LMFDB - Embedded newform 2842.2.a.l.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +2.82843 q^{5} +1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +2.82843 q^{5} +1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +2.82843 q^{10} -2.00000 q^{11} +1.41421 q^{12} +5.65685 q^{13} +4.00000 q^{15} +1.00000 q^{16} +7.07107 q^{17} -1.00000 q^{18} -4.24264 q^{19} +2.82843 q^{20} -2.00000 q^{22} +4.00000 q^{23} +1.41421 q^{24} +3.00000 q^{25} +5.65685 q^{26} -5.65685 q^{27} -1.00000 q^{29} +4.00000 q^{30} -2.82843 q^{31} +1.00000 q^{32} -2.82843 q^{33} +7.07107 q^{34} -1.00000 q^{36} +2.00000 q^{37} -4.24264 q^{38} +8.00000 q^{39} +2.82843 q^{40} -7.07107 q^{41} +2.00000 q^{43} -2.00000 q^{44} -2.82843 q^{45} +4.00000 q^{46} -8.48528 q^{47} +1.41421 q^{48} +3.00000 q^{50} +10.0000 q^{51} +5.65685 q^{52} -2.00000 q^{53} -5.65685 q^{54} -5.65685 q^{55} -6.00000 q^{57} -1.00000 q^{58} +7.07107 q^{59} +4.00000 q^{60} +8.48528 q^{61} -2.82843 q^{62} +1.00000 q^{64} +16.0000 q^{65} -2.82843 q^{66} -4.00000 q^{67} +7.07107 q^{68} +5.65685 q^{69} +12.0000 q^{71} -1.00000 q^{72} +7.07107 q^{73} +2.00000 q^{74} +4.24264 q^{75} -4.24264 q^{76} +8.00000 q^{78} -12.0000 q^{79} +2.82843 q^{80} -5.00000 q^{81} -7.07107 q^{82} -15.5563 q^{83} +20.0000 q^{85} +2.00000 q^{86} -1.41421 q^{87} -2.00000 q^{88} -9.89949 q^{89} -2.82843 q^{90} +4.00000 q^{92} -4.00000 q^{93} -8.48528 q^{94} -12.0000 q^{95} +1.41421 q^{96} +18.3848 q^{97} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 4 q^{11} + 8 q^{15} + 2 q^{16} - 2 q^{18} - 4 q^{22} + 8 q^{23} + 6 q^{25} - 2 q^{29} + 8 q^{30} + 2 q^{32} - 2 q^{36} + 4 q^{37} + 16 q^{39} + 4 q^{43} - 4 q^{44} + 8 q^{46} + 6 q^{50} + 20 q^{51} - 4 q^{53} - 12 q^{57} - 2 q^{58} + 8 q^{60} + 2 q^{64} + 32 q^{65} - 8 q^{67} + 24 q^{71} - 2 q^{72} + 4 q^{74} + 16 q^{78} - 24 q^{79} - 10 q^{81} + 40 q^{85} + 4 q^{86} - 4 q^{88} + 8 q^{92} - 8 q^{93} - 24 q^{95} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 4 * q^11 + 8 * q^15 + 2 * q^16 - 2 * q^18 - 4 * q^22 + 8 * q^23 + 6 * q^25 - 2 * q^29 + 8 * q^30 + 2 * q^32 - 2 * q^36 + 4 * q^37 + 16 * q^39 + 4 * q^43 - 4 * q^44 + 8 * q^46 + 6 * q^50 + 20 * q^51 - 4 * q^53 - 12 * q^57 - 2 * q^58 + 8 * q^60 + 2 * q^64 + 32 * q^65 - 8 * q^67 + 24 * q^71 - 2 * q^72 + 4 * q^74 + 16 * q^78 - 24 * q^79 - 10 * q^81 + 40 * q^85 + 4 * q^86 - 4 * q^88 + 8 * q^92 - 8 * q^93 - 24 * q^95 + 4 * 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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	1.00000	0.500000
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	1.41421	0.577350
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−1.00000	−0.333333
$10$	2.82843	0.894427
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	1.41421	0.408248
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	7.07107	1.71499	0.857493	−	0.514496i	$-0.172021\pi$
$17$	7.07107	1.71499	0.857493	+	0.514496i	$0.172021\pi$
$18$	−1.00000	−0.235702
$19$	−4.24264	−0.973329	−0.486664	−	0.873589i	$-0.661786\pi$
$19$	−4.24264	−0.973329	−0.486664	+	0.873589i	$0.661786\pi$
$20$	2.82843	0.632456
$21$	0	0
$22$	−2.00000	−0.426401
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.41421	0.288675
$25$	3.00000	0.600000
$26$	5.65685	1.10940
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	4.00000	0.730297
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	1.00000	0.176777
$33$	−2.82843	−0.492366
$34$	7.07107	1.21268
$35$	0	0
$36$	−1.00000	−0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−4.24264	−0.688247
$39$	8.00000	1.28103
$40$	2.82843	0.447214
$41$	−7.07107	−1.10432	−0.552158	−	0.833740i	$-0.686195\pi$
$41$	−7.07107	−1.10432	−0.552158	+	0.833740i	$0.686195\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−2.00000	−0.301511
$45$	−2.82843	−0.421637
$46$	4.00000	0.589768
$47$	−8.48528	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$47$	−8.48528	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$48$	1.41421	0.204124
$49$	0	0
$50$	3.00000	0.424264
$51$	10.0000	1.40028
$52$	5.65685	0.784465
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−5.65685	−0.769800
$55$	−5.65685	−0.762770
$56$	0	0
$57$	−6.00000	−0.794719
$58$	−1.00000	−0.131306
$59$	7.07107	0.920575	0.460287	−	0.887770i	$-0.347746\pi$
$59$	7.07107	0.920575	0.460287	+	0.887770i	$0.347746\pi$
$60$	4.00000	0.516398
$61$	8.48528	1.08643	0.543214	−	0.839594i	$-0.317207\pi$
$61$	8.48528	1.08643	0.543214	+	0.839594i	$0.317207\pi$
$62$	−2.82843	−0.359211
$63$	0	0
$64$	1.00000	0.125000
$65$	16.0000	1.98456
$66$	−2.82843	−0.348155
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	7.07107	0.857493
$69$	5.65685	0.681005
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	7.07107	0.827606	0.413803	−	0.910366i	$-0.364200\pi$
$73$	7.07107	0.827606	0.413803	+	0.910366i	$0.364200\pi$
$74$	2.00000	0.232495
$75$	4.24264	0.489898
$76$	−4.24264	−0.486664
$77$	0	0
$78$	8.00000	0.905822
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	2.82843	0.316228
$81$	−5.00000	−0.555556
$82$	−7.07107	−0.780869
$83$	−15.5563	−1.70753	−0.853766	−	0.520658i	$-0.825687\pi$
$83$	−15.5563	−1.70753	−0.853766	+	0.520658i	$0.825687\pi$
$84$	0	0
$85$	20.0000	2.16930
$86$	2.00000	0.215666
$87$	−1.41421	−0.151620
$88$	−2.00000	−0.213201
$89$	−9.89949	−1.04934	−0.524672	−	0.851304i	$-0.675812\pi$
$89$	−9.89949	−1.04934	−0.524672	+	0.851304i	$0.675812\pi$
$90$	−2.82843	−0.298142
$91$	0	0
$92$	4.00000	0.417029
$93$	−4.00000	−0.414781
$94$	−8.48528	−0.875190
$95$	−12.0000	−1.23117
$96$	1.41421	0.144338
$97$	18.3848	1.86669	0.933346	−	0.358979i	$-0.116875\pi$
$97$	18.3848	1.86669	0.933346	+	0.358979i	$0.116875\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	3.00000	0.300000
$101$	2.82843	0.281439	0.140720	−	0.990050i	$-0.455058\pi$
$101$	2.82843	0.281439	0.140720	+	0.990050i	$0.455058\pi$
$102$	10.0000	0.990148
$103$	−14.1421	−1.39347	−0.696733	−	0.717331i	$-0.745364\pi$
$103$	−14.1421	−1.39347	−0.696733	+	0.717331i	$0.745364\pi$
$104$	5.65685	0.554700
$105$	0	0
$106$	−2.00000	−0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−5.65685	−0.544331
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−5.65685	−0.539360
$111$	2.82843	0.268462
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	−6.00000	−0.561951
$115$	11.3137	1.05501
$116$	−1.00000	−0.0928477
$117$	−5.65685	−0.522976
$118$	7.07107	0.650945
$119$	0	0
$120$	4.00000	0.365148
$121$	−7.00000	−0.636364
$122$	8.48528	0.768221
$123$	−10.0000	−0.901670
$124$	−2.82843	−0.254000
$125$	−5.65685	−0.505964
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	2.82843	0.249029
$130$	16.0000	1.40329
$131$	−1.41421	−0.123560	−0.0617802	−	0.998090i	$-0.519678\pi$
$131$	−1.41421	−0.123560	−0.0617802	+	0.998090i	$0.519678\pi$
$132$	−2.82843	−0.246183
$133$	0	0
$134$	−4.00000	−0.345547
$135$	−16.0000	−1.37706
$136$	7.07107	0.606339
$137$	20.0000	1.70872	0.854358	−	0.519685i	$-0.173951\pi$
$137$	20.0000	1.70872	0.854358	+	0.519685i	$0.173951\pi$
$138$	5.65685	0.481543
$139$	−4.24264	−0.359856	−0.179928	−	0.983680i	$-0.557586\pi$
$139$	−4.24264	−0.359856	−0.179928	+	0.983680i	$0.557586\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	12.0000	1.00702
$143$	−11.3137	−0.946100
$144$	−1.00000	−0.0833333
$145$	−2.82843	−0.234888
$146$	7.07107	0.585206
$147$	0	0
$148$	2.00000	0.164399
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	4.24264	0.346410
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−4.24264	−0.344124
$153$	−7.07107	−0.571662
$154$	0	0
$155$	−8.00000	−0.642575
$156$	8.00000	0.640513
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	−12.0000	−0.954669
$159$	−2.82843	−0.224309
$160$	2.82843	0.223607
$161$	0	0
$162$	−5.00000	−0.392837
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	−7.07107	−0.552158
$165$	−8.00000	−0.622799
$166$	−15.5563	−1.20741
$167$	−14.1421	−1.09435	−0.547176	−	0.837018i	$-0.684297\pi$
$167$	−14.1421	−1.09435	−0.547176	+	0.837018i	$0.684297\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	20.0000	1.53393
$171$	4.24264	0.324443
$172$	2.00000	0.152499
$173$	−22.6274	−1.72033	−0.860165	−	0.510015i	$-0.829640\pi$
$173$	−22.6274	−1.72033	−0.860165	+	0.510015i	$0.829640\pi$
$174$	−1.41421	−0.107211
$175$	0	0
$176$	−2.00000	−0.150756
$177$	10.0000	0.751646
$178$	−9.89949	−0.741999
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−2.82843	−0.210819
$181$	5.65685	0.420471	0.210235	−	0.977651i	$-0.432577\pi$
$181$	5.65685	0.420471	0.210235	+	0.977651i	$0.432577\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	4.00000	0.294884
$185$	5.65685	0.415900
$186$	−4.00000	−0.293294
$187$	−14.1421	−1.03418
$188$	−8.48528	−0.618853
$189$	0	0
$190$	−12.0000	−0.870572
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.41421	0.102062
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	18.3848	1.31995
$195$	22.6274	1.62038
$196$	0	0
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	2.00000	0.142134
$199$	−8.48528	−0.601506	−0.300753	−	0.953702i	$-0.597238\pi$
$199$	−8.48528	−0.601506	−0.300753	+	0.953702i	$0.597238\pi$
$200$	3.00000	0.212132
$201$	−5.65685	−0.399004
$202$	2.82843	0.199007
$203$	0	0
$204$	10.0000	0.700140
$205$	−20.0000	−1.39686
$206$	−14.1421	−0.985329
$207$	−4.00000	−0.278019
$208$	5.65685	0.392232
$209$	8.48528	0.586939
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−2.00000	−0.137361
$213$	16.9706	1.16280
$214$	12.0000	0.820303
$215$	5.65685	0.385794
$216$	−5.65685	−0.384900
$217$	0	0
$218$	−2.00000	−0.135457
$219$	10.0000	0.675737
$220$	−5.65685	−0.381385
$221$	40.0000	2.69069
$222$	2.82843	0.189832
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	12.0000	0.798228
$227$	−7.07107	−0.469323	−0.234662	−	0.972077i	$-0.575398\pi$
$227$	−7.07107	−0.469323	−0.234662	+	0.972077i	$0.575398\pi$
$228$	−6.00000	−0.397360
$229$	−28.2843	−1.86908	−0.934539	−	0.355862i	$-0.884187\pi$
$229$	−28.2843	−1.86908	−0.934539	+	0.355862i	$0.884187\pi$
$230$	11.3137	0.746004
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	−5.65685	−0.369800
$235$	−24.0000	−1.56559
$236$	7.07107	0.460287
$237$	−16.9706	−1.10236
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	4.00000	0.258199
$241$	−1.41421	−0.0910975	−0.0455488	−	0.998962i	$-0.514504\pi$
$241$	−1.41421	−0.0910975	−0.0455488	+	0.998962i	$0.514504\pi$
$242$	−7.00000	−0.449977
$243$	9.89949	0.635053
$244$	8.48528	0.543214
$245$	0	0
$246$	−10.0000	−0.637577
$247$	−24.0000	−1.52708
$248$	−2.82843	−0.179605
$249$	−22.0000	−1.39419
$250$	−5.65685	−0.357771
$251$	24.0416	1.51749	0.758747	−	0.651385i	$-0.225812\pi$
$251$	24.0416	1.51749	0.758747	+	0.651385i	$0.225812\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	8.00000	0.501965
$255$	28.2843	1.77123
$256$	1.00000	0.0625000
$257$	−24.0416	−1.49968	−0.749838	−	0.661622i	$-0.769869\pi$
$257$	−24.0416	−1.49968	−0.749838	+	0.661622i	$0.769869\pi$
$258$	2.82843	0.176090
$259$	0	0
$260$	16.0000	0.992278
$261$	1.00000	0.0618984
$262$	−1.41421	−0.0873704
$263$	20.0000	1.23325	0.616626	−	0.787256i	$-0.288499\pi$
$263$	20.0000	1.23325	0.616626	+	0.787256i	$0.288499\pi$
$264$	−2.82843	−0.174078
$265$	−5.65685	−0.347498
$266$	0	0
$267$	−14.0000	−0.856786
$268$	−4.00000	−0.244339
$269$	−11.3137	−0.689809	−0.344904	−	0.938638i	$-0.612089\pi$
$269$	−11.3137	−0.689809	−0.344904	+	0.938638i	$0.612089\pi$
$270$	−16.0000	−0.973729
$271$	28.2843	1.71815	0.859074	−	0.511852i	$-0.171040\pi$
$271$	28.2843	1.71815	0.859074	+	0.511852i	$0.171040\pi$
$272$	7.07107	0.428746
$273$	0	0
$274$	20.0000	1.20824
$275$	−6.00000	−0.361814
$276$	5.65685	0.340503
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	−4.24264	−0.254457
$279$	2.82843	0.169334
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	−12.0000	−0.714590
$283$	−15.5563	−0.924729	−0.462364	−	0.886690i	$-0.652999\pi$
$283$	−15.5563	−0.924729	−0.462364	+	0.886690i	$0.652999\pi$
$284$	12.0000	0.712069
$285$	−16.9706	−1.00525
$286$	−11.3137	−0.668994
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	33.0000	1.94118
$290$	−2.82843	−0.166091
$291$	26.0000	1.52415
$292$	7.07107	0.413803
$293$	31.1127	1.81762	0.908812	−	0.417207i	$-0.136991\pi$
$293$	31.1127	1.81762	0.908812	+	0.417207i	$0.136991\pi$
$294$	0	0
$295$	20.0000	1.16445
$296$	2.00000	0.116248
$297$	11.3137	0.656488
$298$	−14.0000	−0.810998
$299$	22.6274	1.30858
$300$	4.24264	0.244949
$301$	0	0
$302$	−8.00000	−0.460348
$303$	4.00000	0.229794
$304$	−4.24264	−0.243332
$305$	24.0000	1.37424
$306$	−7.07107	−0.404226
$307$	9.89949	0.564994	0.282497	−	0.959268i	$-0.408837\pi$
$307$	9.89949	0.564994	0.282497	+	0.959268i	$0.408837\pi$
$308$	0	0
$309$	−20.0000	−1.13776
$310$	−8.00000	−0.454369
$311$	28.2843	1.60385	0.801927	−	0.597422i	$-0.203808\pi$
$311$	28.2843	1.60385	0.801927	+	0.597422i	$0.203808\pi$
$312$	8.00000	0.452911
$313$	−12.7279	−0.719425	−0.359712	−	0.933063i	$-0.617125\pi$
$313$	−12.7279	−0.719425	−0.359712	+	0.933063i	$0.617125\pi$
$314$	0	0
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−2.82843	−0.158610
$319$	2.00000	0.111979
$320$	2.82843	0.158114
$321$	16.9706	0.947204
$322$	0	0
$323$	−30.0000	−1.66924
$324$	−5.00000	−0.277778
$325$	16.9706	0.941357
$326$	−6.00000	−0.332309
$327$	−2.82843	−0.156412
$328$	−7.07107	−0.390434
$329$	0	0
$330$	−8.00000	−0.440386
$331$	−6.00000	−0.329790	−0.164895	−	0.986311i	$-0.552728\pi$
$331$	−6.00000	−0.329790	−0.164895	+	0.986311i	$0.552728\pi$
$332$	−15.5563	−0.853766
$333$	−2.00000	−0.109599
$334$	−14.1421	−0.773823
$335$	−11.3137	−0.618134
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	19.0000	1.03346
$339$	16.9706	0.921714
$340$	20.0000	1.08465
$341$	5.65685	0.306336
$342$	4.24264	0.229416
$343$	0	0
$344$	2.00000	0.107833
$345$	16.0000	0.861411
$346$	−22.6274	−1.21646
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	−1.41421	−0.0758098
$349$	16.9706	0.908413	0.454207	−	0.890896i	$-0.349923\pi$
$349$	16.9706	0.908413	0.454207	+	0.890896i	$0.349923\pi$
$350$	0	0
$351$	−32.0000	−1.70803
$352$	−2.00000	−0.106600
$353$	1.41421	0.0752710	0.0376355	−	0.999292i	$-0.488017\pi$
$353$	1.41421	0.0752710	0.0376355	+	0.999292i	$0.488017\pi$
$354$	10.0000	0.531494
$355$	33.9411	1.80141
$356$	−9.89949	−0.524672
$357$	0	0
$358$	−4.00000	−0.211407
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−2.82843	−0.149071
$361$	−1.00000	−0.0526316
$362$	5.65685	0.297318
$363$	−9.89949	−0.519589
$364$	0	0
$365$	20.0000	1.04685
$366$	12.0000	0.627250
$367$	−22.6274	−1.18114	−0.590571	−	0.806986i	$-0.701097\pi$
$367$	−22.6274	−1.18114	−0.590571	+	0.806986i	$0.701097\pi$
$368$	4.00000	0.208514
$369$	7.07107	0.368105
$370$	5.65685	0.294086
$371$	0	0
$372$	−4.00000	−0.207390
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	−14.1421	−0.731272
$375$	−8.00000	−0.413118
$376$	−8.48528	−0.437595
$377$	−5.65685	−0.291343
$378$	0	0
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	−12.0000	−0.615587
$381$	11.3137	0.579619
$382$	−12.0000	−0.613973
$383$	25.4558	1.30073	0.650366	−	0.759621i	$-0.274615\pi$
$383$	25.4558	1.30073	0.650366	+	0.759621i	$0.274615\pi$
$384$	1.41421	0.0721688
$385$	0	0
$386$	8.00000	0.407189
$387$	−2.00000	−0.101666
$388$	18.3848	0.933346
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	22.6274	1.14578
$391$	28.2843	1.43040
$392$	0	0
$393$	−2.00000	−0.100887
$394$	−22.0000	−1.10834
$395$	−33.9411	−1.70776
$396$	2.00000	0.100504
$397$	−16.9706	−0.851728	−0.425864	−	0.904787i	$-0.640030\pi$
$397$	−16.9706	−0.851728	−0.425864	+	0.904787i	$0.640030\pi$
$398$	−8.48528	−0.425329
$399$	0	0
$400$	3.00000	0.150000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−5.65685	−0.282138
$403$	−16.0000	−0.797017
$404$	2.82843	0.140720
$405$	−14.1421	−0.702728
$406$	0	0
$407$	−4.00000	−0.198273
$408$	10.0000	0.495074
$409$	12.7279	0.629355	0.314678	−	0.949199i	$-0.398104\pi$
$409$	12.7279	0.629355	0.314678	+	0.949199i	$0.398104\pi$
$410$	−20.0000	−0.987730
$411$	28.2843	1.39516
$412$	−14.1421	−0.696733
$413$	0	0
$414$	−4.00000	−0.196589
$415$	−44.0000	−2.15988
$416$	5.65685	0.277350
$417$	−6.00000	−0.293821
$418$	8.48528	0.415029
$419$	4.24264	0.207267	0.103633	−	0.994616i	$-0.466953\pi$
$419$	4.24264	0.207267	0.103633	+	0.994616i	$0.466953\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	4.00000	0.194717
$423$	8.48528	0.412568
$424$	−2.00000	−0.0971286
$425$	21.2132	1.02899
$426$	16.9706	0.822226
$427$	0	0
$428$	12.0000	0.580042
$429$	−16.0000	−0.772487
$430$	5.65685	0.272798
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	−5.65685	−0.272166
$433$	−21.2132	−1.01944	−0.509721	−	0.860340i	$-0.670251\pi$
$433$	−21.2132	−1.01944	−0.509721	+	0.860340i	$0.670251\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	−2.00000	−0.0957826
$437$	−16.9706	−0.811812
$438$	10.0000	0.477818
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	−5.65685	−0.269680
$441$	0	0
$442$	40.0000	1.90261
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	2.82843	0.134231
$445$	−28.0000	−1.32733
$446$	0	0
$447$	−19.7990	−0.936460
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	−3.00000	−0.141421
$451$	14.1421	0.665927
$452$	12.0000	0.564433
$453$	−11.3137	−0.531564
$454$	−7.07107	−0.331862
$455$	0	0
$456$	−6.00000	−0.280976
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	−28.2843	−1.32164
$459$	−40.0000	−1.86704
$460$	11.3137	0.527504
$461$	−16.9706	−0.790398	−0.395199	−	0.918596i	$-0.629324\pi$
$461$	−16.9706	−0.790398	−0.395199	+	0.918596i	$0.629324\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−1.00000	−0.0464238
$465$	−11.3137	−0.524661
$466$	−8.00000	−0.370593
$467$	1.41421	0.0654420	0.0327210	−	0.999465i	$-0.489583\pi$
$467$	1.41421	0.0654420	0.0327210	+	0.999465i	$0.489583\pi$
$468$	−5.65685	−0.261488
$469$	0	0
$470$	−24.0000	−1.10704
$471$	0	0
$472$	7.07107	0.325472
$473$	−4.00000	−0.183920
$474$	−16.9706	−0.779484
$475$	−12.7279	−0.583997
$476$	0	0
$477$	2.00000	0.0915737
$478$	20.0000	0.914779
$479$	−42.4264	−1.93851	−0.969256	−	0.246054i	$-0.920866\pi$
$479$	−42.4264	−1.93851	−0.969256	+	0.246054i	$0.920866\pi$
$480$	4.00000	0.182574
$481$	11.3137	0.515861
$482$	−1.41421	−0.0644157
$483$	0	0
$484$	−7.00000	−0.318182
$485$	52.0000	2.36120
$486$	9.89949	0.449050
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	8.48528	0.384111
$489$	−8.48528	−0.383718
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−10.0000	−0.450835
$493$	−7.07107	−0.318465
$494$	−24.0000	−1.07981
$495$	5.65685	0.254257
$496$	−2.82843	−0.127000
$497$	0	0
$498$	−22.0000	−0.985844
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	−5.65685	−0.252982
$501$	−20.0000	−0.893534
$502$	24.0416	1.07303
$503$	−33.9411	−1.51336	−0.756680	−	0.653785i	$-0.773180\pi$
$503$	−33.9411	−1.51336	−0.756680	+	0.653785i	$0.773180\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	−8.00000	−0.355643
$507$	26.8701	1.19334
$508$	8.00000	0.354943
$509$	16.9706	0.752207	0.376103	−	0.926578i	$-0.377264\pi$
$509$	16.9706	0.752207	0.376103	+	0.926578i	$0.377264\pi$
$510$	28.2843	1.25245
$511$	0	0
$512$	1.00000	0.0441942
$513$	24.0000	1.05963
$514$	−24.0416	−1.06043
$515$	−40.0000	−1.76261
$516$	2.82843	0.124515
$517$	16.9706	0.746364
$518$	0	0
$519$	−32.0000	−1.40464
$520$	16.0000	0.701646
$521$	−9.89949	−0.433705	−0.216852	−	0.976204i	$-0.569579\pi$
$521$	−9.89949	−0.433705	−0.216852	+	0.976204i	$0.569579\pi$
$522$	1.00000	0.0437688
$523$	−29.6985	−1.29862	−0.649312	−	0.760522i	$-0.724943\pi$
$523$	−29.6985	−1.29862	−0.649312	+	0.760522i	$0.724943\pi$
$524$	−1.41421	−0.0617802
$525$	0	0
$526$	20.0000	0.872041
$527$	−20.0000	−0.871214
$528$	−2.82843	−0.123091
$529$	−7.00000	−0.304348
$530$	−5.65685	−0.245718
$531$	−7.07107	−0.306858
$532$	0	0
$533$	−40.0000	−1.73259
$534$	−14.0000	−0.605839
$535$	33.9411	1.46740
$536$	−4.00000	−0.172774
$537$	−5.65685	−0.244111
$538$	−11.3137	−0.487769
$539$	0	0
$540$	−16.0000	−0.688530
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	28.2843	1.21491
$543$	8.00000	0.343313
$544$	7.07107	0.303170
$545$	−5.65685	−0.242313
$546$	0	0
$547$	46.0000	1.96682	0.983409	−	0.181402i	$-0.0580636\pi$
$547$	46.0000	1.96682	0.983409	+	0.181402i	$0.0580636\pi$
$548$	20.0000	0.854358
$549$	−8.48528	−0.362143
$550$	−6.00000	−0.255841
$551$	4.24264	0.180743
$552$	5.65685	0.240772
$553$	0	0
$554$	6.00000	0.254916
$555$	8.00000	0.339581
$556$	−4.24264	−0.179928
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	2.82843	0.119737
$559$	11.3137	0.478519
$560$	0	0
$561$	−20.0000	−0.844401
$562$	−16.0000	−0.674919
$563$	−32.5269	−1.37085	−0.685423	−	0.728145i	$-0.740383\pi$
$563$	−32.5269	−1.37085	−0.685423	+	0.728145i	$0.740383\pi$
$564$	−12.0000	−0.505291
$565$	33.9411	1.42791
$566$	−15.5563	−0.653882
$567$	0	0
$568$	12.0000	0.503509
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	−16.9706	−0.710819
$571$	38.0000	1.59025	0.795125	−	0.606445i	$-0.207405\pi$
$571$	38.0000	1.59025	0.795125	+	0.606445i	$0.207405\pi$
$572$	−11.3137	−0.473050
$573$	−16.9706	−0.708955
$574$	0	0
$575$	12.0000	0.500435
$576$	−1.00000	−0.0416667
$577$	38.1838	1.58961	0.794805	−	0.606864i	$-0.207573\pi$
$577$	38.1838	1.58961	0.794805	+	0.606864i	$0.207573\pi$
$578$	33.0000	1.37262
$579$	11.3137	0.470182
$580$	−2.82843	−0.117444
$581$	0	0
$582$	26.0000	1.07773
$583$	4.00000	0.165663
$584$	7.07107	0.292603
$585$	−16.0000	−0.661519
$586$	31.1127	1.28525
$587$	−21.2132	−0.875563	−0.437781	−	0.899081i	$-0.644236\pi$
$587$	−21.2132	−0.875563	−0.437781	+	0.899081i	$0.644236\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	20.0000	0.823387
$591$	−31.1127	−1.27981
$592$	2.00000	0.0821995
$593$	−15.5563	−0.638823	−0.319411	−	0.947616i	$-0.603485\pi$
$593$	−15.5563	−0.638823	−0.319411	+	0.947616i	$0.603485\pi$
$594$	11.3137	0.464207
$595$	0	0
$596$	−14.0000	−0.573462
$597$	−12.0000	−0.491127
$598$	22.6274	0.925304
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	4.24264	0.173205
$601$	−12.7279	−0.519183	−0.259591	−	0.965719i	$-0.583588\pi$
$601$	−12.7279	−0.519183	−0.259591	+	0.965719i	$0.583588\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	−8.00000	−0.325515
$605$	−19.7990	−0.804943
$606$	4.00000	0.162489
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−4.24264	−0.172062
$609$	0	0
$610$	24.0000	0.971732
$611$	−48.0000	−1.94187
$612$	−7.07107	−0.285831
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	9.89949	0.399511
$615$	−28.2843	−1.14053
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−20.0000	−0.804518
$619$	38.1838	1.53474	0.767368	−	0.641207i	$-0.221566\pi$
$619$	38.1838	1.53474	0.767368	+	0.641207i	$0.221566\pi$
$620$	−8.00000	−0.321288
$621$	−22.6274	−0.908007
$622$	28.2843	1.13410
$623$	0	0
$624$	8.00000	0.320256
$625$	−31.0000	−1.24000
$626$	−12.7279	−0.508710
$627$	12.0000	0.479234
$628$	0	0
$629$	14.1421	0.563884
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	−12.0000	−0.477334
$633$	5.65685	0.224840
$634$	−6.00000	−0.238290
$635$	22.6274	0.897942
$636$	−2.82843	−0.112154
$637$	0	0
$638$	2.00000	0.0791808
$639$	−12.0000	−0.474713
$640$	2.82843	0.111803
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	16.9706	0.669775
$643$	−4.24264	−0.167313	−0.0836567	−	0.996495i	$-0.526660\pi$
$643$	−4.24264	−0.167313	−0.0836567	+	0.996495i	$0.526660\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	−30.0000	−1.18033
$647$	−31.1127	−1.22317	−0.611583	−	0.791180i	$-0.709467\pi$
$647$	−31.1127	−1.22317	−0.611583	+	0.791180i	$0.709467\pi$
$648$	−5.00000	−0.196419
$649$	−14.1421	−0.555127
$650$	16.9706	0.665640
$651$	0	0
$652$	−6.00000	−0.234978
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	−2.82843	−0.110600
$655$	−4.00000	−0.156293
$656$	−7.07107	−0.276079
$657$	−7.07107	−0.275869
$658$	0	0
$659$	−2.00000	−0.0779089	−0.0389545	−	0.999241i	$-0.512403\pi$
$659$	−2.00000	−0.0779089	−0.0389545	+	0.999241i	$0.512403\pi$
$660$	−8.00000	−0.311400
$661$	8.48528	0.330039	0.165020	−	0.986290i	$-0.447231\pi$
$661$	8.48528	0.330039	0.165020	+	0.986290i	$0.447231\pi$
$662$	−6.00000	−0.233197
$663$	56.5685	2.19694
$664$	−15.5563	−0.603703
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−4.00000	−0.154881
$668$	−14.1421	−0.547176
$669$	0	0
$670$	−11.3137	−0.437087
$671$	−16.9706	−0.655141
$672$	0	0
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	−14.0000	−0.539260
$675$	−16.9706	−0.653197
$676$	19.0000	0.730769
$677$	16.9706	0.652232	0.326116	−	0.945330i	$-0.394260\pi$
$677$	16.9706	0.652232	0.326116	+	0.945330i	$0.394260\pi$
$678$	16.9706	0.651751
$679$	0	0
$680$	20.0000	0.766965
$681$	−10.0000	−0.383201
$682$	5.65685	0.216612
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	4.24264	0.162221
$685$	56.5685	2.16137
$686$	0	0
$687$	−40.0000	−1.52610
$688$	2.00000	0.0762493
$689$	−11.3137	−0.431018
$690$	16.0000	0.609110
$691$	15.5563	0.591791	0.295896	−	0.955220i	$-0.404382\pi$
$691$	15.5563	0.591791	0.295896	+	0.955220i	$0.404382\pi$
$692$	−22.6274	−0.860165
$693$	0	0
$694$	−22.0000	−0.835109
$695$	−12.0000	−0.455186
$696$	−1.41421	−0.0536056
$697$	−50.0000	−1.89389
$698$	16.9706	0.642345
$699$	−11.3137	−0.427924
$700$	0	0
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	−32.0000	−1.20776
$703$	−8.48528	−0.320028
$704$	−2.00000	−0.0753778
$705$	−33.9411	−1.27830
$706$	1.41421	0.0532246
$707$	0	0
$708$	10.0000	0.375823
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	33.9411	1.27379
$711$	12.0000	0.450035
$712$	−9.89949	−0.370999
$713$	−11.3137	−0.423702
$714$	0	0
$715$	−32.0000	−1.19673
$716$	−4.00000	−0.149487
$717$	28.2843	1.05630
$718$	24.0000	0.895672
$719$	−14.1421	−0.527413	−0.263706	−	0.964603i	$-0.584945\pi$
$719$	−14.1421	−0.527413	−0.263706	+	0.964603i	$0.584945\pi$
$720$	−2.82843	−0.105409
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	−2.00000	−0.0743808
$724$	5.65685	0.210235
$725$	−3.00000	−0.111417
$726$	−9.89949	−0.367405
$727$	−42.4264	−1.57351	−0.786754	−	0.617266i	$-0.788240\pi$
$727$	−42.4264	−1.57351	−0.786754	+	0.617266i	$0.788240\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	20.0000	0.740233
$731$	14.1421	0.523066
$732$	12.0000	0.443533
$733$	−8.48528	−0.313411	−0.156706	−	0.987645i	$-0.550087\pi$
$733$	−8.48528	−0.313411	−0.156706	+	0.987645i	$0.550087\pi$
$734$	−22.6274	−0.835193
$735$	0	0
$736$	4.00000	0.147442
$737$	8.00000	0.294684
$738$	7.07107	0.260290
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	5.65685	0.207950
$741$	−33.9411	−1.24686
$742$	0	0
$743$	32.0000	1.17397	0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000	1.17397	0.586983	+	0.809599i	$0.300316\pi$
$744$	−4.00000	−0.146647
$745$	−39.5980	−1.45076
$746$	−14.0000	−0.512576
$747$	15.5563	0.569177
$748$	−14.1421	−0.517088
$749$	0	0
$750$	−8.00000	−0.292119
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	−8.48528	−0.309426
$753$	34.0000	1.23903
$754$	−5.65685	−0.206010
$755$	−22.6274	−0.823496
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	−26.0000	−0.944363
$759$	−11.3137	−0.410662
$760$	−12.0000	−0.435286
$761$	52.3259	1.89681	0.948406	−	0.317058i	$-0.102695\pi$
$761$	52.3259	1.89681	0.948406	+	0.317058i	$0.102695\pi$
$762$	11.3137	0.409852
$763$	0	0
$764$	−12.0000	−0.434145
$765$	−20.0000	−0.723102
$766$	25.4558	0.919757
$767$	40.0000	1.44432
$768$	1.41421	0.0510310
$769$	−12.7279	−0.458981	−0.229490	−	0.973311i	$-0.573706\pi$
$769$	−12.7279	−0.458981	−0.229490	+	0.973311i	$0.573706\pi$
$770$	0	0
$771$	−34.0000	−1.22448
$772$	8.00000	0.287926
$773$	19.7990	0.712120	0.356060	−	0.934463i	$-0.384120\pi$
$773$	19.7990	0.712120	0.356060	+	0.934463i	$0.384120\pi$
$774$	−2.00000	−0.0718885
$775$	−8.48528	−0.304800
$776$	18.3848	0.659975
$777$	0	0
$778$	−14.0000	−0.501924
$779$	30.0000	1.07486
$780$	22.6274	0.810191
$781$	−24.0000	−0.858788
$782$	28.2843	1.01144
$783$	5.65685	0.202159
$784$	0	0
$785$	0	0
$786$	−2.00000	−0.0713376
$787$	52.3259	1.86522	0.932608	−	0.360890i	$-0.117527\pi$
$787$	52.3259	1.86522	0.932608	+	0.360890i	$0.117527\pi$
$788$	−22.0000	−0.783718
$789$	28.2843	1.00695
$790$	−33.9411	−1.20757
$791$	0	0
$792$	2.00000	0.0710669
$793$	48.0000	1.70453
$794$	−16.9706	−0.602263
$795$	−8.00000	−0.283731
$796$	−8.48528	−0.300753
$797$	33.9411	1.20226	0.601128	−	0.799153i	$-0.294718\pi$
$797$	33.9411	1.20226	0.601128	+	0.799153i	$0.294718\pi$
$798$	0	0
$799$	−60.0000	−2.12265
$800$	3.00000	0.106066
$801$	9.89949	0.349781
$802$	−18.0000	−0.635602
$803$	−14.1421	−0.499065
$804$	−5.65685	−0.199502
$805$	0	0
$806$	−16.0000	−0.563576
$807$	−16.0000	−0.563227
$808$	2.82843	0.0995037
$809$	8.00000	0.281265	0.140633	−	0.990062i	$-0.455086\pi$
$809$	8.00000	0.281265	0.140633	+	0.990062i	$0.455086\pi$
$810$	−14.1421	−0.496904
$811$	12.7279	0.446938	0.223469	−	0.974711i	$-0.428262\pi$
$811$	12.7279	0.446938	0.223469	+	0.974711i	$0.428262\pi$
$812$	0	0
$813$	40.0000	1.40286
$814$	−4.00000	−0.140200
$815$	−16.9706	−0.594453
$816$	10.0000	0.350070
$817$	−8.48528	−0.296862
$818$	12.7279	0.445021
$819$	0	0
$820$	−20.0000	−0.698430
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	28.2843	0.986527
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	−14.1421	−0.492665
$825$	−8.48528	−0.295420
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−4.00000	−0.139010
$829$	−2.82843	−0.0982353	−0.0491177	−	0.998793i	$-0.515641\pi$
$829$	−2.82843	−0.0982353	−0.0491177	+	0.998793i	$0.515641\pi$
$830$	−44.0000	−1.52726
$831$	8.48528	0.294351
$832$	5.65685	0.196116
$833$	0	0
$834$	−6.00000	−0.207763
$835$	−40.0000	−1.38426
$836$	8.48528	0.293470
$837$	16.0000	0.553041
$838$	4.24264	0.146560
$839$	19.7990	0.683537	0.341769	−	0.939784i	$-0.388974\pi$
$839$	19.7990	0.683537	0.341769	+	0.939784i	$0.388974\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	38.0000	1.30957
$843$	−22.6274	−0.779330
$844$	4.00000	0.137686
$845$	53.7401	1.84872
$846$	8.48528	0.291730
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	−22.0000	−0.755038
$850$	21.2132	0.727607
$851$	8.00000	0.274236
$852$	16.9706	0.581402
$853$	−33.9411	−1.16212	−0.581061	−	0.813860i	$-0.697362\pi$
$853$	−33.9411	−1.16212	−0.581061	+	0.813860i	$0.697362\pi$
$854$	0	0
$855$	12.0000	0.410391
$856$	12.0000	0.410152
$857$	−18.3848	−0.628012	−0.314006	−	0.949421i	$-0.601671\pi$
$857$	−18.3848	−0.628012	−0.314006	+	0.949421i	$0.601671\pi$
$858$	−16.0000	−0.546231
$859$	26.8701	0.916795	0.458397	−	0.888747i	$-0.348424\pi$
$859$	26.8701	0.916795	0.458397	+	0.888747i	$0.348424\pi$
$860$	5.65685	0.192897
$861$	0	0
$862$	−4.00000	−0.136241
$863$	52.0000	1.77010	0.885050	−	0.465495i	$-0.154124\pi$
$863$	52.0000	1.77010	0.885050	+	0.465495i	$0.154124\pi$
$864$	−5.65685	−0.192450
$865$	−64.0000	−2.17607
$866$	−21.2132	−0.720854
$867$	46.6690	1.58496
$868$	0	0
$869$	24.0000	0.814144
$870$	−4.00000	−0.135613
$871$	−22.6274	−0.766701
$872$	−2.00000	−0.0677285
$873$	−18.3848	−0.622230
$874$	−16.9706	−0.574038
$875$	0	0
$876$	10.0000	0.337869
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	−16.9706	−0.572729
$879$	44.0000	1.48408
$880$	−5.65685	−0.190693
$881$	−24.0416	−0.809983	−0.404992	−	0.914320i	$-0.632726\pi$
$881$	−24.0416	−0.809983	−0.404992	+	0.914320i	$0.632726\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	40.0000	1.34535
$885$	28.2843	0.950765
$886$	12.0000	0.403148
$887$	−14.1421	−0.474846	−0.237423	−	0.971406i	$-0.576303\pi$
$887$	−14.1421	−0.474846	−0.237423	+	0.971406i	$0.576303\pi$
$888$	2.82843	0.0949158
$889$	0	0
$890$	−28.0000	−0.938562
$891$	10.0000	0.335013
$892$	0	0
$893$	36.0000	1.20469
$894$	−19.7990	−0.662177
$895$	−11.3137	−0.378176
$896$	0	0
$897$	32.0000	1.06845
$898$	14.0000	0.467186
$899$	2.82843	0.0943333
$900$	−3.00000	−0.100000
$901$	−14.1421	−0.471143
$902$	14.1421	0.470882
$903$	0	0
$904$	12.0000	0.399114
$905$	16.0000	0.531858
$906$	−11.3137	−0.375873
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−7.07107	−0.234662
$909$	−2.82843	−0.0938130
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−6.00000	−0.198680
$913$	31.1127	1.02968
$914$	8.00000	0.264616
$915$	33.9411	1.12206
$916$	−28.2843	−0.934539
$917$	0	0
$918$	−40.0000	−1.32020
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	11.3137	0.373002
$921$	14.0000	0.461316
$922$	−16.9706	−0.558896
$923$	67.8823	2.23437
$924$	0	0
$925$	6.00000	0.197279
$926$	−24.0000	−0.788689
$927$	14.1421	0.464489
$928$	−1.00000	−0.0328266
$929$	−9.89949	−0.324792	−0.162396	−	0.986726i	$-0.551922\pi$
$929$	−9.89949	−0.324792	−0.162396	+	0.986726i	$0.551922\pi$
$930$	−11.3137	−0.370991
$931$	0	0
$932$	−8.00000	−0.262049
$933$	40.0000	1.30954
$934$	1.41421	0.0462745
$935$	−40.0000	−1.30814
$936$	−5.65685	−0.184900
$937$	−32.5269	−1.06261	−0.531304	−	0.847181i	$-0.678298\pi$
$937$	−32.5269	−1.06261	−0.531304	+	0.847181i	$0.678298\pi$
$938$	0	0
$939$	−18.0000	−0.587408
$940$	−24.0000	−0.782794
$941$	−19.7990	−0.645429	−0.322714	−	0.946496i	$-0.604595\pi$
$941$	−19.7990	−0.645429	−0.322714	+	0.946496i	$0.604595\pi$
$942$	0	0
$943$	−28.2843	−0.921063
$944$	7.07107	0.230144
$945$	0	0
$946$	−4.00000	−0.130051
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	−16.9706	−0.551178
$949$	40.0000	1.29845
$950$	−12.7279	−0.412948
$951$	−8.48528	−0.275154
$952$	0	0
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	2.00000	0.0647524
$955$	−33.9411	−1.09831
$956$	20.0000	0.646846
$957$	2.82843	0.0914301
$958$	−42.4264	−1.37073
$959$	0	0
$960$	4.00000	0.129099
$961$	−23.0000	−0.741935
$962$	11.3137	0.364769
$963$	−12.0000	−0.386695
$964$	−1.41421	−0.0455488
$965$	22.6274	0.728402
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	−7.00000	−0.224989
$969$	−42.4264	−1.36293
$970$	52.0000	1.66962
$971$	46.6690	1.49768	0.748841	−	0.662750i	$-0.230611\pi$
$971$	46.6690	1.49768	0.748841	+	0.662750i	$0.230611\pi$
$972$	9.89949	0.317526
$973$	0	0
$974$	−20.0000	−0.640841
$975$	24.0000	0.768615
$976$	8.48528	0.271607
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	−8.48528	−0.271329
$979$	19.7990	0.632778
$980$	0	0
$981$	2.00000	0.0638551
$982$	20.0000	0.638226
$983$	36.7696	1.17277	0.586383	−	0.810034i	$-0.300551\pi$
$983$	36.7696	1.17277	0.586383	+	0.810034i	$0.300551\pi$
$984$	−10.0000	−0.318788
$985$	−62.2254	−1.98267
$986$	−7.07107	−0.225189
$987$	0	0
$988$	−24.0000	−0.763542
$989$	8.00000	0.254385
$990$	5.65685	0.179787
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−2.82843	−0.0898027
$993$	−8.48528	−0.269272
$994$	0	0
$995$	−24.0000	−0.760851
$996$	−22.0000	−0.697097
$997$	−14.1421	−0.447886	−0.223943	−	0.974602i	$-0.571893\pi$
$997$	−14.1421	−0.447886	−0.223943	+	0.974602i	$0.571893\pi$
$998$	28.0000	0.886325
$999$	−11.3137	−0.357950

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.l.1.1	✓	2	7.6	odd	2	inner
2842.2.a.l.1.2	yes	2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +2.82843 q^{5} +1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +2.82843 q^{5} +1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +2.82843 q^{10} -2.00000 q^{11} +1.41421 q^{12} +5.65685 q^{13} +4.00000 q^{15} +1.00000 q^{16} +7.07107 q^{17} -1.00000 q^{18} -4.24264 q^{19} +2.82843 q^{20} -2.00000 q^{22} +4.00000 q^{23} +1.41421 q^{24} +3.00000 q^{25} +5.65685 q^{26} -5.65685 q^{27} -1.00000 q^{29} +4.00000 q^{30} -2.82843 q^{31} +1.00000 q^{32} -2.82843 q^{33} +7.07107 q^{34} -1.00000 q^{36} +2.00000 q^{37} -4.24264 q^{38} +8.00000 q^{39} +2.82843 q^{40} -7.07107 q^{41} +2.00000 q^{43} -2.00000 q^{44} -2.82843 q^{45} +4.00000 q^{46} -8.48528 q^{47} +1.41421 q^{48} +3.00000 q^{50} +10.0000 q^{51} +5.65685 q^{52} -2.00000 q^{53} -5.65685 q^{54} -5.65685 q^{55} -6.00000 q^{57} -1.00000 q^{58} +7.07107 q^{59} +4.00000 q^{60} +8.48528 q^{61} -2.82843 q^{62} +1.00000 q^{64} +16.0000 q^{65} -2.82843 q^{66} -4.00000 q^{67} +7.07107 q^{68} +5.65685 q^{69} +12.0000 q^{71} -1.00000 q^{72} +7.07107 q^{73} +2.00000 q^{74} +4.24264 q^{75} -4.24264 q^{76} +8.00000 q^{78} -12.0000 q^{79} +2.82843 q^{80} -5.00000 q^{81} -7.07107 q^{82} -15.5563 q^{83} +20.0000 q^{85} +2.00000 q^{86} -1.41421 q^{87} -2.00000 q^{88} -9.89949 q^{89} -2.82843 q^{90} +4.00000 q^{92} -4.00000 q^{93} -8.48528 q^{94} -12.0000 q^{95} +1.41421 q^{96} +18.3848 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 4 q^{11} + 8 q^{15} + 2 q^{16} - 2 q^{18} - 4 q^{22} + 8 q^{23} + 6 q^{25} - 2 q^{29} + 8 q^{30} + 2 q^{32} - 2 q^{36} + 4 q^{37} + 16 q^{39} + 4 q^{43} - 4 q^{44} + 8 q^{46} + 6 q^{50} + 20 q^{51} - 4 q^{53} - 12 q^{57} - 2 q^{58} + 8 q^{60} + 2 q^{64} + 32 q^{65} - 8 q^{67} + 24 q^{71} - 2 q^{72} + 4 q^{74} + 16 q^{78} - 24 q^{79} - 10 q^{81} + 40 q^{85} + 4 q^{86} - 4 q^{88} + 8 q^{92} - 8 q^{93} - 24 q^{95} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 4 * q^11 + 8 * q^15 + 2 * q^16 - 2 * q^18 - 4 * q^22 + 8 * q^23 + 6 * q^25 - 2 * q^29 + 8 * q^30 + 2 * q^32 - 2 * q^36 + 4 * q^37 + 16 * q^39 + 4 * q^43 - 4 * q^44 + 8 * q^46 + 6 * q^50 + 20 * q^51 - 4 * q^53 - 12 * q^57 - 2 * q^58 + 8 * q^60 + 2 * q^64 + 32 * q^65 - 8 * q^67 + 24 * q^71 - 2 * q^72 + 4 * q^74 + 16 * q^78 - 24 * q^79 - 10 * q^81 + 40 * q^85 + 4 * q^86 - 4 * q^88 + 8 * q^92 - 8 * q^93 - 24 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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