756445i	$-0.273065\pi$
$179$	17.5014	1.30811	0.654057	+	0.756445i	$0.273065\pi$
$180$	0	0
$181$	14.9738	1.11299	0.556497	−	0.830850i	$-0.312145\pi$
$181$	14.9738	1.11299	0.556497	+	0.830850i	$0.312145\pi$
$182$	5.42055	0.401798
$183$	0	0
$184$	−2.96772	−0.218783
$185$	6.06373	0.445815
$186$	0	0
$187$	21.3820	1.56361
$188$	−1.34803	−0.0983151
$189$	0	0
$190$	−21.3820	−1.55121
$191$	−15.4395	−1.11716	−0.558582	−	0.829449i	$-0.688654\pi$
$191$	−15.4395	−1.11716	−0.558582	+	0.829449i	$0.688654\pi$
$192$	0	0
$193$	17.4395	1.25532	0.627662	−	0.778486i	$-0.284012\pi$
$193$	17.4395	1.25532	0.627662	+	0.778486i	$0.284012\pi$
$194$	−2.77882	−0.199508
$195$	0	0
$196$	−0.231826	−0.0165590
$197$	−22.1698	−1.57953	−0.789765	−	0.613409i	$-0.789798\pi$
$197$	−22.1698	−1.57953	−0.789765	+	0.613409i	$0.789798\pi$
$198$	0	0
$199$	14.9246	1.05798	0.528989	−	0.848629i	$-0.322571\pi$
$199$	14.9246	1.05798	0.528989	+	0.848629i	$0.322571\pi$
$200$	15.0654	1.06529
$201$	0	0
$202$	17.8598	1.25661
$203$	6.68466	0.469171
$204$	0	0
$205$	−4.36914	−0.305154
$206$	−3.62968	−0.252892
$207$	0	0
$208$	−14.1966	−0.984359
$209$	25.6605	1.77497
$210$	0	0
$211$	−3.70880	−0.255325	−0.127662	−	0.991818i	$-0.540747\pi$
$211$	−3.70880	−0.255325	−0.127662	+	0.991818i	$0.540747\pi$
$212$	−1.52447	−0.104701
$213$	0	0
$214$	−3.14557	−0.215027
$215$	10.7780	0.735053
$216$	0	0
$217$	2.22101	0.150772
$218$	24.0919	1.63171
$219$	0	0
$220$	−3.72776	−0.251325
$221$	−17.2067	−1.15745
$222$	0	0
$223$	5.71883	0.382961	0.191480	−	0.981496i	$-0.438671\pi$
$223$	5.71883	0.382961	0.191480	+	0.981496i	$0.438671\pi$
$224$	1.30452	0.0871621
$225$	0	0
$226$	−8.08475	−0.537790
$227$	−26.5437	−1.76176	−0.880882	−	0.473336i	$-0.843050\pi$
$227$	−26.5437	−1.76176	−0.880882	+	0.473336i	$0.843050\pi$
$228$	0	0
$229$	−3.70818	−0.245044	−0.122522	−	0.992466i	$-0.539098\pi$
$229$	−3.70818	−0.245044	−0.122522	+	0.992466i	$0.539098\pi$
$230$	−4.22101	−0.278325
$231$	0	0
$232$	−19.8382	−1.30244
$233$	−22.1950	−1.45404	−0.727021	−	0.686616i	$-0.759096\pi$
$233$	−22.1950	−1.45404	−0.727021	+	0.686616i	$0.759096\pi$
$234$	0	0
$235$	−18.4583	−1.20409
$236$	−1.31451	−0.0855676
$237$	0	0
$238$	−5.61279	−0.363823
$239$	20.2921	1.31259	0.656293	−	0.754506i	$-0.272124\pi$
$239$	20.2921	1.31259	0.656293	+	0.754506i	$0.272124\pi$
$240$	0	0
$241$	14.5158	0.935043	0.467522	−	0.883982i	$-0.345147\pi$
$241$	14.5158	0.935043	0.467522	+	0.883982i	$0.345147\pi$
$242$	−19.4945	−1.25315
$243$	0	0
$244$	3.35827	0.214991
$245$	−3.17434	−0.202801
$246$	0	0
$247$	−20.6497	−1.31391
$248$	−6.59133	−0.418550
$249$	0	0
$250$	0.322650	0.0204062
$251$	−4.03791	−0.254871	−0.127435	−	0.991847i	$-0.540675\pi$
$251$	−4.03791	−0.254871	−0.127435	+	0.991847i	$0.540675\pi$
$252$	0	0
$253$	5.06562	0.318473
$254$	−7.21727	−0.452852
$255$	0	0
$256$	−5.48617	−0.342886
$257$	16.7676	1.04593	0.522966	−	0.852354i	$-0.324826\pi$
$257$	16.7676	1.04593	0.522966	+	0.852354i	$0.324826\pi$
$258$	0	0
$259$	1.91023	0.118696
$260$	2.99983	0.186041
$261$	0	0
$262$	−0.778818	−0.0481156
$263$	22.0308	1.35848	0.679240	−	0.733917i	$-0.262310\pi$
$263$	22.0308	1.35848	0.679240	+	0.733917i	$0.262310\pi$
$264$	0	0
$265$	−20.8743	−1.28230
$266$	−6.73589	−0.413004
$267$	0	0
$268$	−3.21871	−0.196614
$269$	17.9902	1.09688	0.548442	−	0.836188i	$-0.315221\pi$
$269$	17.9902	1.09688	0.548442	+	0.836188i	$0.315221\pi$
$270$	0	0
$271$	0.940007	0.0571014	0.0285507	−	0.999592i	$-0.490911\pi$
$271$	0.940007	0.0571014	0.0285507	+	0.999592i	$0.490911\pi$
$272$	14.7001	0.891325
$273$	0	0
$274$	26.1431	1.57936
$275$	−25.7153	−1.55069
$276$	0	0
$277$	3.73339	0.224317	0.112159	−	0.993690i	$-0.464223\pi$
$277$	3.73339	0.224317	0.112159	+	0.993690i	$0.464223\pi$
$278$	13.2796	0.796455
$279$	0	0
$280$	9.42055	0.562986
$281$	−21.7180	−1.29559	−0.647794	−	0.761816i	$-0.724308\pi$
$281$	−21.7180	−1.29559	−0.647794	+	0.761816i	$0.724308\pi$
$282$	0	0
$283$	−24.0534	−1.42982	−0.714912	−	0.699215i	$-0.753533\pi$
$283$	−24.0534	−1.42982	−0.714912	+	0.699215i	$0.753533\pi$
$284$	−0.684658	−0.0406270
$285$	0	0
$286$	27.4585	1.62365
$287$	−1.37639	−0.0812459
$288$	0	0
$289$	0.816901	0.0480530
$290$	−28.2160	−1.65690
$291$	0	0
$292$	0.469494	0.0274751
$293$	−24.0021	−1.40222	−0.701109	−	0.713054i	$-0.747312\pi$
$293$	−24.0021	−1.40222	−0.701109	+	0.713054i	$0.747312\pi$
$294$	0	0
$295$	−17.9994	−1.04796
$296$	−5.66904	−0.329506
$297$	0	0
$298$	2.67361	0.154878
$299$	−4.07644	−0.235747
$300$	0	0
$301$	3.39535	0.195705
$302$	9.89251	0.569250
$303$	0	0
$304$	17.6416	1.01181
$305$	45.9840	2.63304
$306$	0	0
$307$	31.9678	1.82450	0.912249	−	0.409637i	$-0.134345\pi$
$307$	31.9678	1.82450	0.912249	+	0.409637i	$0.134345\pi$
$308$	−1.17434	−0.0669143
$309$	0	0
$310$	−9.37489	−0.532458
$311$	21.1889	1.20151	0.600756	−	0.799432i	$-0.294866\pi$
$311$	21.1889	1.20151	0.600756	+	0.799432i	$0.294866\pi$
$312$	0	0
$313$	−14.3935	−0.813567	−0.406783	−	0.913525i	$-0.633350\pi$
$313$	−14.3935	−0.813567	−0.406783	+	0.913525i	$0.633350\pi$
$314$	−23.2756	−1.31352
$315$	0	0
$316$	−1.65716	−0.0932226
$317$	24.3516	1.36772	0.683862	−	0.729612i	$-0.260299\pi$
$317$	24.3516	1.36772	0.683862	+	0.729612i	$0.260299\pi$
$318$	0	0
$319$	33.8619	1.89590
$320$	−27.6164	−1.54380
$321$	0	0
$322$	−1.32973	−0.0741028
$323$	21.3820	1.18973
$324$	0	0
$325$	20.6938	1.14789
$326$	5.25826	0.291228
$327$	0	0
$328$	4.08475	0.225543
$329$	−5.81484	−0.320583
$330$	0	0
$331$	−9.09082	−0.499677	−0.249838	−	0.968288i	$-0.580378\pi$
$331$	−9.09082	−0.499677	−0.249838	+	0.968288i	$0.580378\pi$
$332$	−2.94942	−0.161870
$333$	0	0
$334$	−4.10521	−0.224627
$335$	−44.0732	−2.40798
$336$	0	0
$337$	−35.4620	−1.93174	−0.965870	−	0.259028i	$-0.916598\pi$
$337$	−35.4620	−1.93174	−0.965870	+	0.259028i	$0.916598\pi$
$338$	−4.81010	−0.261635
$339$	0	0
$340$	−3.10621	−0.168458
$341$	11.2508	0.609264
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−10.0764	−0.543285
$345$	0	0
$346$	−31.8157	−1.71042
$347$	0.799628	0.0429263	0.0214631	−	0.999770i	$-0.493168\pi$
$347$	0.799628	0.0429263	0.0214631	+	0.999770i	$0.493168\pi$
$348$	0	0
$349$	1.20311	0.0644011	0.0322006	−	0.999481i	$-0.489748\pi$
$349$	1.20311	0.0644011	0.0322006	+	0.999481i	$0.489748\pi$
$350$	6.75028	0.360818
$351$	0	0
$352$	6.60822	0.352219
$353$	1.03334	0.0549991	0.0274996	−	0.999622i	$-0.491246\pi$
$353$	1.03334	0.0549991	0.0274996	+	0.999622i	$0.491246\pi$
$354$	0	0
$355$	−9.37489	−0.497567
$356$	−3.95563	−0.209648
$357$	0	0
$358$	−23.2721	−1.22997
$359$	−12.2329	−0.645627	−0.322813	−	0.946463i	$-0.604629\pi$
$359$	−12.2329	−0.645627	−0.322813	+	0.946463i	$0.604629\pi$
$360$	0	0
$361$	6.66051	0.350553
$362$	−19.9111	−1.04650
$363$	0	0
$364$	0.945023	0.0495327
$365$	6.42869	0.336493
$366$	0	0
$367$	−20.0388	−1.04602	−0.523008	−	0.852328i	$-0.675190\pi$
$367$	−20.0388	−1.04602	−0.523008	+	0.852328i	$0.675190\pi$
$368$	3.48261	0.181543
$369$	0	0
$370$	−8.06311	−0.419181
$371$	−6.57594	−0.341406
$372$	0	0
$373$	−25.6803	−1.32968	−0.664838	−	0.746988i	$-0.731499\pi$
$373$	−25.6803	−1.32968	−0.664838	+	0.746988i	$0.731499\pi$
$374$	−28.4323	−1.47020
$375$	0	0
$376$	17.2568	0.889952
$377$	−27.2496	−1.40343
$378$	0	0
$379$	10.7782	0.553639	0.276819	−	0.960922i	$-0.410720\pi$
$379$	10.7782	0.553639	0.276819	+	0.960922i	$0.410720\pi$
$380$	−3.72776	−0.191230
$381$	0	0
$382$	20.5303	1.05042
$383$	−16.2877	−0.832262	−0.416131	−	0.909305i	$-0.636614\pi$
$383$	−16.2877	−0.832262	−0.416131	+	0.909305i	$0.636614\pi$
$384$	0	0
$385$	−16.0800	−0.819513
$386$	−23.1898	−1.18033
$387$	0	0
$388$	−0.484461	−0.0245948
$389$	−22.2160	−1.12640	−0.563198	−	0.826322i	$-0.690429\pi$
$389$	−22.2160	−1.12640	−0.563198	+	0.826322i	$0.690429\pi$
$390$	0	0
$391$	4.22101	0.213466
$392$	2.96772	0.149892
$393$	0	0
$394$	29.4797	1.48517
$395$	−22.6912	−1.14172
$396$	0	0
$397$	−25.3518	−1.27237	−0.636185	−	0.771536i	$-0.719489\pi$
$397$	−25.3518	−1.27237	−0.636185	+	0.771536i	$0.719489\pi$
$398$	−19.8457	−0.994774
$399$	0	0
$400$	−17.6792	−0.883962
$401$	37.3622	1.86578	0.932891	−	0.360160i	$-0.117278\pi$
$401$	37.3622	1.86578	0.932891	+	0.360160i	$0.117278\pi$
$402$	0	0
$403$	−9.05380	−0.451002
$404$	3.11370	0.154912
$405$	0	0
$406$	−8.88877	−0.441142
$407$	9.67652	0.479647
$408$	0	0
$409$	−29.7899	−1.47302	−0.736508	−	0.676429i	$-0.763527\pi$
$409$	−29.7899	−1.47302	−0.736508	+	0.676429i	$0.763527\pi$
$410$	5.80977	0.286924
$411$	0	0
$412$	−0.632801	−0.0311759
$413$	−5.67027	−0.279016
$414$	0	0
$415$	−40.3858	−1.98246
$416$	−5.31781	−0.260727
$417$	0	0
$418$	−34.1215	−1.66894
$419$	2.46784	0.120562	0.0602809	−	0.998181i	$-0.480800\pi$
$419$	2.46784	0.120562	0.0602809	+	0.998181i	$0.480800\pi$
$420$	0	0
$421$	−15.1942	−0.740519	−0.370260	−	0.928928i	$-0.620731\pi$
$421$	−15.1942	−0.740519	−0.370260	+	0.928928i	$0.620731\pi$
$422$	4.93170	0.240071
$423$	0	0
$424$	19.5155	0.947758
$425$	−21.4277	−1.03940
$426$	0	0
$427$	14.4862	0.701035
$428$	−0.548401	−0.0265080
$429$	0	0
$430$	−14.3318	−0.691140
$431$	15.3289	0.738369	0.369184	−	0.929356i	$-0.379637\pi$
$431$	15.3289	0.738369	0.369184	+	0.929356i	$0.379637\pi$
$432$	0	0
$433$	−1.93895	−0.0931799	−0.0465899	−	0.998914i	$-0.514835\pi$
$433$	−1.93895	−0.0931799	−0.0465899	+	0.998914i	$0.514835\pi$
$434$	−2.95333	−0.141765
$435$	0	0
$436$	4.20020	0.201153
$437$	5.06562	0.242322
$438$	0	0
$439$	4.04649	0.193129	0.0965643	−	0.995327i	$-0.469215\pi$
$439$	4.04649	0.193129	0.0965643	+	0.995327i	$0.469215\pi$
$440$	47.7209	2.27501
$441$	0	0
$442$	22.8802	1.08830
$443$	−17.8871	−0.849844	−0.424922	−	0.905230i	$-0.639698\pi$
$443$	−17.8871	−0.849844	−0.424922	+	0.905230i	$0.639698\pi$
$444$	0	0
$445$	−54.1636	−2.56760
$446$	−7.60448	−0.360082
$447$	0	0
$448$	−8.69987	−0.411030
$449$	−33.1765	−1.56569	−0.782847	−	0.622214i	$-0.786233\pi$
$449$	−33.1765	−1.56569	−0.782847	+	0.622214i	$0.786233\pi$
$450$	0	0
$451$	−6.97229	−0.328312
$452$	−1.40950	−0.0662974
$453$	0	0
$454$	35.2958	1.65652
$455$	12.9400	0.606637
$456$	0	0
$457$	29.0828	1.36043	0.680217	−	0.733010i	$-0.261885\pi$
$457$	29.0828	1.36043	0.680217	+	0.733010i	$0.261885\pi$
$458$	4.93087	0.230404
$459$	0	0
$460$	−0.735894	−0.0343112
$461$	33.3560	1.55354	0.776772	−	0.629782i	$-0.216856\pi$
$461$	33.3560	1.55354	0.776772	+	0.629782i	$0.216856\pi$
$462$	0	0
$463$	−1.87795	−0.0872759	−0.0436380	−	0.999047i	$-0.513895\pi$
$463$	−1.87795	−0.0872759	−0.0436380	+	0.999047i	$0.513895\pi$
$464$	23.2800	1.08075
$465$	0	0
$466$	29.5133	1.36718
$467$	−12.9275	−0.598214	−0.299107	−	0.954220i	$-0.596689\pi$
$467$	−12.9275	−0.598214	−0.299107	+	0.954220i	$0.596689\pi$
$468$	0	0
$469$	−13.8842	−0.641113
$470$	24.5445	1.13215
$471$	0	0
$472$	16.8278	0.774561
$473$	17.1995	0.790836
$474$	0	0
$475$	−25.7153	−1.17990
$476$	−0.978538	−0.0448512
$477$	0	0
$478$	−26.9829	−1.23417
$479$	43.3983	1.98292	0.991459	−	0.130416i	$-0.0416314\pi$
$479$	43.3983	1.98292	0.991459	+	0.130416i	$0.0416314\pi$
$480$	0	0
$481$	−7.78695	−0.355055
$482$	−19.3020	−0.879183
$483$	0	0
$484$	−3.39868	−0.154486
$485$	−6.63363	−0.301218
$486$	0	0
$487$	16.9992	0.770307	0.385154	−	0.922852i	$-0.374148\pi$
$487$	16.9992	0.770307	0.385154	+	0.922852i	$0.374148\pi$
$488$	−42.9909	−1.94611
$489$	0	0
$490$	4.22101	0.190686
$491$	−10.9480	−0.494075	−0.247037	−	0.969006i	$-0.579457\pi$
$491$	−10.9480	−0.494075	−0.247037	+	0.969006i	$0.579457\pi$
$492$	0	0
$493$	28.2160	1.27078
$494$	27.4585	1.23541
$495$	0	0
$496$	7.73489	0.347307
$497$	−2.95333	−0.132475
$498$	0	0
$499$	30.3588	1.35904	0.679522	−	0.733655i	$-0.262187\pi$
$499$	30.3588	1.35904	0.679522	+	0.733655i	$0.262187\pi$
$500$	0.0562511	0.00251563
$501$	0	0
$502$	5.36932	0.239644
$503$	−25.5951	−1.14123	−0.570614	−	0.821219i	$-0.693295\pi$
$503$	−25.5951	−1.14123	−0.570614	+	0.821219i	$0.693295\pi$
$504$	0	0
$505$	42.6352	1.89724
$506$	−6.73589	−0.299447
$507$	0	0
$508$	−1.25826	−0.0558264
$509$	8.95422	0.396889	0.198444	−	0.980112i	$-0.436411\pi$
$509$	8.95422	0.396889	0.198444	+	0.980112i	$0.436411\pi$
$510$	0	0
$511$	2.02520	0.0895898
$512$	25.2139	1.11431
$513$	0	0
$514$	−22.2963	−0.983446
$515$	−8.66482	−0.381818
$516$	0	0
$517$	−29.4558	−1.29546
$518$	−2.54009	−0.111605
$519$	0	0
$520$	−38.4023	−1.68405
$521$	−15.9405	−0.698364	−0.349182	−	0.937055i	$-0.613541\pi$
$521$	−15.9405	−0.698364	−0.349182	+	0.937055i	$0.613541\pi$
$522$	0	0
$523$	28.1102	1.22917	0.614587	−	0.788849i	$-0.289323\pi$
$523$	28.1102	1.22917	0.614587	+	0.788849i	$0.289323\pi$
$524$	−0.135780	−0.00593157
$525$	0	0
$526$	−29.2950	−1.27732
$527$	9.37489	0.408376
$528$	0	0
$529$	1.00000	0.0434783
$530$	27.7571	1.20569
$531$	0	0
$532$	−1.17434	−0.0509141
$533$	5.61078	0.243030
$534$	0	0
$535$	−7.50914	−0.324648
$536$	41.2044	1.77976
$537$	0	0
$538$	−23.9221	−1.03136
$539$	−5.06562	−0.218192
$540$	0	0
$541$	31.5707	1.35733	0.678666	−	0.734447i	$-0.262558\pi$
$541$	31.5707	1.35733	0.678666	+	0.734447i	$0.262558\pi$
$542$	−1.24995	−0.0536901
$543$	0	0
$544$	5.50640	0.236085
$545$	57.5124	2.46356
$546$	0	0
$547$	−37.7639	−1.61467	−0.807333	−	0.590096i	$-0.799090\pi$
$547$	−37.7639	−1.61467	−0.807333	+	0.590096i	$0.799090\pi$
$548$	4.55781	0.194700
$549$	0	0
$550$	34.1944	1.45805
$551$	33.8619	1.44257
$552$	0	0
$553$	−7.14831	−0.303977
$554$	−4.96438	−0.210916
$555$	0	0
$556$	2.31517	0.0981849
$557$	−11.1660	−0.473120	−0.236560	−	0.971617i	$-0.576020\pi$
$557$	−11.1660	−0.473120	−0.236560	+	0.971617i	$0.576020\pi$
$558$	0	0
$559$	−13.8409	−0.585409
$560$	−11.0550	−0.467158
$561$	0	0
$562$	28.8790	1.21819
$563$	22.5943	0.952235	0.476117	−	0.879382i	$-0.342044\pi$
$563$	22.5943	0.952235	0.476117	+	0.879382i	$0.342044\pi$
$564$	0	0
$565$	−19.3000	−0.811958
$566$	31.9844	1.34440
$567$	0	0
$568$	8.76466	0.367757
$569$	40.4210	1.69454	0.847268	−	0.531166i	$-0.178246\pi$
$569$	40.4210	1.69454	0.847268	+	0.531166i	$0.178246\pi$
$570$	0	0
$571$	−17.1859	−0.719206	−0.359603	−	0.933105i	$-0.617088\pi$
$571$	−17.1859	−0.719206	−0.359603	+	0.933105i	$0.617088\pi$
$572$	4.78713	0.200160
$573$	0	0
$574$	1.83023	0.0763922
$575$	−5.07644	−0.211702
$576$	0	0
$577$	31.1465	1.29665	0.648323	−	0.761365i	$-0.275471\pi$
$577$	31.1465	1.29665	0.648323	+	0.761365i	$0.275471\pi$
$578$	−1.08626	−0.0451823
$579$	0	0
$580$	−4.91920	−0.204259
$581$	−12.7226	−0.527821
$582$	0	0
$583$	−33.3112	−1.37961
$584$	−6.01024	−0.248705
$585$	0	0
$586$	31.9163	1.31845
$587$	11.0144	0.454612	0.227306	−	0.973823i	$-0.427008\pi$
$587$	11.0144	0.454612	0.227306	+	0.973823i	$0.427008\pi$
$588$	0	0
$589$	11.2508	0.463580
$590$	23.9343	0.985358
$591$	0	0
$592$	6.65259	0.273420
$593$	11.6009	0.476392	0.238196	−	0.971217i	$-0.423444\pi$
$593$	11.6009	0.476392	0.238196	+	0.971217i	$0.423444\pi$
$594$	0	0
$595$	−13.3989	−0.549302
$596$	0.466119	0.0190930
$597$	0	0
$598$	5.42055	0.221663
$599$	10.1410	0.414350	0.207175	−	0.978304i	$-0.433573\pi$
$599$	10.1410	0.414350	0.207175	+	0.978304i	$0.433573\pi$
$600$	0	0
$601$	39.1210	1.59578	0.797890	−	0.602803i	$-0.205949\pi$
$601$	39.1210	1.59578	0.797890	+	0.602803i	$0.205949\pi$
$602$	−4.51489	−0.184013
$603$	0	0
$604$	1.72467	0.0701758
$605$	−46.5375	−1.89202
$606$	0	0
$607$	13.8211	0.560981	0.280490	−	0.959857i	$-0.409503\pi$
$607$	13.8211	0.560981	0.280490	+	0.959857i	$0.409503\pi$
$608$	6.60822	0.267999
$609$	0	0
$610$	−61.1462	−2.47574
$611$	23.7038	0.958955
$612$	0	0
$613$	41.6282	1.68135	0.840674	−	0.541541i	$-0.182159\pi$
$613$	41.6282	1.68135	0.840674	+	0.541541i	$0.182159\pi$
$614$	−42.5084	−1.71550
$615$	0	0
$616$	15.0333	0.605711
$617$	−2.31552	−0.0932192	−0.0466096	−	0.998913i	$-0.514842\pi$
$617$	−2.31552	−0.0932192	−0.0466096	+	0.998913i	$0.514842\pi$
$618$	0	0
$619$	18.3318	0.736817	0.368408	−	0.929664i	$-0.379903\pi$
$619$	18.3318	0.736817	0.368408	+	0.929664i	$0.379903\pi$
$620$	−1.63442	−0.0656401
$621$	0	0
$622$	−28.1755	−1.12973
$623$	−17.0629	−0.683612
$624$	0	0
$625$	−24.6120	−0.984478
$626$	19.1394	0.764963
$627$	0	0
$628$	−4.05789	−0.161928
$629$	8.06311	0.321497
$630$	0	0
$631$	37.9644	1.51134	0.755670	−	0.654953i	$-0.227312\pi$
$631$	37.9644	1.51134	0.755670	+	0.654953i	$0.227312\pi$
$632$	21.2142	0.843855
$633$	0	0
$634$	−32.3810	−1.28601
$635$	−17.2291	−0.683718
$636$	0	0
$637$	4.07644	0.161514
$638$	−45.0271	−1.78264
$639$	0	0
$640$	28.4402	1.12420
$641$	−5.55174	−0.219280	−0.109640	−	0.993971i	$-0.534970\pi$
$641$	−5.55174	−0.219280	−0.109640	+	0.993971i	$0.534970\pi$
$642$	0	0
$643$	39.2894	1.54942	0.774711	−	0.632316i	$-0.217895\pi$
$643$	39.2894	1.54942	0.774711	+	0.632316i	$0.217895\pi$
$644$	−0.231826	−0.00913521
$645$	0	0
$646$	−28.4323	−1.11865
$647$	−18.4287	−0.724506	−0.362253	−	0.932080i	$-0.617992\pi$
$647$	−18.4287	−0.724506	−0.362253	+	0.932080i	$0.617992\pi$
$648$	0	0
$649$	−28.7235	−1.12749
$650$	−27.5171	−1.07931
$651$	0	0
$652$	0.916730	0.0359019
$653$	−42.2708	−1.65418	−0.827092	−	0.562067i	$-0.810006\pi$
$653$	−42.2708	−1.65418	−0.827092	+	0.562067i	$0.810006\pi$
$654$	0	0
$655$	−1.85920	−0.0726451
$656$	−4.79344	−0.187152
$657$	0	0
$658$	7.73215	0.301431
$659$	−47.9174	−1.86660	−0.933298	−	0.359103i	$-0.883083\pi$
$659$	−47.9174	−1.86660	−0.933298	+	0.359103i	$0.883083\pi$
$660$	0	0
$661$	17.9334	0.697528	0.348764	−	0.937211i	$-0.386602\pi$
$661$	17.9334	0.697528	0.348764	+	0.937211i	$0.386602\pi$
$662$	12.0883	0.469826
$663$	0	0
$664$	37.7570	1.46526
$665$	−16.0800	−0.623556
$666$	0	0
$667$	6.68466	0.258831
$668$	−0.715705	−0.0276915
$669$	0	0
$670$	58.6053	2.26412
$671$	73.3815	2.83286
$672$	0	0
$673$	−30.4174	−1.17251	−0.586253	−	0.810128i	$-0.699397\pi$
$673$	−30.4174	−1.17251	−0.586253	+	0.810128i	$0.699397\pi$
$674$	47.1548	1.81634
$675$	0	0
$676$	−0.838596	−0.0322537
$677$	−31.1083	−1.19559	−0.597795	−	0.801649i	$-0.703956\pi$
$677$	−31.1083	−1.19559	−0.597795	+	0.801649i	$0.703956\pi$
$678$	0	0
$679$	−2.08977	−0.0801978
$680$	39.7642	1.52489
$681$	0	0
$682$	−14.9605	−0.572866
$683$	−41.1267	−1.57367	−0.786834	−	0.617164i	$-0.788281\pi$
$683$	−41.1267	−1.57367	−0.786834	+	0.617164i	$0.788281\pi$
$684$	0	0
$685$	62.4092	2.38453
$686$	1.32973	0.0507692
$687$	0	0
$688$	11.8247	0.450811
$689$	26.8064	1.02124
$690$	0	0
$691$	−15.1725	−0.577191	−0.288595	−	0.957451i	$-0.593188\pi$
$691$	−15.1725	−0.577191	−0.288595	+	0.957451i	$0.593188\pi$
$692$	−5.54678	−0.210857
$693$	0	0
$694$	−1.06329	−0.0403618
$695$	31.7011	1.20249
$696$	0	0
$697$	−5.80977	−0.220061
$698$	−1.59981	−0.0605537
$699$	0	0
$700$	1.17685	0.0444807
$701$	−6.18062	−0.233439	−0.116719	−	0.993165i	$-0.537238\pi$
$701$	−6.18062	−0.233439	−0.116719	+	0.993165i	$0.537238\pi$
$702$	0	0
$703$	9.67652	0.364957
$704$	−44.0702	−1.66096
$705$	0	0
$706$	−1.37406	−0.0517134
$707$	13.4312	0.505132
$708$	0	0
$709$	16.8330	0.632177	0.316088	−	0.948730i	$-0.397630\pi$
$709$	16.8330	0.632177	0.316088	+	0.948730i	$0.397630\pi$
$710$	12.4660	0.467842
$711$	0	0
$712$	50.6380	1.89774
$713$	2.22101	0.0831774
$714$	0	0
$715$	65.5492	2.45140
$716$	−4.05727	−0.151627
$717$	0	0
$718$	16.2664	0.607057
$719$	36.7510	1.37058	0.685290	−	0.728270i	$-0.259675\pi$
$719$	36.7510	1.37058	0.685290	+	0.728270i	$0.259675\pi$
$720$	0	0
$721$	−2.72964	−0.101657
$722$	−8.85667	−0.329611
$723$	0	0
$724$	−3.47131	−0.129010
$725$	−33.9343	−1.26029
$726$	0	0
$727$	42.5612	1.57851	0.789253	−	0.614068i	$-0.210468\pi$
$727$	42.5612	1.57851	0.789253	+	0.614068i	$0.210468\pi$
$728$	−12.0977	−0.448372
$729$	0	0
$730$	−8.54840	−0.316391
$731$	14.3318	0.530080
$732$	0	0
$733$	−7.67673	−0.283546	−0.141773	−	0.989899i	$-0.545280\pi$
$733$	−7.67673	−0.283546	−0.141773	+	0.989899i	$0.545280\pi$
$734$	26.6461	0.983527
$735$	0	0
$736$	1.30452	0.0480854
$737$	−70.3321	−2.59072
$738$	0	0
$739$	30.0540	1.10555	0.552777	−	0.833329i	$-0.313568\pi$
$739$	30.0540	1.10555	0.552777	+	0.833329i	$0.313568\pi$
$740$	−1.40573	−0.0516756
$741$	0	0
$742$	8.74420	0.321010
$743$	−43.6849	−1.60264	−0.801322	−	0.598234i	$-0.795869\pi$
$743$	−43.6849	−1.60264	−0.801322	+	0.598234i	$0.795869\pi$
$744$	0	0
$745$	6.38247	0.233836
$746$	34.1478	1.25024
$747$	0	0
$748$	−4.95690	−0.181242
$749$	−2.36558	−0.0864362
$750$	0	0
$751$	−35.0236	−1.27803	−0.639014	−	0.769195i	$-0.720658\pi$
$751$	−35.0236	−1.27803	−0.639014	+	0.769195i	$0.720658\pi$
$752$	−20.2508	−0.738471
$753$	0	0
$754$	36.2345	1.31958
$755$	23.6155	0.859457
$756$	0	0
$757$	25.1809	0.915214	0.457607	−	0.889155i	$-0.348707\pi$
$757$	25.1809	0.915214	0.457607	+	0.889155i	$0.348707\pi$
$758$	−14.3321	−0.520564
$759$	0	0
$760$	47.7209	1.73102
$761$	35.7236	1.29498	0.647490	−	0.762074i	$-0.275819\pi$
$761$	35.7236	1.29498	0.647490	+	0.762074i	$0.275819\pi$
$762$	0	0
$763$	18.1179	0.655913
$764$	3.57927	0.129494
$765$	0	0
$766$	21.6582	0.782542
$767$	23.1145	0.834617
$768$	0	0
$769$	−42.1421	−1.51968	−0.759842	−	0.650107i	$-0.774724\pi$
$769$	−42.1421	−1.51968	−0.759842	+	0.650107i	$0.774724\pi$
$770$	21.3820	0.770555
$771$	0	0
$772$	−4.04292	−0.145508
$773$	−12.5030	−0.449702	−0.224851	−	0.974393i	$-0.572190\pi$
$773$	−12.5030	−0.449702	−0.224851	+	0.974393i	$0.572190\pi$
$774$	0	0
$775$	−11.2748	−0.405003
$776$	6.20184	0.222633
$777$	0	0
$778$	29.5412	1.05910
$779$	−6.97229	−0.249808
$780$	0	0
$781$	−14.9605	−0.535328
$782$	−5.61279	−0.200713
$783$	0	0
$784$	−3.48261	−0.124379
$785$	−55.5639	−1.98316
$786$	0	0
$787$	37.5411	1.33820	0.669099	−	0.743174i	$-0.266680\pi$
$787$	37.5411	1.33820	0.669099	+	0.743174i	$0.266680\pi$
$788$	5.13952	0.183088
$789$	0	0
$790$	30.1731	1.07351
$791$	−6.08001	−0.216180
$792$	0	0
$793$	−59.0520	−2.09700
$794$	33.7110	1.19636
$795$	0	0
$796$	−3.45991	−0.122633
$797$	5.97335	0.211587	0.105793	−	0.994388i	$-0.466262\pi$
$797$	5.97335	0.211587	0.105793	+	0.994388i	$0.466262\pi$
$798$	0	0
$799$	−24.5445	−0.868321
$800$	−6.62233	−0.234135
$801$	0	0
$802$	−49.6816	−1.75432
$803$	10.2589	0.362029
$804$	0	0
$805$	−3.17434	−0.111881
$806$	12.0391	0.424059
$807$	0	0
$808$	−39.8600	−1.40227
$809$	−38.3623	−1.34875	−0.674374	−	0.738390i	$-0.735586\pi$
$809$	−38.3623	−1.34875	−0.674374	+	0.738390i	$0.735586\pi$
$810$	0	0
$811$	−17.7595	−0.623619	−0.311810	−	0.950145i	$-0.600935\pi$
$811$	−17.7595	−0.623619	−0.311810	+	0.950145i	$0.600935\pi$
$812$	−1.54968	−0.0543829
$813$	0	0
$814$	−12.8671	−0.450993
$815$	12.5526	0.439698
$816$	0	0
$817$	17.1995	0.601736
$818$	39.6124	1.38502
$819$	0	0
$820$	1.01288	0.0353713
$821$	7.09433	0.247594	0.123797	−	0.992308i	$-0.460493\pi$
$821$	7.09433	0.247594	0.123797	+	0.992308i	$0.460493\pi$
$822$	0	0
$823$	31.6990	1.10496	0.552480	−	0.833526i	$-0.313682\pi$
$823$	31.6990	1.10496	0.552480	+	0.833526i	$0.313682\pi$
$824$	8.10082	0.282205
$825$	0	0
$826$	7.53992	0.262347
$827$	49.0671	1.70623	0.853115	−	0.521722i	$-0.174710\pi$
$827$	49.0671	1.70623	0.853115	+	0.521722i	$0.174710\pi$
$828$	0	0
$829$	16.4384	0.570931	0.285465	−	0.958389i	$-0.407852\pi$
$829$	16.4384	0.570931	0.285465	+	0.958389i	$0.407852\pi$
$830$	53.7020	1.86403
$831$	0	0
$832$	35.4645	1.22951
$833$	−4.22101	−0.146249
$834$	0	0
$835$	−9.80001	−0.339143
$836$	−5.94876	−0.205742
$837$	0	0
$838$	−3.28155	−0.113359
$839$	6.63342	0.229011	0.114506	−	0.993423i	$-0.463472\pi$
$839$	6.63342	0.229011	0.114506	+	0.993423i	$0.463472\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	20.2041	0.696280
$843$	0	0
$844$	0.859796	0.0295954
$845$	−11.4827	−0.395018
$846$	0	0
$847$	−14.6605	−0.503741
$848$	−22.9014	−0.786437
$849$	0	0
$850$	28.4930	0.977301
$851$	1.91023	0.0654820
$852$	0	0
$853$	41.8151	1.43172	0.715861	−	0.698243i	$-0.246034\pi$
$853$	41.8151	1.43172	0.715861	+	0.698243i	$0.246034\pi$
$854$	−19.2627	−0.659155
$855$	0	0
$856$	7.02036	0.239951
$857$	36.4689	1.24575	0.622877	−	0.782319i	$-0.285964\pi$
$857$	36.4689	1.24575	0.622877	+	0.782319i	$0.285964\pi$
$858$	0	0
$859$	−34.5774	−1.17976	−0.589882	−	0.807489i	$-0.700826\pi$
$859$	−34.5774	−1.17976	−0.589882	+	0.807489i	$0.700826\pi$
$860$	−2.49861	−0.0852021
$861$	0	0
$862$	−20.3833	−0.694258
$863$	29.2710	0.996395	0.498198	−	0.867063i	$-0.333995\pi$
$863$	29.2710	0.996395	0.498198	+	0.867063i	$0.333995\pi$
$864$	0	0
$865$	−75.9509	−2.58241
$866$	2.57827	0.0876132
$867$	0	0
$868$	−0.514886	−0.0174764
$869$	−36.2106	−1.22836
$870$	0	0
$871$	56.5981	1.91775
$872$	−53.7689	−1.82084
$873$	0	0
$874$	−6.73589	−0.227845
$875$	0.242644	0.00820287
$876$	0	0
$877$	−4.57140	−0.154365	−0.0771826	−	0.997017i	$-0.524592\pi$
$877$	−4.57140	−0.154365	−0.0771826	+	0.997017i	$0.524592\pi$
$878$	−5.38073	−0.181591
$879$	0	0
$880$	−56.0003	−1.88777
$881$	−11.4285	−0.385036	−0.192518	−	0.981293i	$-0.561665\pi$
$881$	−11.4285	−0.385036	−0.192518	+	0.981293i	$0.561665\pi$
$882$	0	0
$883$	−18.9351	−0.637215	−0.318608	−	0.947887i	$-0.603215\pi$
$883$	−18.9351	−0.637215	−0.318608	+	0.947887i	$0.603215\pi$
$884$	3.98895	0.134163
$885$	0	0
$886$	23.7850	0.799074
$887$	6.96926	0.234005	0.117002	−	0.993132i	$-0.462671\pi$
$887$	6.96926	0.234005	0.117002	+	0.993132i	$0.462671\pi$
$888$	0	0
$889$	−5.42763	−0.182037
$890$	72.0228	2.41421
$891$	0	0
$892$	−1.32577	−0.0443901
$893$	−29.4558	−0.985700
$894$	0	0
$895$	−55.5554	−1.85701
$896$	8.95941	0.299313
$897$	0	0
$898$	44.1156	1.47216
$899$	14.8467	0.495164
$900$	0	0
$901$	−27.7571	−0.924723
$902$	9.27124	0.308699
$903$	0	0
$904$	18.0438	0.600126
$905$	−47.5319	−1.58001
$906$	0	0
$907$	41.9505	1.39294	0.696472	−	0.717584i	$-0.254752\pi$
$907$	41.9505	1.39294	0.696472	+	0.717584i	$0.254752\pi$
$908$	6.15350	0.204211
$909$	0	0
$910$	−17.2067	−0.570396
$911$	10.6040	0.351327	0.175664	−	0.984450i	$-0.443793\pi$
$911$	10.6040	0.351327	0.175664	+	0.984450i	$0.443793\pi$
$912$	0	0
$913$	−64.4477	−2.13291
$914$	−38.6722	−1.27916
$915$	0	0
$916$	0.859652	0.0284037
$917$	−0.585698	−0.0193414
$918$	0	0
$919$	2.20439	0.0727160	0.0363580	−	0.999339i	$-0.488424\pi$
$919$	2.20439	0.0727160	0.0363580	+	0.999339i	$0.488424\pi$
$920$	9.42055	0.310586
$921$	0	0
$922$	−44.3544	−1.46073
$923$	12.0391	0.396271
$924$	0	0
$925$	−9.69719	−0.318842
$926$	2.49717	0.0820620
$927$	0	0
$928$	8.72029	0.286258
$929$	4.27581	0.140285	0.0701424	−	0.997537i	$-0.477655\pi$
$929$	4.27581	0.140285	0.0701424	+	0.997537i	$0.477655\pi$
$930$	0	0
$931$	−5.06562	−0.166019
$932$	5.14536	0.168542
$933$	0	0
$934$	17.1901	0.562476
$935$	−67.8738	−2.21971
$936$	0	0
$937$	−30.6645	−1.00177	−0.500883	−	0.865515i	$-0.666991\pi$
$937$	−30.6645	−1.00177	−0.500883	+	0.865515i	$0.666991\pi$
$938$	18.4622	0.602812
$939$	0	0
$940$	4.27910	0.139569
$941$	−9.24521	−0.301385	−0.150693	−	0.988581i	$-0.548150\pi$
$941$	−9.24521	−0.301385	−0.150693	+	0.988581i	$0.548150\pi$
$942$	0	0
$943$	−1.37639	−0.0448215
$944$	−19.7473	−0.642721
$945$	0	0
$946$	−22.8707	−0.743591
$947$	−21.8800	−0.711005	−0.355502	−	0.934675i	$-0.615690\pi$
$947$	−21.8800	−0.711005	−0.355502	+	0.934675i	$0.615690\pi$
$948$	0	0
$949$	−8.25562	−0.267989
$950$	34.1944	1.10941
$951$	0	0
$952$	12.5268	0.405995
$953$	38.2800	1.24001	0.620005	−	0.784598i	$-0.287130\pi$
$953$	38.2800	1.24001	0.620005	+	0.784598i	$0.287130\pi$
$954$	0	0
$955$	49.0103	1.58593
$956$	−4.70422	−0.152145
$957$	0	0
$958$	−57.7079	−1.86446
$959$	19.6605	0.634871
$960$	0	0
$961$	−26.0671	−0.840875
$962$	10.3545	0.333843
$963$	0	0
$964$	−3.36513	−0.108383
$965$	−55.3589	−1.78207
$966$	0	0
$967$	−15.5397	−0.499723	−0.249862	−	0.968282i	$-0.580385\pi$
$967$	−15.5397	−0.499723	−0.249862	+	0.968282i	$0.580385\pi$
$968$	43.5083	1.39841
$969$	0	0
$970$	8.82092	0.283223
$971$	−23.9345	−0.768094	−0.384047	−	0.923314i	$-0.625470\pi$
$971$	−23.9345	−0.768094	−0.384047	+	0.923314i	$0.625470\pi$
$972$	0	0
$973$	9.98667	0.320158
$974$	−22.6043	−0.724289
$975$	0	0
$976$	50.4496	1.61485
$977$	8.13749	0.260341	0.130171	−	0.991492i	$-0.458447\pi$
$977$	8.13749	0.260341	0.130171	+	0.991492i	$0.458447\pi$
$978$	0	0
$979$	−86.4344	−2.76245
$980$	0.735894	0.0235073
$981$	0	0
$982$	14.5578	0.464558
$983$	−24.2119	−0.772239	−0.386119	−	0.922449i	$-0.626185\pi$
$983$	−24.2119	−0.772239	−0.386119	+	0.922449i	$0.626185\pi$
$984$	0	0
$985$	70.3744	2.24232
$986$	−37.5196	−1.19487
$987$	0	0
$988$	4.78713	0.152299
$989$	3.39535	0.107966
$990$	0	0
$991$	36.1425	1.14810	0.574052	−	0.818819i	$-0.305371\pi$
$991$	36.1425	1.14810	0.574052	+	0.818819i	$0.305371\pi$
$992$	2.89736	0.0919911
$993$	0	0
$994$	3.92713	0.124561
$995$	−47.3758	−1.50191
$996$	0	0
$997$	−0.629234	−0.0199280	−0.00996402	−	0.999950i	$-0.503172\pi$
$997$	−0.629234	−0.0199280	−0.00996402	+	0.999950i	$0.503172\pi$
$998$	−40.3689	−1.27785
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.o.1.2		4
3.2	odd	2		483.2.a.j.1.3	✓	4
12.11	even	2		7728.2.a.ce.1.3		4
21.20	even	2		3381.2.a.x.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.j.1.3	✓	4	3.2	odd	2
1449.2.a.o.1.2		4	1.1	even	1	trivial
3381.2.a.x.1.3		4	21.20	even	2
7728.2.a.ce.1.3		4	12.11	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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q-1.32973 q^{2} -0.231826 q^{4} -3.17434 q^{5} -1.00000 q^{7} +2.96772 q^{8} +O(q^{10})\)	\(q-1.32973 q^{2} -0.231826 q^{4} -3.17434 q^{5} -1.00000 q^{7} +2.96772 q^{8} +4.22101 q^{10} -5.06562 q^{11} +4.07644 q^{13} +1.32973 q^{14} -3.48261 q^{16} -4.22101 q^{17} -5.06562 q^{19} +0.735894 q^{20} +6.73589 q^{22} -1.00000 q^{23} +5.07644 q^{25} -5.42055 q^{26} +0.231826 q^{28} -6.68466 q^{29} -2.22101 q^{31} -1.30452 q^{32} +5.61279 q^{34} +3.17434 q^{35} -1.91023 q^{37} +6.73589 q^{38} -9.42055 q^{40} +1.37639 q^{41} -3.39535 q^{43} +1.17434 q^{44} +1.32973 q^{46} +5.81484 q^{47} +1.00000 q^{49} -6.75028 q^{50} -0.945023 q^{52} +6.57594 q^{53} +16.0800 q^{55} -2.96772 q^{56} +8.88877 q^{58} +5.67027 q^{59} -14.4862 q^{61} +2.95333 q^{62} +8.69987 q^{64} -12.9400 q^{65} +13.8842 q^{67} +0.978538 q^{68} -4.22101 q^{70} +2.95333 q^{71} -2.02520 q^{73} +2.54009 q^{74} +1.17434 q^{76} +5.06562 q^{77} +7.14831 q^{79} +11.0550 q^{80} -1.83023 q^{82} +12.7226 q^{83} +13.3989 q^{85} +4.51489 q^{86} -15.0333 q^{88} +17.0629 q^{89} -4.07644 q^{91} +0.231826 q^{92} -7.73215 q^{94} +16.0800 q^{95} +2.08977 q^{97} -1.32973 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 9 * q^8	\( 4 q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 9 q^{8} + 2 q^{10} - q^{11} + 7 q^{13} + 2 q^{14} + 8 q^{16} - 2 q^{17} - q^{19} - 13 q^{20} + 11 q^{22} - 4 q^{23} + 11 q^{25} + 19 q^{26} - 4 q^{28} - 2 q^{29} + 6 q^{31} - 20 q^{32} + 23 q^{34} + 5 q^{35} + 16 q^{37} + 11 q^{38} + 3 q^{40} - 5 q^{41} + 9 q^{43} - 3 q^{44} + 2 q^{46} + 21 q^{47} + 4 q^{49} + 17 q^{50} - 24 q^{52} - 10 q^{53} + 17 q^{55} + 9 q^{56} + q^{58} + 26 q^{59} + 2 q^{61} + 19 q^{62} + 27 q^{64} - 26 q^{65} + 5 q^{67} - 7 q^{68} - 2 q^{70} + 19 q^{71} + 10 q^{73} - 9 q^{74} - 3 q^{76} + q^{77} - 6 q^{79} + 24 q^{80} - 31 q^{82} + 2 q^{83} - 7 q^{85} + 17 q^{86} - 20 q^{88} + 17 q^{89} - 7 q^{91} - 4 q^{92} - 44 q^{94} + 17 q^{95} + 32 q^{97} - 2 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 9 * q^8 + 2 * q^10 - q^11 + 7 * q^13 + 2 * q^14 + 8 * q^16 - 2 * q^17 - q^19 - 13 * q^20 + 11 * q^22 - 4 * q^23 + 11 * q^25 + 19 * q^26 - 4 * q^28 - 2 * q^29 + 6 * q^31 - 20 * q^32 + 23 * q^34 + 5 * q^35 + 16 * q^37 + 11 * q^38 + 3 * q^40 - 5 * q^41 + 9 * q^43 - 3 * q^44 + 2 * q^46 + 21 * q^47 + 4 * q^49 + 17 * q^50 - 24 * q^52 - 10 * q^53 + 17 * q^55 + 9 * q^56 + q^58 + 26 * q^59 + 2 * q^61 + 19 * q^62 + 27 * q^64 - 26 * q^65 + 5 * q^67 - 7 * q^68 - 2 * q^70 + 19 * q^71 + 10 * q^73 - 9 * q^74 - 3 * q^76 + q^77 - 6 * q^79 + 24 * q^80 - 31 * q^82 + 2 * q^83 - 7 * q^85 + 17 * q^86 - 20 * q^88 + 17 * q^89 - 7 * q^91 - 4 * q^92 - 44 * q^94 + 17 * q^95 + 32 * q^97 - 2 * q^98

Introduction

L-functions

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