LMFDB - Embedded newform 2842.2.a.m.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:	$0$
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:	$\Z[a_1, \ldots, a_{31}]$
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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} +2.82843 q^{5} -2.82843 q^{6} +1.00000 q^{8} +5.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} +2.82843 q^{5} -2.82843 q^{6} +1.00000 q^{8} +5.00000 q^{9} +2.82843 q^{10} +4.00000 q^{11} -2.82843 q^{12} -2.82843 q^{13} -8.00000 q^{15} +1.00000 q^{16} +2.82843 q^{17} +5.00000 q^{18} +2.82843 q^{20} +4.00000 q^{22} +4.00000 q^{23} -2.82843 q^{24} +3.00000 q^{25} -2.82843 q^{26} -5.65685 q^{27} -1.00000 q^{29} -8.00000 q^{30} +1.41421 q^{31} +1.00000 q^{32} -11.3137 q^{33} +2.82843 q^{34} +5.00000 q^{36} +8.00000 q^{37} +8.00000 q^{39} +2.82843 q^{40} +5.65685 q^{41} -4.00000 q^{43} +4.00000 q^{44} +14.1421 q^{45} +4.00000 q^{46} -12.7279 q^{47} -2.82843 q^{48} +3.00000 q^{50} -8.00000 q^{51} -2.82843 q^{52} +10.0000 q^{53} -5.65685 q^{54} +11.3137 q^{55} -1.00000 q^{58} -9.89949 q^{59} -8.00000 q^{60} -12.7279 q^{61} +1.41421 q^{62} +1.00000 q^{64} -8.00000 q^{65} -11.3137 q^{66} +2.00000 q^{67} +2.82843 q^{68} -11.3137 q^{69} -12.0000 q^{71} +5.00000 q^{72} +11.3137 q^{73} +8.00000 q^{74} -8.48528 q^{75} +8.00000 q^{78} -6.00000 q^{79} +2.82843 q^{80} +1.00000 q^{81} +5.65685 q^{82} +1.41421 q^{83} +8.00000 q^{85} -4.00000 q^{86} +2.82843 q^{87} +4.00000 q^{88} +11.3137 q^{89} +14.1421 q^{90} +4.00000 q^{92} -4.00000 q^{93} -12.7279 q^{94} -2.82843 q^{96} +14.1421 q^{97} +20.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 10 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 10 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 10 q^{9} + 8 q^{11} - 16 q^{15} + 2 q^{16} + 10 q^{18} + 8 q^{22} + 8 q^{23} + 6 q^{25} - 2 q^{29} - 16 q^{30} + 2 q^{32} + 10 q^{36} + 16 q^{37} + 16 q^{39} - 8 q^{43} + 8 q^{44} + 8 q^{46} + 6 q^{50} - 16 q^{51} + 20 q^{53} - 2 q^{58} - 16 q^{60} + 2 q^{64} - 16 q^{65} + 4 q^{67} - 24 q^{71} + 10 q^{72} + 16 q^{74} + 16 q^{78} - 12 q^{79} + 2 q^{81} + 16 q^{85} - 8 q^{86} + 8 q^{88} + 8 q^{92} - 8 q^{93} + 40 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 10 * q^9 + 8 * q^11 - 16 * q^15 + 2 * q^16 + 10 * q^18 + 8 * q^22 + 8 * q^23 + 6 * q^25 - 2 * q^29 - 16 * q^30 + 2 * q^32 + 10 * q^36 + 16 * q^37 + 16 * q^39 - 8 * q^43 + 8 * q^44 + 8 * q^46 + 6 * q^50 - 16 * q^51 + 20 * q^53 - 2 * q^58 - 16 * q^60 + 2 * q^64 - 16 * q^65 + 4 * q^67 - 24 * q^71 + 10 * q^72 + 16 * q^74 + 16 * q^78 - 12 * q^79 + 2 * q^81 + 16 * q^85 - 8 * q^86 + 8 * q^88 + 8 * q^92 - 8 * q^93 + 40 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	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	$0.282047\pi$
$6$	−2.82843	−1.15470
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	5.00000	1.66667
$10$	2.82843	0.894427
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−2.82843	−0.816497
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	−8.00000	−2.06559
$16$	1.00000	0.250000
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	5.00000	1.17851
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	2.82843	0.632456
$21$	0	0
$22$	4.00000	0.852803
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−2.82843	−0.577350
$25$	3.00000	0.600000
$26$	−2.82843	−0.554700
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−8.00000	−1.46059
$31$	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
$33$	−11.3137	−1.96946
$34$	2.82843	0.485071
$35$	0	0
$36$	5.00000	0.833333
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	2.82843	0.447214
$41$	5.65685	0.883452	0.441726	−	0.897150i	$-0.354366\pi$
$41$	5.65685	0.883452	0.441726	+	0.897150i	$0.354366\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000	0.603023
$45$	14.1421	2.10819
$46$	4.00000	0.589768
$47$	−12.7279	−1.85656	−0.928279	−	0.371884i	$-0.878712\pi$
$47$	−12.7279	−1.85656	−0.928279	+	0.371884i	$0.878712\pi$
$48$	−2.82843	−0.408248
$49$	0	0
$50$	3.00000	0.424264
$51$	−8.00000	−1.12022
$52$	−2.82843	−0.392232
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	−5.65685	−0.769800
$55$	11.3137	1.52554
$56$	0	0
$57$	0	0
$58$	−1.00000	−0.131306
$59$	−9.89949	−1.28880	−0.644402	−	0.764687i	$-0.722894\pi$
$59$	−9.89949	−1.28880	−0.644402	+	0.764687i	$0.722894\pi$
$60$	−8.00000	−1.03280
$61$	−12.7279	−1.62964	−0.814822	−	0.579712i	$-0.803165\pi$
$61$	−12.7279	−1.62964	−0.814822	+	0.579712i	$0.803165\pi$
$62$	1.41421	0.179605
$63$	0	0
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	−11.3137	−1.39262
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	2.82843	0.342997
$69$	−11.3137	−1.36201
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	5.00000	0.589256
$73$	11.3137	1.32417	0.662085	−	0.749429i	$-0.269672\pi$
$73$	11.3137	1.32417	0.662085	+	0.749429i	$0.269672\pi$
$74$	8.00000	0.929981
$75$	−8.48528	−0.979796
$76$	0	0
$77$	0	0
$78$	8.00000	0.905822
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	2.82843	0.316228
$81$	1.00000	0.111111
$82$	5.65685	0.624695
$83$	1.41421	0.155230	0.0776151	−	0.996983i	$-0.475269\pi$
$83$	1.41421	0.155230	0.0776151	+	0.996983i	$0.475269\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	−4.00000	−0.431331
$87$	2.82843	0.303239
$88$	4.00000	0.426401
$89$	11.3137	1.19925	0.599625	−	0.800281i	$-0.295316\pi$
$89$	11.3137	1.19925	0.599625	+	0.800281i	$0.295316\pi$
$90$	14.1421	1.49071
$91$	0	0
$92$	4.00000	0.417029
$93$	−4.00000	−0.414781
$94$	−12.7279	−1.31278
$95$	0	0
$96$	−2.82843	−0.288675
$97$	14.1421	1.43592	0.717958	−	0.696086i	$-0.245077\pi$
$97$	14.1421	1.43592	0.717958	+	0.696086i	$0.245077\pi$
$98$	0	0
$99$	20.0000	2.01008
$100$	3.00000	0.300000
$101$	7.07107	0.703598	0.351799	−	0.936076i	$-0.385570\pi$
$101$	7.07107	0.703598	0.351799	+	0.936076i	$0.385570\pi$
$102$	−8.00000	−0.792118
$103$	19.7990	1.95085	0.975426	−	0.220326i	$-0.0707122\pi$
$103$	19.7990	1.95085	0.975426	+	0.220326i	$0.0707122\pi$
$104$	−2.82843	−0.277350
$105$	0	0
$106$	10.0000	0.971286
$107$	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	$0.530535\pi$
$110$	11.3137	1.07872
$111$	−22.6274	−2.14770
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	11.3137	1.05501
$116$	−1.00000	−0.0928477
$117$	−14.1421	−1.30744
$118$	−9.89949	−0.911322
$119$	0	0
$120$	−8.00000	−0.730297
$121$	5.00000	0.454545
$122$	−12.7279	−1.15233
$123$	−16.0000	−1.44267
$124$	1.41421	0.127000
$125$	−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
$129$	11.3137	0.996116
$130$	−8.00000	−0.701646
$131$	−5.65685	−0.494242	−0.247121	−	0.968985i	$-0.579484\pi$
$131$	−5.65685	−0.494242	−0.247121	+	0.968985i	$0.579484\pi$
$132$	−11.3137	−0.984732
$133$	0	0
$134$	2.00000	0.172774
$135$	−16.0000	−1.37706
$136$	2.82843	0.242536
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−11.3137	−0.963087
$139$	12.7279	1.07957	0.539784	−	0.841803i	$-0.318506\pi$
$139$	12.7279	1.07957	0.539784	+	0.841803i	$0.318506\pi$
$140$	0	0
$141$	36.0000	3.03175
$142$	−12.0000	−1.00702
$143$	−11.3137	−0.946100
$144$	5.00000	0.416667
$145$	−2.82843	−0.234888
$146$	11.3137	0.936329
$147$	0	0
$148$	8.00000	0.657596
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	−8.48528	−0.692820
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	14.1421	1.14332
$154$	0	0
$155$	4.00000	0.321288
$156$	8.00000	0.640513
$157$	21.2132	1.69300	0.846499	−	0.532390i	$-0.178706\pi$
$157$	21.2132	1.69300	0.846499	+	0.532390i	$0.178706\pi$
$158$	−6.00000	−0.477334
$159$	−28.2843	−2.24309
$160$	2.82843	0.223607
$161$	0	0
$162$	1.00000	0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	5.65685	0.441726
$165$	−32.0000	−2.49120
$166$	1.41421	0.109764
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	8.00000	0.613572
$171$	0	0
$172$	−4.00000	−0.304997
$173$	2.82843	0.215041	0.107521	−	0.994203i	$-0.465709\pi$
$173$	2.82843	0.215041	0.107521	+	0.994203i	$0.465709\pi$
$174$	2.82843	0.214423
$175$	0	0
$176$	4.00000	0.301511
$177$	28.0000	2.10461
$178$	11.3137	0.847998
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	14.1421	1.05409
$181$	−19.7990	−1.47165	−0.735824	−	0.677173i	$-0.763205\pi$
$181$	−19.7990	−1.47165	−0.735824	+	0.677173i	$0.763205\pi$
$182$	0	0
$183$	36.0000	2.66120
$184$	4.00000	0.294884
$185$	22.6274	1.66360
$186$	−4.00000	−0.293294
$187$	11.3137	0.827340
$188$	−12.7279	−0.928279
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	−2.82843	−0.204124
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	14.1421	1.01535
$195$	22.6274	1.62038
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	20.0000	1.42134
$199$	−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
$202$	7.07107	0.497519
$203$	0	0
$204$	−8.00000	−0.560112
$205$	16.0000	1.11749
$206$	19.7990	1.37946
$207$	20.0000	1.39010
$208$	−2.82843	−0.196116
$209$	0	0
$210$	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
$213$	33.9411	2.32561
$214$	12.0000	0.820303
$215$	−11.3137	−0.771589
$216$	−5.65685	−0.384900
$217$	0	0
$218$	−2.00000	−0.135457
$219$	−32.0000	−2.16236
$220$	11.3137	0.762770
$221$	−8.00000	−0.538138
$222$	−22.6274	−1.51865
$223$	25.4558	1.70465	0.852325	−	0.523013i	$-0.175192\pi$
$223$	25.4558	1.70465	0.852325	+	0.523013i	$0.175192\pi$
$224$	0	0
$225$	15.0000	1.00000
$226$	−6.00000	−0.399114
$227$	1.41421	0.0938647	0.0469323	−	0.998898i	$-0.485055\pi$
$227$	1.41421	0.0938647	0.0469323	+	0.998898i	$0.485055\pi$
$228$	0	0
$229$	18.3848	1.21490	0.607450	−	0.794358i	$-0.292192\pi$
$229$	18.3848	1.21490	0.607450	+	0.794358i	$0.292192\pi$
$230$	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	$0.584398\pi$
$234$	−14.1421	−0.924500
$235$	−36.0000	−2.34838
$236$	−9.89949	−0.644402
$237$	16.9706	1.10236
$238$	0	0
$239$	−28.0000	−1.81117	−0.905585	−	0.424165i	$-0.860568\pi$
$239$	−28.0000	−1.81117	−0.905585	+	0.424165i	$0.860568\pi$
$240$	−8.00000	−0.516398
$241$	−1.41421	−0.0910975	−0.0455488	−	0.998962i	$-0.514504\pi$
$241$	−1.41421	−0.0910975	−0.0455488	+	0.998962i	$0.514504\pi$
$242$	5.00000	0.321412
$243$	14.1421	0.907218
$244$	−12.7279	−0.814822
$245$	0	0
$246$	−16.0000	−1.02012
$247$	0	0
$248$	1.41421	0.0898027
$249$	−4.00000	−0.253490
$250$	−5.65685	−0.357771
$251$	−22.6274	−1.42823	−0.714115	−	0.700028i	$-0.753171\pi$
$251$	−22.6274	−1.42823	−0.714115	+	0.700028i	$0.753171\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	8.00000	0.501965
$255$	−22.6274	−1.41698
$256$	1.00000	0.0625000
$257$	26.8701	1.67611	0.838054	−	0.545587i	$-0.183693\pi$
$257$	26.8701	1.67611	0.838054	+	0.545587i	$0.183693\pi$
$258$	11.3137	0.704361
$259$	0	0
$260$	−8.00000	−0.496139
$261$	−5.00000	−0.309492
$262$	−5.65685	−0.349482
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	−11.3137	−0.696311
$265$	28.2843	1.73749
$266$	0	0
$267$	−32.0000	−1.95837
$268$	2.00000	0.122169
$269$	9.89949	0.603583	0.301791	−	0.953374i	$-0.402415\pi$
$269$	9.89949	0.603583	0.301791	+	0.953374i	$0.402415\pi$
$270$	−16.0000	−0.973729
$271$	−26.8701	−1.63224	−0.816120	−	0.577883i	$-0.803879\pi$
$271$	−26.8701	−1.63224	−0.816120	+	0.577883i	$0.803879\pi$
$272$	2.82843	0.171499
$273$	0	0
$274$	2.00000	0.120824
$275$	12.0000	0.723627
$276$	−11.3137	−0.681005
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	12.7279	0.763370
$279$	7.07107	0.423334
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	36.0000	2.14377
$283$	1.41421	0.0840663	0.0420331	−	0.999116i	$-0.486616\pi$
$283$	1.41421	0.0840663	0.0420331	+	0.999116i	$0.486616\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−11.3137	−0.668994
$287$	0	0
$288$	5.00000	0.294628
$289$	−9.00000	−0.529412
$290$	−2.82843	−0.166091
$291$	−40.0000	−2.34484
$292$	11.3137	0.662085
$293$	9.89949	0.578335	0.289167	−	0.957279i	$-0.406622\pi$
$293$	9.89949	0.578335	0.289167	+	0.957279i	$0.406622\pi$
$294$	0	0
$295$	−28.0000	−1.63022
$296$	8.00000	0.464991
$297$	−22.6274	−1.31298
$298$	22.0000	1.27443
$299$	−11.3137	−0.654289
$300$	−8.48528	−0.489898
$301$	0	0
$302$	16.0000	0.920697
$303$	−20.0000	−1.14897
$304$	0	0
$305$	−36.0000	−2.06135
$306$	14.1421	0.808452
$307$	−19.7990	−1.12999	−0.564994	−	0.825095i	$-0.691122\pi$
$307$	−19.7990	−1.12999	−0.564994	+	0.825095i	$0.691122\pi$
$308$	0	0
$309$	−56.0000	−3.18573
$310$	4.00000	0.227185
$311$	7.07107	0.400963	0.200482	−	0.979697i	$-0.435749\pi$
$311$	7.07107	0.400963	0.200482	+	0.979697i	$0.435749\pi$
$312$	8.00000	0.452911
$313$	−4.24264	−0.239808	−0.119904	−	0.992785i	$-0.538259\pi$
$313$	−4.24264	−0.239808	−0.119904	+	0.992785i	$0.538259\pi$
$314$	21.2132	1.19713
$315$	0	0
$316$	−6.00000	−0.337526
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	−28.2843	−1.58610
$319$	−4.00000	−0.223957
$320$	2.82843	0.158114
$321$	−33.9411	−1.89441
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−8.48528	−0.470679
$326$	−12.0000	−0.664619
$327$	5.65685	0.312825
$328$	5.65685	0.312348
$329$	0	0
$330$	−32.0000	−1.76154
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	1.41421	0.0776151
$333$	40.0000	2.19199
$334$	−5.65685	−0.309529
$335$	5.65685	0.309067
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	−5.00000	−0.271964
$339$	16.9706	0.921714
$340$	8.00000	0.433861
$341$	5.65685	0.306336
$342$	0	0
$343$	0	0
$344$	−4.00000	−0.215666
$345$	−32.0000	−1.72282
$346$	2.82843	0.152057
$347$	26.0000	1.39575	0.697877	−	0.716218i	$-0.254128\pi$
$347$	26.0000	1.39575	0.697877	+	0.716218i	$0.254128\pi$
$348$	2.82843	0.151620
$349$	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
$351$	16.0000	0.854017
$352$	4.00000	0.213201
$353$	−24.0416	−1.27961	−0.639803	−	0.768539i	$-0.720984\pi$
$353$	−24.0416	−1.27961	−0.639803	+	0.768539i	$0.720984\pi$
$354$	28.0000	1.48818
$355$	−33.9411	−1.80141
$356$	11.3137	0.599625
$357$	0	0
$358$	−10.0000	−0.528516
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	14.1421	0.745356
$361$	−19.0000	−1.00000
$362$	−19.7990	−1.04061
$363$	−14.1421	−0.742270
$364$	0	0
$365$	32.0000	1.67496
$366$	36.0000	1.88175
$367$	−1.41421	−0.0738213	−0.0369107	−	0.999319i	$-0.511752\pi$
$367$	−1.41421	−0.0738213	−0.0369107	+	0.999319i	$0.511752\pi$
$368$	4.00000	0.208514
$369$	28.2843	1.47242
$370$	22.6274	1.17634
$371$	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	$0.618058\pi$
$374$	11.3137	0.585018
$375$	16.0000	0.826236
$376$	−12.7279	−0.656392
$377$	2.82843	0.145671
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−22.6274	−1.15924
$382$	6.00000	0.306987
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−2.82843	−0.144338
$385$	0	0
$386$	14.0000	0.712581
$387$	−20.0000	−1.01666
$388$	14.1421	0.717958
$389$	28.0000	1.41966	0.709828	−	0.704375i	$-0.248773\pi$
$389$	28.0000	1.41966	0.709828	+	0.704375i	$0.248773\pi$
$390$	22.6274	1.14578
$391$	11.3137	0.572159
$392$	0	0
$393$	16.0000	0.807093
$394$	−10.0000	−0.503793
$395$	−16.9706	−0.853882
$396$	20.0000	1.00504
$397$	−8.48528	−0.425864	−0.212932	−	0.977067i	$-0.568301\pi$
$397$	−8.48528	−0.425864	−0.212932	+	0.977067i	$0.568301\pi$
$398$	−8.48528	−0.425329
$399$	0	0
$400$	3.00000	0.150000
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	−5.65685	−0.282138
$403$	−4.00000	−0.199254
$404$	7.07107	0.351799
$405$	2.82843	0.140546
$406$	0	0
$407$	32.0000	1.58618
$408$	−8.00000	−0.396059
$409$	25.4558	1.25871	0.629355	−	0.777118i	$-0.283319\pi$
$409$	25.4558	1.25871	0.629355	+	0.777118i	$0.283319\pi$
$410$	16.0000	0.790184
$411$	−5.65685	−0.279032
$412$	19.7990	0.975426
$413$	0	0
$414$	20.0000	0.982946
$415$	4.00000	0.196352
$416$	−2.82843	−0.138675
$417$	−36.0000	−1.76293
$418$	0	0
$419$	−38.1838	−1.86540	−0.932700	−	0.360654i	$-0.882553\pi$
$419$	−38.1838	−1.86540	−0.932700	+	0.360654i	$0.882553\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.00000	0.194717
$423$	−63.6396	−3.09426
$424$	10.0000	0.485643
$425$	8.48528	0.411597
$426$	33.9411	1.64445
$427$	0	0
$428$	12.0000	0.580042
$429$	32.0000	1.54497
$430$	−11.3137	−0.545595
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	−5.65685	−0.272166
$433$	33.9411	1.63111	0.815553	−	0.578682i	$-0.196433\pi$
$433$	33.9411	1.63111	0.815553	+	0.578682i	$0.196433\pi$
$434$	0	0
$435$	8.00000	0.383571
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	−32.0000	−1.52902
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	11.3137	0.539360
$441$	0	0
$442$	−8.00000	−0.380521
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	−22.6274	−1.07385
$445$	32.0000	1.51695
$446$	25.4558	1.20537
$447$	−62.2254	−2.94316
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	15.0000	0.707107
$451$	22.6274	1.06548
$452$	−6.00000	−0.282216
$453$	−45.2548	−2.12626
$454$	1.41421	0.0663723
$455$	0	0
$456$	0	0
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	18.3848	0.859064
$459$	−16.0000	−0.746816
$460$	11.3137	0.527504
$461$	−4.24264	−0.197599	−0.0987997	−	0.995107i	$-0.531500\pi$
$461$	−4.24264	−0.197599	−0.0987997	+	0.995107i	$0.531500\pi$
$462$	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
$467$	22.6274	1.04707	0.523536	−	0.852004i	$-0.324613\pi$
$467$	22.6274	1.04707	0.523536	+	0.852004i	$0.324613\pi$
$468$	−14.1421	−0.653720
$469$	0	0
$470$	−36.0000	−1.66056
$471$	−60.0000	−2.76465
$472$	−9.89949	−0.455661
$473$	−16.0000	−0.735681
$474$	16.9706	0.779484
$475$	0	0
$476$	0	0
$477$	50.0000	2.28934
$478$	−28.0000	−1.28069
$479$	−4.24264	−0.193851	−0.0969256	−	0.995292i	$-0.530901\pi$
$479$	−4.24264	−0.193851	−0.0969256	+	0.995292i	$0.530901\pi$
$480$	−8.00000	−0.365148
$481$	−22.6274	−1.03172
$482$	−1.41421	−0.0644157
$483$	0	0
$484$	5.00000	0.227273
$485$	40.0000	1.81631
$486$	14.1421	0.641500
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−12.7279	−0.576166
$489$	33.9411	1.53487
$490$	0	0
$491$	−40.0000	−1.80517	−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000	−1.80517	−0.902587	+	0.430507i	$0.858335\pi$
$492$	−16.0000	−0.721336
$493$	−2.82843	−0.127386
$494$	0	0
$495$	56.5685	2.54257
$496$	1.41421	0.0635001
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−5.65685	−0.252982
$501$	16.0000	0.714827
$502$	−22.6274	−1.00991
$503$	4.24264	0.189170	0.0945850	−	0.995517i	$-0.469848\pi$
$503$	4.24264	0.189170	0.0945850	+	0.995517i	$0.469848\pi$
$504$	0	0
$505$	20.0000	0.889988
$506$	16.0000	0.711287
$507$	14.1421	0.628074
$508$	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	$0.377264\pi$
$510$	−22.6274	−1.00196
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	26.8701	1.18519
$515$	56.0000	2.46765
$516$	11.3137	0.498058
$517$	−50.9117	−2.23909
$518$	0	0
$519$	−8.00000	−0.351161
$520$	−8.00000	−0.350823
$521$	−26.8701	−1.17720	−0.588599	−	0.808425i	$-0.700320\pi$
$521$	−26.8701	−1.17720	−0.588599	+	0.808425i	$0.700320\pi$
$522$	−5.00000	−0.218844
$523$	−21.2132	−0.927589	−0.463794	−	0.885943i	$-0.653512\pi$
$523$	−21.2132	−0.927589	−0.463794	+	0.885943i	$0.653512\pi$
$524$	−5.65685	−0.247121
$525$	0	0
$526$	2.00000	0.0872041
$527$	4.00000	0.174243
$528$	−11.3137	−0.492366
$529$	−7.00000	−0.304348
$530$	28.2843	1.22859
$531$	−49.4975	−2.14801
$532$	0	0
$533$	−16.0000	−0.693037
$534$	−32.0000	−1.38478
$535$	33.9411	1.46740
$536$	2.00000	0.0863868
$537$	28.2843	1.22056
$538$	9.89949	0.426798
$539$	0	0
$540$	−16.0000	−0.688530
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	−26.8701	−1.15417
$543$	56.0000	2.40319
$544$	2.82843	0.121268
$545$	−5.65685	−0.242313
$546$	0	0
$547$	10.0000	0.427569	0.213785	−	0.976881i	$-0.431421\pi$
$547$	10.0000	0.427569	0.213785	+	0.976881i	$0.431421\pi$
$548$	2.00000	0.0854358
$549$	−63.6396	−2.71607
$550$	12.0000	0.511682
$551$	0	0
$552$	−11.3137	−0.481543
$553$	0	0
$554$	−30.0000	−1.27458
$555$	−64.0000	−2.71665
$556$	12.7279	0.539784
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	7.07107	0.299342
$559$	11.3137	0.478519
$560$	0	0
$561$	−32.0000	−1.35104
$562$	−22.0000	−0.928014
$563$	−11.3137	−0.476816	−0.238408	−	0.971165i	$-0.576626\pi$
$563$	−11.3137	−0.476816	−0.238408	+	0.971165i	$0.576626\pi$
$564$	36.0000	1.51587
$565$	−16.9706	−0.713957
$566$	1.41421	0.0594438
$567$	0	0
$568$	−12.0000	−0.503509
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	−11.3137	−0.473050
$573$	−16.9706	−0.708955
$574$	0	0
$575$	12.0000	0.500435
$576$	5.00000	0.208333
$577$	−16.9706	−0.706494	−0.353247	−	0.935530i	$-0.614922\pi$
$577$	−16.9706	−0.706494	−0.353247	+	0.935530i	$0.614922\pi$
$578$	−9.00000	−0.374351
$579$	−39.5980	−1.64564
$580$	−2.82843	−0.117444
$581$	0	0
$582$	−40.0000	−1.65805
$583$	40.0000	1.65663
$584$	11.3137	0.468165
$585$	−40.0000	−1.65380
$586$	9.89949	0.408944
$587$	−29.6985	−1.22579	−0.612894	−	0.790165i	$-0.709995\pi$
$587$	−29.6985	−1.22579	−0.612894	+	0.790165i	$0.709995\pi$
$588$	0	0
$589$	0	0
$590$	−28.0000	−1.15274
$591$	28.2843	1.16346
$592$	8.00000	0.328798
$593$	1.41421	0.0580748	0.0290374	−	0.999578i	$-0.490756\pi$
$593$	1.41421	0.0580748	0.0290374	+	0.999578i	$0.490756\pi$
$594$	−22.6274	−0.928414
$595$	0	0
$596$	22.0000	0.901155
$597$	24.0000	0.982255
$598$	−11.3137	−0.462652
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	−8.48528	−0.346410
$601$	−33.9411	−1.38449	−0.692244	−	0.721664i	$-0.743378\pi$
$601$	−33.9411	−1.38449	−0.692244	+	0.721664i	$0.743378\pi$
$602$	0	0
$603$	10.0000	0.407231
$604$	16.0000	0.651031
$605$	14.1421	0.574960
$606$	−20.0000	−0.812444
$607$	−21.2132	−0.861017	−0.430509	−	0.902586i	$-0.641666\pi$
$607$	−21.2132	−0.861017	−0.430509	+	0.902586i	$0.641666\pi$
$608$	0	0
$609$	0	0
$610$	−36.0000	−1.45760
$611$	36.0000	1.45640
$612$	14.1421	0.571662
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	−19.7990	−0.799022
$615$	−45.2548	−1.82485
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	−56.0000	−2.25265
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	4.00000	0.160644
$621$	−22.6274	−0.908007
$622$	7.07107	0.283524
$623$	0	0
$624$	8.00000	0.320256
$625$	−31.0000	−1.24000
$626$	−4.24264	−0.169570
$627$	0	0
$628$	21.2132	0.846499
$629$	22.6274	0.902214
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	−6.00000	−0.238667
$633$	−11.3137	−0.449680
$634$	−12.0000	−0.476581
$635$	22.6274	0.897942
$636$	−28.2843	−1.12154
$637$	0	0
$638$	−4.00000	−0.158362
$639$	−60.0000	−2.37356
$640$	2.82843	0.111803
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	−33.9411	−1.33955
$643$	−12.7279	−0.501940	−0.250970	−	0.967995i	$-0.580750\pi$
$643$	−12.7279	−0.501940	−0.250970	+	0.967995i	$0.580750\pi$
$644$	0	0
$645$	32.0000	1.26000
$646$	0	0
$647$	−31.1127	−1.22317	−0.611583	−	0.791180i	$-0.709467\pi$
$647$	−31.1127	−1.22317	−0.611583	+	0.791180i	$0.709467\pi$
$648$	1.00000	0.0392837
$649$	−39.5980	−1.55436
$650$	−8.48528	−0.332820
$651$	0	0
$652$	−12.0000	−0.469956
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	5.65685	0.221201
$655$	−16.0000	−0.625172
$656$	5.65685	0.220863
$657$	56.5685	2.20695
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	−32.0000	−1.24560
$661$	8.48528	0.330039	0.165020	−	0.986290i	$-0.447231\pi$
$661$	8.48528	0.330039	0.165020	+	0.986290i	$0.447231\pi$
$662$	12.0000	0.466393
$663$	22.6274	0.878776
$664$	1.41421	0.0548821
$665$	0	0
$666$	40.0000	1.54997
$667$	−4.00000	−0.154881
$668$	−5.65685	−0.218870
$669$	−72.0000	−2.78368
$670$	5.65685	0.218543
$671$	−50.9117	−1.96542
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	10.0000	0.385186
$675$	−16.9706	−0.653197
$676$	−5.00000	−0.192308
$677$	−21.2132	−0.815290	−0.407645	−	0.913141i	$-0.633650\pi$
$677$	−21.2132	−0.815290	−0.407645	+	0.913141i	$0.633650\pi$
$678$	16.9706	0.651751
$679$	0	0
$680$	8.00000	0.306786
$681$	−4.00000	−0.153280
$682$	5.65685	0.216612
$683$	−46.0000	−1.76014	−0.880071	−	0.474843i	$-0.842505\pi$
$683$	−46.0000	−1.76014	−0.880071	+	0.474843i	$0.842505\pi$
$684$	0	0
$685$	5.65685	0.216137
$686$	0	0
$687$	−52.0000	−1.98392
$688$	−4.00000	−0.152499
$689$	−28.2843	−1.07754
$690$	−32.0000	−1.21822
$691$	−9.89949	−0.376595	−0.188297	−	0.982112i	$-0.560297\pi$
$691$	−9.89949	−0.376595	−0.188297	+	0.982112i	$0.560297\pi$
$692$	2.82843	0.107521
$693$	0	0
$694$	26.0000	0.986947
$695$	36.0000	1.36556
$696$	2.82843	0.107211
$697$	16.0000	0.606043
$698$	16.9706	0.642345
$699$	22.6274	0.855848
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	16.0000	0.603881
$703$	0	0
$704$	4.00000	0.150756
$705$	101.823	3.83489
$706$	−24.0416	−0.904819
$707$	0	0
$708$	28.0000	1.05230
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	−33.9411	−1.27379
$711$	−30.0000	−1.12509
$712$	11.3137	0.423999
$713$	5.65685	0.211851
$714$	0	0
$715$	−32.0000	−1.19673
$716$	−10.0000	−0.373718
$717$	79.1960	2.95763
$718$	24.0000	0.895672
$719$	36.7696	1.37127	0.685636	−	0.727944i	$-0.259524\pi$
$719$	36.7696	1.37127	0.685636	+	0.727944i	$0.259524\pi$
$720$	14.1421	0.527046
$721$	0	0
$722$	−19.0000	−0.707107
$723$	4.00000	0.148762
$724$	−19.7990	−0.735824
$725$	−3.00000	−0.111417
$726$	−14.1421	−0.524864
$727$	12.7279	0.472052	0.236026	−	0.971747i	$-0.424155\pi$
$727$	12.7279	0.472052	0.236026	+	0.971747i	$0.424155\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	32.0000	1.18437
$731$	−11.3137	−0.418453
$732$	36.0000	1.33060
$733$	−21.2132	−0.783528	−0.391764	−	0.920066i	$-0.628135\pi$
$733$	−21.2132	−0.783528	−0.391764	+	0.920066i	$0.628135\pi$
$734$	−1.41421	−0.0521996
$735$	0	0
$736$	4.00000	0.147442
$737$	8.00000	0.294684
$738$	28.2843	1.04116
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	22.6274	0.831800
$741$	0	0
$742$	0	0
$743$	2.00000	0.0733729	0.0366864	−	0.999327i	$-0.488320\pi$
$743$	2.00000	0.0733729	0.0366864	+	0.999327i	$0.488320\pi$
$744$	−4.00000	−0.146647
$745$	62.2254	2.27976
$746$	−14.0000	−0.512576
$747$	7.07107	0.258717
$748$	11.3137	0.413670
$749$	0	0
$750$	16.0000	0.584237
$751$	18.0000	0.656829	0.328415	−	0.944534i	$-0.393486\pi$
$751$	18.0000	0.656829	0.328415	+	0.944534i	$0.393486\pi$
$752$	−12.7279	−0.464140
$753$	64.0000	2.33229
$754$	2.82843	0.103005
$755$	45.2548	1.64699
$756$	0	0
$757$	−48.0000	−1.74459	−0.872295	−	0.488980i	$-0.837369\pi$
$757$	−48.0000	−1.74459	−0.872295	+	0.488980i	$0.837369\pi$
$758$	−8.00000	−0.290573
$759$	−45.2548	−1.64265
$760$	0	0
$761$	−49.4975	−1.79428	−0.897141	−	0.441744i	$-0.854360\pi$
$761$	−49.4975	−1.79428	−0.897141	+	0.441744i	$0.854360\pi$
$762$	−22.6274	−0.819705
$763$	0	0
$764$	6.00000	0.217072
$765$	40.0000	1.44620
$766$	0	0
$767$	28.0000	1.01102
$768$	−2.82843	−0.102062
$769$	42.4264	1.52994	0.764968	−	0.644069i	$-0.222755\pi$
$769$	42.4264	1.52994	0.764968	+	0.644069i	$0.222755\pi$
$770$	0	0
$771$	−76.0000	−2.73707
$772$	14.0000	0.503871
$773$	−9.89949	−0.356060	−0.178030	−	0.984025i	$-0.556972\pi$
$773$	−9.89949	−0.356060	−0.178030	+	0.984025i	$0.556972\pi$
$774$	−20.0000	−0.718885
$775$	4.24264	0.152400
$776$	14.1421	0.507673
$777$	0	0
$778$	28.0000	1.00385
$779$	0	0
$780$	22.6274	0.810191
$781$	−48.0000	−1.71758
$782$	11.3137	0.404577
$783$	5.65685	0.202159
$784$	0	0
$785$	60.0000	2.14149
$786$	16.0000	0.570701
$787$	−32.5269	−1.15946	−0.579730	−	0.814809i	$-0.696842\pi$
$787$	−32.5269	−1.15946	−0.579730	+	0.814809i	$0.696842\pi$
$788$	−10.0000	−0.356235
$789$	−5.65685	−0.201389
$790$	−16.9706	−0.603786
$791$	0	0
$792$	20.0000	0.710669
$793$	36.0000	1.27840
$794$	−8.48528	−0.301131
$795$	−80.0000	−2.83731
$796$	−8.48528	−0.300753
$797$	−21.2132	−0.751410	−0.375705	−	0.926739i	$-0.622599\pi$
$797$	−21.2132	−0.751410	−0.375705	+	0.926739i	$0.622599\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	3.00000	0.106066
$801$	56.5685	1.99875
$802$	−36.0000	−1.27120
$803$	45.2548	1.59701
$804$	−5.65685	−0.199502
$805$	0	0
$806$	−4.00000	−0.140894
$807$	−28.0000	−0.985647
$808$	7.07107	0.248759
$809$	14.0000	0.492214	0.246107	−	0.969243i	$-0.420849\pi$
$809$	14.0000	0.492214	0.246107	+	0.969243i	$0.420849\pi$
$810$	2.82843	0.0993808
$811$	−21.2132	−0.744896	−0.372448	−	0.928053i	$-0.621482\pi$
$811$	−21.2132	−0.744896	−0.372448	+	0.928053i	$0.621482\pi$
$812$	0	0
$813$	76.0000	2.66544
$814$	32.0000	1.12160
$815$	−33.9411	−1.18891
$816$	−8.00000	−0.280056
$817$	0	0
$818$	25.4558	0.890043
$819$	0	0
$820$	16.0000	0.558744
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	−5.65685	−0.197305
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	19.7990	0.689730
$825$	−33.9411	−1.18168
$826$	0	0
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	20.0000	0.695048
$829$	−15.5563	−0.540294	−0.270147	−	0.962819i	$-0.587072\pi$
$829$	−15.5563	−0.540294	−0.270147	+	0.962819i	$0.587072\pi$
$830$	4.00000	0.138842
$831$	84.8528	2.94351
$832$	−2.82843	−0.0980581
$833$	0	0
$834$	−36.0000	−1.24658
$835$	−16.0000	−0.553703
$836$	0	0
$837$	−8.00000	−0.276520
$838$	−38.1838	−1.31904
$839$	−1.41421	−0.0488241	−0.0244120	−	0.999702i	$-0.507771\pi$
$839$	−1.41421	−0.0488241	−0.0244120	+	0.999702i	$0.507771\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−10.0000	−0.344623
$843$	62.2254	2.14316
$844$	4.00000	0.137686
$845$	−14.1421	−0.486504
$846$	−63.6396	−2.18797
$847$	0	0
$848$	10.0000	0.343401
$849$	−4.00000	−0.137280
$850$	8.48528	0.291043
$851$	32.0000	1.09695
$852$	33.9411	1.16280
$853$	−29.6985	−1.01686	−0.508428	−	0.861104i	$-0.669773\pi$
$853$	−29.6985	−1.01686	−0.508428	+	0.861104i	$0.669773\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	32.5269	1.11110	0.555549	−	0.831484i	$-0.312508\pi$
$857$	32.5269	1.11110	0.555549	+	0.831484i	$0.312508\pi$
$858$	32.0000	1.09246
$859$	48.0833	1.64058	0.820290	−	0.571948i	$-0.193812\pi$
$859$	48.0833	1.64058	0.820290	+	0.571948i	$0.193812\pi$
$860$	−11.3137	−0.385794
$861$	0	0
$862$	32.0000	1.08992
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	−5.65685	−0.192450
$865$	8.00000	0.272008
$866$	33.9411	1.15337
$867$	25.4558	0.864526
$868$	0	0
$869$	−24.0000	−0.814144
$870$	8.00000	0.271225
$871$	−5.65685	−0.191675
$872$	−2.00000	−0.0677285
$873$	70.7107	2.39319
$874$	0	0
$875$	0	0
$876$	−32.0000	−1.08118
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	16.9706	0.572729
$879$	−28.0000	−0.944417
$880$	11.3137	0.381385
$881$	−19.7990	−0.667045	−0.333522	−	0.942742i	$-0.608237\pi$
$881$	−19.7990	−0.667045	−0.333522	+	0.942742i	$0.608237\pi$
$882$	0	0
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	−8.00000	−0.269069
$885$	79.1960	2.66214
$886$	−24.0000	−0.806296
$887$	24.0416	0.807239	0.403619	−	0.914927i	$-0.367752\pi$
$887$	24.0416	0.807239	0.403619	+	0.914927i	$0.367752\pi$
$888$	−22.6274	−0.759326
$889$	0	0
$890$	32.0000	1.07264
$891$	4.00000	0.134005
$892$	25.4558	0.852325
$893$	0	0
$894$	−62.2254	−2.08113
$895$	−28.2843	−0.945439
$896$	0	0
$897$	32.0000	1.06845
$898$	−10.0000	−0.333704
$899$	−1.41421	−0.0471667
$900$	15.0000	0.500000
$901$	28.2843	0.942286
$902$	22.6274	0.753411
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−56.0000	−1.86150
$906$	−45.2548	−1.50349
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	1.41421	0.0469323
$909$	35.3553	1.17266
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	5.65685	0.187215
$914$	−16.0000	−0.529233
$915$	101.823	3.36618
$916$	18.3848	0.607450
$917$	0	0
$918$	−16.0000	−0.528079
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	11.3137	0.373002
$921$	56.0000	1.84526
$922$	−4.24264	−0.139724
$923$	33.9411	1.11719
$924$	0	0
$925$	24.0000	0.789115
$926$	−24.0000	−0.788689
$927$	98.9949	3.25142
$928$	−1.00000	−0.0328266
$929$	−35.3553	−1.15997	−0.579986	−	0.814627i	$-0.696942\pi$
$929$	−35.3553	−1.15997	−0.579986	+	0.814627i	$0.696942\pi$
$930$	−11.3137	−0.370991
$931$	0	0
$932$	−8.00000	−0.262049
$933$	−20.0000	−0.654771
$934$	22.6274	0.740392
$935$	32.0000	1.04651
$936$	−14.1421	−0.462250
$937$	−15.5563	−0.508204	−0.254102	−	0.967177i	$-0.581780\pi$
$937$	−15.5563	−0.508204	−0.254102	+	0.967177i	$0.581780\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	−36.0000	−1.17419
$941$	−53.7401	−1.75188	−0.875939	−	0.482422i	$-0.839757\pi$
$941$	−53.7401	−1.75188	−0.875939	+	0.482422i	$0.839757\pi$
$942$	−60.0000	−1.95491
$943$	22.6274	0.736850
$944$	−9.89949	−0.322201
$945$	0	0
$946$	−16.0000	−0.520205
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	16.9706	0.551178
$949$	−32.0000	−1.03876
$950$	0	0
$951$	33.9411	1.10062
$952$	0	0
$953$	48.0000	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$953$	48.0000	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$954$	50.0000	1.61881
$955$	16.9706	0.549155
$956$	−28.0000	−0.905585
$957$	11.3137	0.365720
$958$	−4.24264	−0.137073
$959$	0	0
$960$	−8.00000	−0.258199
$961$	−29.0000	−0.935484
$962$	−22.6274	−0.729537
$963$	60.0000	1.93347
$964$	−1.41421	−0.0455488
$965$	39.5980	1.27470
$966$	0	0
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	40.0000	1.28432
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	14.1421	0.453609
$973$	0	0
$974$	−8.00000	−0.256337
$975$	24.0000	0.768615
$976$	−12.7279	−0.407411
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	33.9411	1.08532
$979$	45.2548	1.44635
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−40.0000	−1.27645
$983$	−9.89949	−0.315745	−0.157872	−	0.987460i	$-0.550463\pi$
$983$	−9.89949	−0.315745	−0.157872	+	0.987460i	$0.550463\pi$
$984$	−16.0000	−0.510061
$985$	−28.2843	−0.901212
$986$	−2.82843	−0.0900755
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	56.5685	1.79787
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	1.41421	0.0449013
$993$	−33.9411	−1.07709
$994$	0	0
$995$	−24.0000	−0.760851
$996$	−4.00000	−0.126745
$997$	−26.8701	−0.850983	−0.425492	−	0.904962i	$-0.639899\pi$
$997$	−26.8701	−0.850983	−0.425492	+	0.904962i	$0.639899\pi$
$998$	−20.0000	−0.633089
$999$	−45.2548	−1.43180

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} +2.82843 q^{5} -2.82843 q^{6} +1.00000 q^{8} +5.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} +2.82843 q^{5} -2.82843 q^{6} +1.00000 q^{8} +5.00000 q^{9} +2.82843 q^{10} +4.00000 q^{11} -2.82843 q^{12} -2.82843 q^{13} -8.00000 q^{15} +1.00000 q^{16} +2.82843 q^{17} +5.00000 q^{18} +2.82843 q^{20} +4.00000 q^{22} +4.00000 q^{23} -2.82843 q^{24} +3.00000 q^{25} -2.82843 q^{26} -5.65685 q^{27} -1.00000 q^{29} -8.00000 q^{30} +1.41421 q^{31} +1.00000 q^{32} -11.3137 q^{33} +2.82843 q^{34} +5.00000 q^{36} +8.00000 q^{37} +8.00000 q^{39} +2.82843 q^{40} +5.65685 q^{41} -4.00000 q^{43} +4.00000 q^{44} +14.1421 q^{45} +4.00000 q^{46} -12.7279 q^{47} -2.82843 q^{48} +3.00000 q^{50} -8.00000 q^{51} -2.82843 q^{52} +10.0000 q^{53} -5.65685 q^{54} +11.3137 q^{55} -1.00000 q^{58} -9.89949 q^{59} -8.00000 q^{60} -12.7279 q^{61} +1.41421 q^{62} +1.00000 q^{64} -8.00000 q^{65} -11.3137 q^{66} +2.00000 q^{67} +2.82843 q^{68} -11.3137 q^{69} -12.0000 q^{71} +5.00000 q^{72} +11.3137 q^{73} +8.00000 q^{74} -8.48528 q^{75} +8.00000 q^{78} -6.00000 q^{79} +2.82843 q^{80} +1.00000 q^{81} +5.65685 q^{82} +1.41421 q^{83} +8.00000 q^{85} -4.00000 q^{86} +2.82843 q^{87} +4.00000 q^{88} +11.3137 q^{89} +14.1421 q^{90} +4.00000 q^{92} -4.00000 q^{93} -12.7279 q^{94} -2.82843 q^{96} +14.1421 q^{97} +20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 10 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 10 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 10 q^{9} + 8 q^{11} - 16 q^{15} + 2 q^{16} + 10 q^{18} + 8 q^{22} + 8 q^{23} + 6 q^{25} - 2 q^{29} - 16 q^{30} + 2 q^{32} + 10 q^{36} + 16 q^{37} + 16 q^{39} - 8 q^{43} + 8 q^{44} + 8 q^{46} + 6 q^{50} - 16 q^{51} + 20 q^{53} - 2 q^{58} - 16 q^{60} + 2 q^{64} - 16 q^{65} + 4 q^{67} - 24 q^{71} + 10 q^{72} + 16 q^{74} + 16 q^{78} - 12 q^{79} + 2 q^{81} + 16 q^{85} - 8 q^{86} + 8 q^{88} + 8 q^{92} - 8 q^{93} + 40 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 10 * q^9 + 8 * q^11 - 16 * q^15 + 2 * q^16 + 10 * q^18 + 8 * q^22 + 8 * q^23 + 6 * q^25 - 2 * q^29 - 16 * q^30 + 2 * q^32 + 10 * q^36 + 16 * q^37 + 16 * q^39 - 8 * q^43 + 8 * q^44 + 8 * q^46 + 6 * q^50 - 16 * q^51 + 20 * q^53 - 2 * q^58 - 16 * q^60 + 2 * q^64 - 16 * q^65 + 4 * q^67 - 24 * q^71 + 10 * q^72 + 16 * q^74 + 16 * q^78 - 12 * q^79 + 2 * q^81 + 16 * q^85 - 8 * q^86 + 8 * q^88 + 8 * q^92 - 8 * q^93 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more