LMFDB - Embedded newform 2842.2.a.k.1.1

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:	$1$
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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{5} -1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{5} -1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +1.41421 q^{10} -4.00000 q^{11} -1.41421 q^{12} +1.41421 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.82843 q^{17} -1.00000 q^{18} +4.24264 q^{19} +1.41421 q^{20} -4.00000 q^{22} -4.00000 q^{23} -1.41421 q^{24} -3.00000 q^{25} +1.41421 q^{26} +5.65685 q^{27} +1.00000 q^{29} -2.00000 q^{30} +1.41421 q^{31} +1.00000 q^{32} +5.65685 q^{33} -2.82843 q^{34} -1.00000 q^{36} -10.0000 q^{37} +4.24264 q^{38} -2.00000 q^{39} +1.41421 q^{40} -5.65685 q^{41} +2.00000 q^{43} -4.00000 q^{44} -1.41421 q^{45} -4.00000 q^{46} -4.24264 q^{47} -1.41421 q^{48} -3.00000 q^{50} +4.00000 q^{51} +1.41421 q^{52} -10.0000 q^{53} +5.65685 q^{54} -5.65685 q^{55} -6.00000 q^{57} +1.00000 q^{58} -11.3137 q^{59} -2.00000 q^{60} +1.41421 q^{62} +1.00000 q^{64} +2.00000 q^{65} +5.65685 q^{66} -4.00000 q^{67} -2.82843 q^{68} +5.65685 q^{69} -1.00000 q^{72} +2.82843 q^{73} -10.0000 q^{74} +4.24264 q^{75} +4.24264 q^{76} -2.00000 q^{78} +6.00000 q^{79} +1.41421 q^{80} -5.00000 q^{81} -5.65685 q^{82} +2.82843 q^{83} -4.00000 q^{85} +2.00000 q^{86} -1.41421 q^{87} -4.00000 q^{88} +14.1421 q^{89} -1.41421 q^{90} -4.00000 q^{92} -2.00000 q^{93} -4.24264 q^{94} +6.00000 q^{95} -1.41421 q^{96} -2.82843 q^{97} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 8 q^{11} - 4 q^{15} + 2 q^{16} - 2 q^{18} - 8 q^{22} - 8 q^{23} - 6 q^{25} + 2 q^{29} - 4 q^{30} + 2 q^{32} - 2 q^{36} - 20 q^{37} - 4 q^{39} + 4 q^{43} - 8 q^{44} - 8 q^{46} - 6 q^{50} + 8 q^{51} - 20 q^{53} - 12 q^{57} + 2 q^{58} - 4 q^{60} + 2 q^{64} + 4 q^{65} - 8 q^{67} - 2 q^{72} - 20 q^{74} - 4 q^{78} + 12 q^{79} - 10 q^{81} - 8 q^{85} + 4 q^{86} - 8 q^{88} - 8 q^{92} - 4 q^{93} + 12 q^{95} + 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 8 * q^11 - 4 * q^15 + 2 * q^16 - 2 * q^18 - 8 * q^22 - 8 * q^23 - 6 * q^25 + 2 * q^29 - 4 * q^30 + 2 * q^32 - 2 * q^36 - 20 * q^37 - 4 * q^39 + 4 * q^43 - 8 * q^44 - 8 * q^46 - 6 * q^50 + 8 * q^51 - 20 * q^53 - 12 * q^57 + 2 * q^58 - 4 * q^60 + 2 * q^64 + 4 * q^65 - 8 * q^67 - 2 * q^72 - 20 * q^74 - 4 * q^78 + 12 * q^79 - 10 * q^81 - 8 * q^85 + 4 * q^86 - 8 * q^88 - 8 * q^92 - 4 * q^93 + 12 * q^95 + 8 * 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	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	1.00000	0.500000
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−1.00000	−0.333333
$10$	1.41421	0.447214
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.41421	−0.408248
$13$	1.41421	0.392232	0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421	0.392232	0.196116	+	0.980581i	$0.437167\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	−1.00000	−0.235702
$19$	4.24264	0.973329	0.486664	−	0.873589i	$-0.338214\pi$
$19$	4.24264	0.973329	0.486664	+	0.873589i	$0.338214\pi$
$20$	1.41421	0.316228
$21$	0	0
$22$	−4.00000	−0.852803
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.41421	−0.288675
$25$	−3.00000	−0.600000
$26$	1.41421	0.277350
$27$	5.65685	1.08866
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−2.00000	−0.365148
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	1.00000	0.176777
$33$	5.65685	0.984732
$34$	−2.82843	−0.485071
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	4.24264	0.688247
$39$	−2.00000	−0.320256
$40$	1.41421	0.223607
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−4.00000	−0.603023
$45$	−1.41421	−0.210819
$46$	−4.00000	−0.589768
$47$	−4.24264	−0.618853	−0.309426	−	0.950923i	$-0.600137\pi$
$47$	−4.24264	−0.618853	−0.309426	+	0.950923i	$0.600137\pi$
$48$	−1.41421	−0.204124
$49$	0	0
$50$	−3.00000	−0.424264
$51$	4.00000	0.560112
$52$	1.41421	0.196116
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	5.65685	0.769800
$55$	−5.65685	−0.762770
$56$	0	0
$57$	−6.00000	−0.794719
$58$	1.00000	0.131306
$59$	−11.3137	−1.47292	−0.736460	−	0.676481i	$-0.763504\pi$
$59$	−11.3137	−1.47292	−0.736460	+	0.676481i	$0.763504\pi$
$60$	−2.00000	−0.258199
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	1.41421	0.179605
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	5.65685	0.696311
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−2.82843	−0.342997
$69$	5.65685	0.681005
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	2.82843	0.331042	0.165521	−	0.986206i	$-0.447069\pi$
$73$	2.82843	0.331042	0.165521	+	0.986206i	$0.447069\pi$
$74$	−10.0000	−1.16248
$75$	4.24264	0.489898
$76$	4.24264	0.486664
$77$	0	0
$78$	−2.00000	−0.226455
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	1.41421	0.158114
$81$	−5.00000	−0.555556
$82$	−5.65685	−0.624695
$83$	2.82843	0.310460	0.155230	−	0.987878i	$-0.450388\pi$
$83$	2.82843	0.310460	0.155230	+	0.987878i	$0.450388\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	2.00000	0.215666
$87$	−1.41421	−0.151620
$88$	−4.00000	−0.426401
$89$	14.1421	1.49906	0.749532	−	0.661968i	$-0.230279\pi$
$89$	14.1421	1.49906	0.749532	+	0.661968i	$0.230279\pi$
$90$	−1.41421	−0.149071
$91$	0	0
$92$	−4.00000	−0.417029
$93$	−2.00000	−0.207390
$94$	−4.24264	−0.437595
$95$	6.00000	0.615587
$96$	−1.41421	−0.144338
$97$	−2.82843	−0.287183	−0.143592	−	0.989637i	$-0.545865\pi$
$97$	−2.82843	−0.287183	−0.143592	+	0.989637i	$0.545865\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	−3.00000	−0.300000
$101$	−11.3137	−1.12576	−0.562878	−	0.826540i	$-0.690306\pi$
$101$	−11.3137	−1.12576	−0.562878	+	0.826540i	$0.690306\pi$
$102$	4.00000	0.396059
$103$	11.3137	1.11477	0.557386	−	0.830253i	$-0.311804\pi$
$103$	11.3137	1.11477	0.557386	+	0.830253i	$0.311804\pi$
$104$	1.41421	0.138675
$105$	0	0
$106$	−10.0000	−0.971286
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	5.65685	0.544331
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−5.65685	−0.539360
$111$	14.1421	1.34231
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	−6.00000	−0.561951
$115$	−5.65685	−0.527504
$116$	1.00000	0.0928477
$117$	−1.41421	−0.130744
$118$	−11.3137	−1.04151
$119$	0	0
$120$	−2.00000	−0.182574
$121$	5.00000	0.454545
$122$	0	0
$123$	8.00000	0.721336
$124$	1.41421	0.127000
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	1.00000	0.0883883
$129$	−2.82843	−0.249029
$130$	2.00000	0.175412
$131$	1.41421	0.123560	0.0617802	−	0.998090i	$-0.480322\pi$
$131$	1.41421	0.123560	0.0617802	+	0.998090i	$0.480322\pi$
$132$	5.65685	0.492366
$133$	0	0
$134$	−4.00000	−0.345547
$135$	8.00000	0.688530
$136$	−2.82843	−0.242536
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	5.65685	0.481543
$139$	−16.9706	−1.43942	−0.719712	−	0.694273i	$-0.755726\pi$
$139$	−16.9706	−1.43942	−0.719712	+	0.694273i	$0.755726\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	−5.65685	−0.473050
$144$	−1.00000	−0.0833333
$145$	1.41421	0.117444
$146$	2.82843	0.234082
$147$	0	0
$148$	−10.0000	−0.821995
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	4.24264	0.346410
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	4.24264	0.344124
$153$	2.82843	0.228665
$154$	0	0
$155$	2.00000	0.160644
$156$	−2.00000	−0.160128
$157$	−16.9706	−1.35440	−0.677199	−	0.735800i	$-0.736806\pi$
$157$	−16.9706	−1.35440	−0.677199	+	0.735800i	$0.736806\pi$
$158$	6.00000	0.477334
$159$	14.1421	1.12154
$160$	1.41421	0.111803
$161$	0	0
$162$	−5.00000	−0.392837
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	−5.65685	−0.441726
$165$	8.00000	0.622799
$166$	2.82843	0.219529
$167$	14.1421	1.09435	0.547176	−	0.837018i	$-0.315703\pi$
$167$	14.1421	1.09435	0.547176	+	0.837018i	$0.315703\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	−4.00000	−0.306786
$171$	−4.24264	−0.324443
$172$	2.00000	0.152499
$173$	−15.5563	−1.18273	−0.591364	−	0.806405i	$-0.701410\pi$
$173$	−15.5563	−1.18273	−0.591364	+	0.806405i	$0.701410\pi$
$174$	−1.41421	−0.107211
$175$	0	0
$176$	−4.00000	−0.301511
$177$	16.0000	1.20263
$178$	14.1421	1.06000
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−1.41421	−0.105409
$181$	−15.5563	−1.15629	−0.578147	−	0.815933i	$-0.696224\pi$
$181$	−15.5563	−1.15629	−0.578147	+	0.815933i	$0.696224\pi$
$182$	0	0
$183$	0	0
$184$	−4.00000	−0.294884
$185$	−14.1421	−1.03975
$186$	−2.00000	−0.146647
$187$	11.3137	0.827340
$188$	−4.24264	−0.309426
$189$	0	0
$190$	6.00000	0.435286
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	−1.41421	−0.102062
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	−2.82843	−0.203069
$195$	−2.82843	−0.202548
$196$	0	0
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	4.00000	0.284268
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−3.00000	−0.212132
$201$	5.65685	0.399004
$202$	−11.3137	−0.796030
$203$	0	0
$204$	4.00000	0.280056
$205$	−8.00000	−0.558744
$206$	11.3137	0.788263
$207$	4.00000	0.278019
$208$	1.41421	0.0980581
$209$	−16.9706	−1.17388
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−10.0000	−0.686803
$213$	0	0
$214$	0	0
$215$	2.82843	0.192897
$216$	5.65685	0.384900
$217$	0	0
$218$	10.0000	0.677285
$219$	−4.00000	−0.270295
$220$	−5.65685	−0.381385
$221$	−4.00000	−0.269069
$222$	14.1421	0.949158
$223$	8.48528	0.568216	0.284108	−	0.958792i	$-0.408302\pi$
$223$	8.48528	0.568216	0.284108	+	0.958792i	$0.408302\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	−18.0000	−1.19734
$227$	11.3137	0.750917	0.375459	−	0.926839i	$-0.377485\pi$
$227$	11.3137	0.750917	0.375459	+	0.926839i	$0.377485\pi$
$228$	−6.00000	−0.397360
$229$	5.65685	0.373815	0.186908	−	0.982377i	$-0.440153\pi$
$229$	5.65685	0.373815	0.186908	+	0.982377i	$0.440153\pi$
$230$	−5.65685	−0.373002
$231$	0	0
$232$	1.00000	0.0656532
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	−1.41421	−0.0924500
$235$	−6.00000	−0.391397
$236$	−11.3137	−0.736460
$237$	−8.48528	−0.551178
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	−2.00000	−0.129099
$241$	−9.89949	−0.637683	−0.318841	−	0.947808i	$-0.603294\pi$
$241$	−9.89949	−0.637683	−0.318841	+	0.947808i	$0.603294\pi$
$242$	5.00000	0.321412
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0	0
$246$	8.00000	0.510061
$247$	6.00000	0.381771
$248$	1.41421	0.0898027
$249$	−4.00000	−0.253490
$250$	−11.3137	−0.715542
$251$	18.3848	1.16044	0.580218	−	0.814461i	$-0.302967\pi$
$251$	18.3848	1.16044	0.580218	+	0.814461i	$0.302967\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	−10.0000	−0.627456
$255$	5.65685	0.354246
$256$	1.00000	0.0625000
$257$	7.07107	0.441081	0.220541	−	0.975378i	$-0.429218\pi$
$257$	7.07107	0.441081	0.220541	+	0.975378i	$0.429218\pi$
$258$	−2.82843	−0.176090
$259$	0	0
$260$	2.00000	0.124035
$261$	−1.00000	−0.0618984
$262$	1.41421	0.0873704
$263$	−26.0000	−1.60323	−0.801614	−	0.597841i	$-0.796025\pi$
$263$	−26.0000	−1.60323	−0.801614	+	0.597841i	$0.796025\pi$
$264$	5.65685	0.348155
$265$	−14.1421	−0.868744
$266$	0	0
$267$	−20.0000	−1.22398
$268$	−4.00000	−0.244339
$269$	2.82843	0.172452	0.0862261	−	0.996276i	$-0.472519\pi$
$269$	2.82843	0.172452	0.0862261	+	0.996276i	$0.472519\pi$
$270$	8.00000	0.486864
$271$	7.07107	0.429537	0.214768	−	0.976665i	$-0.431100\pi$
$271$	7.07107	0.429537	0.214768	+	0.976665i	$0.431100\pi$
$272$	−2.82843	−0.171499
$273$	0	0
$274$	−2.00000	−0.120824
$275$	12.0000	0.723627
$276$	5.65685	0.340503
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	−16.9706	−1.01783
$279$	−1.41421	−0.0846668
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	6.00000	0.357295
$283$	−11.3137	−0.672530	−0.336265	−	0.941767i	$-0.609164\pi$
$283$	−11.3137	−0.672530	−0.336265	+	0.941767i	$0.609164\pi$
$284$	0	0
$285$	−8.48528	−0.502625
$286$	−5.65685	−0.334497
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−9.00000	−0.529412
$290$	1.41421	0.0830455
$291$	4.00000	0.234484
$292$	2.82843	0.165521
$293$	19.7990	1.15667	0.578335	−	0.815800i	$-0.303703\pi$
$293$	19.7990	1.15667	0.578335	+	0.815800i	$0.303703\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	−10.0000	−0.581238
$297$	−22.6274	−1.31298
$298$	−16.0000	−0.926855
$299$	−5.65685	−0.327144
$300$	4.24264	0.244949
$301$	0	0
$302$	4.00000	0.230174
$303$	16.0000	0.919176
$304$	4.24264	0.243332
$305$	0	0
$306$	2.82843	0.161690
$307$	−7.07107	−0.403567	−0.201784	−	0.979430i	$-0.564674\pi$
$307$	−7.07107	−0.403567	−0.201784	+	0.979430i	$0.564674\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	2.00000	0.113592
$311$	26.8701	1.52366	0.761831	−	0.647776i	$-0.224301\pi$
$311$	26.8701	1.52366	0.761831	+	0.647776i	$0.224301\pi$
$312$	−2.00000	−0.113228
$313$	12.7279	0.719425	0.359712	−	0.933063i	$-0.382875\pi$
$313$	12.7279	0.719425	0.359712	+	0.933063i	$0.382875\pi$
$314$	−16.9706	−0.957704
$315$	0	0
$316$	6.00000	0.337526
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	14.1421	0.793052
$319$	−4.00000	−0.223957
$320$	1.41421	0.0790569
$321$	0	0
$322$	0	0
$323$	−12.0000	−0.667698
$324$	−5.00000	−0.277778
$325$	−4.24264	−0.235339
$326$	12.0000	0.664619
$327$	−14.1421	−0.782062
$328$	−5.65685	−0.312348
$329$	0	0
$330$	8.00000	0.440386
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	2.82843	0.155230
$333$	10.0000	0.547997
$334$	14.1421	0.773823
$335$	−5.65685	−0.309067
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−11.0000	−0.598321
$339$	25.4558	1.38257
$340$	−4.00000	−0.216930
$341$	−5.65685	−0.306336
$342$	−4.24264	−0.229416
$343$	0	0
$344$	2.00000	0.107833
$345$	8.00000	0.430706
$346$	−15.5563	−0.836315
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	−1.41421	−0.0758098
$349$	12.7279	0.681310	0.340655	−	0.940188i	$-0.389351\pi$
$349$	12.7279	0.681310	0.340655	+	0.940188i	$0.389351\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	−4.00000	−0.213201
$353$	15.5563	0.827981	0.413990	−	0.910281i	$-0.364135\pi$
$353$	15.5563	0.827981	0.413990	+	0.910281i	$0.364135\pi$
$354$	16.0000	0.850390
$355$	0	0
$356$	14.1421	0.749532
$357$	0	0
$358$	4.00000	0.211407
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	−1.41421	−0.0745356
$361$	−1.00000	−0.0526316
$362$	−15.5563	−0.817624
$363$	−7.07107	−0.371135
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	−26.8701	−1.40261	−0.701303	−	0.712864i	$-0.747398\pi$
$367$	−26.8701	−1.40261	−0.701303	+	0.712864i	$0.747398\pi$
$368$	−4.00000	−0.208514
$369$	5.65685	0.294484
$370$	−14.1421	−0.735215
$371$	0	0
$372$	−2.00000	−0.103695
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	11.3137	0.585018
$375$	16.0000	0.826236
$376$	−4.24264	−0.218797
$377$	1.41421	0.0728357
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	6.00000	0.307794
$381$	14.1421	0.724524
$382$	−18.0000	−0.920960
$383$	−8.48528	−0.433578	−0.216789	−	0.976219i	$-0.569558\pi$
$383$	−8.48528	−0.433578	−0.216789	+	0.976219i	$0.569558\pi$
$384$	−1.41421	−0.0721688
$385$	0	0
$386$	26.0000	1.32337
$387$	−2.00000	−0.101666
$388$	−2.82843	−0.143592
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	−2.82843	−0.143223
$391$	11.3137	0.572159
$392$	0	0
$393$	−2.00000	−0.100887
$394$	−8.00000	−0.403034
$395$	8.48528	0.426941
$396$	4.00000	0.201008
$397$	21.2132	1.06466	0.532330	−	0.846537i	$-0.321317\pi$
$397$	21.2132	1.06466	0.532330	+	0.846537i	$0.321317\pi$
$398$	0	0
$399$	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
$403$	2.00000	0.0996271
$404$	−11.3137	−0.562878
$405$	−7.07107	−0.351364
$406$	0	0
$407$	40.0000	1.98273
$408$	4.00000	0.198030
$409$	−25.4558	−1.25871	−0.629355	−	0.777118i	$-0.716681\pi$
$409$	−25.4558	−1.25871	−0.629355	+	0.777118i	$0.716681\pi$
$410$	−8.00000	−0.395092
$411$	2.82843	0.139516
$412$	11.3137	0.557386
$413$	0	0
$414$	4.00000	0.196589
$415$	4.00000	0.196352
$416$	1.41421	0.0693375
$417$	24.0000	1.17529
$418$	−16.9706	−0.830057
$419$	−16.9706	−0.829066	−0.414533	−	0.910034i	$-0.636055\pi$
$419$	−16.9706	−0.829066	−0.414533	+	0.910034i	$0.636055\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	4.00000	0.194717
$423$	4.24264	0.206284
$424$	−10.0000	−0.485643
$425$	8.48528	0.411597
$426$	0	0
$427$	0	0
$428$	0	0
$429$	8.00000	0.386244
$430$	2.82843	0.136399
$431$	4.00000	0.192673	0.0963366	−	0.995349i	$-0.469287\pi$
$431$	4.00000	0.192673	0.0963366	+	0.995349i	$0.469287\pi$
$432$	5.65685	0.272166
$433$	−16.9706	−0.815553	−0.407777	−	0.913082i	$-0.633696\pi$
$433$	−16.9706	−0.815553	−0.407777	+	0.913082i	$0.633696\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	10.0000	0.478913
$437$	−16.9706	−0.811812
$438$	−4.00000	−0.191127
$439$	−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
$442$	−4.00000	−0.190261
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	14.1421	0.671156
$445$	20.0000	0.948091
$446$	8.48528	0.401790
$447$	22.6274	1.07024
$448$	0	0
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	3.00000	0.141421
$451$	22.6274	1.06548
$452$	−18.0000	−0.846649
$453$	−5.65685	−0.265782
$454$	11.3137	0.530979
$455$	0	0
$456$	−6.00000	−0.280976
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	5.65685	0.264327
$459$	−16.0000	−0.746816
$460$	−5.65685	−0.263752
$461$	25.4558	1.18560	0.592798	−	0.805351i	$-0.298023\pi$
$461$	25.4558	1.18560	0.592798	+	0.805351i	$0.298023\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	1.00000	0.0464238
$465$	−2.82843	−0.131165
$466$	−16.0000	−0.741186
$467$	−1.41421	−0.0654420	−0.0327210	−	0.999465i	$-0.510417\pi$
$467$	−1.41421	−0.0654420	−0.0327210	+	0.999465i	$0.510417\pi$
$468$	−1.41421	−0.0653720
$469$	0	0
$470$	−6.00000	−0.276759
$471$	24.0000	1.10586
$472$	−11.3137	−0.520756
$473$	−8.00000	−0.367840
$474$	−8.48528	−0.389742
$475$	−12.7279	−0.583997
$476$	0	0
$477$	10.0000	0.457869
$478$	16.0000	0.731823
$479$	38.1838	1.74466	0.872330	−	0.488917i	$-0.162608\pi$
$479$	38.1838	1.74466	0.872330	+	0.488917i	$0.162608\pi$
$480$	−2.00000	−0.0912871
$481$	−14.1421	−0.644826
$482$	−9.89949	−0.450910
$483$	0	0
$484$	5.00000	0.227273
$485$	−4.00000	−0.181631
$486$	−9.89949	−0.449050
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	0	0
$489$	−16.9706	−0.767435
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	8.00000	0.360668
$493$	−2.82843	−0.127386
$494$	6.00000	0.269953
$495$	5.65685	0.254257
$496$	1.41421	0.0635001
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	−11.3137	−0.505964
$501$	−20.0000	−0.893534
$502$	18.3848	0.820553
$503$	12.7279	0.567510	0.283755	−	0.958897i	$-0.408420\pi$
$503$	12.7279	0.567510	0.283755	+	0.958897i	$0.408420\pi$
$504$	0	0
$505$	−16.0000	−0.711991
$506$	16.0000	0.711287
$507$	15.5563	0.690882
$508$	−10.0000	−0.443678
$509$	38.1838	1.69247	0.846233	−	0.532813i	$-0.178865\pi$
$509$	38.1838	1.69247	0.846233	+	0.532813i	$0.178865\pi$
$510$	5.65685	0.250490
$511$	0	0
$512$	1.00000	0.0441942
$513$	24.0000	1.05963
$514$	7.07107	0.311891
$515$	16.0000	0.705044
$516$	−2.82843	−0.124515
$517$	16.9706	0.746364
$518$	0	0
$519$	22.0000	0.965693
$520$	2.00000	0.0877058
$521$	18.3848	0.805452	0.402726	−	0.915321i	$-0.368063\pi$
$521$	18.3848	0.805452	0.402726	+	0.915321i	$0.368063\pi$
$522$	−1.00000	−0.0437688
$523$	−8.48528	−0.371035	−0.185518	−	0.982641i	$-0.559396\pi$
$523$	−8.48528	−0.371035	−0.185518	+	0.982641i	$0.559396\pi$
$524$	1.41421	0.0617802
$525$	0	0
$526$	−26.0000	−1.13365
$527$	−4.00000	−0.174243
$528$	5.65685	0.246183
$529$	−7.00000	−0.304348
$530$	−14.1421	−0.614295
$531$	11.3137	0.490973
$532$	0	0
$533$	−8.00000	−0.346518
$534$	−20.0000	−0.865485
$535$	0	0
$536$	−4.00000	−0.172774
$537$	−5.65685	−0.244111
$538$	2.82843	0.121942
$539$	0	0
$540$	8.00000	0.344265
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	7.07107	0.303728
$543$	22.0000	0.944110
$544$	−2.82843	−0.121268
$545$	14.1421	0.605783
$546$	0	0
$547$	40.0000	1.71028	0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000	1.71028	0.855138	+	0.518400i	$0.173472\pi$
$548$	−2.00000	−0.0854358
$549$	0	0
$550$	12.0000	0.511682
$551$	4.24264	0.180743
$552$	5.65685	0.240772
$553$	0	0
$554$	12.0000	0.509831
$555$	20.0000	0.848953
$556$	−16.9706	−0.719712
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	−1.41421	−0.0598684
$559$	2.82843	0.119630
$560$	0	0
$561$	−16.0000	−0.675521
$562$	10.0000	0.421825
$563$	15.5563	0.655622	0.327811	−	0.944743i	$-0.393689\pi$
$563$	15.5563	0.655622	0.327811	+	0.944743i	$0.393689\pi$
$564$	6.00000	0.252646
$565$	−25.4558	−1.07094
$566$	−11.3137	−0.475551
$567$	0	0
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	−8.48528	−0.355409
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−5.65685	−0.236525
$573$	25.4558	1.06343
$574$	0	0
$575$	12.0000	0.500435
$576$	−1.00000	−0.0416667
$577$	8.48528	0.353247	0.176623	−	0.984278i	$-0.443483\pi$
$577$	8.48528	0.353247	0.176623	+	0.984278i	$0.443483\pi$
$578$	−9.00000	−0.374351
$579$	−36.7696	−1.52809
$580$	1.41421	0.0587220
$581$	0	0
$582$	4.00000	0.165805
$583$	40.0000	1.65663
$584$	2.82843	0.117041
$585$	−2.00000	−0.0826898
$586$	19.7990	0.817889
$587$	−25.4558	−1.05068	−0.525338	−	0.850894i	$-0.676061\pi$
$587$	−25.4558	−1.05068	−0.525338	+	0.850894i	$0.676061\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	−16.0000	−0.658710
$591$	11.3137	0.465384
$592$	−10.0000	−0.410997
$593$	−26.8701	−1.10342	−0.551711	−	0.834036i	$-0.686025\pi$
$593$	−26.8701	−1.10342	−0.551711	+	0.834036i	$0.686025\pi$
$594$	−22.6274	−0.928414
$595$	0	0
$596$	−16.0000	−0.655386
$597$	0	0
$598$	−5.65685	−0.231326
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	4.24264	0.173205
$601$	42.4264	1.73061	0.865305	−	0.501246i	$-0.167125\pi$
$601$	42.4264	1.73061	0.865305	+	0.501246i	$0.167125\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	4.00000	0.162758
$605$	7.07107	0.287480
$606$	16.0000	0.649956
$607$	21.2132	0.861017	0.430509	−	0.902586i	$-0.358334\pi$
$607$	21.2132	0.861017	0.430509	+	0.902586i	$0.358334\pi$
$608$	4.24264	0.172062
$609$	0	0
$610$	0	0
$611$	−6.00000	−0.242734
$612$	2.82843	0.114332
$613$	32.0000	1.29247	0.646234	−	0.763139i	$-0.276343\pi$
$613$	32.0000	1.29247	0.646234	+	0.763139i	$0.276343\pi$
$614$	−7.07107	−0.285365
$615$	11.3137	0.456213
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−16.0000	−0.643614
$619$	29.6985	1.19368	0.596841	−	0.802359i	$-0.296422\pi$
$619$	29.6985	1.19368	0.596841	+	0.802359i	$0.296422\pi$
$620$	2.00000	0.0803219
$621$	−22.6274	−0.908007
$622$	26.8701	1.07739
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	−1.00000	−0.0400000
$626$	12.7279	0.508710
$627$	24.0000	0.958468
$628$	−16.9706	−0.677199
$629$	28.2843	1.12777
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	6.00000	0.238667
$633$	−5.65685	−0.224840
$634$	6.00000	0.238290
$635$	−14.1421	−0.561214
$636$	14.1421	0.560772
$637$	0	0
$638$	−4.00000	−0.158362
$639$	0	0
$640$	1.41421	0.0559017
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	−12.0000	−0.472134
$647$	−11.3137	−0.444788	−0.222394	−	0.974957i	$-0.571387\pi$
$647$	−11.3137	−0.444788	−0.222394	+	0.974957i	$0.571387\pi$
$648$	−5.00000	−0.196419
$649$	45.2548	1.77641
$650$	−4.24264	−0.166410
$651$	0	0
$652$	12.0000	0.469956
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	−14.1421	−0.553001
$655$	2.00000	0.0781465
$656$	−5.65685	−0.220863
$657$	−2.82843	−0.110347
$658$	0	0
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	8.00000	0.311400
$661$	38.1838	1.48518	0.742588	−	0.669748i	$-0.233598\pi$
$661$	38.1838	1.48518	0.742588	+	0.669748i	$0.233598\pi$
$662$	18.0000	0.699590
$663$	5.65685	0.219694
$664$	2.82843	0.109764
$665$	0	0
$666$	10.0000	0.387492
$667$	−4.00000	−0.154881
$668$	14.1421	0.547176
$669$	−12.0000	−0.463947
$670$	−5.65685	−0.218543
$671$	0	0
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	22.0000	0.847408
$675$	−16.9706	−0.653197
$676$	−11.0000	−0.423077
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	25.4558	0.977626
$679$	0	0
$680$	−4.00000	−0.153393
$681$	−16.0000	−0.613121
$682$	−5.65685	−0.216612
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	−4.24264	−0.162221
$685$	−2.82843	−0.108069
$686$	0	0
$687$	−8.00000	−0.305219
$688$	2.00000	0.0762493
$689$	−14.1421	−0.538772
$690$	8.00000	0.304555
$691$	−48.0833	−1.82917	−0.914587	−	0.404390i	$-0.867484\pi$
$691$	−48.0833	−1.82917	−0.914587	+	0.404390i	$0.867484\pi$
$692$	−15.5563	−0.591364
$693$	0	0
$694$	16.0000	0.607352
$695$	−24.0000	−0.910372
$696$	−1.41421	−0.0536056
$697$	16.0000	0.606043
$698$	12.7279	0.481759
$699$	22.6274	0.855848
$700$	0	0
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	8.00000	0.301941
$703$	−42.4264	−1.60014
$704$	−4.00000	−0.150756
$705$	8.48528	0.319574
$706$	15.5563	0.585471
$707$	0	0
$708$	16.0000	0.601317
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	14.1421	0.529999
$713$	−5.65685	−0.211851
$714$	0	0
$715$	−8.00000	−0.299183
$716$	4.00000	0.149487
$717$	−22.6274	−0.845036
$718$	30.0000	1.11959
$719$	−45.2548	−1.68772	−0.843860	−	0.536563i	$-0.819722\pi$
$719$	−45.2548	−1.68772	−0.843860	+	0.536563i	$0.819722\pi$
$720$	−1.41421	−0.0527046
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	14.0000	0.520666
$724$	−15.5563	−0.578147
$725$	−3.00000	−0.111417
$726$	−7.07107	−0.262432
$727$	46.6690	1.73086	0.865430	−	0.501031i	$-0.167046\pi$
$727$	46.6690	1.73086	0.865430	+	0.501031i	$0.167046\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	4.00000	0.148047
$731$	−5.65685	−0.209226
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	−26.8701	−0.991792
$735$	0	0
$736$	−4.00000	−0.147442
$737$	16.0000	0.589368
$738$	5.65685	0.208232
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	−14.1421	−0.519875
$741$	−8.48528	−0.311715
$742$	0	0
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	−2.00000	−0.0733236
$745$	−22.6274	−0.829004
$746$	16.0000	0.585802
$747$	−2.82843	−0.103487
$748$	11.3137	0.413670
$749$	0	0
$750$	16.0000	0.584237
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	−4.24264	−0.154713
$753$	−26.0000	−0.947493
$754$	1.41421	0.0515026
$755$	5.65685	0.205874
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	−2.00000	−0.0726433
$759$	−22.6274	−0.821323
$760$	6.00000	0.217643
$761$	−26.8701	−0.974039	−0.487019	−	0.873391i	$-0.661916\pi$
$761$	−26.8701	−0.974039	−0.487019	+	0.873391i	$0.661916\pi$
$762$	14.1421	0.512316
$763$	0	0
$764$	−18.0000	−0.651217
$765$	4.00000	0.144620
$766$	−8.48528	−0.306586
$767$	−16.0000	−0.577727
$768$	−1.41421	−0.0510310
$769$	−25.4558	−0.917961	−0.458981	−	0.888446i	$-0.651785\pi$
$769$	−25.4558	−0.917961	−0.458981	+	0.888446i	$0.651785\pi$
$770$	0	0
$771$	−10.0000	−0.360141
$772$	26.0000	0.935760
$773$	−19.7990	−0.712120	−0.356060	−	0.934463i	$-0.615880\pi$
$773$	−19.7990	−0.712120	−0.356060	+	0.934463i	$0.615880\pi$
$774$	−2.00000	−0.0718885
$775$	−4.24264	−0.152400
$776$	−2.82843	−0.101535
$777$	0	0
$778$	2.00000	0.0717035
$779$	−24.0000	−0.859889
$780$	−2.82843	−0.101274
$781$	0	0
$782$	11.3137	0.404577
$783$	5.65685	0.202159
$784$	0	0
$785$	−24.0000	−0.856597
$786$	−2.00000	−0.0713376
$787$	−11.3137	−0.403290	−0.201645	−	0.979459i	$-0.564629\pi$
$787$	−11.3137	−0.403290	−0.201645	+	0.979459i	$0.564629\pi$
$788$	−8.00000	−0.284988
$789$	36.7696	1.30903
$790$	8.48528	0.301893
$791$	0	0
$792$	4.00000	0.142134
$793$	0	0
$794$	21.2132	0.752828
$795$	20.0000	0.709327
$796$	0	0
$797$	16.9706	0.601128	0.300564	−	0.953762i	$-0.402825\pi$
$797$	16.9706	0.601128	0.300564	+	0.953762i	$0.402825\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	−3.00000	−0.106066
$801$	−14.1421	−0.499688
$802$	−18.0000	−0.635602
$803$	−11.3137	−0.399252
$804$	5.65685	0.199502
$805$	0	0
$806$	2.00000	0.0704470
$807$	−4.00000	−0.140807
$808$	−11.3137	−0.398015
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	−7.07107	−0.248452
$811$	50.9117	1.78775	0.893876	−	0.448315i	$-0.147976\pi$
$811$	50.9117	1.78775	0.893876	+	0.448315i	$0.147976\pi$
$812$	0	0
$813$	−10.0000	−0.350715
$814$	40.0000	1.40200
$815$	16.9706	0.594453
$816$	4.00000	0.140028
$817$	8.48528	0.296862
$818$	−25.4558	−0.890043
$819$	0	0
$820$	−8.00000	−0.279372
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	2.82843	0.0986527
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	11.3137	0.394132
$825$	−16.9706	−0.590839
$826$	0	0
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	4.00000	0.139010
$829$	−11.3137	−0.392941	−0.196471	−	0.980510i	$-0.562948\pi$
$829$	−11.3137	−0.392941	−0.196471	+	0.980510i	$0.562948\pi$
$830$	4.00000	0.138842
$831$	−16.9706	−0.588702
$832$	1.41421	0.0490290
$833$	0	0
$834$	24.0000	0.831052
$835$	20.0000	0.692129
$836$	−16.9706	−0.586939
$837$	8.00000	0.276520
$838$	−16.9706	−0.586238
$839$	9.89949	0.341769	0.170884	−	0.985291i	$-0.445338\pi$
$839$	9.89949	0.341769	0.170884	+	0.985291i	$0.445338\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	2.00000	0.0689246
$843$	−14.1421	−0.487081
$844$	4.00000	0.137686
$845$	−15.5563	−0.535155
$846$	4.24264	0.145865
$847$	0	0
$848$	−10.0000	−0.343401
$849$	16.0000	0.549119
$850$	8.48528	0.291043
$851$	40.0000	1.37118
$852$	0	0
$853$	−42.4264	−1.45265	−0.726326	−	0.687350i	$-0.758774\pi$
$853$	−42.4264	−1.45265	−0.726326	+	0.687350i	$0.758774\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	35.3553	1.20772	0.603858	−	0.797092i	$-0.293630\pi$
$857$	35.3553	1.20772	0.603858	+	0.797092i	$0.293630\pi$
$858$	8.00000	0.273115
$859$	−24.0416	−0.820290	−0.410145	−	0.912020i	$-0.634522\pi$
$859$	−24.0416	−0.820290	−0.410145	+	0.912020i	$0.634522\pi$
$860$	2.82843	0.0964486
$861$	0	0
$862$	4.00000	0.136241
$863$	20.0000	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$864$	5.65685	0.192450
$865$	−22.0000	−0.748022
$866$	−16.9706	−0.576683
$867$	12.7279	0.432263
$868$	0	0
$869$	−24.0000	−0.814144
$870$	−2.00000	−0.0678064
$871$	−5.65685	−0.191675
$872$	10.0000	0.338643
$873$	2.82843	0.0957278
$874$	−16.9706	−0.574038
$875$	0	0
$876$	−4.00000	−0.135147
$877$	−56.0000	−1.89099	−0.945493	−	0.325643i	$-0.894419\pi$
$877$	−56.0000	−1.89099	−0.945493	+	0.325643i	$0.894419\pi$
$878$	−16.9706	−0.572729
$879$	−28.0000	−0.944417
$880$	−5.65685	−0.190693
$881$	−22.6274	−0.762337	−0.381169	−	0.924506i	$-0.624478\pi$
$881$	−22.6274	−0.762337	−0.381169	+	0.924506i	$0.624478\pi$
$882$	0	0
$883$	56.0000	1.88455	0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000	1.88455	0.942275	+	0.334840i	$0.108682\pi$
$884$	−4.00000	−0.134535
$885$	22.6274	0.760612
$886$	−6.00000	−0.201574
$887$	−15.5563	−0.522331	−0.261166	−	0.965294i	$-0.584107\pi$
$887$	−15.5563	−0.522331	−0.261166	+	0.965294i	$0.584107\pi$
$888$	14.1421	0.474579
$889$	0	0
$890$	20.0000	0.670402
$891$	20.0000	0.670025
$892$	8.48528	0.284108
$893$	−18.0000	−0.602347
$894$	22.6274	0.756774
$895$	5.65685	0.189088
$896$	0	0
$897$	8.00000	0.267112
$898$	−26.0000	−0.867631
$899$	1.41421	0.0471667
$900$	3.00000	0.100000
$901$	28.2843	0.942286
$902$	22.6274	0.753411
$903$	0	0
$904$	−18.0000	−0.598671
$905$	−22.0000	−0.731305
$906$	−5.65685	−0.187936
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	11.3137	0.375459
$909$	11.3137	0.375252
$910$	0	0
$911$	42.0000	1.39152	0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000	1.39152	0.695761	+	0.718273i	$0.255067\pi$
$912$	−6.00000	−0.198680
$913$	−11.3137	−0.374429
$914$	−10.0000	−0.330771
$915$	0	0
$916$	5.65685	0.186908
$917$	0	0
$918$	−16.0000	−0.528079
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	−5.65685	−0.186501
$921$	10.0000	0.329511
$922$	25.4558	0.838344
$923$	0	0
$924$	0	0
$925$	30.0000	0.986394
$926$	0	0
$927$	−11.3137	−0.371591
$928$	1.00000	0.0328266
$929$	−57.9828	−1.90235	−0.951176	−	0.308648i	$-0.900123\pi$
$929$	−57.9828	−1.90235	−0.951176	+	0.308648i	$0.900123\pi$
$930$	−2.82843	−0.0927478
$931$	0	0
$932$	−16.0000	−0.524097
$933$	−38.0000	−1.24406
$934$	−1.41421	−0.0462745
$935$	16.0000	0.523256
$936$	−1.41421	−0.0462250
$937$	−24.0416	−0.785406	−0.392703	−	0.919665i	$-0.628460\pi$
$937$	−24.0416	−0.785406	−0.392703	+	0.919665i	$0.628460\pi$
$938$	0	0
$939$	−18.0000	−0.587408
$940$	−6.00000	−0.195698
$941$	41.0122	1.33696	0.668480	−	0.743730i	$-0.266945\pi$
$941$	41.0122	1.33696	0.668480	+	0.743730i	$0.266945\pi$
$942$	24.0000	0.781962
$943$	22.6274	0.736850
$944$	−11.3137	−0.368230
$945$	0	0
$946$	−8.00000	−0.260102
$947$	−42.0000	−1.36482	−0.682408	−	0.730971i	$-0.739067\pi$
$947$	−42.0000	−1.36482	−0.682408	+	0.730971i	$0.739067\pi$
$948$	−8.48528	−0.275589
$949$	4.00000	0.129845
$950$	−12.7279	−0.412948
$951$	−8.48528	−0.275154
$952$	0	0
$953$	−12.0000	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$954$	10.0000	0.323762
$955$	−25.4558	−0.823732
$956$	16.0000	0.517477
$957$	5.65685	0.182860
$958$	38.1838	1.23366
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	−29.0000	−0.935484
$962$	−14.1421	−0.455961
$963$	0	0
$964$	−9.89949	−0.318841
$965$	36.7696	1.18365
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	5.00000	0.160706
$969$	16.9706	0.545173
$970$	−4.00000	−0.128432
$971$	−55.1543	−1.76999	−0.884993	−	0.465604i	$-0.845837\pi$
$971$	−55.1543	−1.76999	−0.884993	+	0.465604i	$0.845837\pi$
$972$	−9.89949	−0.317526
$973$	0	0
$974$	−32.0000	−1.02535
$975$	6.00000	0.192154
$976$	0	0
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	−16.9706	−0.542659
$979$	−56.5685	−1.80794
$980$	0	0
$981$	−10.0000	−0.319275
$982$	28.0000	0.893516
$983$	43.8406	1.39830	0.699149	−	0.714976i	$-0.253562\pi$
$983$	43.8406	1.39830	0.699149	+	0.714976i	$0.253562\pi$
$984$	8.00000	0.255031
$985$	−11.3137	−0.360485
$986$	−2.82843	−0.0900755
$987$	0	0
$988$	6.00000	0.190885
$989$	−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	$0.330290\pi$
$992$	1.41421	0.0449013
$993$	−25.4558	−0.807817
$994$	0	0
$995$	0	0
$996$	−4.00000	−0.126745
$997$	11.3137	0.358309	0.179154	−	0.983821i	$-0.442664\pi$
$997$	11.3137	0.358309	0.179154	+	0.983821i	$0.442664\pi$
$998$	−32.0000	−1.01294
$999$	−56.5685	−1.78975

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.k.1.1	✓	2	1.1	even	1	trivial
2842.2.a.k.1.2	yes	2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{5} -1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{5} -1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +1.41421 q^{10} -4.00000 q^{11} -1.41421 q^{12} +1.41421 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.82843 q^{17} -1.00000 q^{18} +4.24264 q^{19} +1.41421 q^{20} -4.00000 q^{22} -4.00000 q^{23} -1.41421 q^{24} -3.00000 q^{25} +1.41421 q^{26} +5.65685 q^{27} +1.00000 q^{29} -2.00000 q^{30} +1.41421 q^{31} +1.00000 q^{32} +5.65685 q^{33} -2.82843 q^{34} -1.00000 q^{36} -10.0000 q^{37} +4.24264 q^{38} -2.00000 q^{39} +1.41421 q^{40} -5.65685 q^{41} +2.00000 q^{43} -4.00000 q^{44} -1.41421 q^{45} -4.00000 q^{46} -4.24264 q^{47} -1.41421 q^{48} -3.00000 q^{50} +4.00000 q^{51} +1.41421 q^{52} -10.0000 q^{53} +5.65685 q^{54} -5.65685 q^{55} -6.00000 q^{57} +1.00000 q^{58} -11.3137 q^{59} -2.00000 q^{60} +1.41421 q^{62} +1.00000 q^{64} +2.00000 q^{65} +5.65685 q^{66} -4.00000 q^{67} -2.82843 q^{68} +5.65685 q^{69} -1.00000 q^{72} +2.82843 q^{73} -10.0000 q^{74} +4.24264 q^{75} +4.24264 q^{76} -2.00000 q^{78} +6.00000 q^{79} +1.41421 q^{80} -5.00000 q^{81} -5.65685 q^{82} +2.82843 q^{83} -4.00000 q^{85} +2.00000 q^{86} -1.41421 q^{87} -4.00000 q^{88} +14.1421 q^{89} -1.41421 q^{90} -4.00000 q^{92} -2.00000 q^{93} -4.24264 q^{94} +6.00000 q^{95} -1.41421 q^{96} -2.82843 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 8 q^{11} - 4 q^{15} + 2 q^{16} - 2 q^{18} - 8 q^{22} - 8 q^{23} - 6 q^{25} + 2 q^{29} - 4 q^{30} + 2 q^{32} - 2 q^{36} - 20 q^{37} - 4 q^{39} + 4 q^{43} - 8 q^{44} - 8 q^{46} - 6 q^{50} + 8 q^{51} - 20 q^{53} - 12 q^{57} + 2 q^{58} - 4 q^{60} + 2 q^{64} + 4 q^{65} - 8 q^{67} - 2 q^{72} - 20 q^{74} - 4 q^{78} + 12 q^{79} - 10 q^{81} - 8 q^{85} + 4 q^{86} - 8 q^{88} - 8 q^{92} - 4 q^{93} + 12 q^{95} + 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 8 * q^11 - 4 * q^15 + 2 * q^16 - 2 * q^18 - 8 * q^22 - 8 * q^23 - 6 * q^25 + 2 * q^29 - 4 * q^30 + 2 * q^32 - 2 * q^36 - 20 * q^37 - 4 * q^39 + 4 * q^43 - 8 * q^44 - 8 * q^46 - 6 * q^50 + 8 * q^51 - 20 * q^53 - 12 * q^57 + 2 * q^58 - 4 * q^60 + 2 * q^64 + 4 * q^65 - 8 * q^67 - 2 * q^72 - 20 * q^74 - 4 * q^78 + 12 * q^79 - 10 * q^81 - 8 * q^85 + 4 * q^86 - 8 * q^88 - 8 * q^92 - 4 * q^93 + 12 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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