LMFDB - Embedded newform 2842.2.a.bc.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2842,2,Mod(1,2842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2842, 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("2842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2842 = 2 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2842.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:	$22.6934842544$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.373409792.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 48x^{2} - 50 $ x^6 - 13x^4 + 48x^2 - 50
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.73329$ of defining polynomial
Character	$\chi$	$=$	2842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.21239 q^{3} +1.00000 q^{4} +0.404393 q^{5} +2.21239 q^{6} +1.00000 q^{8} +1.89468 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.21239 q^{3} +1.00000 q^{4} +0.404393 q^{5} +2.21239 q^{6} +1.00000 q^{8} +1.89468 q^{9} +0.404393 q^{10} +6.04712 q^{11} +2.21239 q^{12} +0.404393 q^{13} +0.894675 q^{15} +1.00000 q^{16} +1.80800 q^{17} +1.89468 q^{18} -5.46659 q^{19} +0.404393 q^{20} +6.04712 q^{22} -2.94179 q^{23} +2.21239 q^{24} -4.83647 q^{25} +0.404393 q^{26} -2.44541 q^{27} +1.00000 q^{29} +0.894675 q^{30} +9.48698 q^{31} +1.00000 q^{32} +13.3786 q^{33} +1.80800 q^{34} +1.89468 q^{36} +11.1524 q^{37} -5.46659 q^{38} +0.894675 q^{39} +0.404393 q^{40} +9.12518 q^{41} -0.894675 q^{43} +6.04712 q^{44} +0.766193 q^{45} -2.94179 q^{46} -10.2958 q^{47} +2.21239 q^{48} -4.83647 q^{50} +4.00000 q^{51} +0.404393 q^{52} +1.10532 q^{53} -2.44541 q^{54} +2.44541 q^{55} -12.0942 q^{57} +1.00000 q^{58} +0.766193 q^{59} +0.894675 q^{60} -9.08258 q^{61} +9.48698 q^{62} +1.00000 q^{64} +0.163534 q^{65} +13.3786 q^{66} +1.78935 q^{67} +1.80800 q^{68} -6.50839 q^{69} -4.73114 q^{71} +1.89468 q^{72} -1.80800 q^{73} +11.1524 q^{74} -10.7002 q^{75} -5.46659 q^{76} +0.894675 q^{78} +3.10532 q^{79} +0.404393 q^{80} -11.0942 q^{81} +9.12518 q^{82} -13.7830 q^{83} +0.731142 q^{85} -0.894675 q^{86} +2.21239 q^{87} +6.04712 q^{88} -17.1660 q^{89} +0.766193 q^{90} -2.94179 q^{92} +20.9889 q^{93} -10.2958 q^{94} -2.21065 q^{95} +2.21239 q^{96} -8.65914 q^{97} +11.4573 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 12 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 12 * q^9	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 12 q^{9} - 2 q^{11} + 6 q^{15} + 6 q^{16} + 12 q^{18} - 2 q^{22} + 20 q^{23} + 8 q^{25} + 6 q^{29} + 6 q^{30} + 6 q^{32} + 12 q^{36} + 28 q^{37} + 6 q^{39} - 6 q^{43} - 2 q^{44} + 20 q^{46} + 8 q^{50} + 24 q^{51} + 6 q^{53} + 4 q^{57} + 6 q^{58} + 6 q^{60} + 6 q^{64} + 38 q^{65} + 12 q^{67} + 8 q^{71} + 12 q^{72} + 28 q^{74} + 6 q^{78} + 18 q^{79} + 10 q^{81} - 32 q^{85} - 6 q^{86} - 2 q^{88} + 20 q^{92} + 50 q^{93} - 12 q^{95} - 48 q^{99}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 12 * q^9 - 2 * q^11 + 6 * q^15 + 6 * q^16 + 12 * q^18 - 2 * q^22 + 20 * q^23 + 8 * q^25 + 6 * q^29 + 6 * q^30 + 6 * q^32 + 12 * q^36 + 28 * q^37 + 6 * q^39 - 6 * q^43 - 2 * q^44 + 20 * q^46 + 8 * q^50 + 24 * q^51 + 6 * q^53 + 4 * q^57 + 6 * q^58 + 6 * q^60 + 6 * q^64 + 38 * q^65 + 12 * q^67 + 8 * q^71 + 12 * q^72 + 28 * q^74 + 6 * q^78 + 18 * q^79 + 10 * q^81 - 32 * q^85 - 6 * q^86 - 2 * q^88 + 20 * q^92 + 50 * q^93 - 12 * q^95 - 48 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.21239	1.27732	0.638662	−	0.769487i	$-0.279488\pi$
$3$	2.21239	1.27732	0.638662	+	0.769487i	$0.279488\pi$
$4$	1.00000	0.500000
$5$	0.404393	0.180850	0.0904250	−	0.995903i	$-0.471177\pi$
$5$	0.404393	0.180850	0.0904250	+	0.995903i	$0.471177\pi$
$6$	2.21239	0.903205
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.89468	0.631558
$10$	0.404393	0.127880
$11$	6.04712	1.82327	0.911637	−	0.410996i	$-0.134819\pi$
$11$	6.04712	1.82327	0.911637	+	0.410996i	$0.134819\pi$
$12$	2.21239	0.638662
$13$	0.404393	0.112158	0.0560792	−	0.998426i	$-0.482140\pi$
$13$	0.404393	0.112158	0.0560792	+	0.998426i	$0.482140\pi$
$14$	0	0
$15$	0.894675	0.231004
$16$	1.00000	0.250000
$17$	1.80800	0.438504	0.219252	−	0.975668i	$-0.429638\pi$
$17$	1.80800	0.438504	0.219252	+	0.975668i	$0.429638\pi$
$18$	1.89468	0.446579
$19$	−5.46659	−1.25412	−0.627061	−	0.778970i	$-0.715742\pi$
$19$	−5.46659	−1.25412	−0.627061	+	0.778970i	$0.715742\pi$
$20$	0.404393	0.0904250
$21$	0	0
$22$	6.04712	1.28925
$23$	−2.94179	−0.613406	−0.306703	−	0.951805i	$-0.599226\pi$
$23$	−2.94179	−0.613406	−0.306703	+	0.951805i	$0.599226\pi$
$24$	2.21239	0.451602
$25$	−4.83647	−0.967293
$26$	0.404393	0.0793080
$27$	−2.44541	−0.470620
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0.894675	0.163345
$31$	9.48698	1.70391	0.851956	−	0.523614i	$-0.175417\pi$
$31$	9.48698	1.70391	0.851956	+	0.523614i	$0.175417\pi$
$32$	1.00000	0.176777
$33$	13.3786	2.32891
$34$	1.80800	0.310069
$35$	0	0
$36$	1.89468	0.315779
$37$	11.1524	1.83345	0.916725	−	0.399519i	$-0.130823\pi$
$37$	11.1524	1.83345	0.916725	+	0.399519i	$0.130823\pi$
$38$	−5.46659	−0.886798
$39$	0.894675	0.143263
$40$	0.404393	0.0639401
$41$	9.12518	1.42511	0.712557	−	0.701615i	$-0.247537\pi$
$41$	9.12518	1.42511	0.712557	+	0.701615i	$0.247537\pi$
$42$	0	0
$43$	−0.894675	−0.136437	−0.0682184	−	0.997670i	$-0.521731\pi$
$43$	−0.894675	−0.136437	−0.0682184	+	0.997670i	$0.521731\pi$
$44$	6.04712	0.911637
$45$	0.766193	0.114217
$46$	−2.94179	−0.433743
$47$	−10.2958	−1.50179	−0.750896	−	0.660421i	$-0.770378\pi$
$47$	−10.2958	−1.50179	−0.750896	+	0.660421i	$0.770378\pi$
$48$	2.21239	0.319331
$49$	0	0
$50$	−4.83647	−0.683980
$51$	4.00000	0.560112
$52$	0.404393	0.0560792
$53$	1.10532	0.151828	0.0759140	−	0.997114i	$-0.475813\pi$
$53$	1.10532	0.151828	0.0759140	+	0.997114i	$0.475813\pi$
$54$	−2.44541	−0.332778
$55$	2.44541	0.329739
$56$	0	0
$57$	−12.0942	−1.60192
$58$	1.00000	0.131306
$59$	0.766193	0.0997499	0.0498749	−	0.998755i	$-0.484118\pi$
$59$	0.766193	0.0997499	0.0498749	+	0.998755i	$0.484118\pi$
$60$	0.894675	0.115502
$61$	−9.08258	−1.16291	−0.581453	−	0.813580i	$-0.697516\pi$
$61$	−9.08258	−1.16291	−0.581453	+	0.813580i	$0.697516\pi$
$62$	9.48698	1.20485
$63$	0	0
$64$	1.00000	0.125000
$65$	0.163534	0.0202838
$66$	13.3786	1.64679
$67$	1.78935	0.218604	0.109302	−	0.994009i	$-0.465138\pi$
$67$	1.78935	0.218604	0.109302	+	0.994009i	$0.465138\pi$
$68$	1.80800	0.219252
$69$	−6.50839	−0.783518
$70$	0	0
$71$	−4.73114	−0.561483	−0.280742	−	0.959783i	$-0.590580\pi$
$71$	−4.73114	−0.561483	−0.280742	+	0.959783i	$0.590580\pi$
$72$	1.89468	0.223290
$73$	−1.80800	−0.211610	−0.105805	−	0.994387i	$-0.533742\pi$
$73$	−1.80800	−0.211610	−0.105805	+	0.994387i	$0.533742\pi$
$74$	11.1524	1.29644
$75$	−10.7002	−1.23555
$76$	−5.46659	−0.627061
$77$	0	0
$78$	0.894675	0.101302
$79$	3.10532	0.349376	0.174688	−	0.984624i	$-0.444108\pi$
$79$	3.10532	0.349376	0.174688	+	0.984624i	$0.444108\pi$
$80$	0.404393	0.0452125
$81$	−11.0942	−1.23269
$82$	9.12518	1.00771
$83$	−13.7830	−1.51288	−0.756439	−	0.654064i	$-0.773063\pi$
$83$	−13.7830	−1.51288	−0.756439	+	0.654064i	$0.773063\pi$
$84$	0	0
$85$	0.731142	0.0793035
$86$	−0.894675	−0.0964753
$87$	2.21239	0.237193
$88$	6.04712	0.644625
$89$	−17.1660	−1.81959	−0.909794	−	0.415060i	$-0.863760\pi$
$89$	−17.1660	−1.81959	−0.909794	+	0.415060i	$0.863760\pi$
$90$	0.766193	0.0807639
$91$	0	0
$92$	−2.94179	−0.306703
$93$	20.9889	2.17645
$94$	−10.2958	−1.06193
$95$	−2.21065	−0.226808
$96$	2.21239	0.225801
$97$	−8.65914	−0.879202	−0.439601	−	0.898193i	$-0.644880\pi$
$97$	−8.65914	−0.879202	−0.439601	+	0.898193i	$0.644880\pi$
$98$	0	0
$99$	11.4573	1.15150
$100$	−4.83647	−0.483647
$101$	−4.19176	−0.417096	−0.208548	−	0.978012i	$-0.566874\pi$
$101$	−4.19176	−0.417096	−0.208548	+	0.978012i	$0.566874\pi$
$102$	4.00000	0.396059
$103$	5.69961	0.561599	0.280799	−	0.959766i	$-0.409400\pi$
$103$	5.69961	0.561599	0.280799	+	0.959766i	$0.409400\pi$
$104$	0.404393	0.0396540
$105$	0	0
$106$	1.10532	0.107359
$107$	5.78935	0.559678	0.279839	−	0.960047i	$-0.409719\pi$
$107$	5.78935	0.559678	0.279839	+	0.960047i	$0.409719\pi$
$108$	−2.44541	−0.235310
$109$	4.68403	0.448648	0.224324	−	0.974515i	$-0.427983\pi$
$109$	4.68403	0.448648	0.224324	+	0.974515i	$0.427983\pi$
$110$	2.44541	0.233161
$111$	24.6736	2.34191
$112$	0	0
$113$	13.6729	1.28624	0.643121	−	0.765765i	$-0.277639\pi$
$113$	13.6729	1.28624	0.643121	+	0.765765i	$0.277639\pi$
$114$	−12.0942	−1.13273
$115$	−1.18964	−0.110934
$116$	1.00000	0.0928477
$117$	0.766193	0.0708346
$118$	0.766193	0.0705338
$119$	0	0
$120$	0.894675	0.0816723
$121$	25.5676	2.32433
$122$	−9.08258	−0.822299
$123$	20.1885	1.82033
$124$	9.48698	0.851956
$125$	−3.97780	−0.355785
$126$	0	0
$127$	1.88358	0.167141	0.0835704	−	0.996502i	$-0.473368\pi$
$127$	1.88358	0.167141	0.0835704	+	0.996502i	$0.473368\pi$
$128$	1.00000	0.0883883
$129$	−1.97937	−0.174274
$130$	0.163534	0.0143428
$131$	14.7822	1.29153	0.645763	−	0.763538i	$-0.276539\pi$
$131$	14.7822	1.29153	0.645763	+	0.763538i	$0.276539\pi$
$132$	13.3786	1.16446
$133$	0	0
$134$	1.78935	0.154576
$135$	−0.988907	−0.0851115
$136$	1.80800	0.155035
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	−6.50839	−0.554031
$139$	2.04102	0.173117	0.0865584	−	0.996247i	$-0.472413\pi$
$139$	2.04102	0.173117	0.0865584	+	0.996247i	$0.472413\pi$
$140$	0	0
$141$	−22.7783	−1.91828
$142$	−4.73114	−0.397029
$143$	2.44541	0.204496
$144$	1.89468	0.157890
$145$	0.404393	0.0335830
$146$	−1.80800	−0.149631
$147$	0	0
$148$	11.1524	0.916725
$149$	−5.19956	−0.425964	−0.212982	−	0.977056i	$-0.568318\pi$
$149$	−5.19956	−0.425964	−0.212982	+	0.977056i	$0.568318\pi$
$150$	−10.7002	−0.873664
$151$	−6.52049	−0.530630	−0.265315	−	0.964162i	$-0.585476\pi$
$151$	−6.52049	−0.530630	−0.265315	+	0.964162i	$0.585476\pi$
$152$	−5.46659	−0.443399
$153$	3.42557	0.276941
$154$	0	0
$155$	3.83647	0.308152
$156$	0.894675	0.0716314
$157$	−8.27380	−0.660321	−0.330161	−	0.943925i	$-0.607103\pi$
$157$	−8.27380	−0.660321	−0.330161	+	0.943925i	$0.607103\pi$
$158$	3.10532	0.247046
$159$	2.44541	0.193934
$160$	0.404393	0.0319701
$161$	0	0
$162$	−11.0942	−0.871645
$163$	19.9307	1.56109	0.780546	−	0.625098i	$-0.214941\pi$
$163$	19.9307	1.56109	0.780546	+	0.625098i	$0.214941\pi$
$164$	9.12518	0.712557
$165$	5.41021	0.421184
$166$	−13.7830	−1.06977
$167$	−19.7827	−1.53083	−0.765417	−	0.643534i	$-0.777467\pi$
$167$	−19.7827	−1.53083	−0.765417	+	0.643534i	$0.777467\pi$
$168$	0	0
$169$	−12.8365	−0.987420
$170$	0.731142	0.0560760
$171$	−10.3574	−0.792051
$172$	−0.894675	−0.0682184
$173$	2.38376	0.181234	0.0906171	−	0.995886i	$-0.471116\pi$
$173$	2.38376	0.181234	0.0906171	+	0.995886i	$0.471116\pi$
$174$	2.21239	0.167721
$175$	0	0
$176$	6.04712	0.455818
$177$	1.69512	0.127413
$178$	−17.1660	−1.28664
$179$	−19.7672	−1.47747	−0.738734	−	0.673998i	$-0.764576\pi$
$179$	−19.7672	−1.47747	−0.738734	+	0.673998i	$0.764576\pi$
$180$	0.766193	0.0571087
$181$	−22.1856	−1.64904	−0.824520	−	0.565833i	$-0.808555\pi$
$181$	−22.1856	−1.64904	−0.824520	+	0.565833i	$0.808555\pi$
$182$	0	0
$183$	−20.0942	−1.48541
$184$	−2.94179	−0.216872
$185$	4.50997	0.331579
$186$	20.9889	1.53898
$187$	10.9332	0.799513
$188$	−10.2958	−0.750896
$189$	0	0
$190$	−2.21065	−0.160377
$191$	10.3049	0.745635	0.372817	−	0.927905i	$-0.378392\pi$
$191$	10.3049	0.745635	0.372817	+	0.927905i	$0.378392\pi$
$192$	2.21239	0.159666
$193$	18.0942	1.30245	0.651226	−	0.758884i	$-0.274255\pi$
$193$	18.0942	1.30245	0.651226	+	0.758884i	$0.274255\pi$
$194$	−8.65914	−0.621690
$195$	0.361800	0.0259091
$196$	0	0
$197$	−8.30488	−0.591698	−0.295849	−	0.955235i	$-0.595603\pi$
$197$	−8.30488	−0.591698	−0.295849	+	0.955235i	$0.595603\pi$
$198$	11.4573	0.814236
$199$	12.9316	0.916697	0.458348	−	0.888773i	$-0.348441\pi$
$199$	12.9316	0.916697	0.458348	+	0.888773i	$0.348441\pi$
$200$	−4.83647	−0.341990
$201$	3.95874	0.279228
$202$	−4.19176	−0.294931
$203$	0	0
$204$	4.00000	0.280056
$205$	3.69016	0.257732
$206$	5.69961	0.397110
$207$	−5.57374	−0.387402
$208$	0.404393	0.0280396
$209$	−33.0571	−2.28661
$210$	0	0
$211$	−16.5676	−1.14056	−0.570281	−	0.821450i	$-0.693166\pi$
$211$	−16.5676	−1.14056	−0.570281	+	0.821450i	$0.693166\pi$
$212$	1.10532	0.0759140
$213$	−10.4671	−0.717196
$214$	5.78935	0.395752
$215$	−0.361800	−0.0246746
$216$	−2.44541	−0.166389
$217$	0	0
$218$	4.68403	0.317242
$219$	−4.00000	−0.270295
$220$	2.44541	0.164870
$221$	0.731142	0.0491819
$222$	24.6736	1.65598
$223$	1.99843	0.133824	0.0669122	−	0.997759i	$-0.478685\pi$
$223$	1.99843	0.133824	0.0669122	+	0.997759i	$0.478685\pi$
$224$	0	0
$225$	−9.16353	−0.610902
$226$	13.6729	0.909510
$227$	−14.1257	−0.937557	−0.468779	−	0.883316i	$-0.655306\pi$
$227$	−14.1257	−0.937557	−0.468779	+	0.883316i	$0.655306\pi$
$228$	−12.0942	−0.800960
$229$	15.9337	1.05293	0.526465	−	0.850197i	$-0.323517\pi$
$229$	15.9337	1.05293	0.526465	+	0.850197i	$0.323517\pi$
$230$	−1.18964	−0.0784425
$231$	0	0
$232$	1.00000	0.0656532
$233$	−10.9889	−0.719907	−0.359954	−	0.932970i	$-0.617207\pi$
$233$	−10.9889	−0.719907	−0.359954	+	0.932970i	$0.617207\pi$
$234$	0.766193	0.0500876
$235$	−4.16353	−0.271599
$236$	0.766193	0.0498749
$237$	6.87019	0.446267
$238$	0	0
$239$	13.8836	0.898054	0.449027	−	0.893518i	$-0.351771\pi$
$239$	13.8836	0.898054	0.449027	+	0.893518i	$0.351771\pi$
$240$	0.894675	0.0577510
$241$	−4.06298	−0.261720	−0.130860	−	0.991401i	$-0.541774\pi$
$241$	−4.06298	−0.261720	−0.130860	+	0.991401i	$0.541774\pi$
$242$	25.5676	1.64355
$243$	−17.2085	−1.10393
$244$	−9.08258	−0.581453
$245$	0	0
$246$	20.1885	1.28717
$247$	−2.21065	−0.140660
$248$	9.48698	0.602424
$249$	−30.4933	−1.93244
$250$	−3.97780	−0.251578
$251$	5.10479	0.322211	0.161106	−	0.986937i	$-0.448494\pi$
$251$	5.10479	0.322211	0.161106	+	0.986937i	$0.448494\pi$
$252$	0	0
$253$	−17.7894	−1.11841
$254$	1.88358	0.118186
$255$	1.61757	0.101296
$256$	1.00000	0.0625000
$257$	13.7213	0.855913	0.427957	−	0.903799i	$-0.359234\pi$
$257$	13.7213	0.855913	0.427957	+	0.903799i	$0.359234\pi$
$258$	−1.97937	−0.123230
$259$	0	0
$260$	0.163534	0.0101419
$261$	1.89468	0.117277
$262$	14.7822	0.913247
$263$	4.89468	0.301819	0.150909	−	0.988548i	$-0.451780\pi$
$263$	4.89468	0.301819	0.150909	+	0.988548i	$0.451780\pi$
$264$	13.3786	0.823395
$265$	0.446985	0.0274581
$266$	0	0
$267$	−37.9778	−2.32420
$268$	1.78935	0.109302
$269$	−19.2922	−1.17626	−0.588132	−	0.808765i	$-0.700136\pi$
$269$	−19.2922	−1.17626	−0.588132	+	0.808765i	$0.700136\pi$
$270$	−0.988907	−0.0601829
$271$	0.0861885	0.00523558	0.00261779	−	0.999997i	$-0.499167\pi$
$271$	0.0861885	0.00523558	0.00261779	+	0.999997i	$0.499167\pi$
$272$	1.80800	0.109626
$273$	0	0
$274$	−2.00000	−0.120824
$275$	−29.2467	−1.76364
$276$	−6.50839	−0.391759
$277$	23.8836	1.43503	0.717513	−	0.696545i	$-0.245281\pi$
$277$	23.8836	1.43503	0.717513	+	0.696545i	$0.245281\pi$
$278$	2.04102	0.122412
$279$	17.9747	1.07612
$280$	0	0
$281$	−0.0471157	−0.00281069	−0.00140534	−	0.999999i	$-0.500447\pi$
$281$	−0.0471157	−0.00281069	−0.00140534	+	0.999999i	$0.500447\pi$
$282$	−22.7783	−1.35643
$283$	8.89216	0.528584	0.264292	−	0.964443i	$-0.414862\pi$
$283$	8.89216	0.528584	0.264292	+	0.964443i	$0.414862\pi$
$284$	−4.73114	−0.280742
$285$	−4.89082	−0.289707
$286$	2.44541	0.144600
$287$	0	0
$288$	1.89468	0.111645
$289$	−13.7311	−0.807714
$290$	0.404393	0.0237468
$291$	−19.1574	−1.12303
$292$	−1.80800	−0.105805
$293$	22.0142	1.28608	0.643041	−	0.765832i	$-0.277672\pi$
$293$	22.0142	1.28608	0.643041	+	0.765832i	$0.277672\pi$
$294$	0	0
$295$	0.309843	0.0180398
$296$	11.1524	0.648222
$297$	−14.7877	−0.858068
$298$	−5.19956	−0.301202
$299$	−1.18964	−0.0687986
$300$	−10.7002	−0.617774
$301$	0	0
$302$	−6.52049	−0.375212
$303$	−9.27382	−0.532767
$304$	−5.46659	−0.313530
$305$	−3.67293	−0.210312
$306$	3.42557	0.195827
$307$	−9.44438	−0.539020	−0.269510	−	0.962998i	$-0.586862\pi$
$307$	−9.44438	−0.539020	−0.269510	+	0.962998i	$0.586862\pi$
$308$	0	0
$309$	12.6098	0.717344
$310$	3.83647	0.217897
$311$	22.3143	1.26533	0.632665	−	0.774426i	$-0.281961\pi$
$311$	22.3143	1.26533	0.632665	+	0.774426i	$0.281961\pi$
$312$	0.894675	0.0506510
$313$	−26.6529	−1.50651	−0.753256	−	0.657727i	$-0.771518\pi$
$313$	−26.6529	−1.50651	−0.753256	+	0.657727i	$0.771518\pi$
$314$	−8.27380	−0.466917
$315$	0	0
$316$	3.10532	0.174688
$317$	29.7672	1.67189	0.835945	−	0.548813i	$-0.184920\pi$
$317$	29.7672	1.67189	0.835945	+	0.548813i	$0.184920\pi$
$318$	2.44541	0.137132
$319$	6.04712	0.338573
$320$	0.404393	0.0226063
$321$	12.8083	0.714890
$322$	0	0
$323$	−9.88358	−0.549937
$324$	−11.0942	−0.616346
$325$	−1.95583	−0.108490
$326$	19.9307	1.10386
$327$	10.3629	0.573070
$328$	9.12518	0.503854
$329$	0	0
$330$	5.41021	0.297822
$331$	−27.1996	−1.49502	−0.747511	−	0.664249i	$-0.768751\pi$
$331$	−27.1996	−1.49502	−0.747511	+	0.664249i	$0.768751\pi$
$332$	−13.7830	−0.756439
$333$	21.1303	1.15793
$334$	−19.7827	−1.08246
$335$	0.723601	0.0395345
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	−12.8365	−0.698212
$339$	30.2499	1.64295
$340$	0.731142	0.0396517
$341$	57.3689	3.10670
$342$	−10.3574	−0.560065
$343$	0	0
$344$	−0.894675	−0.0482377
$345$	−2.63195	−0.141699
$346$	2.38376	0.128152
$347$	−29.9778	−1.60929	−0.804647	−	0.593754i	$-0.797645\pi$
$347$	−29.9778	−1.60929	−0.804647	+	0.593754i	$0.797645\pi$
$348$	2.21239	0.118597
$349$	−30.2263	−1.61798	−0.808989	−	0.587823i	$-0.799985\pi$
$349$	−30.2263	−1.61798	−0.808989	+	0.587823i	$0.799985\pi$
$350$	0	0
$351$	−0.988907	−0.0527839
$352$	6.04712	0.322312
$353$	−19.3167	−1.02812	−0.514062	−	0.857753i	$-0.671860\pi$
$353$	−19.3167	−1.02812	−0.514062	+	0.857753i	$0.671860\pi$
$354$	1.69512	0.0900946
$355$	−1.91324	−0.101544
$356$	−17.1660	−0.909794
$357$	0	0
$358$	−19.7672	−1.04473
$359$	−12.4734	−0.658320	−0.329160	−	0.944274i	$-0.606765\pi$
$359$	−12.4734	−0.658320	−0.329160	+	0.944274i	$0.606765\pi$
$360$	0.766193	0.0403819
$361$	10.8836	0.572820
$362$	−22.1856	−1.16605
$363$	56.5656	2.96892
$364$	0	0
$365$	−0.731142	−0.0382697
$366$	−20.0942	−1.05034
$367$	−22.8656	−1.19357	−0.596786	−	0.802400i	$-0.703556\pi$
$367$	−22.8656	−1.19357	−0.596786	+	0.802400i	$0.703556\pi$
$368$	−2.94179	−0.153351
$369$	17.2892	0.900042
$370$	4.50997	0.234462
$371$	0	0
$372$	20.9889	1.08822
$373$	1.52662	0.0790456	0.0395228	−	0.999219i	$-0.487416\pi$
$373$	1.52662	0.0790456	0.0395228	+	0.999219i	$0.487416\pi$
$374$	10.9332	0.565341
$375$	−8.80044	−0.454453
$376$	−10.2958	−0.530963
$377$	0.404393	0.0208273
$378$	0	0
$379$	−30.3049	−1.55666	−0.778329	−	0.627857i	$-0.783932\pi$
$379$	−30.3049	−1.55666	−0.778329	+	0.627857i	$0.783932\pi$
$380$	−2.21065	−0.113404
$381$	4.16722	0.213493
$382$	10.3049	0.527244
$383$	−7.65993	−0.391404	−0.195702	−	0.980663i	$-0.562699\pi$
$383$	−7.65993	−0.391404	−0.195702	+	0.980663i	$0.562699\pi$
$384$	2.21239	0.112901
$385$	0	0
$386$	18.0942	0.920972
$387$	−1.69512	−0.0861678
$388$	−8.65914	−0.439601
$389$	26.8254	1.36010	0.680050	−	0.733166i	$-0.261958\pi$
$389$	26.8254	1.36010	0.680050	+	0.733166i	$0.261958\pi$
$390$	0.361800	0.0183205
$391$	−5.31875	−0.268981
$392$	0	0
$393$	32.7040	1.64970
$394$	−8.30488	−0.418394
$395$	1.25577	0.0631847
$396$	11.4573	0.575752
$397$	18.5696	0.931980	0.465990	−	0.884790i	$-0.345698\pi$
$397$	18.5696	0.931980	0.465990	+	0.884790i	$0.345698\pi$
$398$	12.9316	0.648203
$399$	0	0
$400$	−4.83647	−0.241823
$401$	−6.89468	−0.344304	−0.172152	−	0.985070i	$-0.555072\pi$
$401$	−6.89468	−0.344304	−0.172152	+	0.985070i	$0.555072\pi$
$402$	3.95874	0.197444
$403$	3.83647	0.191108
$404$	−4.19176	−0.208548
$405$	−4.48643	−0.222932
$406$	0	0
$407$	67.4401	3.34288
$408$	4.00000	0.198030
$409$	−9.59122	−0.474255	−0.237128	−	0.971479i	$-0.576206\pi$
$409$	−9.59122	−0.474255	−0.237128	+	0.971479i	$0.576206\pi$
$410$	3.69016	0.182244
$411$	−4.42478	−0.218258
$412$	5.69961	0.280799
$413$	0	0
$414$	−5.57374	−0.273934
$415$	−5.57374	−0.273604
$416$	0.404393	0.0198270
$417$	4.51553	0.221126
$418$	−33.0571	−1.61688
$419$	28.3322	1.38412	0.692058	−	0.721842i	$-0.256704\pi$
$419$	28.3322	1.38412	0.692058	+	0.721842i	$0.256704\pi$
$420$	0	0
$421$	−7.57374	−0.369121	−0.184561	−	0.982821i	$-0.559086\pi$
$421$	−7.57374	−0.369121	−0.184561	+	0.982821i	$0.559086\pi$
$422$	−16.5676	−0.806499
$423$	−19.5071	−0.948469
$424$	1.10532	0.0536793
$425$	−8.74432	−0.424162
$426$	−10.4671	−0.507134
$427$	0	0
$428$	5.78935	0.279839
$429$	5.41021	0.261207
$430$	−0.361800	−0.0174476
$431$	−12.4213	−0.598313	−0.299156	−	0.954204i	$-0.596705\pi$
$431$	−12.4213	−0.598313	−0.299156	+	0.954204i	$0.596705\pi$
$432$	−2.44541	−0.117655
$433$	34.9503	1.67960	0.839801	−	0.542894i	$-0.182671\pi$
$433$	34.9503	1.67960	0.839801	+	0.542894i	$0.182671\pi$
$434$	0	0
$435$	0.894675	0.0428964
$436$	4.68403	0.224324
$437$	16.0816	0.769285
$438$	−4.00000	−0.191127
$439$	−28.3747	−1.35425	−0.677126	−	0.735867i	$-0.736775\pi$
$439$	−28.3747	−1.35425	−0.677126	+	0.735867i	$0.736775\pi$
$440$	2.44541	0.116580
$441$	0	0
$442$	0.731142	0.0347769
$443$	1.15244	0.0547541	0.0273770	−	0.999625i	$-0.491285\pi$
$443$	1.15244	0.0547541	0.0273770	+	0.999625i	$0.491285\pi$
$444$	24.6736	1.17096
$445$	−6.94179	−0.329072
$446$	1.99843	0.0946282
$447$	−11.5035	−0.544095
$448$	0	0
$449$	−6.51553	−0.307487	−0.153743	−	0.988111i	$-0.549133\pi$
$449$	−6.51553	−0.307487	−0.153743	+	0.988111i	$0.549133\pi$
$450$	−9.16353	−0.431973
$451$	55.1810	2.59837
$452$	13.6729	0.643121
$453$	−14.4259	−0.677787
$454$	−14.1257	−0.662953
$455$	0	0
$456$	−12.0942	−0.566364
$457$	−26.9196	−1.25925	−0.629623	−	0.776901i	$-0.716791\pi$
$457$	−26.9196	−1.25925	−0.629623	+	0.776901i	$0.716791\pi$
$458$	15.9337	0.744534
$459$	−4.42130	−0.206369
$460$	−1.18964	−0.0554672
$461$	−7.89295	−0.367611	−0.183806	−	0.982963i	$-0.558842\pi$
$461$	−7.89295	−0.367611	−0.183806	+	0.982963i	$0.558842\pi$
$462$	0	0
$463$	6.84756	0.318233	0.159117	−	0.987260i	$-0.449135\pi$
$463$	6.84756	0.318233	0.159117	+	0.987260i	$0.449135\pi$
$464$	1.00000	0.0464238
$465$	8.48776	0.393611
$466$	−10.9889	−0.509051
$467$	−31.6535	−1.46475	−0.732374	−	0.680902i	$-0.761588\pi$
$467$	−31.6535	−1.46475	−0.732374	+	0.680902i	$0.761588\pi$
$468$	0.766193	0.0354173
$469$	0	0
$470$	−4.16353	−0.192049
$471$	−18.3049	−0.843444
$472$	0.766193	0.0352669
$473$	−5.41021	−0.248762
$474$	6.87019	0.315558
$475$	26.4390	1.21310
$476$	0	0
$477$	2.09423	0.0958883
$478$	13.8836	0.635020
$479$	34.4181	1.57260	0.786302	−	0.617843i	$-0.211993\pi$
$479$	34.4181	1.57260	0.786302	+	0.617843i	$0.211993\pi$
$480$	0.894675	0.0408362
$481$	4.50997	0.205637
$482$	−4.06298	−0.185064
$483$	0	0
$484$	25.5676	1.16216
$485$	−3.50169	−0.159004
$486$	−17.2085	−0.780596
$487$	−27.7672	−1.25825	−0.629125	−	0.777304i	$-0.716587\pi$
$487$	−27.7672	−1.25825	−0.629125	+	0.777304i	$0.716587\pi$
$488$	−9.08258	−0.411149
$489$	44.0945	1.99402
$490$	0	0
$491$	25.4102	1.14675	0.573373	−	0.819294i	$-0.305635\pi$
$491$	25.4102	1.14675	0.573373	+	0.819294i	$0.305635\pi$
$492$	20.1885	0.910166
$493$	1.80800	0.0814282
$494$	−2.21065	−0.0994618
$495$	4.63326	0.208250
$496$	9.48698	0.425978
$497$	0	0
$498$	−30.4933	−1.36644
$499$	−12.5155	−0.560272	−0.280136	−	0.959960i	$-0.590380\pi$
$499$	−12.5155	−0.560272	−0.280136	+	0.959960i	$0.590380\pi$
$500$	−3.97780	−0.177892
$501$	−43.7672	−1.95537
$502$	5.10479	0.227838
$503$	−19.2305	−0.857446	−0.428723	−	0.903436i	$-0.641036\pi$
$503$	−19.2305	−0.857446	−0.428723	+	0.903436i	$0.641036\pi$
$504$	0	0
$505$	−1.69512	−0.0754318
$506$	−17.7894	−0.790833
$507$	−28.3993	−1.26126
$508$	1.88358	0.0835704
$509$	−16.2284	−0.719311	−0.359655	−	0.933085i	$-0.617106\pi$
$509$	−16.2284	−0.719311	−0.359655	+	0.933085i	$0.617106\pi$
$510$	1.61757	0.0716273
$511$	0	0
$512$	1.00000	0.0441942
$513$	13.3681	0.590214
$514$	13.7213	0.605222
$515$	2.30488	0.101565
$516$	−1.97937	−0.0871370
$517$	−62.2597	−2.73818
$518$	0	0
$519$	5.27382	0.231495
$520$	0.163534	0.00717142
$521$	−30.2689	−1.32611	−0.663053	−	0.748572i	$-0.730740\pi$
$521$	−30.2689	−1.32611	−0.663053	+	0.748572i	$0.730740\pi$
$522$	1.89468	0.0829277
$523$	1.57498	0.0688690	0.0344345	−	0.999407i	$-0.489037\pi$
$523$	1.57498	0.0688690	0.0344345	+	0.999407i	$0.489037\pi$
$524$	14.7822	0.645763
$525$	0	0
$526$	4.89468	0.213418
$527$	17.1524	0.747172
$528$	13.3786	0.582228
$529$	−14.3459	−0.623733
$530$	0.446985	0.0194158
$531$	1.45169	0.0629979
$532$	0	0
$533$	3.69016	0.159838
$534$	−37.9778	−1.64346
$535$	2.34117	0.101218
$536$	1.78935	0.0772882
$537$	−43.7327	−1.88721
$538$	−19.2922	−0.831744
$539$	0	0
$540$	−0.988907	−0.0425558
$541$	32.2106	1.38484	0.692422	−	0.721493i	$-0.256544\pi$
$541$	32.2106	1.38484	0.692422	+	0.721493i	$0.256544\pi$
$542$	0.0861885	0.00370211
$543$	−49.0831	−2.10636
$544$	1.80800	0.0775173
$545$	1.89419	0.0811381
$546$	0	0
$547$	−12.3271	−0.527067	−0.263534	−	0.964650i	$-0.584888\pi$
$547$	−12.3271	−0.527067	−0.263534	+	0.964650i	$0.584888\pi$
$548$	−2.00000	−0.0854358
$549$	−17.2085	−0.734443
$550$	−29.2467	−1.24708
$551$	−5.46659	−0.232884
$552$	−6.50839	−0.277016
$553$	0	0
$554$	23.8836	1.01472
$555$	9.97781	0.423535
$556$	2.04102	0.0865584
$557$	27.8836	1.18147	0.590733	−	0.806867i	$-0.298839\pi$
$557$	27.8836	1.18147	0.590733	+	0.806867i	$0.298839\pi$
$558$	17.9747	0.760932
$559$	−0.361800	−0.0153025
$560$	0	0
$561$	24.1885	1.02124
$562$	−0.0471157	−0.00198746
$563$	25.3536	1.06853	0.534263	−	0.845318i	$-0.320589\pi$
$563$	25.3536	1.06853	0.534263	+	0.845318i	$0.320589\pi$
$564$	−22.7783	−0.959138
$565$	5.52924	0.232617
$566$	8.89216	0.373765
$567$	0	0
$568$	−4.73114	−0.198514
$569$	−30.1885	−1.26557	−0.632783	−	0.774329i	$-0.718088\pi$
$569$	−30.1885	−1.26557	−0.632783	+	0.774329i	$0.718088\pi$
$570$	−4.89082	−0.204854
$571$	−40.5155	−1.69552	−0.847761	−	0.530378i	$-0.822050\pi$
$571$	−40.5155	−1.69552	−0.847761	+	0.530378i	$0.822050\pi$
$572$	2.44541	0.102248
$573$	22.7984	0.952418
$574$	0	0
$575$	14.2279	0.593343
$576$	1.89468	0.0789448
$577$	33.7987	1.40706	0.703530	−	0.710666i	$-0.251606\pi$
$577$	33.7987	1.40706	0.703530	+	0.710666i	$0.251606\pi$
$578$	−13.7311	−0.571140
$579$	40.0315	1.66365
$580$	0.404393	0.0167915
$581$	0	0
$582$	−19.1574	−0.794100
$583$	6.68403	0.276824
$584$	−1.80800	−0.0748155
$585$	0.309843	0.0128104
$586$	22.0142	0.909398
$587$	23.9926	0.990279	0.495139	−	0.868814i	$-0.335117\pi$
$587$	23.9926	0.990279	0.495139	+	0.868814i	$0.335117\pi$
$588$	0	0
$589$	−51.8614	−2.13691
$590$	0.309843	0.0127560
$591$	−18.3736	−0.755791
$592$	11.1524	0.458362
$593$	−34.3510	−1.41063	−0.705313	−	0.708896i	$-0.749193\pi$
$593$	−34.3510	−1.41063	−0.705313	+	0.708896i	$0.749193\pi$
$594$	−14.7877	−0.606746
$595$	0	0
$596$	−5.19956	−0.212982
$597$	28.6098	1.17092
$598$	−1.18964	−0.0486480
$599$	16.1463	0.659720	0.329860	−	0.944030i	$-0.392998\pi$
$599$	16.1463	0.659720	0.329860	+	0.944030i	$0.392998\pi$
$600$	−10.7002	−0.436832
$601$	16.3572	0.667223	0.333611	−	0.942711i	$-0.391733\pi$
$601$	16.3572	0.667223	0.333611	+	0.942711i	$0.391733\pi$
$602$	0	0
$603$	3.39024	0.138061
$604$	−6.52049	−0.265315
$605$	10.3394	0.420355
$606$	−9.27382	−0.376723
$607$	38.6705	1.56959	0.784794	−	0.619757i	$-0.212769\pi$
$607$	38.6705	1.56959	0.784794	+	0.619757i	$0.212769\pi$
$608$	−5.46659	−0.221699
$609$	0	0
$610$	−3.67293	−0.148713
$611$	−4.16353	−0.168439
$612$	3.42557	0.138470
$613$	9.01109	0.363955	0.181977	−	0.983303i	$-0.441750\pi$
$613$	9.01109	0.363955	0.181977	+	0.983303i	$0.441750\pi$
$614$	−9.44438	−0.381144
$615$	8.16407	0.329207
$616$	0	0
$617$	−7.97781	−0.321175	−0.160587	−	0.987022i	$-0.551339\pi$
$617$	−7.97781	−0.321175	−0.160587	+	0.987022i	$0.551339\pi$
$618$	12.6098	0.507239
$619$	−2.55514	−0.102700	−0.0513498	−	0.998681i	$-0.516352\pi$
$619$	−2.55514	−0.102700	−0.0513498	+	0.998681i	$0.516352\pi$
$620$	3.83647	0.154076
$621$	7.19389	0.288681
$622$	22.3143	0.894724
$623$	0	0
$624$	0.894675	0.0358157
$625$	22.5737	0.902950
$626$	−26.6529	−1.06527
$627$	−73.1352	−2.92074
$628$	−8.27380	−0.330161
$629$	20.1636	0.803975
$630$	0	0
$631$	−45.2294	−1.80056	−0.900278	−	0.435316i	$-0.856637\pi$
$631$	−45.2294	−1.80056	−0.900278	+	0.435316i	$0.856637\pi$
$632$	3.10532	0.123523
$633$	−36.6540	−1.45687
$634$	29.7672	1.18221
$635$	0.761707	0.0302274
$636$	2.44541	0.0969668
$637$	0	0
$638$	6.04712	0.239408
$639$	−8.96398	−0.354609
$640$	0.404393	0.0159850
$641$	3.97781	0.157114	0.0785571	−	0.996910i	$-0.474969\pi$
$641$	3.97781	0.157114	0.0785571	+	0.996910i	$0.474969\pi$
$642$	12.8083	0.505504
$643$	−1.23223	−0.0485945	−0.0242972	−	0.999705i	$-0.507735\pi$
$643$	−1.23223	−0.0485945	−0.0242972	+	0.999705i	$0.507735\pi$
$644$	0	0
$645$	−0.800444	−0.0315175
$646$	−9.88358	−0.388864
$647$	30.6307	1.20422	0.602109	−	0.798414i	$-0.294327\pi$
$647$	30.6307	1.20422	0.602109	+	0.798414i	$0.294327\pi$
$648$	−11.0942	−0.435823
$649$	4.63326	0.181871
$650$	−1.95583	−0.0767141
$651$	0	0
$652$	19.9307	0.780546
$653$	−28.8204	−1.12783	−0.563915	−	0.825833i	$-0.690705\pi$
$653$	−28.8204	−1.12783	−0.563915	+	0.825833i	$0.690705\pi$
$654$	10.3629	0.405221
$655$	5.97781	0.233572
$656$	9.12518	0.356278
$657$	−3.42557	−0.133644
$658$	0	0
$659$	18.1413	0.706687	0.353343	−	0.935494i	$-0.385045\pi$
$659$	18.1413	0.706687	0.353343	+	0.935494i	$0.385045\pi$
$660$	5.41021	0.210592
$661$	37.0965	1.44289	0.721444	−	0.692473i	$-0.243479\pi$
$661$	37.0965	1.44289	0.721444	+	0.692473i	$0.243479\pi$
$662$	−27.1996	−1.05714
$663$	1.61757	0.0628213
$664$	−13.7830	−0.534883
$665$	0	0
$666$	21.1303	0.818781
$667$	−2.94179	−0.113907
$668$	−19.7827	−0.765417
$669$	4.42130	0.170937
$670$	0.723601	0.0279551
$671$	−54.9234	−2.12030
$672$	0	0
$673$	1.93070	0.0744229	0.0372115	−	0.999307i	$-0.488152\pi$
$673$	1.93070	0.0744229	0.0372115	+	0.999307i	$0.488152\pi$
$674$	−6.00000	−0.231111
$675$	11.8271	0.455227
$676$	−12.8365	−0.493710
$677$	42.5205	1.63420	0.817098	−	0.576498i	$-0.195581\pi$
$677$	42.5205	1.63420	0.817098	+	0.576498i	$0.195581\pi$
$678$	30.2499	1.16174
$679$	0	0
$680$	0.731142	0.0280380
$681$	−31.2516	−1.19757
$682$	57.3689	2.19677
$683$	28.4213	1.08751	0.543755	−	0.839244i	$-0.317002\pi$
$683$	28.4213	1.08751	0.543755	+	0.839244i	$0.317002\pi$
$684$	−10.3574	−0.396025
$685$	−0.808786	−0.0309021
$686$	0	0
$687$	35.2516	1.34493
$688$	−0.894675	−0.0341092
$689$	0.446985	0.0170288
$690$	−2.63195	−0.100197
$691$	17.3990	0.661888	0.330944	−	0.943650i	$-0.392633\pi$
$691$	17.3990	0.661888	0.330944	+	0.943650i	$0.392633\pi$
$692$	2.38376	0.0906171
$693$	0	0
$694$	−29.9778	−1.13794
$695$	0.825373	0.0313082
$696$	2.21239	0.0838605
$697$	16.4983	0.624918
$698$	−30.2263	−1.14408
$699$	−24.3118	−0.919555
$700$	0	0
$701$	40.8725	1.54373	0.771866	−	0.635785i	$-0.219323\pi$
$701$	40.8725	1.54373	0.771866	+	0.635785i	$0.219323\pi$
$702$	−0.988907	−0.0373239
$703$	−60.9658	−2.29937
$704$	6.04712	0.227909
$705$	−9.21137	−0.346920
$706$	−19.3167	−0.726994
$707$	0	0
$708$	1.69512	0.0637065
$709$	−49.7151	−1.86709	−0.933545	−	0.358461i	$-0.883302\pi$
$709$	−49.7151	−1.86709	−0.933545	+	0.358461i	$0.883302\pi$
$710$	−1.91324	−0.0718026
$711$	5.88358	0.220652
$712$	−17.1660	−0.643321
$713$	−27.9087	−1.04519
$714$	0	0
$715$	0.988907	0.0369830
$716$	−19.7672	−0.738734
$717$	30.7159	1.14711
$718$	−12.4734	−0.465502
$719$	4.04393	0.150813	0.0754066	−	0.997153i	$-0.475975\pi$
$719$	4.04393	0.150813	0.0754066	+	0.997153i	$0.475975\pi$
$720$	0.766193	0.0285543
$721$	0	0
$722$	10.8836	0.405045
$723$	−8.98891	−0.334301
$724$	−22.1856	−0.824520
$725$	−4.83647	−0.179622
$726$	56.5656	2.09934
$727$	38.5663	1.43034	0.715172	−	0.698949i	$-0.246348\pi$
$727$	38.5663	1.43034	0.715172	+	0.698949i	$0.246348\pi$
$728$	0	0
$729$	−4.78935	−0.177383
$730$	−0.731142	−0.0270608
$731$	−1.61757	−0.0598280
$732$	−20.0942	−0.742704
$733$	48.1820	1.77964	0.889822	−	0.456308i	$-0.150828\pi$
$733$	48.1820	1.77964	0.889822	+	0.456308i	$0.150828\pi$
$734$	−22.8656	−0.843983
$735$	0	0
$736$	−2.94179	−0.108436
$737$	10.8204	0.398575
$738$	17.2892	0.636426
$739$	9.08314	0.334129	0.167064	−	0.985946i	$-0.446571\pi$
$739$	9.08314	0.334129	0.167064	+	0.985946i	$0.446571\pi$
$740$	4.50997	0.165790
$741$	−4.89082	−0.179669
$742$	0	0
$743$	38.3049	1.40527	0.702635	−	0.711551i	$-0.252007\pi$
$743$	38.3049	1.40527	0.702635	+	0.711551i	$0.252007\pi$
$744$	20.9889	0.769491
$745$	−2.10266	−0.0770356
$746$	1.52662	0.0558936
$747$	−26.1143	−0.955471
$748$	10.9332	0.399756
$749$	0	0
$750$	−8.80044	−0.321347
$751$	1.88358	0.0687329	0.0343664	−	0.999409i	$-0.489059\pi$
$751$	1.88358	0.0687329	0.0343664	+	0.999409i	$0.489059\pi$
$752$	−10.2958	−0.375448
$753$	11.2938	0.411568
$754$	0.404393	0.0147271
$755$	−2.63684	−0.0959645
$756$	0	0
$757$	28.2877	1.02813	0.514066	−	0.857751i	$-0.328139\pi$
$757$	28.2877	1.02813	0.514066	+	0.857751i	$0.328139\pi$
$758$	−30.3049	−1.10072
$759$	−39.3570	−1.42857
$760$	−2.21065	−0.0801887
$761$	−27.9468	−1.01307	−0.506536	−	0.862219i	$-0.669074\pi$
$761$	−27.9468	−1.01307	−0.506536	+	0.862219i	$0.669074\pi$
$762$	4.16722	0.150962
$763$	0	0
$764$	10.3049	0.372817
$765$	1.38528	0.0500848
$766$	−7.65993	−0.276764
$767$	0.309843	0.0111878
$768$	2.21239	0.0798328
$769$	−9.59122	−0.345868	−0.172934	−	0.984933i	$-0.555325\pi$
$769$	−9.59122	−0.345868	−0.172934	+	0.984933i	$0.555325\pi$
$770$	0	0
$771$	30.3570	1.09328
$772$	18.0942	0.651226
$773$	4.27695	0.153831	0.0769156	−	0.997038i	$-0.475493\pi$
$773$	4.27695	0.153831	0.0769156	+	0.997038i	$0.475493\pi$
$774$	−1.69512	−0.0609298
$775$	−45.8834	−1.64818
$776$	−8.65914	−0.310845
$777$	0	0
$778$	26.8254	0.961736
$779$	−49.8836	−1.78726
$780$	0.361800	0.0129545
$781$	−28.6098	−1.02374
$782$	−5.31875	−0.190198
$783$	−2.44541	−0.0873918
$784$	0	0
$785$	−3.34587	−0.119419
$786$	32.7040	1.16651
$787$	−25.0970	−0.894612	−0.447306	−	0.894381i	$-0.647616\pi$
$787$	−25.0970	−0.894612	−0.447306	+	0.894381i	$0.647616\pi$
$788$	−8.30488	−0.295849
$789$	10.8289	0.385520
$790$	1.25577	0.0446783
$791$	0	0
$792$	11.4573	0.407118
$793$	−3.67293	−0.130430
$794$	18.5696	0.659009
$795$	0.988907	0.0350729
$796$	12.9316	0.458348
$797$	−4.10658	−0.145462	−0.0727312	−	0.997352i	$-0.523172\pi$
$797$	−4.10658	−0.145462	−0.0727312	+	0.997352i	$0.523172\pi$
$798$	0	0
$799$	−18.6147	−0.658542
$800$	−4.83647	−0.170995
$801$	−32.5239	−1.14918
$802$	−6.89468	−0.243459
$803$	−10.9332	−0.385823
$804$	3.95874	0.139614
$805$	0	0
$806$	3.83647	0.135134
$807$	−42.6818	−1.50247
$808$	−4.19176	−0.147466
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	−4.48643	−0.157637
$811$	−26.5061	−0.930755	−0.465378	−	0.885112i	$-0.654081\pi$
$811$	−26.5061	−0.930755	−0.465378	+	0.885112i	$0.654081\pi$
$812$	0	0
$813$	0.190683	0.00668753
$814$	67.4401	2.36377
$815$	8.05983	0.282324
$816$	4.00000	0.140028
$817$	4.89082	0.171108
$818$	−9.59122	−0.335349
$819$	0	0
$820$	3.69016	0.128866
$821$	17.6209	0.614972	0.307486	−	0.951553i	$-0.400512\pi$
$821$	17.6209	0.614972	0.307486	+	0.951553i	$0.400512\pi$
$822$	−4.42478	−0.154332
$823$	−18.3049	−0.638068	−0.319034	−	0.947743i	$-0.603358\pi$
$823$	−18.3049	−0.638068	−0.319034	+	0.947743i	$0.603358\pi$
$824$	5.69961	0.198555
$825$	−64.7051	−2.25274
$826$	0	0
$827$	6.37418	0.221652	0.110826	−	0.993840i	$-0.464650\pi$
$827$	6.37418	0.221652	0.110826	+	0.993840i	$0.464650\pi$
$828$	−5.57374	−0.193701
$829$	6.27537	0.217953	0.108976	−	0.994044i	$-0.465243\pi$
$829$	6.27537	0.217953	0.108976	+	0.994044i	$0.465243\pi$
$830$	−5.57374	−0.193467
$831$	52.8398	1.83299
$832$	0.404393	0.0140198
$833$	0	0
$834$	4.51553	0.156360
$835$	−8.00000	−0.276851
$836$	−33.0571	−1.14330
$837$	−23.1996	−0.801894
$838$	28.3322	0.978718
$839$	−16.7190	−0.577203	−0.288601	−	0.957449i	$-0.593190\pi$
$839$	−16.7190	−0.577203	−0.288601	+	0.957449i	$0.593190\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−7.57374	−0.261008
$843$	−0.104238	−0.00359016
$844$	−16.5676	−0.570281
$845$	−5.19098	−0.178575
$846$	−19.5071	−0.670669
$847$	0	0
$848$	1.10532	0.0379570
$849$	19.6729	0.675173
$850$	−8.74432	−0.299928
$851$	−32.8081	−1.12465
$852$	−10.4671	−0.358598
$853$	39.7985	1.36267	0.681337	−	0.731970i	$-0.261399\pi$
$853$	39.7985	1.36267	0.681337	+	0.731970i	$0.261399\pi$
$854$	0	0
$855$	−4.18846	−0.143242
$856$	5.78935	0.197876
$857$	33.8849	1.15749	0.578744	−	0.815510i	$-0.303543\pi$
$857$	33.8849	1.15749	0.578744	+	0.815510i	$0.303543\pi$
$858$	5.41021	0.184701
$859$	−40.5882	−1.38485	−0.692426	−	0.721489i	$-0.743458\pi$
$859$	−40.5882	−1.38485	−0.692426	+	0.721489i	$0.743458\pi$
$860$	−0.361800	−0.0123373
$861$	0	0
$862$	−12.4213	−0.423071
$863$	−11.7844	−0.401145	−0.200573	−	0.979679i	$-0.564280\pi$
$863$	−11.7844	−0.401145	−0.200573	+	0.979679i	$0.564280\pi$
$864$	−2.44541	−0.0831946
$865$	0.963978	0.0327762
$866$	34.9503	1.18766
$867$	−30.3787	−1.03171
$868$	0	0
$869$	18.7783	0.637009
$870$	0.894675	0.0303323
$871$	0.723601	0.0245183
$872$	4.68403	0.158621
$873$	−16.4063	−0.555268
$874$	16.0816	0.543967
$875$	0	0
$876$	−4.00000	−0.135147
$877$	36.1241	1.21982	0.609912	−	0.792469i	$-0.291205\pi$
$877$	36.1241	1.21982	0.609912	+	0.792469i	$0.291205\pi$
$878$	−28.3747	−0.957601
$879$	48.7040	1.64274
$880$	2.44541	0.0824348
$881$	1.38007	0.0464956	0.0232478	−	0.999730i	$-0.492599\pi$
$881$	1.38007	0.0464956	0.0232478	+	0.999730i	$0.492599\pi$
$882$	0	0
$883$	46.5876	1.56780	0.783898	−	0.620889i	$-0.213228\pi$
$883$	46.5876	1.56780	0.783898	+	0.620889i	$0.213228\pi$
$884$	0.731142	0.0245910
$885$	0.685494	0.0230426
$886$	1.15244	0.0387170
$887$	13.1882	0.442815	0.221408	−	0.975181i	$-0.428935\pi$
$887$	13.1882	0.442815	0.221408	+	0.975181i	$0.428935\pi$
$888$	24.6736	0.827991
$889$	0	0
$890$	−6.94179	−0.232689
$891$	−67.0881	−2.24754
$892$	1.99843	0.0669122
$893$	56.2827	1.88343
$894$	−11.5035	−0.384733
$895$	−7.99370	−0.267200
$896$	0	0
$897$	−2.63195	−0.0878782
$898$	−6.51553	−0.217426
$899$	9.48698	0.316408
$900$	−9.16353	−0.305451
$901$	1.99843	0.0665772
$902$	55.1810	1.83733
$903$	0	0
$904$	13.6729	0.454755
$905$	−8.97168	−0.298229
$906$	−14.4259	−0.479268
$907$	−33.1524	−1.10081	−0.550404	−	0.834898i	$-0.685526\pi$
$907$	−33.1524	−1.10081	−0.550404	+	0.834898i	$0.685526\pi$
$908$	−14.1257	−0.468779
$909$	−7.94203	−0.263421
$910$	0	0
$911$	48.5676	1.60912	0.804558	−	0.593874i	$-0.202402\pi$
$911$	48.5676	1.60912	0.804558	+	0.593874i	$0.202402\pi$
$912$	−12.0942	−0.400480
$913$	−83.3473	−2.75839
$914$	−26.9196	−0.890421
$915$	−8.12596	−0.268636
$916$	15.9337	0.526465
$917$	0	0
$918$	−4.42130	−0.145925
$919$	−18.9418	−0.624832	−0.312416	−	0.949945i	$-0.601138\pi$
$919$	−18.9418	−0.624832	−0.312416	+	0.949945i	$0.601138\pi$
$920$	−1.18964	−0.0392213
$921$	−20.8947	−0.688503
$922$	−7.89295	−0.259940
$923$	−1.91324	−0.0629751
$924$	0	0
$925$	−53.9384	−1.77348
$926$	6.84756	0.225025
$927$	10.7989	0.354683
$928$	1.00000	0.0328266
$929$	27.1851	0.891914	0.445957	−	0.895054i	$-0.352863\pi$
$929$	27.1851	0.891914	0.445957	+	0.895054i	$0.352863\pi$
$930$	8.48776	0.278325
$931$	0	0
$932$	−10.9889	−0.359954
$933$	49.3681	1.61624
$934$	−31.6535	−1.03573
$935$	4.42130	0.144592
$936$	0.766193	0.0250438
$937$	54.1998	1.77063	0.885316	−	0.464990i	$-0.153942\pi$
$937$	54.1998	1.77063	0.885316	+	0.464990i	$0.153942\pi$
$938$	0	0
$939$	−58.9667	−1.92431
$940$	−4.16353	−0.135799
$941$	47.6298	1.55269	0.776344	−	0.630310i	$-0.217072\pi$
$941$	47.6298	1.55269	0.776344	+	0.630310i	$0.217072\pi$
$942$	−18.3049	−0.596405
$943$	−26.8444	−0.874173
$944$	0.766193	0.0249375
$945$	0	0
$946$	−5.41021	−0.175901
$947$	34.8725	1.13320	0.566602	−	0.823992i	$-0.308258\pi$
$947$	34.8725	1.13320	0.566602	+	0.823992i	$0.308258\pi$
$948$	6.87019	0.223134
$949$	−0.731142	−0.0237339
$950$	26.4390	0.857793
$951$	65.8566	2.13555
$952$	0	0
$953$	−30.8675	−0.999897	−0.499949	−	0.866055i	$-0.666648\pi$
$953$	−30.8675	−0.999897	−0.499949	+	0.866055i	$0.666648\pi$
$954$	2.09423	0.0678032
$955$	4.16722	0.134848
$956$	13.8836	0.449027
$957$	13.3786	0.432468
$958$	34.4181	1.11200
$959$	0	0
$960$	0.894675	0.0288755
$961$	59.0027	1.90331
$962$	4.50997	0.145407
$963$	10.9689	0.353469
$964$	−4.06298	−0.130860
$965$	7.31718	0.235548
$966$	0	0
$967$	−42.5454	−1.36817	−0.684084	−	0.729403i	$-0.739798\pi$
$967$	−42.5454	−1.36817	−0.684084	+	0.729403i	$0.739798\pi$
$968$	25.5676	0.821774
$969$	−21.8664	−0.702448
$970$	−3.50169	−0.112433
$971$	41.3309	1.32637	0.663186	−	0.748455i	$-0.269204\pi$
$971$	41.3309	1.32637	0.663186	+	0.748455i	$0.269204\pi$
$972$	−17.2085	−0.551964
$973$	0	0
$974$	−27.7672	−0.889717
$975$	−4.32707	−0.138577
$976$	−9.08258	−0.290726
$977$	6.16353	0.197189	0.0985945	−	0.995128i	$-0.468565\pi$
$977$	6.16353	0.197189	0.0985945	+	0.995128i	$0.468565\pi$
$978$	44.0945	1.40999
$979$	−103.805	−3.31761
$980$	0	0
$981$	8.87471	0.283348
$982$	25.4102	0.810872
$983$	−28.8037	−0.918695	−0.459347	−	0.888257i	$-0.651917\pi$
$983$	−28.8037	−0.918695	−0.459347	+	0.888257i	$0.651917\pi$
$984$	20.1885	0.643585
$985$	−3.35843	−0.107009
$986$	1.80800	0.0575784
$987$	0	0
$988$	−2.21065	−0.0703301
$989$	2.63195	0.0836911
$990$	4.63326	0.147255
$991$	38.3991	1.21979	0.609894	−	0.792483i	$-0.291212\pi$
$991$	38.3991	1.21979	0.609894	+	0.792483i	$0.291212\pi$
$992$	9.48698	0.301212
$993$	−60.1761	−1.90963
$994$	0	0
$995$	5.22945	0.165785
$996$	−30.4933	−0.966219
$997$	56.3461	1.78450	0.892249	−	0.451543i	$-0.149126\pi$
$997$	56.3461	1.78450	0.892249	+	0.451543i	$0.149126\pi$
$998$	−12.5155	−0.396172
$999$	−27.2723	−0.862857

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2842.2.a.bc.1.5	yes	6
7.6	odd	2	inner	2842.2.a.bc.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.bc.1.2	✓	6	7.6	odd	2	inner
2842.2.a.bc.1.5	yes	6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2842.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.21239 q^{3} +1.00000 q^{4} +0.404393 q^{5} +2.21239 q^{6} +1.00000 q^{8} +1.89468 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.21239 q^{3} +1.00000 q^{4} +0.404393 q^{5} +2.21239 q^{6} +1.00000 q^{8} +1.89468 q^{9} +0.404393 q^{10} +6.04712 q^{11} +2.21239 q^{12} +0.404393 q^{13} +0.894675 q^{15} +1.00000 q^{16} +1.80800 q^{17} +1.89468 q^{18} -5.46659 q^{19} +0.404393 q^{20} +6.04712 q^{22} -2.94179 q^{23} +2.21239 q^{24} -4.83647 q^{25} +0.404393 q^{26} -2.44541 q^{27} +1.00000 q^{29} +0.894675 q^{30} +9.48698 q^{31} +1.00000 q^{32} +13.3786 q^{33} +1.80800 q^{34} +1.89468 q^{36} +11.1524 q^{37} -5.46659 q^{38} +0.894675 q^{39} +0.404393 q^{40} +9.12518 q^{41} -0.894675 q^{43} +6.04712 q^{44} +0.766193 q^{45} -2.94179 q^{46} -10.2958 q^{47} +2.21239 q^{48} -4.83647 q^{50} +4.00000 q^{51} +0.404393 q^{52} +1.10532 q^{53} -2.44541 q^{54} +2.44541 q^{55} -12.0942 q^{57} +1.00000 q^{58} +0.766193 q^{59} +0.894675 q^{60} -9.08258 q^{61} +9.48698 q^{62} +1.00000 q^{64} +0.163534 q^{65} +13.3786 q^{66} +1.78935 q^{67} +1.80800 q^{68} -6.50839 q^{69} -4.73114 q^{71} +1.89468 q^{72} -1.80800 q^{73} +11.1524 q^{74} -10.7002 q^{75} -5.46659 q^{76} +0.894675 q^{78} +3.10532 q^{79} +0.404393 q^{80} -11.0942 q^{81} +9.12518 q^{82} -13.7830 q^{83} +0.731142 q^{85} -0.894675 q^{86} +2.21239 q^{87} +6.04712 q^{88} -17.1660 q^{89} +0.766193 q^{90} -2.94179 q^{92} +20.9889 q^{93} -10.2958 q^{94} -2.21065 q^{95} +2.21239 q^{96} -8.65914 q^{97} +11.4573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 12 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 12 * q^9	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 12 q^{9} - 2 q^{11} + 6 q^{15} + 6 q^{16} + 12 q^{18} - 2 q^{22} + 20 q^{23} + 8 q^{25} + 6 q^{29} + 6 q^{30} + 6 q^{32} + 12 q^{36} + 28 q^{37} + 6 q^{39} - 6 q^{43} - 2 q^{44} + 20 q^{46} + 8 q^{50} + 24 q^{51} + 6 q^{53} + 4 q^{57} + 6 q^{58} + 6 q^{60} + 6 q^{64} + 38 q^{65} + 12 q^{67} + 8 q^{71} + 12 q^{72} + 28 q^{74} + 6 q^{78} + 18 q^{79} + 10 q^{81} - 32 q^{85} - 6 q^{86} - 2 q^{88} + 20 q^{92} + 50 q^{93} - 12 q^{95} - 48 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 12 * q^9 - 2 * q^11 + 6 * q^15 + 6 * q^16 + 12 * q^18 - 2 * q^22 + 20 * q^23 + 8 * q^25 + 6 * q^29 + 6 * q^30 + 6 * q^32 + 12 * q^36 + 28 * q^37 + 6 * q^39 - 6 * q^43 - 2 * q^44 + 20 * q^46 + 8 * q^50 + 24 * q^51 + 6 * q^53 + 4 * q^57 + 6 * q^58 + 6 * q^60 + 6 * q^64 + 38 * q^65 + 12 * q^67 + 8 * q^71 + 12 * q^72 + 28 * q^74 + 6 * q^78 + 18 * q^79 + 10 * q^81 - 32 * q^85 - 6 * q^86 - 2 * q^88 + 20 * q^92 + 50 * q^93 - 12 * q^95 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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