LMFDB - Embedded newform 2667.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2667,2,Mod(1,2667)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2667, 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("2667.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2667 = 3 \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2667.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:	$21.2961022191$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	2667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{2} +1.00000 q^{3} +4.00000 q^{4} +2.44949 q^{5} +2.44949 q^{6} +1.00000 q^{7} +4.89898 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.44949 q^{2} +1.00000 q^{3} +4.00000 q^{4} +2.44949 q^{5} +2.44949 q^{6} +1.00000 q^{7} +4.89898 q^{8} +1.00000 q^{9} +6.00000 q^{10} +4.00000 q^{12} -2.89898 q^{13} +2.44949 q^{14} +2.44949 q^{15} +4.00000 q^{16} -5.44949 q^{17} +2.44949 q^{18} +4.44949 q^{19} +9.79796 q^{20} +1.00000 q^{21} +6.00000 q^{23} +4.89898 q^{24} +1.00000 q^{25} -7.10102 q^{26} +1.00000 q^{27} +4.00000 q^{28} -7.89898 q^{29} +6.00000 q^{30} +8.00000 q^{31} -13.3485 q^{34} +2.44949 q^{35} +4.00000 q^{36} -7.00000 q^{37} +10.8990 q^{38} -2.89898 q^{39} +12.0000 q^{40} +5.44949 q^{41} +2.44949 q^{42} +2.00000 q^{43} +2.44949 q^{45} +14.6969 q^{46} -12.0000 q^{47} +4.00000 q^{48} +1.00000 q^{49} +2.44949 q^{50} -5.44949 q^{51} -11.5959 q^{52} +1.89898 q^{53} +2.44949 q^{54} +4.89898 q^{56} +4.44949 q^{57} -19.3485 q^{58} -2.44949 q^{59} +9.79796 q^{60} +4.44949 q^{61} +19.5959 q^{62} +1.00000 q^{63} -8.00000 q^{64} -7.10102 q^{65} -10.2474 q^{67} -21.7980 q^{68} +6.00000 q^{69} +6.00000 q^{70} +3.55051 q^{71} +4.89898 q^{72} +3.34847 q^{73} -17.1464 q^{74} +1.00000 q^{75} +17.7980 q^{76} -7.10102 q^{78} +3.89898 q^{79} +9.79796 q^{80} +1.00000 q^{81} +13.3485 q^{82} +6.00000 q^{83} +4.00000 q^{84} -13.3485 q^{85} +4.89898 q^{86} -7.89898 q^{87} -9.79796 q^{89} +6.00000 q^{90} -2.89898 q^{91} +24.0000 q^{92} +8.00000 q^{93} -29.3939 q^{94} +10.8990 q^{95} -2.34847 q^{97} +2.44949 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 8 q^{4} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 8 * q^4 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 8 q^{4} + 2 q^{7} + 2 q^{9} + 12 q^{10} + 8 q^{12} + 4 q^{13} + 8 q^{16} - 6 q^{17} + 4 q^{19} + 2 q^{21} + 12 q^{23} + 2 q^{25} - 24 q^{26} + 2 q^{27} + 8 q^{28} - 6 q^{29} + 12 q^{30} + 16 q^{31} - 12 q^{34} + 8 q^{36} - 14 q^{37} + 12 q^{38} + 4 q^{39} + 24 q^{40} + 6 q^{41} + 4 q^{43} - 24 q^{47} + 8 q^{48} + 2 q^{49} - 6 q^{51} + 16 q^{52} - 6 q^{53} + 4 q^{57} - 24 q^{58} + 4 q^{61} + 2 q^{63} - 16 q^{64} - 24 q^{65} + 4 q^{67} - 24 q^{68} + 12 q^{69} + 12 q^{70} + 12 q^{71} - 8 q^{73} + 2 q^{75} + 16 q^{76} - 24 q^{78} - 2 q^{79} + 2 q^{81} + 12 q^{82} + 12 q^{83} + 8 q^{84} - 12 q^{85} - 6 q^{87} + 12 q^{90} + 4 q^{91} + 48 q^{92} + 16 q^{93} + 12 q^{95} + 10 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 8 * q^4 + 2 * q^7 + 2 * q^9 + 12 * q^10 + 8 * q^12 + 4 * q^13 + 8 * q^16 - 6 * q^17 + 4 * q^19 + 2 * q^21 + 12 * q^23 + 2 * q^25 - 24 * q^26 + 2 * q^27 + 8 * q^28 - 6 * q^29 + 12 * q^30 + 16 * q^31 - 12 * q^34 + 8 * q^36 - 14 * q^37 + 12 * q^38 + 4 * q^39 + 24 * q^40 + 6 * q^41 + 4 * q^43 - 24 * q^47 + 8 * q^48 + 2 * q^49 - 6 * q^51 + 16 * q^52 - 6 * q^53 + 4 * q^57 - 24 * q^58 + 4 * q^61 + 2 * q^63 - 16 * q^64 - 24 * q^65 + 4 * q^67 - 24 * q^68 + 12 * q^69 + 12 * q^70 + 12 * q^71 - 8 * q^73 + 2 * q^75 + 16 * q^76 - 24 * q^78 - 2 * q^79 + 2 * q^81 + 12 * q^82 + 12 * q^83 + 8 * q^84 - 12 * q^85 - 6 * q^87 + 12 * q^90 + 4 * q^91 + 48 * q^92 + 16 * q^93 + 12 * q^95 + 10 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.44949	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$2$	2.44949	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.00000	2.00000
$5$	2.44949	1.09545	0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949	1.09545	0.547723	+	0.836660i	$0.315495\pi$
$6$	2.44949	1.00000
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	4.89898	1.73205
$9$	1.00000	0.333333
$10$	6.00000	1.89737
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	4.00000	1.15470
$13$	−2.89898	−0.804032	−0.402016	−	0.915633i	$-0.631690\pi$
$13$	−2.89898	−0.804032	−0.402016	+	0.915633i	$0.631690\pi$
$14$	2.44949	0.654654
$15$	2.44949	0.632456
$16$	4.00000	1.00000
$17$	−5.44949	−1.32170	−0.660848	−	0.750520i	$-0.729803\pi$
$17$	−5.44949	−1.32170	−0.660848	+	0.750520i	$0.729803\pi$
$18$	2.44949	0.577350
$19$	4.44949	1.02078	0.510391	−	0.859942i	$-0.329501\pi$
$19$	4.44949	1.02078	0.510391	+	0.859942i	$0.329501\pi$
$20$	9.79796	2.19089
$21$	1.00000	0.218218
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	4.89898	1.00000
$25$	1.00000	0.200000
$26$	−7.10102	−1.39262
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	−7.89898	−1.46680	−0.733402	−	0.679795i	$-0.762069\pi$
$29$	−7.89898	−1.46680	−0.733402	+	0.679795i	$0.762069\pi$
$30$	6.00000	1.09545
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	0	0
$34$	−13.3485	−2.28924
$35$	2.44949	0.414039
$36$	4.00000	0.666667
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	10.8990	1.76805
$39$	−2.89898	−0.464208
$40$	12.0000	1.89737
$41$	5.44949	0.851067	0.425534	−	0.904943i	$-0.360086\pi$
$41$	5.44949	0.851067	0.425534	+	0.904943i	$0.360086\pi$
$42$	2.44949	0.377964
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	2.44949	0.365148
$46$	14.6969	2.16695
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	4.00000	0.577350
$49$	1.00000	0.142857
$50$	2.44949	0.346410
$51$	−5.44949	−0.763081
$52$	−11.5959	−1.60806
$53$	1.89898	0.260845	0.130422	−	0.991459i	$-0.458367\pi$
$53$	1.89898	0.260845	0.130422	+	0.991459i	$0.458367\pi$
$54$	2.44949	0.333333
$55$	0	0
$56$	4.89898	0.654654
$57$	4.44949	0.589349
$58$	−19.3485	−2.54058
$59$	−2.44949	−0.318896	−0.159448	−	0.987206i	$-0.550971\pi$
$59$	−2.44949	−0.318896	−0.159448	+	0.987206i	$0.550971\pi$
$60$	9.79796	1.26491
$61$	4.44949	0.569699	0.284849	−	0.958572i	$-0.408056\pi$
$61$	4.44949	0.569699	0.284849	+	0.958572i	$0.408056\pi$
$62$	19.5959	2.48868
$63$	1.00000	0.125988
$64$	−8.00000	−1.00000
$65$	−7.10102	−0.880773
$66$	0	0
$67$	−10.2474	−1.25193	−0.625963	−	0.779853i	$-0.715294\pi$
$67$	−10.2474	−1.25193	−0.625963	+	0.779853i	$0.715294\pi$
$68$	−21.7980	−2.64339
$69$	6.00000	0.722315
$70$	6.00000	0.717137
$71$	3.55051	0.421368	0.210684	−	0.977554i	$-0.432431\pi$
$71$	3.55051	0.421368	0.210684	+	0.977554i	$0.432431\pi$
$72$	4.89898	0.577350
$73$	3.34847	0.391909	0.195954	−	0.980613i	$-0.437220\pi$
$73$	3.34847	0.391909	0.195954	+	0.980613i	$0.437220\pi$
$74$	−17.1464	−1.99323
$75$	1.00000	0.115470
$76$	17.7980	2.04157
$77$	0	0
$78$	−7.10102	−0.804032
$79$	3.89898	0.438669	0.219335	−	0.975650i	$-0.429611\pi$
$79$	3.89898	0.438669	0.219335	+	0.975650i	$0.429611\pi$
$80$	9.79796	1.09545
$81$	1.00000	0.111111
$82$	13.3485	1.47409
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	4.00000	0.436436
$85$	−13.3485	−1.44784
$86$	4.89898	0.528271
$87$	−7.89898	−0.846859
$88$	0	0
$89$	−9.79796	−1.03858	−0.519291	−	0.854598i	$-0.673804\pi$
$89$	−9.79796	−1.03858	−0.519291	+	0.854598i	$0.673804\pi$
$90$	6.00000	0.632456
$91$	−2.89898	−0.303896
$92$	24.0000	2.50217
$93$	8.00000	0.829561
$94$	−29.3939	−3.03175
$95$	10.8990	1.11821
$96$	0	0
$97$	−2.34847	−0.238451	−0.119225	−	0.992867i	$-0.538041\pi$
$97$	−2.34847	−0.238451	−0.119225	+	0.992867i	$0.538041\pi$
$98$	2.44949	0.247436
$99$	0	0
$100$	4.00000	0.400000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−13.3485	−1.32170
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−14.2020	−1.39262
$105$	2.44949	0.239046
$106$	4.65153	0.451797
$107$	7.34847	0.710403	0.355202	−	0.934790i	$-0.384412\pi$
$107$	7.34847	0.710403	0.355202	+	0.934790i	$0.384412\pi$
$108$	4.00000	0.384900
$109$	−5.34847	−0.512290	−0.256145	−	0.966638i	$-0.582453\pi$
$109$	−5.34847	−0.512290	−0.256145	+	0.966638i	$0.582453\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	4.00000	0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	10.8990	1.02078
$115$	14.6969	1.37050
$116$	−31.5959	−2.93361
$117$	−2.89898	−0.268011
$118$	−6.00000	−0.552345
$119$	−5.44949	−0.499554
$120$	12.0000	1.09545
$121$	−11.0000	−1.00000
$122$	10.8990	0.986747
$123$	5.44949	0.491364
$124$	32.0000	2.87368
$125$	−9.79796	−0.876356
$126$	2.44949	0.218218
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−19.5959	−1.73205
$129$	2.00000	0.176090
$130$	−17.3939	−1.52554
$131$	−4.34847	−0.379928	−0.189964	−	0.981791i	$-0.560837\pi$
$131$	−4.34847	−0.379928	−0.189964	+	0.981791i	$0.560837\pi$
$132$	0	0
$133$	4.44949	0.385820
$134$	−25.1010	−2.16840
$135$	2.44949	0.210819
$136$	−26.6969	−2.28924
$137$	2.20204	0.188133	0.0940665	−	0.995566i	$-0.470013\pi$
$137$	2.20204	0.188133	0.0940665	+	0.995566i	$0.470013\pi$
$138$	14.6969	1.25109
$139$	−5.65153	−0.479357	−0.239678	−	0.970852i	$-0.577042\pi$
$139$	−5.65153	−0.479357	−0.239678	+	0.970852i	$0.577042\pi$
$140$	9.79796	0.828079
$141$	−12.0000	−1.01058
$142$	8.69694	0.729831
$143$	0	0
$144$	4.00000	0.333333
$145$	−19.3485	−1.60680
$146$	8.20204	0.678806
$147$	1.00000	0.0824786
$148$	−28.0000	−2.30159
$149$	−20.6969	−1.69556	−0.847780	−	0.530349i	$-0.822061\pi$
$149$	−20.6969	−1.69556	−0.847780	+	0.530349i	$0.822061\pi$
$150$	2.44949	0.200000
$151$	9.34847	0.760768	0.380384	−	0.924829i	$-0.375792\pi$
$151$	9.34847	0.760768	0.380384	+	0.924829i	$0.375792\pi$
$152$	21.7980	1.76805
$153$	−5.44949	−0.440565
$154$	0	0
$155$	19.5959	1.57398
$156$	−11.5959	−0.928416
$157$	−1.55051	−0.123744	−0.0618721	−	0.998084i	$-0.519707\pi$
$157$	−1.55051	−0.123744	−0.0618721	+	0.998084i	$0.519707\pi$
$158$	9.55051	0.759798
$159$	1.89898	0.150599
$160$	0	0
$161$	6.00000	0.472866
$162$	2.44949	0.192450
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	21.7980	1.70213
$165$	0	0
$166$	14.6969	1.14070
$167$	23.1464	1.79112	0.895562	−	0.444936i	$-0.146774\pi$
$167$	23.1464	1.79112	0.895562	+	0.444936i	$0.146774\pi$
$168$	4.89898	0.377964
$169$	−4.59592	−0.353532
$170$	−32.6969	−2.50774
$171$	4.44949	0.340261
$172$	8.00000	0.609994
$173$	18.2474	1.38733	0.693664	−	0.720299i	$-0.255995\pi$
$173$	18.2474	1.38733	0.693664	+	0.720299i	$0.255995\pi$
$174$	−19.3485	−1.46680
$175$	1.00000	0.0755929
$176$	0	0
$177$	−2.44949	−0.184115
$178$	−24.0000	−1.79888
$179$	15.7980	1.18079	0.590397	−	0.807113i	$-0.298971\pi$
$179$	15.7980	1.18079	0.590397	+	0.807113i	$0.298971\pi$
$180$	9.79796	0.730297
$181$	10.1464	0.754178	0.377089	−	0.926177i	$-0.376925\pi$
$181$	10.1464	0.754178	0.377089	+	0.926177i	$0.376925\pi$
$182$	−7.10102	−0.526363
$183$	4.44949	0.328916
$184$	29.3939	2.16695
$185$	−17.1464	−1.26063
$186$	19.5959	1.43684
$187$	0	0
$188$	−48.0000	−3.50076
$189$	1.00000	0.0727393
$190$	26.6969	1.93680
$191$	−13.3485	−0.965861	−0.482931	−	0.875659i	$-0.660428\pi$
$191$	−13.3485	−0.965861	−0.482931	+	0.875659i	$0.660428\pi$
$192$	−8.00000	−0.577350
$193$	5.55051	0.399534	0.199767	−	0.979843i	$-0.435981\pi$
$193$	5.55051	0.399534	0.199767	+	0.979843i	$0.435981\pi$
$194$	−5.75255	−0.413009
$195$	−7.10102	−0.508515
$196$	4.00000	0.285714
$197$	8.44949	0.602001	0.301001	−	0.953624i	$-0.402679\pi$
$197$	8.44949	0.602001	0.301001	+	0.953624i	$0.402679\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	4.89898	0.346410
$201$	−10.2474	−0.722799
$202$	0	0
$203$	−7.89898	−0.554400
$204$	−21.7980	−1.52616
$205$	13.3485	0.932298
$206$	−9.79796	−0.682656
$207$	6.00000	0.417029
$208$	−11.5959	−0.804032
$209$	0	0
$210$	6.00000	0.414039
$211$	−24.6969	−1.70021	−0.850104	−	0.526615i	$-0.823461\pi$
$211$	−24.6969	−1.70021	−0.850104	+	0.526615i	$0.823461\pi$
$212$	7.59592	0.521690
$213$	3.55051	0.243277
$214$	18.0000	1.23045
$215$	4.89898	0.334108
$216$	4.89898	0.333333
$217$	8.00000	0.543075
$218$	−13.1010	−0.887313
$219$	3.34847	0.226269
$220$	0	0
$221$	15.7980	1.06269
$222$	−17.1464	−1.15079
$223$	2.55051	0.170795	0.0853974	−	0.996347i	$-0.472784\pi$
$223$	2.55051	0.170795	0.0853974	+	0.996347i	$0.472784\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−14.6969	−0.977626
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	17.7980	1.17870
$229$	7.44949	0.492276	0.246138	−	0.969235i	$-0.420838\pi$
$229$	7.44949	0.492276	0.246138	+	0.969235i	$0.420838\pi$
$230$	36.0000	2.37377
$231$	0	0
$232$	−38.6969	−2.54058
$233$	−7.89898	−0.517479	−0.258740	−	0.965947i	$-0.583307\pi$
$233$	−7.89898	−0.517479	−0.258740	+	0.965947i	$0.583307\pi$
$234$	−7.10102	−0.464208
$235$	−29.3939	−1.91745
$236$	−9.79796	−0.637793
$237$	3.89898	0.253266
$238$	−13.3485	−0.865253
$239$	3.00000	0.194054	0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000	0.194054	0.0970269	+	0.995282i	$0.469067\pi$
$240$	9.79796	0.632456
$241$	5.24745	0.338018	0.169009	−	0.985615i	$-0.445943\pi$
$241$	5.24745	0.338018	0.169009	+	0.985615i	$0.445943\pi$
$242$	−26.9444	−1.73205
$243$	1.00000	0.0641500
$244$	17.7980	1.13940
$245$	2.44949	0.156492
$246$	13.3485	0.851067
$247$	−12.8990	−0.820742
$248$	39.1918	2.48868
$249$	6.00000	0.380235
$250$	−24.0000	−1.51789
$251$	4.34847	0.274473	0.137236	−	0.990538i	$-0.456178\pi$
$251$	4.34847	0.274473	0.137236	+	0.990538i	$0.456178\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	2.44949	0.153695
$255$	−13.3485	−0.835914
$256$	−32.0000	−2.00000
$257$	−11.1464	−0.695295	−0.347648	−	0.937625i	$-0.613019\pi$
$257$	−11.1464	−0.695295	−0.347648	+	0.937625i	$0.613019\pi$
$258$	4.89898	0.304997
$259$	−7.00000	−0.434959
$260$	−28.4041	−1.76155
$261$	−7.89898	−0.488935
$262$	−10.6515	−0.658054
$263$	−23.1464	−1.42727	−0.713635	−	0.700518i	$-0.752952\pi$
$263$	−23.1464	−1.42727	−0.713635	+	0.700518i	$0.752952\pi$
$264$	0	0
$265$	4.65153	0.285741
$266$	10.8990	0.668259
$267$	−9.79796	−0.599625
$268$	−40.9898	−2.50385
$269$	27.2474	1.66131	0.830653	−	0.556790i	$-0.187967\pi$
$269$	27.2474	1.66131	0.830653	+	0.556790i	$0.187967\pi$
$270$	6.00000	0.365148
$271$	−30.9444	−1.87974	−0.939869	−	0.341536i	$-0.889053\pi$
$271$	−30.9444	−1.87974	−0.939869	+	0.341536i	$0.889053\pi$
$272$	−21.7980	−1.32170
$273$	−2.89898	−0.175454
$274$	5.39388	0.325856
$275$	0	0
$276$	24.0000	1.44463
$277$	10.2020	0.612981	0.306491	−	0.951874i	$-0.400845\pi$
$277$	10.2020	0.612981	0.306491	+	0.951874i	$0.400845\pi$
$278$	−13.8434	−0.830270
$279$	8.00000	0.478947
$280$	12.0000	0.717137
$281$	−5.20204	−0.310328	−0.155164	−	0.987889i	$-0.549591\pi$
$281$	−5.20204	−0.310328	−0.155164	+	0.987889i	$0.549591\pi$
$282$	−29.3939	−1.75038
$283$	24.3485	1.44737	0.723683	−	0.690132i	$-0.242448\pi$
$283$	24.3485	1.44737	0.723683	+	0.690132i	$0.242448\pi$
$284$	14.2020	0.842736
$285$	10.8990	0.645600
$286$	0	0
$287$	5.44949	0.321673
$288$	0	0
$289$	12.6969	0.746879
$290$	−47.3939	−2.78306
$291$	−2.34847	−0.137670
$292$	13.3939	0.783817
$293$	26.4495	1.54520	0.772598	−	0.634896i	$-0.218957\pi$
$293$	26.4495	1.54520	0.772598	+	0.634896i	$0.218957\pi$
$294$	2.44949	0.142857
$295$	−6.00000	−0.349334
$296$	−34.2929	−1.99323
$297$	0	0
$298$	−50.6969	−2.93679
$299$	−17.3939	−1.00591
$300$	4.00000	0.230940
$301$	2.00000	0.115278
$302$	22.8990	1.31769
$303$	0	0
$304$	17.7980	1.02078
$305$	10.8990	0.624074
$306$	−13.3485	−0.763081
$307$	20.5505	1.17288	0.586440	−	0.809993i	$-0.300529\pi$
$307$	20.5505	1.17288	0.586440	+	0.809993i	$0.300529\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	48.0000	2.72622
$311$	−17.1464	−0.972285	−0.486142	−	0.873880i	$-0.661596\pi$
$311$	−17.1464	−0.972285	−0.486142	+	0.873880i	$0.661596\pi$
$312$	−14.2020	−0.804032
$313$	25.4495	1.43849	0.719245	−	0.694756i	$-0.244488\pi$
$313$	25.4495	1.43849	0.719245	+	0.694756i	$0.244488\pi$
$314$	−3.79796	−0.214331
$315$	2.44949	0.138013
$316$	15.5959	0.877339
$317$	−19.1010	−1.07282	−0.536410	−	0.843957i	$-0.680220\pi$
$317$	−19.1010	−1.07282	−0.536410	+	0.843957i	$0.680220\pi$
$318$	4.65153	0.260845
$319$	0	0
$320$	−19.5959	−1.09545
$321$	7.34847	0.410152
$322$	14.6969	0.819028
$323$	−24.2474	−1.34916
$324$	4.00000	0.222222
$325$	−2.89898	−0.160806
$326$	41.6413	2.30630
$327$	−5.34847	−0.295771
$328$	26.6969	1.47409
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−18.4495	−1.01408	−0.507038	−	0.861924i	$-0.669260\pi$
$331$	−18.4495	−1.01408	−0.507038	+	0.861924i	$0.669260\pi$
$332$	24.0000	1.31717
$333$	−7.00000	−0.383598
$334$	56.6969	3.10232
$335$	−25.1010	−1.37142
$336$	4.00000	0.218218
$337$	−23.3485	−1.27187	−0.635936	−	0.771742i	$-0.719386\pi$
$337$	−23.3485	−1.27187	−0.635936	+	0.771742i	$0.719386\pi$
$338$	−11.2577	−0.612336
$339$	−6.00000	−0.325875
$340$	−53.3939	−2.89569
$341$	0	0
$342$	10.8990	0.589349
$343$	1.00000	0.0539949
$344$	9.79796	0.528271
$345$	14.6969	0.791257
$346$	44.6969	2.40292
$347$	−17.6969	−0.950021	−0.475011	−	0.879980i	$-0.657556\pi$
$347$	−17.6969	−0.950021	−0.475011	+	0.879980i	$0.657556\pi$
$348$	−31.5959	−1.69372
$349$	−17.0454	−0.912420	−0.456210	−	0.889872i	$-0.650793\pi$
$349$	−17.0454	−0.912420	−0.456210	+	0.889872i	$0.650793\pi$
$350$	2.44949	0.130931
$351$	−2.89898	−0.154736
$352$	0	0
$353$	16.8990	0.899442	0.449721	−	0.893169i	$-0.351523\pi$
$353$	16.8990	0.899442	0.449721	+	0.893169i	$0.351523\pi$
$354$	−6.00000	−0.318896
$355$	8.69694	0.461586
$356$	−39.1918	−2.07716
$357$	−5.44949	−0.288418
$358$	38.6969	2.04520
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	12.0000	0.632456
$361$	0.797959	0.0419978
$362$	24.8536	1.30627
$363$	−11.0000	−0.577350
$364$	−11.5959	−0.607791
$365$	8.20204	0.429314
$366$	10.8990	0.569699
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	24.0000	1.25109
$369$	5.44949	0.283689
$370$	−42.0000	−2.18348
$371$	1.89898	0.0985901
$372$	32.0000	1.65912
$373$	15.1010	0.781901	0.390951	−	0.920412i	$-0.372146\pi$
$373$	15.1010	0.781901	0.390951	+	0.920412i	$0.372146\pi$
$374$	0	0
$375$	−9.79796	−0.505964
$376$	−58.7878	−3.03175
$377$	22.8990	1.17936
$378$	2.44949	0.125988
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	43.5959	2.23642
$381$	1.00000	0.0512316
$382$	−32.6969	−1.67292
$383$	−21.2474	−1.08569	−0.542847	−	0.839832i	$-0.682654\pi$
$383$	−21.2474	−1.08569	−0.542847	+	0.839832i	$0.682654\pi$
$384$	−19.5959	−1.00000
$385$	0	0
$386$	13.5959	0.692014
$387$	2.00000	0.101666
$388$	−9.39388	−0.476902
$389$	−34.0454	−1.72617	−0.863085	−	0.505058i	$-0.831471\pi$
$389$	−34.0454	−1.72617	−0.863085	+	0.505058i	$0.831471\pi$
$390$	−17.3939	−0.880773
$391$	−32.6969	−1.65356
$392$	4.89898	0.247436
$393$	−4.34847	−0.219351
$394$	20.6969	1.04270
$395$	9.55051	0.480538
$396$	0	0
$397$	−6.69694	−0.336110	−0.168055	−	0.985778i	$-0.553749\pi$
$397$	−6.69694	−0.336110	−0.168055	+	0.985778i	$0.553749\pi$
$398$	48.9898	2.45564
$399$	4.44949	0.222753
$400$	4.00000	0.200000
$401$	−0.303062	−0.0151342	−0.00756709	−	0.999971i	$-0.502409\pi$
$401$	−0.303062	−0.0151342	−0.00756709	+	0.999971i	$0.502409\pi$
$402$	−25.1010	−1.25193
$403$	−23.1918	−1.15527
$404$	0	0
$405$	2.44949	0.121716
$406$	−19.3485	−0.960248
$407$	0	0
$408$	−26.6969	−1.32170
$409$	16.6969	0.825610	0.412805	−	0.910819i	$-0.364549\pi$
$409$	16.6969	0.825610	0.412805	+	0.910819i	$0.364549\pi$
$410$	32.6969	1.61479
$411$	2.20204	0.108619
$412$	−16.0000	−0.788263
$413$	−2.44949	−0.120532
$414$	14.6969	0.722315
$415$	14.6969	0.721444
$416$	0	0
$417$	−5.65153	−0.276757
$418$	0	0
$419$	−31.5959	−1.54356	−0.771781	−	0.635889i	$-0.780634\pi$
$419$	−31.5959	−1.54356	−0.771781	+	0.635889i	$0.780634\pi$
$420$	9.79796	0.478091
$421$	6.04541	0.294635	0.147318	−	0.989089i	$-0.452936\pi$
$421$	6.04541	0.294635	0.147318	+	0.989089i	$0.452936\pi$
$422$	−60.4949	−2.94485
$423$	−12.0000	−0.583460
$424$	9.30306	0.451797
$425$	−5.44949	−0.264339
$426$	8.69694	0.421368
$427$	4.44949	0.215326
$428$	29.3939	1.42081
$429$	0	0
$430$	12.0000	0.578691
$431$	1.34847	0.0649535	0.0324767	−	0.999472i	$-0.489661\pi$
$431$	1.34847	0.0649535	0.0324767	+	0.999472i	$0.489661\pi$
$432$	4.00000	0.192450
$433$	0.651531	0.0313106	0.0156553	−	0.999877i	$-0.495017\pi$
$433$	0.651531	0.0313106	0.0156553	+	0.999877i	$0.495017\pi$
$434$	19.5959	0.940634
$435$	−19.3485	−0.927688
$436$	−21.3939	−1.02458
$437$	26.6969	1.27709
$438$	8.20204	0.391909
$439$	−14.3485	−0.684815	−0.342408	−	0.939552i	$-0.611242\pi$
$439$	−14.3485	−0.684815	−0.342408	+	0.939552i	$0.611242\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	38.6969	1.84063
$443$	21.5505	1.02390	0.511948	−	0.859017i	$-0.328924\pi$
$443$	21.5505	1.02390	0.511948	+	0.859017i	$0.328924\pi$
$444$	−28.0000	−1.32882
$445$	−24.0000	−1.13771
$446$	6.24745	0.295825
$447$	−20.6969	−0.978932
$448$	−8.00000	−0.377964
$449$	19.1010	0.901433	0.450716	−	0.892667i	$-0.351169\pi$
$449$	19.1010	0.901433	0.450716	+	0.892667i	$0.351169\pi$
$450$	2.44949	0.115470
$451$	0	0
$452$	−24.0000	−1.12887
$453$	9.34847	0.439229
$454$	29.3939	1.37952
$455$	−7.10102	−0.332901
$456$	21.7980	1.02078
$457$	29.4949	1.37971	0.689857	−	0.723946i	$-0.257674\pi$
$457$	29.4949	1.37971	0.689857	+	0.723946i	$0.257674\pi$
$458$	18.2474	0.852647
$459$	−5.44949	−0.254360
$460$	58.7878	2.74099
$461$	−5.14643	−0.239693	−0.119847	−	0.992792i	$-0.538240\pi$
$461$	−5.14643	−0.239693	−0.119847	+	0.992792i	$0.538240\pi$
$462$	0	0
$463$	−0.202041	−0.00938964	−0.00469482	−	0.999989i	$-0.501494\pi$
$463$	−0.202041	−0.00938964	−0.00469482	+	0.999989i	$0.501494\pi$
$464$	−31.5959	−1.46680
$465$	19.5959	0.908739
$466$	−19.3485	−0.896301
$467$	31.5959	1.46208	0.731042	−	0.682332i	$-0.239034\pi$
$467$	31.5959	1.46208	0.731042	+	0.682332i	$0.239034\pi$
$468$	−11.5959	−0.536021
$469$	−10.2474	−0.473183
$470$	−72.0000	−3.32111
$471$	−1.55051	−0.0714437
$472$	−12.0000	−0.552345
$473$	0	0
$474$	9.55051	0.438669
$475$	4.44949	0.204157
$476$	−21.7980	−0.999108
$477$	1.89898	0.0869483
$478$	7.34847	0.336111
$479$	3.24745	0.148380	0.0741899	−	0.997244i	$-0.476363\pi$
$479$	3.24745	0.148380	0.0741899	+	0.997244i	$0.476363\pi$
$480$	0	0
$481$	20.2929	0.925275
$482$	12.8536	0.585464
$483$	6.00000	0.273009
$484$	−44.0000	−2.00000
$485$	−5.75255	−0.261210
$486$	2.44949	0.111111
$487$	−30.9444	−1.40222	−0.701112	−	0.713051i	$-0.747313\pi$
$487$	−30.9444	−1.40222	−0.701112	+	0.713051i	$0.747313\pi$
$488$	21.7980	0.986747
$489$	17.0000	0.768767
$490$	6.00000	0.271052
$491$	−13.1010	−0.591241	−0.295620	−	0.955305i	$-0.595526\pi$
$491$	−13.1010	−0.591241	−0.295620	+	0.955305i	$0.595526\pi$
$492$	21.7980	0.982728
$493$	43.0454	1.93867
$494$	−31.5959	−1.42157
$495$	0	0
$496$	32.0000	1.43684
$497$	3.55051	0.159262
$498$	14.6969	0.658586
$499$	6.04541	0.270630	0.135315	−	0.990803i	$-0.456795\pi$
$499$	6.04541	0.270630	0.135315	+	0.990803i	$0.456795\pi$
$500$	−39.1918	−1.75271
$501$	23.1464	1.03411
$502$	10.6515	0.475401
$503$	29.4495	1.31309	0.656544	−	0.754288i	$-0.272018\pi$
$503$	29.4495	1.31309	0.656544	+	0.754288i	$0.272018\pi$
$504$	4.89898	0.218218
$505$	0	0
$506$	0	0
$507$	−4.59592	−0.204112
$508$	4.00000	0.177471
$509$	−7.04541	−0.312282	−0.156141	−	0.987735i	$-0.549905\pi$
$509$	−7.04541	−0.312282	−0.156141	+	0.987735i	$0.549905\pi$
$510$	−32.6969	−1.44784
$511$	3.34847	0.148128
$512$	−39.1918	−1.73205
$513$	4.44949	0.196450
$514$	−27.3031	−1.20429
$515$	−9.79796	−0.431750
$516$	8.00000	0.352180
$517$	0	0
$518$	−17.1464	−0.753371
$519$	18.2474	0.800974
$520$	−34.7878	−1.52554
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−19.3485	−0.846859
$523$	−28.2474	−1.23517	−0.617587	−	0.786502i	$-0.711890\pi$
$523$	−28.2474	−1.23517	−0.617587	+	0.786502i	$0.711890\pi$
$524$	−17.3939	−0.759855
$525$	1.00000	0.0436436
$526$	−56.6969	−2.47210
$527$	−43.5959	−1.89907
$528$	0	0
$529$	13.0000	0.565217
$530$	11.3939	0.494918
$531$	−2.44949	−0.106299
$532$	17.7980	0.771639
$533$	−15.7980	−0.684286
$534$	−24.0000	−1.03858
$535$	18.0000	0.778208
$536$	−50.2020	−2.16840
$537$	15.7980	0.681732
$538$	66.7423	2.87747
$539$	0	0
$540$	9.79796	0.421637
$541$	−8.89898	−0.382597	−0.191299	−	0.981532i	$-0.561270\pi$
$541$	−8.89898	−0.382597	−0.191299	+	0.981532i	$0.561270\pi$
$542$	−75.7980	−3.25580
$543$	10.1464	0.435425
$544$	0	0
$545$	−13.1010	−0.561186
$546$	−7.10102	−0.303896
$547$	−1.79796	−0.0768752	−0.0384376	−	0.999261i	$-0.512238\pi$
$547$	−1.79796	−0.0768752	−0.0384376	+	0.999261i	$0.512238\pi$
$548$	8.80816	0.376266
$549$	4.44949	0.189900
$550$	0	0
$551$	−35.1464	−1.49729
$552$	29.3939	1.25109
$553$	3.89898	0.165801
$554$	24.9898	1.06171
$555$	−17.1464	−0.727825
$556$	−22.6061	−0.958713
$557$	21.7980	0.923609	0.461805	−	0.886982i	$-0.347202\pi$
$557$	21.7980	0.923609	0.461805	+	0.886982i	$0.347202\pi$
$558$	19.5959	0.829561
$559$	−5.79796	−0.245228
$560$	9.79796	0.414039
$561$	0	0
$562$	−12.7423	−0.537503
$563$	34.2929	1.44527	0.722636	−	0.691229i	$-0.242930\pi$
$563$	34.2929	1.44527	0.722636	+	0.691229i	$0.242930\pi$
$564$	−48.0000	−2.02116
$565$	−14.6969	−0.618305
$566$	59.6413	2.50691
$567$	1.00000	0.0419961
$568$	17.3939	0.729831
$569$	−30.4949	−1.27841	−0.639206	−	0.769035i	$-0.720737\pi$
$569$	−30.4949	−1.27841	−0.639206	+	0.769035i	$0.720737\pi$
$570$	26.6969	1.11821
$571$	−45.1464	−1.88932	−0.944660	−	0.328052i	$-0.893608\pi$
$571$	−45.1464	−1.88932	−0.944660	+	0.328052i	$0.893608\pi$
$572$	0	0
$573$	−13.3485	−0.557640
$574$	13.3485	0.557154
$575$	6.00000	0.250217
$576$	−8.00000	−0.333333
$577$	−20.6515	−0.859734	−0.429867	−	0.902892i	$-0.641440\pi$
$577$	−20.6515	−0.859734	−0.429867	+	0.902892i	$0.641440\pi$
$578$	31.1010	1.29363
$579$	5.55051	0.230671
$580$	−77.3939	−3.21361
$581$	6.00000	0.248922
$582$	−5.75255	−0.238451
$583$	0	0
$584$	16.4041	0.678806
$585$	−7.10102	−0.293591
$586$	64.7878	2.67636
$587$	−5.44949	−0.224925	−0.112462	−	0.993656i	$-0.535874\pi$
$587$	−5.44949	−0.224925	−0.112462	+	0.993656i	$0.535874\pi$
$588$	4.00000	0.164957
$589$	35.5959	1.46670
$590$	−14.6969	−0.605063
$591$	8.44949	0.347566
$592$	−28.0000	−1.15079
$593$	−9.79796	−0.402354	−0.201177	−	0.979555i	$-0.564477\pi$
$593$	−9.79796	−0.402354	−0.201177	+	0.979555i	$0.564477\pi$
$594$	0	0
$595$	−13.3485	−0.547234
$596$	−82.7878	−3.39112
$597$	20.0000	0.818546
$598$	−42.6061	−1.74229
$599$	−33.0000	−1.34834	−0.674172	−	0.738575i	$-0.735499\pi$
$599$	−33.0000	−1.34834	−0.674172	+	0.738575i	$0.735499\pi$
$600$	4.89898	0.200000
$601$	−5.65153	−0.230531	−0.115265	−	0.993335i	$-0.536772\pi$
$601$	−5.65153	−0.230531	−0.115265	+	0.993335i	$0.536772\pi$
$602$	4.89898	0.199667
$603$	−10.2474	−0.417308
$604$	37.3939	1.52154
$605$	−26.9444	−1.09545
$606$	0	0
$607$	9.34847	0.379443	0.189721	−	0.981838i	$-0.439242\pi$
$607$	9.34847	0.379443	0.189721	+	0.981838i	$0.439242\pi$
$608$	0	0
$609$	−7.89898	−0.320083
$610$	26.6969	1.08093
$611$	34.7878	1.40736
$612$	−21.7980	−0.881130
$613$	38.4949	1.55479	0.777397	−	0.629010i	$-0.216540\pi$
$613$	38.4949	1.55479	0.777397	+	0.629010i	$0.216540\pi$
$614$	50.3383	2.03149
$615$	13.3485	0.538262
$616$	0	0
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	−9.79796	−0.394132
$619$	18.3485	0.737487	0.368744	−	0.929531i	$-0.379788\pi$
$619$	18.3485	0.737487	0.368744	+	0.929531i	$0.379788\pi$
$620$	78.3837	3.14796
$621$	6.00000	0.240772
$622$	−42.0000	−1.68405
$623$	−9.79796	−0.392547
$624$	−11.5959	−0.464208
$625$	−29.0000	−1.16000
$626$	62.3383	2.49154
$627$	0	0
$628$	−6.20204	−0.247488
$629$	38.1464	1.52100
$630$	6.00000	0.239046
$631$	47.7980	1.90281	0.951403	−	0.307948i	$-0.0996421\pi$
$631$	47.7980	1.90281	0.951403	+	0.307948i	$0.0996421\pi$
$632$	19.1010	0.759798
$633$	−24.6969	−0.981615
$634$	−46.7878	−1.85818
$635$	2.44949	0.0972050
$636$	7.59592	0.301198
$637$	−2.89898	−0.114862
$638$	0	0
$639$	3.55051	0.140456
$640$	−48.0000	−1.89737
$641$	1.89898	0.0750052	0.0375026	−	0.999297i	$-0.488060\pi$
$641$	1.89898	0.0750052	0.0375026	+	0.999297i	$0.488060\pi$
$642$	18.0000	0.710403
$643$	9.34847	0.368668	0.184334	−	0.982864i	$-0.440987\pi$
$643$	9.34847	0.368668	0.184334	+	0.982864i	$0.440987\pi$
$644$	24.0000	0.945732
$645$	4.89898	0.192897
$646$	−59.3939	−2.33682
$647$	44.6969	1.75722	0.878609	−	0.477542i	$-0.158472\pi$
$647$	44.6969	1.75722	0.878609	+	0.477542i	$0.158472\pi$
$648$	4.89898	0.192450
$649$	0	0
$650$	−7.10102	−0.278525
$651$	8.00000	0.313545
$652$	68.0000	2.66309
$653$	26.4495	1.03505	0.517524	−	0.855669i	$-0.326854\pi$
$653$	26.4495	1.03505	0.517524	+	0.855669i	$0.326854\pi$
$654$	−13.1010	−0.512290
$655$	−10.6515	−0.416190
$656$	21.7980	0.851067
$657$	3.34847	0.130636
$658$	−29.3939	−1.14589
$659$	−13.1010	−0.510343	−0.255172	−	0.966896i	$-0.582132\pi$
$659$	−13.1010	−0.510343	−0.255172	+	0.966896i	$0.582132\pi$
$660$	0	0
$661$	−20.6515	−0.803251	−0.401626	−	0.915804i	$-0.631555\pi$
$661$	−20.6515	−0.803251	−0.401626	+	0.915804i	$0.631555\pi$
$662$	−45.1918	−1.75643
$663$	15.7980	0.613542
$664$	29.3939	1.14070
$665$	10.8990	0.422644
$666$	−17.1464	−0.664411
$667$	−47.3939	−1.83510
$668$	92.5857	3.58225
$669$	2.55051	0.0986084
$670$	−61.4847	−2.37536
$671$	0	0
$672$	0	0
$673$	43.6969	1.68439	0.842197	−	0.539171i	$-0.181262\pi$
$673$	43.6969	1.68439	0.842197	+	0.539171i	$0.181262\pi$
$674$	−57.1918	−2.20295
$675$	1.00000	0.0384900
$676$	−18.3837	−0.707064
$677$	−23.4495	−0.901237	−0.450619	−	0.892717i	$-0.648797\pi$
$677$	−23.4495	−0.901237	−0.450619	+	0.892717i	$0.648797\pi$
$678$	−14.6969	−0.564433
$679$	−2.34847	−0.0901260
$680$	−65.3939	−2.50774
$681$	12.0000	0.459841
$682$	0	0
$683$	−25.5959	−0.979401	−0.489700	−	0.871891i	$-0.662894\pi$
$683$	−25.5959	−0.979401	−0.489700	+	0.871891i	$0.662894\pi$
$684$	17.7980	0.680522
$685$	5.39388	0.206089
$686$	2.44949	0.0935220
$687$	7.44949	0.284216
$688$	8.00000	0.304997
$689$	−5.50510	−0.209728
$690$	36.0000	1.37050
$691$	32.5505	1.23828	0.619140	−	0.785281i	$-0.287481\pi$
$691$	32.5505	1.23828	0.619140	+	0.785281i	$0.287481\pi$
$692$	72.9898	2.77466
$693$	0	0
$694$	−43.3485	−1.64549
$695$	−13.8434	−0.525109
$696$	−38.6969	−1.46680
$697$	−29.6969	−1.12485
$698$	−41.7526	−1.58036
$699$	−7.89898	−0.298767
$700$	4.00000	0.151186
$701$	−24.4949	−0.925160	−0.462580	−	0.886578i	$-0.653076\pi$
$701$	−24.4949	−0.925160	−0.462580	+	0.886578i	$0.653076\pi$
$702$	−7.10102	−0.268011
$703$	−31.1464	−1.17471
$704$	0	0
$705$	−29.3939	−1.10704
$706$	41.3939	1.55788
$707$	0	0
$708$	−9.79796	−0.368230
$709$	16.3939	0.615685	0.307842	−	0.951437i	$-0.400393\pi$
$709$	16.3939	0.615685	0.307842	+	0.951437i	$0.400393\pi$
$710$	21.3031	0.799490
$711$	3.89898	0.146223
$712$	−48.0000	−1.79888
$713$	48.0000	1.79761
$714$	−13.3485	−0.499554
$715$	0	0
$716$	63.1918	2.36159
$717$	3.00000	0.112037
$718$	36.7423	1.37121
$719$	−38.1464	−1.42262	−0.711311	−	0.702878i	$-0.751898\pi$
$719$	−38.1464	−1.42262	−0.711311	+	0.702878i	$0.751898\pi$
$720$	9.79796	0.365148
$721$	−4.00000	−0.148968
$722$	1.95459	0.0727424
$723$	5.24745	0.195155
$724$	40.5857	1.50836
$725$	−7.89898	−0.293361
$726$	−26.9444	−1.00000
$727$	52.6413	1.95236	0.976179	−	0.216965i	$-0.0696158\pi$
$727$	52.6413	1.95236	0.976179	+	0.216965i	$0.0696158\pi$
$728$	−14.2020	−0.526363
$729$	1.00000	0.0370370
$730$	20.0908	0.743594
$731$	−10.8990	−0.403113
$732$	17.7980	0.657831
$733$	−29.3485	−1.08401	−0.542005	−	0.840375i	$-0.682335\pi$
$733$	−29.3485	−1.08401	−0.542005	+	0.840375i	$0.682335\pi$
$734$	4.89898	0.180825
$735$	2.44949	0.0903508
$736$	0	0
$737$	0	0
$738$	13.3485	0.491364
$739$	−4.30306	−0.158291	−0.0791453	−	0.996863i	$-0.525219\pi$
$739$	−4.30306	−0.158291	−0.0791453	+	0.996863i	$0.525219\pi$
$740$	−68.5857	−2.52126
$741$	−12.8990	−0.473856
$742$	4.65153	0.170763
$743$	−23.6969	−0.869356	−0.434678	−	0.900586i	$-0.643138\pi$
$743$	−23.6969	−0.869356	−0.434678	+	0.900586i	$0.643138\pi$
$744$	39.1918	1.43684
$745$	−50.6969	−1.85739
$746$	36.9898	1.35429
$747$	6.00000	0.219529
$748$	0	0
$749$	7.34847	0.268507
$750$	−24.0000	−0.876356
$751$	43.6413	1.59249	0.796247	−	0.604971i	$-0.206815\pi$
$751$	43.6413	1.59249	0.796247	+	0.604971i	$0.206815\pi$
$752$	−48.0000	−1.75038
$753$	4.34847	0.158467
$754$	56.0908	2.04271
$755$	22.8990	0.833379
$756$	4.00000	0.145479
$757$	43.0908	1.56616	0.783081	−	0.621920i	$-0.213647\pi$
$757$	43.0908	1.56616	0.783081	+	0.621920i	$0.213647\pi$
$758$	48.9898	1.77939
$759$	0	0
$760$	53.3939	1.93680
$761$	−53.6413	−1.94450	−0.972248	−	0.233952i	$-0.924834\pi$
$761$	−53.6413	−1.94450	−0.972248	+	0.233952i	$0.924834\pi$
$762$	2.44949	0.0887357
$763$	−5.34847	−0.193628
$764$	−53.3939	−1.93172
$765$	−13.3485	−0.482615
$766$	−52.0454	−1.88048
$767$	7.10102	0.256403
$768$	−32.0000	−1.15470
$769$	32.5505	1.17380	0.586901	−	0.809659i	$-0.300348\pi$
$769$	32.5505	1.17380	0.586901	+	0.809659i	$0.300348\pi$
$770$	0	0
$771$	−11.1464	−0.401429
$772$	22.2020	0.799069
$773$	−51.2474	−1.84324	−0.921621	−	0.388090i	$-0.873135\pi$
$773$	−51.2474	−1.84324	−0.921621	+	0.388090i	$0.873135\pi$
$774$	4.89898	0.176090
$775$	8.00000	0.287368
$776$	−11.5051	−0.413009
$777$	−7.00000	−0.251124
$778$	−83.3939	−2.98982
$779$	24.2474	0.868755
$780$	−28.4041	−1.01703
$781$	0	0
$782$	−80.0908	−2.86404
$783$	−7.89898	−0.282286
$784$	4.00000	0.142857
$785$	−3.79796	−0.135555
$786$	−10.6515	−0.379928
$787$	55.1464	1.96576	0.982879	−	0.184253i	$-0.0589864\pi$
$787$	55.1464	1.96576	0.982879	+	0.184253i	$0.0589864\pi$
$788$	33.7980	1.20400
$789$	−23.1464	−0.824035
$790$	23.3939	0.832317
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−12.8990	−0.458056
$794$	−16.4041	−0.582159
$795$	4.65153	0.164973
$796$	80.0000	2.83552
$797$	8.20204	0.290531	0.145266	−	0.989393i	$-0.453596\pi$
$797$	8.20204	0.290531	0.145266	+	0.989393i	$0.453596\pi$
$798$	10.8990	0.385820
$799$	65.3939	2.31347
$800$	0	0
$801$	−9.79796	−0.346194
$802$	−0.742346	−0.0262132
$803$	0	0
$804$	−40.9898	−1.44560
$805$	14.6969	0.517999
$806$	−56.8082	−2.00098
$807$	27.2474	0.959156
$808$	0	0
$809$	11.1464	0.391888	0.195944	−	0.980615i	$-0.437223\pi$
$809$	11.1464	0.391888	0.195944	+	0.980615i	$0.437223\pi$
$810$	6.00000	0.210819
$811$	44.4949	1.56243	0.781214	−	0.624264i	$-0.214601\pi$
$811$	44.4949	1.56243	0.781214	+	0.624264i	$0.214601\pi$
$812$	−31.5959	−1.10880
$813$	−30.9444	−1.08527
$814$	0	0
$815$	41.6413	1.45863
$816$	−21.7980	−0.763081
$817$	8.89898	0.311336
$818$	40.8990	1.43000
$819$	−2.89898	−0.101299
$820$	53.3939	1.86460
$821$	6.79796	0.237250	0.118625	−	0.992939i	$-0.462151\pi$
$821$	6.79796	0.237250	0.118625	+	0.992939i	$0.462151\pi$
$822$	5.39388	0.188133
$823$	−19.0000	−0.662298	−0.331149	−	0.943578i	$-0.607436\pi$
$823$	−19.0000	−0.662298	−0.331149	+	0.943578i	$0.607436\pi$
$824$	−19.5959	−0.682656
$825$	0	0
$826$	−6.00000	−0.208767
$827$	−24.7980	−0.862310	−0.431155	−	0.902278i	$-0.641894\pi$
$827$	−24.7980	−0.862310	−0.431155	+	0.902278i	$0.641894\pi$
$828$	24.0000	0.834058
$829$	32.4949	1.12859	0.564297	−	0.825572i	$-0.309147\pi$
$829$	32.4949	1.12859	0.564297	+	0.825572i	$0.309147\pi$
$830$	36.0000	1.24958
$831$	10.2020	0.353905
$832$	23.1918	0.804032
$833$	−5.44949	−0.188814
$834$	−13.8434	−0.479357
$835$	56.6969	1.96208
$836$	0	0
$837$	8.00000	0.276520
$838$	−77.3939	−2.67353
$839$	−37.8434	−1.30650	−0.653249	−	0.757143i	$-0.726595\pi$
$839$	−37.8434	−1.30650	−0.653249	+	0.757143i	$0.726595\pi$
$840$	12.0000	0.414039
$841$	33.3939	1.15151
$842$	14.8082	0.510323
$843$	−5.20204	−0.179168
$844$	−98.7878	−3.40041
$845$	−11.2577	−0.387275
$846$	−29.3939	−1.01058
$847$	−11.0000	−0.377964
$848$	7.59592	0.260845
$849$	24.3485	0.835637
$850$	−13.3485	−0.457849
$851$	−42.0000	−1.43974
$852$	14.2020	0.486554
$853$	17.7423	0.607486	0.303743	−	0.952754i	$-0.401764\pi$
$853$	17.7423	0.607486	0.303743	+	0.952754i	$0.401764\pi$
$854$	10.8990	0.372955
$855$	10.8990	0.372737
$856$	36.0000	1.23045
$857$	32.6969	1.11691	0.558453	−	0.829536i	$-0.311395\pi$
$857$	32.6969	1.11691	0.558453	+	0.829536i	$0.311395\pi$
$858$	0	0
$859$	−36.1464	−1.23330	−0.616650	−	0.787237i	$-0.711511\pi$
$859$	−36.1464	−1.23330	−0.616650	+	0.787237i	$0.711511\pi$
$860$	19.5959	0.668215
$861$	5.44949	0.185718
$862$	3.30306	0.112503
$863$	−37.8990	−1.29010	−0.645048	−	0.764142i	$-0.723163\pi$
$863$	−37.8990	−1.29010	−0.645048	+	0.764142i	$0.723163\pi$
$864$	0	0
$865$	44.6969	1.51974
$866$	1.59592	0.0542315
$867$	12.6969	0.431211
$868$	32.0000	1.08615
$869$	0	0
$870$	−47.3939	−1.60680
$871$	29.7071	1.00659
$872$	−26.2020	−0.887313
$873$	−2.34847	−0.0794836
$874$	65.3939	2.21198
$875$	−9.79796	−0.331231
$876$	13.3939	0.452537
$877$	14.3031	0.482980	0.241490	−	0.970403i	$-0.422364\pi$
$877$	14.3031	0.482980	0.241490	+	0.970403i	$0.422364\pi$
$878$	−35.1464	−1.18613
$879$	26.4495	0.892119
$880$	0	0
$881$	−17.7526	−0.598099	−0.299049	−	0.954238i	$-0.596670\pi$
$881$	−17.7526	−0.598099	−0.299049	+	0.954238i	$0.596670\pi$
$882$	2.44949	0.0824786
$883$	−0.202041	−0.00679922	−0.00339961	−	0.999994i	$-0.501082\pi$
$883$	−0.202041	−0.00679922	−0.00339961	+	0.999994i	$0.501082\pi$
$884$	63.1918	2.12537
$885$	−6.00000	−0.201688
$886$	52.7878	1.77344
$887$	4.89898	0.164492	0.0822458	−	0.996612i	$-0.473791\pi$
$887$	4.89898	0.164492	0.0822458	+	0.996612i	$0.473791\pi$
$888$	−34.2929	−1.15079
$889$	1.00000	0.0335389
$890$	−58.7878	−1.97057
$891$	0	0
$892$	10.2020	0.341590
$893$	−53.3939	−1.78676
$894$	−50.6969	−1.69556
$895$	38.6969	1.29350
$896$	−19.5959	−0.654654
$897$	−17.3939	−0.580765
$898$	46.7878	1.56133
$899$	−63.1918	−2.10757
$900$	4.00000	0.133333
$901$	−10.3485	−0.344757
$902$	0	0
$903$	2.00000	0.0665558
$904$	−29.3939	−0.977626
$905$	24.8536	0.826161
$906$	22.8990	0.760768
$907$	32.7980	1.08904	0.544519	−	0.838748i	$-0.316712\pi$
$907$	32.7980	1.08904	0.544519	+	0.838748i	$0.316712\pi$
$908$	48.0000	1.59294
$909$	0	0
$910$	−17.3939	−0.576601
$911$	−20.6969	−0.685720	−0.342860	−	0.939386i	$-0.611396\pi$
$911$	−20.6969	−0.685720	−0.342860	+	0.939386i	$0.611396\pi$
$912$	17.7980	0.589349
$913$	0	0
$914$	72.2474	2.38973
$915$	10.8990	0.360309
$916$	29.7980	0.984552
$917$	−4.34847	−0.143599
$918$	−13.3485	−0.440565
$919$	14.7980	0.488140	0.244070	−	0.969758i	$-0.421517\pi$
$919$	14.7980	0.488140	0.244070	+	0.969758i	$0.421517\pi$
$920$	72.0000	2.37377
$921$	20.5505	0.677162
$922$	−12.6061	−0.415161
$923$	−10.2929	−0.338793
$924$	0	0
$925$	−7.00000	−0.230159
$926$	−0.494897	−0.0162633
$927$	−4.00000	−0.131377
$928$	0	0
$929$	46.2929	1.51882	0.759410	−	0.650613i	$-0.225488\pi$
$929$	46.2929	1.51882	0.759410	+	0.650613i	$0.225488\pi$
$930$	48.0000	1.57398
$931$	4.44949	0.145826
$932$	−31.5959	−1.03496
$933$	−17.1464	−0.561349
$934$	77.3939	2.53241
$935$	0	0
$936$	−14.2020	−0.464208
$937$	−13.3031	−0.434592	−0.217296	−	0.976106i	$-0.569724\pi$
$937$	−13.3031	−0.434592	−0.217296	+	0.976106i	$0.569724\pi$
$938$	−25.1010	−0.819577
$939$	25.4495	0.830513
$940$	−117.576	−3.83489
$941$	34.8434	1.13586	0.567931	−	0.823076i	$-0.307744\pi$
$941$	34.8434	1.13586	0.567931	+	0.823076i	$0.307744\pi$
$942$	−3.79796	−0.123744
$943$	32.6969	1.06476
$944$	−9.79796	−0.318896
$945$	2.44949	0.0796819
$946$	0	0
$947$	45.9898	1.49447	0.747234	−	0.664561i	$-0.231382\pi$
$947$	45.9898	1.49447	0.747234	+	0.664561i	$0.231382\pi$
$948$	15.5959	0.506532
$949$	−9.70714	−0.315107
$950$	10.8990	0.353610
$951$	−19.1010	−0.619393
$952$	−26.6969	−0.865253
$953$	−1.34847	−0.0436812	−0.0218406	−	0.999761i	$-0.506953\pi$
$953$	−1.34847	−0.0436812	−0.0218406	+	0.999761i	$0.506953\pi$
$954$	4.65153	0.150599
$955$	−32.6969	−1.05805
$956$	12.0000	0.388108
$957$	0	0
$958$	7.95459	0.257001
$959$	2.20204	0.0711076
$960$	−19.5959	−0.632456
$961$	33.0000	1.06452
$962$	49.7071	1.60262
$963$	7.34847	0.236801
$964$	20.9898	0.676036
$965$	13.5959	0.437668
$966$	14.6969	0.472866
$967$	10.9444	0.351948	0.175974	−	0.984395i	$-0.443693\pi$
$967$	10.9444	0.351948	0.175974	+	0.984395i	$0.443693\pi$
$968$	−53.8888	−1.73205
$969$	−24.2474	−0.778940
$970$	−14.0908	−0.452429
$971$	37.5959	1.20651	0.603255	−	0.797548i	$-0.293870\pi$
$971$	37.5959	1.20651	0.603255	+	0.797548i	$0.293870\pi$
$972$	4.00000	0.128300
$973$	−5.65153	−0.181180
$974$	−75.7980	−2.42872
$975$	−2.89898	−0.0928416
$976$	17.7980	0.569699
$977$	−57.7980	−1.84912	−0.924560	−	0.381036i	$-0.875567\pi$
$977$	−57.7980	−1.84912	−0.924560	+	0.381036i	$0.875567\pi$
$978$	41.6413	1.33154
$979$	0	0
$980$	9.79796	0.312984
$981$	−5.34847	−0.170763
$982$	−32.0908	−1.02406
$983$	11.9444	0.380967	0.190483	−	0.981690i	$-0.438994\pi$
$983$	11.9444	0.380967	0.190483	+	0.981690i	$0.438994\pi$
$984$	26.6969	0.851067
$985$	20.6969	0.659459
$986$	105.439	3.35787
$987$	−12.0000	−0.381964
$988$	−51.5959	−1.64148
$989$	12.0000	0.381578
$990$	0	0
$991$	10.4495	0.331939	0.165969	−	0.986131i	$-0.446925\pi$
$991$	10.4495	0.331939	0.165969	+	0.986131i	$0.446925\pi$
$992$	0	0
$993$	−18.4495	−0.585477
$994$	8.69694	0.275850
$995$	48.9898	1.55308
$996$	24.0000	0.760469
$997$	7.50510	0.237689	0.118844	−	0.992913i	$-0.462081\pi$
$997$	7.50510	0.237689	0.118844	+	0.992913i	$0.462081\pi$
$998$	14.8082	0.468744
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2667.2.a.h.1.2	✓	2
3.2	odd	2		8001.2.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.h.1.2	✓	2	1.1	even	1	trivial
8001.2.a.k.1.1		2	3.2	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:	\( N \)	\(=\)	\( 2667 = 3 \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2667.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{2} +1.00000 q^{3} +4.00000 q^{4} +2.44949 q^{5} +2.44949 q^{6} +1.00000 q^{7} +4.89898 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.44949 q^{2} +1.00000 q^{3} +4.00000 q^{4} +2.44949 q^{5} +2.44949 q^{6} +1.00000 q^{7} +4.89898 q^{8} +1.00000 q^{9} +6.00000 q^{10} +4.00000 q^{12} -2.89898 q^{13} +2.44949 q^{14} +2.44949 q^{15} +4.00000 q^{16} -5.44949 q^{17} +2.44949 q^{18} +4.44949 q^{19} +9.79796 q^{20} +1.00000 q^{21} +6.00000 q^{23} +4.89898 q^{24} +1.00000 q^{25} -7.10102 q^{26} +1.00000 q^{27} +4.00000 q^{28} -7.89898 q^{29} +6.00000 q^{30} +8.00000 q^{31} -13.3485 q^{34} +2.44949 q^{35} +4.00000 q^{36} -7.00000 q^{37} +10.8990 q^{38} -2.89898 q^{39} +12.0000 q^{40} +5.44949 q^{41} +2.44949 q^{42} +2.00000 q^{43} +2.44949 q^{45} +14.6969 q^{46} -12.0000 q^{47} +4.00000 q^{48} +1.00000 q^{49} +2.44949 q^{50} -5.44949 q^{51} -11.5959 q^{52} +1.89898 q^{53} +2.44949 q^{54} +4.89898 q^{56} +4.44949 q^{57} -19.3485 q^{58} -2.44949 q^{59} +9.79796 q^{60} +4.44949 q^{61} +19.5959 q^{62} +1.00000 q^{63} -8.00000 q^{64} -7.10102 q^{65} -10.2474 q^{67} -21.7980 q^{68} +6.00000 q^{69} +6.00000 q^{70} +3.55051 q^{71} +4.89898 q^{72} +3.34847 q^{73} -17.1464 q^{74} +1.00000 q^{75} +17.7980 q^{76} -7.10102 q^{78} +3.89898 q^{79} +9.79796 q^{80} +1.00000 q^{81} +13.3485 q^{82} +6.00000 q^{83} +4.00000 q^{84} -13.3485 q^{85} +4.89898 q^{86} -7.89898 q^{87} -9.79796 q^{89} +6.00000 q^{90} -2.89898 q^{91} +24.0000 q^{92} +8.00000 q^{93} -29.3939 q^{94} +10.8990 q^{95} -2.34847 q^{97} +2.44949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 8 q^{4} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 8 * q^4 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 8 q^{4} + 2 q^{7} + 2 q^{9} + 12 q^{10} + 8 q^{12} + 4 q^{13} + 8 q^{16} - 6 q^{17} + 4 q^{19} + 2 q^{21} + 12 q^{23} + 2 q^{25} - 24 q^{26} + 2 q^{27} + 8 q^{28} - 6 q^{29} + 12 q^{30} + 16 q^{31} - 12 q^{34} + 8 q^{36} - 14 q^{37} + 12 q^{38} + 4 q^{39} + 24 q^{40} + 6 q^{41} + 4 q^{43} - 24 q^{47} + 8 q^{48} + 2 q^{49} - 6 q^{51} + 16 q^{52} - 6 q^{53} + 4 q^{57} - 24 q^{58} + 4 q^{61} + 2 q^{63} - 16 q^{64} - 24 q^{65} + 4 q^{67} - 24 q^{68} + 12 q^{69} + 12 q^{70} + 12 q^{71} - 8 q^{73} + 2 q^{75} + 16 q^{76} - 24 q^{78} - 2 q^{79} + 2 q^{81} + 12 q^{82} + 12 q^{83} + 8 q^{84} - 12 q^{85} - 6 q^{87} + 12 q^{90} + 4 q^{91} + 48 q^{92} + 16 q^{93} + 12 q^{95} + 10 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 8 * q^4 + 2 * q^7 + 2 * q^9 + 12 * q^10 + 8 * q^12 + 4 * q^13 + 8 * q^16 - 6 * q^17 + 4 * q^19 + 2 * q^21 + 12 * q^23 + 2 * q^25 - 24 * q^26 + 2 * q^27 + 8 * q^28 - 6 * q^29 + 12 * q^30 + 16 * q^31 - 12 * q^34 + 8 * q^36 - 14 * q^37 + 12 * q^38 + 4 * q^39 + 24 * q^40 + 6 * q^41 + 4 * q^43 - 24 * q^47 + 8 * q^48 + 2 * q^49 - 6 * q^51 + 16 * q^52 - 6 * q^53 + 4 * q^57 - 24 * q^58 + 4 * q^61 + 2 * q^63 - 16 * q^64 - 24 * q^65 + 4 * q^67 - 24 * q^68 + 12 * q^69 + 12 * q^70 + 12 * q^71 - 8 * q^73 + 2 * q^75 + 16 * q^76 - 24 * q^78 - 2 * q^79 + 2 * q^81 + 12 * q^82 + 12 * q^83 + 8 * q^84 - 12 * q^85 - 6 * q^87 + 12 * q^90 + 4 * q^91 + 48 * q^92 + 16 * q^93 + 12 * q^95 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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