LMFDB - Embedded newform 322.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [322,2,Mod(1,322)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(322, 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("322.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 322 = 2 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	322.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:	$2.57118294509$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	322.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.23607 q^{3} +1.00000 q^{4} -3.23607 q^{5} -1.23607 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.47214 q^{9} +3.23607 q^{10} +1.23607 q^{12} -6.47214 q^{13} +1.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -2.76393 q^{17} +1.47214 q^{18} +4.47214 q^{19} -3.23607 q^{20} -1.23607 q^{21} -1.00000 q^{23} -1.23607 q^{24} +5.47214 q^{25} +6.47214 q^{26} -5.52786 q^{27} -1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{30} +3.23607 q^{31} -1.00000 q^{32} +2.76393 q^{34} +3.23607 q^{35} -1.47214 q^{36} +6.00000 q^{37} -4.47214 q^{38} -8.00000 q^{39} +3.23607 q^{40} -10.0000 q^{41} +1.23607 q^{42} +2.47214 q^{43} +4.76393 q^{45} +1.00000 q^{46} -11.2361 q^{47} +1.23607 q^{48} +1.00000 q^{49} -5.47214 q^{50} -3.41641 q^{51} -6.47214 q^{52} -6.00000 q^{53} +5.52786 q^{54} +1.00000 q^{56} +5.52786 q^{57} +2.00000 q^{58} +6.76393 q^{59} -4.00000 q^{60} -1.70820 q^{61} -3.23607 q^{62} +1.47214 q^{63} +1.00000 q^{64} +20.9443 q^{65} +4.00000 q^{67} -2.76393 q^{68} -1.23607 q^{69} -3.23607 q^{70} +6.47214 q^{71} +1.47214 q^{72} +13.4164 q^{73} -6.00000 q^{74} +6.76393 q^{75} +4.47214 q^{76} +8.00000 q^{78} -8.94427 q^{79} -3.23607 q^{80} -2.41641 q^{81} +10.0000 q^{82} +10.9443 q^{83} -1.23607 q^{84} +8.94427 q^{85} -2.47214 q^{86} -2.47214 q^{87} +6.18034 q^{89} -4.76393 q^{90} +6.47214 q^{91} -1.00000 q^{92} +4.00000 q^{93} +11.2361 q^{94} -14.4721 q^{95} -1.23607 q^{96} -14.1803 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{10} - 2 q^{12} - 4 q^{13} + 2 q^{14} - 8 q^{15} + 2 q^{16} - 10 q^{17} - 6 q^{18} - 2 q^{20} + 2 q^{21} - 2 q^{23}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 6 * q^9 + 2 * q^10 - 2 * q^12 - 4 * q^13 + 2 * q^14 - 8 * q^15 + 2 * q^16 - 10 * q^17 - 6 * q^18 - 2 * q^20 + 2 * q^21 - 2 * q^23 + 2 * q^24 + 2 * q^25 + 4 * q^26 - 20 * q^27 - 2 * q^28 - 4 * q^29 + 8 * q^30 + 2 * q^31 - 2 * q^32 + 10 * q^34 + 2 * q^35 + 6 * q^36 + 12 * q^37 - 16 * q^39 + 2 * q^40 - 20 * q^41 - 2 * q^42 - 4 * q^43 + 14 * q^45 + 2 * q^46 - 18 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^50 + 20 * q^51 - 4 * q^52 - 12 * q^53 + 20 * q^54 + 2 * q^56 + 20 * q^57 + 4 * q^58 + 18 * q^59 - 8 * q^60 + 10 * q^61 - 2 * q^62 - 6 * q^63 + 2 * q^64 + 24 * q^65 + 8 * q^67 - 10 * q^68 + 2 * q^69 - 2 * q^70 + 4 * q^71 - 6 * q^72 - 12 * q^74 + 18 * q^75 + 16 * q^78 - 2 * q^80 + 22 * q^81 + 20 * q^82 + 4 * q^83 + 2 * q^84 + 4 * q^86 + 4 * q^87 - 10 * q^89 - 14 * q^90 + 4 * q^91 - 2 * q^92 + 8 * q^93 + 18 * q^94 - 20 * q^95 + 2 * q^96 - 6 * q^97 - 2 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	1.00000	0.500000
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	−1.23607	−0.504623
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−1.47214	−0.490712
$10$	3.23607	1.02333
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.23607	0.356822
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	1.00000	0.267261
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	−2.76393	−0.670352	−0.335176	−	0.942156i	$-0.608796\pi$
$17$	−2.76393	−0.670352	−0.335176	+	0.942156i	$0.608796\pi$
$18$	1.47214	0.346986
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	−3.23607	−0.723607
$21$	−1.23607	−0.269732
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.23607	−0.252311
$25$	5.47214	1.09443
$26$	6.47214	1.26929
$27$	−5.52786	−1.06384
$28$	−1.00000	−0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	4.00000	0.730297
$31$	3.23607	0.581215	0.290607	−	0.956842i	$-0.406143\pi$
$31$	3.23607	0.581215	0.290607	+	0.956842i	$0.406143\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.76393	0.474010
$35$	3.23607	0.546995
$36$	−1.47214	−0.245356
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.47214	−0.725476
$39$	−8.00000	−1.28103
$40$	3.23607	0.511667
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	1.23607	0.190729
$43$	2.47214	0.376997	0.188499	−	0.982073i	$-0.439638\pi$
$43$	2.47214	0.376997	0.188499	+	0.982073i	$0.439638\pi$
$44$	0	0
$45$	4.76393	0.710165
$46$	1.00000	0.147442
$47$	−11.2361	−1.63895	−0.819474	−	0.573116i	$-0.805735\pi$
$47$	−11.2361	−1.63895	−0.819474	+	0.573116i	$0.805735\pi$
$48$	1.23607	0.178411
$49$	1.00000	0.142857
$50$	−5.47214	−0.773877
$51$	−3.41641	−0.478393
$52$	−6.47214	−0.897524
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	5.52786	0.752247
$55$	0	0
$56$	1.00000	0.133631
$57$	5.52786	0.732183
$58$	2.00000	0.262613
$59$	6.76393	0.880589	0.440294	−	0.897853i	$-0.354874\pi$
$59$	6.76393	0.880589	0.440294	+	0.897853i	$0.354874\pi$
$60$	−4.00000	−0.516398
$61$	−1.70820	−0.218713	−0.109357	−	0.994003i	$-0.534879\pi$
$61$	−1.70820	−0.218713	−0.109357	+	0.994003i	$0.534879\pi$
$62$	−3.23607	−0.410981
$63$	1.47214	0.185472
$64$	1.00000	0.125000
$65$	20.9443	2.59782
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−2.76393	−0.335176
$69$	−1.23607	−0.148805
$70$	−3.23607	−0.386784
$71$	6.47214	0.768101	0.384051	−	0.923312i	$-0.374529\pi$
$71$	6.47214	0.768101	0.384051	+	0.923312i	$0.374529\pi$
$72$	1.47214	0.173493
$73$	13.4164	1.57027	0.785136	−	0.619324i	$-0.212593\pi$
$73$	13.4164	1.57027	0.785136	+	0.619324i	$0.212593\pi$
$74$	−6.00000	−0.697486
$75$	6.76393	0.781032
$76$	4.47214	0.512989
$77$	0	0
$78$	8.00000	0.905822
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	−3.23607	−0.361803
$81$	−2.41641	−0.268490
$82$	10.0000	1.10432
$83$	10.9443	1.20129	0.600645	−	0.799516i	$-0.294911\pi$
$83$	10.9443	1.20129	0.600645	+	0.799516i	$0.294911\pi$
$84$	−1.23607	−0.134866
$85$	8.94427	0.970143
$86$	−2.47214	−0.266577
$87$	−2.47214	−0.265041
$88$	0	0
$89$	6.18034	0.655115	0.327557	−	0.944831i	$-0.393775\pi$
$89$	6.18034	0.655115	0.327557	+	0.944831i	$0.393775\pi$
$90$	−4.76393	−0.502163
$91$	6.47214	0.678464
$92$	−1.00000	−0.104257
$93$	4.00000	0.414781
$94$	11.2361	1.15891
$95$	−14.4721	−1.48481
$96$	−1.23607	−0.126156
$97$	−14.1803	−1.43980	−0.719898	−	0.694080i	$-0.755811\pi$
$97$	−14.1803	−1.43980	−0.719898	+	0.694080i	$0.755811\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	5.47214	0.547214
$101$	−5.52786	−0.550043	−0.275022	−	0.961438i	$-0.588685\pi$
$101$	−5.52786	−0.550043	−0.275022	+	0.961438i	$0.588685\pi$
$102$	3.41641	0.338275
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	6.47214	0.634645
$105$	4.00000	0.390360
$106$	6.00000	0.582772
$107$	−1.52786	−0.147704	−0.0738521	−	0.997269i	$-0.523529\pi$
$107$	−1.52786	−0.147704	−0.0738521	+	0.997269i	$0.523529\pi$
$108$	−5.52786	−0.531919
$109$	12.4721	1.19461	0.597307	−	0.802013i	$-0.296237\pi$
$109$	12.4721	1.19461	0.597307	+	0.802013i	$0.296237\pi$
$110$	0	0
$111$	7.41641	0.703934
$112$	−1.00000	−0.0944911
$113$	12.4721	1.17328	0.586640	−	0.809848i	$-0.300450\pi$
$113$	12.4721	1.17328	0.586640	+	0.809848i	$0.300450\pi$
$114$	−5.52786	−0.517732
$115$	3.23607	0.301765
$116$	−2.00000	−0.185695
$117$	9.52786	0.880851
$118$	−6.76393	−0.622670
$119$	2.76393	0.253369
$120$	4.00000	0.365148
$121$	−11.0000	−1.00000
$122$	1.70820	0.154654
$123$	−12.3607	−1.11452
$124$	3.23607	0.290607
$125$	−1.52786	−0.136656
$126$	−1.47214	−0.131148
$127$	−19.4164	−1.72293	−0.861464	−	0.507819i	$-0.830452\pi$
$127$	−19.4164	−1.72293	−0.861464	+	0.507819i	$0.830452\pi$
$128$	−1.00000	−0.0883883
$129$	3.05573	0.269042
$130$	−20.9443	−1.83693
$131$	−17.2361	−1.50592	−0.752961	−	0.658065i	$-0.771375\pi$
$131$	−17.2361	−1.50592	−0.752961	+	0.658065i	$0.771375\pi$
$132$	0	0
$133$	−4.47214	−0.387783
$134$	−4.00000	−0.345547
$135$	17.8885	1.53960
$136$	2.76393	0.237005
$137$	−9.41641	−0.804498	−0.402249	−	0.915530i	$-0.631771\pi$
$137$	−9.41641	−0.804498	−0.402249	+	0.915530i	$0.631771\pi$
$138$	1.23607	0.105221
$139$	−23.1246	−1.96140	−0.980702	−	0.195509i	$-0.937364\pi$
$139$	−23.1246	−1.96140	−0.980702	+	0.195509i	$0.937364\pi$
$140$	3.23607	0.273498
$141$	−13.8885	−1.16963
$142$	−6.47214	−0.543130
$143$	0	0
$144$	−1.47214	−0.122678
$145$	6.47214	0.537482
$146$	−13.4164	−1.11035
$147$	1.23607	0.101949
$148$	6.00000	0.493197
$149$	21.4164	1.75450	0.877250	−	0.480033i	$-0.159375\pi$
$149$	21.4164	1.75450	0.877250	+	0.480033i	$0.159375\pi$
$150$	−6.76393	−0.552273
$151$	−9.52786	−0.775367	−0.387683	−	0.921793i	$-0.626725\pi$
$151$	−9.52786	−0.775367	−0.387683	+	0.921793i	$0.626725\pi$
$152$	−4.47214	−0.362738
$153$	4.06888	0.328950
$154$	0	0
$155$	−10.4721	−0.841142
$156$	−8.00000	−0.640513
$157$	−18.6525	−1.48863	−0.744315	−	0.667829i	$-0.767224\pi$
$157$	−18.6525	−1.48863	−0.744315	+	0.667829i	$0.767224\pi$
$158$	8.94427	0.711568
$159$	−7.41641	−0.588159
$160$	3.23607	0.255834
$161$	1.00000	0.0788110
$162$	2.41641	0.189851
$163$	−6.47214	−0.506937	−0.253468	−	0.967344i	$-0.581571\pi$
$163$	−6.47214	−0.506937	−0.253468	+	0.967344i	$0.581571\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−10.9443	−0.849440
$167$	−6.65248	−0.514784	−0.257392	−	0.966307i	$-0.582863\pi$
$167$	−6.65248	−0.514784	−0.257392	+	0.966307i	$0.582863\pi$
$168$	1.23607	0.0953647
$169$	28.8885	2.22220
$170$	−8.94427	−0.685994
$171$	−6.58359	−0.503460
$172$	2.47214	0.188499
$173$	3.05573	0.232323	0.116161	−	0.993230i	$-0.462941\pi$
$173$	3.05573	0.232323	0.116161	+	0.993230i	$0.462941\pi$
$174$	2.47214	0.187412
$175$	−5.47214	−0.413655
$176$	0	0
$177$	8.36068	0.628427
$178$	−6.18034	−0.463236
$179$	16.9443	1.26647	0.633237	−	0.773958i	$-0.281726\pi$
$179$	16.9443	1.26647	0.633237	+	0.773958i	$0.281726\pi$
$180$	4.76393	0.355083
$181$	8.76393	0.651418	0.325709	−	0.945470i	$-0.394397\pi$
$181$	8.76393	0.651418	0.325709	+	0.945470i	$0.394397\pi$
$182$	−6.47214	−0.479747
$183$	−2.11146	−0.156083
$184$	1.00000	0.0737210
$185$	−19.4164	−1.42752
$186$	−4.00000	−0.293294
$187$	0	0
$188$	−11.2361	−0.819474
$189$	5.52786	0.402093
$190$	14.4721	1.04992
$191$	3.05573	0.221105	0.110552	−	0.993870i	$-0.464738\pi$
$191$	3.05573	0.221105	0.110552	+	0.993870i	$0.464738\pi$
$192$	1.23607	0.0892055
$193$	−5.05573	−0.363919	−0.181960	−	0.983306i	$-0.558244\pi$
$193$	−5.05573	−0.363919	−0.181960	+	0.983306i	$0.558244\pi$
$194$	14.1803	1.01809
$195$	25.8885	1.85392
$196$	1.00000	0.0714286
$197$	−19.8885	−1.41700	−0.708500	−	0.705711i	$-0.750628\pi$
$197$	−19.8885	−1.41700	−0.708500	+	0.705711i	$0.750628\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	−5.47214	−0.386938
$201$	4.94427	0.348742
$202$	5.52786	0.388939
$203$	2.00000	0.140372
$204$	−3.41641	−0.239196
$205$	32.3607	2.26017
$206$	4.00000	0.278693
$207$	1.47214	0.102321
$208$	−6.47214	−0.448762
$209$	0	0
$210$	−4.00000	−0.276026
$211$	16.9443	1.16649	0.583246	−	0.812296i	$-0.301782\pi$
$211$	16.9443	1.16649	0.583246	+	0.812296i	$0.301782\pi$
$212$	−6.00000	−0.412082
$213$	8.00000	0.548151
$214$	1.52786	0.104443
$215$	−8.00000	−0.545595
$216$	5.52786	0.376124
$217$	−3.23607	−0.219679
$218$	−12.4721	−0.844720
$219$	16.5836	1.12062
$220$	0	0
$221$	17.8885	1.20331
$222$	−7.41641	−0.497757
$223$	−10.6525	−0.713343	−0.356671	−	0.934230i	$-0.616088\pi$
$223$	−10.6525	−0.713343	−0.356671	+	0.934230i	$0.616088\pi$
$224$	1.00000	0.0668153
$225$	−8.05573	−0.537049
$226$	−12.4721	−0.829634
$227$	−21.4164	−1.42146	−0.710728	−	0.703466i	$-0.751635\pi$
$227$	−21.4164	−1.42146	−0.710728	+	0.703466i	$0.751635\pi$
$228$	5.52786	0.366092
$229$	−15.2361	−1.00683	−0.503414	−	0.864045i	$-0.667923\pi$
$229$	−15.2361	−1.00683	−0.503414	+	0.864045i	$0.667923\pi$
$230$	−3.23607	−0.213380
$231$	0	0
$232$	2.00000	0.131306
$233$	−16.4721	−1.07913	−0.539563	−	0.841945i	$-0.681410\pi$
$233$	−16.4721	−1.07913	−0.539563	+	0.841945i	$0.681410\pi$
$234$	−9.52786	−0.622856
$235$	36.3607	2.37191
$236$	6.76393	0.440294
$237$	−11.0557	−0.718147
$238$	−2.76393	−0.179159
$239$	6.47214	0.418648	0.209324	−	0.977846i	$-0.432874\pi$
$239$	6.47214	0.418648	0.209324	+	0.977846i	$0.432874\pi$
$240$	−4.00000	−0.258199
$241$	0.291796	0.0187962	0.00939812	−	0.999956i	$-0.497008\pi$
$241$	0.291796	0.0187962	0.00939812	+	0.999956i	$0.497008\pi$
$242$	11.0000	0.707107
$243$	13.5967	0.872232
$244$	−1.70820	−0.109357
$245$	−3.23607	−0.206745
$246$	12.3607	0.788088
$247$	−28.9443	−1.84168
$248$	−3.23607	−0.205491
$249$	13.5279	0.857294
$250$	1.52786	0.0966306
$251$	−4.47214	−0.282279	−0.141139	−	0.989990i	$-0.545077\pi$
$251$	−4.47214	−0.282279	−0.141139	+	0.989990i	$0.545077\pi$
$252$	1.47214	0.0927358
$253$	0	0
$254$	19.4164	1.21829
$255$	11.0557	0.692337
$256$	1.00000	0.0625000
$257$	−26.9443	−1.68074	−0.840369	−	0.542015i	$-0.817662\pi$
$257$	−26.9443	−1.68074	−0.840369	+	0.542015i	$0.817662\pi$
$258$	−3.05573	−0.190241
$259$	−6.00000	−0.372822
$260$	20.9443	1.29891
$261$	2.94427	0.182246
$262$	17.2361	1.06485
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	19.4164	1.19274
$266$	4.47214	0.274204
$267$	7.63932	0.467519
$268$	4.00000	0.244339
$269$	3.05573	0.186311	0.0931555	−	0.995652i	$-0.470305\pi$
$269$	3.05573	0.186311	0.0931555	+	0.995652i	$0.470305\pi$
$270$	−17.8885	−1.08866
$271$	−15.2361	−0.925525	−0.462763	−	0.886482i	$-0.653142\pi$
$271$	−15.2361	−0.925525	−0.462763	+	0.886482i	$0.653142\pi$
$272$	−2.76393	−0.167588
$273$	8.00000	0.484182
$274$	9.41641	0.568866
$275$	0	0
$276$	−1.23607	−0.0744025
$277$	19.8885	1.19499	0.597493	−	0.801874i	$-0.296163\pi$
$277$	19.8885	1.19499	0.597493	+	0.801874i	$0.296163\pi$
$278$	23.1246	1.38692
$279$	−4.76393	−0.285209
$280$	−3.23607	−0.193392
$281$	5.41641	0.323116	0.161558	−	0.986863i	$-0.448348\pi$
$281$	5.41641	0.323116	0.161558	+	0.986863i	$0.448348\pi$
$282$	13.8885	0.827051
$283$	4.47214	0.265841	0.132920	−	0.991127i	$-0.457565\pi$
$283$	4.47214	0.265841	0.132920	+	0.991127i	$0.457565\pi$
$284$	6.47214	0.384051
$285$	−17.8885	−1.05963
$286$	0	0
$287$	10.0000	0.590281
$288$	1.47214	0.0867464
$289$	−9.36068	−0.550628
$290$	−6.47214	−0.380057
$291$	−17.5279	−1.02750
$292$	13.4164	0.785136
$293$	−29.1246	−1.70148	−0.850739	−	0.525588i	$-0.823845\pi$
$293$	−29.1246	−1.70148	−0.850739	+	0.525588i	$0.823845\pi$
$294$	−1.23607	−0.0720889
$295$	−21.8885	−1.27440
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−21.4164	−1.24062
$299$	6.47214	0.374293
$300$	6.76393	0.390516
$301$	−2.47214	−0.142492
$302$	9.52786	0.548267
$303$	−6.83282	−0.392535
$304$	4.47214	0.256495
$305$	5.52786	0.316525
$306$	−4.06888	−0.232603
$307$	29.2361	1.66859	0.834295	−	0.551318i	$-0.185875\pi$
$307$	29.2361	1.66859	0.834295	+	0.551318i	$0.185875\pi$
$308$	0	0
$309$	−4.94427	−0.281270
$310$	10.4721	0.594777
$311$	25.1246	1.42469	0.712343	−	0.701831i	$-0.247634\pi$
$311$	25.1246	1.42469	0.712343	+	0.701831i	$0.247634\pi$
$312$	8.00000	0.452911
$313$	26.1803	1.47980	0.739900	−	0.672717i	$-0.234873\pi$
$313$	26.1803	1.47980	0.739900	+	0.672717i	$0.234873\pi$
$314$	18.6525	1.05262
$315$	−4.76393	−0.268417
$316$	−8.94427	−0.503155
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	7.41641	0.415892
$319$	0	0
$320$	−3.23607	−0.180902
$321$	−1.88854	−0.105408
$322$	−1.00000	−0.0557278
$323$	−12.3607	−0.687767
$324$	−2.41641	−0.134245
$325$	−35.4164	−1.96455
$326$	6.47214	0.358458
$327$	15.4164	0.852529
$328$	10.0000	0.552158
$329$	11.2361	0.619464
$330$	0	0
$331$	−30.4721	−1.67490	−0.837450	−	0.546514i	$-0.815955\pi$
$331$	−30.4721	−1.67490	−0.837450	+	0.546514i	$0.815955\pi$
$332$	10.9443	0.600645
$333$	−8.83282	−0.484035
$334$	6.65248	0.364007
$335$	−12.9443	−0.707221
$336$	−1.23607	−0.0674330
$337$	−35.8885	−1.95497	−0.977487	−	0.210997i	$-0.932329\pi$
$337$	−35.8885	−1.95497	−0.977487	+	0.210997i	$0.932329\pi$
$338$	−28.8885	−1.57133
$339$	15.4164	0.837304
$340$	8.94427	0.485071
$341$	0	0
$342$	6.58359	0.356000
$343$	−1.00000	−0.0539949
$344$	−2.47214	−0.133289
$345$	4.00000	0.215353
$346$	−3.05573	−0.164277
$347$	24.3607	1.30775	0.653875	−	0.756603i	$-0.273142\pi$
$347$	24.3607	1.30775	0.653875	+	0.756603i	$0.273142\pi$
$348$	−2.47214	−0.132520
$349$	−3.05573	−0.163569	−0.0817847	−	0.996650i	$-0.526062\pi$
$349$	−3.05573	−0.163569	−0.0817847	+	0.996650i	$0.526062\pi$
$350$	5.47214	0.292498
$351$	35.7771	1.90964
$352$	0	0
$353$	0.472136	0.0251293	0.0125646	−	0.999921i	$-0.496000\pi$
$353$	0.472136	0.0251293	0.0125646	+	0.999921i	$0.496000\pi$
$354$	−8.36068	−0.444365
$355$	−20.9443	−1.11161
$356$	6.18034	0.327557
$357$	3.41641	0.180815
$358$	−16.9443	−0.895533
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	−4.76393	−0.251081
$361$	1.00000	0.0526316
$362$	−8.76393	−0.460622
$363$	−13.5967	−0.713644
$364$	6.47214	0.339232
$365$	−43.4164	−2.27252
$366$	2.11146	0.110368
$367$	24.3607	1.27162	0.635809	−	0.771847i	$-0.280667\pi$
$367$	24.3607	1.27162	0.635809	+	0.771847i	$0.280667\pi$
$368$	−1.00000	−0.0521286
$369$	14.7214	0.766363
$370$	19.4164	1.00941
$371$	6.00000	0.311504
$372$	4.00000	0.207390
$373$	−10.5836	−0.547998	−0.273999	−	0.961730i	$-0.588346\pi$
$373$	−10.5836	−0.547998	−0.273999	+	0.961730i	$0.588346\pi$
$374$	0	0
$375$	−1.88854	−0.0975240
$376$	11.2361	0.579456
$377$	12.9443	0.666664
$378$	−5.52786	−0.284323
$379$	−17.8885	−0.918873	−0.459436	−	0.888211i	$-0.651949\pi$
$379$	−17.8885	−0.918873	−0.459436	+	0.888211i	$0.651949\pi$
$380$	−14.4721	−0.742405
$381$	−24.0000	−1.22956
$382$	−3.05573	−0.156345
$383$	−0.583592	−0.0298202	−0.0149101	−	0.999889i	$-0.504746\pi$
$383$	−0.583592	−0.0298202	−0.0149101	+	0.999889i	$0.504746\pi$
$384$	−1.23607	−0.0630778
$385$	0	0
$386$	5.05573	0.257330
$387$	−3.63932	−0.184997
$388$	−14.1803	−0.719898
$389$	9.41641	0.477431	0.238715	−	0.971090i	$-0.423274\pi$
$389$	9.41641	0.477431	0.238715	+	0.971090i	$0.423274\pi$
$390$	−25.8885	−1.31092
$391$	2.76393	0.139778
$392$	−1.00000	−0.0505076
$393$	−21.3050	−1.07469
$394$	19.8885	1.00197
$395$	28.9443	1.45634
$396$	0	0
$397$	14.4721	0.726336	0.363168	−	0.931724i	$-0.381695\pi$
$397$	14.4721	0.726336	0.363168	+	0.931724i	$0.381695\pi$
$398$	12.0000	0.601506
$399$	−5.52786	−0.276739
$400$	5.47214	0.273607
$401$	−37.7771	−1.88650	−0.943249	−	0.332087i	$-0.892247\pi$
$401$	−37.7771	−1.88650	−0.943249	+	0.332087i	$0.892247\pi$
$402$	−4.94427	−0.246598
$403$	−20.9443	−1.04331
$404$	−5.52786	−0.275022
$405$	7.81966	0.388562
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	3.41641	0.169137
$409$	28.4721	1.40786	0.703928	−	0.710271i	$-0.251428\pi$
$409$	28.4721	1.40786	0.703928	+	0.710271i	$0.251428\pi$
$410$	−32.3607	−1.59818
$411$	−11.6393	−0.574125
$412$	−4.00000	−0.197066
$413$	−6.76393	−0.332831
$414$	−1.47214	−0.0723515
$415$	−35.4164	−1.73852
$416$	6.47214	0.317323
$417$	−28.5836	−1.39974
$418$	0	0
$419$	34.3607	1.67863	0.839315	−	0.543646i	$-0.182957\pi$
$419$	34.3607	1.67863	0.839315	+	0.543646i	$0.182957\pi$
$420$	4.00000	0.195180
$421$	35.8885	1.74910	0.874550	−	0.484935i	$-0.161157\pi$
$421$	35.8885	1.74910	0.874550	+	0.484935i	$0.161157\pi$
$422$	−16.9443	−0.824834
$423$	16.5410	0.804252
$424$	6.00000	0.291386
$425$	−15.1246	−0.733651
$426$	−8.00000	−0.387601
$427$	1.70820	0.0826658
$428$	−1.52786	−0.0738521
$429$	0	0
$430$	8.00000	0.385794
$431$	20.9443	1.00885	0.504425	−	0.863455i	$-0.331704\pi$
$431$	20.9443	1.00885	0.504425	+	0.863455i	$0.331704\pi$
$432$	−5.52786	−0.265959
$433$	6.76393	0.325054	0.162527	−	0.986704i	$-0.448036\pi$
$433$	6.76393	0.325054	0.162527	+	0.986704i	$0.448036\pi$
$434$	3.23607	0.155336
$435$	8.00000	0.383571
$436$	12.4721	0.597307
$437$	−4.47214	−0.213931
$438$	−16.5836	−0.792395
$439$	−8.76393	−0.418280	−0.209140	−	0.977886i	$-0.567066\pi$
$439$	−8.76393	−0.418280	−0.209140	+	0.977886i	$0.567066\pi$
$440$	0	0
$441$	−1.47214	−0.0701017
$442$	−17.8885	−0.850871
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	7.41641	0.351967
$445$	−20.0000	−0.948091
$446$	10.6525	0.504409
$447$	26.4721	1.25209
$448$	−1.00000	−0.0472456
$449$	9.41641	0.444388	0.222194	−	0.975003i	$-0.428678\pi$
$449$	9.41641	0.444388	0.222194	+	0.975003i	$0.428678\pi$
$450$	8.05573	0.379751
$451$	0	0
$452$	12.4721	0.586640
$453$	−11.7771	−0.553336
$454$	21.4164	1.00512
$455$	−20.9443	−0.981883
$456$	−5.52786	−0.258866
$457$	22.3607	1.04599	0.522994	−	0.852336i	$-0.324815\pi$
$457$	22.3607	1.04599	0.522994	+	0.852336i	$0.324815\pi$
$458$	15.2361	0.711935
$459$	15.2786	0.713146
$460$	3.23607	0.150882
$461$	3.05573	0.142319	0.0711597	−	0.997465i	$-0.477330\pi$
$461$	3.05573	0.142319	0.0711597	+	0.997465i	$0.477330\pi$
$462$	0	0
$463$	−11.0557	−0.513803	−0.256902	−	0.966438i	$-0.582702\pi$
$463$	−11.0557	−0.513803	−0.256902	+	0.966438i	$0.582702\pi$
$464$	−2.00000	−0.0928477
$465$	−12.9443	−0.600276
$466$	16.4721	0.763057
$467$	−11.5279	−0.533446	−0.266723	−	0.963773i	$-0.585941\pi$
$467$	−11.5279	−0.533446	−0.266723	+	0.963773i	$0.585941\pi$
$468$	9.52786	0.440426
$469$	−4.00000	−0.184703
$470$	−36.3607	−1.67719
$471$	−23.0557	−1.06235
$472$	−6.76393	−0.311335
$473$	0	0
$474$	11.0557	0.507806
$475$	24.4721	1.12286
$476$	2.76393	0.126685
$477$	8.83282	0.404427
$478$	−6.47214	−0.296029
$479$	−34.8328	−1.59155	−0.795776	−	0.605591i	$-0.792937\pi$
$479$	−34.8328	−1.59155	−0.795776	+	0.605591i	$0.792937\pi$
$480$	4.00000	0.182574
$481$	−38.8328	−1.77062
$482$	−0.291796	−0.0132909
$483$	1.23607	0.0562430
$484$	−11.0000	−0.500000
$485$	45.8885	2.08369
$486$	−13.5967	−0.616761
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	1.70820	0.0773268
$489$	−8.00000	−0.361773
$490$	3.23607	0.146191
$491$	−7.05573	−0.318421	−0.159210	−	0.987245i	$-0.550895\pi$
$491$	−7.05573	−0.318421	−0.159210	+	0.987245i	$0.550895\pi$
$492$	−12.3607	−0.557262
$493$	5.52786	0.248962
$494$	28.9443	1.30226
$495$	0	0
$496$	3.23607	0.145304
$497$	−6.47214	−0.290315
$498$	−13.5279	−0.606198
$499$	−8.94427	−0.400401	−0.200200	−	0.979755i	$-0.564159\pi$
$499$	−8.94427	−0.400401	−0.200200	+	0.979755i	$0.564159\pi$
$500$	−1.52786	−0.0683282
$501$	−8.22291	−0.367373
$502$	4.47214	0.199601
$503$	−4.94427	−0.220454	−0.110227	−	0.993906i	$-0.535158\pi$
$503$	−4.94427	−0.220454	−0.110227	+	0.993906i	$0.535158\pi$
$504$	−1.47214	−0.0655741
$505$	17.8885	0.796030
$506$	0	0
$507$	35.7082	1.58586
$508$	−19.4164	−0.861464
$509$	33.8885	1.50208	0.751042	−	0.660255i	$-0.229552\pi$
$509$	33.8885	1.50208	0.751042	+	0.660255i	$0.229552\pi$
$510$	−11.0557	−0.489556
$511$	−13.4164	−0.593507
$512$	−1.00000	−0.0441942
$513$	−24.7214	−1.09147
$514$	26.9443	1.18846
$515$	12.9443	0.570393
$516$	3.05573	0.134521
$517$	0	0
$518$	6.00000	0.263625
$519$	3.77709	0.165796
$520$	−20.9443	−0.918467
$521$	11.7082	0.512946	0.256473	−	0.966551i	$-0.417440\pi$
$521$	11.7082	0.512946	0.256473	+	0.966551i	$0.417440\pi$
$522$	−2.94427	−0.128867
$523$	10.5836	0.462788	0.231394	−	0.972860i	$-0.425671\pi$
$523$	10.5836	0.462788	0.231394	+	0.972860i	$0.425671\pi$
$524$	−17.2361	−0.752961
$525$	−6.76393	−0.295202
$526$	−12.0000	−0.523225
$527$	−8.94427	−0.389619
$528$	0	0
$529$	1.00000	0.0434783
$530$	−19.4164	−0.843395
$531$	−9.95743	−0.432116
$532$	−4.47214	−0.193892
$533$	64.7214	2.80339
$534$	−7.63932	−0.330586
$535$	4.94427	0.213760
$536$	−4.00000	−0.172774
$537$	20.9443	0.903812
$538$	−3.05573	−0.131742
$539$	0	0
$540$	17.8885	0.769800
$541$	3.88854	0.167182	0.0835908	−	0.996500i	$-0.473361\pi$
$541$	3.88854	0.167182	0.0835908	+	0.996500i	$0.473361\pi$
$542$	15.2361	0.654445
$543$	10.8328	0.464881
$544$	2.76393	0.118503
$545$	−40.3607	−1.72886
$546$	−8.00000	−0.342368
$547$	−22.4721	−0.960839	−0.480420	−	0.877039i	$-0.659516\pi$
$547$	−22.4721	−0.960839	−0.480420	+	0.877039i	$0.659516\pi$
$548$	−9.41641	−0.402249
$549$	2.51471	0.107325
$550$	0	0
$551$	−8.94427	−0.381039
$552$	1.23607	0.0526105
$553$	8.94427	0.380349
$554$	−19.8885	−0.844983
$555$	−24.0000	−1.01874
$556$	−23.1246	−0.980702
$557$	−17.4164	−0.737957	−0.368978	−	0.929438i	$-0.620292\pi$
$557$	−17.4164	−0.737957	−0.368978	+	0.929438i	$0.620292\pi$
$558$	4.76393	0.201673
$559$	−16.0000	−0.676728
$560$	3.23607	0.136749
$561$	0	0
$562$	−5.41641	−0.228477
$563$	6.58359	0.277465	0.138733	−	0.990330i	$-0.455697\pi$
$563$	6.58359	0.277465	0.138733	+	0.990330i	$0.455697\pi$
$564$	−13.8885	−0.584813
$565$	−40.3607	−1.69799
$566$	−4.47214	−0.187978
$567$	2.41641	0.101480
$568$	−6.47214	−0.271565
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	17.8885	0.749269
$571$	−33.3050	−1.39377	−0.696884	−	0.717183i	$-0.745431\pi$
$571$	−33.3050	−1.39377	−0.696884	+	0.717183i	$0.745431\pi$
$572$	0	0
$573$	3.77709	0.157790
$574$	−10.0000	−0.417392
$575$	−5.47214	−0.228204
$576$	−1.47214	−0.0613390
$577$	−25.4164	−1.05810	−0.529049	−	0.848591i	$-0.677451\pi$
$577$	−25.4164	−1.05810	−0.529049	+	0.848591i	$0.677451\pi$
$578$	9.36068	0.389353
$579$	−6.24922	−0.259709
$580$	6.47214	0.268741
$581$	−10.9443	−0.454045
$582$	17.5279	0.726553
$583$	0	0
$584$	−13.4164	−0.555175
$585$	−30.8328	−1.27478
$586$	29.1246	1.20313
$587$	39.7082	1.63893	0.819466	−	0.573127i	$-0.194270\pi$
$587$	39.7082	1.63893	0.819466	+	0.573127i	$0.194270\pi$
$588$	1.23607	0.0509746
$589$	14.4721	0.596314
$590$	21.8885	0.901137
$591$	−24.5836	−1.01123
$592$	6.00000	0.246598
$593$	−38.0000	−1.56047	−0.780236	−	0.625485i	$-0.784901\pi$
$593$	−38.0000	−1.56047	−0.780236	+	0.625485i	$0.784901\pi$
$594$	0	0
$595$	−8.94427	−0.366679
$596$	21.4164	0.877250
$597$	−14.8328	−0.607067
$598$	−6.47214	−0.264665
$599$	38.8328	1.58667	0.793333	−	0.608788i	$-0.208344\pi$
$599$	38.8328	1.58667	0.793333	+	0.608788i	$0.208344\pi$
$600$	−6.76393	−0.276136
$601$	−4.47214	−0.182422	−0.0912111	−	0.995832i	$-0.529074\pi$
$601$	−4.47214	−0.182422	−0.0912111	+	0.995832i	$0.529074\pi$
$602$	2.47214	0.100757
$603$	−5.88854	−0.239800
$604$	−9.52786	−0.387683
$605$	35.5967	1.44721
$606$	6.83282	0.277564
$607$	0.763932	0.0310070	0.0155035	−	0.999880i	$-0.495065\pi$
$607$	0.763932	0.0310070	0.0155035	+	0.999880i	$0.495065\pi$
$608$	−4.47214	−0.181369
$609$	2.47214	0.100176
$610$	−5.52786	−0.223817
$611$	72.7214	2.94199
$612$	4.06888	0.164475
$613$	−36.2492	−1.46409	−0.732046	−	0.681255i	$-0.761434\pi$
$613$	−36.2492	−1.46409	−0.732046	+	0.681255i	$0.761434\pi$
$614$	−29.2361	−1.17987
$615$	40.0000	1.61296
$616$	0	0
$617$	9.05573	0.364570	0.182285	−	0.983246i	$-0.441651\pi$
$617$	9.05573	0.364570	0.182285	+	0.983246i	$0.441651\pi$
$618$	4.94427	0.198888
$619$	1.05573	0.0424333	0.0212166	−	0.999775i	$-0.493246\pi$
$619$	1.05573	0.0424333	0.0212166	+	0.999775i	$0.493246\pi$
$620$	−10.4721	−0.420571
$621$	5.52786	0.221826
$622$	−25.1246	−1.00741
$623$	−6.18034	−0.247610
$624$	−8.00000	−0.320256
$625$	−22.4164	−0.896656
$626$	−26.1803	−1.04638
$627$	0	0
$628$	−18.6525	−0.744315
$629$	−16.5836	−0.661231
$630$	4.76393	0.189800
$631$	−2.11146	−0.0840557	−0.0420279	−	0.999116i	$-0.513382\pi$
$631$	−2.11146	−0.0840557	−0.0420279	+	0.999116i	$0.513382\pi$
$632$	8.94427	0.355784
$633$	20.9443	0.832460
$634$	−10.0000	−0.397151
$635$	62.8328	2.49344
$636$	−7.41641	−0.294080
$637$	−6.47214	−0.256435
$638$	0	0
$639$	−9.52786	−0.376916
$640$	3.23607	0.127917
$641$	34.9443	1.38022	0.690108	−	0.723707i	$-0.257563\pi$
$641$	34.9443	1.38022	0.690108	+	0.723707i	$0.257563\pi$
$642$	1.88854	0.0745349
$643$	44.2492	1.74502	0.872510	−	0.488597i	$-0.162491\pi$
$643$	44.2492	1.74502	0.872510	+	0.488597i	$0.162491\pi$
$644$	1.00000	0.0394055
$645$	−9.88854	−0.389361
$646$	12.3607	0.486324
$647$	1.34752	0.0529766	0.0264883	−	0.999649i	$-0.491568\pi$
$647$	1.34752	0.0529766	0.0264883	+	0.999649i	$0.491568\pi$
$648$	2.41641	0.0949255
$649$	0	0
$650$	35.4164	1.38915
$651$	−4.00000	−0.156772
$652$	−6.47214	−0.253468
$653$	29.7771	1.16527	0.582634	−	0.812735i	$-0.302022\pi$
$653$	29.7771	1.16527	0.582634	+	0.812735i	$0.302022\pi$
$654$	−15.4164	−0.602829
$655$	55.7771	2.17939
$656$	−10.0000	−0.390434
$657$	−19.7508	−0.770551
$658$	−11.2361	−0.438028
$659$	−46.2492	−1.80161	−0.900807	−	0.434220i	$-0.857024\pi$
$659$	−46.2492	−1.80161	−0.900807	+	0.434220i	$0.857024\pi$
$660$	0	0
$661$	5.12461	0.199324	0.0996621	−	0.995021i	$-0.468224\pi$
$661$	5.12461	0.199324	0.0996621	+	0.995021i	$0.468224\pi$
$662$	30.4721	1.18433
$663$	22.1115	0.858738
$664$	−10.9443	−0.424720
$665$	14.4721	0.561205
$666$	8.83282	0.342265
$667$	2.00000	0.0774403
$668$	−6.65248	−0.257392
$669$	−13.1672	−0.509073
$670$	12.9443	0.500081
$671$	0	0
$672$	1.23607	0.0476824
$673$	−29.4164	−1.13392	−0.566960	−	0.823746i	$-0.691880\pi$
$673$	−29.4164	−1.13392	−0.566960	+	0.823746i	$0.691880\pi$
$674$	35.8885	1.38238
$675$	−30.2492	−1.16429
$676$	28.8885	1.11110
$677$	23.2361	0.893035	0.446517	−	0.894775i	$-0.352664\pi$
$677$	23.2361	0.893035	0.446517	+	0.894775i	$0.352664\pi$
$678$	−15.4164	−0.592064
$679$	14.1803	0.544191
$680$	−8.94427	−0.342997
$681$	−26.4721	−1.01441
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	−6.58359	−0.251730
$685$	30.4721	1.16428
$686$	1.00000	0.0381802
$687$	−18.8328	−0.718517
$688$	2.47214	0.0942493
$689$	38.8328	1.47941
$690$	−4.00000	−0.152277
$691$	49.0132	1.86455	0.932274	−	0.361753i	$-0.117821\pi$
$691$	49.0132	1.86455	0.932274	+	0.361753i	$0.117821\pi$
$692$	3.05573	0.116161
$693$	0	0
$694$	−24.3607	−0.924719
$695$	74.8328	2.83857
$696$	2.47214	0.0937061
$697$	27.6393	1.04691
$698$	3.05573	0.115661
$699$	−20.3607	−0.770112
$700$	−5.47214	−0.206827
$701$	−21.0557	−0.795264	−0.397632	−	0.917545i	$-0.630168\pi$
$701$	−21.0557	−0.795264	−0.397632	+	0.917545i	$0.630168\pi$
$702$	−35.7771	−1.35032
$703$	26.8328	1.01202
$704$	0	0
$705$	44.9443	1.69270
$706$	−0.472136	−0.0177691
$707$	5.52786	0.207897
$708$	8.36068	0.314214
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	20.9443	0.786025
$711$	13.1672	0.493808
$712$	−6.18034	−0.231618
$713$	−3.23607	−0.121192
$714$	−3.41641	−0.127856
$715$	0	0
$716$	16.9443	0.633237
$717$	8.00000	0.298765
$718$	−12.0000	−0.447836
$719$	−32.5410	−1.21358	−0.606788	−	0.794864i	$-0.707542\pi$
$719$	−32.5410	−1.21358	−0.606788	+	0.794864i	$0.707542\pi$
$720$	4.76393	0.177541
$721$	4.00000	0.148968
$722$	−1.00000	−0.0372161
$723$	0.360680	0.0134138
$724$	8.76393	0.325709
$725$	−10.9443	−0.406460
$726$	13.5967	0.504623
$727$	7.05573	0.261682	0.130841	−	0.991403i	$-0.458232\pi$
$727$	7.05573	0.261682	0.130841	+	0.991403i	$0.458232\pi$
$728$	−6.47214	−0.239873
$729$	24.0557	0.890953
$730$	43.4164	1.60691
$731$	−6.83282	−0.252721
$732$	−2.11146	−0.0780417
$733$	3.59675	0.132849	0.0664245	−	0.997791i	$-0.478841\pi$
$733$	3.59675	0.132849	0.0664245	+	0.997791i	$0.478841\pi$
$734$	−24.3607	−0.899169
$735$	−4.00000	−0.147542
$736$	1.00000	0.0368605
$737$	0	0
$738$	−14.7214	−0.541901
$739$	24.3607	0.896122	0.448061	−	0.894003i	$-0.352115\pi$
$739$	24.3607	0.896122	0.448061	+	0.894003i	$0.352115\pi$
$740$	−19.4164	−0.713761
$741$	−35.7771	−1.31430
$742$	−6.00000	−0.220267
$743$	−48.9443	−1.79559	−0.897796	−	0.440412i	$-0.854832\pi$
$743$	−48.9443	−1.79559	−0.897796	+	0.440412i	$0.854832\pi$
$744$	−4.00000	−0.146647
$745$	−69.3050	−2.53914
$746$	10.5836	0.387493
$747$	−16.1115	−0.589487
$748$	0	0
$749$	1.52786	0.0558269
$750$	1.88854	0.0689599
$751$	−40.9443	−1.49408	−0.747039	−	0.664780i	$-0.768525\pi$
$751$	−40.9443	−1.49408	−0.747039	+	0.664780i	$0.768525\pi$
$752$	−11.2361	−0.409737
$753$	−5.52786	−0.201447
$754$	−12.9443	−0.471403
$755$	30.8328	1.12212
$756$	5.52786	0.201046
$757$	23.5279	0.855135	0.427567	−	0.903983i	$-0.359371\pi$
$757$	23.5279	0.855135	0.427567	+	0.903983i	$0.359371\pi$
$758$	17.8885	0.649741
$759$	0	0
$760$	14.4721	0.524960
$761$	−0.472136	−0.0171149	−0.00855746	−	0.999963i	$-0.502724\pi$
$761$	−0.472136	−0.0171149	−0.00855746	+	0.999963i	$0.502724\pi$
$762$	24.0000	0.869428
$763$	−12.4721	−0.451522
$764$	3.05573	0.110552
$765$	−13.1672	−0.476061
$766$	0.583592	0.0210860
$767$	−43.7771	−1.58070
$768$	1.23607	0.0446028
$769$	21.2361	0.765792	0.382896	−	0.923791i	$-0.374927\pi$
$769$	21.2361	0.765792	0.382896	+	0.923791i	$0.374927\pi$
$770$	0	0
$771$	−33.3050	−1.19945
$772$	−5.05573	−0.181960
$773$	16.1803	0.581966	0.290983	−	0.956728i	$-0.406018\pi$
$773$	16.1803	0.581966	0.290983	+	0.956728i	$0.406018\pi$
$774$	3.63932	0.130813
$775$	17.7082	0.636097
$776$	14.1803	0.509045
$777$	−7.41641	−0.266062
$778$	−9.41641	−0.337595
$779$	−44.7214	−1.60231
$780$	25.8885	0.926959
$781$	0	0
$782$	−2.76393	−0.0988380
$783$	11.0557	0.395099
$784$	1.00000	0.0357143
$785$	60.3607	2.15437
$786$	21.3050	0.759922
$787$	−35.8885	−1.27929	−0.639644	−	0.768671i	$-0.720918\pi$
$787$	−35.8885	−1.27929	−0.639644	+	0.768671i	$0.720918\pi$
$788$	−19.8885	−0.708500
$789$	14.8328	0.528062
$790$	−28.9443	−1.02979
$791$	−12.4721	−0.443458
$792$	0	0
$793$	11.0557	0.392600
$794$	−14.4721	−0.513597
$795$	24.0000	0.851192
$796$	−12.0000	−0.425329
$797$	−13.3475	−0.472794	−0.236397	−	0.971657i	$-0.575967\pi$
$797$	−13.3475	−0.472794	−0.236397	+	0.971657i	$0.575967\pi$
$798$	5.52786	0.195684
$799$	31.0557	1.09867
$800$	−5.47214	−0.193469
$801$	−9.09830	−0.321473
$802$	37.7771	1.33396
$803$	0	0
$804$	4.94427	0.174371
$805$	−3.23607	−0.114056
$806$	20.9443	0.737731
$807$	3.77709	0.132960
$808$	5.52786	0.194470
$809$	−32.8328	−1.15434	−0.577170	−	0.816624i	$-0.695843\pi$
$809$	−32.8328	−1.15434	−0.577170	+	0.816624i	$0.695843\pi$
$810$	−7.81966	−0.274755
$811$	21.5967	0.758364	0.379182	−	0.925322i	$-0.376205\pi$
$811$	21.5967	0.758364	0.379182	+	0.925322i	$0.376205\pi$
$812$	2.00000	0.0701862
$813$	−18.8328	−0.660496
$814$	0	0
$815$	20.9443	0.733646
$816$	−3.41641	−0.119598
$817$	11.0557	0.386791
$818$	−28.4721	−0.995505
$819$	−9.52786	−0.332931
$820$	32.3607	1.13008
$821$	−2.94427	−0.102756	−0.0513779	−	0.998679i	$-0.516361\pi$
$821$	−2.94427	−0.102756	−0.0513779	+	0.998679i	$0.516361\pi$
$822$	11.6393	0.405968
$823$	−11.4164	−0.397951	−0.198975	−	0.980004i	$-0.563761\pi$
$823$	−11.4164	−0.397951	−0.198975	+	0.980004i	$0.563761\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	6.76393	0.235347
$827$	−12.5836	−0.437574	−0.218787	−	0.975773i	$-0.570210\pi$
$827$	−12.5836	−0.437574	−0.218787	+	0.975773i	$0.570210\pi$
$828$	1.47214	0.0511603
$829$	13.8885	0.482369	0.241185	−	0.970479i	$-0.422464\pi$
$829$	13.8885	0.482369	0.241185	+	0.970479i	$0.422464\pi$
$830$	35.4164	1.22932
$831$	24.5836	0.852795
$832$	−6.47214	−0.224381
$833$	−2.76393	−0.0957646
$834$	28.5836	0.989769
$835$	21.5279	0.745002
$836$	0	0
$837$	−17.8885	−0.618319
$838$	−34.3607	−1.18697
$839$	28.3607	0.979119	0.489560	−	0.871970i	$-0.337158\pi$
$839$	28.3607	0.979119	0.489560	+	0.871970i	$0.337158\pi$
$840$	−4.00000	−0.138013
$841$	−25.0000	−0.862069
$842$	−35.8885	−1.23680
$843$	6.69505	0.230590
$844$	16.9443	0.583246
$845$	−93.4853	−3.21599
$846$	−16.5410	−0.568692
$847$	11.0000	0.377964
$848$	−6.00000	−0.206041
$849$	5.52786	0.189716
$850$	15.1246	0.518770
$851$	−6.00000	−0.205677
$852$	8.00000	0.274075
$853$	11.4164	0.390890	0.195445	−	0.980715i	$-0.437385\pi$
$853$	11.4164	0.390890	0.195445	+	0.980715i	$0.437385\pi$
$854$	−1.70820	−0.0584535
$855$	21.3050	0.728614
$856$	1.52786	0.0522213
$857$	−11.8885	−0.406105	−0.203052	−	0.979168i	$-0.565086\pi$
$857$	−11.8885	−0.406105	−0.203052	+	0.979168i	$0.565086\pi$
$858$	0	0
$859$	2.76393	0.0943041	0.0471521	−	0.998888i	$-0.484985\pi$
$859$	2.76393	0.0943041	0.0471521	+	0.998888i	$0.484985\pi$
$860$	−8.00000	−0.272798
$861$	12.3607	0.421251
$862$	−20.9443	−0.713365
$863$	−3.41641	−0.116296	−0.0581479	−	0.998308i	$-0.518520\pi$
$863$	−3.41641	−0.116296	−0.0581479	+	0.998308i	$0.518520\pi$
$864$	5.52786	0.188062
$865$	−9.88854	−0.336221
$866$	−6.76393	−0.229848
$867$	−11.5704	−0.392953
$868$	−3.23607	−0.109839
$869$	0	0
$870$	−8.00000	−0.271225
$871$	−25.8885	−0.877200
$872$	−12.4721	−0.422360
$873$	20.8754	0.706525
$874$	4.47214	0.151272
$875$	1.52786	0.0516512
$876$	16.5836	0.560308
$877$	26.9443	0.909843	0.454922	−	0.890531i	$-0.349667\pi$
$877$	26.9443	0.909843	0.454922	+	0.890531i	$0.349667\pi$
$878$	8.76393	0.295768
$879$	−36.0000	−1.21425
$880$	0	0
$881$	−21.2361	−0.715461	−0.357731	−	0.933825i	$-0.616449\pi$
$881$	−21.2361	−0.715461	−0.357731	+	0.933825i	$0.616449\pi$
$882$	1.47214	0.0495694
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	17.8885	0.601657
$885$	−27.0557	−0.909468
$886$	−4.00000	−0.134383
$887$	−35.5967	−1.19522	−0.597611	−	0.801786i	$-0.703883\pi$
$887$	−35.5967	−1.19522	−0.597611	+	0.801786i	$0.703883\pi$
$888$	−7.41641	−0.248878
$889$	19.4164	0.651205
$890$	20.0000	0.670402
$891$	0	0
$892$	−10.6525	−0.356671
$893$	−50.2492	−1.68153
$894$	−26.4721	−0.885361
$895$	−54.8328	−1.83286
$896$	1.00000	0.0334077
$897$	8.00000	0.267112
$898$	−9.41641	−0.314230
$899$	−6.47214	−0.215858
$900$	−8.05573	−0.268524
$901$	16.5836	0.552480
$902$	0	0
$903$	−3.05573	−0.101688
$904$	−12.4721	−0.414817
$905$	−28.3607	−0.942741
$906$	11.7771	0.391268
$907$	−1.52786	−0.0507319	−0.0253659	−	0.999678i	$-0.508075\pi$
$907$	−1.52786	−0.0507319	−0.0253659	+	0.999678i	$0.508075\pi$
$908$	−21.4164	−0.710728
$909$	8.13777	0.269913
$910$	20.9443	0.694296
$911$	21.8885	0.725200	0.362600	−	0.931945i	$-0.381889\pi$
$911$	21.8885	0.725200	0.362600	+	0.931945i	$0.381889\pi$
$912$	5.52786	0.183046
$913$	0	0
$914$	−22.3607	−0.739626
$915$	6.83282	0.225886
$916$	−15.2361	−0.503414
$917$	17.2361	0.569185
$918$	−15.2786	−0.504270
$919$	−10.1115	−0.333546	−0.166773	−	0.985995i	$-0.553335\pi$
$919$	−10.1115	−0.333546	−0.166773	+	0.985995i	$0.553335\pi$
$920$	−3.23607	−0.106690
$921$	36.1378	1.19078
$922$	−3.05573	−0.100635
$923$	−41.8885	−1.37878
$924$	0	0
$925$	32.8328	1.07954
$926$	11.0557	0.363314
$927$	5.88854	0.193405
$928$	2.00000	0.0656532
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	12.9443	0.424459
$931$	4.47214	0.146568
$932$	−16.4721	−0.539563
$933$	31.0557	1.01672
$934$	11.5279	0.377203
$935$	0	0
$936$	−9.52786	−0.311428
$937$	50.7639	1.65839	0.829193	−	0.558963i	$-0.188801\pi$
$937$	50.7639	1.65839	0.829193	+	0.558963i	$0.188801\pi$
$938$	4.00000	0.130605
$939$	32.3607	1.05605
$940$	36.3607	1.18595
$941$	19.5967	0.638836	0.319418	−	0.947614i	$-0.396513\pi$
$941$	19.5967	0.638836	0.319418	+	0.947614i	$0.396513\pi$
$942$	23.0557	0.751196
$943$	10.0000	0.325645
$944$	6.76393	0.220147
$945$	−17.8885	−0.581914
$946$	0	0
$947$	42.2492	1.37292	0.686458	−	0.727170i	$-0.259165\pi$
$947$	42.2492	1.37292	0.686458	+	0.727170i	$0.259165\pi$
$948$	−11.0557	−0.359073
$949$	−86.8328	−2.81871
$950$	−24.4721	−0.793981
$951$	12.3607	0.400823
$952$	−2.76393	−0.0895796
$953$	45.7771	1.48287	0.741433	−	0.671027i	$-0.234147\pi$
$953$	45.7771	1.48287	0.741433	+	0.671027i	$0.234147\pi$
$954$	−8.83282	−0.285973
$955$	−9.88854	−0.319986
$956$	6.47214	0.209324
$957$	0	0
$958$	34.8328	1.12540
$959$	9.41641	0.304072
$960$	−4.00000	−0.129099
$961$	−20.5279	−0.662189
$962$	38.8328	1.25202
$963$	2.24922	0.0724802
$964$	0.291796	0.00939812
$965$	16.3607	0.526669
$966$	−1.23607	−0.0397698
$967$	−9.88854	−0.317994	−0.158997	−	0.987279i	$-0.550826\pi$
$967$	−9.88854	−0.317994	−0.158997	+	0.987279i	$0.550826\pi$
$968$	11.0000	0.353553
$969$	−15.2786	−0.490821
$970$	−45.8885	−1.47339
$971$	−35.8885	−1.15172	−0.575859	−	0.817549i	$-0.695332\pi$
$971$	−35.8885	−1.15172	−0.575859	+	0.817549i	$0.695332\pi$
$972$	13.5967	0.436116
$973$	23.1246	0.741341
$974$	0	0
$975$	−43.7771	−1.40199
$976$	−1.70820	−0.0546783
$977$	−12.4721	−0.399019	−0.199509	−	0.979896i	$-0.563935\pi$
$977$	−12.4721	−0.399019	−0.199509	+	0.979896i	$0.563935\pi$
$978$	8.00000	0.255812
$979$	0	0
$980$	−3.23607	−0.103372
$981$	−18.3607	−0.586211
$982$	7.05573	0.225157
$983$	−13.5279	−0.431472	−0.215736	−	0.976452i	$-0.569215\pi$
$983$	−13.5279	−0.431472	−0.215736	+	0.976452i	$0.569215\pi$
$984$	12.3607	0.394044
$985$	64.3607	2.05070
$986$	−5.52786	−0.176043
$987$	13.8885	0.442077
$988$	−28.9443	−0.920840
$989$	−2.47214	−0.0786094
$990$	0	0
$991$	21.3050	0.676774	0.338387	−	0.941007i	$-0.390119\pi$
$991$	21.3050	0.676774	0.338387	+	0.941007i	$0.390119\pi$
$992$	−3.23607	−0.102745
$993$	−37.6656	−1.19528
$994$	6.47214	0.205284
$995$	38.8328	1.23108
$996$	13.5279	0.428647
$997$	10.8328	0.343079	0.171539	−	0.985177i	$-0.445126\pi$
$997$	10.8328	0.343079	0.171539	+	0.985177i	$0.445126\pi$
$998$	8.94427	0.283126
$999$	−33.1672	−1.04936

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.a.e.1.2	✓	2
3.2	odd	2		2898.2.a.bd.1.2		2
4.3	odd	2		2576.2.a.t.1.1		2
5.4	even	2		8050.2.a.bf.1.1		2
7.6	odd	2		2254.2.a.k.1.1		2
23.22	odd	2		7406.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.a.e.1.2	✓	2	1.1	even	1	trivial
2254.2.a.k.1.1		2	7.6	odd	2
2576.2.a.t.1.1		2	4.3	odd	2
2898.2.a.bd.1.2		2	3.2	odd	2
7406.2.a.j.1.2		2	23.22	odd	2
8050.2.a.bf.1.1		2	5.4	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:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	322.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.23607 q^{3} +1.00000 q^{4} -3.23607 q^{5} -1.23607 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.47214 q^{9} +3.23607 q^{10} +1.23607 q^{12} -6.47214 q^{13} +1.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -2.76393 q^{17} +1.47214 q^{18} +4.47214 q^{19} -3.23607 q^{20} -1.23607 q^{21} -1.00000 q^{23} -1.23607 q^{24} +5.47214 q^{25} +6.47214 q^{26} -5.52786 q^{27} -1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{30} +3.23607 q^{31} -1.00000 q^{32} +2.76393 q^{34} +3.23607 q^{35} -1.47214 q^{36} +6.00000 q^{37} -4.47214 q^{38} -8.00000 q^{39} +3.23607 q^{40} -10.0000 q^{41} +1.23607 q^{42} +2.47214 q^{43} +4.76393 q^{45} +1.00000 q^{46} -11.2361 q^{47} +1.23607 q^{48} +1.00000 q^{49} -5.47214 q^{50} -3.41641 q^{51} -6.47214 q^{52} -6.00000 q^{53} +5.52786 q^{54} +1.00000 q^{56} +5.52786 q^{57} +2.00000 q^{58} +6.76393 q^{59} -4.00000 q^{60} -1.70820 q^{61} -3.23607 q^{62} +1.47214 q^{63} +1.00000 q^{64} +20.9443 q^{65} +4.00000 q^{67} -2.76393 q^{68} -1.23607 q^{69} -3.23607 q^{70} +6.47214 q^{71} +1.47214 q^{72} +13.4164 q^{73} -6.00000 q^{74} +6.76393 q^{75} +4.47214 q^{76} +8.00000 q^{78} -8.94427 q^{79} -3.23607 q^{80} -2.41641 q^{81} +10.0000 q^{82} +10.9443 q^{83} -1.23607 q^{84} +8.94427 q^{85} -2.47214 q^{86} -2.47214 q^{87} +6.18034 q^{89} -4.76393 q^{90} +6.47214 q^{91} -1.00000 q^{92} +4.00000 q^{93} +11.2361 q^{94} -14.4721 q^{95} -1.23607 q^{96} -14.1803 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{10} - 2 q^{12} - 4 q^{13} + 2 q^{14} - 8 q^{15} + 2 q^{16} - 10 q^{17} - 6 q^{18} - 2 q^{20} + 2 q^{21} - 2 q^{23}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 6 * q^9 + 2 * q^10 - 2 * q^12 - 4 * q^13 + 2 * q^14 - 8 * q^15 + 2 * q^16 - 10 * q^17 - 6 * q^18 - 2 * q^20 + 2 * q^21 - 2 * q^23 + 2 * q^24 + 2 * q^25 + 4 * q^26 - 20 * q^27 - 2 * q^28 - 4 * q^29 + 8 * q^30 + 2 * q^31 - 2 * q^32 + 10 * q^34 + 2 * q^35 + 6 * q^36 + 12 * q^37 - 16 * q^39 + 2 * q^40 - 20 * q^41 - 2 * q^42 - 4 * q^43 + 14 * q^45 + 2 * q^46 - 18 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^50 + 20 * q^51 - 4 * q^52 - 12 * q^53 + 20 * q^54 + 2 * q^56 + 20 * q^57 + 4 * q^58 + 18 * q^59 - 8 * q^60 + 10 * q^61 - 2 * q^62 - 6 * q^63 + 2 * q^64 + 24 * q^65 + 8 * q^67 - 10 * q^68 + 2 * q^69 - 2 * q^70 + 4 * q^71 - 6 * q^72 - 12 * q^74 + 18 * q^75 + 16 * q^78 - 2 * q^80 + 22 * q^81 + 20 * q^82 + 4 * q^83 + 2 * q^84 + 4 * q^86 + 4 * q^87 - 10 * q^89 - 14 * q^90 + 4 * q^91 - 2 * q^92 + 8 * q^93 + 18 * q^94 - 20 * q^95 + 2 * q^96 - 6 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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