LMFDB - Embedded newform 2842.2.a.bc.1.3

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			1.3
Root			$-1.33784$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.833497 q^{3} +1.00000 q^{4} +3.96556 q^{5} -0.833497 q^{6} +1.00000 q^{8} -2.30528 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.833497 q^{3} +1.00000 q^{4} +3.96556 q^{5} -0.833497 q^{6} +1.00000 q^{8} -2.30528 q^{9} +3.96556 q^{10} -1.11509 q^{11} -0.833497 q^{12} +3.96556 q^{13} -3.30528 q^{15} +1.00000 q^{16} -4.79906 q^{17} -2.30528 q^{18} -2.67568 q^{19} +3.96556 q^{20} -1.11509 q^{22} +8.42037 q^{23} -0.833497 q^{24} +10.7257 q^{25} +3.96556 q^{26} +4.42194 q^{27} +1.00000 q^{29} -3.30528 q^{30} -2.95687 q^{31} +1.00000 q^{32} +0.929421 q^{33} -4.79906 q^{34} -2.30528 q^{36} +8.19020 q^{37} -2.67568 q^{38} -3.30528 q^{39} +3.96556 q^{40} +10.1504 q^{41} +3.30528 q^{43} -1.11509 q^{44} -9.14173 q^{45} +8.42037 q^{46} -4.97424 q^{47} -0.833497 q^{48} +10.7257 q^{50} +4.00000 q^{51} +3.96556 q^{52} +5.30528 q^{53} +4.42194 q^{54} -4.42194 q^{55} +2.23017 q^{57} +1.00000 q^{58} -9.14173 q^{59} -3.30528 q^{60} +6.92243 q^{61} -2.95687 q^{62} +1.00000 q^{64} +15.7257 q^{65} +0.929421 q^{66} -6.61056 q^{67} -4.79906 q^{68} -7.01835 q^{69} +15.0309 q^{71} -2.30528 q^{72} +4.79906 q^{73} +8.19020 q^{74} -8.93980 q^{75} -2.67568 q^{76} -3.30528 q^{78} +7.30528 q^{79} +3.96556 q^{80} +3.23017 q^{81} +10.1504 q^{82} -4.89498 q^{83} -19.0309 q^{85} +3.30528 q^{86} -0.833497 q^{87} -1.11509 q^{88} +1.11469 q^{89} -9.14173 q^{90} +8.42037 q^{92} +2.46455 q^{93} -4.97424 q^{94} -10.6106 q^{95} -0.833497 q^{96} -17.3273 q^{97} +2.57059 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.833497	−0.481220	−0.240610	−	0.970622i	$-0.577347\pi$
$3$	−0.833497	−0.481220	−0.240610	+	0.970622i	$0.577347\pi$
$4$	1.00000	0.500000
$5$	3.96556	1.77345	0.886726	−	0.462296i	$-0.152974\pi$
$5$	3.96556	1.77345	0.886726	+	0.462296i	$0.152974\pi$
$6$	−0.833497	−0.340274
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.30528	−0.768427
$10$	3.96556	1.25402
$11$	−1.11509	−0.336211	−0.168106	−	0.985769i	$-0.553765\pi$
$11$	−1.11509	−0.336211	−0.168106	+	0.985769i	$0.553765\pi$
$12$	−0.833497	−0.240610
$13$	3.96556	1.09985	0.549924	−	0.835215i	$-0.314657\pi$
$13$	3.96556	1.09985	0.549924	+	0.835215i	$0.314657\pi$
$14$	0	0
$15$	−3.30528	−0.853420
$16$	1.00000	0.250000
$17$	−4.79906	−1.16394	−0.581971	−	0.813210i	$-0.697718\pi$
$17$	−4.79906	−1.16394	−0.581971	+	0.813210i	$0.697718\pi$
$18$	−2.30528	−0.543360
$19$	−2.67568	−0.613843	−0.306922	−	0.951735i	$-0.599299\pi$
$19$	−2.67568	−0.613843	−0.306922	+	0.951735i	$0.599299\pi$
$20$	3.96556	0.886726
$21$	0	0
$22$	−1.11509	−0.237737
$23$	8.42037	1.75577	0.877884	−	0.478873i	$-0.158955\pi$
$23$	8.42037	1.75577	0.877884	+	0.478873i	$0.158955\pi$
$24$	−0.833497	−0.170137
$25$	10.7257	2.14513
$26$	3.96556	0.777710
$27$	4.42194	0.851002
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−3.30528	−0.603459
$31$	−2.95687	−0.531070	−0.265535	−	0.964101i	$-0.585549\pi$
$31$	−2.95687	−0.531070	−0.265535	+	0.964101i	$0.585549\pi$
$32$	1.00000	0.176777
$33$	0.929421	0.161791
$34$	−4.79906	−0.823031
$35$	0	0
$36$	−2.30528	−0.384214
$37$	8.19020	1.34646	0.673230	−	0.739433i	$-0.264906\pi$
$37$	8.19020	1.34646	0.673230	+	0.739433i	$0.264906\pi$
$38$	−2.67568	−0.434053
$39$	−3.30528	−0.529269
$40$	3.96556	0.627010
$41$	10.1504	1.58523	0.792614	−	0.609723i	$-0.208719\pi$
$41$	10.1504	1.58523	0.792614	+	0.609723i	$0.208719\pi$
$42$	0	0
$43$	3.30528	0.504051	0.252025	−	0.967721i	$-0.418903\pi$
$43$	3.30528	0.504051	0.252025	+	0.967721i	$0.418903\pi$
$44$	−1.11509	−0.168106
$45$	−9.14173	−1.36277
$46$	8.42037	1.24152
$47$	−4.97424	−0.725568	−0.362784	−	0.931873i	$-0.618174\pi$
$47$	−4.97424	−0.725568	−0.362784	+	0.931873i	$0.618174\pi$
$48$	−0.833497	−0.120305
$49$	0	0
$50$	10.7257	1.51684
$51$	4.00000	0.560112
$52$	3.96556	0.549924
$53$	5.30528	0.728737	0.364368	−	0.931255i	$-0.381285\pi$
$53$	5.30528	0.728737	0.364368	+	0.931255i	$0.381285\pi$
$54$	4.42194	0.601750
$55$	−4.42194	−0.596254
$56$	0	0
$57$	2.23017	0.295394
$58$	1.00000	0.131306
$59$	−9.14173	−1.19015	−0.595076	−	0.803669i	$-0.702878\pi$
$59$	−9.14173	−1.19015	−0.595076	+	0.803669i	$0.702878\pi$
$60$	−3.30528	−0.426710
$61$	6.92243	0.886326	0.443163	−	0.896441i	$-0.353856\pi$
$61$	6.92243	0.886326	0.443163	+	0.896441i	$0.353856\pi$
$62$	−2.95687	−0.375523
$63$	0	0
$64$	1.00000	0.125000
$65$	15.7257	1.95053
$66$	0.929421	0.114404
$67$	−6.61056	−0.807609	−0.403804	−	0.914845i	$-0.632312\pi$
$67$	−6.61056	−0.807609	−0.403804	+	0.914845i	$0.632312\pi$
$68$	−4.79906	−0.581971
$69$	−7.01835	−0.844911
$70$	0	0
$71$	15.0309	1.78384	0.891922	−	0.452190i	$-0.149357\pi$
$71$	15.0309	1.78384	0.891922	+	0.452190i	$0.149357\pi$
$72$	−2.30528	−0.271680
$73$	4.79906	0.561687	0.280843	−	0.959754i	$-0.409386\pi$
$73$	4.79906	0.561687	0.280843	+	0.959754i	$0.409386\pi$
$74$	8.19020	0.952091
$75$	−8.93980	−1.03228
$76$	−2.67568	−0.306922
$77$	0	0
$78$	−3.30528	−0.374250
$79$	7.30528	0.821908	0.410954	−	0.911656i	$-0.365196\pi$
$79$	7.30528	0.821908	0.410954	+	0.911656i	$0.365196\pi$
$80$	3.96556	0.443363
$81$	3.23017	0.358908
$82$	10.1504	1.12093
$83$	−4.89498	−0.537294	−0.268647	−	0.963239i	$-0.586576\pi$
$83$	−4.89498	−0.537294	−0.268647	+	0.963239i	$0.586576\pi$
$84$	0	0
$85$	−19.0309	−2.06419
$86$	3.30528	0.356418
$87$	−0.833497	−0.0893603
$88$	−1.11509	−0.118869
$89$	1.11469	0.118157	0.0590785	−	0.998253i	$-0.481184\pi$
$89$	1.11469	0.118157	0.0590785	+	0.998253i	$0.481184\pi$
$90$	−9.14173	−0.963623
$91$	0	0
$92$	8.42037	0.877884
$93$	2.46455	0.255561
$94$	−4.97424	−0.513054
$95$	−10.6106	−1.08862
$96$	−0.833497	−0.0850685
$97$	−17.3273	−1.75932	−0.879660	−	0.475603i	$-0.842230\pi$
$97$	−17.3273	−1.75932	−0.879660	+	0.475603i	$0.842230\pi$
$98$	0	0
$99$	2.57059	0.258354
$100$	10.7257	1.07257
$101$	−1.92145	−0.191191	−0.0955955	−	0.995420i	$-0.530476\pi$
$101$	−1.92145	−0.191191	−0.0955955	+	0.995420i	$0.530476\pi$
$102$	4.00000	0.396059
$103$	−0.912761	−0.0899370	−0.0449685	−	0.998988i	$-0.514319\pi$
$103$	−0.912761	−0.0899370	−0.0449685	+	0.998988i	$0.514319\pi$
$104$	3.96556	0.388855
$105$	0	0
$106$	5.30528	0.515295
$107$	−2.61056	−0.252373	−0.126186	−	0.992007i	$-0.540274\pi$
$107$	−2.61056	−0.252373	−0.126186	+	0.992007i	$0.540274\pi$
$108$	4.42194	0.425501
$109$	−7.91585	−0.758201	−0.379100	−	0.925356i	$-0.623766\pi$
$109$	−7.91585	−0.758201	−0.379100	+	0.925356i	$0.623766\pi$
$110$	−4.42194	−0.421615
$111$	−6.82651	−0.647943
$112$	0	0
$113$	−17.4513	−1.64168	−0.820840	−	0.571158i	$-0.806494\pi$
$113$	−17.4513	−1.64168	−0.820840	+	0.571158i	$0.806494\pi$
$114$	2.23017	0.208875
$115$	33.3915	3.11377
$116$	1.00000	0.0928477
$117$	−9.14173	−0.845153
$118$	−9.14173	−0.841564
$119$	0	0
$120$	−3.30528	−0.301730
$121$	−9.75658	−0.886962
$122$	6.92243	0.626727
$123$	−8.46034	−0.762844
$124$	−2.95687	−0.265535
$125$	22.7054	2.03083
$126$	0	0
$127$	−20.8407	−1.84932	−0.924658	−	0.380798i	$-0.875649\pi$
$127$	−20.8407	−1.84932	−0.924658	+	0.380798i	$0.875649\pi$
$128$	1.00000	0.0883883
$129$	−2.75494	−0.242559
$130$	15.7257	1.37923
$131$	−7.83519	−0.684564	−0.342282	−	0.939597i	$-0.611200\pi$
$131$	−7.83519	−0.684564	−0.342282	+	0.939597i	$0.611200\pi$
$132$	0.929421	0.0808957
$133$	0	0
$134$	−6.61056	−0.571066
$135$	17.5355	1.50921
$136$	−4.79906	−0.411516
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	−7.01835	−0.597442
$139$	−8.38750	−0.711418	−0.355709	−	0.934597i	$-0.615761\pi$
$139$	−8.38750	−0.711418	−0.355709	+	0.934597i	$0.615761\pi$
$140$	0	0
$141$	4.14602	0.349158
$142$	15.0309	1.26137
$143$	−4.42194	−0.369781
$144$	−2.30528	−0.192107
$145$	3.96556	0.329322
$146$	4.79906	0.397173
$147$	0	0
$148$	8.19020	0.673230
$149$	4.92489	0.403463	0.201731	−	0.979441i	$-0.435343\pi$
$149$	4.92489	0.403463	0.201731	+	0.979441i	$0.435343\pi$
$150$	−8.93980	−0.729932
$151$	21.6415	1.76116	0.880580	−	0.473897i	$-0.157153\pi$
$151$	21.6415	1.76116	0.880580	+	0.473897i	$0.157153\pi$
$152$	−2.67568	−0.217026
$153$	11.0632	0.894405
$154$	0	0
$155$	−11.7257	−0.941827
$156$	−3.30528	−0.264634
$157$	14.8535	1.18544	0.592721	−	0.805408i	$-0.298054\pi$
$157$	14.8535	1.18544	0.592721	+	0.805408i	$0.298054\pi$
$158$	7.30528	0.581177
$159$	−4.42194	−0.350683
$160$	3.96556	0.313505
$161$	0	0
$162$	3.23017	0.253786
$163$	−9.95582	−0.779800	−0.389900	−	0.920857i	$-0.627490\pi$
$163$	−9.95582	−0.779800	−0.389900	+	0.920857i	$0.627490\pi$
$164$	10.1504	0.792614
$165$	3.68567	0.286929
$166$	−4.89498	−0.379924
$167$	−2.01737	−0.156109	−0.0780544	−	0.996949i	$-0.524871\pi$
$167$	−2.01737	−0.156109	−0.0780544	+	0.996949i	$0.524871\pi$
$168$	0	0
$169$	2.72565	0.209665
$170$	−19.0309	−1.45961
$171$	6.16820	0.471694
$172$	3.30528	0.252025
$173$	6.72050	0.510950	0.255475	−	0.966816i	$-0.417768\pi$
$173$	6.72050	0.510950	0.255475	+	0.966816i	$0.417768\pi$
$174$	−0.833497	−0.0631873
$175$	0	0
$176$	−1.11509	−0.0840528
$177$	7.61961	0.572725
$178$	1.11469	0.0835496
$179$	25.6815	1.91952	0.959762	−	0.280816i	$-0.0906050\pi$
$179$	25.6815	1.91952	0.959762	+	0.280816i	$0.0906050\pi$
$180$	−9.14173	−0.681384
$181$	19.4774	1.44774	0.723872	−	0.689934i	$-0.242360\pi$
$181$	19.4774	1.44774	0.723872	+	0.689934i	$0.242360\pi$
$182$	0	0
$183$	−5.76983	−0.426518
$184$	8.42037	0.620758
$185$	32.4787	2.38788
$186$	2.46455	0.180709
$187$	5.35136	0.391330
$188$	−4.97424	−0.362784
$189$	0	0
$190$	−10.6106	−0.769771
$191$	4.38039	0.316954	0.158477	−	0.987363i	$-0.449342\pi$
$191$	4.38039	0.316954	0.158477	+	0.987363i	$0.449342\pi$
$192$	−0.833497	−0.0601525
$193$	3.76983	0.271358	0.135679	−	0.990753i	$-0.456678\pi$
$193$	3.76983	0.271358	0.135679	+	0.990753i	$0.456678\pi$
$194$	−17.3273	−1.24403
$195$	−13.1073	−0.938632
$196$	0	0
$197$	−2.38039	−0.169596	−0.0847980	−	0.996398i	$-0.527024\pi$
$197$	−2.38039	−0.169596	−0.0847980	+	0.996398i	$0.527024\pi$
$198$	2.57059	0.182684
$199$	−20.1090	−1.42549	−0.712744	−	0.701424i	$-0.752548\pi$
$199$	−20.1090	−1.42549	−0.712744	+	0.701424i	$0.752548\pi$
$200$	10.7257	0.758418
$201$	5.50989	0.388638
$202$	−1.92145	−0.135193
$203$	0	0
$204$	4.00000	0.280056
$205$	40.2521	2.81133
$206$	−0.912761	−0.0635951
$207$	−19.4113	−1.34918
$208$	3.96556	0.274962
$209$	2.98361	0.206381
$210$	0	0
$211$	18.7566	1.29126	0.645628	−	0.763652i	$-0.276596\pi$
$211$	18.7566	1.29126	0.645628	+	0.763652i	$0.276596\pi$
$212$	5.30528	0.364368
$213$	−12.5282	−0.858421
$214$	−2.61056	−0.178455
$215$	13.1073	0.893910
$216$	4.42194	0.300875
$217$	0	0
$218$	−7.91585	−0.536129
$219$	−4.00000	−0.270295
$220$	−4.42194	−0.298127
$221$	−19.0309	−1.28016
$222$	−6.82651	−0.458165
$223$	−25.4603	−1.70495	−0.852475	−	0.522767i	$-0.824900\pi$
$223$	−25.4603	−1.70495	−0.852475	+	0.522767i	$0.824900\pi$
$224$	0	0
$225$	−24.7257	−1.64838
$226$	−17.4513	−1.16084
$227$	−20.0030	−1.32764	−0.663822	−	0.747890i	$-0.731067\pi$
$227$	−20.0030	−1.32764	−0.663822	+	0.747890i	$0.731067\pi$
$228$	2.23017	0.147697
$229$	15.2039	1.00470	0.502352	−	0.864663i	$-0.332468\pi$
$229$	15.2039	1.00470	0.502352	+	0.864663i	$0.332468\pi$
$230$	33.3915	2.20177
$231$	0	0
$232$	1.00000	0.0656532
$233$	7.53545	0.493664	0.246832	−	0.969058i	$-0.420610\pi$
$233$	7.53545	0.493664	0.246832	+	0.969058i	$0.420610\pi$
$234$	−9.14173	−0.597614
$235$	−19.7257	−1.28676
$236$	−9.14173	−0.595076
$237$	−6.08893	−0.395519
$238$	0	0
$239$	−8.84074	−0.571860	−0.285930	−	0.958251i	$-0.592302\pi$
$239$	−8.84074	−0.571860	−0.285930	+	0.958251i	$0.592302\pi$
$240$	−3.30528	−0.213355
$241$	−11.4403	−0.736934	−0.368467	−	0.929641i	$-0.620117\pi$
$241$	−11.4403	−0.736934	−0.368467	+	0.929641i	$0.620117\pi$
$242$	−9.75658	−0.627177
$243$	−15.9582	−1.02372
$244$	6.92243	0.443163
$245$	0	0
$246$	−8.46034	−0.539412
$247$	−10.6106	−0.675134
$248$	−2.95687	−0.187762
$249$	4.07995	0.258556
$250$	22.7054	1.43602
$251$	15.7830	0.996212	0.498106	−	0.867116i	$-0.334029\pi$
$251$	15.7830	0.996212	0.498106	+	0.867116i	$0.334029\pi$
$252$	0	0
$253$	−9.38944	−0.590309
$254$	−20.8407	−1.30766
$255$	15.8622	0.993332
$256$	1.00000	0.0625000
$257$	16.0374	1.00039	0.500193	−	0.865914i	$-0.333262\pi$
$257$	16.0374	1.00039	0.500193	+	0.865914i	$0.333262\pi$
$258$	−2.75494	−0.171515
$259$	0	0
$260$	15.7257	0.975263
$261$	−2.30528	−0.142693
$262$	−7.83519	−0.484060
$263$	0.694718	0.0428381	0.0214191	−	0.999771i	$-0.493182\pi$
$263$	0.694718	0.0428381	0.0214191	+	0.999771i	$0.493182\pi$
$264$	0.929421	0.0572019
$265$	21.0384	1.29238
$266$	0	0
$267$	−0.929091	−0.0568595
$268$	−6.61056	−0.403804
$269$	−24.6435	−1.50254	−0.751271	−	0.659994i	$-0.770559\pi$
$269$	−24.6435	−1.50254	−0.751271	+	0.659994i	$0.770559\pi$
$270$	17.5355	1.06717
$271$	−26.5917	−1.61533	−0.807665	−	0.589641i	$-0.799269\pi$
$271$	−26.5917	−1.61533	−0.807665	+	0.589641i	$0.799269\pi$
$272$	−4.79906	−0.290985
$273$	0	0
$274$	−2.00000	−0.120824
$275$	−11.9600	−0.721217
$276$	−7.01835	−0.422455
$277$	1.15926	0.0696534	0.0348267	−	0.999393i	$-0.488912\pi$
$277$	1.15926	0.0696534	0.0348267	+	0.999393i	$0.488912\pi$
$278$	−8.38750	−0.503049
$279$	6.81643	0.408089
$280$	0	0
$281$	7.11509	0.424450	0.212225	−	0.977221i	$-0.431929\pi$
$281$	7.11509	0.424450	0.212225	+	0.977221i	$0.431929\pi$
$282$	4.14602	0.246892
$283$	13.7389	0.816690	0.408345	−	0.912828i	$-0.366106\pi$
$283$	13.7389	0.816690	0.408345	+	0.912828i	$0.366106\pi$
$284$	15.0309	0.891922
$285$	8.84388	0.523866
$286$	−4.42194	−0.261475
$287$	0	0
$288$	−2.30528	−0.135840
$289$	6.03093	0.354761
$290$	3.96556	0.232866
$291$	14.4423	0.846620
$292$	4.79906	0.280843
$293$	−27.0314	−1.57919	−0.789596	−	0.613627i	$-0.789710\pi$
$293$	−27.0314	−1.57919	−0.789596	+	0.613627i	$0.789710\pi$
$294$	0	0
$295$	−36.2521	−2.11068
$296$	8.19020	0.476045
$297$	−4.93084	−0.286116
$298$	4.92489	0.285291
$299$	33.3915	1.93108
$300$	−8.93980	−0.516140
$301$	0	0
$302$	21.6415	1.24533
$303$	1.60152	0.0920050
$304$	−2.67568	−0.153461
$305$	27.4513	1.57186
$306$	11.0632	0.632440
$307$	20.0297	1.14316	0.571578	−	0.820548i	$-0.306331\pi$
$307$	20.0297	1.14316	0.571578	+	0.820548i	$0.306331\pi$
$308$	0	0
$309$	0.760784	0.0432795
$310$	−11.7257	−0.665972
$311$	−28.9963	−1.64423	−0.822114	−	0.569324i	$-0.807205\pi$
$311$	−28.9963	−1.64423	−0.822114	+	0.569324i	$0.807205\pi$
$312$	−3.30528	−0.187125
$313$	4.07156	0.230138	0.115069	−	0.993357i	$-0.463291\pi$
$313$	4.07156	0.230138	0.115069	+	0.993357i	$0.463291\pi$
$314$	14.8535	0.838234
$315$	0	0
$316$	7.30528	0.410954
$317$	−15.6815	−0.880759	−0.440380	−	0.897812i	$-0.645156\pi$
$317$	−15.6815	−0.880759	−0.440380	+	0.897812i	$0.645156\pi$
$318$	−4.42194	−0.247970
$319$	−1.11509	−0.0624328
$320$	3.96556	0.221681
$321$	2.17590	0.121447
$322$	0	0
$323$	12.8407	0.714478
$324$	3.23017	0.179454
$325$	42.5332	2.35932
$326$	−9.95582	−0.551402
$327$	6.59784	0.364861
$328$	10.1504	0.560463
$329$	0	0
$330$	3.68567	0.202890
$331$	−17.0751	−0.938533	−0.469266	−	0.883057i	$-0.655482\pi$
$331$	−17.0751	−0.938533	−0.469266	+	0.883057i	$0.655482\pi$
$332$	−4.89498	−0.268647
$333$	−18.8807	−1.03466
$334$	−2.01737	−0.110386
$335$	−26.2146	−1.43226
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	2.72565	0.148256
$339$	14.5456	0.790009
$340$	−19.0309	−1.03210
$341$	3.29717	0.178552
$342$	6.16820	0.333538
$343$	0	0
$344$	3.30528	0.178209
$345$	−27.8317	−1.49841
$346$	6.72050	0.361296
$347$	7.07091	0.379586	0.189793	−	0.981824i	$-0.439218\pi$
$347$	7.07091	0.379586	0.189793	+	0.981824i	$0.439218\pi$
$348$	−0.833497	−0.0446801
$349$	30.7425	1.64561	0.822805	−	0.568324i	$-0.192408\pi$
$349$	30.7425	1.64561	0.822805	+	0.568324i	$0.192408\pi$
$350$	0	0
$351$	17.5355	0.935973
$352$	−1.11509	−0.0594343
$353$	−9.19425	−0.489361	−0.244680	−	0.969604i	$-0.578683\pi$
$353$	−9.19425	−0.489361	−0.244680	+	0.969604i	$0.578683\pi$
$354$	7.61961	0.404978
$355$	59.6060	3.16356
$356$	1.11469	0.0590785
$357$	0	0
$358$	25.6815	1.35731
$359$	8.52641	0.450007	0.225003	−	0.974358i	$-0.427761\pi$
$359$	8.52641	0.450007	0.225003	+	0.974358i	$0.427761\pi$
$360$	−9.14173	−0.481811
$361$	−11.8407	−0.623197
$362$	19.4774	1.02371
$363$	8.13209	0.426824
$364$	0	0
$365$	19.0309	0.996125
$366$	−5.76983	−0.301594
$367$	2.02745	0.105832	0.0529161	−	0.998599i	$-0.483148\pi$
$367$	2.02745	0.105832	0.0529161	+	0.998599i	$0.483148\pi$
$368$	8.42037	0.438942
$369$	−23.3996	−1.21813
$370$	32.4787	1.68849
$371$	0	0
$372$	2.46455	0.127781
$373$	22.5264	1.16637	0.583187	−	0.812338i	$-0.301806\pi$
$373$	22.5264	1.16637	0.583187	+	0.812338i	$0.301806\pi$
$374$	5.35136	0.276712
$375$	−18.9249	−0.977277
$376$	−4.97424	−0.256527
$377$	3.96556	0.204237
$378$	0	0
$379$	−24.3804	−1.25234	−0.626168	−	0.779688i	$-0.715378\pi$
$379$	−24.3804	−1.25234	−0.626168	+	0.779688i	$0.715378\pi$
$380$	−10.6106	−0.544310
$381$	17.3707	0.889928
$382$	4.38039	0.224120
$383$	−30.0575	−1.53586	−0.767932	−	0.640531i	$-0.778714\pi$
$383$	−30.0575	−1.53586	−0.767932	+	0.640531i	$0.778714\pi$
$384$	−0.833497	−0.0425342
$385$	0	0
$386$	3.76983	0.191879
$387$	−7.61961	−0.387326
$388$	−17.3273	−0.879660
$389$	−7.26110	−0.368153	−0.184076	−	0.982912i	$-0.558929\pi$
$389$	−7.26110	−0.368153	−0.184076	+	0.982912i	$0.558929\pi$
$390$	−13.1073	−0.663713
$391$	−40.4098	−2.04361
$392$	0	0
$393$	6.53061	0.329426
$394$	−2.38039	−0.119922
$395$	28.9695	1.45761
$396$	2.57059	0.129177
$397$	−9.87930	−0.495828	−0.247914	−	0.968782i	$-0.579745\pi$
$397$	−9.87930	−0.495828	−0.247914	+	0.968782i	$0.579745\pi$
$398$	−20.1090	−1.00797
$399$	0	0
$400$	10.7257	0.536283
$401$	−2.69472	−0.134568	−0.0672839	−	0.997734i	$-0.521433\pi$
$401$	−2.69472	−0.134568	−0.0672839	+	0.997734i	$0.521433\pi$
$402$	5.50989	0.274808
$403$	−11.7257	−0.584096
$404$	−1.92145	−0.0955955
$405$	12.8094	0.636506
$406$	0	0
$407$	−9.13277	−0.452695
$408$	4.00000	0.198030
$409$	−2.97353	−0.147032	−0.0735159	−	0.997294i	$-0.523422\pi$
$409$	−2.97353	−0.147032	−0.0735159	+	0.997294i	$0.523422\pi$
$410$	40.2521	1.98791
$411$	1.66699	0.0822268
$412$	−0.912761	−0.0449685
$413$	0	0
$414$	−19.4113	−0.954015
$415$	−19.4113	−0.952864
$416$	3.96556	0.194427
$417$	6.99096	0.342349
$418$	2.98361	0.145933
$419$	0.648229	0.0316680	0.0158340	−	0.999875i	$-0.494960\pi$
$419$	0.648229	0.0316680	0.0158340	+	0.999875i	$0.494960\pi$
$420$	0	0
$421$	−21.4113	−1.04352	−0.521762	−	0.853091i	$-0.674725\pi$
$421$	−21.4113	−1.04352	−0.521762	+	0.853091i	$0.674725\pi$
$422$	18.7566	0.913056
$423$	11.4670	0.557546
$424$	5.30528	0.257647
$425$	−51.4730	−2.49681
$426$	−12.5282	−0.606995
$427$	0	0
$428$	−2.61056	−0.126186
$429$	3.68567	0.177946
$430$	13.1073	0.632090
$431$	−29.2211	−1.40753	−0.703766	−	0.710432i	$-0.748500\pi$
$431$	−29.2211	−1.40753	−0.703766	+	0.710432i	$0.748500\pi$
$432$	4.42194	0.212751
$433$	26.3630	1.26693	0.633463	−	0.773773i	$-0.281633\pi$
$433$	26.3630	1.26693	0.633463	+	0.773773i	$0.281633\pi$
$434$	0	0
$435$	−3.30528	−0.158476
$436$	−7.91585	−0.379100
$437$	−22.5302	−1.07777
$438$	−4.00000	−0.191127
$439$	−17.7211	−0.845781	−0.422890	−	0.906181i	$-0.638984\pi$
$439$	−17.7211	−0.845781	−0.422890	+	0.906181i	$0.638984\pi$
$440$	−4.42194	−0.210808
$441$	0	0
$442$	−19.0309	−0.905209
$443$	−1.80980	−0.0859864	−0.0429932	−	0.999075i	$-0.513689\pi$
$443$	−1.80980	−0.0859864	−0.0429932	+	0.999075i	$0.513689\pi$
$444$	−6.82651	−0.323972
$445$	4.42037	0.209546
$446$	−25.4603	−1.20558
$447$	−4.10488	−0.194154
$448$	0	0
$449$	−8.99096	−0.424309	−0.212155	−	0.977236i	$-0.568048\pi$
$449$	−8.99096	−0.424309	−0.212155	+	0.977236i	$0.568048\pi$
$450$	−24.7257	−1.16558
$451$	−11.3186	−0.532971
$452$	−17.4513	−0.820840
$453$	−18.0381	−0.847505
$454$	−20.0030	−0.938786
$455$	0	0
$456$	2.23017	0.104437
$457$	21.4913	1.00532	0.502660	−	0.864484i	$-0.332355\pi$
$457$	21.4913	1.00532	0.502660	+	0.864484i	$0.332355\pi$
$458$	15.2039	0.710433
$459$	−21.2211	−0.990517
$460$	33.3915	1.55688
$461$	−26.4690	−1.23279	−0.616393	−	0.787439i	$-0.711407\pi$
$461$	−26.4690	−1.23279	−0.616393	+	0.787439i	$0.711407\pi$
$462$	0	0
$463$	9.80980	0.455900	0.227950	−	0.973673i	$-0.426798\pi$
$463$	9.80980	0.455900	0.227950	+	0.973673i	$0.426798\pi$
$464$	1.00000	0.0464238
$465$	9.77330	0.453226
$466$	7.53545	0.349073
$467$	−5.78100	−0.267513	−0.133756	−	0.991014i	$-0.542704\pi$
$467$	−5.78100	−0.267513	−0.133756	+	0.991014i	$0.542704\pi$
$468$	−9.14173	−0.422577
$469$	0	0
$470$	−19.7257	−0.909876
$471$	−12.3804	−0.570458
$472$	−9.14173	−0.420782
$473$	−3.68567	−0.169467
$474$	−6.08893	−0.279674
$475$	−28.6984	−1.31677
$476$	0	0
$477$	−12.2302	−0.559981
$478$	−8.84074	−0.404366
$479$	−28.8211	−1.31687	−0.658434	−	0.752638i	$-0.728781\pi$
$479$	−28.8211	−1.31687	−0.658434	+	0.752638i	$0.728781\pi$
$480$	−3.30528	−0.150865
$481$	32.4787	1.48090
$482$	−11.4403	−0.521091
$483$	0	0
$484$	−9.75658	−0.443481
$485$	−68.7124	−3.12007
$486$	−15.9582	−0.723877
$487$	17.6815	0.801224	0.400612	−	0.916248i	$-0.368798\pi$
$487$	17.6815	0.801224	0.400612	+	0.916248i	$0.368798\pi$
$488$	6.92243	0.313364
$489$	8.29815	0.375255
$490$	0	0
$491$	23.6857	1.06892	0.534460	−	0.845194i	$-0.320515\pi$
$491$	23.6857	1.06892	0.534460	+	0.845194i	$0.320515\pi$
$492$	−8.46034	−0.381422
$493$	−4.79906	−0.216139
$494$	−10.6106	−0.477392
$495$	10.1938	0.458178
$496$	−2.95687	−0.132768
$497$	0	0
$498$	4.07995	0.182827
$499$	−14.9910	−0.671087	−0.335544	−	0.942025i	$-0.608920\pi$
$499$	−14.9910	−0.671087	−0.335544	+	0.942025i	$0.608920\pi$
$500$	22.7054	1.01542
$501$	1.68147	0.0751227
$502$	15.7830	0.704428
$503$	−35.7859	−1.59562	−0.797808	−	0.602911i	$-0.794007\pi$
$503$	−35.7859	−1.59562	−0.797808	+	0.602911i	$0.794007\pi$
$504$	0	0
$505$	−7.61961	−0.339068
$506$	−9.38944	−0.417411
$507$	−2.27182	−0.100895
$508$	−20.8407	−0.924658
$509$	−0.473041	−0.0209672	−0.0104836	−	0.999945i	$-0.503337\pi$
$509$	−0.473041	−0.0209672	−0.0104836	+	0.999945i	$0.503337\pi$
$510$	15.8622	0.702391
$511$	0	0
$512$	1.00000	0.0441942
$513$	−11.8317	−0.522382
$514$	16.0374	0.707380
$515$	−3.61961	−0.159499
$516$	−2.75494	−0.121280
$517$	5.54671	0.243944
$518$	0	0
$519$	−5.60152	−0.245879
$520$	15.7257	0.689615
$521$	13.6697	0.598879	0.299440	−	0.954115i	$-0.403200\pi$
$521$	13.6697	0.598879	0.299440	+	0.954115i	$0.403200\pi$
$522$	−2.30528	−0.100899
$523$	−1.21061	−0.0529365	−0.0264682	−	0.999650i	$-0.508426\pi$
$523$	−1.21061	−0.0529365	−0.0264682	+	0.999650i	$0.508426\pi$
$524$	−7.83519	−0.342282
$525$	0	0
$526$	0.694718	0.0302911
$527$	14.1902	0.618135
$528$	0.929421	0.0404479
$529$	47.9026	2.08272
$530$	21.0384	0.913850
$531$	21.0743	0.914545
$532$	0	0
$533$	40.2521	1.74351
$534$	−0.929091	−0.0402057
$535$	−10.3523	−0.447571
$536$	−6.61056	−0.285533
$537$	−21.4054	−0.923713
$538$	−24.6435	−1.06246
$539$	0	0
$540$	17.5355	0.754606
$541$	40.6106	1.74598	0.872992	−	0.487734i	$-0.162176\pi$
$541$	40.6106	1.74598	0.872992	+	0.487734i	$0.162176\pi$
$542$	−26.5917	−1.14221
$543$	−16.2344	−0.696684
$544$	−4.79906	−0.205758
$545$	−31.3907	−1.34463
$546$	0	0
$547$	−43.4513	−1.85784	−0.928922	−	0.370276i	$-0.879263\pi$
$547$	−43.4513	−1.85784	−0.928922	+	0.370276i	$0.879263\pi$
$548$	−2.00000	−0.0854358
$549$	−15.9582	−0.681078
$550$	−11.9600	−0.509977
$551$	−2.67568	−0.113988
$552$	−7.01835	−0.298721
$553$	0	0
$554$	1.15926	0.0492524
$555$	−27.0709	−1.14910
$556$	−8.38750	−0.355709
$557$	5.15926	0.218605	0.109303	−	0.994009i	$-0.465138\pi$
$557$	5.15926	0.218605	0.109303	+	0.994009i	$0.465138\pi$
$558$	6.81643	0.288562
$559$	13.1073	0.554379
$560$	0	0
$561$	−4.46034	−0.188316
$562$	7.11509	0.300132
$563$	10.6235	0.447725	0.223863	−	0.974621i	$-0.428133\pi$
$563$	10.6235	0.447725	0.223863	+	0.974621i	$0.428133\pi$
$564$	4.14602	0.174579
$565$	−69.2041	−2.91144
$566$	13.7389	0.577487
$567$	0	0
$568$	15.0309	0.630684
$569$	−1.53966	−0.0645457	−0.0322729	−	0.999479i	$-0.510275\pi$
$569$	−1.53966	−0.0645457	−0.0322729	+	0.999479i	$0.510275\pi$
$570$	8.84388	0.370429
$571$	−42.9910	−1.79912	−0.899558	−	0.436802i	$-0.856111\pi$
$571$	−42.9910	−1.79912	−0.899558	+	0.436802i	$0.856111\pi$
$572$	−4.42194	−0.184891
$573$	−3.65105	−0.152525
$574$	0	0
$575$	90.3139	3.76635
$576$	−2.30528	−0.0960534
$577$	3.32391	0.138376	0.0691881	−	0.997604i	$-0.477959\pi$
$577$	3.32391	0.138376	0.0691881	+	0.997604i	$0.477959\pi$
$578$	6.03093	0.250854
$579$	−3.14214	−0.130583
$580$	3.96556	0.164661
$581$	0	0
$582$	14.4423	0.598651
$583$	−5.91585	−0.245009
$584$	4.79906	0.198586
$585$	−36.2521	−1.49884
$586$	−27.0314	−1.11666
$587$	36.4609	1.50490	0.752452	−	0.658648i	$-0.228871\pi$
$587$	36.4609	1.50490	0.752452	+	0.658648i	$0.228871\pi$
$588$	0	0
$589$	7.91165	0.325994
$590$	−36.2521	−1.49247
$591$	1.98405	0.0816129
$592$	8.19020	0.336615
$593$	30.4447	1.25021	0.625106	−	0.780540i	$-0.285056\pi$
$593$	30.4447	1.25021	0.625106	+	0.780540i	$0.285056\pi$
$594$	−4.93084	−0.202315
$595$	0	0
$596$	4.92489	0.201731
$597$	16.7608	0.685973
$598$	33.3915	1.36548
$599$	−35.9777	−1.47001	−0.735005	−	0.678062i	$-0.762820\pi$
$599$	−35.9777	−1.47001	−0.735005	+	0.678062i	$0.762820\pi$
$600$	−8.93980	−0.364966
$601$	−9.04581	−0.368986	−0.184493	−	0.982834i	$-0.559064\pi$
$601$	−9.04581	−0.368986	−0.184493	+	0.982834i	$0.559064\pi$
$602$	0	0
$603$	15.2392	0.620589
$604$	21.6415	0.880580
$605$	−38.6903	−1.57298
$606$	1.60152	0.0650573
$607$	22.6953	0.921175	0.460587	−	0.887614i	$-0.347639\pi$
$607$	22.6953	0.921175	0.460587	+	0.887614i	$0.347639\pi$
$608$	−2.67568	−0.108513
$609$	0	0
$610$	27.4513	1.11147
$611$	−19.7257	−0.798014
$612$	11.0632	0.447202
$613$	27.5355	1.11215	0.556073	−	0.831133i	$-0.312307\pi$
$613$	27.5355	1.11215	0.556073	+	0.831133i	$0.312307\pi$
$614$	20.0297	0.808334
$615$	−33.5500	−1.35287
$616$	0	0
$617$	29.0709	1.17035	0.585175	−	0.810907i	$-0.301026\pi$
$617$	29.0709	1.17035	0.585175	+	0.810907i	$0.301026\pi$
$618$	0.760784	0.0306032
$619$	−14.2745	−0.573741	−0.286870	−	0.957969i	$-0.592615\pi$
$619$	−14.2745	−0.573741	−0.286870	+	0.957969i	$0.592615\pi$
$620$	−11.7257	−0.470913
$621$	37.2344	1.49416
$622$	−28.9963	−1.16264
$623$	0	0
$624$	−3.30528	−0.132317
$625$	36.4113	1.45645
$626$	4.07156	0.162732
$627$	−2.48683	−0.0993146
$628$	14.8535	0.592721
$629$	−39.3052	−1.56720
$630$	0	0
$631$	39.7433	1.58216	0.791079	−	0.611714i	$-0.209520\pi$
$631$	39.7433	1.58216	0.791079	+	0.611714i	$0.209520\pi$
$632$	7.30528	0.290589
$633$	−15.6336	−0.621378
$634$	−15.6815	−0.622791
$635$	−82.6451	−3.27967
$636$	−4.42194	−0.175341
$637$	0	0
$638$	−1.11509	−0.0441467
$639$	−34.6505	−1.37075
$640$	3.96556	0.156752
$641$	−33.0709	−1.30622	−0.653111	−	0.757262i	$-0.726537\pi$
$641$	−33.0709	−1.30622	−0.653111	+	0.757262i	$0.726537\pi$
$642$	2.17590	0.0858759
$643$	16.3186	0.643543	0.321772	−	0.946817i	$-0.395722\pi$
$643$	16.3186	0.643543	0.321772	+	0.946817i	$0.395722\pi$
$644$	0	0
$645$	−10.9249	−0.430167
$646$	12.8407	0.505212
$647$	−26.7770	−1.05271	−0.526355	−	0.850265i	$-0.676442\pi$
$647$	−26.7770	−1.05271	−0.526355	+	0.850265i	$0.676442\pi$
$648$	3.23017	0.126893
$649$	10.1938	0.400142
$650$	42.5332	1.66829
$651$	0	0
$652$	−9.95582	−0.389900
$653$	−25.3713	−0.992858	−0.496429	−	0.868077i	$-0.665356\pi$
$653$	−25.3713	−0.992858	−0.496429	+	0.868077i	$0.665356\pi$
$654$	6.59784	0.257996
$655$	−31.0709	−1.21404
$656$	10.1504	0.396307
$657$	−11.0632	−0.431616
$658$	0	0
$659$	−3.34526	−0.130313	−0.0651564	−	0.997875i	$-0.520755\pi$
$659$	−3.34526	−0.130313	−0.0651564	+	0.997875i	$0.520755\pi$
$660$	3.68567	0.143465
$661$	−36.8315	−1.43258	−0.716289	−	0.697804i	$-0.754161\pi$
$661$	−36.8315	−1.43258	−0.716289	+	0.697804i	$0.754161\pi$
$662$	−17.0751	−0.663643
$663$	15.8622	0.616038
$664$	−4.89498	−0.189962
$665$	0	0
$666$	−18.8807	−0.731613
$667$	8.42037	0.326038
$668$	−2.01737	−0.0780544
$669$	21.2211	0.820456
$670$	−26.2146	−1.01276
$671$	−7.71911	−0.297993
$672$	0	0
$673$	−27.9558	−1.07762	−0.538809	−	0.842428i	$-0.681126\pi$
$673$	−27.9558	−1.07762	−0.538809	+	0.842428i	$0.681126\pi$
$674$	−6.00000	−0.231111
$675$	47.4282	1.82551
$676$	2.72565	0.104833
$677$	−51.2286	−1.96888	−0.984438	−	0.175733i	$-0.943771\pi$
$677$	−51.2286	−1.96888	−0.984438	+	0.175733i	$0.943771\pi$
$678$	14.5456	0.558621
$679$	0	0
$680$	−19.0309	−0.729803
$681$	16.6724	0.638889
$682$	3.29717	0.126255
$683$	45.2211	1.73034	0.865169	−	0.501480i	$-0.167211\pi$
$683$	45.2211	1.73034	0.865169	+	0.501480i	$0.167211\pi$
$684$	6.16820	0.235847
$685$	−7.93112	−0.303032
$686$	0	0
$687$	−12.6724	−0.483483
$688$	3.30528	0.126013
$689$	21.0384	0.801499
$690$	−27.8317	−1.05953
$691$	−4.70313	−0.178916	−0.0894578	−	0.995991i	$-0.528513\pi$
$691$	−4.70313	−0.178916	−0.0894578	+	0.995991i	$0.528513\pi$
$692$	6.72050	0.255475
$693$	0	0
$694$	7.07091	0.268408
$695$	−33.2611	−1.26167
$696$	−0.833497	−0.0315936
$697$	−48.7124	−1.84511
$698$	30.7425	1.16362
$699$	−6.28078	−0.237561
$700$	0	0
$701$	−0.376191	−0.0142085	−0.00710427	−	0.999975i	$-0.502261\pi$
$701$	−0.376191	−0.0142085	−0.00710427	+	0.999975i	$0.502261\pi$
$702$	17.5355	0.661833
$703$	−21.9143	−0.826515
$704$	−1.11509	−0.0420264
$705$	16.4413	0.619214
$706$	−9.19425	−0.346030
$707$	0	0
$708$	7.61961	0.286362
$709$	−42.0661	−1.57982	−0.789912	−	0.613220i	$-0.789874\pi$
$709$	−42.0661	−1.57982	−0.789912	+	0.613220i	$0.789874\pi$
$710$	59.6060	2.23697
$711$	−16.8407	−0.631577
$712$	1.11469	0.0417748
$713$	−24.8980	−0.932436
$714$	0	0
$715$	−17.5355	−0.655789
$716$	25.6815	0.959762
$717$	7.36873	0.275190
$718$	8.52641	0.318203
$719$	39.6556	1.47890	0.739452	−	0.673210i	$-0.235085\pi$
$719$	39.6556	1.47890	0.739452	+	0.673210i	$0.235085\pi$
$720$	−9.14173	−0.340692
$721$	0	0
$722$	−11.8407	−0.440667
$723$	9.53545	0.354627
$724$	19.4774	0.723872
$725$	10.7257	0.398341
$726$	8.13209	0.301810
$727$	16.7649	0.621776	0.310888	−	0.950447i	$-0.399374\pi$
$727$	16.7649	0.621776	0.310888	+	0.950447i	$0.399374\pi$
$728$	0	0
$729$	3.61056	0.133725
$730$	19.0309	0.704366
$731$	−15.8622	−0.586686
$732$	−5.76983	−0.213259
$733$	4.28919	0.158425	0.0792125	−	0.996858i	$-0.474759\pi$
$733$	4.28919	0.158425	0.0792125	+	0.996858i	$0.474759\pi$
$734$	2.02745	0.0748346
$735$	0	0
$736$	8.42037	0.310379
$737$	7.37135	0.271527
$738$	−23.3996	−0.861350
$739$	−23.7656	−0.874233	−0.437116	−	0.899405i	$-0.644000\pi$
$739$	−23.7656	−0.874233	−0.437116	+	0.899405i	$0.644000\pi$
$740$	32.4787	1.19394
$741$	8.84388	0.324888
$742$	0	0
$743$	32.3804	1.18792	0.593961	−	0.804494i	$-0.297563\pi$
$743$	32.3804	1.18792	0.593961	+	0.804494i	$0.297563\pi$
$744$	2.46455	0.0903546
$745$	19.5299	0.715522
$746$	22.5264	0.824750
$747$	11.2843	0.412871
$748$	5.35136	0.195665
$749$	0	0
$750$	−18.9249	−0.691039
$751$	−20.8407	−0.760489	−0.380245	−	0.924886i	$-0.624160\pi$
$751$	−20.8407	−0.760489	−0.380245	+	0.924886i	$0.624160\pi$
$752$	−4.97424	−0.181392
$753$	−13.1551	−0.479397
$754$	3.96556	0.144417
$755$	85.8206	3.12333
$756$	0	0
$757$	−45.3230	−1.64729	−0.823646	−	0.567105i	$-0.808063\pi$
$757$	−45.3230	−1.64729	−0.823646	+	0.567105i	$0.808063\pi$
$758$	−24.3804	−0.885536
$759$	7.82607	0.284068
$760$	−10.6106	−0.384886
$761$	31.5326	1.14306	0.571528	−	0.820582i	$-0.306351\pi$
$761$	31.5326	1.14306	0.571528	+	0.820582i	$0.306351\pi$
$762$	17.3707	0.629274
$763$	0	0
$764$	4.38039	0.158477
$765$	43.8717	1.58618
$766$	−30.0575	−1.08602
$767$	−36.2521	−1.30899
$768$	−0.833497	−0.0300762
$769$	−2.97353	−0.107228	−0.0536142	−	0.998562i	$-0.517074\pi$
$769$	−2.97353	−0.107228	−0.0536142	+	0.998562i	$0.517074\pi$
$770$	0	0
$771$	−13.3671	−0.481406
$772$	3.76983	0.135679
$773$	36.0671	1.29724	0.648622	−	0.761110i	$-0.275345\pi$
$773$	36.0671	1.29724	0.648622	+	0.761110i	$0.275345\pi$
$774$	−7.61961	−0.273881
$775$	−31.7144	−1.13921
$776$	−17.3273	−0.622014
$777$	0	0
$778$	−7.26110	−0.260323
$779$	−27.1593	−0.973082
$780$	−13.1073	−0.469316
$781$	−16.7608	−0.599748
$782$	−40.4098	−1.44505
$783$	4.42194	0.158027
$784$	0	0
$785$	58.9026	2.10232
$786$	6.53061	0.232939
$787$	31.0762	1.10775	0.553874	−	0.832600i	$-0.313149\pi$
$787$	31.0762	1.10775	0.553874	+	0.832600i	$0.313149\pi$
$788$	−2.38039	−0.0847980
$789$	−0.579045	−0.0206146
$790$	28.9695	1.03069
$791$	0	0
$792$	2.57059	0.0913419
$793$	27.4513	0.974824
$794$	−9.87930	−0.350603
$795$	−17.5355	−0.621918
$796$	−20.1090	−0.712744
$797$	32.2242	1.14144	0.570721	−	0.821144i	$-0.306664\pi$
$797$	32.2242	1.14144	0.570721	+	0.821144i	$0.306664\pi$
$798$	0	0
$799$	23.8717	0.844519
$800$	10.7257	0.379209
$801$	−2.56968	−0.0907950
$802$	−2.69472	−0.0951538
$803$	−5.35136	−0.188845
$804$	5.50989	0.194319
$805$	0	0
$806$	−11.7257	−0.413018
$807$	20.5403	0.723053
$808$	−1.92145	−0.0675963
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	12.8094	0.450078
$811$	27.0748	0.950725	0.475363	−	0.879790i	$-0.342317\pi$
$811$	27.0748	0.950725	0.475363	+	0.879790i	$0.342317\pi$
$812$	0	0
$813$	22.1641	0.777329
$814$	−9.13277	−0.320104
$815$	−39.4804	−1.38294
$816$	4.00000	0.140028
$817$	−8.84388	−0.309408
$818$	−2.97353	−0.103967
$819$	0	0
$820$	40.2521	1.40566
$821$	24.2962	0.847945	0.423972	−	0.905675i	$-0.360635\pi$
$821$	24.2962	0.847945	0.423972	+	0.905675i	$0.360635\pi$
$822$	1.66699	0.0581431
$823$	−12.3804	−0.431553	−0.215777	−	0.976443i	$-0.569228\pi$
$823$	−12.3804	−0.431553	−0.215777	+	0.976443i	$0.569228\pi$
$824$	−0.912761	−0.0317975
$825$	9.96865	0.347064
$826$	0	0
$827$	30.3362	1.05489	0.527447	−	0.849588i	$-0.323149\pi$
$827$	30.3362	1.05489	0.527447	+	0.849588i	$0.323149\pi$
$828$	−19.4113	−0.674590
$829$	10.6068	0.368389	0.184195	−	0.982890i	$-0.441032\pi$
$829$	10.6068	0.368389	0.184195	+	0.982890i	$0.441032\pi$
$830$	−19.4113	−0.673777
$831$	−0.966243	−0.0335186
$832$	3.96556	0.137481
$833$	0	0
$834$	6.99096	0.242077
$835$	−8.00000	−0.276851
$836$	2.98361	0.103190
$837$	−13.0751	−0.451942
$838$	0.648229	0.0223927
$839$	22.1531	0.764810	0.382405	−	0.923995i	$-0.375096\pi$
$839$	22.1531	0.764810	0.382405	+	0.923995i	$0.375096\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−21.4113	−0.737883
$843$	−5.93041	−0.204254
$844$	18.7566	0.645628
$845$	10.8087	0.371831
$846$	11.4670	0.394245
$847$	0	0
$848$	5.30528	0.182184
$849$	−11.4513	−0.393008
$850$	−51.4730	−1.76551
$851$	68.9645	2.36407
$852$	−12.5282	−0.429211
$853$	0.446300	0.0152810	0.00764051	−	0.999971i	$-0.497568\pi$
$853$	0.446300	0.0152810	0.00764051	+	0.999971i	$0.497568\pi$
$854$	0	0
$855$	24.4603	0.836526
$856$	−2.61056	−0.0892273
$857$	−23.2678	−0.794812	−0.397406	−	0.917643i	$-0.630090\pi$
$857$	−23.2678	−0.794812	−0.397406	+	0.917643i	$0.630090\pi$
$858$	3.68567	0.125827
$859$	−36.5927	−1.24853	−0.624263	−	0.781214i	$-0.714601\pi$
$859$	−36.5927	−1.24853	−0.624263	+	0.781214i	$0.714601\pi$
$860$	13.1073	0.446955
$861$	0	0
$862$	−29.2211	−0.995276
$863$	−34.0219	−1.15812	−0.579059	−	0.815285i	$-0.696580\pi$
$863$	−34.0219	−1.15812	−0.579059	+	0.815285i	$0.696580\pi$
$864$	4.42194	0.150437
$865$	26.6505	0.906146
$866$	26.3630	0.895852
$867$	−5.02677	−0.170718
$868$	0	0
$869$	−8.14602	−0.276335
$870$	−3.30528	−0.112060
$871$	−26.2146	−0.888247
$872$	−7.91585	−0.268064
$873$	39.9443	1.35191
$874$	−22.5302	−0.762096
$875$	0	0
$876$	−4.00000	−0.135147
$877$	−53.0486	−1.79132	−0.895662	−	0.444735i	$-0.853298\pi$
$877$	−53.0486	−1.79132	−0.895662	+	0.444735i	$0.853298\pi$
$878$	−17.7211	−0.598057
$879$	22.5306	0.759939
$880$	−4.42194	−0.149064
$881$	−54.0527	−1.82108	−0.910542	−	0.413417i	$-0.864335\pi$
$881$	−54.0527	−1.82108	−0.910542	+	0.413417i	$0.864335\pi$
$882$	0	0
$883$	−2.31012	−0.0777419	−0.0388709	−	0.999244i	$-0.512376\pi$
$883$	−2.31012	−0.0777419	−0.0388709	+	0.999244i	$0.512376\pi$
$884$	−19.0309	−0.640080
$885$	30.2160	1.01570
$886$	−1.80980	−0.0608016
$887$	21.5907	0.724945	0.362473	−	0.931994i	$-0.381933\pi$
$887$	21.5907	0.724945	0.362473	+	0.931994i	$0.381933\pi$
$888$	−6.82651	−0.229083
$889$	0	0
$890$	4.42037	0.148171
$891$	−3.60192	−0.120669
$892$	−25.4603	−0.852475
$893$	13.3095	0.445385
$894$	−4.10488	−0.137288
$895$	101.841	3.40418
$896$	0	0
$897$	−27.8317	−0.929273
$898$	−8.99096	−0.300032
$899$	−2.95687	−0.0986172
$900$	−24.7257	−0.824188
$901$	−25.4603	−0.848207
$902$	−11.3186	−0.376868
$903$	0	0
$904$	−17.4513	−0.580422
$905$	77.2388	2.56751
$906$	−18.0381	−0.599277
$907$	−30.1902	−1.00245	−0.501225	−	0.865317i	$-0.667117\pi$
$907$	−30.1902	−1.00245	−0.501225	+	0.865317i	$0.667117\pi$
$908$	−20.0030	−0.663822
$909$	4.42948	0.146916
$910$	0	0
$911$	13.2434	0.438774	0.219387	−	0.975638i	$-0.429594\pi$
$911$	13.2434	0.438774	0.219387	+	0.975638i	$0.429594\pi$
$912$	2.23017	0.0738484
$913$	5.45832	0.180644
$914$	21.4913	0.710868
$915$	−22.8806	−0.756409
$916$	15.2039	0.502352
$917$	0	0
$918$	−21.2211	−0.700402
$919$	−7.57963	−0.250029	−0.125014	−	0.992155i	$-0.539898\pi$
$919$	−7.57963	−0.250029	−0.125014	+	0.992155i	$0.539898\pi$
$920$	33.3915	1.10088
$921$	−16.6947	−0.550110
$922$	−26.4690	−0.871711
$923$	59.6060	1.96196
$924$	0	0
$925$	87.8452	2.88833
$926$	9.80980	0.322370
$927$	2.10417	0.0691101
$928$	1.00000	0.0328266
$929$	51.1125	1.67695	0.838474	−	0.544942i	$-0.183448\pi$
$929$	51.1125	1.67695	0.838474	+	0.544942i	$0.183448\pi$
$930$	9.77330	0.320479
$931$	0	0
$932$	7.53545	0.246832
$933$	24.1683	0.791235
$934$	−5.78100	−0.189160
$935$	21.2211	0.694005
$936$	−9.14173	−0.298807
$937$	33.9337	1.10857	0.554283	−	0.832329i	$-0.312993\pi$
$937$	33.9337	1.10857	0.554283	+	0.832329i	$0.312993\pi$
$938$	0	0
$939$	−3.39364	−0.110747
$940$	−19.7257	−0.643380
$941$	38.0578	1.24065	0.620324	−	0.784346i	$-0.287001\pi$
$941$	38.0578	1.24065	0.620324	+	0.784346i	$0.287001\pi$
$942$	−12.3804	−0.403375
$943$	85.4702	2.78329
$944$	−9.14173	−0.297538
$945$	0	0
$946$	−3.68567	−0.119832
$947$	−6.37619	−0.207198	−0.103599	−	0.994619i	$-0.533036\pi$
$947$	−6.37619	−0.207198	−0.103599	+	0.994619i	$0.533036\pi$
$948$	−6.08893	−0.197759
$949$	19.0309	0.617770
$950$	−28.6984	−0.931099
$951$	13.0705	0.423839
$952$	0	0
$953$	−20.2563	−0.656165	−0.328082	−	0.944649i	$-0.606402\pi$
$953$	−20.2563	−0.656165	−0.328082	+	0.944649i	$0.606402\pi$
$954$	−12.2302	−0.395966
$955$	17.3707	0.562103
$956$	−8.84074	−0.285930
$957$	0.929421	0.0300439
$958$	−28.8211	−0.931167
$959$	0	0
$960$	−3.30528	−0.106678
$961$	−22.2569	−0.717965
$962$	32.4787	1.04716
$963$	6.01809	0.193930
$964$	−11.4403	−0.368467
$965$	14.9495	0.481240
$966$	0	0
$967$	29.8275	0.959187	0.479594	−	0.877491i	$-0.340784\pi$
$967$	29.8275	0.959187	0.479594	+	0.877491i	$0.340784\pi$
$968$	−9.75658	−0.313588
$969$	−10.7027	−0.343821
$970$	−68.7124	−2.20622
$971$	−17.8372	−0.572422	−0.286211	−	0.958167i	$-0.592396\pi$
$971$	−17.8372	−0.572422	−0.286211	+	0.958167i	$0.592396\pi$
$972$	−15.9582	−0.511858
$973$	0	0
$974$	17.6815	0.566551
$975$	−35.4513	−1.13535
$976$	6.92243	0.221582
$977$	21.7257	0.695065	0.347533	−	0.937668i	$-0.387020\pi$
$977$	21.7257	0.695065	0.347533	+	0.937668i	$0.387020\pi$
$978$	8.29815	0.265346
$979$	−1.24298	−0.0397257
$980$	0	0
$981$	18.2483	0.582622
$982$	23.6857	0.755840
$983$	−6.23738	−0.198942	−0.0994708	−	0.995040i	$-0.531715\pi$
$983$	−6.23738	−0.198942	−0.0994708	+	0.995040i	$0.531715\pi$
$984$	−8.46034	−0.269706
$985$	−9.43958	−0.300770
$986$	−4.79906	−0.152833
$987$	0	0
$988$	−10.6106	−0.337567
$989$	27.8317	0.884996
$990$	10.1938	0.323981
$991$	18.1502	0.576561	0.288280	−	0.957546i	$-0.406916\pi$
$991$	18.1502	0.576561	0.288280	+	0.957546i	$0.406916\pi$
$992$	−2.95687	−0.0938808
$993$	14.2321	0.451641
$994$	0	0
$995$	−79.7433	−2.52803
$996$	4.07995	0.129278
$997$	−29.2608	−0.926699	−0.463349	−	0.886176i	$-0.653352\pi$
$997$	−29.2608	−0.926699	−0.463349	+	0.886176i	$0.653352\pi$
$998$	−14.9910	−0.474530
$999$	36.2165	1.14584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2842.2.a.bc.1.3	✓	6
7.6	odd	2	inner	2842.2.a.bc.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.bc.1.3	✓	6	1.1	even	1	trivial
2842.2.a.bc.1.4	yes	6	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.833497 q^{3} +1.00000 q^{4} +3.96556 q^{5} -0.833497 q^{6} +1.00000 q^{8} -2.30528 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.833497 q^{3} +1.00000 q^{4} +3.96556 q^{5} -0.833497 q^{6} +1.00000 q^{8} -2.30528 q^{9} +3.96556 q^{10} -1.11509 q^{11} -0.833497 q^{12} +3.96556 q^{13} -3.30528 q^{15} +1.00000 q^{16} -4.79906 q^{17} -2.30528 q^{18} -2.67568 q^{19} +3.96556 q^{20} -1.11509 q^{22} +8.42037 q^{23} -0.833497 q^{24} +10.7257 q^{25} +3.96556 q^{26} +4.42194 q^{27} +1.00000 q^{29} -3.30528 q^{30} -2.95687 q^{31} +1.00000 q^{32} +0.929421 q^{33} -4.79906 q^{34} -2.30528 q^{36} +8.19020 q^{37} -2.67568 q^{38} -3.30528 q^{39} +3.96556 q^{40} +10.1504 q^{41} +3.30528 q^{43} -1.11509 q^{44} -9.14173 q^{45} +8.42037 q^{46} -4.97424 q^{47} -0.833497 q^{48} +10.7257 q^{50} +4.00000 q^{51} +3.96556 q^{52} +5.30528 q^{53} +4.42194 q^{54} -4.42194 q^{55} +2.23017 q^{57} +1.00000 q^{58} -9.14173 q^{59} -3.30528 q^{60} +6.92243 q^{61} -2.95687 q^{62} +1.00000 q^{64} +15.7257 q^{65} +0.929421 q^{66} -6.61056 q^{67} -4.79906 q^{68} -7.01835 q^{69} +15.0309 q^{71} -2.30528 q^{72} +4.79906 q^{73} +8.19020 q^{74} -8.93980 q^{75} -2.67568 q^{76} -3.30528 q^{78} +7.30528 q^{79} +3.96556 q^{80} +3.23017 q^{81} +10.1504 q^{82} -4.89498 q^{83} -19.0309 q^{85} +3.30528 q^{86} -0.833497 q^{87} -1.11509 q^{88} +1.11469 q^{89} -9.14173 q^{90} +8.42037 q^{92} +2.46455 q^{93} -4.97424 q^{94} -10.6106 q^{95} -0.833497 q^{96} -17.3273 q^{97} +2.57059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 12 q^{9}+O(q^{10}) \) 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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