LMFDB - Embedded newform 2842.2.a.bb.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:	$6$
Coefficient field:	6.6.401917952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 14x^{4} + 49x^{2} - 18 $ x^6 - 14x^4 + 49x^2 - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.90839$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.90839 q^{3} +1.00000 q^{4} -1.49418 q^{5} -2.90839 q^{6} +1.00000 q^{8} +5.45876 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.90839 q^{3} +1.00000 q^{4} -1.49418 q^{5} -2.90839 q^{6} +1.00000 q^{8} +5.45876 q^{9} -1.49418 q^{10} +1.65433 q^{11} -2.90839 q^{12} +4.81145 q^{13} +4.34567 q^{15} +1.00000 q^{16} +0.925367 q^{17} +5.45876 q^{18} -4.40257 q^{19} -1.49418 q^{20} +1.65433 q^{22} -0.691333 q^{23} -2.90839 q^{24} -2.76742 q^{25} +4.81145 q^{26} -7.15103 q^{27} -1.00000 q^{29} +4.34567 q^{30} -7.15103 q^{31} +1.00000 q^{32} -4.81145 q^{33} +0.925367 q^{34} +5.45876 q^{36} -2.00000 q^{37} -4.40257 q^{38} -13.9936 q^{39} -1.49418 q^{40} +4.73149 q^{41} +7.07609 q^{43} +1.65433 q^{44} -8.15637 q^{45} -0.691333 q^{46} -7.15103 q^{47} -2.90839 q^{48} -2.76742 q^{50} -2.69133 q^{51} +4.81145 q^{52} +2.34567 q^{53} -7.15103 q^{54} -2.47187 q^{55} +12.8044 q^{57} -1.00000 q^{58} +7.87978 q^{59} +4.34567 q^{60} +10.4960 q^{61} -7.15103 q^{62} +1.00000 q^{64} -7.18918 q^{65} -4.81145 q^{66} -8.91751 q^{67} +0.925367 q^{68} +2.01067 q^{69} +8.00000 q^{71} +5.45876 q^{72} +7.87978 q^{73} -2.00000 q^{74} +8.04876 q^{75} -4.40257 q^{76} -13.9936 q^{78} +14.4588 q^{79} -1.49418 q^{80} +4.42176 q^{81} +4.73149 q^{82} -4.73149 q^{83} -1.38267 q^{85} +7.07609 q^{86} +2.90839 q^{87} +1.65433 q^{88} +10.5483 q^{89} -8.15637 q^{90} -0.691333 q^{92} +20.7980 q^{93} -7.15103 q^{94} +6.57824 q^{95} -2.90839 q^{96} +19.1935 q^{97} +9.03060 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 10 * q^9	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9} + 8 q^{11} + 28 q^{15} + 6 q^{16} + 10 q^{18} + 8 q^{22} - 8 q^{23} + 10 q^{25} - 6 q^{29} + 28 q^{30} + 6 q^{32} + 10 q^{36} - 12 q^{37} - 8 q^{39} + 12 q^{43} + 8 q^{44} - 8 q^{46} + 10 q^{50} - 20 q^{51} + 16 q^{53} + 56 q^{57} - 6 q^{58} + 28 q^{60} + 6 q^{64} + 12 q^{65} - 8 q^{67} + 48 q^{71} + 10 q^{72} - 12 q^{74} - 8 q^{78} + 64 q^{79} - 2 q^{81} - 16 q^{85} + 12 q^{86} + 8 q^{88} - 8 q^{92} + 28 q^{93} + 68 q^{95} - 16 q^{99}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 10 * q^9 + 8 * q^11 + 28 * q^15 + 6 * q^16 + 10 * q^18 + 8 * q^22 - 8 * q^23 + 10 * q^25 - 6 * q^29 + 28 * q^30 + 6 * q^32 + 10 * q^36 - 12 * q^37 - 8 * q^39 + 12 * q^43 + 8 * q^44 - 8 * q^46 + 10 * q^50 - 20 * q^51 + 16 * q^53 + 56 * q^57 - 6 * q^58 + 28 * q^60 + 6 * q^64 + 12 * q^65 - 8 * q^67 + 48 * q^71 + 10 * q^72 - 12 * q^74 - 8 * q^78 + 64 * q^79 - 2 * q^81 - 16 * q^85 + 12 * q^86 + 8 * q^88 - 8 * q^92 + 28 * q^93 + 68 * q^95 - 16 * 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
$3$	−2.90839	−1.67916	−0.839581	−	0.543234i	$-0.817200\pi$
$3$	−2.90839	−1.67916	−0.839581	+	0.543234i	$0.817200\pi$
$4$	1.00000	0.500000
$5$	−1.49418	−0.668218	−0.334109	−	0.942534i	$-0.608435\pi$
$5$	−1.49418	−0.668218	−0.334109	+	0.942534i	$0.608435\pi$
$6$	−2.90839	−1.18735
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	5.45876	1.81959
$10$	−1.49418	−0.472501
$11$	1.65433	0.498800	0.249400	−	0.968401i	$-0.419767\pi$
$11$	1.65433	0.498800	0.249400	+	0.968401i	$0.419767\pi$
$12$	−2.90839	−0.839581
$13$	4.81145	1.33446	0.667229	−	0.744853i	$-0.267480\pi$
$13$	4.81145	1.33446	0.667229	+	0.744853i	$0.267480\pi$
$14$	0	0
$15$	4.34567	1.12205
$16$	1.00000	0.250000
$17$	0.925367	0.224435	0.112217	−	0.993684i	$-0.464205\pi$
$17$	0.925367	0.224435	0.112217	+	0.993684i	$0.464205\pi$
$18$	5.45876	1.28664
$19$	−4.40257	−1.01002	−0.505010	−	0.863114i	$-0.668511\pi$
$19$	−4.40257	−1.01002	−0.505010	+	0.863114i	$0.668511\pi$
$20$	−1.49418	−0.334109
$21$	0	0
$22$	1.65433	0.352705
$23$	−0.691333	−0.144153	−0.0720764	−	0.997399i	$-0.522963\pi$
$23$	−0.691333	−0.144153	−0.0720764	+	0.997399i	$0.522963\pi$
$24$	−2.90839	−0.593673
$25$	−2.76742	−0.553485
$26$	4.81145	0.943604
$27$	−7.15103	−1.37622
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	4.34567	0.793407
$31$	−7.15103	−1.28436	−0.642182	−	0.766552i	$-0.721971\pi$
$31$	−7.15103	−1.28436	−0.642182	+	0.766552i	$0.721971\pi$
$32$	1.00000	0.176777
$33$	−4.81145	−0.837567
$34$	0.925367	0.158699
$35$	0	0
$36$	5.45876	0.909793
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−4.40257	−0.714192
$39$	−13.9936	−2.24077
$40$	−1.49418	−0.236251
$41$	4.73149	0.738934	0.369467	−	0.929244i	$-0.379540\pi$
$41$	4.73149	0.738934	0.369467	+	0.929244i	$0.379540\pi$
$42$	0	0
$43$	7.07609	1.07909	0.539547	−	0.841956i	$-0.318596\pi$
$43$	7.07609	1.07909	0.539547	+	0.841956i	$0.318596\pi$
$44$	1.65433	0.249400
$45$	−8.15637	−1.21588
$46$	−0.691333	−0.101931
$47$	−7.15103	−1.04309	−0.521543	−	0.853225i	$-0.674643\pi$
$47$	−7.15103	−1.04309	−0.521543	+	0.853225i	$0.674643\pi$
$48$	−2.90839	−0.419791
$49$	0	0
$50$	−2.76742	−0.391373
$51$	−2.69133	−0.376862
$52$	4.81145	0.667229
$53$	2.34567	0.322202	0.161101	−	0.986938i	$-0.448495\pi$
$53$	2.34567	0.322202	0.161101	+	0.986938i	$0.448495\pi$
$54$	−7.15103	−0.973133
$55$	−2.47187	−0.333307
$56$	0	0
$57$	12.8044	1.69599
$58$	−1.00000	−0.131306
$59$	7.87978	1.02586	0.512930	−	0.858430i	$-0.328560\pi$
$59$	7.87978	1.02586	0.512930	+	0.858430i	$0.328560\pi$
$60$	4.34567	0.561023
$61$	10.4960	1.34387	0.671934	−	0.740611i	$-0.265464\pi$
$61$	10.4960	1.34387	0.671934	+	0.740611i	$0.265464\pi$
$62$	−7.15103	−0.908182
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.18918	−0.891708
$66$	−4.81145	−0.592249
$67$	−8.91751	−1.08945	−0.544724	−	0.838615i	$-0.683366\pi$
$67$	−8.91751	−1.08945	−0.544724	+	0.838615i	$0.683366\pi$
$68$	0.925367	0.112217
$69$	2.01067	0.242056
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	5.45876	0.643321
$73$	7.87978	0.922259	0.461129	−	0.887333i	$-0.347444\pi$
$73$	7.87978	0.922259	0.461129	+	0.887333i	$0.347444\pi$
$74$	−2.00000	−0.232495
$75$	8.04876	0.929391
$76$	−4.40257	−0.505010
$77$	0	0
$78$	−13.9936	−1.58446
$79$	14.4588	1.62674	0.813369	−	0.581749i	$-0.197631\pi$
$79$	14.4588	1.62674	0.813369	+	0.581749i	$0.197631\pi$
$80$	−1.49418	−0.167054
$81$	4.42176	0.491306
$82$	4.73149	0.522505
$83$	−4.73149	−0.519348	−0.259674	−	0.965696i	$-0.583615\pi$
$83$	−4.73149	−0.519348	−0.259674	+	0.965696i	$0.583615\pi$
$84$	0	0
$85$	−1.38267	−0.149971
$86$	7.07609	0.763035
$87$	2.90839	0.311813
$88$	1.65433	0.176353
$89$	10.5483	1.11811	0.559057	−	0.829129i	$-0.311163\pi$
$89$	10.5483	1.11811	0.559057	+	0.829129i	$0.311163\pi$
$90$	−8.15637	−0.859757
$91$	0	0
$92$	−0.691333	−0.0720764
$93$	20.7980	2.15666
$94$	−7.15103	−0.737573
$95$	6.57824	0.674913
$96$	−2.90839	−0.296837
$97$	19.1935	1.94880	0.974402	−	0.224813i	$-0.0721772\pi$
$97$	19.1935	1.94880	0.974402	+	0.224813i	$0.0721772\pi$
$98$	0	0
$99$	9.03060	0.907610
$100$	−2.76742	−0.276742
$101$	−7.50759	−0.747033	−0.373517	−	0.927624i	$-0.621848\pi$
$101$	−7.50759	−0.747033	−0.373517	+	0.927624i	$0.621848\pi$
$102$	−2.69133	−0.266482
$103$	8.48528	0.836080	0.418040	−	0.908429i	$-0.362717\pi$
$103$	8.48528	0.836080	0.418040	+	0.908429i	$0.362717\pi$
$104$	4.81145	0.471802
$105$	0	0
$106$	2.34567	0.227831
$107$	−3.30867	−0.319861	−0.159930	−	0.987128i	$-0.551127\pi$
$107$	−3.30867	−0.319861	−0.159930	+	0.987128i	$0.551127\pi$
$108$	−7.15103	−0.688109
$109$	0.963001	0.0922387	0.0461194	−	0.998936i	$-0.485315\pi$
$109$	0.963001	0.0922387	0.0461194	+	0.998936i	$0.485315\pi$
$110$	−2.47187	−0.235684
$111$	5.81679	0.552105
$112$	0	0
$113$	−8.45236	−0.795131	−0.397566	−	0.917574i	$-0.630145\pi$
$113$	−8.45236	−0.795131	−0.397566	+	0.917574i	$0.630145\pi$
$114$	12.8044	1.19924
$115$	1.03298	0.0963255
$116$	−1.00000	−0.0928477
$117$	26.2646	2.42816
$118$	7.87978	0.725393
$119$	0	0
$120$	4.34567	0.396703
$121$	−8.26318	−0.751198
$122$	10.4960	0.950259
$123$	−13.7610	−1.24079
$124$	−7.15103	−0.642182
$125$	11.6059	1.03807
$126$	0	0
$127$	8.91751	0.791301	0.395651	−	0.918401i	$-0.370519\pi$
$127$	8.91751	0.791301	0.395651	+	0.918401i	$0.370519\pi$
$128$	1.00000	0.0883883
$129$	−20.5801	−1.81197
$130$	−7.18918	−0.630533
$131$	17.9915	1.57193	0.785964	−	0.618272i	$-0.212167\pi$
$131$	17.9915	1.57193	0.785964	+	0.618272i	$0.212167\pi$
$132$	−4.81145	−0.418783
$133$	0	0
$134$	−8.91751	−0.770356
$135$	10.6849	0.919613
$136$	0.925367	0.0793496
$137$	18.5264	1.58281	0.791407	−	0.611290i	$-0.209349\pi$
$137$	18.5264	1.58281	0.791407	+	0.611290i	$0.209349\pi$
$138$	2.01067	0.171160
$139$	−13.3767	−1.13460	−0.567299	−	0.823512i	$-0.692011\pi$
$139$	−13.3767	−1.13460	−0.567299	+	0.823512i	$0.692011\pi$
$140$	0	0
$141$	20.7980	1.75151
$142$	8.00000	0.671345
$143$	7.95975	0.665628
$144$	5.45876	0.454896
$145$	1.49418	0.124085
$146$	7.87978	0.652135
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−8.45876	−0.692968	−0.346484	−	0.938056i	$-0.612625\pi$
$149$	−8.45876	−0.692968	−0.346484	+	0.938056i	$0.612625\pi$
$150$	8.04876	0.657179
$151$	7.53485	0.613177	0.306589	−	0.951842i	$-0.400812\pi$
$151$	7.53485	0.613177	0.306589	+	0.951842i	$0.400812\pi$
$152$	−4.40257	−0.357096
$153$	5.05136	0.408378
$154$	0	0
$155$	10.6849	0.858235
$156$	−13.9936	−1.12039
$157$	1.29756	0.103557	0.0517783	−	0.998659i	$-0.483511\pi$
$157$	1.29756	0.103557	0.0517783	+	0.998659i	$0.483511\pi$
$158$	14.4588	1.15028
$159$	−6.82212	−0.541030
$160$	−1.49418	−0.118125
$161$	0	0
$162$	4.42176	0.347406
$163$	−11.4894	−0.899916	−0.449958	−	0.893050i	$-0.648561\pi$
$163$	−11.4894	−0.899916	−0.449958	+	0.893050i	$0.648561\pi$
$164$	4.73149	0.369467
$165$	7.18918	0.559677
$166$	−4.73149	−0.367235
$167$	14.4620	1.11910	0.559552	−	0.828795i	$-0.310973\pi$
$167$	14.4620	1.11910	0.559552	+	0.828795i	$0.310973\pi$
$168$	0	0
$169$	10.1501	0.780776
$170$	−1.38267	−0.106046
$171$	−24.0326	−1.83782
$172$	7.07609	0.539547
$173$	7.99644	0.607958	0.303979	−	0.952679i	$-0.401685\pi$
$173$	7.99644	0.607958	0.303979	+	0.952679i	$0.401685\pi$
$174$	2.90839	0.220485
$175$	0	0
$176$	1.65433	0.124700
$177$	−22.9175	−1.72259
$178$	10.5483	0.790627
$179$	−21.8350	−1.63203	−0.816013	−	0.578033i	$-0.803820\pi$
$179$	−21.8350	−1.63203	−0.816013	+	0.578033i	$0.803820\pi$
$180$	−8.15637	−0.607940
$181$	−22.5907	−1.67916	−0.839578	−	0.543239i	$-0.817198\pi$
$181$	−22.5907	−1.67916	−0.839578	+	0.543239i	$0.817198\pi$
$182$	0	0
$183$	−30.5264	−2.25657
$184$	−0.691333	−0.0509657
$185$	2.98836	0.219709
$186$	20.7980	1.52499
$187$	1.53087	0.111948
$188$	−7.15103	−0.521543
$189$	0	0
$190$	6.57824	0.477236
$191$	−4.91751	−0.355819	−0.177909	−	0.984047i	$-0.556933\pi$
$191$	−4.91751	−0.355819	−0.177909	+	0.984047i	$0.556933\pi$
$192$	−2.90839	−0.209895
$193$	5.83503	0.420015	0.210007	−	0.977700i	$-0.432651\pi$
$193$	5.83503	0.420015	0.210007	+	0.977700i	$0.432651\pi$
$194$	19.1935	1.37801
$195$	20.9090	1.49732
$196$	0	0
$197$	14.6913	1.04671	0.523357	−	0.852113i	$-0.324679\pi$
$197$	14.6913	1.04671	0.523357	+	0.852113i	$0.324679\pi$
$198$	9.03060	0.641777
$199$	3.14830	0.223177	0.111588	−	0.993755i	$-0.464406\pi$
$199$	3.14830	0.223177	0.111588	+	0.993755i	$0.464406\pi$
$200$	−2.76742	−0.195686
$201$	25.9356	1.82936
$202$	−7.50759	−0.528232
$203$	0	0
$204$	−2.69133	−0.188431
$205$	−7.06970	−0.493769
$206$	8.48528	0.591198
$207$	−3.77382	−0.262299
$208$	4.81145	0.333614
$209$	−7.28333	−0.503798
$210$	0	0
$211$	8.79803	0.605681	0.302841	−	0.953041i	$-0.402065\pi$
$211$	8.79803	0.605681	0.302841	+	0.953041i	$0.402065\pi$
$212$	2.34567	0.161101
$213$	−23.2672	−1.59424
$214$	−3.30867	−0.226176
$215$	−10.5730	−0.721070
$216$	−7.15103	−0.486566
$217$	0	0
$218$	0.963001	0.0652226
$219$	−22.9175	−1.54862
$220$	−2.47187	−0.166654
$221$	4.45236	0.299498
$222$	5.81679	0.390397
$223$	−19.2458	−1.28880	−0.644398	−	0.764691i	$-0.722892\pi$
$223$	−19.2458	−1.28880	−0.644398	+	0.764691i	$0.722892\pi$
$224$	0	0
$225$	−15.1067	−1.00711
$226$	−8.45236	−0.562243
$227$	−2.93604	−0.194872	−0.0974358	−	0.995242i	$-0.531064\pi$
$227$	−2.93604	−0.194872	−0.0974358	+	0.995242i	$0.531064\pi$
$228$	12.8044	0.847994
$229$	13.0045	0.859362	0.429681	−	0.902981i	$-0.358626\pi$
$229$	13.0045	0.859362	0.429681	+	0.902981i	$0.358626\pi$
$230$	1.03298	0.0681124
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	24.2938	1.59154	0.795769	−	0.605600i	$-0.207067\pi$
$233$	24.2938	1.59154	0.795769	+	0.605600i	$0.207067\pi$
$234$	26.2646	1.71697
$235$	10.6849	0.697009
$236$	7.87978	0.512930
$237$	−42.0518	−2.73156
$238$	0	0
$239$	13.8350	0.894914	0.447457	−	0.894306i	$-0.352330\pi$
$239$	13.8350	0.894914	0.447457	+	0.894306i	$0.352330\pi$
$240$	4.34567	0.280512
$241$	4.91906	0.316865	0.158432	−	0.987370i	$-0.449356\pi$
$241$	4.91906	0.316865	0.158432	+	0.987370i	$0.449356\pi$
$242$	−8.26318	−0.531177
$243$	8.59289	0.551234
$244$	10.4960	0.671934
$245$	0	0
$246$	−13.7610	−0.877371
$247$	−21.1828	−1.34783
$248$	−7.15103	−0.454091
$249$	13.7610	0.872070
$250$	11.6059	0.734024
$251$	18.8369	1.18898	0.594488	−	0.804104i	$-0.297355\pi$
$251$	18.8369	1.18898	0.594488	+	0.804104i	$0.297355\pi$
$252$	0	0
$253$	−1.14370	−0.0719035
$254$	8.91751	0.559535
$255$	4.02134	0.251826
$256$	1.00000	0.0625000
$257$	−4.75913	−0.296866	−0.148433	−	0.988922i	$-0.547423\pi$
$257$	−4.75913	−0.296866	−0.148433	+	0.988922i	$0.547423\pi$
$258$	−20.5801	−1.28126
$259$	0	0
$260$	−7.18918	−0.445854
$261$	−5.45876	−0.337889
$262$	17.9915	1.11152
$263$	14.9111	0.919459	0.459730	−	0.888059i	$-0.347946\pi$
$263$	14.9111	0.919459	0.459730	+	0.888059i	$0.347946\pi$
$264$	−4.81145	−0.296125
$265$	−3.50485	−0.215301
$266$	0	0
$267$	−30.6785	−1.87750
$268$	−8.91751	−0.544724
$269$	27.6264	1.68441	0.842207	−	0.539154i	$-0.181256\pi$
$269$	27.6264	1.68441	0.842207	+	0.539154i	$0.181256\pi$
$270$	10.6849	0.650265
$271$	2.80079	0.170136	0.0850678	−	0.996375i	$-0.472889\pi$
$271$	2.80079	0.170136	0.0850678	+	0.996375i	$0.472889\pi$
$272$	0.925367	0.0561086
$273$	0	0
$274$	18.5264	1.11922
$275$	−4.57824	−0.276078
$276$	2.01067	0.121028
$277$	28.9787	1.74116	0.870581	−	0.492024i	$-0.163743\pi$
$277$	28.9787	1.74116	0.870581	+	0.492024i	$0.163743\pi$
$278$	−13.3767	−0.802281
$279$	−39.0358	−2.33701
$280$	0	0
$281$	−17.7155	−1.05682	−0.528410	−	0.848989i	$-0.677212\pi$
$281$	−17.7155	−1.05682	−0.528410	+	0.848989i	$0.677212\pi$
$282$	20.7980	1.23850
$283$	−18.3757	−1.09232	−0.546162	−	0.837680i	$-0.683912\pi$
$283$	−18.3757	−1.09232	−0.546162	+	0.837680i	$0.683912\pi$
$284$	8.00000	0.474713
$285$	−19.1321	−1.13329
$286$	7.95975	0.470670
$287$	0	0
$288$	5.45876	0.321660
$289$	−16.1437	−0.949629
$290$	1.49418	0.0877413
$291$	−55.8222	−3.27236
$292$	7.87978	0.461129
$293$	−6.79448	−0.396938	−0.198469	−	0.980107i	$-0.563597\pi$
$293$	−6.79448	−0.396938	−0.198469	+	0.980107i	$0.563597\pi$
$294$	0	0
$295$	−11.7738	−0.685498
$296$	−2.00000	−0.116248
$297$	−11.8302	−0.686458
$298$	−8.45876	−0.490003
$299$	−3.32632	−0.192366
$300$	8.04876	0.464695
$301$	0	0
$302$	7.53485	0.433582
$303$	21.8350	1.25439
$304$	−4.40257	−0.252505
$305$	−15.6828	−0.897997
$306$	5.05136	0.288767
$307$	−8.46060	−0.482872	−0.241436	−	0.970417i	$-0.577618\pi$
$307$	−8.46060	−0.482872	−0.241436	+	0.970417i	$0.577618\pi$
$308$	0	0
$309$	−24.6785	−1.40391
$310$	10.6849	0.606864
$311$	22.2985	1.26443	0.632216	−	0.774792i	$-0.282145\pi$
$311$	22.2985	1.26443	0.632216	+	0.774792i	$0.282145\pi$
$312$	−13.9936	−0.792232
$313$	28.2999	1.59961	0.799803	−	0.600263i	$-0.204937\pi$
$313$	28.2999	1.59961	0.799803	+	0.600263i	$0.204937\pi$
$314$	1.29756	0.0732256
$315$	0	0
$316$	14.4588	0.813369
$317$	−10.6785	−0.599767	−0.299883	−	0.953976i	$-0.596948\pi$
$317$	−10.6785	−0.599767	−0.299883	+	0.953976i	$0.596948\pi$
$318$	−6.82212	−0.382566
$319$	−1.65433	−0.0926249
$320$	−1.49418	−0.0835272
$321$	9.62291	0.537098
$322$	0	0
$323$	−4.07400	−0.226683
$324$	4.42176	0.245653
$325$	−13.3153	−0.738602
$326$	−11.4894	−0.636337
$327$	−2.80079	−0.154884
$328$	4.73149	0.261253
$329$	0	0
$330$	7.18918	0.395751
$331$	−30.6109	−1.68253	−0.841265	−	0.540623i	$-0.818188\pi$
$331$	−30.6109	−1.68253	−0.841265	+	0.540623i	$0.818188\pi$
$332$	−4.73149	−0.259674
$333$	−10.9175	−0.598276
$334$	14.4620	0.791326
$335$	13.3244	0.727989
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	10.1501	0.552092
$339$	24.5828	1.33515
$340$	−1.38267	−0.0749856
$341$	−11.8302	−0.640641
$342$	−24.0326	−1.29953
$343$	0	0
$344$	7.07609	0.381517
$345$	−3.00430	−0.161746
$346$	7.99644	0.429891
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	2.90839	0.155906
$349$	−22.2709	−1.19213	−0.596066	−	0.802935i	$-0.703270\pi$
$349$	−22.2709	−1.19213	−0.596066	+	0.802935i	$0.703270\pi$
$350$	0	0
$351$	−34.4069	−1.83650
$352$	1.65433	0.0881763
$353$	26.2122	1.39514	0.697568	−	0.716518i	$-0.254265\pi$
$353$	26.2122	1.39514	0.697568	+	0.716518i	$0.254265\pi$
$354$	−22.9175	−1.21805
$355$	−11.9534	−0.634423
$356$	10.5483	0.559057
$357$	0	0
$358$	−21.8350	−1.15402
$359$	3.84142	0.202743	0.101371	−	0.994849i	$-0.467677\pi$
$359$	3.84142	0.202743	0.101371	+	0.994849i	$0.467677\pi$
$360$	−8.15637	−0.429878
$361$	0.382666	0.0201403
$362$	−22.5907	−1.18734
$363$	24.0326	1.26138
$364$	0	0
$365$	−11.7738	−0.616270
$366$	−30.5264	−1.59564
$367$	−22.6184	−1.18067	−0.590335	−	0.807159i	$-0.701004\pi$
$367$	−22.6184	−1.18067	−0.590335	+	0.807159i	$0.701004\pi$
$368$	−0.691333	−0.0360382
$369$	25.8280	1.34455
$370$	2.98836	0.155358
$371$	0	0
$372$	20.7980	1.07833
$373$	17.6934	0.916131	0.458065	−	0.888918i	$-0.348543\pi$
$373$	17.6934	0.916131	0.458065	+	0.888918i	$0.348543\pi$
$374$	1.53087	0.0791592
$375$	−33.7546	−1.74308
$376$	−7.15103	−0.368786
$377$	−4.81145	−0.247802
$378$	0	0
$379$	20.0612	1.03048	0.515238	−	0.857047i	$-0.327704\pi$
$379$	20.0612	1.03048	0.515238	+	0.857047i	$0.327704\pi$
$380$	6.57824	0.337457
$381$	−25.9356	−1.32872
$382$	−4.91751	−0.251602
$383$	−33.1234	−1.69253	−0.846263	−	0.532766i	$-0.821153\pi$
$383$	−33.1234	−1.69253	−0.846263	+	0.532766i	$0.821153\pi$
$384$	−2.90839	−0.148418
$385$	0	0
$386$	5.83503	0.296995
$387$	38.6267	1.96350
$388$	19.1935	0.974402
$389$	20.7525	1.05220	0.526098	−	0.850424i	$-0.323655\pi$
$389$	20.7525	1.05220	0.526098	+	0.850424i	$0.323655\pi$
$390$	20.9090	1.05877
$391$	−0.639737	−0.0323529
$392$	0	0
$393$	−52.3265	−2.63952
$394$	14.6913	0.740139
$395$	−21.6040	−1.08701
$396$	9.03060	0.453805
$397$	16.1804	0.812073	0.406037	−	0.913857i	$-0.366911\pi$
$397$	16.1804	0.812073	0.406037	+	0.913857i	$0.366911\pi$
$398$	3.14830	0.157810
$399$	0	0
$400$	−2.76742	−0.138371
$401$	22.5718	1.12718	0.563592	−	0.826053i	$-0.309419\pi$
$401$	22.5718	1.12718	0.563592	+	0.826053i	$0.309419\pi$
$402$	25.9356	1.29355
$403$	−34.4069	−1.71393
$404$	−7.50759	−0.373517
$405$	−6.60690	−0.328300
$406$	0	0
$407$	−3.30867	−0.164005
$408$	−2.69133	−0.133241
$409$	19.8513	0.981584	0.490792	−	0.871277i	$-0.336707\pi$
$409$	19.8513	0.981584	0.490792	+	0.871277i	$0.336707\pi$
$410$	−7.06970	−0.349147
$411$	−53.8820	−2.65780
$412$	8.48528	0.418040
$413$	0	0
$414$	−3.77382	−0.185473
$415$	7.06970	0.347038
$416$	4.81145	0.235901
$417$	38.9047	1.90517
$418$	−7.28333	−0.356239
$419$	23.3195	1.13923	0.569616	−	0.821911i	$-0.307092\pi$
$419$	23.3195	1.13923	0.569616	+	0.821911i	$0.307092\pi$
$420$	0	0
$421$	−37.4439	−1.82490	−0.912451	−	0.409185i	$-0.865813\pi$
$421$	−37.4439	−1.82490	−0.912451	+	0.409185i	$0.865813\pi$
$422$	8.79803	0.428281
$423$	−39.0358	−1.89798
$424$	2.34567	0.113916
$425$	−2.56088	−0.124221
$426$	−23.2672	−1.12730
$427$	0	0
$428$	−3.30867	−0.159930
$429$	−23.1501	−1.11770
$430$	−10.5730	−0.509873
$431$	15.7738	0.759798	0.379899	−	0.925028i	$-0.375959\pi$
$431$	15.7738	0.759798	0.379899	+	0.925028i	$0.375959\pi$
$432$	−7.15103	−0.344054
$433$	10.8681	0.522290	0.261145	−	0.965300i	$-0.415900\pi$
$433$	10.8681	0.522290	0.261145	+	0.965300i	$0.415900\pi$
$434$	0	0
$435$	−4.34567	−0.208359
$436$	0.963001	0.0461194
$437$	3.04365	0.145597
$438$	−22.9175	−1.09504
$439$	−38.5469	−1.83974	−0.919872	−	0.392219i	$-0.871708\pi$
$439$	−38.5469	−1.83974	−0.919872	+	0.392219i	$0.871708\pi$
$440$	−2.47187	−0.117842
$441$	0	0
$442$	4.45236	0.211777
$443$	30.6045	1.45407	0.727033	−	0.686603i	$-0.240899\pi$
$443$	30.6045	1.45407	0.727033	+	0.686603i	$0.240899\pi$
$444$	5.81679	0.276053
$445$	−15.7610	−0.747144
$446$	−19.2458	−0.911316
$447$	24.6014	1.16361
$448$	0	0
$449$	30.2134	1.42586	0.712929	−	0.701236i	$-0.247368\pi$
$449$	30.2134	1.42586	0.712929	+	0.701236i	$0.247368\pi$
$450$	−15.1067	−0.712136
$451$	7.82746	0.368581
$452$	−8.45236	−0.397566
$453$	−21.9143	−1.02962
$454$	−2.93604	−0.137795
$455$	0	0
$456$	12.8044	0.599622
$457$	33.5961	1.57156	0.785779	−	0.618508i	$-0.212262\pi$
$457$	33.5961	1.57156	0.785779	+	0.618508i	$0.212262\pi$
$458$	13.0045	0.607661
$459$	−6.61733	−0.308871
$460$	1.03298	0.0481628
$461$	−27.9463	−1.30159	−0.650795	−	0.759254i	$-0.725564\pi$
$461$	−27.9463	−1.30159	−0.650795	+	0.759254i	$0.725564\pi$
$462$	0	0
$463$	28.4524	1.32229	0.661147	−	0.750257i	$-0.270070\pi$
$463$	28.4524	1.32229	0.661147	+	0.750257i	$0.270070\pi$
$464$	−1.00000	−0.0464238
$465$	−31.0760	−1.44112
$466$	24.2938	1.12539
$467$	−36.0227	−1.66693	−0.833466	−	0.552570i	$-0.813647\pi$
$467$	−36.0227	−1.66693	−0.833466	+	0.552570i	$0.813647\pi$
$468$	26.2646	1.21408
$469$	0	0
$470$	10.6849	0.492859
$471$	−3.77382	−0.173888
$472$	7.87978	0.362696
$473$	11.7062	0.538252
$474$	−42.0518	−1.93150
$475$	12.1838	0.559031
$476$	0	0
$477$	12.8044	0.586274
$478$	13.8350	0.632799
$479$	32.2046	1.47147	0.735733	−	0.677272i	$-0.236838\pi$
$479$	32.2046	1.47147	0.735733	+	0.677272i	$0.236838\pi$
$480$	4.34567	0.198352
$481$	−9.62291	−0.438767
$482$	4.91906	0.224057
$483$	0	0
$484$	−8.26318	−0.375599
$485$	−28.6785	−1.30223
$486$	8.59289	0.389781
$487$	10.0612	0.455917	0.227958	−	0.973671i	$-0.426795\pi$
$487$	10.0612	0.455917	0.227958	+	0.973671i	$0.426795\pi$
$488$	10.4960	0.475129
$489$	33.4156	1.51111
$490$	0	0
$491$	3.56336	0.160812	0.0804061	−	0.996762i	$-0.474378\pi$
$491$	3.56336	0.160812	0.0804061	+	0.996762i	$0.474378\pi$
$492$	−13.7610	−0.620395
$493$	−0.925367	−0.0416764
$494$	−21.1828	−0.953059
$495$	−13.4934	−0.606481
$496$	−7.15103	−0.321091
$497$	0	0
$498$	13.7610	0.616646
$499$	−2.85630	−0.127866	−0.0639329	−	0.997954i	$-0.520364\pi$
$499$	−2.85630	−0.127866	−0.0639329	+	0.997954i	$0.520364\pi$
$500$	11.6059	0.519033
$501$	−42.0612	−1.87916
$502$	18.8369	0.840734
$503$	−8.83279	−0.393835	−0.196917	−	0.980420i	$-0.563093\pi$
$503$	−8.83279	−0.393835	−0.196917	+	0.980420i	$0.563093\pi$
$504$	0	0
$505$	11.2177	0.499181
$506$	−1.14370	−0.0508435
$507$	−29.5205	−1.31105
$508$	8.91751	0.395651
$509$	−1.99207	−0.0882970	−0.0441485	−	0.999025i	$-0.514057\pi$
$509$	−1.99207	−0.0882970	−0.0441485	+	0.999025i	$0.514057\pi$
$510$	4.02134	0.178068
$511$	0	0
$512$	1.00000	0.0441942
$513$	31.4830	1.39001
$514$	−4.75913	−0.209916
$515$	−12.6785	−0.558683
$516$	−20.5801	−0.905987
$517$	−11.8302	−0.520291
$518$	0	0
$519$	−23.2568	−1.02086
$520$	−7.18918	−0.315266
$521$	−31.8324	−1.39460	−0.697301	−	0.716778i	$-0.745616\pi$
$521$	−31.8324	−1.39460	−0.697301	+	0.716778i	$0.745616\pi$
$522$	−5.45876	−0.238923
$523$	20.9889	0.917782	0.458891	−	0.888492i	$-0.348247\pi$
$523$	20.9889	0.917782	0.458891	+	0.888492i	$0.348247\pi$
$524$	17.9915	0.785964
$525$	0	0
$526$	14.9111	0.650156
$527$	−6.61733	−0.288256
$528$	−4.81145	−0.209392
$529$	−22.5221	−0.979220
$530$	−3.50485	−0.152241
$531$	43.0138	1.86664
$532$	0	0
$533$	22.7653	0.986076
$534$	−30.6785	−1.32759
$535$	4.94375	0.213737
$536$	−8.91751	−0.385178
$537$	63.5049	2.74044
$538$	27.6264	1.19106
$539$	0	0
$540$	10.6849	0.459807
$541$	−36.3784	−1.56403	−0.782014	−	0.623261i	$-0.785807\pi$
$541$	−36.3784	−1.56403	−0.782014	+	0.623261i	$0.785807\pi$
$542$	2.80079	0.120304
$543$	65.7028	2.81958
$544$	0.925367	0.0396748
$545$	−1.43890	−0.0616356
$546$	0	0
$547$	8.06121	0.344672	0.172336	−	0.985038i	$-0.444868\pi$
$547$	8.06121	0.344672	0.172336	+	0.985038i	$0.444868\pi$
$548$	18.5264	0.791407
$549$	57.2948	2.44528
$550$	−4.57824	−0.195217
$551$	4.40257	0.187556
$552$	2.01067	0.0855798
$553$	0	0
$554$	28.9787	1.23119
$555$	−8.69133	−0.368927
$556$	−13.3767	−0.567299
$557$	−5.06970	−0.214810	−0.107405	−	0.994215i	$-0.534254\pi$
$557$	−5.06970	−0.214810	−0.107405	+	0.994215i	$0.534254\pi$
$558$	−39.0358	−1.65252
$559$	34.0463	1.44000
$560$	0	0
$561$	−4.45236	−0.187979
$562$	−17.7155	−0.747285
$563$	−21.8343	−0.920208	−0.460104	−	0.887865i	$-0.652188\pi$
$563$	−21.8343	−0.920208	−0.460104	+	0.887865i	$0.652188\pi$
$564$	20.7980	0.875755
$565$	12.6294	0.531321
$566$	−18.3757	−0.772390
$567$	0	0
$568$	8.00000	0.335673
$569$	−2.67854	−0.112290	−0.0561452	−	0.998423i	$-0.517881\pi$
$569$	−2.67854	−0.112290	−0.0561452	+	0.998423i	$0.517881\pi$
$570$	−19.1321	−0.801356
$571$	−45.4439	−1.90177	−0.950883	−	0.309549i	$-0.899822\pi$
$571$	−45.4439	−1.90177	−0.950883	+	0.309549i	$0.899822\pi$
$572$	7.95975	0.332814
$573$	14.3021	0.597478
$574$	0	0
$575$	1.91321	0.0797864
$576$	5.45876	0.227448
$577$	14.0164	0.583512	0.291756	−	0.956493i	$-0.405760\pi$
$577$	14.0164	0.583512	0.291756	+	0.956493i	$0.405760\pi$
$578$	−16.1437	−0.671489
$579$	−16.9706	−0.705273
$580$	1.49418	0.0620425
$581$	0	0
$582$	−55.8222	−2.31391
$583$	3.88051	0.160714
$584$	7.87978	0.326068
$585$	−39.2440	−1.62254
$586$	−6.79448	−0.280677
$587$	−29.2409	−1.20690	−0.603451	−	0.797400i	$-0.706208\pi$
$587$	−29.2409	−1.20690	−0.603451	+	0.797400i	$0.706208\pi$
$588$	0	0
$589$	31.4830	1.29723
$590$	−11.7738	−0.484720
$591$	−42.7282	−1.75760
$592$	−2.00000	−0.0821995
$593$	−4.54391	−0.186596	−0.0932980	−	0.995638i	$-0.529741\pi$
$593$	−4.54391	−0.186596	−0.0932980	+	0.995638i	$0.529741\pi$
$594$	−11.8302	−0.485399
$595$	0	0
$596$	−8.45876	−0.346484
$597$	−9.15648	−0.374750
$598$	−3.32632	−0.136023
$599$	−42.9111	−1.75330	−0.876650	−	0.481128i	$-0.840227\pi$
$599$	−42.9111	−1.75330	−0.876650	+	0.481128i	$0.840227\pi$
$600$	8.04876	0.328589
$601$	−26.8057	−1.09343	−0.546714	−	0.837319i	$-0.684122\pi$
$601$	−26.8057	−1.09343	−0.546714	+	0.837319i	$0.684122\pi$
$602$	0	0
$603$	−48.6785	−1.98234
$604$	7.53485	0.306589
$605$	12.3467	0.501964
$606$	21.8350	0.886987
$607$	−12.1591	−0.493523	−0.246761	−	0.969076i	$-0.579366\pi$
$607$	−12.1591	−0.493523	−0.246761	+	0.969076i	$0.579366\pi$
$608$	−4.40257	−0.178548
$609$	0	0
$610$	−15.6828	−0.634980
$611$	−34.4069	−1.39195
$612$	5.05136	0.204189
$613$	−44.1288	−1.78235	−0.891173	−	0.453664i	$-0.850117\pi$
$613$	−44.1288	−1.78235	−0.891173	+	0.453664i	$0.850117\pi$
$614$	−8.46060	−0.341442
$615$	20.5615	0.829118
$616$	0	0
$617$	1.53485	0.0617907	0.0308953	−	0.999523i	$-0.490164\pi$
$617$	1.53485	0.0617907	0.0308953	+	0.999523i	$0.490164\pi$
$618$	−24.6785	−0.992717
$619$	−6.81012	−0.273722	−0.136861	−	0.990590i	$-0.543701\pi$
$619$	−6.81012	−0.273722	−0.136861	+	0.990590i	$0.543701\pi$
$620$	10.6849	0.429117
$621$	4.94375	0.198386
$622$	22.2985	0.894089
$623$	0	0
$624$	−13.9936	−0.560193
$625$	−3.50424	−0.140170
$626$	28.2999	1.13109
$627$	21.1828	0.845959
$628$	1.29756	0.0517783
$629$	−1.85073	−0.0737936
$630$	0	0
$631$	0.226181	0.00900413	0.00450207	−	0.999990i	$-0.498567\pi$
$631$	0.226181	0.00900413	0.00450207	+	0.999990i	$0.498567\pi$
$632$	14.4588	0.575138
$633$	−25.5881	−1.01704
$634$	−10.6785	−0.424099
$635$	−13.3244	−0.528762
$636$	−6.82212	−0.270515
$637$	0	0
$638$	−1.65433	−0.0654957
$639$	43.6701	1.72756
$640$	−1.49418	−0.0590627
$641$	26.2134	1.03537	0.517683	−	0.855572i	$-0.326795\pi$
$641$	26.2134	1.03537	0.517683	+	0.855572i	$0.326795\pi$
$642$	9.62291	0.379786
$643$	−6.02905	−0.237762	−0.118881	−	0.992908i	$-0.537931\pi$
$643$	−6.02905	−0.237762	−0.118881	+	0.992908i	$0.537931\pi$
$644$	0	0
$645$	30.7503	1.21079
$646$	−4.07400	−0.160289
$647$	28.3708	1.11537	0.557686	−	0.830052i	$-0.311689\pi$
$647$	28.3708	1.11537	0.557686	+	0.830052i	$0.311689\pi$
$648$	4.42176	0.173703
$649$	13.0358	0.511699
$650$	−13.3153	−0.522270
$651$	0	0
$652$	−11.4894	−0.449958
$653$	4.85630	0.190042	0.0950209	−	0.995475i	$-0.469708\pi$
$653$	4.85630	0.190042	0.0950209	+	0.995475i	$0.469708\pi$
$654$	−2.80079	−0.109519
$655$	−26.8826	−1.05039
$656$	4.73149	0.184734
$657$	43.0138	1.67813
$658$	0	0
$659$	−16.6331	−0.647932	−0.323966	−	0.946069i	$-0.605016\pi$
$659$	−16.6331	−0.647932	−0.323966	+	0.946069i	$0.605016\pi$
$660$	7.18918	0.279839
$661$	−21.4807	−0.835504	−0.417752	−	0.908561i	$-0.637182\pi$
$661$	−21.4807	−0.835504	−0.417752	+	0.908561i	$0.637182\pi$
$662$	−30.6109	−1.18973
$663$	−12.9492	−0.502906
$664$	−4.73149	−0.183617
$665$	0	0
$666$	−10.9175	−0.423045
$667$	0.691333	0.0267685
$668$	14.4620	0.559552
$669$	55.9744	2.16410
$670$	13.3244	0.514766
$671$	17.3638	0.670322
$672$	0	0
$673$	−4.33288	−0.167020	−0.0835101	−	0.996507i	$-0.526613\pi$
$673$	−4.33288	−0.167020	−0.0835101	+	0.996507i	$0.526613\pi$
$674$	10.0000	0.385186
$675$	19.7899	0.761715
$676$	10.1501	0.390388
$677$	−30.8794	−1.18679	−0.593396	−	0.804911i	$-0.702213\pi$
$677$	−30.8794	−1.18679	−0.593396	+	0.804911i	$0.702213\pi$
$678$	24.5828	0.944097
$679$	0	0
$680$	−1.38267	−0.0530228
$681$	8.53915	0.327221
$682$	−11.8302	−0.453002
$683$	32.6785	1.25041	0.625205	−	0.780461i	$-0.285015\pi$
$683$	32.6785	1.25041	0.625205	+	0.780461i	$0.285015\pi$
$684$	−24.0326	−0.918909
$685$	−27.6817	−1.05766
$686$	0	0
$687$	−37.8222	−1.44301
$688$	7.07609	0.269773
$689$	11.2861	0.429965
$690$	−3.00430	−0.114372
$691$	−39.7369	−1.51166	−0.755831	−	0.654767i	$-0.772767\pi$
$691$	−39.7369	−1.51166	−0.755831	+	0.654767i	$0.772767\pi$
$692$	7.99644	0.303979
$693$	0	0
$694$	6.00000	0.227757
$695$	19.9872	0.758158
$696$	2.90839	0.110242
$697$	4.37836	0.165842
$698$	−22.2709	−0.842965
$699$	−70.6559	−2.67245
$700$	0	0
$701$	25.3635	0.957965	0.478983	−	0.877824i	$-0.341006\pi$
$701$	25.3635	0.957965	0.478983	+	0.877824i	$0.341006\pi$
$702$	−34.4069	−1.29860
$703$	8.80515	0.332093
$704$	1.65433	0.0623500
$705$	−31.0760	−1.17039
$706$	26.2122	0.986510
$707$	0	0
$708$	−22.9175	−0.861293
$709$	−42.5591	−1.59834	−0.799169	−	0.601106i	$-0.794727\pi$
$709$	−42.5591	−1.59834	−0.799169	+	0.601106i	$0.794727\pi$
$710$	−11.9534	−0.448605
$711$	78.9268	2.95999
$712$	10.5483	0.395313
$713$	4.94375	0.185145
$714$	0	0
$715$	−11.8933	−0.444784
$716$	−21.8350	−0.816013
$717$	−40.2377	−1.50271
$718$	3.84142	0.143361
$719$	37.3359	1.39239	0.696197	−	0.717850i	$-0.254874\pi$
$719$	37.3359	1.39239	0.696197	+	0.717850i	$0.254874\pi$
$720$	−8.15637	−0.303970
$721$	0	0
$722$	0.382666	0.0142414
$723$	−14.3066	−0.532067
$724$	−22.5907	−0.839578
$725$	2.76742	0.102780
$726$	24.0326	0.891933
$727$	−9.77380	−0.362490	−0.181245	−	0.983438i	$-0.558013\pi$
$727$	−9.77380	−0.362490	−0.181245	+	0.983438i	$0.558013\pi$
$728$	0	0
$729$	−38.2568	−1.41692
$730$	−11.7738	−0.435769
$731$	6.54798	0.242186
$732$	−30.5264	−1.12829
$733$	42.3530	1.56435	0.782173	−	0.623062i	$-0.214112\pi$
$733$	42.3530	1.56435	0.782173	+	0.623062i	$0.214112\pi$
$734$	−22.6184	−0.834859
$735$	0	0
$736$	−0.691333	−0.0254829
$737$	−14.7525	−0.543417
$738$	25.8280	0.950743
$739$	−40.4460	−1.48783	−0.743914	−	0.668275i	$-0.767033\pi$
$739$	−40.4460	−1.48783	−0.743914	+	0.668275i	$0.767033\pi$
$740$	2.98836	0.109854
$741$	61.6079	2.26322
$742$	0	0
$743$	−23.5221	−0.862941	−0.431470	−	0.902127i	$-0.642005\pi$
$743$	−23.5221	−0.862941	−0.431470	+	0.902127i	$0.642005\pi$
$744$	20.7980	0.762493
$745$	12.6389	0.463054
$746$	17.6934	0.647802
$747$	−25.8280	−0.944998
$748$	1.53087	0.0559740
$749$	0	0
$750$	−33.7546	−1.23254
$751$	−35.0697	−1.27971	−0.639856	−	0.768495i	$-0.721006\pi$
$751$	−35.0697	−1.27971	−0.639856	+	0.768495i	$0.721006\pi$
$752$	−7.15103	−0.260771
$753$	−54.7852	−1.99649
$754$	−4.81145	−0.175223
$755$	−11.2584	−0.409736
$756$	0	0
$757$	−23.8222	−0.865834	−0.432917	−	0.901434i	$-0.642516\pi$
$757$	−23.8222	−0.865834	−0.432917	+	0.901434i	$0.642516\pi$
$758$	20.0612	0.728656
$759$	3.32632	0.120738
$760$	6.57824	0.238618
$761$	−39.6965	−1.43900	−0.719499	−	0.694493i	$-0.755629\pi$
$761$	−39.6965	−1.43900	−0.719499	+	0.694493i	$0.755629\pi$
$762$	−25.9356	−0.939549
$763$	0	0
$764$	−4.91751	−0.177909
$765$	−7.54764	−0.272885
$766$	−33.1234	−1.19680
$767$	37.9132	1.36897
$768$	−2.90839	−0.104948
$769$	−20.4911	−0.738926	−0.369463	−	0.929245i	$-0.620458\pi$
$769$	−20.4911	−0.738926	−0.369463	+	0.929245i	$0.620458\pi$
$770$	0	0
$771$	13.8414	0.498487
$772$	5.83503	0.210007
$773$	40.7909	1.46715	0.733573	−	0.679610i	$-0.237851\pi$
$773$	40.7909	1.46715	0.733573	+	0.679610i	$0.237851\pi$
$774$	38.6267	1.38841
$775$	19.7899	0.710876
$776$	19.1935	0.689006
$777$	0	0
$778$	20.7525	0.744015
$779$	−20.8307	−0.746338
$780$	20.9090	0.748661
$781$	13.2347	0.473574
$782$	−0.639737	−0.0228769
$783$	7.15103	0.255557
$784$	0	0
$785$	−1.93879	−0.0691984
$786$	−52.3265	−1.86642
$787$	−20.2758	−0.722755	−0.361378	−	0.932420i	$-0.617694\pi$
$787$	−20.2758	−0.722755	−0.361378	+	0.932420i	$0.617694\pi$
$788$	14.6913	0.523357
$789$	−43.3674	−1.54392
$790$	−21.6040	−0.768636
$791$	0	0
$792$	9.03060	0.320889
$793$	50.5008	1.79334
$794$	16.1804	0.574223
$795$	10.1935	0.361526
$796$	3.14830	0.111588
$797$	−9.22967	−0.326932	−0.163466	−	0.986549i	$-0.552267\pi$
$797$	−9.22967	−0.326932	−0.163466	+	0.986549i	$0.552267\pi$
$798$	0	0
$799$	−6.61733	−0.234104
$800$	−2.76742	−0.0978432
$801$	57.5805	2.03451
$802$	22.5718	0.797040
$803$	13.0358	0.460023
$804$	25.9356	0.914680
$805$	0	0
$806$	−34.4069	−1.21193
$807$	−80.3486	−2.82840
$808$	−7.50759	−0.264116
$809$	−11.3087	−0.397592	−0.198796	−	0.980041i	$-0.563703\pi$
$809$	−11.3087	−0.397592	−0.198796	+	0.980041i	$0.563703\pi$
$810$	−6.60690	−0.232143
$811$	−7.71985	−0.271081	−0.135540	−	0.990772i	$-0.543277\pi$
$811$	−7.71985	−0.271081	−0.135540	+	0.990772i	$0.543277\pi$
$812$	0	0
$813$	−8.14579	−0.285685
$814$	−3.30867	−0.115969
$815$	17.1672	0.601340
$816$	−2.69133	−0.0942155
$817$	−31.1530	−1.08991
$818$	19.8513	0.694085
$819$	0	0
$820$	−7.06970	−0.246885
$821$	−22.5591	−0.787317	−0.393658	−	0.919257i	$-0.628791\pi$
$821$	−22.5591	−0.787317	−0.393658	+	0.919257i	$0.628791\pi$
$822$	−53.8820	−1.87935
$823$	43.6701	1.52224	0.761121	−	0.648610i	$-0.224649\pi$
$823$	43.6701	1.52224	0.761121	+	0.648610i	$0.224649\pi$
$824$	8.48528	0.295599
$825$	13.3153	0.463580
$826$	0	0
$827$	12.5328	0.435807	0.217903	−	0.975970i	$-0.430078\pi$
$827$	12.5328	0.435807	0.217903	+	0.975970i	$0.430078\pi$
$828$	−3.77382	−0.131149
$829$	24.7980	0.861271	0.430635	−	0.902526i	$-0.358289\pi$
$829$	24.7980	0.861271	0.430635	+	0.902526i	$0.358289\pi$
$830$	7.06970	0.245393
$831$	−84.2816	−2.92369
$832$	4.81145	0.166807
$833$	0	0
$834$	38.9047	1.34716
$835$	−21.6088	−0.747805
$836$	−7.28333	−0.251899
$837$	51.1373	1.76756
$838$	23.3195	0.805558
$839$	28.5031	0.984037	0.492019	−	0.870585i	$-0.336259\pi$
$839$	28.5031	0.984037	0.492019	+	0.870585i	$0.336259\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−37.4439	−1.29040
$843$	51.5238	1.77457
$844$	8.79803	0.302841
$845$	−15.1661	−0.521729
$846$	−39.0358	−1.34208
$847$	0	0
$848$	2.34567	0.0805505
$849$	53.4439	1.83419
$850$	−2.56088	−0.0878376
$851$	1.38267	0.0473972
$852$	−23.2672	−0.797120
$853$	17.2171	0.589501	0.294750	−	0.955574i	$-0.404763\pi$
$853$	17.2171	0.589501	0.294750	+	0.955574i	$0.404763\pi$
$854$	0	0
$855$	35.9090	1.22806
$856$	−3.30867	−0.113088
$857$	20.4634	0.699017	0.349508	−	0.936933i	$-0.386349\pi$
$857$	20.4634	0.699017	0.349508	+	0.936933i	$0.386349\pi$
$858$	−23.1501	−0.790331
$859$	38.5403	1.31498	0.657489	−	0.753464i	$-0.271618\pi$
$859$	38.5403	1.31498	0.657489	+	0.753464i	$0.271618\pi$
$860$	−10.5730	−0.360535
$861$	0	0
$862$	15.7738	0.537258
$863$	51.9872	1.76967	0.884833	−	0.465909i	$-0.154273\pi$
$863$	51.9872	1.76967	0.884833	+	0.465909i	$0.154273\pi$
$864$	−7.15103	−0.243283
$865$	−11.9481	−0.406248
$866$	10.8681	0.369315
$867$	46.9522	1.59458
$868$	0	0
$869$	23.9196	0.811417
$870$	−4.34567	−0.147332
$871$	−42.9062	−1.45382
$872$	0.963001	0.0326113
$873$	104.773	3.54602
$874$	3.04365	0.102953
$875$	0	0
$876$	−22.9175	−0.774311
$877$	14.1586	0.478101	0.239051	−	0.971007i	$-0.423164\pi$
$877$	14.1586	0.478101	0.239051	+	0.971007i	$0.423164\pi$
$878$	−38.5469	−1.30090
$879$	19.7610	0.666523
$880$	−2.47187	−0.0833268
$881$	−56.7627	−1.91238	−0.956192	−	0.292740i	$-0.905433\pi$
$881$	−56.7627	−1.91238	−0.956192	+	0.292740i	$0.905433\pi$
$882$	0	0
$883$	17.9260	0.603258	0.301629	−	0.953425i	$-0.402470\pi$
$883$	17.9260	0.603258	0.301629	+	0.953425i	$0.402470\pi$
$884$	4.45236	0.149749
$885$	34.2429	1.15106
$886$	30.6045	1.02818
$887$	6.43793	0.216164	0.108082	−	0.994142i	$-0.465529\pi$
$887$	6.43793	0.216164	0.108082	+	0.994142i	$0.465529\pi$
$888$	5.81679	0.195199
$889$	0	0
$890$	−15.7610	−0.528311
$891$	7.31506	0.245064
$892$	−19.2458	−0.644398
$893$	31.4830	1.05354
$894$	24.6014	0.822794
$895$	32.6255	1.09055
$896$	0	0
$897$	9.67424	0.323014
$898$	30.2134	1.00823
$899$	7.15103	0.238500
$900$	−15.1067	−0.503557
$901$	2.17060	0.0723133
$902$	7.82746	0.260626
$903$	0	0
$904$	−8.45236	−0.281121
$905$	33.7546	1.12204
$906$	−21.9143	−0.728054
$907$	−20.7695	−0.689640	−0.344820	−	0.938669i	$-0.612060\pi$
$907$	−20.7695	−0.689640	−0.344820	+	0.938669i	$0.612060\pi$
$908$	−2.93604	−0.0974358
$909$	−40.9821	−1.35929
$910$	0	0
$911$	35.2283	1.16716	0.583582	−	0.812054i	$-0.301651\pi$
$911$	35.2283	1.16716	0.583582	+	0.812054i	$0.301651\pi$
$912$	12.8044	0.423997
$913$	−7.82746	−0.259051
$914$	33.5961	1.11126
$915$	45.6119	1.50788
$916$	13.0045	0.429681
$917$	0	0
$918$	−6.61733	−0.218405
$919$	−28.1352	−0.928095	−0.464047	−	0.885810i	$-0.653603\pi$
$919$	−28.1352	−0.928095	−0.464047	+	0.885810i	$0.653603\pi$
$920$	1.03298	0.0340562
$921$	24.6068	0.810820
$922$	−27.9463	−0.920363
$923$	38.4916	1.26697
$924$	0	0
$925$	5.53485	0.181985
$926$	28.4524	0.935003
$927$	46.3191	1.52132
$928$	−1.00000	−0.0328266
$929$	16.3247	0.535597	0.267799	−	0.963475i	$-0.413704\pi$
$929$	16.3247	0.535597	0.267799	+	0.963475i	$0.413704\pi$
$930$	−31.0760	−1.01902
$931$	0	0
$932$	24.2938	0.795769
$933$	−64.8528	−2.12319
$934$	−36.0227	−1.17870
$935$	−2.28739	−0.0748057
$936$	26.2646	0.858484
$937$	44.1424	1.44207	0.721035	−	0.692899i	$-0.243667\pi$
$937$	44.1424	1.44207	0.721035	+	0.692899i	$0.243667\pi$
$938$	0	0
$939$	−82.3073	−2.68600
$940$	10.6849	0.348504
$941$	−0.420881	−0.0137203	−0.00686017	−	0.999976i	$-0.502184\pi$
$941$	−0.420881	−0.0137203	−0.00686017	+	0.999976i	$0.502184\pi$
$942$	−3.77382	−0.122958
$943$	−3.27103	−0.106519
$944$	7.87978	0.256465
$945$	0	0
$946$	11.7062	0.380602
$947$	14.6109	0.474792	0.237396	−	0.971413i	$-0.423706\pi$
$947$	14.6109	0.474792	0.237396	+	0.971413i	$0.423706\pi$
$948$	−42.0518	−1.36578
$949$	37.9132	1.23071
$950$	12.1838	0.395294
$951$	31.0574	1.00711
$952$	0	0
$953$	−0.219786	−0.00711958	−0.00355979	−	0.999994i	$-0.501133\pi$
$953$	−0.219786	−0.00711958	−0.00355979	+	0.999994i	$0.501133\pi$
$954$	12.8044	0.414559
$955$	7.34765	0.237765
$956$	13.8350	0.447457
$957$	4.81145	0.155532
$958$	32.2046	1.04048
$959$	0	0
$960$	4.34567	0.140256
$961$	20.1373	0.649590
$962$	−9.62291	−0.310255
$963$	−18.0612	−0.582014
$964$	4.91906	0.158432
$965$	−8.71859	−0.280661
$966$	0	0
$967$	20.8108	0.669231	0.334615	−	0.942355i	$-0.391394\pi$
$967$	20.8108	0.669231	0.334615	+	0.942355i	$0.391394\pi$
$968$	−8.26318	−0.265589
$969$	11.8488	0.380638
$970$	−28.6785	−0.920813
$971$	−51.5947	−1.65575	−0.827876	−	0.560910i	$-0.810451\pi$
$971$	−51.5947	−1.65575	−0.827876	+	0.560910i	$0.810451\pi$
$972$	8.59289	0.275617
$973$	0	0
$974$	10.0612	0.322382
$975$	38.7262	1.24023
$976$	10.4960	0.335967
$977$	−8.66282	−0.277148	−0.138574	−	0.990352i	$-0.544252\pi$
$977$	−8.66282	−0.277148	−0.138574	+	0.990352i	$0.544252\pi$
$978$	33.4156	1.06851
$979$	17.4504	0.557716
$980$	0	0
$981$	5.25679	0.167836
$982$	3.56336	0.113711
$983$	12.3190	0.392916	0.196458	−	0.980512i	$-0.437056\pi$
$983$	12.3190	0.392916	0.196458	+	0.980512i	$0.437056\pi$
$984$	−13.7610	−0.438686
$985$	−21.9515	−0.699433
$986$	−0.925367	−0.0294697
$987$	0	0
$988$	−21.1828	−0.673914
$989$	−4.89194	−0.155554
$990$	−13.4934	−0.428847
$991$	−39.7610	−1.26305	−0.631525	−	0.775355i	$-0.717571\pi$
$991$	−39.7610	−1.26305	−0.631525	+	0.775355i	$0.717571\pi$
$992$	−7.15103	−0.227046
$993$	89.0287	2.82524
$994$	0	0
$995$	−4.70412	−0.149131
$996$	13.7610	0.436035
$997$	3.86140	0.122292	0.0611459	−	0.998129i	$-0.480524\pi$
$997$	3.86140	0.122292	0.0611459	+	0.998129i	$0.480524\pi$
$998$	−2.85630	−0.0904147
$999$	14.3021	0.452497

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2842.2.a.bb.1.1	✓	6
7.6	odd	2	inner	2842.2.a.bb.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.bb.1.1	✓	6	1.1	even	1	trivial
2842.2.a.bb.1.6	yes	6	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.90839 q^{3} +1.00000 q^{4} -1.49418 q^{5} -2.90839 q^{6} +1.00000 q^{8} +5.45876 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.90839 q^{3} +1.00000 q^{4} -1.49418 q^{5} -2.90839 q^{6} +1.00000 q^{8} +5.45876 q^{9} -1.49418 q^{10} +1.65433 q^{11} -2.90839 q^{12} +4.81145 q^{13} +4.34567 q^{15} +1.00000 q^{16} +0.925367 q^{17} +5.45876 q^{18} -4.40257 q^{19} -1.49418 q^{20} +1.65433 q^{22} -0.691333 q^{23} -2.90839 q^{24} -2.76742 q^{25} +4.81145 q^{26} -7.15103 q^{27} -1.00000 q^{29} +4.34567 q^{30} -7.15103 q^{31} +1.00000 q^{32} -4.81145 q^{33} +0.925367 q^{34} +5.45876 q^{36} -2.00000 q^{37} -4.40257 q^{38} -13.9936 q^{39} -1.49418 q^{40} +4.73149 q^{41} +7.07609 q^{43} +1.65433 q^{44} -8.15637 q^{45} -0.691333 q^{46} -7.15103 q^{47} -2.90839 q^{48} -2.76742 q^{50} -2.69133 q^{51} +4.81145 q^{52} +2.34567 q^{53} -7.15103 q^{54} -2.47187 q^{55} +12.8044 q^{57} -1.00000 q^{58} +7.87978 q^{59} +4.34567 q^{60} +10.4960 q^{61} -7.15103 q^{62} +1.00000 q^{64} -7.18918 q^{65} -4.81145 q^{66} -8.91751 q^{67} +0.925367 q^{68} +2.01067 q^{69} +8.00000 q^{71} +5.45876 q^{72} +7.87978 q^{73} -2.00000 q^{74} +8.04876 q^{75} -4.40257 q^{76} -13.9936 q^{78} +14.4588 q^{79} -1.49418 q^{80} +4.42176 q^{81} +4.73149 q^{82} -4.73149 q^{83} -1.38267 q^{85} +7.07609 q^{86} +2.90839 q^{87} +1.65433 q^{88} +10.5483 q^{89} -8.15637 q^{90} -0.691333 q^{92} +20.7980 q^{93} -7.15103 q^{94} +6.57824 q^{95} -2.90839 q^{96} +19.1935 q^{97} +9.03060 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 10 * q^9	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9} + 8 q^{11} + 28 q^{15} + 6 q^{16} + 10 q^{18} + 8 q^{22} - 8 q^{23} + 10 q^{25} - 6 q^{29} + 28 q^{30} + 6 q^{32} + 10 q^{36} - 12 q^{37} - 8 q^{39} + 12 q^{43} + 8 q^{44} - 8 q^{46} + 10 q^{50} - 20 q^{51} + 16 q^{53} + 56 q^{57} - 6 q^{58} + 28 q^{60} + 6 q^{64} + 12 q^{65} - 8 q^{67} + 48 q^{71} + 10 q^{72} - 12 q^{74} - 8 q^{78} + 64 q^{79} - 2 q^{81} - 16 q^{85} + 12 q^{86} + 8 q^{88} - 8 q^{92} + 28 q^{93} + 68 q^{95} - 16 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 10 * q^9 + 8 * q^11 + 28 * q^15 + 6 * q^16 + 10 * q^18 + 8 * q^22 - 8 * q^23 + 10 * q^25 - 6 * q^29 + 28 * q^30 + 6 * q^32 + 10 * q^36 - 12 * q^37 - 8 * q^39 + 12 * q^43 + 8 * q^44 - 8 * q^46 + 10 * q^50 - 20 * q^51 + 16 * q^53 + 56 * q^57 - 6 * q^58 + 28 * q^60 + 6 * q^64 + 12 * q^65 - 8 * q^67 + 48 * q^71 + 10 * q^72 - 12 * q^74 - 8 * q^78 + 64 * q^79 - 2 * q^81 - 16 * q^85 + 12 * q^86 + 8 * q^88 - 8 * q^92 + 28 * q^93 + 68 * q^95 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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